**4. Cooperative wavelet communication scheme**

114 Ultra Wideband – Current Status and Future Trends

to make *<sup>e</sup> P* reach maximum.

cooperation is shown in the flow chart of Fig. 3.

, <sup>2</sup> *h <sup>c</sup> K J* γ

(22)

discard this AF relay

no

γ

can be

γ

the last hops, or the channels between the relay nodes and destination node.

divide AF relays from DF relays by threshold

start

*Pe* of this AF relay added smaller than without

adopt this AF relay

yes

exist next relay

yes

end

no

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

test next AF relay

where *K* and *J* are the numbers of the DF relays and AF relays, respectively. *<sup>h</sup>*

<sup>=</sup> + ×

obtained from Eq. (6) In the DF protocol, due to the reliable detection, we can only consider

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

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 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 direct link between *S* and *D*.

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 the hybrid DF/AF case.

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* and *Q*, i.e., <sup>1</sup> SNR*SQ* > ··· > SNR *<sup>R</sup> SQ* , where SNR *<sup>r</sup> SQ* denotes the *r*-th largest SNR between *<sup>S</sup>* and *Q*.

**Figure 4.** Cooperative wavelet communication scheme with dynamic optimal selection of DF relays in wideband multi-scale and multi-lag channel (*S*: Source, *D*: Destination, *Qr*: *r*-th Relay).

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

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

1

, ,

∏ ∏∏ (27)

Test next DF relay

yes

*Pe* of this DF relay added smaller than without

> Adopt this DF relay

yes

Discard this DF relay

no

( ) ,1 ( )

*h ml* denotes the power gains corresponding to the *m-*th scale and *l-*th lag, of

*<sup>M</sup> <sup>M</sup> L m QQD*

,

*E h ml N*

*<sup>D</sup> Qr r*

,0

*Q mM l o*

<sup>=</sup>

the channel from the *r*-th relay to the destination in DF protocol. Combining Eq. (25), (26) and (27), we derive the analytical expression of the BER performance for the proposed DF

As the average probability of error *<sup>e</sup> P* is a precise indication, we can use it to predict the comprehensive transmission performance, only given the channel gains and SNRs at the destination. Consequently, we propose a dynamic optimal selection strategy for the cooperative multi-scale and multi-lag communications. In this algorithm the proper relays are selected to make *<sup>e</sup> P* reach the minimum. First of all, relays are ordered according to the descending order of the SNR between source and relays, as shown in the Fig. 4. According to the proposed SNR threshold, we pick up the DF relays whose SNR is above

DF 0

∈= =

*r Qr*

*c*

γ

where ( ) , , *Q Dr*

the threshold.

cooperative wideband network.

Remove unsuitable relays by threshold

Predict *Pe* of the First DF relay

Is *Pe* satisfied by the requirement?

End

yes

communications

in the flow chart of Fig. 5.

no

Next DF Relay exist?

**Figure 5.** Flow chart of the dynamic optimal selection strategy for the cooperative wideband

Then, we proceed with the relay selection to maximize the entire BER performance and try to satisfy the *<sup>e</sup> P* requirement, where the inappropriate DF relays are removed. The whole dynamic optimal selection strategy for the cooperative wideband communication is shown

no

Start

where *B* is the target rate of link between source node and relay, and , *<sup>r</sup> S Q <sup>h</sup>* denotes the power gain of the channel from source node to the *r*-th relay. Therefore, the relays with SNR below the threshold will be removed first, as shown with the gray circles in Fig. 4. The other *RD* relays shown with hexagons are DF relays. According to the dynamic optimal selection strategy, which will be proposed in the next section, we select proper DF relays for cooperation. A one bit feedback channel from destination to relay is used for removing the unsuitable DF relays.

