**2. Non-coherent detection in multipath AWGN**

2 Will-be-set-by-IN-TECH

Unfortunately, a very large signal bandwidth is also associated with some serious problems. These problems are related to the transceiver components itself (availability of broadband antennas, amplifiers etc.), and to the technical effort which is required for synchronization, channel estimation and interference rejection. Since UWB networks operate in frequency bands already assigned to other RF-systems, the probability that narrowband interference

Furthermore, by increasing the bandwidth, more and more multipath arrivals with different path gains and delays are resolvable at the receiver, which makes it more difficult to collect the multipath energy coherently — although the power at the receive antenna does not suffer from the fading effect. Fig. 1 shows an example that one may lose 10 dB and more, if an UWB-receiver uses only the strongest signal echo. Thus, especially in non-LOS scenarios, a coherent RAKE receiver requires a very large number of RAKE fingers and a precise channel knowledge to efficiently capture the multipath energy. Such a coherent RAKE receiver will be very complex and costly, such that the hardware itself may consume a lot of power. This fact is the major motivation for systems using non-coherent detection, which are discussed in this

1 10 100 500 2000 7500

signal bandwidth *B* [MHz]

**Figure 1.** Received energy *E*rx normalized by the transmitted symbol's energy *E*tx in dB, versus different signal bandwidths. The thickness of the curves indicates LOS or non-LOS regimes. The curves without markers show *E*rx/*E*tx averaged across all *x*-*y*-positions within an rectangular area of 30 cm × 40 cm (1 cm grid, data from [7]), if an ideal full RAKE-receiver is used. The curves marked with triangles show the minimum value of *E*rx/*E*tx which occurs within these positions, again assuming a full RAKE. Thus the small-scale fading effect becomes visible. The curves with circles depict the normalized receive energy for a receiver which exploits only the strongest propagation path, i.e., a single correlator is applied. Transmitters-receiver separation is about 3 m, the carrier frequency is always set to 6.85 GHz. Non-coherent UWB transmission is an attractive approach especially if simple and robust implementations with a small power consumption are required. Main application fields are low data rate sensor or personal area networks, which require low cost devices and a long battery life time. It should be noted that the current IEEE802.15.4a UWB-PHY for low data rate communications enables non-coherent detection, too [18]. The main advantage of a

NLOS, full RAKE, averaged NLOS, full RAKE, min. value NLOS, single Corr., min. value LOS, full RAKE, averaged LOS, full RAKE, min. value LOS, single Corr., min. value

occurs at all increases with the bandwidth, too.

−110 −105 −100 −95 −90 −85 −80 −75 −70 −65 −60

*E*rx/*E*tx [dB]

chapter.

Although non-coherent detection is not restricted to low data rates — even orthogonal frequency division multiplex (OFDM) can be combined with non-coherent modulation and detection [19] — we focus our attention on low data rate single carrier transmission.

Non-coherent detection can either be based on envelope detection or on differential detection. In the simplest case, path-diversity combining can be achieved by means of a single, analog integrate and dump filter, see Fig. 3 and Fig. 4. The integration effectively provides a binary weighting of the multipath arrivals: all components inside the integration window of size *T*int are weighted with "1", while all the others are weighted with "0". Regardless of whether envelope or differential detection is chosen, we assume that the receiver uses a quadrature

4 Will-be-set-by-IN-TECH 112 Ultra-Wideband Radio Technologies for Communications, Localization and Sensor Applications Non-Coherent UWB Communications <sup>5</sup>

pulse with a duration of 2 ns (or less) or a burst of up to 128 such pulses with a scrambled

*bn* ∈ {0, 1}, denotes differentially encoded bits.

