**3. Cooperative DF and AF relay communication**

#### **3.1. Cooperative DF/AF system model**

108 Ultra Wideband – Current Status and Future Trends

scale and *l-*th lag, the detection statistic

*m l*

Maximum Ratio Combination (MRC).

offset [31].

λ

signal ( ) *m l*, *x t* . At the receiver side, for the diversity component corresponding to the *m-*th

is the correlator output of the received signal *y*( )*t* and the basic waveform

lag samples of the discrete wavelet transform of *y*( )*t* associated with the wavelet function *x t*( ) , which forms a scale-lag RAKE receiver. Then, the channel gain is combined

( ) <sup>1</sup>

0 0 , 1 ˆ sign Re , . *M L m*

We note that this coherent detection of the scale-lag RAKE receiver corresponds to a

Wideband LTV multi-scale and multi-lag channels are of interest in a variety of wireless communication scenarios including wideband terrestrial radio frequency systems such as spread-spectrum systems or Ultra Wideband (UWB) systems and underwater acoustic systems. Due to the nature of wideband propagation, such wideband multi-scale and multilag channels exhibit some fundamental differences compared to the so-called narrowband channels. In particular, these multi-scale, multi-lag channel descriptions offer improved modeling of LTV wideband channels over multi-Doppler-shift, multi-lag models [10-12]. Orthogonal Frequency Division Multiplexing (OFDM) technology has been introduced and examined for wideband LTV channels. Approaches include splitting the wideband LTV channel into parallel narrowband LTV channels [30] or assuming a simplified model which reduces the wideband LTV channel to a narrowband LTV channel with a carrier frequency

Receivers for single-scaled wavelet-based pulses for wideband multi-scale, multi-lag channels are presented in [10, 11], and a similar waveform is adopted in spread-spectrum systems [32] over wideband channels modeled by wavelet transforms; while [33] considers equalizers for block transmissions in wideband multi-scale, multi-lag channels. In order to achieve better realistic channel matching, single-scaled rational wavelet modulation was designed in [34]. The above mentioned schemes all employ single-scale modulation and thus do not maximize the spectral efficiency. In order to exploit the frequency diversity, a new form of Orthogonal Wavelet Division Multiplexing (OWDM) has been previously

However, no cooperative schemes for multi-scale, multi-lag channels have been exploited to provide further increased performance for wideband systems. In this chapter, we will design a cooperative wavelet communication scheme to exploit the joint scale-lag diversity

examined in [35] for additive white Gaussian noise channels.

= = <sup>=</sup>

*mM l b h ml*

*<sup>y</sup> t xt y t x t dt W W*

−∞

λ

( )

*m l*

(5)

λ<sup>∗</sup>

*l l*

<sup>=</sup> − = <sup>−</sup> (4)

can be obtained by the dyadic scale-

( ) ( ) \* , ,2 2 2 2 *m m m m*

<sup>∞</sup>

coherently to obtain the estimate of the transmitted information symbol 0*b* as

2 2( ) *m m x t lW* <sup>−</sup> . Therefore, the detection statistic *m l*,

In the cooperative communications system, DF relaying performs better than AF relaying, due to reducing the effects of noise and interference at the fully decoding relay. However, in some case, DF relaying entails the possibility of forwarding erroneously detected signals to the destination as well; causing error propagation that can diminish the performance of the system. The mutual information between the source and the destination is limited by the mutual information of the weakest link between the source–relay and the combined channel from the source-destination and relay-destination.

Since the reliable decoding is not always available, which also means DF protocol is not always suitable for all relaying situations. The tradeoff between the time-consuming decoding, and a better cooperative transmission, finding the appropriate hybrid cooperative schemes, which include both DF and AF for specific situations, is a critical issue for the cooperative relaying networks design.

In this Section, we review the cooperative strategy with the combination of the DF and AF relay as shown in Fig. 2, where we transmit data from source node *S* to destination node *D* through *R* relays, without the direct link between *S* and *D*. This relay structure is called 2-hop relay system, i.e., first hop from source node to relay, and second hop from relay to destination. The channel fading for different links are assumed to be identical and statistically independent, quasi-statistic, i.e., channels are constant within several symbol durations. This is a reasonable assumption as the relays are usually spatially well separated and in a slow changing environment. We assume that the channels are well known at the corresponding receiver sides, and a one bit feedback channel from destination to relay is used for removing the unsuitable AF relays. All the Additive White Gaussian Noise (AWGN) terms have equal variance *N0*. Relays are re-ordered according to the descending order of the Signal-to-Noise Ratio (SNR) between *S* and *Q*, i.e., <sup>1</sup> SNR*SQ* > ··· > <sup>R</sup> SNR*SQ* , where SNR *<sup>r</sup> SQ* denotes the *r*-th largest SNR between *S* and *Q*.