Haar wavelet signaling is adopted in the cooperative wideband system to transfer the multiscale and multi-lag channel into the total *MD* flat-fading channels

$$\mathcal{M}\_D = \sum\_{r=1}^{R\_D} \sum\_{m=M\_{Q\_r,0}}^{M\_{Q\_r,1}} \left( L\{m\} \right)\_{\prime} \tag{24}$$

where *L m*( ) denotes the number of the multilag for corresponding scaling index *m* , for the Doppler scale index *m* with spread , , 1 0 *Q Q r r M M*− , at *r*-th cooperative link. For capturing the multi-scale and multi-lag diversity in the wideband channel, other wavelets, such as Daubechielkhs wavelets, Symlets, etc., have the same capability, since they all possess orthogonality in both scale and lag domain. Rational orthogonal wavelets can be adopted for the scale factor of <sup>0</sup> *<sup>m</sup> <sup>a</sup>* , 0 1 2 ≤ ≤ *<sup>a</sup>* , which is more suitable for the practical scenario [34]. However, wavelet selection problem is beyond the scope of this thesis. In this chapter, we focus on the multi-relay, multi-scale, and multi-lag diversity issue of the cooperative wideband system.

#### **5. Dynamic optimal relay selection strategy**

In the maximum ratio combining, the transmitted signal from *RD* cooperative relays nodes over all multi-scale and multi-lag channel, which underwent independent identically distributed (i.i.d.) complex Gaussian fading, are forwarded to the destination node and combined. In this case, the average probability of error can be found in the closed form as

$$P\_e = \left(\frac{1-\mu}{2}\right)^{M\_D} \sum\_{k=0}^{M\_D-1} \binom{M\_D-1+k}{k} \left(\frac{1+\mu}{2}\right)^{M\_D} \tag{25}$$

where

$$
\mu = \sqrt{\frac{\overline{\mathcal{P}\_c}}{1 + \overline{\mathcal{P}\_c}}},\tag{26}
$$

In the proposed DF cooperative wideband network, because of the fully decoding at the relays, we only consider the link between relays and destination. Therefore, the average SNR per channel *<sup>c</sup>* γcan be derived as

$$\overline{\mathcal{N}\_c} = \left( \prod\_{Q\_r \in \text{DF}} \prod\_{m=M\_{Q\_r,0}}^{M\_{Q\_{r,1}}} \prod\_{l=0}^{L(m)} \frac{E\_Q h\_{Q\_r,D}(m\_l l)}{N\_o} \right)^{\mathcal{W}M\_D} \tag{27}$$

where ( ) , , *Q Dr h ml* denotes the power gains corresponding to the *m-*th scale and *l-*th lag, of the channel from the *r*-th relay to the destination in DF protocol. Combining Eq. (25), (26) and (27), we derive the analytical expression of the BER performance for the proposed DF cooperative wideband network.

116 Ultra Wideband – Current Status and Future Trends

unsuitable DF relays.

scale factor of <sup>0</sup>

where

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

γ

, 2 1 , *r*

<sup>−</sup> <sup>≥</sup> (23)

<sup>=</sup> (24)

*M M*− , at *r*-th cooperative link. For capturing the

*S Q h*

where *B* is the target rate of link between source node and relay, and , *<sup>r</sup> S Q <sup>h</sup>* denotes the power gain of the channel from source node to the *r*-th relay. Therefore, the relays with SNR below the threshold will be removed first, as shown with the gray circles in Fig. 4. The other *RD* relays shown with hexagons are DF relays. According to the dynamic optimal selection strategy, which will be proposed in the next section, we select proper DF relays for cooperation. A one bit feedback channel from destination to relay is used for removing the

Haar wavelet signaling is adopted in the cooperative wideband system to transfer the multi-

,0 1

where *L m*( ) denotes the number of the multilag for corresponding scaling index *m* , for the

multi-scale and multi-lag diversity in the wideband channel, other wavelets, such as Daubechielkhs wavelets, Symlets, etc., have the same capability, since they all possess orthogonality in both scale and lag domain. Rational orthogonal wavelets can be adopted for the

wavelet selection problem is beyond the scope of this thesis. In this chapter, we focus on the multi-relay, multi-scale, and multi-lag diversity issue of the cooperative wideband system.

