**4.1. Interference robustness of energy detection**

A basic issue of the MIR-UWB's energy detection receiver is its high sensitivity with respect to interferences passing the analogue front-end. A significant reduction of the instantaneous Signal-to-Interference-and Noise Ratio (SINR) can occur so that a reliable communication is not guaranteed.

For this reason it is required to investigate the interference robustness of an OOK and BPPM specific energy detection [7, 9]. The analysis bases on an analytical investigation of the interference robustness of an energy detector within an arbitrary but fixed subband. Thereby, dependencies between system- and interference specific parameters can be identified which

#### 24 Will-be-set-by-IN-TECH 24 Ultra-Wideband Radio Technologies for Communications, Localization and Sensor Applications MIRA – Physical Layer Optimisation for the Multiband Impulse Radio UWB Architecture <sup>25</sup>

promise an increase of interference robustness. The analysis bases on one hand on [53]. Therein, the performance of a BPPM specific energy detector is analysed in presence of out-of-band interference. On the other hand it relies on [60] in which the performance of a BPPM specific correlation receiver is investigated in presence of interference.

### *4.1.1. Signal model*

Binary data transmission within an MIR-UWB subband of bandwidth *B* is considered. Thereby, based on OOK/BPPM the rectangular pulse

$$p\left(t\right) = \begin{cases} \sqrt{\frac{2}{T\_p}} \cos\left(2\pi f\_c t\right) & , 0 < t < T\_{\text{Pl}}\\ 0 & , \text{else} \end{cases} \tag{53}$$

The received signal of (56) is first bandpass filtered and afterwards subjected to non-coherent energy detection with integration time *T*p. At its input stage the available SINR is given by

which is identical for OOK/BPPM. In (58) *P*<sup>N</sup> stands for the mean noise power of the passband noise signal, which is modelled as band-limited wide-sense stationary, time-continuous zero

> sin (*πBτ*) *πBτ*

*y*<sup>2</sup> (*t*) d*t* = *x*<sup>O</sup>

0 , *bn* = 0

0 , *bn* = 0

In contrast to OOK the BPPM specific decision variable at the output of energy detection is

*y*2

<sup>P</sup> (*t*) <sup>d</sup>*<sup>t</sup>* <sup>=</sup> *<sup>x</sup>*<sup>P</sup>

*T*b <sup>2</sup> <sup>+</sup>*T*<sup>p</sup> �

> *T*b 2

<sup>b</sup>. The component <sup>Δ</sup>*x*<sup>O</sup> <sup>=</sup> *<sup>x</sup>*<sup>O</sup>

*T*b � *P*<sup>J</sup> + *P*<sup>N</sup>

*E*b

MIRA – Physical Layer Optimisation for the Multiband Impulse Radio UWB Architecture 25

� , (58)

cos (2*π f*c*τ*). (59)

<sup>s</sup> + <sup>Δ</sup>*x*<sup>O</sup> (60)

sjn <sup>+</sup> *<sup>x</sup>*<sup>O</sup>

<sup>s</sup> + <sup>Δ</sup>*x*P. (64)

jn contains the

<sup>2</sup>*E*<sup>b</sup> , *bn* <sup>=</sup> <sup>1</sup> (61)

cos (2*<sup>π</sup> <sup>f</sup>*c*t*) (*<sup>n</sup>* (*t*) <sup>+</sup> *<sup>j</sup>*(*t*)) <sup>d</sup>*t*d*<sup>t</sup>* , *bn* <sup>=</sup> <sup>1</sup> (62)

(*n* (*t*) + *j*(*t*))<sup>2</sup> d*t* , *bn* = 0, 1 (63)

SINRE = 10 log10

At the output of energy detection the decision variable differs for OOK/BPPM.

*T*p �

0

occurs. The resulting energy value *x*<sup>O</sup> consists of a deterministic signal-only part

*R*<sup>N</sup> (*τ*) = *P*<sup>N</sup>

*x*<sup>O</sup> =

*x*O <sup>s</sup> = �

mean Gaussian process *N* (*t*): (*τ* = *t*<sup>1</sup> − *t*2)

For OOK, the asymmetric decision variable

of mean *E*<sup>b</sup> and second order moment 2*E*<sup>2</sup>

*x*O sjn =

due to noise and interference-only.

as well as the contribution

**Energy detection of BPPM**

symmetric:

mixed signal-noise and signal-interference term

⎧ ⎪⎨

⎪⎩

2 �2*E*<sup>O</sup> p *T*p *T*p � 0

*x*<sup>P</sup> =

*T*p �

0 *y*2

*x*O jn = *T*p �

0

<sup>P</sup> (*t*) d*t* −

**Energy detection of OOK**

with carrier frequency *f*<sup>c</sup> and pulse duration *T*<sup>p</sup> is emitted with energy 1 (*f*<sup>c</sup> � 1/*T*p). The resulting signal to be transmitted conducts to

$$s\_{\mathcal{O}}(t) = \sqrt{E\_{\mathbb{P}}^{\mathcal{O}}} \sum\_{n=-\infty}^{\infty} b\_n p\_i \left( t - nT\_{\mathcal{b}} \right) \tag{54}$$

for OOK and

$$s\_{\rm P}(t) = \sqrt{E\_{\rm P}^{\rm P}} \sum\_{n=-\infty}^{\infty} p\_i \left( t - nT\_{\rm b} - b\_n \frac{T\_{\rm b}}{2} \right) \tag{55}$$

for BPPM. The uniformly distributed data bit *bn* ∈ {0, 1} is characterised by bit energy *E*<sup>b</sup> as well as by bit duration *<sup>T</sup>*<sup>b</sup> <sup>=</sup> *<sup>T</sup>*<sup>p</sup> *<sup>d</sup>*<sup>s</sup> , *<sup>d</sup>*<sup>s</sup> <sup>&</sup>gt; 0 with duty cycle *<sup>d</sup>*<sup>s</sup> <sup>≤</sup> <sup>1</sup> <sup>2</sup> . Finally, *<sup>E</sup><sup>i</sup>* p, *i* ∈ {O, P} stands for the modulation specific pulse energy which equals *E*<sup>O</sup> <sup>p</sup> = 2*E*<sup>P</sup> <sup>p</sup> = 2*E*b.

Assuming perfect synchronisation between the transmitter and the receiver, the signal *si* (*t*), *i* ∈ {O, P} is superposed with zero mean white Gaussian noise *n* (*t*) of two-sided spectral density *<sup>N</sup>*<sup>0</sup> <sup>2</sup> and interference *j*(*t*) leading to (no fading)

$$y\_i(t) = s\_i\left(t\right) + n\left(t\right) + j\left(t\right). \tag{56}$$

Interference is described as band-limited wide-sense stationary, time-continuous zero mean Gaussian process *J* (*t*) characterised by the autocorrelation function (*τ* = *t*<sup>1</sup> − *t*2)

$$R\_{\mathbf{J}}\left(\tau\right) = P\_{\mathbf{J}} \frac{\sin\left(\pi B\_{\mathbf{J}}\tau\right)}{\pi B\_{\mathbf{J}}\tau} \cos\left(2\pi f\_{\mathbf{J}}\tau\right). \tag{57}$$

It depends on the mean interference power *P*<sup>J</sup> determined by the ratio of the interferer's bit energy *E*b,J and bit duration *T*b,J = *qT*b, *q* > 0. Further parameters are the interference center frequency *f*<sup>J</sup> as well as its bandwidth *B*J. <sup>1</sup> The resulting interferer's signal duration *<sup>T</sup>*p,J <sup>≈</sup> 1 *<sup>B</sup>*<sup>J</sup> <sup>≤</sup> *<sup>T</sup>*b,J leads to an interference duty cycle of *<sup>d</sup>*<sup>J</sup> <sup>=</sup> *<sup>T</sup>*p,J *<sup>T</sup>*b,J <sup>=</sup> *<sup>d</sup>*s*T*p,J *qT*<sup>p</sup> .

<sup>1</sup> Eq. (57) holds if the interference source is completely inside the MIR-UWB subband. In case *B*<sup>J</sup> overlaps completely or only partially with the subband *f*J, *B*<sup>J</sup> and *P*<sup>J</sup> have to be properly modified. However, as it can be ascribed to (57) the following investigations focus solely on an interference source being completely inside the subband.

The received signal of (56) is first bandpass filtered and afterwards subjected to non-coherent energy detection with integration time *T*p. At its input stage the available SINR is given by

$$\text{SINR}\_{\text{E}} = 10 \log\_{10} \frac{E\_{\text{b}}}{T\_{\text{b}} \left(P\_{\text{I}} + P\_{\text{N}}\right)} \,\text{}\tag{58}$$

which is identical for OOK/BPPM. In (58) *P*<sup>N</sup> stands for the mean noise power of the passband noise signal, which is modelled as band-limited wide-sense stationary, time-continuous zero mean Gaussian process *N* (*t*): (*τ* = *t*<sup>1</sup> − *t*2)

$$R\_{\rm N} \left( \tau \right) = P\_{\rm N} \frac{\sin \left( \pi B \tau \right)}{\pi B \tau} \cos \left( 2 \pi f\_{\rm c} \tau \right) \,. \tag{59}$$

At the output of energy detection the decision variable differs for OOK/BPPM.

### **Energy detection of OOK**

24 Will-be-set-by-IN-TECH

promise an increase of interference robustness. The analysis bases on one hand on [53]. Therein, the performance of a BPPM specific energy detector is analysed in presence of out-of-band interference. On the other hand it relies on [60] in which the performance of a

Binary data transmission within an MIR-UWB subband of bandwidth *B* is considered.

0 , else

with carrier frequency *f*<sup>c</sup> and pulse duration *T*<sup>p</sup> is emitted with energy 1 (*f*<sup>c</sup> � 1/*T*p). The

∞ ∑ *n*=−∞

for BPPM. The uniformly distributed data bit *bn* ∈ {0, 1} is characterised by bit energy *E*<sup>b</sup> as

*<sup>d</sup>*<sup>s</sup> , *<sup>d</sup>*<sup>s</sup> <sup>&</sup>gt; 0 with duty cycle *<sup>d</sup>*<sup>s</sup> <sup>≤</sup> <sup>1</sup>

Assuming perfect synchronisation between the transmitter and the receiver, the signal *si* (*t*), *i* ∈ {O, P} is superposed with zero mean white Gaussian noise *n* (*t*) of two-sided

Interference is described as band-limited wide-sense stationary, time-continuous zero mean

*πB*J*τ*

It depends on the mean interference power *P*<sup>J</sup> determined by the ratio of the interferer's bit energy *E*b,J and bit duration *T*b,J = *qT*b, *q* > 0. Further parameters are the interference center

<sup>1</sup> Eq. (57) holds if the interference source is completely inside the MIR-UWB subband. In case *B*<sup>J</sup> overlaps completely or only partially with the subband *f*J, *B*<sup>J</sup> and *P*<sup>J</sup> have to be properly modified. However, as it can be ascribed to (57)

the following investigations focus solely on an interference source being completely inside the subband.

cos �

2*π f*J*τ* �

*<sup>T</sup>*b,J <sup>=</sup> *<sup>d</sup>*s*T*p,J *qT*<sup>p</sup> .

*<sup>T</sup>*<sup>p</sup> cos (2*π f*c*t*) , 0 < *t* < *T*<sup>p</sup>

*t* − *nT*<sup>b</sup> − *bn*

*T*b 2 �

<sup>p</sup> = 2*E*<sup>P</sup>

*yi* (*t*) = *si* (*t*) + *n* (*t*) + *j*(*t*). (56)

<sup>2</sup> . Finally, *<sup>E</sup><sup>i</sup>*

<sup>p</sup> = 2*E*b.

<sup>1</sup> The resulting interferer's signal duration *<sup>T</sup>*p,J <sup>≈</sup>

*bn pi* (*t* − *nT*b) (54)

(53)

(55)

p, *i* ∈ {O, P} stands

. (57)

BPPM specific correlation receiver is investigated in presence of interference.

Thereby, based on OOK/BPPM the rectangular pulse

resulting signal to be transmitted conducts to

*p* (*t*) =

⎧ ⎨ ⎩

*s*<sup>O</sup> (*t*) =

� *E*P p

*s*<sup>P</sup> (*t*) =

for the modulation specific pulse energy which equals *E*<sup>O</sup>

� 2

� *E*<sup>O</sup> p

> ∞ <sup>∑</sup>*n*=−<sup>∞</sup> *pi* �

<sup>2</sup> and interference *j*(*t*) leading to (no fading)

Gaussian process *J* (*t*) characterised by the autocorrelation function (*τ* = *t*<sup>1</sup> − *t*2)

sin � *πB*J*τ* �

*R*<sup>J</sup> (*τ*) = *P*<sup>J</sup>

*4.1.1. Signal model*

for OOK and

spectral density *<sup>N</sup>*<sup>0</sup>

1

well as by bit duration *<sup>T</sup>*<sup>b</sup> <sup>=</sup> *<sup>T</sup>*<sup>p</sup>

frequency *f*<sup>J</sup> as well as its bandwidth *B*J.

*<sup>B</sup>*<sup>J</sup> <sup>≤</sup> *<sup>T</sup>*b,J leads to an interference duty cycle of *<sup>d</sup>*<sup>J</sup> <sup>=</sup> *<sup>T</sup>*p,J

For OOK, the asymmetric decision variable

$$\mathbf{x}^{\bullet \bullet} = \int\_{0}^{T\_{\rm P}} \mathbf{y}^{2} \left( t \right) \mathbf{d}t = \mathbf{x}\_{\rm s}^{\bullet \bullet} + \Delta \mathbf{x}^{\bullet \bullet} \tag{60}$$

occurs. The resulting energy value *x*<sup>O</sup> consists of a deterministic signal-only part

$$\mathbf{x}\_{\mathbf{s}}^{\mathcal{O}} = \begin{cases} 0 & \text{, } b\_{\mathcal{U}} = 0 \\ 2E\_{\mathbf{b}} & \text{, } b\_{\mathcal{U}} = 1 \end{cases} \tag{61}$$

of mean *E*<sup>b</sup> and second order moment 2*E*<sup>2</sup> <sup>b</sup>. The component <sup>Δ</sup>*x*<sup>O</sup> <sup>=</sup> *<sup>x</sup>*<sup>O</sup> sjn <sup>+</sup> *<sup>x</sup>*<sup>O</sup> jn contains the mixed signal-noise and signal-interference term

$$\mathbf{x}\_{\text{spin}}^{\text{O}} = \begin{cases} 0 & \text{, } b\_{\text{ll}} = 0\\ 2\sqrt{\frac{2E\_{\text{p}}^{\text{O}}}{T\_{\text{p}}}} \int\_{0}^{T\_{\text{p}}} \cos\left(2\pi f\_{\text{c}}t\right) \left(n\left(t\right) + j\left(t\right)\right) \,\text{dtdt} & \text{, } b\_{\text{ll}} = 1 \end{cases} \tag{62}$$

as well as the contribution

$$\mathbf{x}\_{\rm in}^{\rm O} = \int\_0^{T\_{\rm P}} \left( n\left(t\right) + j\left(t\right) \right)^2 \mathbf{d}t \quad , b\_{\rm n} = 0, 1 \tag{63}$$

due to noise and interference-only.

### **Energy detection of BPPM**

In contrast to OOK the BPPM specific decision variable at the output of energy detection is symmetric:

$$\mathbf{x}^{\mathbf{P}} = \int\_{0}^{T\_{\mathrm{p}}} y\_{\mathrm{P}}^{2}(t) \, \mathrm{d}t - \int\_{\frac{T\_{\mathrm{p}}}{2}}^{\frac{T\_{\mathrm{p}}}{2} + T\_{\mathrm{p}}} y\_{\mathrm{P}}^{2}(t) \, \mathrm{d}t = \mathbf{x}\_{\mathrm{s}}^{\mathbf{P}} + \Delta \mathbf{x}^{\mathbf{P}}.\tag{64}$$

#### 26 Will-be-set-by-IN-TECH 26 Ultra-Wideband Radio Technologies for Communications, Localization and Sensor Applications MIRA – Physical Layer Optimisation for the Multiband Impulse Radio UWB Architecture <sup>27</sup>

The decision variable *x*<sup>P</sup> compares energy values within two observation intervals of duration *T*p. <sup>2</sup> It is composed of a signal-only contribution

$$\mathbf{x}\_{\mathbf{s}}^{\mathbf{P}} = \begin{cases} E\_{\mathbf{b}} & \text{, } b\_{\mathbf{n}} = 0 \\ -E\_{\mathbf{b}} & \text{, } b\_{\mathbf{n}} = 1 \end{cases} \tag{65}$$

and vice versa. Hence, the smaller *Q<sup>i</sup>*

*Q*<sup>O</sup> <sup>1</sup> =

> *Q*<sup>O</sup> <sup>1</sup> <sup>=</sup> *<sup>E</sup>*<sup>O</sup> <sup>p</sup> *P*<sup>J</sup> ∞ ∑ *n*=0

> > + *E*<sup>O</sup> <sup>p</sup> *P*<sup>N</sup> 2*π f*c

*<sup>r</sup><sup>ν</sup>* <sup>=</sup> <sup>1</sup> *B*J

*wn*,*<sup>l</sup>* =

*zn*,*<sup>l</sup>* =

cosine tone, *r<sup>ν</sup>* has to be replaced with *r<sup>m</sup>*

Eq. (71) shows that *Q*<sup>O</sup>

*vn*,*<sup>l</sup>* <sup>=</sup> *zn*,*<sup>l</sup>*

cos

modulation schemes.

A general solution of *Q*<sup>O</sup>

part *x*<sup>O</sup>

error detection probability. In the following, *Q<sup>i</sup>*

sjn can be formulated as (*τ* = *t*<sup>1</sup> − *t*2)

*<sup>T</sup>*p−*t*<sup>1</sup>

*T*p

*f*J| � *B*<sup>J</sup> and 4 *f*<sup>c</sup> � *B*. This leads to the closed-form expression

<sup>−</sup> <sup>2</sup>*πT*p*r*2*n*+<sup>2</sup> <sup>∑</sup>2*n*+<sup>2</sup>

∞ ∑ *n*=0

4*π f*c*T*p

whereas, with Δ*f*c,J = *f*<sup>c</sup> − *f*J, the following notations are used:

*B*<sup>J</sup>

(2*n* + 1 − *l*)!

sin

*un*,*<sup>l</sup>* <sup>=</sup> *wn*,*<sup>l</sup>* <sup>+</sup> (−1)

cos

(2*n* + 1 − *l*)!

(−1) *n* 2*πT*p <sup>2</sup>*<sup>n</sup>*

> (−1) *n πT*p*B*<sup>T</sup>

> > 2*n*+1 ∑ *l*=0

<sup>2</sup> <sup>+</sup> <sup>Δ</sup>*f*c,J*<sup>ν</sup>*

4*π f*c*T*<sup>p</sup> + <sup>1</sup>

 4*π f*c*T*p *l* ,

(2*n* + 1 − *l*)!

4*π f*c*T*<sup>p</sup> + <sup>1</sup>

(2*<sup>n</sup>* <sup>+</sup> <sup>2</sup> <sup>−</sup> *<sup>l</sup>*) <sup>−</sup> (−1)

 4*π f*c*T*p *l* ,

*l*

0

−*t*<sup>1</sup>

<sup>1</sup> and *<sup>Q</sup><sup>i</sup>*

*R*<sup>J</sup> (*τ*) + *R*<sup>N</sup> (*τ*)

*<sup>l</sup>*=<sup>0</sup> *vn*,*<sup>l</sup>* <sup>2</sup>*<sup>π</sup> <sup>f</sup>*<sup>c</sup> (2*<sup>n</sup>* <sup>+</sup> <sup>2</sup>) <sup>+</sup>

(2*n* + 1)

− − *B*J

> *l* ∑ *k*=0

sin

*l*

(2*n* + 2 − *l*)!

<sup>1</sup> is depending from the system parameters *<sup>E</sup>*<sup>O</sup>

*<sup>ν</sup>* = lim *B*J→0

the interference parameters *P*J, *B*J, *f*J. In addition, concerning the special case *B*<sup>J</sup> → 0, e.g., a

(*l* − *k*)! 4*π f*c*T*p

> *l* ∑ *k*=0

*r<sup>ν</sup>* = *ν*Δ*ν*−<sup>1</sup> *f*c,J

<sup>2</sup> *lπ* 

<sup>2</sup> *lπ* 

**OOK:** For OOK the second order moment of the signal-noise and signal-interference

<sup>1</sup> can be found using Parseval's theorem under the assumptions 2| *f*<sup>c</sup> +

MIRA – Physical Layer Optimisation for the Multiband Impulse Radio UWB Architecture 27

*<sup>r</sup>*2*n*+<sup>1</sup> <sup>∑</sup>2*n*+<sup>1</sup>

2*π f*<sup>c</sup> (2*n* + 1)

4*T*p*r*2*n*+<sup>1</sup> (2*n* + 2)! (2*n* + 1)

> 8*πT*<sup>p</sup> *f*<sup>c</sup> (2*n* + 2)!

