**1. Introduction**

102 Ultra Wideband – Current Status and Future Trends

Yazdandoost K. & Sayrafian-Pour K. (2009). Channel Model for Body Area Network (BAN),

*Report to the IEEE*, P802.15. ID: IEEE 802.15-08-0780-02-0006, April 2009.

The development of wireless communication applications in the last few years is unprecedented. Wireless communication has evolved in various ways. The next generation of wireless systems should service more users while supporting mobility and high data rates. These requirements necessitate efficient use of available resources to provide acceptable service quality.

In the wireless channel, fading can be coped with by using diversity techniques or by transmitting the signal over several independently fading channels and combining different signal at the receiver before demodulation and detection. Spatial diversity techniques are known to increase the system reliability without sacrificing time and bandwidth efficiency. However, due to the limitation of the diversity order and correlated channel, multiple antenna diversity is not always practically feasible.

Spatial diversity has been studied intensively in the context of Multiple-Input-Multiple-Output (MIMO) systems [1]. It has been shown that utilizing MIMO systems can significantly improve the system throughput and reliability [2]. However, MIMO gains hinge on the independence of the paths between transmit and receive antennas, for which one must guarantee antenna element separation several times the wavelength, a requirement difficult to meet with the small-size terminals. To overcome this problem, and to benefit from the performance enhanced by MIMO systems, cooperative diversity schemes for the relay transmission have been introduced in [3-5].

Cooperative diversity [6] is an alternative way to achieve spatial diversity when the multiple antenna structure is not an option. Cooperative communications offer diversity based on the fact that other users in the cooperative network are able to overhear the transmitted signal and forward the information to the destination through different paths. Cooperative

communications have been receiving a lot of attention recently as an attractive way to combat frequency-selectivity of wireless channel, as they consume neither time nor frequency resources. Furthermore, cooperative communications are promising techniques to increase the transmission reliability, since they can achieve spatial diversity by using the relay nodes as virtual antennas, and mitigating fading effects. By adopting relay to forward information, we can increase the capacity, lower the bit-error rate, and increase the achievable transmission range.

Cooperative Communication over Multi-Scale and Multi-Lag Wireless Channels 105

branches of the model are orthogonal when a single wavelet pulse is transmitted. The single pulse case is examined in paper [10]. Multi-scale multi-lag wavelet signaling is possible as well [12], [13], although inter-scale and inter-delay interference results. In paper [12], multiple receiver designs to combat such interference are provided exploiting the banded

Note that scale-spreading arises from the same fundamental mechanism that causes Doppler spreading. This scale-lag diversity is better described by the wavelet transform than the conventional time-frequency representation for the narrowband linear time-varying (LTV) system, and is so called the wideband LTV representation [10, 11]. Wideband LTV representation has been proven and verified for many applications in terms of high data rate wireless communications [14-17], high-speed underwater acoustic communications [18-20], vehicle-to-vehicle (V2V) wideband communication systems [21, 22], and radar/sonar systems [23]. In general, the transmit waveform could be designed to optimally enable the

Doppler scaling and multipath spread in the wideband system implementations are usually treated as distortions rather than potential diversity sources, and always compensated after estimation. In this chapter, Doppler scaling and multipath spread are utilized to obtain a joint scale-lag diversity with the discrete multi-scale and multi-lag wireless channel model by properly designing signaling and reception schemes using the discrete wavelet transform. The wavelet technique used in the wideband system is well motivated since wideband processing is intimately related to the wavelet theory [24-26]. The wideband LTV representation has proven useful in many applications as noted above. However, no cooperative wavelet implementations have been exploited to provide further increased

In this chapter, we will design a cooperative wavelet communication scheme to exploit the joint scale-lag diversity in the wideband LTV system. Furthermore, we propose the analytical Bit Error Rate (BER) expression for the cooperative wideband system, and provide a dynamic optimal selection strategy for relay selection to gain from multi-relay, multi-scaling, and multi-lag diversity, and maximize the whole system transmission