In the maximum ratio combining, the transmitted signal from *RD* cooperative relays nodes over all multi-scale and multi-lag channel, which underwent independent identically distributed (i.i.d.) complex Gaussian fading, are forwarded to the destination node and combined. In this case, the average probability of error can be found in the closed form as

> 1 0

*M D*

*D*

=

μ

*e k*

can be derived as

μ

1 1 <sup>1</sup> , 2 2 *D D*

> , <sup>1</sup> *c c* γ

In the proposed DF cooperative wideband network, because of the fully decoding at the relays, we only consider the link between relays and destination. Therefore, the average

γ<sup>=</sup> <sup>+</sup>

*M M*

*M k*

−

 μ

(26)

− + − + <sup>=</sup> (25)

*k*

*R M*

*r mM M L m* = =

1 0 *Q Q r r*

( ) ( ) ,1

*<sup>m</sup> <sup>a</sup>* , 0 1 2 ≤ ≤ *<sup>a</sup>* , which is more suitable for the practical scenario [34]. However,

, *<sup>D</sup> Qr Qr*

scale and multi-lag channel into the total *MD* flat-fading channels

Doppler scale index *m* with spread , ,

**5. Dynamic optimal relay selection strategy** 

*P*

*D*

*B*

*th*

γ

As the average probability of error *<sup>e</sup> P* is a precise indication, we can use it to predict the comprehensive transmission performance, only given the channel gains and SNRs at the destination. Consequently, we propose a dynamic optimal selection strategy for the cooperative multi-scale and multi-lag communications. In this algorithm the proper relays are selected to make *<sup>e</sup> P* reach the minimum. First of all, relays are ordered according to the descending order of the SNR between source and relays, as shown in the Fig. 4. According to the proposed SNR threshold, we pick up the DF relays whose SNR is above the threshold.

**Figure 5.** Flow chart of the dynamic optimal selection strategy for the cooperative wideband communications

Then, we proceed with the relay selection to maximize the entire BER performance and try to satisfy the *<sup>e</sup> P* requirement, where the inappropriate DF relays are removed. The whole dynamic optimal selection strategy for the cooperative wideband communication is shown in the flow chart of Fig. 5.

**Figure 7.** BER performance for DF or AF cooperation.

direct transmission (sim) direct transmission (theory) 2-DF cooperation (sim) 2-DF cooperation (theory) 3-DF cooperation (sim) 3-DF cooperation (theory) 2-AF cooperation (sim) 2-AF cooperation (theory) 3-AF cooperation (sim) 3-AF cooperation (theory)

Obviously, the distribution of combined SNR (i.e., *<sup>c</sup>*

distribution giving rise to this slight difference.

these relay nodes.

10-5

10-4

10-3

Bit Error Rate

10-2

10-1

*channels):* 

Fig. 8 shows the BER performance for hybrid DF-AF cooperation. For the DF-dominant hybrid cooperation, the theoretical curves exhibit a good match with the Monte Carlo simulation results curves. The slight gap between theoretical and simulation BER results for the hybrid case of 1-DF + multi-AF can be explained by the AF relay fading which was considered as a double Gaussian channel, a product of two complex Gaussian channel [49].

0 2 4 6 8 10 12 14 16

BER for BPSK modulation with DF or AF cooperation in Rayleigh channel

Eb/No, dB

Comparing 2-DF to 2-AF in Fig. 7, or 2-DF plus 1-AF to 1-DF plus 2-AF in Fig. 8, or other hybrid DF-AF protocols with the same *R*, we can see that the fully decoded DF protocols always show a better BER performance than AF protocols. Therefore, DF protocols with a reliable decoding play a more important role in hybrid cooperative networks than AF protocols. Meanwhile, we can see from the figure that, changing to the AF scheme for the relay nodes with SNR below the threshold also improves the BER performance, as well as the diversity gain of the whole network. In fact, this is a better way than just discarding

*Test Case 2 (Relay selection for cooperative communications over multi-scale and multi-lag wireless* 

γ

) will no longer follow the chi-square

**Figure 6.** Relay selection in the cooperative wavelet wideband wireless transmission strategy (top: source, middle: cooperative relay, bottom: destination)

The wavelet signaling and transceiver design are shown in the Fig. 6. Before the transmission, Haar wavelet signaling is adopted to capture the multi-scale and multi-lag diversity in the wideband channel. In the relaying section, we first remove un-decodable relays by SNR threshold. Then, those relays which undergo the deep fading between relaydestination links will be removed by using the dynamic optimal selection strategy, in order to meet the *<sup>e</sup> P* requirement. After recoding again, Haar wavelet signaling is applied again on the signal. At the destination node, after inverse wavelet transformation, the resulting signals are used for the combination and detection.