> > 4*π f*c*T*p 2*n*+1 ∑ *l*=0 *zn*,*<sup>l</sup>*

> > > ,

<sup>2</sup> *kπ* 

*<sup>k</sup>* ,

4*π f*c*T*<sup>p</sup> + <sup>1</sup>

<sup>2</sup> *kπ* 

*<sup>k</sup>* .

. Note that this result is consistent

<sup>p</sup> , *T*p, *f*c, *B* as well as from

<sup>2</sup> <sup>+</sup> <sup>Δ</sup>*f*c,J*ν*

4*π f*c*T*<sup>p</sup> + <sup>1</sup>

cos

(*l* − *k*)! 4*π f*c*T*p

<sup>2</sup>*<sup>n</sup>*

*wn*,*<sup>l</sup>* <sup>−</sup> sin

*<sup>l</sup>*=<sup>0</sup> *un*,*<sup>l</sup>*

+ (1+

, (71)

<sup>1</sup> and *<sup>Q</sup><sup>i</sup>*

<sup>2</sup>, *i* ∈ {O, P} the lower the modulation related

· *p* (*t*1) · *p* (*t*<sup>1</sup> + *τ*) d*t*1d*τ*. (70)

<sup>2</sup>, *i* ∈ {O, P} are determined for both

which is characterised by mean zero and second order moment *E*<sup>2</sup> <sup>b</sup>. The additional term <sup>Δ</sup>*x*<sup>P</sup> <sup>=</sup> *x*P sjn <sup>+</sup> *<sup>x</sup>*<sup>P</sup> jn is on one hand composed of a mixed signal-noise and signal-interference component

$$\mathbf{x}\_{\text{spin}}^{\text{p}} = \begin{cases} a \int\_{0}^{T\_{\text{p}}} \cos \left(2\pi f\_{\text{c}}t\right) \left(n\left(t\right) + j\left(t\right)\right) \,\mathrm{d}t & \text{, } b\_{\text{n}} = 0\\ 0 & \frac{\tau\_{\text{p}}}{2} + T\_{\text{p}}\\ -a \int\_{0}^{T\_{\text{p}}} \cos \left(2\pi f\_{\text{c}} \left(t - \frac{T\_{\text{b}}}{2}\right)\right) \left(n\left(t\right) + j\left(t\right)\right) \,\mathrm{d}t & \text{, } b\_{\text{n}} = 1 \end{cases} \tag{66}$$

with *a* = 2 � 2*E*<sup>P</sup> p/*T*p. On the other hand it consists of the noise and interference-only part

$$\mathbf{x}\_{\rm in}^{\rm P} = \int\_{0}^{T\_{\rm P}} \left( n\left(t\right) + j\left(t\right) \right)^{2} \mathbf{d}t - \int\_{\frac{T\_{\rm P}}{2}}^{\frac{T\_{\rm P}}{2} + T\_{\rm P}} \left( n\left(t\right) + j\left(t\right) \right)^{2} \mathbf{d}t.\tag{67}$$

### *4.1.2. Statistical analysis of interference robustness*

To make statements on the interference robustness of an OOK and BPPM specific energy detection a proper quality criterion has to be introduced. A possible measure is the processing gain (PG) of the energy detector. It refers the available SINR at its output to the SINRE at its input. For OOK this can be described as

$$\mathrm{PG}^{\mathrm{O}} = 10 \log\_{10} \left( \frac{2E\_{\mathrm{b}}^{2}}{0.5Q\_{1}^{\mathrm{O}} + Q\_{2}^{\mathrm{O}}} \right) - 10 \log\_{10} \left( \mathrm{SINR}\_{\mathrm{E}} \right), \tag{68}$$

which differs from the PG of the BPPM based energy detection receiver expressed as

$$\mathrm{PG}^{\mathrm{P}} = 10 \log\_{10} \left( \frac{E\_{\mathrm{b}}^{2}}{Q\_{1}^{\mathrm{P}} + Q\_{2}^{\mathrm{P}}} \right) - 10 \log\_{10} \left( \mathrm{SINR}\_{\mathrm{E}} \right) \,. \tag{69}$$

In (68) and (69) *Q<sup>i</sup>* <sup>1</sup>, *i* ∈ {O, P} stands for the second order moment of the mixed signal-noise and signal-interference component *x<sup>i</sup>* sjn, *<sup>i</sup>* ∈ {O, P}. In contrast, *<sup>Q</sup><sup>i</sup>* <sup>2</sup>, *i* ∈ {O, P} describes the second order moment of the noise and interference-only part *x<sup>i</sup>* jn, *i* ∈ {O, P}.

Based on PG, separate statements on the detection performance can be made for each modulation scheme, i.e., a low modulation specific PG indicates an increased error probability

<sup>2</sup> It is assumed that the pulse of duration *<sup>T</sup>*<sup>p</sup> <sup>≤</sup> *<sup>T</sup>*<sup>b</sup> <sup>2</sup> occurs at the beginning of an interval of duration *<sup>T</sup>*<sup>b</sup> <sup>2</sup> . Hence, the position of a pulse within the interval is perfectly known.

and vice versa. Hence, the smaller *Q<sup>i</sup>* <sup>1</sup> and *<sup>Q</sup><sup>i</sup>* <sup>2</sup>, *i* ∈ {O, P} the lower the modulation related error detection probability. In the following, *Q<sup>i</sup>* <sup>1</sup> and *<sup>Q</sup><sup>i</sup>* <sup>2</sup>, *i* ∈ {O, P} are determined for both modulation schemes.

**OOK:** For OOK the second order moment of the signal-noise and signal-interference part *x*<sup>O</sup> sjn can be formulated as (*τ* = *t*<sup>1</sup> − *t*2)

$$\mathbf{Q}\_{1}^{\rm O} = \int\_{-t\_{1}}^{T\_{\rm P} - t\_{1}} \int\_{0}^{T\_{\rm P}} \left( \mathbf{R}\_{\rm I}(\tau) + \mathbf{R}\_{\rm N}(\tau) \right) \cdot p\left(t\_{1}\right) \cdot p\left(t\_{1} + \tau\right) \,\mathrm{d}t\_{1} \mathrm{d}\tau. \tag{70}$$

A general solution of *Q*<sup>O</sup> <sup>1</sup> can be found using Parseval's theorem under the assumptions 2| *f*<sup>c</sup> + *f*J| � *B*<sup>J</sup> and 4 *f*<sup>c</sup> � *B*. This leads to the closed-form expression

$$\begin{split} \mathbf{Q}\_{1}^{\rm O} &= \mathbf{E}\_{\rm P}^{\rm O} P\_{\rm f} \sum\_{n=0}^{\infty} (-1)^{n} \left( 2\pi T\_{\rm P} \right)^{2n} \left( \frac{r\_{2n+1} \sum\_{l=0}^{2n+1} u\_{n,l}}{2\pi f\_{\rm c} \left( 2n+1 \right)} \right. \\ &\left. - \frac{2\pi T\_{\rm P} r\_{2n+2} \sum\_{l=0}^{2n+2} v\_{n,l}}{2\pi f\_{\rm c} \left( 2n+2 \right)} + \frac{4T\_{\rm P} r\_{2n+1}}{(2n+2)! \left( 2n+1 \right)} \right) \\ &+ \frac{E\_{\rm P}^{\rm O} P\_{\rm N}}{2\pi f\_{\rm c}} \sum\_{n=0}^{\infty} \frac{\left( -1 \right)^{n} \left( \pi T\_{\rm P} B\_{\rm T} \right)^{2n}}{(2n+1)} \left( \frac{8\pi T\_{\rm P} f\_{\rm c}}{(2n+2)!} + (1+ \\ &\cos \left( 4\pi f\_{\rm c} t\_{\rm P} \right) \sum\_{l=0}^{2n+1} w\_{n,l} - \sin \left( 4\pi f\_{\rm c} t\_{\rm P} \right) \sum\_{l=0}^{2n+1} z\_{n,l} \right), \end{split} \tag{71}$$

whereas, with Δ*f*c,J = *f*<sup>c</sup> − *f*J, the following notations are used:

26 Will-be-set-by-IN-TECH

The decision variable *x*<sup>P</sup> compares energy values within two observation intervals of duration

*E*<sup>b</sup> , *bn* = 0

jn is on one hand composed of a mixed signal-noise and signal-interference component

cos (2*π f*c*t*) (*n* (*t*) + *j*(*t*)) d*t* , *bn* = 0

p/*T*p. On the other hand it consists of the noise and interference-only part

*T*b <sup>2</sup> <sup>+</sup>*T*<sup>p</sup> �

> *T*b 2

> > �

To make statements on the interference robustness of an OOK and BPPM specific energy detection a proper quality criterion has to be introduced. A possible measure is the processing gain (PG) of the energy detector. It refers the available SINR at its output to the SINRE at its

b

<sup>1</sup> <sup>+</sup> *<sup>Q</sup>*<sup>O</sup> 2

Based on PG, separate statements on the detection performance can be made for each modulation scheme, i.e., a low modulation specific PG indicates an increased error probability

�

<sup>1</sup>, *i* ∈ {O, P} stands for the second order moment of the mixed signal-noise

<sup>2</sup> occurs at the beginning of an interval of duration *<sup>T</sup>*<sup>b</sup>

sjn, *<sup>i</sup>* ∈ {O, P}. In contrast, *<sup>Q</sup><sup>i</sup>*

<sup>−</sup>*E*<sup>b</sup> , *bn* <sup>=</sup> 1, (65)

(*n* (*t*) + *j*(*t*)) d*t* , *bn* = 1

(*n* (*t*) + *j*(*t*))<sup>2</sup> d*t*. (67)

− 10 log10 (SINRE), (68)

− 10 log10 (SINRE). (69)

jn, *i* ∈ {O, P}.

<sup>2</sup>, *i* ∈ {O, P} describes the

<sup>2</sup> . Hence, the

<sup>b</sup>. The additional term <sup>Δ</sup>*x*<sup>P</sup> <sup>=</sup>

(66)

*T*p. <sup>2</sup> It is composed of a signal-only contribution

*x*P sjn <sup>+</sup> *<sup>x</sup>*<sup>P</sup>

with *a* = 2

*x*P sjn =

� 2*E*<sup>P</sup>

In (68) and (69) *Q<sup>i</sup>*

⎧ ⎪⎪⎪⎪⎪⎨ *a T*p � 0

−*a*

*T*b <sup>2</sup> <sup>+</sup>*T*<sup>p</sup> � *T*b 2

*T*p �

0

PG<sup>O</sup> <sup>=</sup> 10 log10

PG<sup>P</sup> <sup>=</sup> 10 log10

second order moment of the noise and interference-only part *x<sup>i</sup>*

⎪⎪⎪⎪⎪⎩

*x*P jn =

*4.1.2. Statistical analysis of interference robustness*

input. For OOK this can be described as

and signal-interference component *x<sup>i</sup>*

<sup>2</sup> It is assumed that the pulse of duration *<sup>T</sup>*<sup>p</sup> <sup>≤</sup> *<sup>T</sup>*<sup>b</sup>

position of a pulse within the interval is perfectly known.

*x*P *<sup>s</sup>* = �

which is characterised by mean zero and second order moment *E*<sup>2</sup>

cos � 2*π f*c � *<sup>t</sup>* <sup>−</sup> *<sup>T</sup>*<sup>b</sup> 2 ��

(*<sup>n</sup>* (*t*) <sup>+</sup> *<sup>j</sup>*(*t*))<sup>2</sup> <sup>d</sup>*<sup>t</sup>* <sup>−</sup>

� 2*E*<sup>2</sup>

0, 5*Q*<sup>O</sup>

which differs from the PG of the BPPM based energy detection receiver expressed as

� *E*<sup>2</sup> b *Q*<sup>P</sup> <sup>1</sup> <sup>+</sup> *<sup>Q</sup>*<sup>P</sup> 2

*<sup>r</sup><sup>ν</sup>* <sup>=</sup> <sup>1</sup> *B*J *B*<sup>J</sup> <sup>2</sup> <sup>+</sup> <sup>Δ</sup>*f*c,J*<sup>ν</sup>* − − *B*J <sup>2</sup> <sup>+</sup> <sup>Δ</sup>*f*c,J*ν* , *wn*,*<sup>l</sup>* = sin 4*π f*c*T*<sup>p</sup> + <sup>1</sup> <sup>2</sup> *lπ* (2*n* + 1 − *l*)! 4*π f*c*T*p *l* , *un*,*<sup>l</sup>* <sup>=</sup> *wn*,*<sup>l</sup>* <sup>+</sup> (−1) *l* (2*n* + 1 − *l*)! *l* ∑ *k*=0 sin 4*π f*c*T*<sup>p</sup> + <sup>1</sup> <sup>2</sup> *kπ* (*l* − *k*)! 4*π f*c*T*p *<sup>k</sup>* , *zn*,*<sup>l</sup>* = cos 4*π f*c*T*<sup>p</sup> + <sup>1</sup> <sup>2</sup> *lπ* (2*n* + 1 − *l*)! 4*π f*c*T*p *l* , *vn*,*<sup>l</sup>* <sup>=</sup> *zn*,*<sup>l</sup>* (2*<sup>n</sup>* <sup>+</sup> <sup>2</sup> <sup>−</sup> *<sup>l</sup>*) <sup>−</sup> (−1) *l* (2*n* + 2 − *l*)! *l* ∑ *k*=0 cos 4*π f*c*T*<sup>p</sup> + <sup>1</sup> <sup>2</sup> *kπ* (*l* − *k*)! 4*π f*c*T*p *<sup>k</sup>* .

Eq. (71) shows that *Q*<sup>O</sup> <sup>1</sup> is depending from the system parameters *<sup>E</sup>*<sup>O</sup> <sup>p</sup> , *T*p, *f*c, *B* as well as from the interference parameters *P*J, *B*J, *f*J. In addition, concerning the special case *B*<sup>J</sup> → 0, e.g., a cosine tone, *r<sup>ν</sup>* has to be replaced with *r<sup>m</sup> <sup>ν</sup>* = lim *B*J→0 *r<sup>ν</sup>* = *ν*Δ*ν*−<sup>1</sup> *f*c,J . Note that this result is consistent

#### 28 Will-be-set-by-IN-TECH 28 Ultra-Wideband Radio Technologies for Communications, Localization and Sensor Applications MIRA – Physical Layer Optimisation for the Multiband Impulse Radio UWB Architecture <sup>29</sup>

to [9] if *P*<sup>N</sup> = 0. In this case (71) simplifies to

$$\begin{split} Q\_1^{\rm O} &= \frac{E\_\mathrm{p}^{\rm O} P\_\mathrm{f}}{T\_\mathrm{P} \pi^2 \left(f\_\mathrm{c}^2 - f\_\mathrm{f}^2\right)^2} \left[f\_\mathrm{c}^2 + 3f\_\mathrm{f}^2 + \left(f\_\mathrm{f}^2 - f\_\mathrm{c}^2\right) \cos\left(4\pi f\_\mathrm{c} T\_\mathrm{P}\right)\right] \\ &- 2f\_\mathrm{f} \left(f\_\mathrm{f} + f\_\mathrm{c}\right) \cos\left(2\pi \left(f\_\mathrm{c} - f\_\mathrm{f}\right) T\_\mathrm{P}\right) \\ &- 2f\_\mathrm{f} \left(f\_\mathrm{f} - f\_\mathrm{c}\right) \cos\left(2\pi \left(f\_\mathrm{c} + f\_\mathrm{f}\right) T\_\mathrm{P}\right) \, . \end{split} \tag{72}$$

cannot be reduced via *E*<sup>O</sup>

signal-interference part *x*<sup>P</sup>

interference-only part *x*<sup>P</sup>

*Q*<sup>P</sup> <sup>2</sup> = 2

> + ∞ ∑ *k*=2

+

*πB*<sup>T</sup> *f*<sup>p</sup> − *f*<sup>m</sup> ∞ ∑ *k*=0

*Q*<sup>P</sup> <sup>2</sup> = 4

<sup>2</sup> <sup>+</sup> <sup>Δ</sup>*f*c,J <sup>−</sup> *<sup>B</sup>*

*<sup>f</sup>*p<sup>−</sup> *<sup>f</sup>*<sup>m</sup> <sup>=</sup> (2*<sup>k</sup>* <sup>+</sup> <sup>2</sup>)

 − *B*

> p*P*<sup>2</sup> <sup>J</sup> +

> > <sup>1</sup> <sup>=</sup> <sup>1</sup>

of the noise and interference specific autocorrelation functions: (*τ* = *t*<sup>1</sup> − *t*2)

<sup>J</sup> (*τ*) <sup>+</sup> *<sup>R</sup>*<sup>2</sup>

<sup>J</sup> (*τ*) <sup>+</sup> *<sup>R</sup>*<sup>2</sup>

(2*k* + 1)! (2*k* + 1) (*k* + 1)

(−1)

(2*k*)!(2*k*)

*<sup>k</sup>* (2*π*)

contrast, the larger the data rate the higher its impact, e.g., for *T*<sup>b</sup> = 2*T*<sup>p</sup> *g<sup>ν</sup>* conducts to *g<sup>ν</sup>* =

<sup>2</sup> <sup>−</sup> <sup>Δ</sup>*f*c,J<sup>2</sup>*<sup>k</sup>*

sjn is: *<sup>Q</sup>*<sup>P</sup>

*Q*<sup>O</sup> <sup>2</sup> <sup>=</sup> <sup>2</sup>*T*<sup>2</sup>

*<sup>T</sup>*p−*t*<sup>1</sup>

*T*p

 *R*2

*<sup>T</sup>*p+ *<sup>T</sup>*<sup>b</sup> <sup>2</sup>

 *R*2

Therefore, using the theorem of Parseval for 2 *f*<sup>J</sup> � *B*J, the closed-form result

2*k*−2 *P*2 <sup>J</sup> *<sup>B</sup>*2*k*−<sup>2</sup> <sup>J</sup> <sup>+</sup> *<sup>P</sup>*<sup>2</sup>

*<sup>T</sup>*<sup>p</sup> <sup>−</sup> *<sup>T</sup>*<sup>b</sup> 2 *ν* + 2 *T*<sup>b</sup> 2 *ν* − *T*<sup>p</sup> + *T*b 2 *ν*

*T*b 2

*<sup>k</sup>* (2*π*) 2*k P*2 <sup>J</sup> *<sup>B</sup>*2*<sup>k</sup>* <sup>J</sup> <sup>+</sup> *<sup>P</sup>*<sup>2</sup>

*<sup>k</sup>* 22*<sup>k</sup>* (*π*)

0

−*t*<sup>1</sup>

−*t*<sup>1</sup>

(−1)

(−1)

2*P*J*P*<sup>N</sup>

*g<sup>ν</sup>* = 2*T<sup>ν</sup>*

interference parameters *P*J, *B*J, *f*J. Similar to *Q*<sup>O</sup>

that for low data rates (*T*<sup>b</sup> <sup>→</sup> <sup>∞</sup>) *<sup>g</sup><sup>ν</sup>* <sup>≈</sup> <sup>2</sup>*T<sup>ν</sup>*

p − 

*<sup>T</sup>*p−*t*<sup>1</sup>

− 4

∞ ∑ *k*=1 <sup>2</sup> <sup>−</sup> <sup>Δ</sup>*f*c,J<sup>2</sup>*k*+<sup>1</sup>

*P*2 J 8*π*<sup>2</sup> *f* <sup>2</sup> J 

**BPPM:** Considering BPPM the second order moment of the signal-noise and

two which can be ascribed to the reduced modulation specific pulse energy. In contrast to

<sup>1</sup> there is a significant difference concerning the second order moment of the noise and

<sup>2</sup> *<sup>Q</sup>*<sup>O</sup> <sup>1</sup> . *<sup>Q</sup>*<sup>P</sup>

<sup>J</sup> <sup>=</sup> 0, *<sup>f</sup>*1(*k*)

*<sup>B</sup>*

*P*2 <sup>J</sup> *<sup>B</sup>*2*k*−<sup>2</sup>

*Q*<sup>P</sup>

with

4*T<sup>ν</sup>* <sup>p</sup> <sup>−</sup> 2*T*p *ν*

can be found. *Q*<sup>P</sup>

(2*k* + 1)

<sup>p</sup> . Eq. (74) simplifies for *<sup>B</sup>*<sup>J</sup> <sup>→</sup> 0 due to *<sup>P</sup>*<sup>2</sup>

MIRA – Physical Layer Optimisation for the Multiband Impulse Radio UWB Architecture 29

. Assuming *P*<sup>N</sup> = 0 (74) equals the result of [9]:

4*π f*J*T*<sup>p</sup>

<sup>1</sup> differs from *<sup>Q</sup>*<sup>O</sup>

 d*t*1d*τ*

jn. With the theorem of Price this can be generally described in terms

<sup>N</sup> (*τ*) + *R*<sup>J</sup> (*τ*) *R*<sup>N</sup> (*τ*)