The rest of chapter is organized as follows. In section 2, an overview of the multi-scale and multi-lag diversity in wideband system is provided. In section 3, we investigate the general hybrid cooperative scheme that includes both DF and AF relays, and review the SNR thresholding scheme as well as dynamic optimal combination strategy for the hybrid DF-AF cooperation to achieve the optimal system BER performance. In section 4, we construct the cooperative wavelet wideband transmission strategy, and derive the analytical BER expression for the cooperative wavelet communications in the multi-relay, multi-scale and multi-lag channel. In Section 5, we represent the dynamic optimal selection strategy for the relay selection. Simulations results are provided in Section 6 and are compared to the analytical formulas. The relay selection is also illustrated in this section. Finally, Section 7

nature of the resulting interference.

scale-lag diversity in the wideband LTV system.

performance for wideband systems.

performance.

concludes the chapter.

There are mainly two relaying protocols in cooperative communications: Amplify-and-Forward (AF) and Decode-and-Forward (DF). In AF, the received signal is amplified and retransmitted to the destination. The advantage of this protocol is its simplicity and low cost implementation. However, the noise is also amplified at the relay. In the DF, the relay attempts to decode the received signals. If successful, it re-encodes the information and retransmits it. Therefore, DF relaying usually enjoys a better transmission performance than the AF relaying. The time-consuming decoding tradeoff for a better cooperative transmission, and finding the optimum hybrid cooperative schemes, that include both DF and AF for different situations, is an important issue for the cooperative networks design.

In this chapter, we investigate the performance and relay selection issue in cooperative wideband communication systems. Wideband communication systems are defined as having a fractional bandwidth—the ratio of single-sided bandwidth to center frequency that exceeds 0.2 [7]. Wideband channels are of interest in a variety of wireless communication scenarios including underwater acoustic systems and wideband terrestrial radio frequency systems such as spread-spectrum or ultra wideband. Due to the nature of wideband propagation, such channels exhibit some fundamental differences relative to socalled narrowband channels. In the wideband systems, the effects of mobility in the multipath mobile environment are not well described by frequency-domain spreading, but rather by time-domain scale spreading. More specifically, in narrowband channels, the transmitted signal experiences multiple propagation paths each with a possibly distinct Doppler frequency shift, and thus these channels are also known as multi-Doppler shift, multi-lag channels. For wideband channels, on the other hand, each propagation path experiences a distinct Doppler scale, hence the term, multi-scale, multi-lag channel. For both types of wideband and narrowband time-varying channels, so-called canonical channel models have been proposed [8-11], limiting the number of channel coefficients required to represent the channel.

In particular, there has been significant success in the application of canonical models to narrowband time-varying channels [8]. For wideband time-varying channels a canonical model has been proposed in [9-11], which is also dubbed as the scale-lag canonical model. This model has been adopted for direct sequence spread spectrum (DSSS) communication systems [11] to develop a scale-lag RAKE receiver to collect the diversity inherent in the multi-scale multi-lag channel. In addition, this model has spurred the use of wavelet signaling due to the fact that when the wavelets are "matched" to the scale-lag model, the receiver structure is greatly simplified – the signals corresponding to different scale-lag branches of the model are orthogonal when a single wavelet pulse is transmitted. The single pulse case is examined in paper [10]. Multi-scale multi-lag wavelet signaling is possible as well [12], [13], although inter-scale and inter-delay interference results. In paper [12], multiple receiver designs to combat such interference are provided exploiting the banded nature of the resulting interference.

104 Ultra Wideband – Current Status and Future Trends

achievable transmission range.

networks design.

represent the channel.

communications have been receiving a lot of attention recently as an attractive way to combat frequency-selectivity of wireless channel, as they consume neither time nor frequency resources. Furthermore, cooperative communications are promising techniques to increase the transmission reliability, since they can achieve spatial diversity by using the relay nodes as virtual antennas, and mitigating fading effects. By adopting relay to forward information, we can increase the capacity, lower the bit-error rate, and increase the