By increasing the transmission bandwidth *B*, more and more multipath arrivals are resolvable at the receiver. For example, with *B* = 500 MHz and an assumed excess delay of 50 ns, about 25 individual paths are resolvable in the time domain. In contrast to a channel matched filter (or its RAKE receiver equivalent), which combines all these arrivals coherently and with an appropriate weighting, the integrate and dump filter shown in Fig. 3 and Fig. 4 clearly allows only a non-coherent combining. This suboptimal combining leads to a performance loss, which increases with the product *B* · *T*int, where it is assumed that the whole signal energy is contained within the integration interval *T*int. If it is further assumed that *B* · *T*int is an

> *N*−1 ∑ *i*=0

1 2*i* <sup>L</sup>*N*−<sup>1</sup> *i*

 ,

*<sup>i</sup>* is a generalized Laguerre polynomial. The second expression corresponds to a

 <sup>−</sup> *<sup>E</sup>*<sup>b</sup> 2*N*<sup>0</sup> (3)

*bn* − 1)*ψ*1(*t* − *nT*b), (2)

Non-Coherent UWB Communications 113

A (single carrier) DPSK signal given in the complex baseband can be written as

integer *N* ≥ 1, the BER *p*<sup>b</sup> for 2-PPM (energy detection) can be estimated as [5]

<sup>−</sup> *<sup>E</sup>*<sup>b</sup> 2*N*<sup>0</sup>

*E*b/*N*<sup>0</sup>

*<sup>E</sup>*b/*N*<sup>0</sup> + *<sup>N</sup>*

Gaussian approximation which can be used for *N* > 15. Eqn. (3) is only valid if the integration interval *T*int contains the whole bit energy *E*b, and if no ISI occurs, which is the case if the channel excess delay is small compared to *T*b. The compact solution (3) has its origin in the fact that the samples *y*<sup>0</sup> and *y*<sup>1</sup> (see Fig. 3) are *χ*2-distributed with 2*N* degrees of freedom [10]. For DPSK and differential detection, the statistical description is very similar. Thus, it is only required to substitute *N*<sup>0</sup> by *N*0/2 in (3), i.e., the *E*b/*N*0-performance differs by exactly 3 dB

Fig. 5 shows the penalty with respect to the required *E*b/*N*0, if we switch from — rather hypothetical — coherent channel matched filter detection to non-coherent detection with equal gain single-window combining (SinW-C). A value of 1.2 dB at *N* = 1 just corresponds

The results must be interpreted very carefully, since it is assumed that the whole bit energy is concentrated within the interval *T*int. In reality, this is surely not the case and an appropriate *T*int must be found. If *T*int is increased, more and more noise is integrated which leads to the

From the energy efficiency point of view, any non-coherent combining should take place with respect to the multipath energy only. If chirp or direct sequence spread spectrum (DSSS) signals are used, where the signal energy is spread over time even at the transmitter, it is

∞ <sup>∑</sup>*n*=−<sup>∞</sup> (2˜

*<sup>s</sup>*(*t*) = *<sup>E</sup>*<sup>b</sup>

**2.1. Performance estimation for single-window combining**

*<sup>p</sup>*<sup>b</sup> <sup>=</sup> <sup>1</sup>

≈ 1 2

to the *E*b/*N*0-penalty in AWGN (*p*<sup>b</sup> = 10−3).

<sup>2</sup>*<sup>N</sup>* exp

erfc

loss shown in Fig. 5, but more signal energy may be collected as well.

2

polarity.

where ˜

where <sup>L</sup>*N*−<sup>1</sup>

in favor of DPSK.

*bn* <sup>=</sup> *bn* <sup>⊕</sup> ˜

*bn*−1, ˜

**Figure 3.** Envelope detection of a 2-PPM signal using a quadrature down-conversion stage. The analog integrate and dump filter (integrator) is required to capture the multipath energy.

down-conversion stage, since an ECC-conform UWB-signal is a 'truth' band-pass signal with a maximum bandwidth of 2.5 GHz and a lower cut-on frequency of 6 GHz. It is also assumed that each the inphase and the quadrature branch contain a low-pass filter, whose impulse response *g*T(*t*) is matched to the transmitted pulse *ψ*1(*t*). Without multipath, this ensures that the energy of the received signal is focused at the sampling times. For example, according to the IEEE 802.15.4a UWB PHY standard a receiver needs to perform a matched de-chirp operation, if the optional "chirp on UWB" pulse is used.