N*B*2*k*−<sup>2</sup> T

<sup>2</sup>*k*+<sup>1</sup> *<sup>g</sup>*2*k*+<sup>2</sup> (2*<sup>k</sup>* + <sup>2</sup>)! ·

<sup>2</sup> is influenced by the system parameters *T*p, *T*b, *f*c, *B* as well as by the

. In addition, for *B*<sup>J</sup> → 0 and *P*<sup>N</sup> = 0, (77) allows the same simplifications as for

<sup>2</sup> it cannot be reduced via *<sup>E</sup>*<sup>P</sup>

<sup>p</sup> resulting in a negligible influence of *Q*<sup>P</sup>

 *g*2*<sup>k</sup>*

*f*<sup>1</sup> (*k*)

<sup>2</sup>*<sup>k</sup>* <sup>+</sup> <sup>2</sup> <sup>+</sup> *<sup>f</sup>*<sup>2</sup> (*k*)

2*k* + 1

p. Eq. (77) reveals

(77)

<sup>2</sup> . In

<sup>N</sup> (*τ*) + *R*<sup>J</sup> (*τ*) *R*<sup>N</sup> (*τ*)

N*B*2*<sup>k</sup>* T *g*2*k*+<sup>2</sup>

<sup>2</sup> <sup>+</sup> <sup>Δ</sup>*f*c,J<sup>2</sup>*k*+<sup>1</sup>

+ − *B*

<sup>1</sup> <sup>−</sup> cos

<sup>J</sup> *<sup>B</sup>*2*<sup>k</sup>* <sup>J</sup> =

*<sup>f</sup>*p<sup>−</sup> *<sup>f</sup>*<sup>m</sup> <sup>=</sup>

as well as *<sup>f</sup>*<sup>2</sup> (*k*)

. (75)

<sup>1</sup> solely in a factor of

d*t*1d*τ*. (76)

The second order moment of the noise and interference-only part *x*<sup>O</sup> jn can be generally described as

$$\mathbf{Q}\_2^{\mathbf{O}} = \int\_{-t\_1}^{T\_\mathrm{p} - t\_1} \int\_0^{T\_\mathrm{p}} \left[ \left( P\_\mathrm{N} + P\_\mathrm{I} \right)^2 + 2 \left( R\_\mathrm{N} \left( \tau \right) + R\_\mathrm{I} \left( \tau \right) \right)^2 \right] \mathrm{d}t\_1 \mathrm{d}\tau. \tag{73}$$

Thereby, using the theorem of Price [44], (73) can be written in terms of the noise and interference related autocorrelation functions. With Parseval and the assumptions 2 *f*<sup>c</sup> � *B*, <sup>2</sup> *<sup>f</sup>*<sup>J</sup> � *<sup>B</sup>*<sup>J</sup> and <sup>|</sup> *<sup>f</sup>*<sup>c</sup> <sup>+</sup> *<sup>f</sup>*J| � � *<sup>B</sup>*<sup>J</sup> or � *B* − *B*<sup>J</sup> �� (73) results in

$$\begin{split} Q\_{2}^{\mathcal{O}} &= 2T\_{\mathcal{P}}^{2} \left[ P\_{\mathcal{I}}^{2} + P\_{\mathcal{I}} p\_{\mathcal{N}} + P\_{\mathcal{N}}^{2} + \sum\_{k=1}^{\infty} \frac{(-1)^{k} \left( 2\pi T\_{\mathcal{P}} \right)^{2k} \left( P\_{\mathcal{I}}^{2} B\_{\mathcal{I}}^{2k} + P\_{\mathcal{N}}^{2} B\_{\mathcal{I}}^{2k} \right)}{\left( 2k+1 \right)! \left( 2k+1 \right) \left( k+1 \right)} \\ &+ \sum\_{k=2}^{\infty} \frac{(-1)^{k} \left( 2\pi T\_{\mathcal{P}} \right)^{2k-2} \left( P\_{\mathcal{I}}^{2} B\_{\mathcal{I}}^{2k-2} + P\_{\mathcal{N}}^{2} B\_{\mathcal{I}}^{2k-2} \right)}{\left( k \right) \left( 2k \right)!} \right] \\ &+ \frac{2 P\_{\mathcal{I}} p\_{\mathcal{N}}}{\pi B\_{\mathcal{I}} \left( f\_{\mathcal{I}} - f\_{\mathcal{m}} \right)} \sum\_{k=0}^{\infty} \frac{(-1)^{k} \left( 2\pi \right)^{2k+1} T\_{\mathcal{P}}^{2k+2}}{\left( 2k+2 \right)!} \cdot \left( \frac{f\_{1}\left( k\right)}{2k+2} + \frac{f\_{2}\left( k\right)}{2k+1} \right) . \end{split} \tag{74}$$

Thereby, with *f*<sup>p</sup> = *<sup>B</sup>* <sup>2</sup> <sup>+</sup> *<sup>B</sup>*<sup>J</sup> <sup>2</sup> and *<sup>f</sup>*<sup>m</sup> <sup>=</sup> *<sup>B</sup>* <sup>2</sup> <sup>−</sup> *<sup>B</sup>*<sup>J</sup> <sup>2</sup> , *f*<sup>1</sup> (*k*) and *f*<sup>2</sup> (*k*) are defined as:

$$\begin{aligned} f\_1(k) &= \left(-f\_{\mathrm{m}} - \Delta\_{f\_{\mathrm{cJ}}}\right)^{2k+2} - \left(-f\_{\mathrm{P}} - \Delta\_{f\_{\mathrm{cJ}}}\right)^{2k+2} \\ &+ \left(-f\_{\mathrm{m}} + \Delta\_{f\_{\mathrm{cJ}}}\right)^{2k+2} - \left(-f\_{\mathrm{P}} + \Delta\_{f\_{\mathrm{cJ}}}\right)^{2k+2} \end{aligned}$$

and

$$\begin{split} f\_{2}\left(k\right) &= \left(f\_{\sf P} + \Delta\_{f\_{\sf cJ}}\right) \left[ \left(-f\_{\sf m} - \Delta\_{f\_{\sf cJ}}\right)^{2k+1} - \left(-f\_{\sf P} - \Delta\_{f\_{\sf cJ}}\right)^{2k+1} \right] \\ &+ \left(f\_{\sf P} - \Delta\_{f\_{\sf cJ}}\right) \left[ \left(-f\_{\sf m} + \Delta\_{f\_{\sf cJ}}\right)^{2k+1} - \left(-f\_{\sf P} + \Delta\_{f\_{\sf cJ}}\right)^{2k+1} \right] \\ &+ \left(f\_{\sf P} - f\_{\sf m}\right) \left[ \left(f\_{\sf m} - \Delta\_{f\_{\sf cJ}}\right)^{2k+1} - \left(-f\_{\sf m} - \Delta\_{f\_{\sf cJ}}\right)^{2k+1} \right]. \end{split}$$

*Q*<sup>O</sup> <sup>2</sup> is influenced by the system parameters *T*p, *f*c, *B* as well as by the interference parameters *P*J, *B*J, *f*J. However, in contrast to the second order moment *Q*<sup>O</sup> <sup>1</sup> it

$$\begin{array}{ll}\text{cancel} & \text{reduced via } E\_{\text{p}}^{\text{O}}. & \text{Eq.} & \text{(74) simplifies for } B\_{\text{l}} \to 0 \text{ due to } P\_{\text{l}}^{2}B\_{\text{l}}^{2k} = 0. \\\ P\_{\text{f}}^{2}B\_{\text{l}}^{2k-2} = 0, \frac{f\_{\text{f}}(k)}{f\_{\text{p}} - f\_{\text{m}}} = (2k+2) \left[ \left( -\frac{\text{B}}{2} - \Delta f\_{\text{cJ}} \right)^{2k+1} + \left( -\frac{\text{B}}{2} + \Delta f\_{\text{cJ}} \right)^{2k+1} \right] & \text{as well as } \frac{f\_{\text{f}}(k)}{f\_{\text{p}} - f\_{\text{m}}} = (2k+1) \left( \frac{\text{B}}{2} + \Delta f\_{\text{cJ}} \right)^{2k}. & \text{Assuming } P\_{\text{N}} = 0 \text{ (74) equals the result of [9]:} \end{array}$$

$$Q\_2^{\rm O} = 2T\_\text{P}^2 P\_\text{J}^2 + \frac{P\_\text{J}^2}{8\pi^2 f\_\text{J}^2} \left[1 - \cos\left(4\pi f\_\text{J} T\_\text{P}\right)\right].\tag{75}$$

**BPPM:** Considering BPPM the second order moment of the signal-noise and signal-interference part *x*<sup>P</sup> sjn is: *<sup>Q</sup>*<sup>P</sup> <sup>1</sup> <sup>=</sup> <sup>1</sup> <sup>2</sup> *<sup>Q</sup>*<sup>O</sup> <sup>1</sup> . *<sup>Q</sup>*<sup>P</sup> <sup>1</sup> differs from *<sup>Q</sup>*<sup>O</sup> <sup>1</sup> solely in a factor of two which can be ascribed to the reduced modulation specific pulse energy. In contrast to *Q*<sup>P</sup> <sup>1</sup> there is a significant difference concerning the second order moment of the noise and interference-only part *x*<sup>P</sup> jn. With the theorem of Price this can be generally described in terms of the noise and interference specific autocorrelation functions: (*τ* = *t*<sup>1</sup> − *t*2)

$$\mathbf{Q}\_{2}^{\mathbf{P}} = \mathbf{4} \int\_{-t\_{1}}^{T\_{\mathrm{p}} - t\_{1}} \int\_{0}^{T\_{\mathrm{p}}} \left[ R\_{\mathrm{f}}^{2} \left( \tau \right) + R\_{\mathrm{N}}^{2} \left( \tau \right) + R\_{\mathrm{J}} \left( \tau \right) R\_{\mathrm{N}} \left( \tau \right) \right] \, \mathrm{d}t\_{1} \mathrm{d}\tau$$

$$\begin{aligned} & - \mathbf{4} \int\_{-t\_{1}}^{T\_{\mathrm{p}} - t\_{1}} \int\_{\frac{\tau\_{\mathrm{p}}}{2}}^{T\_{\mathrm{p}}} \left[ R\_{\mathrm{J}}^{2} \left( \tau \right) + R\_{\mathrm{N}}^{2} \left( \tau \right) + R\_{\mathrm{J}} \left( \tau \right) R\_{\mathrm{N}} \left( \tau \right) \right] \, \mathrm{d}t\_{1} \mathrm{d}\tau. \end{aligned} \tag{76}$$

Therefore, using the theorem of Parseval for 2 *f*<sup>J</sup> � *B*J, the closed-form result

$$\begin{split} \mathbf{Q}\_{2}^{\rm P} &= 2 \sum\_{k=1}^{\infty} \frac{(-1)^{k} \left(2\pi\right)^{2k} \left(P\_{\sf f}^{2} B\_{\sf f}^{2k} + P\_{\sf N}^{2} B\_{\sf T}^{2k}\right) \mathcal{g}\_{2k+2}}{(2k+1)! \left(2k+1\right) \left(k+1\right)} \\ &+ \sum\_{k=2}^{\infty} \frac{(-1)^{k} 2^{2k} \left(\pi\right)^{2k-2} \left(P\_{\sf f}^{2} B\_{\sf f}^{2k-2} + P\_{\sf N}^{2} B\_{\sf T}^{2k-2}\right) \mathcal{g}\_{2k}}{(2k)! \left(2k\right)} \\ &+ \frac{2P\_{\sf P} P\_{\sf N}}{\pi B\_{\sf T} \left(f\_{\sf P} - f\_{\sf m}\right)} \sum\_{k=0}^{\infty} \frac{(-1)^{k} \left(2\pi\right)^{2k+1} \mathcal{g}\_{2k+2}}{(2k+2)!} \cdot \left(\frac{f\_{1}\left(k\right)}{2k+2} + \frac{f\_{2}\left(k\right)}{2k+1}\right) \end{split} \tag{77}$$

with

28 Will-be-set-by-IN-TECH

*f*<sup>c</sup> − *f*<sup>J</sup> � *T*p �

*f*<sup>c</sup> + *f*<sup>J</sup> � *T*p

�<sup>2</sup> + 2 �

Thereby, using the theorem of Price [44], (73) can be written in terms of the noise and interference related autocorrelation functions. With Parseval and the assumptions 2 *f*<sup>c</sup> � *B*,

�� (73) results in

(−1) *k* � 2*πT*p

∞ ∑ *k*=1

*P*2 <sup>J</sup> *<sup>B</sup>*2*k*−<sup>2</sup> <sup>J</sup> <sup>+</sup> *<sup>P</sup>*<sup>2</sup>

(*k*) (2*k*)!

*<sup>k</sup>* (2*π*)

�2*k*+<sup>2</sup> − �

�2*k*+<sup>2</sup> − �

− *f*<sup>m</sup> − Δ*f*c,J

− *f*<sup>m</sup> + Δ*f*c,J

*f*<sup>m</sup> − Δ*f*c,J

*R*<sup>N</sup> (*τ*) + *R*<sup>J</sup> (*τ*)

�2 �

�2*<sup>k</sup>* � *P*2 <sup>J</sup> *<sup>B</sup>*2*<sup>k</sup>* <sup>J</sup> <sup>+</sup> *<sup>P</sup>*<sup>2</sup>

N*B*2*k*−<sup>2</sup> T

<sup>2</sup> , *f*<sup>1</sup> (*k*) and *f*<sup>2</sup> (*k*) are defined as:

− *f*<sup>p</sup> − Δ*f*c,J

− *f*<sup>p</sup> + Δ*f*c,J

<sup>2</sup>*k*+<sup>1</sup> *T*2*k*+<sup>2</sup> p (2*<sup>k</sup>* + <sup>2</sup>)! ·

> �2*k*+<sup>1</sup> − �

> �2*k*+<sup>1</sup> − �

�2*k*+<sup>1</sup> − �

<sup>2</sup> is influenced by the system parameters *T*p, *f*c, *B* as well as by the interference parameters *P*J, *B*J, *f*J. However, in contrast to the second order moment *Q*<sup>O</sup>

(2*k* + 1)!(2*k* + 1) (*k* + 1)

�

⎤ ⎦

� *f*<sup>1</sup> (*k*)

<sup>2</sup>*<sup>k</sup>* <sup>+</sup> <sup>2</sup> <sup>+</sup> *<sup>f</sup>*<sup>2</sup> (*k*)

�2*k*+<sup>2</sup>

�2*k*+<sup>2</sup>

− *f*<sup>p</sup> − Δ*f*c,J

− *f*<sup>p</sup> + Δ*f*c,J

− *f*<sup>m</sup> − Δ*f*c,J

2*k* + 1

�2*k*+<sup>1</sup> �

�2*k*+<sup>1</sup> �

�2*k*+<sup>1</sup> � . �

. (74)

<sup>1</sup> it

4*π f*c*T*p �

�� . (72)

jn can be generally

d*t*1d*τ*. (73)

N*B*2*<sup>k</sup>* T �

to [9] if *P*<sup>N</sup> = 0. In this case (71) simplifies to

<sup>1</sup> <sup>=</sup> *<sup>E</sup>*<sup>O</sup>

− 2 *f*<sup>J</sup> � *f*<sup>J</sup> + *f*<sup>c</sup> � cos � 2*π* �

− 2 *f*<sup>J</sup> � *f*<sup>J</sup> − *f*<sup>c</sup> � cos � 2*π* �

*T*p*π*<sup>2</sup> � *f* 2 <sup>c</sup> <sup>−</sup> *<sup>f</sup>* <sup>2</sup> J �2 � *f* 2 <sup>c</sup> + <sup>3</sup> *<sup>f</sup>* <sup>2</sup> <sup>J</sup> + � *f* 2 <sup>J</sup> <sup>−</sup> *<sup>f</sup>* <sup>2</sup> c � cos �

*<sup>T</sup>*p−*t*<sup>1</sup> �

*T*p �

��

<sup>J</sup> <sup>+</sup> *<sup>P</sup>*J*P*<sup>N</sup> <sup>+</sup> *<sup>P</sup>*<sup>2</sup>

<sup>2</sup> and *<sup>f</sup>*<sup>m</sup> <sup>=</sup> *<sup>B</sup>*

�

+ �

*f*<sup>p</sup> + Δ*f*c,J

*f*<sup>p</sup> − Δ*f*c,J

*f*<sup>p</sup> − *f*<sup>m</sup>

0

*<sup>B</sup>*<sup>J</sup> or �

−*t*<sup>1</sup>

(−1) *k* � 2*πT*p

2*P*J*P*<sup>N</sup>

*f*<sup>1</sup> (*k*) =

�

+ �

+ �

<sup>p</sup> *P*<sup>J</sup>

The second order moment of the noise and interference-only part *x*<sup>O</sup>

*P*<sup>N</sup> + *P*<sup>J</sup>

*B* − *B*<sup>J</sup>

<sup>N</sup> +

�2*k*−<sup>2</sup> �

(−1)

<sup>2</sup> <sup>−</sup> *<sup>B</sup>*<sup>J</sup>

− *f*<sup>m</sup> − Δ*f*c,J

− *f*<sup>m</sup> + Δ*f*c,J

� ��

� ��

� ��

*Q*<sup>O</sup>

*Q*<sup>O</sup> <sup>2</sup> =

<sup>2</sup> *<sup>f</sup>*<sup>J</sup> � *<sup>B</sup>*<sup>J</sup> and <sup>|</sup> *<sup>f</sup>*<sup>c</sup> <sup>+</sup> *<sup>f</sup>*J| � �

*Q*<sup>O</sup> <sup>2</sup> <sup>=</sup> <sup>2</sup>*T*<sup>2</sup> p ⎡ <sup>⎣</sup>*P*<sup>2</sup>

> + ∞ ∑ *k*=2

+

Thereby, with *f*<sup>p</sup> = *<sup>B</sup>*

and

*Q*<sup>O</sup>

*πB* � *f*<sup>p</sup> − *f*<sup>m</sup> � ∞ ∑ *k*=0

*f*<sup>2</sup> (*k*) =

<sup>2</sup> <sup>+</sup> *<sup>B</sup>*<sup>J</sup>

described as

$$\mathcal{g}\_{\boldsymbol{V}} = 2T\_{\rm p}^{\boldsymbol{V}} - \left(T\_{\rm p} - \frac{T\_{\rm b}}{2}\right)^{\boldsymbol{\nu}} + 2\left(\frac{T\_{\rm b}}{2}\right)^{\boldsymbol{\nu}} - \left(T\_{\rm p} + \frac{T\_{\rm b}}{2}\right)^{\boldsymbol{\nu}}$$

can be found. *Q*<sup>P</sup> <sup>2</sup> is influenced by the system parameters *T*p, *T*b, *f*c, *B* as well as by the interference parameters *P*J, *B*J, *f*J. Similar to *Q*<sup>O</sup> <sup>2</sup> it cannot be reduced via *<sup>E</sup>*<sup>P</sup> p. Eq. (77) reveals that for low data rates (*T*<sup>b</sup> <sup>→</sup> <sup>∞</sup>) *<sup>g</sup><sup>ν</sup>* <sup>≈</sup> <sup>2</sup>*T<sup>ν</sup>* <sup>p</sup> resulting in a negligible influence of *Q*<sup>P</sup> <sup>2</sup> . In contrast, the larger the data rate the higher its impact, e.g., for *T*<sup>b</sup> = 2*T*<sup>p</sup> *g<sup>ν</sup>* conducts to *g<sup>ν</sup>* = 4*T<sup>ν</sup>* <sup>p</sup> <sup>−</sup> 2*T*p *ν* . In addition, for *B*<sup>J</sup> → 0 and *P*<sup>N</sup> = 0, (77) allows the same simplifications as for

#### 30 Will-be-set-by-IN-TECH 30 Ultra-Wideband Radio Technologies for Communications, Localization and Sensor Applications MIRA – Physical Layer Optimisation for the Multiband Impulse Radio UWB Architecture <sup>31</sup>

*Q*<sup>O</sup> <sup>2</sup> . In this case (77) conducts to

$$Q\_2^{\mathbb{P}} = \frac{P\_\mathcal{I}^2}{8\pi^2 f\_\mathcal{I}^2} \left[ 2 - 2\cos\left(2\pi f\_\mathcal{I} T\_\mathcal{b} \right) + \cos\left(2\pi f\_\mathcal{I} \left(T\_\mathcal{b} - 2T\_\mathcal{P}\right)\right) \right. \\ \left. \left. \left. 2\pi f\_\mathcal{I} \left(T\_\mathcal{b} + 2T\_\mathcal{P}\right) \right) \right] \\ \left. \left. 2\pi f\_\mathcal{I} \left(T\_\mathcal{b} + 2T\_\mathcal{P}\right) \right) \right] . \tag{78}$$

**Figure 18.** Processing gain of an OOK and BPPM specific energy detection vs. SINRE with

and *d*<sup>J</sup> (*B*J,2) = 0.0244 for *T*<sup>b</sup> = 2*T*<sup>p</sup> (a) and *T*<sup>b</sup> = 4*T*<sup>p</sup> (b).