In this model, relays can determine whether the received signals are decoded correctly or not, just simply by comparing the SNR to the threshold, which will be elaborated in Section 3.2. Therefore, the relays with SNR above the threshold will be chosen to decode and forward the data to the destination, as shown with the white hexagons in Fig. 2. The white circle is the removed AF relay according to the dynamic optimal combination strategy which will be described in Section 3.3. The rest of the relays follow the AF protocol, as shown with the white hexagons in Fig. 2 [36].

**Figure 2.** Cooperation communications with dynamic optimal combination of DF/AF relays (*S*: Source, *D*: Destination, *Qr*: *r*-th Relay)

The received SNR at the destination in the hybrid cooperative network can be denoted as

$$\mathcal{Y}\_h = \sum\_{Q\_i \in \text{DF}} \frac{E\_Q h\_{Q\_i, D}}{N\_0} + \sum\_{Q\_j \in \text{AF}} \frac{\frac{E\_S h\_{S, Q\_j}}{N\_0} \frac{E\_Q h\_{Q\_j, D}}{N\_0}}{\frac{E\_S h\_{S, Q\_j}}{N\_0} + \frac{E\_Q h\_{Q\_j, D}}{N\_0} + 1} \tag{6}$$

where , *Q Di <sup>h</sup>* , ,Q*<sup>j</sup> Sh* and , *Q Dj h* denote the power gains of the channel from the *i*-th relay to the destination in DF protocol, source node to the *j*-th relay in AF protocol and *j*-th relay to the destination in AF protocol, respectively. *ES* and *EQ* in Eq. (6) are the average transmission energy at the source node and at the relays, respectively. By choosing the amplification factor *Qj A* in the AF protocol as

$$A\_{Q\_j}^2 = \frac{E\_S}{E\_S h\_{S, Q\_j} + N\_0} \, \, \, \tag{7}$$

Cooperative Communication over Multi-Scale and Multi-Lag Wireless Channels 111

( )

\* \* , , , \* \* DF AF , , , ,, ,

*QD Q SQ Q Q D Q*

*Q Q QD QD SQ Q Q D SQ Q Q D*

In the hybrid DF/AF cooperative network, DF plays a dominant role in the whole system. However, switching to AF scheme for the relay nodes with SNR below the threshold often improves the total transmission performance, and accordingly AF plays a positive

**3.2. SNR thresholding scheme for DF relays Cooperative DF/AF system model** 

In general, mutual information *I* is the upper bound of the target rate *B* bit/s/Hz, i.e., the spectral efficiency attempted by the transmitting terminal. Normally, *B* ≤ *I*, and the case *B* > *I* is known as the outage event. Meanwhile, channel capacity, *C*, is also regarded as the

Conventionally, the maximum average mutual information of the direct transmission between source and destination, i.e., *ID*, achieved by independent and identically distributed

as a function of the power gain over source and destination, *S D*, *h* . According to the

2 1 SNR . *B S D h*

Then, we suppose all of the *X* relays adopt the DF cooperative transmission without direct transmission. The maximum average mutual information for DF cooperation *DF co* \_ *I* is

*DF\_co r r S Q Q D I hh*

which is a function of the channel power gains. Here, *R* denotes the number of the relays.

For the *r*-th DF link, requiring both the relay and destination to decode perfectly, the

,

{ <sup>2</sup> ( 1 1 , 2 ) ( , )} <sup>1</sup> min log 1 SNR ,log 1 SNR , *r r R R*

*<sup>X</sup>* = = =+ + (11)

*I hh DF li* \_ 2 ,2 , =+ + min log 1 SNR ,log 1 SNR . { ( )( ) *S Qr r Q D* } (12)

(i.i.d) zero-mean, circularly symmetric complex Gaussian inputs, is given by

inequality *B* ≤ *I*, we can derive the SNR threshold for the full decoding as

maximum average mutual information *DF li* \_ *I* can be shown as

∈ ∈ *H H H AH H AH*

<sup>⋅</sup> denotes the conjugate operation. , *Q Di*

*H Y H AH Y*

*i i jj j jj j*

, *jj j j i i*

= + (8)

*Y* are the received signal from DF *i*-th relay and AF *j*-th relays,

( ) ( )

*<sup>H</sup>* , , *<sup>j</sup>*

*<sup>D</sup>* log 1 SNR , 2 , ( ) *S D I h* = + (9)

<sup>−</sup> <sup>≥</sup> (10)

*HS Q* and , *Q Dj*

*H* are

( )

*i j*

frequency response of the channel power gains, respectively.