There are mainly two relaying protocols in cooperative communications: Amplify-and-Forward (AF) and Decode-and-Forward (DF). In AF, the received signal is amplified and retransmitted to the destination. The advantage of this protocol is its simplicity and low cost implementation. However, the noise is also amplified at the relay. In the DF, the relay attempts to decode the received signals. If successful, it re-encodes the information and retransmits it. Therefore, DF relaying usually enjoys a better transmission performance than the AF relaying. The time-consuming decoding tradeoff for a better cooperative transmission, and finding the optimum hybrid cooperative schemes, that include both DF and AF for different situations, is an important issue for the cooperative

In this chapter, we investigate the performance and relay selection issue in cooperative wideband communication systems. Wideband communication systems are defined as having a fractional bandwidth—the ratio of single-sided bandwidth to center frequency that exceeds 0.2 [7]. Wideband channels are of interest in a variety of wireless communication scenarios including underwater acoustic systems and wideband terrestrial radio frequency systems such as spread-spectrum or ultra wideband. Due to the nature of wideband propagation, such channels exhibit some fundamental differences relative to socalled narrowband channels. In the wideband systems, the effects of mobility in the multipath mobile environment are not well described by frequency-domain spreading, but rather by time-domain scale spreading. More specifically, in narrowband channels, the transmitted signal experiences multiple propagation paths each with a possibly distinct Doppler frequency shift, and thus these channels are also known as multi-Doppler shift, multi-lag channels. For wideband channels, on the other hand, each propagation path experiences a distinct Doppler scale, hence the term, multi-scale, multi-lag channel. For both types of wideband and narrowband time-varying channels, so-called canonical channel models have been proposed [8-11], limiting the number of channel coefficients required to

In particular, there has been significant success in the application of canonical models to narrowband time-varying channels [8]. For wideband time-varying channels a canonical model has been proposed in [9-11], which is also dubbed as the scale-lag canonical model. This model has been adopted for direct sequence spread spectrum (DSSS) communication systems [11] to develop a scale-lag RAKE receiver to collect the diversity inherent in the multi-scale multi-lag channel. In addition, this model has spurred the use of wavelet signaling due to the fact that when the wavelets are "matched" to the scale-lag model, the receiver structure is greatly simplified – the signals corresponding to different scale-lag Note that scale-spreading arises from the same fundamental mechanism that causes Doppler spreading. This scale-lag diversity is better described by the wavelet transform than the conventional time-frequency representation for the narrowband linear time-varying (LTV) system, and is so called the wideband LTV representation [10, 11]. Wideband LTV representation has been proven and verified for many applications in terms of high data rate wireless communications [14-17], high-speed underwater acoustic communications [18-20], vehicle-to-vehicle (V2V) wideband communication systems [21, 22], and radar/sonar systems [23]. In general, the transmit waveform could be designed to optimally enable the scale-lag diversity in the wideband LTV system.

Doppler scaling and multipath spread in the wideband system implementations are usually treated as distortions rather than potential diversity sources, and always compensated after estimation. In this chapter, Doppler scaling and multipath spread are utilized to obtain a joint scale-lag diversity with the discrete multi-scale and multi-lag wireless channel model by properly designing signaling and reception schemes using the discrete wavelet transform. The wavelet technique used in the wideband system is well motivated since wideband processing is intimately related to the wavelet theory [24-26]. The wideband LTV representation has proven useful in many applications as noted above. However, no cooperative wavelet implementations 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 in the wideband LTV system. Furthermore, we propose the analytical Bit Error Rate (BER) expression for the cooperative wideband system, and provide a dynamic optimal selection strategy for relay selection to gain from multi-relay, multi-scaling, and multi-lag diversity, and maximize the whole system transmission performance.

The rest of chapter is organized as follows. In section 2, an overview of the multi-scale and multi-lag diversity in wideband system is provided. In section 3, we investigate the general hybrid cooperative scheme that includes both DF and AF relays, and review the SNR thresholding scheme as well as dynamic optimal combination strategy for the hybrid DF-AF cooperation to achieve the optimal system BER performance. In section 4, we construct the cooperative wavelet wideband transmission strategy, and derive the analytical BER expression for the cooperative wavelet communications in the multi-relay, multi-scale and multi-lag channel. In Section 5, we represent the dynamic optimal selection strategy for the relay selection. Simulations results are provided in Section 6 and are compared to the analytical formulas. The relay selection is also illustrated in this section. Finally, Section 7 concludes the chapter.