**4.2. Coexistence-based approaches**

and easy-to-realise interference mitigation.

increase of the modulation specific PG can be achieved for *d*<sup>s</sup> = <sup>1</sup>

*B*J,1 = 20 MHz and *B*J,2 = 400 MHz, *f*<sup>J</sup> = *f*<sup>c</sup> + 50 MHz, SNRE = 10 dB, *T*b,J = 102.4 ns, *d*<sup>J</sup> (*B*J,1) = 0.4883

MIRA – Physical Layer Optimisation for the Multiband Impulse Radio UWB Architecture 31

well as for *B*J,1 = 20 MHz and *B*J,2 = 400 MHz. As long as the interference is completely inside the subband *T*p,J, *d*<sup>J</sup> and hence *P*<sup>J</sup> are fix. In particular, the modulation specific PG at *f*<sup>J</sup> = *f*<sup>c</sup> + 50 MHz coincides with the one of Fig. 18. In addition, both modulation schemes show an increase of PG the more the interference is located at the subband's boundary (*f*<sup>J</sup> <sup>=</sup> *<sup>f</sup>*<sup>c</sup> <sup>±</sup> <sup>1</sup>

This can be on one hand ascribed to the subband pulse's sinc spectrum which is zero at the subband's boundary. On the other hand, the more *f*<sup>J</sup> is located at the subband's boundary the minor the interference bandwidth falling into the subband. In case interference overlaps with the subband's boundary, the effective interference parameters *B*J, *f*<sup>J</sup> and *d*<sup>J</sup> changes resulting in a reduction of the actual interference power *P*J. Fig. 19 (b) shows again that a significant

pulse shaping the increase depends strongly on the interferer's position inside the subband.

The MIR-UWB system has no exclusive frequency range within the available transmission bandwidth. For this reason there is an increased interference potential from a possible large number of radio systems operating in the same frequency domain. As shown in [12] the impact of interference can result in a significant decrease of the bit error rate (BER) performance of the MIR-UWB system. To maintain system performance, the following section gives a short overview of various coexistence-based approaches with respect to an efficient

Coexistence-based approaches aim at the reliable on-line mitigation of interferences which occur in the environment of the MIR-UWB system. Thus, a best possible trade-off between a maximum data rate and a minimum BER can be obtained for arbitrary interference situations. In this context, an essential requirement is the integration of coexistence-based approaches into the existing MIR-UWB system with only minor complexity increase. Thus, the MIR-UWB system configuration should not be changed in presence of interference on one hand; on

*<sup>T</sup>*<sup>p</sup> ).

<sup>4</sup> . However, without proper

### *4.1.3. Results*

Based on the previous analysis the interference robustness of an OOK/BPPM based energy detection receiver can be identified. Thereby, assuming regulation of ECC [14] an MIR-UWB system with four subbands of equal bandwidth *B* = 625 MHz is taken into account. Without loss of generality, the analysis focuses solely on the first subband located at *f*<sup>c</sup> = 6.3125 GHz. However, an extension to other subbands or other MIR-UWB system configurations, which are possibly based on other frequency masks is easily possible. Further common system parameters used in the following are the pulse duration *T*<sup>p</sup> = 3.2 ns, a mean transmit power normalized to one, the modulation specific pulse energy *E<sup>i</sup>* p, *i* ∈ {O, P} as well as a constant SNRE = 10 dB at the input of the energy detector. Fixed interference parameter is the interference specific bit duration *T*b,J = 16*T*<sup>b</sup> = 102.4 ns.

In Fig. 18 (a) the PG is plotted vs. the SINRE for the duty cycle *d*<sup>s</sup> = <sup>1</sup> <sup>2</sup> . An interference with the two bandwidths *B*J,1 = 20 MHz or *B*J,2 = 400 MHz located at *f*<sup>J</sup> = *f*<sup>c</sup> + 50 MHz is considered leading to the fixed duty cycles *d*J,1 = <sup>1</sup> *<sup>B</sup>*J,1*T*b,J <sup>=</sup> 0.4883 or *<sup>d</sup>*J,2 <sup>=</sup> <sup>1</sup> *<sup>B</sup>*J,2*T*b,J = 0.0244. For OOK/BPPM, the PG increases with higher SINRE up to the interference-free PG at SNRE = 10 dB. Furthermore, it can be observed that the OOK/BPPM based PG varies with the interference bandwidth. For OOK, the PG increases with a larger interference bandwidth because of the minor impact of the mixed signal-interference as well as the interference-only component involved in the energy detection. A PG of energy detection can be achieved from an SINRE = −3.5 dB (*B*J,1 = 20 MHz) and from an SINRE = −5.5 dB (*B*J,2 = 400 MHz), respectively. For strong interference no PG results as the energy detector's decision variable is significantly corrupted. In contrast, for BPPM a PG can be achieved for small interference bandwidths, e.g., *B*J,1 = 20 MHz, over nearly the complete SINRE range. For *B*J,2 = 400 MHz a PG occurs from SINRE = −2 dB. The reason for this behaviour lies in a different amount of energy resulting from the mixed signal-interference and interference-only term within the two observation periods of duration *T*p. Finally, the consideration of OOK/BPPM with respect to their relative PG shows that for strong NBI BPPM is more robust whereas OOK is more robust for mean and low interference.

The detector efficiency in terms of PG can be increased by increasing *T*b, e.g., with multiples of *T*p. This is illustrated in Fig. 18 (b) for *d*<sup>s</sup> = <sup>1</sup> <sup>4</sup> . The enlargement of *T*<sup>b</sup> via *d*<sup>s</sup> results in an increase of *E*<sup>b</sup> for fixed signal power. Thereby, the interference related second order moments *Qi* <sup>1</sup> and *<sup>Q</sup><sup>i</sup>* <sup>2</sup>, *i* ∈ {O, P} will be reduced as the interferer's energy is only collected during integration time *T*<sup>p</sup> within *T*b. Larger *T*<sup>b</sup> can be implemented into an MIR-UWB transmitter with minor complexity. However, the trade-off to increase the detector's interference robustness is a reduction of data rate.

Fig. 19 (a) shows the PG of an OOK and BPPM specific energy detection vs. *f*<sup>J</sup> for *d*<sup>s</sup> = <sup>1</sup> 2 , which varies from *<sup>f</sup>*<sup>c</sup> <sup>−</sup> *<sup>B</sup>* <sup>2</sup> to *<sup>f</sup>*<sup>c</sup> <sup>+</sup> *<sup>B</sup>* <sup>2</sup> for fixed SINRE = 0 dB, SNRE = 10 dB, *T*b,J = 102.4 ns as

**Figure 18.** Processing gain of an OOK and BPPM specific energy detection vs. SINRE with *B*J,1 = 20 MHz and *B*J,2 = 400 MHz, *f*<sup>J</sup> = *f*<sup>c</sup> + 50 MHz, SNRE = 10 dB, *T*b,J = 102.4 ns, *d*<sup>J</sup> (*B*J,1) = 0.4883 and *d*<sup>J</sup> (*B*J,2) = 0.0244 for *T*<sup>b</sup> = 2*T*<sup>p</sup> (a) and *T*<sup>b</sup> = 4*T*<sup>p</sup> (b).

well as for *B*J,1 = 20 MHz and *B*J,2 = 400 MHz. As long as the interference is completely inside the subband *T*p,J, *d*<sup>J</sup> and hence *P*<sup>J</sup> are fix. In particular, the modulation specific PG at *f*<sup>J</sup> = *f*<sup>c</sup> + 50 MHz coincides with the one of Fig. 18. In addition, both modulation schemes show an increase of PG the more the interference is located at the subband's boundary (*f*<sup>J</sup> <sup>=</sup> *<sup>f</sup>*<sup>c</sup> <sup>±</sup> <sup>1</sup> *<sup>T</sup>*<sup>p</sup> ). This can be on one hand ascribed to the subband pulse's sinc spectrum which is zero at the subband's boundary. On the other hand, the more *f*<sup>J</sup> is located at the subband's boundary the minor the interference bandwidth falling into the subband. In case interference overlaps with the subband's boundary, the effective interference parameters *B*J, *f*<sup>J</sup> and *d*<sup>J</sup> changes resulting in a reduction of the actual interference power *P*J. Fig. 19 (b) shows again that a significant increase of the modulation specific PG can be achieved for *d*<sup>s</sup> = <sup>1</sup> <sup>4</sup> . However, without proper pulse shaping the increase depends strongly on the interferer's position inside the subband.

### **4.2. Coexistence-based approaches**

30 Will-be-set-by-IN-TECH

2*π f*J*T*<sup>b</sup> + cos

> 2*π f*<sup>J</sup>

Based on the previous analysis the interference robustness of an OOK/BPPM based energy detection receiver can be identified. Thereby, assuming regulation of ECC [14] an MIR-UWB system with four subbands of equal bandwidth *B* = 625 MHz is taken into account. Without loss of generality, the analysis focuses solely on the first subband located at *f*<sup>c</sup> = 6.3125 GHz. However, an extension to other subbands or other MIR-UWB system configurations, which are possibly based on other frequency masks is easily possible. Further common system parameters used in the following are the pulse duration *T*<sup>p</sup> = 3.2 ns, a mean transmit power

SNRE = 10 dB at the input of the energy detector. Fixed interference parameter is the

with the two bandwidths *B*J,1 = 20 MHz or *B*J,2 = 400 MHz located at *f*<sup>J</sup> = *f*<sup>c</sup> + 50 MHz is

For OOK/BPPM, the PG increases with higher SINRE up to the interference-free PG at SNRE = 10 dB. Furthermore, it can be observed that the OOK/BPPM based PG varies with the interference bandwidth. For OOK, the PG increases with a larger interference bandwidth because of the minor impact of the mixed signal-interference as well as the interference-only component involved in the energy detection. A PG of energy detection can be achieved from an SINRE = −3.5 dB (*B*J,1 = 20 MHz) and from an SINRE = −5.5 dB (*B*J,2 = 400 MHz), respectively. For strong interference no PG results as the energy detector's decision variable is significantly corrupted. In contrast, for BPPM a PG can be achieved for small interference bandwidths, e.g., *B*J,1 = 20 MHz, over nearly the complete SINRE range. For *B*J,2 = 400 MHz a PG occurs from SINRE = −2 dB. The reason for this behaviour lies in a different amount of energy resulting from the mixed signal-interference and interference-only term within the two observation periods of duration *T*p. Finally, the consideration of OOK/BPPM with respect to their relative PG shows that for strong NBI BPPM is more robust whereas OOK is more robust

The detector efficiency in terms of PG can be increased by increasing *T*b, e.g., with multiples

increase of *E*<sup>b</sup> for fixed signal power. Thereby, the interference related second order moments

integration time *T*<sup>p</sup> within *T*b. Larger *T*<sup>b</sup> can be implemented into an MIR-UWB transmitter with minor complexity. However, the trade-off to increase the detector's interference

Fig. 19 (a) shows the PG of an OOK and BPPM specific energy detection vs. *f*<sup>J</sup> for *d*<sup>s</sup> = <sup>1</sup>

<sup>2</sup>, *i* ∈ {O, P} will be reduced as the interferer's energy is only collected during

2*π f*<sup>J</sup> 

*T*<sup>b</sup> + 2*T*<sup>p</sup>

*T*<sup>b</sup> − 2*T*<sup>p</sup>

*<sup>B</sup>*J,1*T*b,J <sup>=</sup> 0.4883 or *<sup>d</sup>*J,2 <sup>=</sup> <sup>1</sup>

<sup>4</sup> . The enlargement of *T*<sup>b</sup> via *d*<sup>s</sup> results in an

<sup>2</sup> for fixed SINRE = 0 dB, SNRE = 10 dB, *T*b,J = 102.4 ns as

. (78)

p, *i* ∈ {O, P} as well as a constant

<sup>2</sup> . An interference

*<sup>B</sup>*J,2*T*b,J = 0.0244.

2 ,

<sup>2</sup> <sup>−</sup> 2 cos

4*π f*J*T*<sup>p</sup> + cos

*Q*<sup>O</sup>

*4.1.3. Results*

<sup>2</sup> . In this case (77) conducts to

*Q*<sup>P</sup>

<sup>2</sup> <sup>=</sup> *<sup>P</sup>*<sup>2</sup> J 8*π*<sup>2</sup> *f* <sup>2</sup> J 

<sup>−</sup> 2 cos

normalized to one, the modulation specific pulse energy *E<sup>i</sup>*

In Fig. 18 (a) the PG is plotted vs. the SINRE for the duty cycle *d*<sup>s</sup> = <sup>1</sup>

interference specific bit duration *T*b,J = 16*T*<sup>b</sup> = 102.4 ns.

considered leading to the fixed duty cycles *d*J,1 = <sup>1</sup>

for mean and low interference.

robustness is a reduction of data rate.

which varies from *<sup>f</sup>*<sup>c</sup> <sup>−</sup> *<sup>B</sup>*

*Qi*

<sup>1</sup> and *<sup>Q</sup><sup>i</sup>*

of *T*p. This is illustrated in Fig. 18 (b) for *d*<sup>s</sup> = <sup>1</sup>

<sup>2</sup> to *<sup>f</sup>*<sup>c</sup> <sup>+</sup> *<sup>B</sup>*

The MIR-UWB system has no exclusive frequency range within the available transmission bandwidth. For this reason there is an increased interference potential from a possible large number of radio systems operating in the same frequency domain. As shown in [12] the impact of interference can result in a significant decrease of the bit error rate (BER) performance of the MIR-UWB system. To maintain system performance, the following section gives a short overview of various coexistence-based approaches with respect to an efficient and easy-to-realise interference mitigation.

Coexistence-based approaches aim at the reliable on-line mitigation of interferences which occur in the environment of the MIR-UWB system. Thus, a best possible trade-off between a maximum data rate and a minimum BER can be obtained for arbitrary interference situations. In this context, an essential requirement is the integration of coexistence-based approaches into the existing MIR-UWB system with only minor complexity increase. Thus, the MIR-UWB system configuration should not be changed in presence of interference on one hand; on

#### 32 Will-be-set-by-IN-TECH 32 Ultra-Wideband Radio Technologies for Communications, Localization and Sensor Applications MIRA – Physical Layer Optimisation for the Multiband Impulse Radio UWB Architecture <sup>33</sup>

a characteristic weighting factor *k* is used to manipulate the instantaneous energy value's influence on the recursive estimation. Simulation results [6] show a robust interference

MIRA – Physical Layer Optimisation for the Multiband Impulse Radio UWB Architecture 33

In [10], a simple cluster-based coexistence approach (coexistence approach 1) is analysed resulting in a decision threshold th with respect to interfered or not interfered noise energy values. Based on the image-based thresholding method of [41] occupied time-frequency slots can be automatically detected and recorded within an extended time-frequency band plan

Assuming knowledge of the interferers' periodicity and perfect synchronisation, *m*<sup>0</sup> binary zeros are sent over each subband *i* ∈ {1, . . . , *N*sub}. The measured energy values are written

quantized energy levels *E*min = <sup>1</sup> < <sup>2</sup> < ... < *<sup>U</sup>* = *E*max. Thereby, *E*min and *E*max stands for the minimum and the maximum occurring energy value in *X*. The allocation leads to an

To obtain a separation between interfered and not interfered energy values, two energy classes *C*<sup>0</sup> = {1,..., *<sup>u</sup>*} and *C*<sup>1</sup> = {*<sup>u</sup>*+1,..., *<sup>U</sup>*} can be defined, which include interfered and not

**Figure 20.** Energy classes, threshold determination. Sub-figure (a) shows the average of all values,

sub-figure (b) optimizes according to eq. (81) and sub-figure (c) according to eq. (84).

*pi* <sup>=</sup> *ni m*0*N*sub

, *j* = 1, . . . , *m*0, *i* = 1, . . . , *N*sub and assigned to their nearest

, *i* ∈ 1, . . . , *U*. (79)

detection with only a marginal BER performance loss for *k* = 10.

*4.2.3. Image-based thresholding*

as an energy matrix *X* =

used for initialisation or data transmission.

energy distribution, which is described as

interfered energy values (Fig. 20 (a))

 *xj*,*<sup>i</sup>* 

**Figure 19.** Processing gain of an OOK and BPPM specific energy detection vs. *f*<sup>J</sup> for SINRE = 0 dB, SNR = 10 dB, *B*J,1 = 20 MHz, *B*J,2 = 400 MHz and *T*b,J = 102.4 ns for *T*<sup>b</sup> = 2*T*<sup>p</sup> (a) as well as *T*<sup>b</sup> = 4*T*<sup>p</sup> (b).

the other hand it should be possible to realise coexistence without complex estimations of interference specific parameters such as the interference power, the interference bandwidth, the interference carrier frequency or the number of instantaneous available interferences.

### *4.2.1. Static coexistence approach*

From a complexity point of view a static coexistence approach should be used. Thereby interfered subbands are deactivated by means of the system specific band plan before any data transmission occurs. Best trade-off between system performance and effort for interference handling will be achieved. However, this approach does not consider dynamic interference situations and hence does not contribute to efficient spectrum usage. Therefrom, the necessity for further efficient and low complex, but more flexible alternatives arises.

### *4.2.2. Detect and Avoid (DAA)*

In [6, 34] an easy-to-realise DAA approach is presented allowing a reliable detection of temporary NBI after system initialisation or within data transmission.

For this purpose, the regularly transmitted preamble is adjusted to simultaneous subband specific signal and noise energy estimation used by the DAA approach. Thereby, a static interference-free working point (WP) is defined, ensuring a determinated BER in each subband. After initialisation, the estimated SNRs are compared with the WP, leading to subband deactivation, if the SNR is lower than the WP. Otherwise, the subband is (re)enabled for data transmission.

During data transmission, the initially estimated signal and noise energy values of enabled subbands are recursively updated in dependence of the actual subband specific bit decision to adapt the initial decision threshold. In addition, this process allows the possibility of fast and reliable detection of suddenly weak or strong interfered subbands. For this reason, a characteristic weighting factor *k* is used to manipulate the instantaneous energy value's influence on the recursive estimation. Simulation results [6] show a robust interference detection with only a marginal BER performance loss for *k* = 10.

### *4.2.3. Image-based thresholding*

32 Will-be-set-by-IN-TECH

**Figure 19.** Processing gain of an OOK and BPPM specific energy detection vs. *f*<sup>J</sup> for SINRE = 0 dB, SNR = 10 dB, *B*J,1 = 20 MHz, *B*J,2 = 400 MHz and *T*b,J = 102.4 ns for *T*<sup>b</sup> = 2*T*<sup>p</sup> (a) as well as *T*<sup>b</sup> = 4*T*<sup>p</sup> (b). the other hand it should be possible to realise coexistence without complex estimations of interference specific parameters such as the interference power, the interference bandwidth, the interference carrier frequency or the number of instantaneous available interferences.

From a complexity point of view a static coexistence approach should be used. Thereby interfered subbands are deactivated by means of the system specific band plan before any data transmission occurs. Best trade-off between system performance and effort for interference handling will be achieved. However, this approach does not consider dynamic interference situations and hence does not contribute to efficient spectrum usage. Therefrom, the necessity

In [6, 34] an easy-to-realise DAA approach is presented allowing a reliable detection of

For this purpose, the regularly transmitted preamble is adjusted to simultaneous subband specific signal and noise energy estimation used by the DAA approach. Thereby, a static interference-free working point (WP) is defined, ensuring a determinated BER in each subband. After initialisation, the estimated SNRs are compared with the WP, leading to subband deactivation, if the SNR is lower than the WP. Otherwise, the subband is (re)enabled

During data transmission, the initially estimated signal and noise energy values of enabled subbands are recursively updated in dependence of the actual subband specific bit decision to adapt the initial decision threshold. In addition, this process allows the possibility of fast and reliable detection of suddenly weak or strong interfered subbands. For this reason,

for further efficient and low complex, but more flexible alternatives arises.

temporary NBI after system initialisation or within data transmission.

*4.2.1. Static coexistence approach*

*4.2.2. Detect and Avoid (DAA)*

for data transmission.

In [10], a simple cluster-based coexistence approach (coexistence approach 1) is analysed resulting in a decision threshold th with respect to interfered or not interfered noise energy values. Based on the image-based thresholding method of [41] occupied time-frequency slots can be automatically detected and recorded within an extended time-frequency band plan used for initialisation or data transmission.

Assuming knowledge of the interferers' periodicity and perfect synchronisation, *m*<sup>0</sup> binary zeros are sent over each subband *i* ∈ {1, . . . , *N*sub}. The measured energy values are written as an energy matrix *X* = *xj*,*<sup>i</sup>* , *j* = 1, . . . , *m*0, *i* = 1, . . . , *N*sub and assigned to their nearest quantized energy levels *E*min = <sup>1</sup> < <sup>2</sup> < ... < *<sup>U</sup>* = *E*max. Thereby, *E*min and *E*max stands for the minimum and the maximum occurring energy value in *X*. The allocation leads to an energy distribution, which is described as

$$p\_{\dot{i}} = \frac{n\_{\dot{i}}}{m\_0 N\_{\text{sub}}}, i \in 1, \dots, U. \tag{79}$$

To obtain a separation between interfered and not interfered energy values, two energy classes *C*<sup>0</sup> = {1,..., *<sup>u</sup>*} and *C*<sup>1</sup> = {*<sup>u</sup>*+1,..., *<sup>U</sup>*} can be defined, which include interfered and not interfered energy values (Fig. 20 (a))

**Figure 20.** Energy classes, threshold determination. Sub-figure (a) shows the average of all values, sub-figure (b) optimizes according to eq. (81) and sub-figure (c) according to eq. (84).