**Figure 4.** Differential detection for DPSK (*τ* = *T*b) or transmitted-reference PSK (*τ* < *T*b).

In the following, we consider binary pulse-position modulation (2-PPM) and differential phase-shift keying (DPSK) as straightforward modulation schemes to be combined with either envelope detection or with differential detection.

If *E*<sup>b</sup> denotes the mean energy per bit, a 2-PPM signal (single carrier with fixed carrier frequency) given in the complex baseband can be written as

$$s(t) = \sqrt{E\_{\mathbf{b}}} \sum\_{n = -\infty}^{\infty} (1 - b\_n) \psi\_1(t - nT\_{\mathbf{b}}) + b\_n \psi\_1(t - nT\_{\mathbf{b}} - T\_{\mathbf{b}}/2),\tag{1}$$

where *ψ*1(*t*) needs to be orthogonal to *ψ*1(*t* − *T*b/2). *T*<sup>b</sup> is the bit interval, which is split into two subintervals each of length *T*b/2. Depending on the binary information *bn*, *bn* ∈ {0, 1}, to be transmitted, the waveform <sup>√</sup>*E*b*ψ*1(*t*) is generated either at the time *nT*<sup>b</sup> or *<sup>T</sup>*b/2 seconds later. For single carrier transmission with a fixed carrier frequency, the unit energy basis function *ψ*1(*t*) needs to exhibit a bandwidth of at least 500 MHz, i.e., it is a spread spectrum waveform. For example, according to the IEEE 802.15.4a UWB PHY, *ψ*1(*t*) consists of a single pulse with a duration of 2 ns (or less) or a burst of up to 128 such pulses with a scrambled polarity.

A (single carrier) DPSK signal given in the complex baseband can be written as

$$s(t) = \sqrt{E\_\mathbf{b}} \sum\_{n = -\infty}^{\infty} (2\breve{b}\_n - 1)\psi\_1(t - nT\_\mathbf{b}).\tag{2}$$

where ˜ *bn* <sup>=</sup> *bn* <sup>⊕</sup> ˜ *bn*−1, ˜ *bn* ∈ {0, 1}, denotes differentially encoded bits.

### **2.1. Performance estimation for single-window combining**

4 Will-be-set-by-IN-TECH

 *t <sup>t</sup>*−*T*int · <sup>d</sup>*<sup>τ</sup>*

**Figure 3.** Envelope detection of a 2-PPM signal using a quadrature down-conversion stage. The analog

down-conversion stage, since an ECC-conform UWB-signal is a 'truth' band-pass signal with a maximum bandwidth of 2.5 GHz and a lower cut-on frequency of 6 GHz. It is also assumed that each the inphase and the quadrature branch contain a low-pass filter, whose impulse response *g*T(*t*) is matched to the transmitted pulse *ψ*1(*t*). Without multipath, this ensures that the energy of the received signal is focused at the sampling times. For example, according to the IEEE 802.15.4a UWB PHY standard a receiver needs to perform a matched de-chirp

*yD* <sup>≈</sup>

In the following, we consider binary pulse-position modulation (2-PPM) and differential phase-shift keying (DPSK) as straightforward modulation schemes to be combined with either

If *E*<sup>b</sup> denotes the mean energy per bit, a 2-PPM signal (single carrier with fixed carrier

where *ψ*1(*t*) needs to be orthogonal to *ψ*1(*t* − *T*b/2). *T*<sup>b</sup> is the bit interval, which is split into two subintervals each of length *T*b/2. Depending on the binary information *bn*, *bn* ∈ {0, 1}, to be transmitted, the waveform <sup>√</sup>*E*b*ψ*1(*t*) is generated either at the time *nT*<sup>b</sup> or *<sup>T</sup>*b/2 seconds later. For single carrier transmission with a fixed carrier frequency, the unit energy basis function *ψ*1(*t*) needs to exhibit a bandwidth of at least 500 MHz, i.e., it is a spread spectrum waveform. For example, according to the IEEE 802.15.4a UWB PHY, *ψ*1(*t*) consists of a single