## **6. Simulation results and analysis**

*Test Case 1 (BER performance based Relay selection for cooperative communications):* 

In this example, first, we simulated BPSK modulation, Rayleigh channel, flat fading, without OFDM, and supposed the SNR threshold for correct decoding is 4*Eb*/*N0*, then we assumed ,, , <sup>1</sup> *Q D SQ Q D i jj h hh* == = , for all branches, to verify proposed analytical BER expression. The resulting average BER were plotted against the transmit SNR defined as SNR = *Eb*/*N0*. As shown in the Fig. 7, the theoretical curves of multi-DF cooperation derived from our analytical closed-form BER expression clearly agree with the Monte Carlo simulated curves, while the theoretical curves of 2-AF and 3-AF cooperation match the simulation result only at the low SNR region.

**Figure 7.** BER performance for DF or AF cooperation.

Dynamic optimal selection

source, middle: cooperative relay, bottom: destination)

signals are used for the combination and detection.

**6. Simulation results and analysis** 

,, , <sup>1</sup> *Q D SQ Q D i jj*

at the low SNR region.

Transmit data

SNR vs threshold Coding Wavelet

source

cooperative relay

destination

The wavelet signaling and transceiver design are shown in the Fig. 6. Before the transmission, Haar wavelet signaling is adopted to capture the multi-scale and multi-lag diversity in the wideband channel. In the relaying section, we first remove un-decodable relays by SNR threshold. Then, those relays which undergo the deep fading between relaydestination links will be removed by using the dynamic optimal selection strategy, in order to meet the *<sup>e</sup> P* requirement. After recoding again, Haar wavelet signaling is applied again on the signal. At the destination node, after inverse wavelet transformation, the resulting

In this example, first, we simulated BPSK modulation, Rayleigh channel, flat fading, without OFDM, and supposed the SNR threshold for correct decoding is 4*Eb*/*N0*, then we assumed

*h hh* == = , for all branches, to verify proposed analytical BER expression. The resulting average BER were plotted against the transmit SNR defined as SNR = *Eb*/*N0*. As shown in the Fig. 7, the theoretical curves of multi-DF cooperation derived from our analytical closed-form BER expression clearly agree with the Monte Carlo simulated curves, while the theoretical curves of 2-AF and 3-AF cooperation match the simulation result only

**Figure 6.** Relay selection in the cooperative wavelet wideband wireless transmission strategy (top:

*Test Case 1 (BER performance based Relay selection for cooperative communications):* 

transform Decoding Coding

Decoding

Receive data

Wavelet signaling

Wavelet

Wavelet transform signaling

Fig. 8 shows the BER performance for hybrid DF-AF cooperation. For the DF-dominant hybrid cooperation, the theoretical curves exhibit a good match with the Monte Carlo simulation results curves. The slight gap between theoretical and simulation BER results for the hybrid case of 1-DF + multi-AF can be explained by the AF relay fading which was considered as a double Gaussian channel, a product of two complex Gaussian channel [49]. Obviously, the distribution of combined SNR (i.e., *<sup>c</sup>* γ ) will no longer follow the chi-square distribution giving rise to this slight difference.

Comparing 2-DF to 2-AF in Fig. 7, or 2-DF plus 1-AF to 1-DF plus 2-AF in Fig. 8, or other hybrid DF-AF protocols with the same *R*, we can see that the fully decoded DF protocols always show a better BER performance than AF protocols. Therefore, DF protocols with a reliable decoding play a more important role in hybrid cooperative networks than AF protocols. Meanwhile, we can see from the figure that, changing to the AF scheme for the relay nodes with SNR below the threshold also improves the BER performance, as well as the diversity gain of the whole network. In fact, this is a better way than just discarding these relay nodes.