#### 34 Will-be-set-by-IN-TECH 34 Ultra-Wideband Radio Technologies for Communications, Localization and Sensor Applications MIRA – Physical Layer Optimisation for the Multiband Impulse Radio UWB Architecture <sup>35</sup>

In order to classify energy values, index *u* has to be determined using two optimisation criteria.

The first optimisation criterion consists of the minimisation of the combined empirical energy class variance

$$s\_{\rm W}^2(\mu) = s\_{\rm C\_0}^2(\mu) P\_{\rm C\_0}(\mu) + s\_{\rm C\_1}^2(\mu) P\_{\rm C\_1}(\mu) \,. \tag{80}$$

received energy values *xj*,*i*, *j* = 1, . . . , *m*0, *i* = 1, . . . , *N*sub, which are logged in band plan *L*. In

MIRA – Physical Layer Optimisation for the Multiband Impulse Radio UWB Architecture 35

**Figure 21.** Schematic procedure of global iterative method (coexistence approach 2) and local

*�*th ?

Since the main drawback of the global coexistence approach is its reduced efficiency in case of simultaneously operating broad- or narrowband interference, it should be improved by using iterative coexistence methods [12]. In a global iterative method (coexistence approach 2, left side of Fig. 21) the plausibility of the resulting interference detection threshold is verified via

for which Gaussian distribution as well as an interference-free noise source *n* (*t*) with mean *μ*<sup>n</sup> and standard deviation *σ*<sup>n</sup> is assumed at the receiver side. If *�*th exceeds the confidence

Simultaneously, all labelled possibly corrupted noise energy values are replaced with artificially generated noise energy values resulting from the noise source mentioned above. This procedure is iteratively repeated until the termination criterion is achieved. In this case, the final iterative band plan is delivered and used for initialisation and data transmission. In contrast, a local hierarchical iterative method (coexistence approach 3, right side of Fig. 21) handles every subband individually to achieve a local based interference detection threshold. The subband with maximum occurring energy value is identified and delivered to receiver side interference detection in conjunction with a sufficient number of additional artificial noise energy values of the above mentioned noise source. For the detection of the coexistence approach's termination, the 3*σn* standard deviation is used again. If the interference threshold lies above the confidence interval, detected noise cells of the considered subband are labelled

≥ *μ*<sup>n</sup> + 3*σ*<sup>n</sup> (86)

hierarchical method (coexistence approach 3).

the 3*σ*n standard deviation termination criterion

interval, deactivation flags are logged within the band plan.

this context, we define an energy cell as interfered if *xj*,*<sup>i</sup>* ≥ *�*th.

This step aims at an adjustment of the initially set index *u*, whereas a correct allocation of energy outliers to the corresponding other class is achieved (Fig. 20 (b)). It is based on a weighted sum of the classes' occurrence probabilities

$$P\_{\mathbb{C}\_0}(\boldsymbol{\mu}) = \sum\_{i=1}^{\mu} p\_{i\prime} \qquad \qquad P\_{\mathbb{C}\_1}(\boldsymbol{\mu}) = 1 - P\_{\mathbb{C}\_0}(\boldsymbol{\mu}) \, \, \, \, \tag{81}$$

as well as on the empirical energy classes' variances

$$s\_{\mathbb{C}\_l}^2(u) = \begin{cases} \sum\_{i=1}^{\mathcal{U}} \left(\mathfrak{c}\_i - \mathfrak{x}\_{\mathbb{C}\_l}(u)\right)^2 \frac{p\_l}{\mathbb{P}\_{\mathbb{C}\_l}(u)}, & l=0\\ \sum\_{i=u+1}^{\mathcal{U}} \left(\mathfrak{c}\_i - \mathfrak{x}\_{\mathbb{C}\_l}(u)\right)^2 \frac{p\_l}{\mathbb{P}\_{\mathbb{C}\_l}(u)}, & l=1 \end{cases} \tag{82}$$

with

$$\bar{\mathbf{x}}\_{\mathbb{C}\_{l}}(u) = \begin{cases} \sum\_{i=1}^{u} \frac{\varepsilon\_{i} p\_{i}}{P\_{\mathbb{C}\_{l}}(u)}, & l = 0 \\ \sum\_{i=u+1}^{U} \frac{\varepsilon\_{i} p\_{i}}{P\_{\mathbb{C}\_{l}}(u)}, & l = 1 \end{cases} \tag{83}$$

describing the empirical energy class mean levels.

The second optimisation criterion considers the maximisation of the empirical variance between both energy classes

$$s\_{\mathbf{b}}^2(\boldsymbol{\mu}) = \left(\boldsymbol{\overline{x}\_{\mathbb{C}\_0}}(\boldsymbol{\mu}) - \boldsymbol{\overline{x}\_{\text{tot}}}\right)^2 P\_{\mathbb{C}\_0}(\boldsymbol{\mu}) + \left(\boldsymbol{\overline{x}\_{\mathbb{C}\_1}}(\boldsymbol{\mu}) - \boldsymbol{\overline{x}\_{\text{tot}}}\right)^2 P\_{\mathbb{C}\_1}(\boldsymbol{\mu}) \tag{84}$$

standing for the weighted variance of the energy class means *x*¯*Ci* , *i* = 0, 1 themselves around the total mean *x*¯tot = *x*¯*C*<sup>0</sup> + *x*¯*C*<sup>1</sup> of the time-frequency pattern (Fig. 20 (c)). Hence, a separation of both classes with respect to the mean value of the total time-frequency pattern is obtained, leading to a more accurate adaptation of index *u*.

As both optimisation criteria have opposing effects with respect to the best index *u*, they are combined into one characteristic optimisation criterion which is defined as [41]:

$$
\mu^\star = \arg\max\_{\boldsymbol{u}=1,\ldots,\boldsymbol{U}} \frac{s\_\mathbf{b}^2(\boldsymbol{u})}{s\_\mathbf{w}^2(\boldsymbol{u})}.\tag{85}
$$

This leads to an adjustment of the initially arbitrary index *u*. Thereby, a correct allocation of energy outliers to the corresponding energy classes is achieved. Evaluated index *u* leads to interference detection threshold th = *<sup>u</sup>* . Afterwards a binary decision has to be done for all received energy values *xj*,*i*, *j* = 1, . . . , *m*0, *i* = 1, . . . , *N*sub, which are logged in band plan *L*. In this context, we define an energy cell as interfered if *xj*,*<sup>i</sup>* ≥ *�*th.

34 Will-be-set-by-IN-TECH

In order to classify energy values, index *u* has to be determined using two optimisation

The first optimisation criterion consists of the minimisation of the combined empirical energy

This step aims at an adjustment of the initially set index *u*, whereas a correct allocation of energy outliers to the corresponding other class is achieved (Fig. 20 (b)). It is based on a

2

�<sup>2</sup> *pi PCl*

> �<sup>2</sup> *pi PCl*

(*u*), *l* = 0

�<sup>2</sup> *PC*<sup>0</sup> (*u*) <sup>+</sup> (*x*¯*C*<sup>1</sup> (*u*) <sup>−</sup> *<sup>x</sup>*¯*tot*)

*s*2 <sup>b</sup>(*u*) *s*2 <sup>w</sup> (*u*)

(*u*), *l* = 1

*<sup>C</sup>*<sup>1</sup> (*u*) *PC*<sup>1</sup> (*u*). (80)

(82)

(83)

<sup>2</sup> *PC*<sup>1</sup> (*u*) (84)

, *i* = 0, 1 themselves around

. (85)

*pi*, *PC*<sup>1</sup> (*u*) = 1 − *PC*<sup>0</sup> (*u*), (81)

(*u*), *l* = 0

(*u*), *l* = 1

*<sup>C</sup>*<sup>0</sup> (*u*) *PC*<sup>0</sup> (*u*) + *s*

*<sup>i</sup>* − *x*¯*Cl* (*u*)

*<sup>i</sup>* − *x*¯*Cl* (*u*)

*<sup>i</sup> pi PCl*

> *<sup>i</sup> pi PCl*

criteria.

with

class variance

*s* 2 <sup>w</sup> (*u*) = *s*

weighted sum of the classes' occurrence probabilities

as well as on the empirical energy classes' variances

*s*2 *Cl* (*u*) =

describing the empirical energy class mean levels.

leading to a more accurate adaptation of index *u*.

between both energy classes

*s* 2 <sup>b</sup> (*u*) <sup>=</sup> �

*PC*<sup>0</sup> (*u*) =

2

*u* ∑ *i*=1

⎧ ⎪⎪⎨

*u* ∑ *i*=1 �

*U* ∑ *i*=*u*+1 �

> ⎧ ⎪⎪⎨

*u* ∑ *i*=1

*U* ∑ *i*=*u*+1

The second optimisation criterion considers the maximisation of the empirical variance

the total mean *x*¯tot = *x*¯*C*<sup>0</sup> + *x*¯*C*<sup>1</sup> of the time-frequency pattern (Fig. 20 (c)). Hence, a separation of both classes with respect to the mean value of the total time-frequency pattern is obtained,

As both optimisation criteria have opposing effects with respect to the best index *u*, they are

This leads to an adjustment of the initially arbitrary index *u*. Thereby, a correct allocation of energy outliers to the corresponding energy classes is achieved. Evaluated index *u* leads to interference detection threshold th = *<sup>u</sup>* . Afterwards a binary decision has to be done for all

combined into one characteristic optimisation criterion which is defined as [41]:

*<sup>u</sup>* <sup>=</sup> arg max *<sup>u</sup>*=1,...,*<sup>U</sup>*

⎪⎪⎩

⎪⎪⎩

*x*¯*Cl* (*u*) =

*x*¯*C*<sup>0</sup> (*u*) − *x*¯*tot*

standing for the weighted variance of the energy class means *x*¯*Ci*

**Figure 21.** Schematic procedure of global iterative method (coexistence approach 2) and local hierarchical method (coexistence approach 3).

Since the main drawback of the global coexistence approach is its reduced efficiency in case of simultaneously operating broad- or narrowband interference, it should be improved by using iterative coexistence methods [12]. In a global iterative method (coexistence approach 2, left side of Fig. 21) the plausibility of the resulting interference detection threshold is verified via the 3*σ*n standard deviation termination criterion

$$
\epsilon\_{\rm th} \stackrel{?}{\geq} \mu\_{\rm n} + \mathfrak{z}\sigma\_{\rm n} \tag{86}
$$

for which Gaussian distribution as well as an interference-free noise source *n* (*t*) with mean *μ*<sup>n</sup> and standard deviation *σ*<sup>n</sup> is assumed at the receiver side. If *�*th exceeds the confidence interval, deactivation flags are logged within the band plan.

Simultaneously, all labelled possibly corrupted noise energy values are replaced with artificially generated noise energy values resulting from the noise source mentioned above. This procedure is iteratively repeated until the termination criterion is achieved. In this case, the final iterative band plan is delivered and used for initialisation and data transmission.

In contrast, a local hierarchical iterative method (coexistence approach 3, right side of Fig. 21) handles every subband individually to achieve a local based interference detection threshold. The subband with maximum occurring energy value is identified and delivered to receiver side interference detection in conjunction with a sufficient number of additional artificial noise energy values of the above mentioned noise source. For the detection of the coexistence approach's termination, the 3*σn* standard deviation is used again. If the interference threshold lies above the confidence interval, detected noise cells of the considered subband are labelled

#### 36 Will-be-set-by-IN-TECH 36 Ultra-Wideband Radio Technologies for Communications, Localization and Sensor Applications MIRA – Physical Layer Optimisation for the Multiband Impulse Radio UWB Architecture <sup>37</sup>

*4.3.1. Teager-Kaiser operation*

<sup>0</sup>

than the one of an 1000 Hz signal.

Ψ *K*<sup>S</sup> ∑ *i*=1

Ψc 

cross component

Ψ (*x* (*t*)) = *A*2*ω*<sup>2</sup>

0 

as the resulting spectrum at the output of TK operation (b).

0

resulting signal at the output of TK operation conducts to

*xi* (*t*) = *K*S ∑ *i*=1

*xj* (*t*), *xi* (*t*)

 = Ψc*<sup>j</sup>* 

 A 2

be described as

The continuous TK operation is a non-linear differential operator of order two defined as

To illustrate the effectiveness of the TK operation the harmonic oscillation *x* (*t*) = *A* cos (*ω*0*t* + *φ*0) is considered. Using (87) the signal at the output of the TK operation can

Hence, for the special case of a simple harmonic oscillation the TK operation leads to a frequency shift to DC. Fig. 23 highlights the spectrum of a harmonic oscillation (a) as well

0 0

(a) (b) **Figure 23.** Effectiveness of TK operation for the harmonic oscillation *x* (*t*) = *A* cos (*ω*0*t* + *φ*0).

An interpretation of this behaviour is given in [22]. In case of a harmonic oscillation the output of the TK operation describes the required energy to generate the oscillation. Considering (88) the energy depends not only by the amplitude *A* but also by the oscillation frequency *ω*0. Thus, for a constant *A* the required energy to generate an exemplary 10 Hz signal is lower

When using the continuous TK operation of (87) basic operator specific rules have to be generally considered. A detailed description can be found in [23]. In context with interference mitigation the most important property can be the behaviour of the TK operation in presence of several overlapping signals. In case of *K*<sup>S</sup> overlapping signals *x*<sup>1</sup> (*t*), *x*<sup>2</sup> (*t*), . . ., *xK*<sup>S</sup> (*t*) the

It consists on one hand of *K*<sup>S</sup> summands describing the TK operation of the signals *x*<sup>1</sup> (*t*), *x*<sup>2</sup> (*t*), . . ., *xK*<sup>S</sup> (*t*). Furthermore, due to the non-linearity of the TK operation the additional

*xj* (*t*), *xi* (*t*)

*K*S−1 ∑ *j*=1

*K*S ∑ *i*=*j*+1

> + Ψc*<sup>i</sup>*

Ψc 

*xj* (*t*), *xi* (*t*)

*xi* (*t*), *xj* (*t*)

= 2*x*˙*<sup>j</sup>* (*t*) *x*˙*<sup>i</sup>* (*t*) − *xj* (*t*) *x*¨*<sup>i</sup>* (*t*) − *xi* (*t*) *x*¨*<sup>j</sup>* (*t*) (89)

. (88)

Ψ (*xi* (*t*)) +

sin2 (*ω*0*t* + *φ*0) + cos2 (*ω*0*t* + *φ*0)

F {x(t)} F { (x(t))}

<sup>2</sup> (*t*) <sup>−</sup> *<sup>x</sup>* (*t*) *<sup>x</sup>*¨ (*t*). (87)

MIRA – Physical Layer Optimisation for the Multiband Impulse Radio UWB Architecture 37

= *A*2*ω*<sup>2</sup> 0.

> <sup>2</sup> (A ) <sup>2</sup> 0

Ψ (*x* (*t*)) = *x*˙

**Figure 22.** BER vs. *Eb*/*N*<sup>0</sup> for AWGN and fixed SIRs of -10 dB (IEEE 802.11a WLAN [47]), 0 dB and 5 dB (IEEE 802.15.3a MB OFDM UWB 1 and 2 [2]) regarding an interferer related band plan, no coexistence approach and the coexistence approaches 1 to 3 [12].

within the band plan. In parallel, the corresponding subband is deactivated and the procedure is iteratively repeated. If the coexistence approach terminates, the binary band plan is allocated to the transmitter for initialisation and data transmission.

Performance analysis [12] demonstrates the high capability of both iterative coexistence methods in presence of multiple interferers having the same or different interference powers (Fig. 22).

### **4.3. Narrowband interference mitigation**

The MIR-UWB system is characterised by a particular high vulnerability to NBI as all interferences inside the passband of the analogue front-end are considered by the energy detection receiver. Its performance can decrease significantly resulting in an increase of the error probability. Hence, a crucial issue concerns the efficient mitigation of NBI [58].

This section analyses the potential of the non-linear Teager-Kaiser (TK) operation [22] to mitigate NBI without the knowledge of the interference related carrier frequency. Based on the definition of the TK operation and some of its most important properties the mitigation potential of a TK based energy detector is analysed. In this context a modified TK operation is introduced and compared with the traditional TK operation [8]. The analysis of the traditional and the modified TK operation considers first one narrowband signal in the baseband. Finally, the analysis is extended to the bandpass domain for one and multiple NBI [5]. Thereby, the potential to integrate the TK operation into the existing MIR-UWB system is discussed for one NBI. Based on the proposal of [42] it is shown that the integration can be realised with only minor complexity increase.

### *4.3.1. Teager-Kaiser operation*

36 Will-be-set-by-IN-TECH

**Figure 22.** BER vs. *Eb*/*N*<sup>0</sup> for AWGN and fixed SIRs of -10 dB (IEEE 802.11a WLAN [47]), 0 dB and 5 dB (IEEE 802.15.3a MB OFDM UWB 1 and 2 [2]) regarding an interferer related band plan, no coexistence

within the band plan. In parallel, the corresponding subband is deactivated and the procedure is iteratively repeated. If the coexistence approach terminates, the binary band plan is

Performance analysis [12] demonstrates the high capability of both iterative coexistence methods in presence of multiple interferers having the same or different interference powers

The MIR-UWB system is characterised by a particular high vulnerability to NBI as all interferences inside the passband of the analogue front-end are considered by the energy detection receiver. Its performance can decrease significantly resulting in an increase of the

This section analyses the potential of the non-linear Teager-Kaiser (TK) operation [22] to mitigate NBI without the knowledge of the interference related carrier frequency. Based on the definition of the TK operation and some of its most important properties the mitigation potential of a TK based energy detector is analysed. In this context a modified TK operation is introduced and compared with the traditional TK operation [8]. The analysis of the traditional and the modified TK operation considers first one narrowband signal in the baseband. Finally, the analysis is extended to the bandpass domain for one and multiple NBI [5]. Thereby, the potential to integrate the TK operation into the existing MIR-UWB system is discussed for one NBI. Based on the proposal of [42] it is shown that the integration can be realised with only

error probability. Hence, a crucial issue concerns the efficient mitigation of NBI [58].

approach and the coexistence approaches 1 to 3 [12].

**4.3. Narrowband interference mitigation**

minor complexity increase.

(Fig. 22).

allocated to the transmitter for initialisation and data transmission.

The continuous TK operation is a non-linear differential operator of order two defined as

$$\Psi\left(\mathbf{x}\left(t\right)\right) = \dot{\mathbf{x}}^2\left(t\right) - \mathbf{x}\left(t\right)\ddot{\mathbf{x}}\left(t\right). \tag{87}$$

To illustrate the effectiveness of the TK operation the harmonic oscillation *x* (*t*) = *A* cos (*ω*0*t* + *φ*0) is considered. Using (87) the signal at the output of the TK operation can be described as

$$\mathbb{Y}\left(\mathbf{x}\left(t\right)\right) = A^2 \omega\_0^2 \left(\sin^2\left(\omega\_0 t + \phi\_0\right) + \cos^2\left(\omega\_0 t + \phi\_0\right)\right) = A^2 \omega\_0^2.$$

Hence, for the special case of a simple harmonic oscillation the TK operation leads to a frequency shift to DC. Fig. 23 highlights the spectrum of a harmonic oscillation (a) as well as the resulting spectrum at the output of TK operation (b).

**Figure 23.** Effectiveness of TK operation for the harmonic oscillation *x* (*t*) = *A* cos (*ω*0*t* + *φ*0).

An interpretation of this behaviour is given in [22]. In case of a harmonic oscillation the output of the TK operation describes the required energy to generate the oscillation. Considering (88) the energy depends not only by the amplitude *A* but also by the oscillation frequency *ω*0. Thus, for a constant *A* the required energy to generate an exemplary 10 Hz signal is lower than the one of an 1000 Hz signal.