*i* ∈ {0, 1}

 *t <sup>t</sup>*−*T*int · <sup>d</sup>*<sup>τ</sup>*

(1 − *bn*)*ψ*1(*t* − *nT*b) + *bnψ*1(*t* − *nT*<sup>b</sup> − *T*b/2), (1)

*t* = *T*<sup>1</sup> + *iT*b/2

*yi*

*y*<sup>0</sup> − *y*<sup>1</sup>

*t* = *T*<sup>1</sup> + *nT*<sup>b</sup>

*g*T(*t*)

<sup>√</sup>2 sin(2*<sup>π</sup> <sup>f</sup>*c*t*)

(·)<sup>2</sup>

operation, if the optional "chirp on UWB" pulse is used.

Delay

*τ*

Delay

*τ*

**Figure 4.** Differential detection for DPSK (*τ* = *T*b) or transmitted-reference PSK (*τ* < *T*b).

*g*T(*t*)

<sup>√</sup>2 cos(2*<sup>π</sup> <sup>f</sup>*c*t*)

−

*g*T(*t*)

envelope detection or with differential detection.

*<sup>s</sup>*(*t*) = *<sup>E</sup>*<sup>b</sup>

frequency) given in the complex baseband can be written as

∞ ∑ *n*=−∞

<sup>√</sup>2 sin(2*<sup>π</sup> <sup>f</sup>*c*t*)

integrate and dump filter (integrator) is required to capture the multipath energy.

(·)<sup>2</sup>

*g*T(*t*)

<sup>√</sup>2 cos(2*<sup>π</sup> <sup>f</sup>*c*t*)

−

≈

By increasing the transmission bandwidth *B*, more and more multipath arrivals are resolvable at the receiver. For example, with *B* = 500 MHz and an assumed excess delay of 50 ns, about 25 individual paths are resolvable in the time domain. In contrast to a channel matched filter (or its RAKE receiver equivalent), which combines all these arrivals coherently and with an appropriate weighting, the integrate and dump filter shown in Fig. 3 and Fig. 4 clearly allows only a non-coherent combining. This suboptimal combining leads to a performance loss, which increases with the product *B* · *T*int, where it is assumed that the whole signal energy is contained within the integration interval *T*int. If it is further assumed that *B* · *T*int is an integer *N* ≥ 1, the BER *p*<sup>b</sup> for 2-PPM (energy detection) can be estimated as [5]

$$p\_{\rm b} = \frac{1}{2^N} \exp\left(-\frac{E\_{\rm b}}{2N\_0}\right) \sum\_{i=0}^{N-1} \frac{1}{2^i} \mathcal{L}\_i^{N-1} \left(-\frac{E\_{\rm b}}{2N\_0}\right) \tag{3}$$

$$\approx \frac{1}{2} \text{erfc}\left(\frac{E\_{\rm b}/N\_0}{2\sqrt{E\_{\rm b}/N\_0 + N}}\right) .$$

where <sup>L</sup>*N*−<sup>1</sup> *<sup>i</sup>* is a generalized Laguerre polynomial. The second expression corresponds to a Gaussian approximation which can be used for *N* > 15. Eqn. (3) is only valid if the integration interval *T*int contains the whole bit energy *E*b, and if no ISI occurs, which is the case if the channel excess delay is small compared to *T*b. The compact solution (3) has its origin in the fact that the samples *y*<sup>0</sup> and *y*<sup>1</sup> (see Fig. 3) are *χ*2-distributed with 2*N* degrees of freedom [10].

For DPSK and differential detection, the statistical description is very similar. Thus, it is only required to substitute *N*<sup>0</sup> by *N*0/2 in (3), i.e., the *E*b/*N*0-performance differs by exactly 3 dB in favor of DPSK.