### **2. Wideband multi-scale and multi-lag representation**

Multi-scale and multi-lag representation is suitable for the wideband systems to satisfy either of the two conditions, i.e., absolute condition or relative condition. First is the absolute condition, which requires the signal fractional bandwidth (ratio of bandwidth to center frequency) to be larger than 0.2. Second one is the relative condition, i.e., the motion velocity *v* , the propagation speed *c* and the signal time bandwidth (TB) product should satisfy 2 1 *v c TW* ( ) , where *T* stands for the transmitted signal duration and *W* denotes the transmitted signal bandwidth. Therefore, the multi-scale and multi-lag system can be defined as a system that operates at high fractional bandwidths or large TB products or when the *v c* ratio is large.

For example, an ultra wideband (UWB) system transmits signals with high fractional bandwidths (> 0.2) or large TB products (105- 106) to improve resolution capacity and increase noise immunity [27]. Or an underwater acoustic environment with fast moving objects could result in a large ratio due to the relatively low speed of sound [28]. In these situations, multi-scale and multi-lag representation is needed to account for the Doppler scale effects, but not Doppler shift.

Under the wideband background mentioned above, we consider a multi-scale and multi-lag system, that is, a signal, *x* ( )*t* , transmitted over a wideband propagation medium is received as

$$y(t) = \int\_{A\_l}^{A\_u} \int\_0^{T\_d} h(\tau, a) \sqrt{a} \propto \left(a(t - \tau)\right) d\tau da + n\left(t\right),\tag{1}$$

Cooperative Communication over Multi-Scale and Multi-Lag Wireless Channels 107

(2)

<sup>=</sup> (3)

τ

*l*

( ) <sup>1</sup>

*y t b h ml x t nt* = = *W* <sup>=</sup> − +

where *L m*( ) denotes the number of the multilag for corresponding scaling index *m* , as shown to be the number of cross points on each row in Fig. 1. *M*0 and *M*<sup>1</sup> are lower and upper bounds of *m*, respectively. In fact, the multilag resolution in a wideband channel is 1 / ( ) *aW* if the signal is scaled by *a* . When the number of scatterers contributing to the discrete channel gain *h ml* ( ) , is exceedingly large, the random variables *h ml* ( ) , can be

Consequently, the inverse discrete wavelet transform description in Eq. (2) effectively

0

orthogonal, flat-fading channels. This results in a potential joint scale-lag diversity order *M*

**multilag**

**Figure 1.** Dyadic sampling in the multi-scale and multi-lag plane, the dyadic scale is 2*<sup>m</sup> a* = and for the given 0 1 *<sup>M</sup>* ≤ ≤ *m M* , 0 1 *mM M* , , are all integer. The multi-lag resolution is 1/ 2( ) *mW* , given the signal

In order for a scale-lag RAKE receiver to collect the aforementioned diversity components, transmitted signal should be designed as a wavelet-based waveform. Wideband multi-scale and multi-lag channel performs the inverse discrete wavelet transform on the transmitted

1 2*mW* *M*

*m M M Lm* =

( ) ( ) <sup>1</sup>

,

*m m*

, 22 ,

( ) ( ) ( )

*M L m*

0

assumed Gaussian and therefore independent.

that can be exploited to increase system performance.

*a*

decomposes the wideband channel into

**multiscale**

<sup>0</sup> 2*<sup>M</sup>*

bandwidth *W* .

<sup>1</sup> 2*<sup>M</sup>*

0

*mM l*

1

where *a vc* ≈ +1 2 is the Doppler scale, results in a time compression or expansion of the waveform caused by a relative velocity *v* between transmitter and scatterer. When the Doppler scale is such that 1 *a* > , then the scatterer is approaching the transmitter and the transmitted signal is compressed with respect to time; in contrast, when 0 1 < <*a* , the received signal is dilated and the scatterer is moving away from the transmitter. τ is the propagation delay due to reflections of *x t*( ) by scatterers in the medium. Channel gain *h a* ( ) τ , can be modeled as a stochastic process, when the system is randomly varying [10]. Due to physical restrictions on the system, we can assume that *h a* ( ) τ , is effectively nonzero only when 0 *Al u* < ≤≤*a A* and 0 *<sup>d</sup>* ≤ ≤ τ *T* , where *Au l* − *A* is the Doppler scale spread and *<sup>d</sup> T* is the multipath delay spread. The noise process, *n t*( ) is modeled as a white Gaussian random process.