When using the continuous TK operation of (87) basic operator specific rules have to be generally considered. A detailed description can be found in [23]. In context with interference mitigation the most important property can be the behaviour of the TK operation in presence of several overlapping signals. In case of *K*<sup>S</sup> overlapping signals *x*<sup>1</sup> (*t*), *x*<sup>2</sup> (*t*), . . ., *xK*<sup>S</sup> (*t*) the resulting signal at the output of TK operation conducts to

$$\Psi\left(\sum\_{i=1}^{K\_{\mathbb{S}}} \boldsymbol{\boldsymbol{x}}\_{i}\left(t\right)\right) = \sum\_{i=1}^{K\_{\mathbb{S}}} \Psi\left(\boldsymbol{x}\_{i}\left(t\right)\right) + \sum\_{j=1}^{K\_{\mathbb{S}}-1} \sum\_{i=j+1}^{K\_{\mathbb{S}}} \Psi\_{\mathbb{C}}\left(\boldsymbol{x}\_{j}\left(t\right), \boldsymbol{x}\_{i}\left(t\right)\right). \tag{88}$$

It consists on one hand of *K*<sup>S</sup> summands describing the TK operation of the signals *x*<sup>1</sup> (*t*), *x*<sup>2</sup> (*t*), . . ., *xK*<sup>S</sup> (*t*). Furthermore, due to the non-linearity of the TK operation the additional cross component

$$\begin{split} \Psi\_{\mathbf{c}}\left(\mathbf{x}\_{\circ}(t), \mathbf{x}\_{i}(t)\right) &= \Psi\_{\mathbf{c}\_{\circ}}\left(\mathbf{x}\_{\circ}(t), \mathbf{x}\_{i}(t)\right) + \Psi\_{\mathbf{c}\_{\circ}}\left(\mathbf{x}\_{i}(t), \mathbf{x}\_{\circ}(t)\right) \\ &= 2\dot{\mathbf{x}}\_{\circ}(t)\,\dot{\mathbf{x}}\_{i}\left(t\right) - \mathbf{x}\_{\circ}\left(t\right)\,\ddot{\mathbf{x}}\_{i}\left(t\right) - \mathbf{x}\_{i}\left(t\right)\,\ddot{\mathbf{x}}\_{\circ}\left(t\right) \end{split} \tag{89}$$

#### 38 Will-be-set-by-IN-TECH 38 Ultra-Wideband Radio Technologies for Communications, Localization and Sensor Applications MIRA – Physical Layer Optimisation for the Multiband Impulse Radio UWB Architecture <sup>39</sup>

occurs. This component considers the mutual influence of the two signals *xj* (*t*) and *xi* (*t*). It is composed of the generally non-symmetric signal parts Ψc1 (*x*<sup>1</sup> (*t*), *x*<sup>2</sup> (*t*)) = *x*˙1 (*t*) *x*˙2 (*t*) − *x*<sup>1</sup> (*t*) *x*¨2 (*t*) and Ψc2 (*x*<sup>2</sup> (*t*), *x*<sup>1</sup> (*t*)) = *x*˙2 (*t*) *x*˙1 (*t*) − *x*<sup>2</sup> (*t*) *x*¨1 (*t*). From (89) two special cases can be immediately concluded. Firstly, the cross component Ψ<sup>c</sup> (*x* (*t*), *x* (*t*)) = 2Ψ (*x* (*t*)) if *x* (*t*) is overlapped with itself. Secondly, Ψ<sup>c</sup> (*a*, *x* (*t*)) = −*ax*¨ (*t*). The cross component of a constant *a* and a signal *x* (*t*) can be expressed as the product of *a* with the second derivative of *x* (*t*). In particular, the cross component disappears completely, if *a* = 0.

The TK operation of (87) can be modified with the weighting factor *k* �= 0 as follows [8]:

$$\Psi\_k\left(\mathbf{x}\left(t\right)\right) = k\dot{\mathbf{x}}\left(t\right)^2 - \mathbf{x}\left(t\right)\ddot{\mathbf{x}}\left(t\right). \tag{90}$$

In contrast to the traditional TK operation (Fig. 24 (b)) additional spectral parts at 2*ω*<sup>0</sup> can be observed for *k* � 1 (Fig. 24 (d)). For the special case *k* = −1 the complete energy is even shifted to this frequency (Fig. 24 (c)). Hence, it can be concluded that a modified TK operation

Due to the modification of the TK operation the property of overlapping signals changes. If two signals *x*<sup>1</sup> (*t*) and *x*<sup>2</sup> (*t*) occur at the input of the TK operation its output conducts to [8]:

<sup>2</sup> (*t*) − *x*<sup>1</sup> (*t*) *x*¨2 (*t*) − *x*<sup>2</sup> (*t*) *x*¨

MIRA – Physical Layer Optimisation for the Multiband Impulse Radio UWB Architecture 39

*<sup>k</sup>* (*x*<sup>1</sup> (*t*), *x*<sup>2</sup> (*t*)), (93)

<sup>1</sup> (*t*) (94)

<sup>1</sup> (*p* (*t*), *n* (*t*)) a

<sup>2</sup> the

<sup>Ψ</sup>*<sup>k</sup>* (*x*<sup>1</sup> (*t*) + *<sup>x</sup>*<sup>2</sup> (*t*)) = <sup>Ψ</sup>*<sup>k</sup>* (*x*<sup>1</sup> (*t*)) + <sup>Ψ</sup>*<sup>k</sup>* (*x*<sup>2</sup> (*t*)) + <sup>Ψ</sup><sup>c</sup>

<sup>1</sup> (*t*) *x*˙

**Figure 25.** Normalized amplitude spectrum at the output of the traditional TK operation for *k* = 1 (a)

Fig. 25 shows the difference of the traditional (a) and the modified TK operation (b) for OOK in case of a binary one. Thereby, Fig. 25 (a) highlights the normalized amplitude spectrum of the occurring components for the interference-free AWGN case for *k* = 1 and *E*b/*N*<sup>0</sup> = 12 dB. The considered pulse is a cosine-shaped pulse with pulse duration *T* = 12.8 ns and carrier frequency *f*<sup>c</sup> = 5.13 GHz [8]. Obviously, the TK operation mixes a large signal part of the pulse to around DC. However, a minor contribution around 2 *f*c can also be observed. As this

In contrast, Fig. 25 (b) illustrates that the modification of the TK operation with, e.g. *k* = 1.5, has the potential to improve the detection for AWGN. Also in this case large pulse related spectral contributions around DC occur. Furthermore, a significant increase of ≈ 30 dB of the pulse's amplitude spectrum F {Ψ1,5 (*p* (*t*))} occurs around 2 *f*c. The resulting amplitude spectrum of the pulse is both around DC and around 2 *f*c larger than the contributions of

The reason for this behaviour can be ascribed to the modified TK operation of the pulse. Assuming negligible influences of the first and the second derivative of *<sup>p</sup>* (*t*) at <sup>|</sup>*t*<sup>|</sup> <sup>=</sup> *<sup>T</sup>*

1.5 (*p* (*t*), *n* (*t*)). Hence, an improved detection performance can be expected

contribution is much lower than the spectral contributions of Ψ<sup>1</sup> (*n* (*t*)) and Ψ<sup>c</sup>

possible detection may be limited to the frequency range around DC.

for AWGN if frequencies around DC as well as around 2 *f*c are considered.

output of the TK operation on *p* (*t*) can be described as [8]:

operates not only as DC frequency shifter.

Ψc

and *k* = 1.5 (b), *E*b/*N*<sup>0</sup> = 12 dB.

Ψ1.5 (*n* (*t*)) and Ψ<sup>c</sup>

*<sup>k</sup>* (*x*<sup>1</sup> (*t*), *x*<sup>2</sup> (*t*)) = 2*kx*˙

stands for the modified cross component of *x*<sup>1</sup> (*t*) and *x*<sup>2</sup> (*t*).

where

This definition contains the traditional TK operation if *k* = 1. Thereby, the Fourier transform of Ψ*<sup>k</sup>* (*x* (*t*)) of a signal *x* (*t*) is generally given as

$$\mathcal{F}\{\Psi\_k\left(\mathbf{x}\left(t\right)\right)\} = 4\pi^2 \left(\mathbf{X}\left(f\right) \* f^2 \mathbf{X}\left(f\right) - kf \mathbf{X}\left(f\right) \* f \mathbf{X}\left(f\right)\right). \tag{91}$$

Considering again the special case of a harmonic oscillation *x* (*t*) = *A* cos (*ω*0*t*) the Fourier transform conducts to

$$\mathcal{F}\{\Psi\_k\left(\mathbf{x}\left(t\right)\right)\} = \frac{1}{2}A^2 \left(2\pi f\_0\right)^2 \left(\left(k+1\right)\delta\left(f\right) + \frac{1}{2}\left(1-k\right)\left(\delta\left(f+2f\_0\right) + \delta\left(f-2f\_0\right)\right)\right). \tag{92}$$

The effectiveness of the modified TK operation is highlighted in Fig. 24.

**Figure 24.** Effectiveness of the traditional and the modified TK operation for *x* (*t*) = *A* cos (*ω*0*t*).

In contrast to the traditional TK operation (Fig. 24 (b)) additional spectral parts at 2*ω*<sup>0</sup> can be observed for *k* � 1 (Fig. 24 (d)). For the special case *k* = −1 the complete energy is even shifted to this frequency (Fig. 24 (c)). Hence, it can be concluded that a modified TK operation operates not only as DC frequency shifter.

Due to the modification of the TK operation the property of overlapping signals changes. If two signals *x*<sup>1</sup> (*t*) and *x*<sup>2</sup> (*t*) occur at the input of the TK operation its output conducts to [8]:

$$\Psi\_k\left(\mathbf{x}\_1\left(t\right) + \mathbf{x}\_2\left(t\right)\right) = \Psi\_k\left(\mathbf{x}\_1\left(t\right)\right) + \Psi\_k\left(\mathbf{x}\_2\left(t\right)\right) + \Psi\_k^c\left(\mathbf{x}\_1\left(t\right), \mathbf{x}\_2\left(t\right)\right),\tag{93}$$

where

38 Will-be-set-by-IN-TECH

occurs. This component considers the mutual influence of the two signals *xj* (*t*) and *xi* (*t*). It is composed of the generally non-symmetric signal parts Ψc1 (*x*<sup>1</sup> (*t*), *x*<sup>2</sup> (*t*)) = *x*˙1 (*t*) *x*˙2 (*t*) − *x*<sup>1</sup> (*t*) *x*¨2 (*t*) and Ψc2 (*x*<sup>2</sup> (*t*), *x*<sup>1</sup> (*t*)) = *x*˙2 (*t*) *x*˙1 (*t*) − *x*<sup>2</sup> (*t*) *x*¨1 (*t*). From (89) two special cases can be immediately concluded. Firstly, the cross component Ψ<sup>c</sup> (*x* (*t*), *x* (*t*)) = 2Ψ (*x* (*t*)) if *x* (*t*) is overlapped with itself. Secondly, Ψ<sup>c</sup> (*a*, *x* (*t*)) = −*ax*¨ (*t*). The cross component of a constant *a* and a signal *x* (*t*) can be expressed as the product of *a* with the second derivative

The TK operation of (87) can be modified with the weighting factor *k* �= 0 as follows [8]:

This definition contains the traditional TK operation if *k* = 1. Thereby, the Fourier transform

Considering again the special case of a harmonic oscillation *x* (*t*) = *A* cos (*ω*0*t*) the Fourier

1

F {(x(t)} F { (x(t))} <sup>1</sup>


A 2 4 2 <sup>0</sup> (1 k)

2<sup>0</sup> 2<sup>0</sup>

(d)

(b)

A 2 2 2 <sup>0</sup> (k 1)

*<sup>X</sup>* (*f*) <sup>∗</sup> *<sup>f</sup>* <sup>2</sup>*<sup>X</sup>* (*f*) <sup>−</sup> *kfX* (*f*) <sup>∗</sup> *f X* (*f*)

<sup>2</sup> <sup>−</sup> *<sup>x</sup>* (*t*) *<sup>x</sup>*¨ (*t*). (90)

<sup>2</sup> (<sup>1</sup> <sup>−</sup> *<sup>k</sup>*) (*<sup>δ</sup>* (*<sup>f</sup>* <sup>+</sup> <sup>2</sup> *<sup>f</sup>*0) <sup>+</sup> *<sup>δ</sup>* (*<sup>f</sup>* <sup>−</sup> <sup>2</sup> *<sup>f</sup>*0))

. (91)

<sup>2</sup> (A ) <sup>2</sup> 0

. (92)

of *x* (*t*). In particular, the cross component disappears completely, if *a* = 0.

of Ψ*<sup>k</sup>* (*x* (*t*)) of a signal *x* (*t*) is generally given as

transform conducts to

F {Ψ*<sup>k</sup>* (*<sup>x</sup>* (*t*))} <sup>=</sup> <sup>1</sup>

<sup>0</sup>

F {Ψ*<sup>k</sup>* (*<sup>x</sup>* (*t*))} <sup>=</sup> <sup>4</sup>*π*<sup>2</sup>

<sup>2</sup> *<sup>A</sup>*<sup>2</sup> (2*<sup>π</sup> <sup>f</sup>*0)

0

<sup>2</sup><sup>0</sup> <sup>2</sup><sup>0</sup>

(a)

(c)

2 

The effectiveness of the modified TK operation is highlighted in Fig. 24.

A 2

Ψ*<sup>k</sup>* (*x* (*t*)) = *kx*˙ (*t*)

(*k* + 1) *δ* (*f*) +

0 0

0 0

**Figure 24.** Effectiveness of the traditional and the modified TK operation for *x* (*t*) = *A* cos (*ω*0*t*).

A

$$\mathbf{x}\_{k}^{\mathbb{C}}(\mathbf{x}\_{1}(t), \mathbf{x}\_{2}(t)) = 2k\dot{\mathbf{x}}\_{1}(t)\dot{\mathbf{x}}\_{2}(t) - \mathbf{x}\_{1}(t)\ddot{\mathbf{x}}\_{2}(t) - \mathbf{x}\_{2}(t)\ddot{\mathbf{x}}\_{1}(t) \tag{94}$$

stands for the modified cross component of *x*<sup>1</sup> (*t*) and *x*<sup>2</sup> (*t*).

**Figure 25.** Normalized amplitude spectrum at the output of the traditional TK operation for *k* = 1 (a) and *k* = 1.5 (b), *E*b/*N*<sup>0</sup> = 12 dB.

Fig. 25 shows the difference of the traditional (a) and the modified TK operation (b) for OOK in case of a binary one. Thereby, Fig. 25 (a) highlights the normalized amplitude spectrum of the occurring components for the interference-free AWGN case for *k* = 1 and *E*b/*N*<sup>0</sup> = 12 dB. The considered pulse is a cosine-shaped pulse with pulse duration *T* = 12.8 ns and carrier frequency *f*<sup>c</sup> = 5.13 GHz [8]. Obviously, the TK operation mixes a large signal part of the pulse to around DC. However, a minor contribution around 2 *f*c can also be observed. As this contribution is much lower than the spectral contributions of Ψ<sup>1</sup> (*n* (*t*)) and Ψ<sup>c</sup> <sup>1</sup> (*p* (*t*), *n* (*t*)) a possible detection may be limited to the frequency range around DC.

In contrast, Fig. 25 (b) illustrates that the modification of the TK operation with, e.g. *k* = 1.5, has the potential to improve the detection for AWGN. Also in this case large pulse related spectral contributions around DC occur. Furthermore, a significant increase of ≈ 30 dB of the pulse's amplitude spectrum F {Ψ1,5 (*p* (*t*))} occurs around 2 *f*c. The resulting amplitude spectrum of the pulse is both around DC and around 2 *f*c larger than the contributions of Ψ1.5 (*n* (*t*)) and Ψ<sup>c</sup> 1.5 (*p* (*t*), *n* (*t*)). Hence, an improved detection performance can be expected for AWGN if frequencies around DC as well as around 2 *f*c are considered.

The reason for this behaviour can be ascribed to the modified TK operation of the pulse. Assuming negligible influences of the first and the second derivative of *<sup>p</sup>* (*t*) at <sup>|</sup>*t*<sup>|</sup> <sup>=</sup> *<sup>T</sup>* <sup>2</sup> the output of the TK operation on *p* (*t*) can be described as [8]:

#### 40 Will-be-set-by-IN-TECH 40 Ultra-Wideband Radio Technologies for Communications, Localization and Sensor Applications MIRA – Physical Layer Optimisation for the Multiband Impulse Radio UWB Architecture <sup>41</sup>

$$\begin{aligned} \Psi\_{k}\left(p\left(t\right)\right) &= \frac{2}{3T} \left[0.5\left(k+1\right)\left(\omega\_{\rm T}^{2} + 3\omega\_{\rm c}^{2}\right)\right] \\ &\quad - 0.25\left(k-1\right)\left(\omega\_{\rm T} - \omega\_{\rm c}\right)^{2}\cos\left(2\left(\omega\_{\rm c} - \omega\_{\rm T}\right)t\right) \\ &\quad + 0.5\left(\left(k+1\right)\omega\_{\rm T}^{2} - 3\left(k-1\right)\omega\_{\rm c}^{2}\right)\cos\left(2\omega\_{\rm c}t\right) \\ &\quad + \left(0.5\omega\_{\rm T}^{2} + \left(k-1\right)\left(\omega\_{\rm c}\omega\_{\rm T} - \omega\_{\rm c}^{2}\right)\right)\cos\left(\left(2\omega\_{\rm c} - \omega\_{\rm T}\right)t\right) \\ &\quad + \left(0.5\omega\_{\rm T}^{2} - \left(k-1\right)\left(\omega\_{\rm c}\omega\_{\rm T} + \omega\_{\rm c}^{2}\right)\right)\cos\left(\left(2\omega\_{\rm c} + \omega\_{\rm T}\right)t\right) \\ &\quad - 0.25\left(k-1\right)\left(\omega\_{\rm T} + \omega\_{\rm c}\right)^{2}\cos\left(2\left(\omega\_{\rm c} + \omega\_{\rm T}\right)t\right) \\ &\quad + 0.5\left(\omega\_{\rm c}^{2}\left(k+1\right) - \omega\_{\rm T}^{2}\left(k-1\right)\right)\cos\left(2\omega\_{\rm T}t\right) \\ &\quad + \left(2\omega\_{\rm c}^{2}\left(k+1\right) + \omega\_{\rm T}^{2}\right)\cos\left(\omega\_{\rm r}t\right) \end{aligned} \tag{95}$$

with

and

*C*<sup>1</sup> = ∑*<sup>N</sup>*

*<sup>i</sup>*=<sup>1</sup> *<sup>A</sup>*<sup>2</sup> *<sup>i</sup> <sup>ω</sup>*<sup>2</sup>

concentration reduces.

**Bandpass Domain**

*ωl*,2 < ... < *ωl*,*Nl*

can be described as

with the bandwidth *Bl* = 2*ωNl*

quantity depends on the two factors *z*

*jl* (*t*) = 2*Al*

= *Al*

*Ck* (*t*) <sup>=</sup> <sup>1</sup>

*z j*

*zi*

and *ω<sup>i</sup>* − *ω<sup>j</sup>* are influenced with larger weighting factors *z*

*Nl* ∑ *i*=1

*Nl* ∑ *i*=1

.

 *K*<sup>S</sup> ∑ *l*=1

Ψ*k*

sin *ωl*,*it* 

sin *<sup>ω</sup>*c*<sup>l</sup>* + *<sup>ω</sup>l*,*<sup>i</sup>*

 *αl*(*t*)

> *jl* (*t*) = *K*S ∑ *l*=1

It consists of the components Ψ*<sup>k</sup>* (*jl* (*t*)) = Ψ*<sup>k</sup>* (*α<sup>l</sup>* (*t*)) + Ψ*<sup>k</sup>* (*β<sup>l</sup>* (*t*)) + Ψ<sup>c</sup>

2

*N* ∑ *i*=1 *A*2 *<sup>i</sup> <sup>ω</sup>*<sup>2</sup>

*<sup>i</sup>* = *kAiAjωiω<sup>j</sup>* +

*<sup>j</sup>* <sup>=</sup> *kAiAjωiω<sup>j</sup>* <sup>−</sup> <sup>1</sup>

*j*

1 <sup>2</sup> *AiAjω*<sup>2</sup>

For the traditional TK operation (*k* = 1) the resulting signal is composed of the DC component

Thereby, a higher energy concentration can be observed for lower frequencies. In contrast, the modified TK operation (*k* �= 1) shows spectral contributions around DC, around *ω<sup>i</sup>* + *ω<sup>j</sup>* and *ω<sup>i</sup>* − *ω<sup>j</sup>* as well as around 2*ωi*, *i* = 1, . . . , *N*. In particular, the contributions at *ω<sup>i</sup>* + *ω<sup>j</sup>*

TK operation a high energy concentration still occurs for low frequencies. However, due to the additional spectral components at 2*ωi*, *i* = 1, . . . , *N* the relative difference of energy

NBI influencing the MIR-UWB system operates in the bandpass domain. Based on the insight for a baseband signal an analytical description of the effectiveness of the traditional and the modified TK operation is done for one or more narrowband bandpass signals. Due to

cos (*ω*c*lt*)

 *t* 

The bandpass signal *jl* (*t*) of amplitude 2*Al* consists of *Nl* sinusoids of frequencies *ωl*,1 <

If *K*<sup>S</sup> bandpass signals *jl* (*t*), *l* = 1, . . . , *K*<sup>S</sup> are present at the input of TK operation its output

1, . . . , *K*<sup>S</sup> which always occur in presence of one bandpass signal. Assuming *ω*c*<sup>l</sup>* � *Bl* the *l*th

− *Al*

*Nl* ∑ *i*=1

, *ωl*,*<sup>i</sup>* = 2*π fl*,*i*, *i* = 1, . . . , *Nl* located around the carrier frequency *ω*c*<sup>l</sup>* = 2*π f*c*<sup>l</sup>*

<sup>Ψ</sup>*<sup>k</sup>* (*jl* (*t*)) + <sup>Ψ</sup><sup>c</sup>

sin *<sup>ω</sup>*c*<sup>l</sup>* <sup>−</sup> *<sup>ω</sup>l*,*<sup>i</sup>*

 *βl*(*t*)

 *t* 

*<sup>k</sup>*,*<sup>m</sup>* (*t*). (99)

*<sup>k</sup>* (*α<sup>l</sup>* (*t*), *β<sup>l</sup>* (*t*)), *l* =

. (98)

analytical tractability it is assumed that the *l*th bandpass signal can be described as

*<sup>i</sup>* = 0, 5*AiAj*

*<sup>i</sup>* [(*k* + 1) + (*k* − 1) cos (2*ωit*)] ,

MIRA – Physical Layer Optimisation for the Multiband Impulse Radio UWB Architecture 41

*<sup>i</sup>* <sup>−</sup> <sup>1</sup>

2

*j <sup>i</sup>* and *<sup>z</sup><sup>i</sup> j*

<sup>2</sup> *AiAjω*<sup>2</sup> *j* .

and *z<sup>i</sup>*

*<sup>j</sup>* = −0, 5*AiAj*

. Hence, using the modified

*ω<sup>i</sup>* − *ω<sup>j</sup>*

2 .