( )

maximum achievable spectral efficiency, i.e., *B* ≤ *C*.

*u*

*<sup>Y</sup>* and *Qj*

respectively, and ( )\*

compensating role.

shown [3] to be

where *Qi*

and forcing the *EQ* in DF equal to *ES*, it will be convenient to maintain constant average transmission energy at relays, equal to the original transmitted energy at the source node.

The receiver at the destination collects the data from DF and AF relays with a MRC. Because of the amplification in the intermediate stage in the AF protocol, the overall channel gain of the AF protocol should include the source to relay, relay to destination channels gains and amplification factor. The decision variable *u* at the MRC output is given by

$$\mu = \sum\_{Q\_i \in \text{DF}} \frac{\left(H\_{Q\_i, D}\right)^\* Y\_{Q\_i}}{\left(H\_{Q\_i, D}\right)^\* H\_{Q\_i, D}} + \sum\_{Q\_j \in \text{AF}} \frac{\left(H\_{S, Q\_j} A\_{Q\_j} H\_{Q\_j, D}\right)^\* Y\_{Q\_j}}{\left(H\_{S, Q\_j} A\_{Q\_j} H\_{Q\_j, D}\right)^\* \left(H\_{S, Q\_j} A\_{Q\_j} H\_{Q\_j, D}\right)^\*} \tag{8}$$

where *Qi <sup>Y</sup>* and *Qj Y* are the received signal from DF *i*-th relay and AF *j*-th relays, respectively, and ( )\* <sup>⋅</sup> denotes the conjugate operation. , *Q Di <sup>H</sup>* , , *<sup>j</sup> HS Q* and , *Q Dj H* are frequency response of the channel power gains, respectively.

110 Ultra Wideband – Current Status and Future Trends

SNR threshold

*D*: Destination, *Qr*: *r*-th Relay)

where , *Q Di*

factor *Qj*

node.

*Q1*

*Q2*

*Qr S D*

···

*QR*

*i*

*QQD*

∈ ∈ = +

2

*<sup>E</sup> <sup>A</sup>*

*Q*

amplification factor. The decision variable *u* at the MRC output is given by

*h*

γ

*<sup>h</sup>* , ,Q*<sup>j</sup> Sh* and , *Q Dj*

*A* in the AF protocol as

**Figure 2.** Cooperation communications with dynamic optimal combination of DF/AF relays (*S*: Source,

The received SNR at the destination in the hybrid cooperative network can be denoted as

, 0 0

, 0 , *<sup>j</sup> j S*

DF 0 AF , ,

*E h N N N Eh E h*

*i j j j*

*Q Q S SQ Q Q D*

the destination in DF protocol, source node to the *j*-th relay in AF protocol and *j*-th relay to the destination in AF protocol, respectively. *ES* and *EQ* in Eq. (6) are the average transmission energy at the source node and at the relays, respectively. By choosing the amplification

*S SQ*

and forcing the *EQ* in DF equal to *ES*, it will be convenient to maintain constant average transmission energy at relays, equal to the original transmitted energy at the source

The receiver at the destination collects the data from DF and AF relays with a MRC. Because of the amplification in the intermediate stage in the AF protocol, the overall channel gain of the AF protocol should include the source to relay, relay to destination channels gains and

*Qr+2*

*Q*

DF relay

AF relay

Removed AF relay

*Q*

*Q*

, 1

*Eh N* <sup>=</sup> <sup>+</sup> (7)

, ,

*S SQ Q Q D*

*Eh E h*

*j j*

+ + (6)

0 0

*h* denote the power gains of the channel from the *i*-th relay to

*N N*

···

*Qr+1*

In the hybrid DF/AF cooperative network, DF plays a dominant role in the whole system. However, switching to AF scheme for the relay nodes with SNR below the threshold often improves the total transmission performance, and accordingly AF plays a positive compensating role.