Note that regardless the noise term, Eq. (1) is in the form of an inverse wavelet transform with *x t*( ) acting as the wavelet. Therefore, according to the wavelet theory, we sample the multi-scale and multi-lag plane in a dyadic lattice as shown in Fig. 1 [10, 29].

Without loss the generality, we consider BPSK modulation, the information-bearing symbol of the transmitted signal is 0*b* = ±1 . From the multi-scale and multi-lag channel defined in Fig. 1, the overall baseband signal at the receiver can be rewritten as:

$$y\left(t\right) = b\_0 \sum\_{m=M\_0}^{M\_1} \sum\_{l=1}^{L(m)} h\left(m, l\right) \sqrt{2^m} \ge \left(2^m t - \frac{l}{W}\right) + n\left(t\right),\tag{2}$$

where *L m*( ) denotes the number of the multilag for corresponding scaling index *m* , as shown to be the number of cross points on each row in Fig. 1. *M*0 and *M*<sup>1</sup> are lower and upper bounds of *m*, respectively. In fact, the multilag resolution in a wideband channel is 1 / ( ) *aW* if the signal is scaled by *a* . When the number of scatterers contributing to the discrete channel gain *h ml* ( ) , is exceedingly large, the random variables *h ml* ( ) , can be assumed Gaussian and therefore independent.

106 Ultra Wideband – Current Status and Future Trends

when the *v c* ratio is large.

scale effects, but not Doppler shift.

only when 0 *Al u* < ≤≤*a A* and 0 *<sup>d</sup>* ≤ ≤

as

*h a* ( ) τ

process.

**2. Wideband multi-scale and multi-lag representation** 

Multi-scale and multi-lag representation is suitable for the wideband systems to satisfy either of the two conditions, i.e., absolute condition or relative condition. First is the absolute condition, which requires the signal fractional bandwidth (ratio of bandwidth to center frequency) to be larger than 0.2. Second one is the relative condition, i.e., the motion velocity *v* , the propagation speed *c* and the signal time bandwidth (TB) product should satisfy 2 1 *v c TW* ( ) , where *T* stands for the transmitted signal duration and *W* denotes the transmitted signal bandwidth. Therefore, the multi-scale and multi-lag system can be defined as a system that operates at high fractional bandwidths or large TB products or

For example, an ultra wideband (UWB) system transmits signals with high fractional bandwidths (> 0.2) or large TB products (105- 106) to improve resolution capacity and increase noise immunity [27]. Or an underwater acoustic environment with fast moving objects could result in a large ratio due to the relatively low speed of sound [28]. In these situations, multi-scale and multi-lag representation is needed to account for the Doppler

Under the wideband background mentioned above, we consider a multi-scale and multi-lag system, that is, a signal, *x* ( )*t* , transmitted over a wideband propagation medium is received

() ( ) ( ) ( ) ( ) <sup>0</sup> , , *u d*

where *a vc* ≈ +1 2 is the Doppler scale, results in a time compression or expansion of the waveform caused by a relative velocity *v* between transmitter and scatterer. When the Doppler scale is such that 1 *a* > , then the scatterer is approaching the transmitter and the transmitted signal is compressed with respect to time; in contrast, when 0 1 < <*a* , the

propagation delay due to reflections of *x t*( ) by scatterers in the medium. Channel gain

the multipath delay spread. The noise process, *n t*( ) is modeled as a white Gaussian random

Note that regardless the noise term, Eq. (1) is in the form of an inverse wavelet transform with *x t*( ) acting as the wavelet. Therefore, according to the wavelet theory, we sample the

Without loss the generality, we consider BPSK modulation, the information-bearing symbol of the transmitted signal is 0*b* = ±1 . From the multi-scale and multi-lag channel defined in

, can be modeled as a stochastic process, when the system is randomly varying [10].