*<sup>i</sup>* + 1 <sup>2</sup> *AiAjω*<sup>2</sup> *j*

*<sup>i</sup>* . In addition, further components around *ω<sup>i</sup>* − *ω<sup>j</sup>* and *ω<sup>i</sup>* + *ω<sup>j</sup>* occur. Its

<sup>2</sup> *AiAjω*<sup>2</sup>

 *ω<sup>i</sup>* + *ω<sup>j</sup>*

with *ω*<sup>T</sup> = <sup>2</sup>*<sup>π</sup>* <sup>T</sup> and *ω*<sup>c</sup> = 2*π f*c. Obviously, a modification of the TK operation leads to additional spectral parts at its output. Hence, it is possible to influence the distribution of pulse energy via a simple weighting of the TK operation. It should be noted that the traditional as well as the modified TK operation acts not only as a frequency-to-DC shifter due to additional spectral parts around 2 *f*c. For the traditional TK operation (*k* = 1) the two spectral components at 2 (*ω*<sup>c</sup> − *ω*T) and 2 (*ω*<sup>c</sup> + *ω*T) disappear completely resulting in a lower energy concentration around 2 *f*c.

### *4.3.2. Mitigation potential of Teager-Kaiser operation*

Assuming negligible noise this section analyses the potential of the traditional and the modified TK operation to mitigate NBI. For this the TK operation's effectiveness is first described in the baseband. Afterwards it is extended to the bandpass domain.

### **Baseband Domain**

To make statements on the effectiveness of the TK operation for a narrowband signal we first consider a narrowband baseband signal of bandwidth 2*ω<sup>N</sup>* which can be modelled as

$$j\left(t\right) = \sum\_{i=1}^{N} A\_i \sin\left(\omega\_i t + \phi\_i\right). \tag{96}$$

It consists of *N* overlapping sinusoids of amplitudes *Ai*, *i* = 1, . . . , *N*, of phases *φi*, *i* = 1, . . . , *N* as well as of frequencies *ω<sup>i</sup>* = 2*π fi*, *i* = 1, . . . , *N*. Given to the modified TK operation its output can be described for *φ<sup>i</sup>* = 0, *i* = 1, . . . , *N* as<sup>3</sup>

$$\Psi\_k\left(j\left(t\right)\right) = \mathbb{C}\_k\left(t\right) + \sum\_{i=1}^N \sum\_{j>i}^N z\_i^j \cos\left(\left(\omega\_i - \omega\_j\right)t\right) + z\_j^i \cos\left(\left(\omega\_i + \omega\_j\right)t\right),\tag{97}$$

<sup>3</sup> The TK operation is generally characterised by a strong phase dependency. To simplify the descriptions we only consider the special case *φ<sup>i</sup>* = 0, *i* = 1, . . . , *N* in the following.

with

40 Will-be-set-by-IN-TECH

<sup>T</sup> <sup>−</sup> <sup>3</sup> (*<sup>k</sup>* <sup>−</sup> <sup>1</sup>) *<sup>ω</sup>*<sup>2</sup>

*<sup>ω</sup>*c*ω*<sup>T</sup> <sup>−</sup> *<sup>ω</sup>*<sup>2</sup>

*ω*c*ω*<sup>T</sup> + *ω*<sup>2</sup>

<sup>T</sup> (*k* − 1)

cos (*ω*T*t*)

<sup>T</sup> and *ω*<sup>c</sup> = 2*π f*c. Obviously, a modification of the TK operation leads to

T 

additional spectral parts at its output. Hence, it is possible to influence the distribution of pulse energy via a simple weighting of the TK operation. It should be noted that the traditional as well as the modified TK operation acts not only as a frequency-to-DC shifter due to additional spectral parts around 2 *f*c. For the traditional TK operation (*k* = 1) the two spectral components at 2 (*ω*<sup>c</sup> − *ω*T) and 2 (*ω*<sup>c</sup> + *ω*T) disappear completely resulting in a

Assuming negligible noise this section analyses the potential of the traditional and the modified TK operation to mitigate NBI. For this the TK operation's effectiveness is first

To make statements on the effectiveness of the TK operation for a narrowband signal we first

It consists of *N* overlapping sinusoids of amplitudes *Ai*, *i* = 1, . . . , *N*, of phases *φi*, *i* = 1, . . . , *N* as well as of frequencies *ω<sup>i</sup>* = 2*π fi*, *i* = 1, . . . , *N*. Given to the modified TK operation its

<sup>3</sup> The TK operation is generally characterised by a strong phase dependency. To simplify the descriptions we only

*ω<sup>i</sup>* − *ω<sup>j</sup>*

 *t* + *z<sup>i</sup> <sup>j</sup>* cos 

consider a narrowband baseband signal of bandwidth 2*ω<sup>N</sup>* which can be modelled as

*N* ∑ *i*=1 <sup>2</sup> cos (<sup>2</sup> (*ω*<sup>c</sup> <sup>−</sup> *<sup>ω</sup>*T)*t*)

cos (2*ω*c*t*)

cos ((2*ω*<sup>c</sup> − *ω*T)*t*)

cos ((2*ω*<sup>c</sup> + *ω*T)*t*)

*Ai* sin (*ωit* + *φi*). (96)

*ω<sup>i</sup>* + *ω<sup>j</sup>*

 *t* 

, (97)

(95)

c 

> c

> c

<sup>2</sup> cos (<sup>2</sup> (*ω*<sup>c</sup> + *<sup>ω</sup>*T)*t*)

cos (2*ω*T*t*)

 *ω*2 <sup>T</sup> <sup>+</sup> <sup>3</sup>*ω*<sup>2</sup> c 

<sup>Ψ</sup>*<sup>k</sup>* (*<sup>p</sup>* (*t*)) <sup>=</sup> <sup>2</sup>

3*T* 

+ 0.5 

+ 0.5*ω*<sup>2</sup>

+ 0.5*ω*<sup>2</sup>

+ 2*ω*<sup>2</sup>

lower energy concentration around 2 *f*c.

*4.3.2. Mitigation potential of Teager-Kaiser operation*

output can be described for *φ<sup>i</sup>* = 0, *i* = 1, . . . , *N* as<sup>3</sup>

consider the special case *φ<sup>i</sup>* = 0, *i* = 1, . . . , *N* in the following.

Ψ*<sup>k</sup>* (*j*(*t*)) = *Ck* (*t*) +

with *ω*<sup>T</sup> = <sup>2</sup>*<sup>π</sup>*

**Baseband Domain**

+ 0.5 *ω*2

0.5 (*k* + 1)

− 0.25 (*k* − 1) (*ω*<sup>T</sup> − *ω*c)

(*k* + 1) *ω*<sup>2</sup>

<sup>T</sup> + (*k* − 1)

<sup>T</sup> − (*k* − 1)

<sup>c</sup> (*<sup>k</sup>* + <sup>1</sup>) + *<sup>ω</sup>*<sup>2</sup>

described in the baseband. Afterwards it is extended to the bandpass domain.

*j*(*t*) =

*N* ∑ *i*=1

*N* ∑ *j*>*i z j <sup>i</sup>* cos 

<sup>c</sup> (*<sup>k</sup>* <sup>+</sup> <sup>1</sup>) <sup>−</sup> *<sup>ω</sup>*<sup>2</sup>

− 0.25 (*k* − 1) (*ω*<sup>T</sup> + *ω*c)

$$\mathcal{C}\_{k}\left(t\right) = \frac{1}{2} \sum\_{i=1}^{N} A\_{i}^{2} \omega\_{i}^{2} \left[ (k+1) + (k-1) \cos\left(2\omega\_{i}t\right) \right] \omega\_{i}$$

$$z\_{i}^{j} = k A\_{i} A\_{j} \omega\_{i} \omega\_{j} + \frac{1}{2} A\_{i} A\_{j} \omega\_{i}^{2} + \frac{1}{2} A\_{i} A\_{j} \omega\_{j}^{2}$$

and

$$z\_j^i = k A\_i A\_j \omega\_i \omega\_j - \frac{1}{2} A\_i A\_j \omega\_i^2 - \frac{1}{2} A\_i A\_j \omega\_j^2.$$

For the traditional TK operation (*k* = 1) the resulting signal is composed of the DC component *C*<sup>1</sup> = ∑*<sup>N</sup> <sup>i</sup>*=<sup>1</sup> *<sup>A</sup>*<sup>2</sup> *<sup>i</sup> <sup>ω</sup>*<sup>2</sup> *<sup>i</sup>* . In addition, further components around *ω<sup>i</sup>* − *ω<sup>j</sup>* and *ω<sup>i</sup>* + *ω<sup>j</sup>* occur. Its quantity depends on the two factors *z j <sup>i</sup>* = 0, 5*AiAj ω<sup>i</sup>* + *ω<sup>j</sup>* 2 and *z<sup>i</sup> <sup>j</sup>* = −0, 5*AiAj ω<sup>i</sup>* − *ω<sup>j</sup>* 2 . Thereby, a higher energy concentration can be observed for lower frequencies. In contrast, the modified TK operation (*k* �= 1) shows spectral contributions around DC, around *ω<sup>i</sup>* + *ω<sup>j</sup>* and *ω<sup>i</sup>* − *ω<sup>j</sup>* as well as around 2*ωi*, *i* = 1, . . . , *N*. In particular, the contributions at *ω<sup>i</sup>* + *ω<sup>j</sup>* and *ω<sup>i</sup>* − *ω<sup>j</sup>* are influenced with larger weighting factors *z j <sup>i</sup>* and *<sup>z</sup><sup>i</sup> j* . Hence, using the modified TK operation a high energy concentration still occurs for low frequencies. However, due to the additional spectral components at 2*ωi*, *i* = 1, . . . , *N* the relative difference of energy concentration reduces.

### **Bandpass Domain**

NBI influencing the MIR-UWB system operates in the bandpass domain. Based on the insight for a baseband signal an analytical description of the effectiveness of the traditional and the modified TK operation is done for one or more narrowband bandpass signals. Due to analytical tractability it is assumed that the *l*th bandpass signal can be described as

$$\begin{split} j\_l(t) &= 2A\_l \sum\_{i=1}^{N\_l} \sin\left(\omega\_{l,i} t\right) \cos\left(\omega\_{l,i} t\right) \\ &= \underbrace{A\_l \sum\_{i=1}^{N\_l} \sin\left(\left(\omega\_{\mathbf{c}\_l} + \omega\_{l,i}\right)t\right)}\_{a\_l(t)} - \underbrace{A\_l \sum\_{i=1}^{N\_l} \sin\left(\left(\omega\_{\mathbf{c}\_l} - \omega\_{l,i}\right)t\right)}\_{\beta\_l(t)}.\end{split} \tag{98}$$

The bandpass signal *jl* (*t*) of amplitude 2*Al* consists of *Nl* sinusoids of frequencies *ωl*,1 < *ωl*,2 < ... < *ωl*,*Nl* , *ωl*,*<sup>i</sup>* = 2*π fl*,*i*, *i* = 1, . . . , *Nl* located around the carrier frequency *ω*c*<sup>l</sup>* = 2*π f*c*<sup>l</sup>* with the bandwidth *Bl* = 2*ωNl* .

If *K*<sup>S</sup> bandpass signals *jl* (*t*), *l* = 1, . . . , *K*<sup>S</sup> are present at the input of TK operation its output can be described as

$$\Psi\_k\left(\sum\_{l=1}^{K\_\mathbb{S}} j\_l\left(t\right)\right) = \sum\_{l=1}^{K\_\mathbb{S}} \Psi\_k\left(j\_l\left(t\right)\right) + \Psi\_{k,m}^\mathbb{C}\left(t\right). \tag{99}$$

It consists of the components Ψ*<sup>k</sup>* (*jl* (*t*)) = Ψ*<sup>k</sup>* (*α<sup>l</sup>* (*t*)) + Ψ*<sup>k</sup>* (*β<sup>l</sup>* (*t*)) + Ψ<sup>c</sup> *<sup>k</sup>* (*α<sup>l</sup>* (*t*), *β<sup>l</sup>* (*t*)), *l* = 1, . . . , *K*<sup>S</sup> which always occur in presence of one bandpass signal. Assuming *ω*c*<sup>l</sup>* � *Bl* the *l*th

#### 42 Will-be-set-by-IN-TECH 42 Ultra-Wideband Radio Technologies for Communications, Localization and Sensor Applications MIRA – Physical Layer Optimisation for the Multiband Impulse Radio UWB Architecture <sup>43</sup>

bandpass signal *jl* (*t*), *l* = 1, . . . , *K*<sup>S</sup> results in [8]

$$\begin{split} \Psi\_{k}\left(j\_{l}\left(t\right)\right) &\approx \\ \quad \quad \quad A\_{l}^{2}\omega\_{\mathbb{C}}^{2}\sum\_{u=1}^{N\_{l}}\left[\left(k+1\right) + 0.5\left(k-1\right)\cdot\left[\cos\left(2\left(\omega\_{\mathbb{C}}-\omega\_{l,u}\right)t\right) + \cos\left(2\left(\omega\_{\mathbb{C}}+\omega\_{l,u}\right)t\right)\right]\right] \\ &+ \left.A\_{l}^{2}\omega\_{\mathbb{C}}^{2}\sum\_{u=1}^{N\_{l}}\sum\_{v>u}^{N\_{l}}\left[2\left(k+1\right)\cos\left(\left(\omega\_{l,u}-\omega\_{l,v}\right)t\right) + \left(k-1\right)\cos\left(\left(2\omega\_{\mathbb{C}\_{l}}+\omega\_{l,u}+\omega\_{l,v}\right)t\right)\right] \\ &+ \left.(k-1)\cos\left(\left(2\omega\_{\mathbb{C}\_{l}}-\omega\_{l,u}-\omega\_{l,v}\right)t\right)\right] \\ &- A\_{l}^{2}\omega\_{\mathbb{C}\_{l}}^{2}\sum\_{u=1}^{N\_{l}}\sum\_{v=1}^{N\_{l}}\left[\left(k+1\right)\cos\left(\left(\omega\_{l,u}+\omega\_{l,v}\right)t\right) - \left(k-1\right)\cos\left(\left(2\omega\_{\mathbb{C}\_{l}}+\omega\_{l,u}-\omega\_{l,v}\right)t\right)\right] \tag{100} \end{split}$$

Finally, the third cross component Ψ<sup>c</sup>

*<sup>k</sup>* (*α<sup>r</sup>* (*t*), *<sup>β</sup><sup>l</sup>* (*t*)) <sup>=</sup> <sup>Ψ</sup>*<sup>c</sup>*

−*Z*1,*<sup>k</sup>* 

−*Z*2,*<sup>k</sup>* 

center frequency 2*ω*c critical.

presence of three narrowband bandpass signals.

*Nl* ∑ *v*=1

Ψc

≈ *Nr* ∑ *u*=1 *<sup>k</sup>* (*α<sup>r</sup>* (*t*), *β<sup>l</sup>* (*t*)) is given for *r* �= *l* as

 *t* 

 *t* 

Hence, in presence of more than one bandpass signal additional spectral components occur arround |*ω*c*<sup>r</sup>* − *ω*c*<sup>l</sup>* | and *ω*c*<sup>r</sup>* + *ω*c*<sup>l</sup>* . The frequency parts depend on the carrier frequencies of the bandpass signals. The spectral components are influenced by the weighting factor *k*. E.g., for the traditional TK operation (*k* = 1) the spectral component around |*ω*c*<sup>r</sup>* − *ω*c*<sup>l</sup>* | dominates. In contrast for the modified TK operation (*k* �= 1) additional relevant spectral components can be identified around *ω*c*<sup>r</sup>* + *ω*c*<sup>l</sup>* making the usage of the frequency at twice the subband's

To verify the results a subband of bandwidth 625 MHz and carrier frequency 5.2 GHz is considered for *k* = 1. *K*<sup>S</sup> = 3 bandpass signals of amplitudes *A*<sup>1</sup> = 1, *A*<sup>2</sup> = 1/3 and *A*<sup>3</sup> = 2/3, of bandwidths *B*<sup>1</sup> = 5 MHz, *B*<sup>2</sup> = 10 MHz and *B*<sup>3</sup> = 1 MHz as well as of carrier frequencies *f*c1 = 4.98 GHz, *f*c2 = 5.04 GHz and *f*c3 = 5.28 GHz are assumed at the input of TK operation. Fig. 26 shows the positive frequency range for the resulting to one

**Figure 26.** Normalized amplitude spectrum at the output of the traditional TK operation (*k* = 1) in

becomes inefficient to efficiently mitigate all occurring interferences.

normalized baseband spectrum at the output of the traditional TK operation. Simulation as well as analytical results show the spectral contributions of the three signals occurring at *f*c2 − *f*c1 = 60 MHz, *f*c3 − *f*c2 = 240 MHz and *f*c3 − *f*c1 = 300 MHz with the bandwidths 30 MHz, 22 MHz and 12 MHz. The spectral components are distributed over the complete bandwidth of the subband. For this reason the consideration of the TK operation with an additional filtering operation is critical. In particular, the mitigation scheme proposed in [42]

*<sup>k</sup>*,*l*>*<sup>r</sup>* (*α<sup>r</sup>* (*t*), *β<sup>l</sup>* (*t*))

MIRA – Physical Layer Optimisation for the Multiband Impulse Radio UWB Architecture 43

<sup>+</sup> cos *<sup>ω</sup>*c*<sup>r</sup>* <sup>−</sup> *<sup>ω</sup>*c*<sup>l</sup>* <sup>+</sup> *<sup>ω</sup>r*,*<sup>u</sup>* <sup>+</sup> *<sup>ω</sup>l*,*<sup>v</sup>*

<sup>+</sup> cos *<sup>ω</sup>*c*<sup>r</sup>* <sup>+</sup> *<sup>ω</sup>*c*<sup>l</sup>* <sup>+</sup> *<sup>ω</sup>r*,*<sup>u</sup>* <sup>−</sup> *<sup>ω</sup>l*,*<sup>v</sup>*

 *t* 

 *t* (103) .

*<sup>k</sup>*,*r*>*<sup>l</sup>* (*α<sup>r</sup>* (*t*), *<sup>β</sup><sup>l</sup>* (*t*)) <sup>+</sup> <sup>Ψ</sup>*<sup>c</sup>*

cos *<sup>ω</sup>*c*<sup>r</sup>* <sup>−</sup> *<sup>ω</sup>*c*<sup>l</sup>* <sup>−</sup> *<sup>ω</sup>r*,*<sup>u</sup>* <sup>−</sup> *<sup>ω</sup>l*,*<sup>v</sup>*

cos *<sup>ω</sup>*c*<sup>r</sup>* <sup>+</sup> *<sup>ω</sup>*c*<sup>l</sup>* <sup>−</sup> *<sup>ω</sup>r*,*<sup>u</sup>* <sup>+</sup> *<sup>ω</sup>l*,*<sup>v</sup>*

This result reveals that the traditional TK operation (*k* = 1) acts as a frequency-to-DC shifter for each bandpass signal *jl* (*t*), *l* = 1, . . . , *K*S. In this case the corresponding spectral range goes from DC to the largest occurring bandwidth *Bl*, *l* = 1, . . . , *K*<sup>S</sup> of the *K*<sup>S</sup> bandpass signals. For the modified TK operation additional spectral components around 2*ω*c,*l*, *l* = 1, . . . , *K*<sup>S</sup> occur. For this reason a mitigation of NBI by the modified TK operation is critical. Finally, for the special case *k* = −1 the complete energy is shifted to 2*ω*c,*l*, *l* = 1, . . . , *K*S. This confirms the statement that energy parts can be shifted between frequency ranges with a modified TK operation.