#### **3.2. SNR thresholding scheme for DF relays Cooperative DF/AF system model**

In general, mutual information *I* is the upper bound of the target rate *B* bit/s/Hz, i.e., the spectral efficiency attempted by the transmitting terminal. Normally, *B* ≤ *I*, and the case *B* > *I* is known as the outage event. Meanwhile, channel capacity, *C*, is also regarded as the maximum achievable spectral efficiency, i.e., *B* ≤ *C*.

Conventionally, the maximum average mutual information of the direct transmission between source and destination, i.e., *ID*, achieved by independent and identically distributed (i.i.d) zero-mean, circularly symmetric complex Gaussian inputs, is given by

$$I\_D = \log\_2\left(1 + \text{SNR } h\_{S,D}\right)\_\prime \tag{9}$$

as a function of the power gain over source and destination, *S D*, *h* . According to the inequality *B* ≤ *I*, we can derive the SNR threshold for the full decoding as

$$\text{SNR} \ge \frac{2^B - 1}{h\_{\text{S},D}}.\tag{10}$$

Then, we suppose all of the *X* relays adopt the DF cooperative transmission without direct transmission. The maximum average mutual information for DF cooperation *DF co* \_ *I* is shown [3] to be

$$I\_{D\text{F\\_}o} = \frac{1}{X} \min \left[ \log\_2 \left( 1 + \sum\_{r=1}^{R} \text{SNR} \, h\_{\text{S},Q\_r} \right) , \log\_2 \left( 1 + \sum\_{r=1}^{R} \text{SNR} \, h\_{Q\_r,D} \right) \right] \tag{11}$$

which is a function of the channel power gains. Here, *R* denotes the number of the relays.

For the *r*-th DF link, requiring both the relay and destination to decode perfectly, the maximum average mutual information *DF li* \_ *I* can be shown as

$$I\_{D\text{F\\_}li} = \min\left[\log\_2\left(1 + \text{SNR}\,h\_{\text{S},Q\_r}\right), \log\_2\left(1 + \text{SNR}\,h\_{Q\_r,D}\right)\right].\tag{12}$$

The first term in Eq. (12) represents the maximum rate at which the relay can reliably decode the source message, while the second term in Eq. (12) represents the maximum rate at which the destination can reliably decode the message forwarded from relay. We note that such mutual information forms are typical of relay channel with full decoding at the relay [37]. The SNR threshold of this DF link for target rate *B* is given by *IDF\_li* ≥ *B* which is derived as

$$\text{SNR} \ge \frac{2^B - 1}{\min\left(h\_{\mathbb{S}, \mathbb{Q}\_r}, h\_{\mathbb{Q}\_r, D}\right)}.\tag{13}$$

Cooperative Communication over Multi-Scale and Multi-Lag Wireless Channels 113

γ

γ:

<sup>−</sup> = (17)

will have a Chi-square distribution with 2*R* degrees of

<sup>−</sup> <sup>−</sup> <sup>=</sup> − (18)

<sup>=</sup> (19)

 μ

− + − + <sup>=</sup> (20)

(21)

*c c*

γ γ has an

because the DF protocol plays a dominant role in the hybrid cooperation strategy, and accordingly we want to find the lower bound which provides as much as possible DF relays. We will elaborate this issue later. Fully decoding check can also be guaranteed by employing the error detection code, such as cyclic redundancy check. However, it will

In the maximum ratio combining the transmitted signal from *R* cooperative relays nodes, which underwent independent identically distributed Rayleigh fading, and forwarded to

> ( ) <sup>1</sup> / . *<sup>r</sup> <sup>r</sup> <sup>r</sup> p e*

γ

( ) <sup>1</sup> <sup>1</sup> / , ( 1)!