Fig. 5 shows the penalty with respect to the required *E*b/*N*0, if we switch from — rather hypothetical — coherent channel matched filter detection to non-coherent detection with equal gain single-window combining (SinW-C). A value of 1.2 dB at *N* = 1 just corresponds to the *E*b/*N*0-penalty in AWGN (*p*<sup>b</sup> = 10−3).

The results must be interpreted very carefully, since it is assumed that the whole bit energy is concentrated within the interval *T*int. In reality, this is surely not the case and an appropriate *T*int must be found. If *T*int is increased, more and more noise is integrated which leads to the loss shown in Fig. 5, but more signal energy may be collected as well.

From the energy efficiency point of view, any non-coherent combining should take place with respect to the multipath energy only. If chirp or direct sequence spread spectrum (DSSS) signals are used, where the signal energy is spread over time even at the transmitter, it is preferable to equip the receiver with a matched filter *g*T(*t*) which focuses the energy of the chirp or DSSS-waveform before the non-coherent processing takes place, as it was assumed in Fig. 3 and Fig. 4.

Fig. 6 shows the *E*b/*N*0-performance of 2-PPM in the case of a non-LOS indoor channel, where *E*<sup>b</sup> is interpreted as the bit energy available at the receiver's antenna output. For the results shown, we have used measured UWB channels (5m×5m×2.6m office) including the antennas. The measurements were carried out by the IMST GmbH [6, 7]. *B*<sup>6</sup> is chosen to be 500 MHz.

The results show that the BER strongly depends on *T*int, whereas the position of the integration window is always chosen optimally. For the reference indoor channel considered here [10], the optimal value of *T*int is about 20 ns. At *T*int = 20 ns, SinW-C loses additionally 1.5 dB compared to an optimal non-coherent detector, cf. Section 2.3.

Bound *T*int =10 ns *T*int =20 ns *T*int =50 ns *T*int =80 ns *T*int =100 ns *T*int =120 ns

500 MHz. A measured indoor non-LOS channel is used to obtain the results.

*p*b

**2.3. Performance limit**

channels, cf. Fig. 6 (black, dotted curve).

**3. Analog receiver implementations**

**3.1. Feasibility of analog differential detection**

*E*b/*N*<sup>0</sup> (dB)

Non-Coherent UWB Communications 115

**Figure 6.** BER performance of non-coherent 2-PPM detection for SinW-C and a 6 dB signal bandwidth of

If the sampling rate of the WSubW-C is equal to the signal bandwidth *B*, i.e., *T*sub = 1/*B*, each resolvable multipath component2 can be weighted according to its energy. In [10] we have shown that this approach ensures the benchmark performance, if non-coherent combining is used. The advantage compared to perfectly synchronized SinW-C is about 1.5 - 2 dB for indoor

ISI will degrade the BER performance of DPSK systems, if the symbol interval is considerably smaller than the channel excess delay. However, it is crucial to realize delays on the order of 50 ns or more in the analog domain, if the ultra-wideband nature of the signals is taken into account. Fig. 7 shows the normalized group delay of a Bessel-Thomson all-pass filter with a maximum flat group delay *t*g(*f*). If a 5 % group delay error is chosen to define the cut-off frequency, it is clear that a 5th order filter can provide a usable frequency range of ≈ 1/*t*g(0), i.e., for a desired cut-off frequency of *f*<sup>g</sup> = *B*6/2 = 250 MHz, the delay is only 4 ns. Even a huge and completely unrealistic filter order of 20 could only provide a delay of 5.6 · 4 = 22 ns, if *f*<sup>g</sup> = 250 MHz. It should be noted that two of these analog delay lines have to be implemented, if a quadrature down-conversion stage as shown in Fig. 4 is used.

A basic motivation of impulse radio based on transmitted reference (TR) signaling is that shorter delays can be used. This is possible, since the autocorrelation does not take place with the previous modulated symbol but rather with an additional reference pulse. Our results show that the performance of TR-signaling varies extremely from channel realization

<sup>2</sup> At a total transmission bandwidth *B*, multipath components can be resolved down to 1/*B* in the time domain.