*Test Case 2 (Relay selection for cooperative communications over multi-scale and multi-lag wireless channels):* 

BER for BPSK modulation with DF-AF hybrid cooperation in Rayleigh channel

h1=4,1-DF sim h1=4,1-DF theory h1=4,h2=1,2-DF sim h1=4,h2=1,2-DF theory h1=4,h2=1,h3=1,3-DF sim h1=4,h2=1,h3=1,3-DF theory

from our analytical closed-form BER expression clearly agree with the Monte Carlo simulated

BER for cooperative wavelet communications with the scale-lag channel

**Figure 9.** BER performance for cooperative wavelet wideband communications

In the second example, we illustrate how to exploit the proposed analytical BER expression together with dynamic optimal selection strategy to select relays for the cooperative wideband communications. We suppose the target *<sup>e</sup> <sup>P</sup>* at SNR *Eb*/*No* = 10dB is <sup>4</sup> <sup>10</sup><sup>−</sup> . In the original state, we suppose that we already have 1-Relay with 1-scale and 2-lag diversity components, with power gains 1 *h* = 4 and 2 *h* = 1 . The BER performance is shown by the triangle marked curve in Fig. 10. The *<sup>e</sup> P* requirement is not met by the original state, so we expect to cooperate with more relays, to gain from more diversity components. For the test case 1, we test and combine with a deep fading relay with only one scale-lag diversity, power gain 3 *h* = 0.04 . Analytical BER expression predicts that adding this deep fading relay deteriorates the BER performance. Therefore, we discard this relay. For the test case 2, we

0 5 10 15 20 25

Eb/No, dB

curves.

10-6

10-5

10-4

10-3

Bit Error Rate

10-2

10-1

**Figure 8.** BER performance for hybrid DF+AF cooperation.

In this case, we use the simulation results to verify our theoretical claims on the analytical BER expression and illustrate the dynamic optimal selection strategy.

In the first example, simulation results justify the proposed analytical BER expression for cooperative wavelet communications with multi-scale and multi-lag wireless channel, i.e., the combination of Eq. (25), (26) and (27), which can be used to predict the transmission performance and enable the dynamic optimal selection strategy as shown in the Fig. 5. BPSK is adopted as the modulation scheme, and the 2-decomposition level Haar wavelet transform is adopted as a RAKE receiver to capture the multi-scale and multi-lag diversity components, and transfer the multi-relay, multi-scale and multi-lag channel into the orthogonal flat-fading channels. Therefore, we consider 2-relay three orthogonal channels in this simulation. Relay 1 has 1-scale and 2-lag diversity components, the power gains are <sup>1</sup> *h* = 4 and 2 *h* = 1 . Relay 2 has 1-scale and 1-lag diversity component, the power gain is <sup>3</sup> *h* = 1 . The resulting average BER are plotted against the transmit SNR defined as SNR = *Eb*/*No*. As shown in the Fig. 9, the theoretical curves of different diversities derived from our analytical closed-form BER expression clearly agree with the Monte Carlo simulated curves.

120 Ultra Wideband – Current Status and Future Trends

**Figure 8.** BER performance for hybrid DF+AF cooperation.

10-5

10-4

10-3

Bit Error Rate

10-2

10-1

BER expression and illustrate the dynamic optimal selection strategy.

In this case, we use the simulation results to verify our theoretical claims on the analytical

Eb/No, dB

BER for BPSK modulation with DF-AF hybrid cooperation in Rayleigh channel

1-DF+1-AF cooperation (sim) 1-DF+1-AF cooperation (theory) 1-DF+2-AF cooperation (sim) 1-DF+2-AF cooperation (theory) 2-DF+1-AF cooperation (sim) 2-DF+1-AF cooperation (theory) 2-DF+2-AF cooperation (sim) 2-DF+2-AF cooperation (theory) 3-DF-1-AF cooperation (sim) 3-DF-1-AF cooperation (theory)