γ

, the average probability of error *<sup>e</sup> P* can be obtained as

( ) ( ) <sup>0</sup> 2 , *<sup>c</sup> <sup>e</sup> c cc P Q gp d* γ γ

where *g* = 1 for coherent Binary Phase Shift Keying (BPSK), *g* = ½ for coherent orthogonal BFSK, *g* = 0.715 for coherent BFSK with minimum correlation, and *Q*( )⋅ is the Gaussian *Q*-

> 1 0 1 1 <sup>1</sup> , 2 2 *R R*

*R*

μ

=

<sup>∞</sup>

*c*

 γ

is the average SNR per channel, then by integrating the conditional error

γ γ

= − . For the BPSK case, the average probability of

*R k*

−

*k*

. <sup>1</sup> *c c* γ

In the hybrid DF/AF cooperative network with two hops in each AF relay, the average SNR

γ<sup>=</sup> <sup>+</sup>

*c c R*

*p e <sup>R</sup>*

*R*

Since the fading on the *R* paths is identical and mutually statistically independent, the SNR

γ γ

the destination node are combined. In this case the SNR per bit per relay link *<sup>r</sup>*

**3.3. Error probability for hybrid DF/AF cooperative transmission** 

exponential probability density function (PDF) with average SNR per bit

γ

*c*

*t dt* <sup>∞</sup>

error can be found in the closed form by successive integration by parts, i.e.,

*e k*

μ

γ γ γ γ

increase the system complexity [39].

per bit of the combined SNR *<sup>c</sup>*

γ

function, i.e., ( ) ( ) <sup>2</sup> 1 2 exp 2 *<sup>x</sup> Q x*

π

*P*

can be approximately derived as

γ γis

freedom. The PDF ( ) *<sup>c</sup> <sup>c</sup> <sup>p</sup>*

where *<sup>c</sup>* γ

where

per channel *<sup>c</sup>*

γ

probability over *<sup>c</sup>*

In the proposed hybrid DF/AF cooperative transmission, we only consider that a relay can fully decode the signal transmitted over the source-relay link, but not the whole DF link, thus, the SNR threshold for the full decoding at the *r*-th relay reaches its lower bound as

$$\mathcal{Y}\_{th} \geq \frac{2^B - 1}{h\_{S, Q\_r}}.\tag{14}$$

For the DF protocol, let *R* denotes the number of the total relays, *M* denotes the set of participating relays, whose SNRS are above the SNR threshold, and the reliably decoding is available. The achievable channel capacity, *CDF*, with SNR threshold is calculated as

$$C\_{\rm DF} = \sum\_{M} \frac{1}{R} \mathbf{E} \left( \log\_2 \left( 1 + y \middle| M \right) \right) \text{Pr} \left( M \right) \,\tag{15}$$

where E( )⋅ denotes the expectation operator, ( ) *S D*, *Q M Q D*, *yM R K* γ γ <sup>∈</sup> =− + denotes the instantaneous received SNR at the destination given set *M* with *K* participating relays, where *n m*, γ denotes the instantaneous received SNR at node *m*, which is directly transmitted from *n* to *m*. Since *y M* is the weighted sum of independent exponential random variables [38], the probability density function (PDF) of *y M* can be obtained using its moment generating function (MGF) and partial fraction technique for evaluation of the inverse Laplace transform, see Eq. (8d) and Eq. (8e) in paper [38]. Pr( ) *M* in Eq. (15) is the probability of a particular set of participating relays which are obtained as

$$\Pr\left(M\right) = \prod\_{Q \in M} \exp\left(-\frac{R\mathcal{Y}\_{th}}{\Gamma\_{S, Q \in M}}\right) \prod\_{Q \in M} \left(1 \cdot \exp\left(-\frac{R\mathcal{Y}\_{th}}{\Gamma\_{S, Q \in M}}\right)\right) \tag{16}$$

where *u v*, Γ denotes the average SNR over the link between nodes *u* and *v*.

Combining Eq. (11), Eq. (15) and Eq. (16) with the inequality *IDF\_co* ≤ *CDF*, since the maximum average mutual information, *I*, is upper bound by the achievable channel capacity, *C*, we can calculate the upper bound of SNR threshold *th* γfor fully decoding in the DF protocol.

Now, we can obtain the upper bound and the lower bound of the SNR threshold *th* γ for hybrid DF/AF cooperation. However, compared to the upper bound, the lower bound as shown in the Eq. (14) is more crucial for improving the transmission performance. This is because the DF protocol plays a dominant role in the hybrid cooperation strategy, and accordingly we want to find the lower bound which provides as much as possible DF relays. We will elaborate this issue later. Fully decoding check can also be guaranteed by employing the error detection code, such as cyclic redundancy check. However, it will increase the system complexity [39].