 ττ

− + (1)

τ

*T* , where *Au l* − *A* is the Doppler scale spread and *<sup>d</sup> T* is

τis the

, is effectively nonzero

*<sup>A</sup> <sup>y</sup> t h a ax a t d da n t* <sup>=</sup>

τ

received signal is dilated and the scatterer is moving away from the transmitter.

*l A T*

Due to physical restrictions on the system, we can assume that *h a* ( )

multi-scale and multi-lag plane in a dyadic lattice as shown in Fig. 1 [10, 29].

τ

Fig. 1, the overall baseband signal at the receiver can be rewritten as:

Consequently, the inverse discrete wavelet transform description in Eq. (2) effectively decomposes the wideband channel into

$$M = \sum\_{m=M\_0}^{M\_1} \left( L\left(m\right) \right),\tag{3}$$

orthogonal, flat-fading channels. This results in a potential joint scale-lag diversity order *M* that can be exploited to increase system performance.

**Figure 1.** Dyadic sampling in the multi-scale and multi-lag plane, the dyadic scale is 2*<sup>m</sup> a* = and for the given 0 1 *<sup>M</sup>* ≤ ≤ *m M* , 0 1 *mM M* , , are all integer. The multi-lag resolution is 1/ 2( ) *mW* , given the signal bandwidth *W* .

In order for a scale-lag RAKE receiver to collect the aforementioned diversity components, transmitted signal should be designed as a wavelet-based waveform. Wideband multi-scale and multi-lag channel performs the inverse discrete wavelet transform on the transmitted signal ( ) *m l*, *x t* . At the receiver side, for the diversity component corresponding to the *m-*th scale and *l-*th lag, the detection statistic

$$\mathcal{A}\_{m,l} = \left\langle \boldsymbol{y}\left(t\right), \sqrt{2^{m}} \,\mathrm{x}\left(2^{m}t - \frac{l}{W}\right) \right\rangle = \int\_{-\infty}^{\infty} \boldsymbol{y}\left(t\right) \sqrt{2^{m}} \,\mathrm{x}^{\*}\left(2^{m}t - \frac{l}{W}\right) dt\tag{4}$$

Cooperative Communication over Multi-Scale and Multi-Lag Wireless Channels 109

SNR*SQ* > ··· > <sup>R</sup>

SNR*SQ* ,

in the wideband LTV system. Furthermore, we propose the analytical BER expression for the cooperative wideband system, and provide a dynamic optimal selection strategy for relay selection to gain from multi-relay, multi-scaling, and multi-lag diversity, and

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

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

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

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

order of the Signal-to-Noise Ratio (SNR) between *S* and *Q*, i.e., <sup>1</sup>

where SNR *<sup>r</sup> SQ* denotes the *r*-th largest SNR between *S* and *Q*.

shown with the white hexagons in Fig. 2 [36].

maximize the whole system transmission performance.

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

from the source-destination and relay-destination.

cooperative relaying networks design.

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

is the correlator output of the received signal *y*( )*t* and the basic waveform 2 2( ) *m m x t lW* <sup>−</sup> . Therefore, the detection statistic *m l*, λ can be obtained by the dyadic scalelag 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 coherently to obtain the estimate of the transmitted information symbol 0*b* as

$$\hat{\boldsymbol{\theta}}\_{0} = \text{sign}\left\{ \text{Re}\left(\sum\_{m=M\_{0}}^{M\_{1}} \sum\_{l=1}^{L(m)} \boldsymbol{h}^{\*}\left(m,l\right) \boldsymbol{\mathcal{A}}\_{m,l}\right) \right\}.\tag{5}$$

We note that this coherent detection of the scale-lag RAKE receiver corresponds to a Maximum Ratio Combination (MRC).

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 offset [31].

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

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 in the wideband LTV system. Furthermore, we propose the analytical BER expression for the cooperative wideband system, and provide a dynamic optimal selection strategy for relay selection to gain from multi-relay, multi-scaling, and multi-lag diversity, and maximize the whole system transmission performance.