0 5 10 15 20

In the first example, simulation results justify the proposed analytical BER expression for cooperative wavelet communications with multi-scale and multi-lag wireless channel, i.e., the combination of Eq. (25), (26) and (27), which can be used to predict the transmission performance and enable the dynamic optimal selection strategy as shown in the Fig. 5. BPSK is adopted as the modulation scheme, and the 2-decomposition level Haar wavelet transform is adopted as a RAKE receiver to capture the multi-scale and multi-lag diversity components, and transfer the multi-relay, multi-scale and multi-lag channel into the orthogonal flat-fading channels. Therefore, we consider 2-relay three orthogonal channels in this simulation. Relay 1 has 1-scale and 2-lag diversity components, the power gains are <sup>1</sup> *h* = 4 and 2 *h* = 1 . Relay 2 has 1-scale and 1-lag diversity component, the power gain is <sup>3</sup> *h* = 1 . The resulting average BER are plotted against the transmit SNR defined as SNR = *Eb*/*No*. As shown in the Fig. 9, the theoretical curves of different diversities derived

BER for cooperative wavelet communications with the scale-lag channel

**Figure 9.** BER performance for cooperative wavelet wideband communications

In the second example, we illustrate how to exploit the proposed analytical BER expression together with dynamic optimal selection strategy to select relays for the cooperative wideband communications. We suppose the target *<sup>e</sup> <sup>P</sup>* at SNR *Eb*/*No* = 10dB is <sup>4</sup> <sup>10</sup><sup>−</sup> . In the original state, we suppose that we already have 1-Relay with 1-scale and 2-lag diversity components, with power gains 1 *h* = 4 and 2 *h* = 1 . The BER performance is shown by the triangle marked curve in Fig. 10. The *<sup>e</sup> P* requirement is not met by the original state, so we expect to cooperate with more relays, to gain from more diversity components. For the test case 1, we test and combine with a deep fading relay with only one scale-lag diversity, power gain 3 *h* = 0.04 . Analytical BER expression predicts that adding this deep fading relay deteriorates the BER performance. Therefore, we discard this relay. For the test case 2, we test and combine with a relay with one scale-lag diversity, power gain 3 *h* = 4 , which improves the BER performance, and satisfies the *<sup>e</sup> P* requirement. Therefore, we adopt this relay.

Cooperative Communication over Multi-Scale and Multi-Lag Wireless Channels 123

curves and numerical simulated results shows that the derived analytical BER expression is suitable for the performance prediction of cooperation wavelet wideband communication. The compact and closed-form BER expression can easily provide an insight into the results as well as a heuristic help for the design of future cooperative wavelet wideband systems. For the suggested cooperative wavelet protocol, we also presented a dynamic optimal selection strategy for the optimal relay selection, which maximizes the whole system

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description, *IEEE Trans. Commun.*, vol. 51, no. 11, Nov. 2003, pp. 1927–1938.

*International Research Centre for Telecommunications and Radar (IRCTR),* 

*Dept. EEMCS, Delft University of Technology, Delft,* 

*Dept. EEMCS, Delft University of Technology, Delft,* 

*Ph.D. dissertation*, MIT. Cambridge, MA, Sep. 2002.

transmission performance.

**Author details** 

*The Netherlands* 

*The Netherlands* 

**8. References** 

House, 2005.

1998, pp. 311-335.

Nov. 2003, pp. 1939–1948.

Cambridge University Press, 2009

153, FCC 02-48, Feb. 14, 2002, pp. 1–118.

T. Xu

H. Lu and H. Nikookar

*Circuits and Systems Group (CAS),* 

**Figure 10.** Relay selection for cooperative 2-decomposition level wavelet wideband communication

#### **7. Conclusions**

Wideband scale-lag channels can be found in many applications, including ultra-wideband communications and underwater acoustic communications. Signaling and reception schemes using the wavelet theory enable the multi-scale and multi-lag diversity in the wideband system. In this chapter, we designed a cooperative wavelet system to capture the joint cooperative-scale-lag diversity. We proposed the analytical BER expression for the cooperative wavelet wideband communication. The agreement between the analytical curves and numerical simulated results shows that the derived analytical BER expression is suitable for the performance prediction of cooperation wavelet wideband communication. The compact and closed-form BER expression can easily provide an insight into the results as well as a heuristic help for the design of future cooperative wavelet wideband systems. For the suggested cooperative wavelet protocol, we also presented a dynamic optimal selection strategy for the optimal relay selection, which maximizes the whole system transmission performance.