#### **3.3. Error probability for hybrid DF/AF cooperative transmission**

In the maximum ratio combining the transmitted signal from *R* cooperative relays nodes, which underwent independent identically distributed Rayleigh fading, and forwarded to the destination node are combined. In this case the SNR per bit per relay link *<sup>r</sup>* γ has an exponential probability density function (PDF) with average SNR per bit γ:

$$p\_{\mathcal{I}\_r} \left( \mathcal{Y}\_r \right) = \frac{1}{\mathcal{\overline{\mathcal{Y}}}} e^{-\mathcal{Y}\_r/\overline{\mathcal{Y}}}.\tag{17}$$

Since the fading on the *R* paths is identical and mutually statistically independent, the SNR per bit of the combined SNR *<sup>c</sup>* γ will have a Chi-square distribution with 2*R* degrees of freedom. The PDF ( ) *<sup>c</sup> <sup>c</sup> <sup>p</sup>*γγis

$$p\_{\mathcal{Y}\_c} \left( \mathcal{Y}\_c \right) = \frac{1}{(R-1)! \overline{\mathcal{Y}}\_c^R} \mathcal{Y}\_c^{R-1} e^{-\mathcal{Y}\_c / \overline{\mathcal{Y}}\_c} \, \, \, \tag{18}$$

where *<sup>c</sup>* γ is the average SNR per channel, then by integrating the conditional error probability over *<sup>c</sup>* γ, the average probability of error *<sup>e</sup> P* can be obtained as

$$P\_c = \int\_0^\infty \hat{Q}\left(\sqrt{2\,\text{g}\,\text{y}\_c}\right) p\_{\mathcal{I}\_c}\left(\text{y}\_c\right) d\mathcal{\gamma}\_{c'} \tag{19}$$

where *g* = 1 for coherent Binary Phase Shift Keying (BPSK), *g* = ½ for coherent orthogonal BFSK, *g* = 0.715 for coherent BFSK with minimum correlation, and *Q*( )⋅ is the Gaussian *Q*function, i.e., ( ) ( ) <sup>2</sup> 1 2 exp 2 *<sup>x</sup> Q x* π *t dt* <sup>∞</sup> = − . For the BPSK case, the average probability of error can be found in the closed form by successive integration by parts, i.e.,

$$P\_e = \left(\frac{1-\mu}{2}\right)^R \sum\_{k=0}^{R-1} \binom{R-1+k}{k} \left(\frac{1+\mu}{2}\right)^R \,\_{\prime} \tag{20}$$

where

112 Ultra Wideband – Current Status and Future Trends

where *n m*, γ

The first term in Eq. (12) represents the maximum rate at which the relay can reliably decode the source message, while the second term in Eq. (12) represents the maximum rate at which the destination can reliably decode the message forwarded from relay. We note that such mutual information forms are typical of relay channel with full decoding at the relay [37]. The SNR threshold of this DF link for target rate *B* is given by *IDF\_li* ≥ *B* which is derived as

> 2 1 SNR . min , *r r*

In the proposed hybrid DF/AF cooperative transmission, we only consider that a relay can fully decode the signal transmitted over the source-relay link, but not the whole DF link, thus, the SNR threshold for the full decoding at the *r*-th relay reaches its lower bound as

> , 2 1. *r*

( ) ( ) ( ) <sup>2</sup>

*S Q h*

For the DF protocol, let *R* denotes the number of the total relays, *M* denotes the set of participating relays, whose SNRS are above the SNR threshold, and the reliably decoding is

<sup>1</sup> log 1 Pr , *DF*

instantaneous received SNR at the destination given set *M* with *K* participating relays,

from *n* to *m*. Since *y M* is the weighted sum of independent exponential random variables [38], the probability density function (PDF) of *y M* can be obtained using its moment generating function (MGF) and partial fraction technique for evaluation of the inverse Laplace transform, see Eq. (8d) and Eq. (8e) in paper [38]. Pr( ) *M* in Eq. (15) is the

> Pr exp 1-exp , *th th Q M SQ M Q M SQ M*

Combining Eq. (11), Eq. (15) and Eq. (16) with the inequality *IDF\_co* ≤ *CDF*, since the maximum average mutual information, *I*, is upper bound by the achievable channel capacity, *C*, we can

γ

hybrid DF/AF cooperation. However, compared to the upper bound, the lower bound as shown in the Eq. (14) is more crucial for improving the transmission performance. This is

Now, we can obtain the upper bound and the lower bound of the SNR threshold *th*

∈ ∉ ∈ ∉ =− −

γ

denotes the instantaneous received SNR at node *m*, which is directly transmitted

, ,

Γ Γ

*R R*

*<sup>C</sup> yM M <sup>R</sup>*

*B*

*th*

available. The achievable channel capacity, *CDF*, with SNR threshold is calculated as

*M*

probability of a particular set of participating relays which are obtained as

where *u v*, Γ denotes the average SNR over the link between nodes *u* and *v*.