If the output of the traditional or the modified TK operation would only consist of components from Ψ*<sup>k</sup>* (*jl* (*t*)), *l* = 1, . . . , *K*<sup>S</sup> a mitigation of the *K*<sup>S</sup> bandpass signal could be possible as *B*<sup>T</sup> � *Bl*, *l* = 1, . . . , *K*S. However, as can be seen in (99), the additional signal component

$$\mathbf{V}\_{k,m}^{\mathbb{C}}\left(t\right) = \sum\_{r=1}^{K\_{\mathbb{S}}-1} \sum\_{l=r+1}^{K\_{\mathbb{S}}} \left[\mathbf{\varPsi}\_{k}^{\mathbb{C}}\left(\mathbf{a}\_{r}\left(t\right), \mathbf{a}\_{l}\left(t\right)\right) + \mathbf{\varPsi}\_{k}^{\mathbb{C}}\left(\boldsymbol{\upbeta}\_{r}\left(t\right), \boldsymbol{\upbeta}\_{l}\left(t\right)\right)\right] + \sum\_{r=1}^{K\_{\mathbb{S}}} \sum\_{l \neq r} \mathbf{\varPsi}\_{k}^{\mathbb{C}}\left(\mathbf{a}\_{r}\left(t\right), \boldsymbol{\upbeta}\_{l}\left(t\right)\right) \text{(101)}$$

occurs in case of at least two bandpass signals. The component Ψ<sup>c</sup> *<sup>k</sup>*,*<sup>m</sup>* (*t*) describes the cross components between different bandpass signals *jl* (*t*) and *jr* (*t*), *l* �= *r*. Thereby, assuming *ω*c*<sup>l</sup>* � *Bl* and *ω*c*<sup>r</sup>* � *Br* the two cross components can be described as

$$\begin{split} & \mathbf{F}\_{k}^{\mathbf{c}} \left( \boldsymbol{a}\_{r} \left( \boldsymbol{t} \right), \boldsymbol{a}\_{l} \left( \boldsymbol{t} \right) \right) + \mathbf{F}\_{k}^{\mathbf{c}} \left( \boldsymbol{\beta}\_{r} \left( \boldsymbol{t} \right), \boldsymbol{\beta}\_{l} \left( \boldsymbol{t} \right) \right) \approx \\ & \sum\_{\begin{subarray}{c} \mathbf{u} = 1 \ \mathbf{v} = 1 \end{subarray}}^{\mathbf{N}\_{l}} \mathbf{Z}\_{1,k} \left[ \cos \left( \left( \boldsymbol{\omega}\_{\mathbf{c}\_{r}} - \boldsymbol{\omega}\_{\mathbf{c}\_{l}} + \boldsymbol{\omega}\_{\mathbf{r},\mu} - \boldsymbol{\omega}\_{l,\mathbf{r}} \right) \boldsymbol{t} \right) + \cos \left( \left( \boldsymbol{\omega}\_{\mathbf{c}\_{r}} - \boldsymbol{\omega}\_{\mathbf{c}\_{l}} - \boldsymbol{\omega}\_{r,\mu} + \boldsymbol{\omega}\_{l,\nu} \right) \boldsymbol{t} \right) \right] \\ & \qquad \qquad + \mathbf{Z}\_{2,k} \left[ \cos \left( \left( \boldsymbol{\omega}\_{\mathbf{c}\_{r}} + \boldsymbol{\omega}\_{\mathbf{c}\_{l}} + \boldsymbol{\omega}\_{r,\mu} + \boldsymbol{\omega}\_{l,\nu} \right) \boldsymbol{t} \right) + \cos \left( \left( \boldsymbol{\omega}\_{\mathbf{c}\_{r}} + \boldsymbol{\omega}\_{\mathbf{c}\_{l}} - \boldsymbol{\omega}\_{r,\mu} - \boldsymbol{\omega}\_{l,\nu} \right) \boldsymbol{t} \right) \right] (102) \end{split}$$

with the amplitudes

$$Z\_{1,k} = \frac{A\_r A\_l}{2} \left(\omega\_{\text{c}\_l}^2 + 2k\omega\_{\text{c}\_l}\omega\_{\text{c}\_l} + \omega\_{\text{c}\_l}^2\right)^2$$

and

$$Z\_{2,k} = -\frac{A\_r A\_l}{2} \left(\omega\_{\mathbf{c}\_r}^2 - 2k\omega\_{\mathbf{c}\_l}\omega\_{\mathbf{c}\_l} + \omega\_{\mathbf{c}\_l}^2\right).$$

Finally, the third cross component Ψ<sup>c</sup> *<sup>k</sup>* (*α<sup>r</sup>* (*t*), *β<sup>l</sup>* (*t*)) is given for *r* �= *l* as

42 Will-be-set-by-IN-TECH

*ω*c*<sup>l</sup>* − *ωl*,*<sup>u</sup>*

 *t* 

 *t* 

 *t* 

This result reveals that the traditional TK operation (*k* = 1) acts as a frequency-to-DC shifter for each bandpass signal *jl* (*t*), *l* = 1, . . . , *K*S. In this case the corresponding spectral range goes from DC to the largest occurring bandwidth *Bl*, *l* = 1, . . . , *K*<sup>S</sup> of the *K*<sup>S</sup> bandpass signals. For the modified TK operation additional spectral components around 2*ω*c,*l*, *l* = 1, . . . , *K*<sup>S</sup> occur. For this reason a mitigation of NBI by the modified TK operation is critical. Finally, for the special case *k* = −1 the complete energy is shifted to 2*ω*c,*l*, *l* = 1, . . . , *K*S. This confirms the statement that energy parts can be shifted between frequency ranges with a modified TK

If the output of the traditional or the modified TK operation would only consist of components from Ψ*<sup>k</sup>* (*jl* (*t*)), *l* = 1, . . . , *K*<sup>S</sup> a mitigation of the *K*<sup>S</sup> bandpass signal could be possible as *B*<sup>T</sup> � *Bl*, *l* = 1, . . . , *K*S. However, as can be seen in (99), the additional signal component

components between different bandpass signals *jl* (*t*) and *jr* (*t*), *l* �= *r*. Thereby, assuming

 *t* 

> *t*

<sup>c</sup>*<sup>r</sup>* <sup>+</sup> <sup>2</sup>*kω*c*rω*c*<sup>l</sup>* <sup>+</sup> *<sup>ω</sup>*<sup>2</sup>

<sup>c</sup>*<sup>r</sup>* <sup>−</sup> <sup>2</sup>*kω*c*rω*c*<sup>l</sup>* <sup>+</sup> *<sup>ω</sup>*<sup>2</sup>

*<sup>k</sup>* (*β<sup>r</sup>* (*t*), *<sup>β</sup><sup>l</sup>* (*t*))

+ *K*S ∑ *r*=1 ∑ *l*�=*r* Ψc

<sup>+</sup> cos *<sup>ω</sup>*c*<sup>r</sup>* <sup>−</sup> *<sup>ω</sup>*c*<sup>l</sup>* <sup>−</sup> *<sup>ω</sup>r*,*<sup>u</sup>* <sup>+</sup> *<sup>ω</sup>l*,*<sup>v</sup>*

c*l* 

> c*l* .

<sup>+</sup> cos *<sup>ω</sup>*c*<sup>r</sup>* <sup>+</sup> *<sup>ω</sup>*c*<sup>l</sup>* <sup>−</sup> *<sup>ω</sup>r*,*<sup>u</sup>* <sup>−</sup> *<sup>ω</sup>l*,*<sup>v</sup>*

 *t* + cos 2 

*ω*c*<sup>l</sup>* + *ωl*,*<sup>u</sup>*

<sup>+</sup> (*<sup>k</sup>* <sup>−</sup> <sup>1</sup>) cos <sup>2</sup>*ω*c*<sup>l</sup>* <sup>+</sup> *<sup>ω</sup>l*,*<sup>u</sup>* <sup>+</sup> *<sup>ω</sup>l*,*<sup>v</sup>*

<sup>−</sup> (*<sup>k</sup>* <sup>−</sup> <sup>1</sup>) cos <sup>2</sup>*ω*c*<sup>l</sup>* <sup>+</sup> *<sup>ω</sup>l*,*<sup>u</sup>* <sup>−</sup> *<sup>ω</sup>l*,*<sup>v</sup>*

 *t* 

> *t*

 *t* .(100)

*<sup>k</sup>* (*α<sup>r</sup>* (*t*), *β<sup>l</sup>* (*t*))(101)

 *t* 

> *t* (102)

*<sup>k</sup>*,*<sup>m</sup>* (*t*) describes the cross

 cos 2 

bandpass signal *jl* (*t*), *l* = 1, . . . , *K*<sup>S</sup> results in [8]

*Nl* ∑*v*>*u* 

*Nl* ∑ *v*=1 

(*k* + 1) + 0.5 (*k* − 1) ·

<sup>2</sup> (*<sup>k</sup>* <sup>+</sup> <sup>1</sup>) cos *<sup>ω</sup>l*,*<sup>u</sup>* <sup>−</sup> *<sup>ω</sup>l*,*<sup>v</sup>*

(*<sup>k</sup>* + <sup>1</sup>) cos *<sup>ω</sup>l*,*<sup>u</sup>* + *<sup>ω</sup>l*,*<sup>v</sup>*

*<sup>k</sup>* (*α<sup>r</sup>* (*t*), *<sup>α</sup><sup>l</sup>* (*t*)) <sup>+</sup> <sup>Ψ</sup><sup>c</sup>

occurs in case of at least two bandpass signals. The component Ψ<sup>c</sup>

*ω*c*<sup>l</sup>* � *Bl* and *ω*c*<sup>r</sup>* � *Br* the two cross components can be described as

*<sup>k</sup>* (*β<sup>r</sup>* (*t*), *β<sup>l</sup>* (*t*)) ≈

cos *<sup>ω</sup>*c*<sup>r</sup>* <sup>−</sup> *<sup>ω</sup>*c*<sup>l</sup>* <sup>+</sup> *<sup>ω</sup>r*,*<sup>u</sup>* <sup>−</sup> *<sup>ω</sup>l*,*<sup>v</sup>*

cos *<sup>ω</sup>*c*<sup>r</sup>* + *<sup>ω</sup>*c*<sup>l</sup>* + *<sup>ω</sup>r*,*<sup>u</sup>* + *<sup>ω</sup>l*,*<sup>v</sup>*

*<sup>Z</sup>*1,*<sup>k</sup>* <sup>=</sup> *ArAl* 2

*<sup>Z</sup>*2,*<sup>k</sup>* <sup>=</sup> <sup>−</sup> *ArAl*

2

 *ω*2

> *ω*2

<sup>+</sup> (*<sup>k</sup>* <sup>−</sup> <sup>1</sup>) cos <sup>2</sup>*ω*c*<sup>l</sup>* <sup>−</sup> *<sup>ω</sup>l*,*<sup>u</sup>* <sup>−</sup> *<sup>ω</sup>l*,*<sup>v</sup>*

Ψ*<sup>k</sup>* (*jl* (*t*)) ≈

*A*2 *<sup>l</sup> <sup>ω</sup>*<sup>2</sup> c*l Nl* ∑ *u*=1 

+ *A*<sup>2</sup> *<sup>l</sup> <sup>ω</sup>*<sup>2</sup> c*l Nl* ∑ *u*=1

<sup>−</sup> *<sup>A</sup>*<sup>2</sup> *<sup>l</sup> <sup>ω</sup>*<sup>2</sup> c1 *Nl* ∑ *u*=1

operation.

Ψc *<sup>k</sup>*,*<sup>m</sup>* (*t*) =

Ψc

and

*Nr* ∑ *u*=1

with the amplitudes

*Nl* ∑ *v*=1 *Z*1,*<sup>k</sup>* 

*K*S−1 ∑ *r*=1

*<sup>k</sup>* (*α<sup>r</sup>* (*t*), *<sup>α</sup><sup>l</sup>* (*t*)) <sup>+</sup> <sup>Ψ</sup><sup>c</sup>

+*Z*2,*<sup>k</sup>* 

*K*S ∑ *l*=*r*+1 Ψc

$$\begin{split} \mathbf{F}\_{k}^{\mathbf{c}} \left( \mathfrak{a}\_{r} \left( \boldsymbol{t} \right), \boldsymbol{\beta}\_{l} \left( t \right) \right) &= \mathbf{F}\_{k,r>l}^{\mathbf{c}} \left( \mathfrak{a}\_{r} \left( \boldsymbol{t} \right), \boldsymbol{\beta}\_{l} \left( t \right) \right) + \mathbf{F}\_{k,l>r}^{\mathbf{c}} \left( \mathfrak{a}\_{r} \left( \mathfrak{a}\_{l} \left( \boldsymbol{t} \right), \boldsymbol{\beta}\_{l} \left( t \right) \right) \\ \approx \sum\_{u=1}^{N\_{r}} \sum\_{v=1}^{N\_{l}} -\mathbf{Z}\_{1,k} \left[ \cos \left( \left( \omega\_{\mathsf{c}\_{r}} - \omega\_{\mathsf{c}\_{l}} - \omega\_{r,\mathsf{u}} - \omega\_{l,\mathsf{p}} \right) t \right) + \cos \left( \left( \omega\_{\mathsf{c}\_{r}} - \omega\_{\mathsf{c}\_{l}} + \omega\_{r,\mathsf{u}} + \omega\_{l,\mathsf{p}} \right) t \right) \right] \\ &- \mathbf{Z}\_{2,k} \left[ \cos \left( \left( \omega\_{\mathsf{c}\_{r}} + \omega\_{\mathsf{c}\_{l}} - \omega\_{r,\mathsf{u}} + \omega\_{l,\mathsf{p}} \right) t \right) + \cos \left( \left( \omega\_{\mathsf{c}\_{r}} + \omega\_{\mathsf{c}\_{l}} + \omega\_{r,\mathsf{u}} - \omega\_{l,\mathsf{p}} \right) t \right) \right] \tag{103} \end{split}$$

Hence, in presence of more than one bandpass signal additional spectral components occur arround |*ω*c*<sup>r</sup>* − *ω*c*<sup>l</sup>* | and *ω*c*<sup>r</sup>* + *ω*c*<sup>l</sup>* . The frequency parts depend on the carrier frequencies of the bandpass signals. The spectral components are influenced by the weighting factor *k*. E.g., for the traditional TK operation (*k* = 1) the spectral component around |*ω*c*<sup>r</sup>* − *ω*c*<sup>l</sup>* | dominates. In contrast for the modified TK operation (*k* �= 1) additional relevant spectral components can be identified around *ω*c*<sup>r</sup>* + *ω*c*<sup>l</sup>* making the usage of the frequency at twice the subband's center frequency 2*ω*c critical.

To verify the results a subband of bandwidth 625 MHz and carrier frequency 5.2 GHz is considered for *k* = 1. *K*<sup>S</sup> = 3 bandpass signals of amplitudes *A*<sup>1</sup> = 1, *A*<sup>2</sup> = 1/3 and *A*<sup>3</sup> = 2/3, of bandwidths *B*<sup>1</sup> = 5 MHz, *B*<sup>2</sup> = 10 MHz and *B*<sup>3</sup> = 1 MHz as well as of carrier frequencies *f*c1 = 4.98 GHz, *f*c2 = 5.04 GHz and *f*c3 = 5.28 GHz are assumed at the input of TK operation. Fig. 26 shows the positive frequency range for the resulting to one

**Figure 26.** Normalized amplitude spectrum at the output of the traditional TK operation (*k* = 1) in presence of three narrowband bandpass signals.

normalized baseband spectrum at the output of the traditional TK operation. Simulation as well as analytical results show the spectral contributions of the three signals occurring at *f*c2 − *f*c1 = 60 MHz, *f*c3 − *f*c2 = 240 MHz and *f*c3 − *f*c1 = 300 MHz with the bandwidths 30 MHz, 22 MHz and 12 MHz. The spectral components are distributed over the complete bandwidth of the subband. For this reason the consideration of the TK operation with an additional filtering operation is critical. In particular, the mitigation scheme proposed in [42] becomes inefficient to efficiently mitigate all occurring interferences.

#### 44 Will-be-set-by-IN-TECH 44 Ultra-Wideband Radio Technologies for Communications, Localization and Sensor Applications MIRA – Physical Layer Optimisation for the Multiband Impulse Radio UWB Architecture <sup>45</sup>

### *4.3.3. Integration of Teager-Kaiser operation*

As illustrated in the previous section an integration of TK operation into the MIR-UWB system is possible with only minor complexity increase if at most one NBI occurs in each subband. In this case the approach of [42] can be used. It bases on the interplay of the TK operation with a highpass filtering. As illustrated in Fig. 27 only two additional analogue components have to be integrated into each subband of the existing non-coherent MIR-UWB receiver. Thereby, received subband signals are given to TK operation which acts as a frequency-to-DC shifter. The resulting low-frequency signal is afterwards highpass filtered to mitigate interfered signal components without any a priori information of the interference specific carrier frequency. As the bandwidth of the subband signal is larger than the interference bandwidth energy detection might be possible.

**Figure 28.** Normalized amplitude spectrum at the output of TK operation (a) and after highpass

amplitude spectrum which ranges from 50 MHz to 162.5 MHz. As this spectrum dominates the amplitude spectra of the occurring cross components energy detection is possible.

MIRA – Physical Layer Optimisation for the Multiband Impulse Radio UWB Architecture 45

This chapter deals with an easy-to-realise non-coherent MIR-UWB system which is a promising approach for high data rate and energy efficient communication over short distances. Due to its low complexity the MIR-UWB system is an alternative to already existing

The MIR-UWB system is based on an energy detection receiver. Thus the first part of this chapter deals with the performance of this component. To understand the energy detection receiver we look at the bit and symbol error probability in different wireless channels.

First we introduce a closed form expression of the SEP for an energy detection receiver with *M*-PAM in the AWGN channel. Based on this result, we optimise the interval thresholds to minimise the SEP. Optimal interval thresholds guarantee a minimal SEP for *M*-PAM. In the next step we look into the optimal amplitudes for *M*-PAM using an energy detection receiver. This approach enables to reduce the SEP for *M*-PAM with medium to large degrees

To understand the characteristics of the energy detection receiver in fading channels we look into different approaches to model the energy at the receiver. It has been shown, that the flat fading channel model can be used to model the energy at a receiver for a receiver bandwidth *B* > 100 MHz. Based on this assumption we introduce closed form expressions for the SEP of the energy detection receiver with *M*-PAM for different fading statistics such as *Rayleigh*, *Rice* and *Nakagami*-*m*. We also analyse the SEP of an multichannel receiver using different combining techniques. Square Law Combing and Square Law Selection are possible combining schemes for an energy detection receiver. A closed form solution for SLC and SLS is introduced for the AWGN and for the *Rayleigh* fading channel including i.i.d. and correlated

The first part ends with the analysis of the SEP in a frequency selective fading channel. Based on *Rayleigh* distributed fading gains, representing a non line-of-sight channel (NLOS), we

UWB systems for high data rate applications such as Multiband OFDM UWB.

filtering (b), SNR = 11 dB.

**5. Summary**

of freedom.

fading gains.

**Figure 27.** Integration of TK operation into the existing non-coherent MIR-UWB receiver.

In the following the potential of interference mitigation with the TK operation is shown for OOK in case of a binary one. An MIR-UWB subband of carrier frequency 5.13 GHz and effective bandwidth 162.5 MHz is considered for SNR = 11 dB. It is assumed that an IEEE 802.11a WLAN signal [47] of bandwidth 20 MHz and of carrier frequency 5.14 GHz interferes the MIR-UWB subband with an SIR of −5 dB.

Fig. 28 (a) shows the to one normalized amplitude spectrum of all occurring signal components at the output of TK operation. Thereby, the UWB signal spectrum ranges from DC to 162.5 MHz whereas the lower frequency regions have a higher energy concentration. A similar behaviour occurs for the narrowband WLAN signal. Its corresponding amplitude spectrum ranges from DC to 20 MHz whereas energy is strongly distributed around DC. Furthermore, additional spectral cross components between signal, noise and interference occur which can be ascribed to the non-linearity of the TK operation. To mitigate the WLAN signal highpass filtering is done after the TK operation.

Fig. 28 (b) illustrates the to one normalized amplitude spectrum after highpass filtering. The used highpass filter is characterised by the order six, a passband ripple of 0.1 dB, a 50 dB stopband attenuation as well as a 50 MHz wide stopband. Obviously, the narrowband WLAN signal is mitigated after highpass filtering. In contrast the subband signal has an

**Figure 28.** Normalized amplitude spectrum at the output of TK operation (a) and after highpass filtering (b), SNR = 11 dB.

amplitude spectrum which ranges from 50 MHz to 162.5 MHz. As this spectrum dominates the amplitude spectra of the occurring cross components energy detection is possible.