**Figure 5.** Power penalty due to non-coherent combining as a function of the time bandwidth product *N* (*p*<sup>b</sup> = 10−3). Coherent matched filter detection acts as the reference.

### **2.2. Weighted sub-window combining**

In the case of SinW-C, at least two parameters need to be adjusted, the window size *T*int and its position *T*<sup>1</sup> (synchronization). Since the BER over *T*int performance of SinW-C may also exhibit several local minima, the practical determination of appropriate *T*int and *T*<sup>1</sup> values may be more difficult than it seems. These problems are reduced, if weighted sub-window combining (WSubW-C) is used, where the whole integration window is divided into a number of *N*sub sub-windows of size *T*sub. From the *E*b/*N*<sup>0</sup> performance point of view, it is preferable to choose the *N*sub weighting coefficients according to the sub-window energies. In [10] we have shown that WSubW-C with *T*sub = 4 ns (which corresponds to a sampling rate of 250 MHz) outperforms SinW-C (with optimum synchronization) by about 0.5 dB, if indoor channels are considered.

**Figure 6.** BER performance of non-coherent 2-PPM detection for SinW-C and a 6 dB signal bandwidth of 500 MHz. A measured indoor non-LOS channel is used to obtain the results.

### **2.3. Performance limit**

6 Will-be-set-by-IN-TECH

preferable to equip the receiver with a matched filter *g*T(*t*) which focuses the energy of the chirp or DSSS-waveform before the non-coherent processing takes place, as it was assumed in

Fig. 6 shows the *E*b/*N*0-performance of 2-PPM in the case of a non-LOS indoor channel, where *E*<sup>b</sup> is interpreted as the bit energy available at the receiver's antenna output. For the results shown, we have used measured UWB channels (5m×5m×2.6m office) including the antennas. The measurements were carried out by the IMST GmbH [6, 7]. *B*<sup>6</sup> is chosen to be 500 MHz. The results show that the BER strongly depends on *T*int, whereas the position of the integration window is always chosen optimally. For the reference indoor channel considered here [10], the optimal value of *T*int is about 20 ns. At *T*int = 20 ns, SinW-C loses additionally 1.5 dB

Number of combined paths N

**Figure 5.** Power penalty due to non-coherent combining as a function of the time bandwidth product *N*

In the case of SinW-C, at least two parameters need to be adjusted, the window size *T*int and its position *T*<sup>1</sup> (synchronization). Since the BER over *T*int performance of SinW-C may also exhibit several local minima, the practical determination of appropriate *T*int and *T*<sup>1</sup> values may be more difficult than it seems. These problems are reduced, if weighted sub-window combining (WSubW-C) is used, where the whole integration window is divided into a number of *N*sub sub-windows of size *T*sub. From the *E*b/*N*<sup>0</sup> performance point of view, it is preferable to choose the *N*sub weighting coefficients according to the sub-window energies. In [10] we have shown that WSubW-C with *T*sub = 4 ns (which corresponds to a sampling rate of 250 MHz) outperforms SinW-C (with optimum synchronization) by about 0.5 dB, if indoor channels are

compared to an optimal non-coherent detector, cf. Section 2.3.

(*p*<sup>b</sup> = 10−3). Coherent matched filter detection acts as the reference.

**2.2. Weighted sub-window combining**

energy det. (2-PPM) differential det. (DPSK)

Fig. 3 and Fig. 4.

loss compared

considered.

 to ideal coherent

 det. [dB]

> If the sampling rate of the WSubW-C is equal to the signal bandwidth *B*, i.e., *T*sub = 1/*B*, each resolvable multipath component2 can be weighted according to its energy. In [10] we have shown that this approach ensures the benchmark performance, if non-coherent combining is used. The advantage compared to perfectly synchronized SinW-C is about 1.5 - 2 dB for indoor channels, cf. Fig. 6 (black, dotted curve).