( )

calculate the upper bound of SNR threshold *th*

*M*

where E( )⋅ denotes the expectation operator, ( ) *S D*, *Q M Q D*, *yM R K*

γ

*B*

( ) , ,

<sup>−</sup> <sup>≥</sup> (13)

<sup>−</sup> <sup>≥</sup> (14)

= + **<sup>E</sup>** (15)

γ

 γ

for fully decoding in the DF protocol.

γfor

∏ ∏ (16)

 γ<sup>∈</sup> =− + denotes the

*SQ Q D h h*

$$
\mu = \sqrt{\frac{\overline{\mathcal{Y}\_c}}{1 + \overline{\mathcal{Y}\_c}}}.\tag{21}
$$

In the hybrid DF/AF cooperative network with two hops in each AF relay, the average SNR per channel *<sup>c</sup>* γcan be approximately derived as

$$\overline{\mathcal{N}}\_c = \frac{\mathcal{N}\_h}{\mathcal{K} + 2 \times \mathcal{J}} \,' \tag{22}$$

**4. Cooperative wavelet communication scheme** 

direct link between *S* and *D*.

the hybrid DF/AF case.

and *Q*, i.e., <sup>1</sup>

*S*

SNR threshold

Relay removed by Dynamic Optimal

Selection Strategy (DOSS) minimizing the system bit error rate

and *Q*.

In this section, by taking advantage of the MRC property of the above mentioned multiscale and multi-lag wideband channel and wavelet transceiver model, we consider a wideband cooperative wavelet communication scheme as shown in Fig. 4 [40], where we transmit data from source node *S* to destination node *D* through *R* DF relays, without the

We only consider and illustrate DF relay case; it is because only DF relays strictly fulfill the MRC property. The hybrid DF/AF scheme can approximately fulfill the MRC property and with some errors. This can be shown by the simulation results as well. If error requirement is not very demanding, the relay selection strategy for DF relay case can easily extended for

Different relays operate at different frequency bands and all relay links undergo multi-scale and multi-lag wideband channel. We assume that the channels are well known at the corresponding receiver sides. All the AWGN terms have equal variance *No* . Relays are re-ordered according to the descending order of the (Signal to Noise Ratio) SNR between *S*

···

*QRD*

*Qr*

*Q1*

···

···

*QR*

**Figure 4.** Cooperative wavelet communication scheme with dynamic optimal selection of DF relays in

In this model, relays can determine whether the received signals are decoded correctly or not, just simply compare its received SNR to the threshold. The SNR threshold for the full

wideband multi-scale and multi-lag channel (*S*: Source, *D*: Destination, *Qr*: *r*-th Relay).

decoding at the *r*-th relay reaches its lower bound as

*QRD+1*

*Q*

Cooperative relay

Relay removed by threshold

*D*

Relay removed by DOSS

*Q*

*Q*

SNR*SQ* > ··· > SNR *<sup>R</sup> SQ* , where SNR *<sup>r</sup> SQ* denotes the *r*-th largest SNR between *<sup>S</sup>*

where *K* and *J* are the numbers of the DF relays and AF relays, respectively. *<sup>h</sup>* γ can be obtained from Eq. (6) In the DF protocol, due to the reliable detection, we can only consider the last hops, or the channels between the relay nodes and destination node.

As the average probability of error *<sup>e</sup> P* is a precise indication for the transmission performance, we consequently propose a dynamic optimal combination strategy for the hybrid DF/AF cooperative transmission. In this algorithm the proper AF relays are selected to make *<sup>e</sup> P* reach maximum.

First of all, like aforementioned procedure, relays are reordered according to the descending order of the SNR between source and relays, as shown in the Fig. 2. According to the proposed SNR threshold, we pick up the DF relays having SNR greater than threshold. Then, we proceed with the AF relay selection scheme, where the inappropriate AF relays are removed. The whole dynamic optimal combination strategy for the hybrid DF/AF cooperation is shown in the flow chart of Fig. 3.

**Figure 3.** Flow chart of the dynamic optimal combination strategy for the hybrid DF/AF cooperation
