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**Chapter 4 Diversity analysis in frequency selective fading channels**

**Introduction**

In wireless communication systems multipath channels severely degrade the performance of the system. In frequency selective channels these multipath components extend beyond the symbol duration and cause intersymbol interferences [75]. This intersymbol interference significantly deteriorates the performance of the system as compared to flat fading envi-ronments where the performance degradation happens due to vector sum of the received multipath component within the symbol duration. Diversity techniques, spread spectrum techniques and channel estimation algorithms are widely used method to mitigate the effect of fading [74].

In the last decade chaotic signals are studied intensively. The chaotic signals generated from the same chaotic map with different initial condition are uncorrelated. Therefore an infinite number of chaotic signals can be generated from the same chaotic map. Due to these properties chaos-based multiuser DS-CDMA systems gain significant interest among the researchers [39, 91–93].

Lots of research has been done in the chaos-based CDMA field [10, 49–51, 58, 92]. In these research works chaos-based CDMA systems are studied in flat fading channels. Furthermore, the fading coefficients are assumed to be real and positive. A complex flat fading channel with known channel coefficients is considered in [52].

The objective of this chapter is to investigate the effect of imperfect channel estimation in frequency selective fading channels for downlink systems. A slowly time varying chan-nel is considered such that fading coefficients are constant for one symbol duration. BER expression in closed form is derived for a BPSK system with receiver antenna diversity. MRC technique is used to optimize the decision variable. Further BER expressions show that the probability of error depends on the cosine of the estimated phase error. Various estimation algorithms are shown in [32] but we have used LMS algorithm to estimate the channel coefficients because of its simplicity, and results are compared with perfect channel estimation. We have shown that the BER expression in [10, 49] is a special case of BER ex-pression derived in this chapter. Perfect synchronization is assumed in this chapter. Various synchronization techniques are discussed in [49, 84, 85].

This chapter is organized as follows. A chaos-based adaptive CDMA system with re-ceiver diversity is shown in Section 4.2. Section 4.3 presents the analytical performance of the system. Simulation results are shown in Section 4.4. Section 4.5 reports some conclud-ing remarks and possible future works.

**System model**

Baseband representation of the chaos-based CDMA system is shown in Fig. 4.1. The baseband signal s^{g}_{k} for g^{th} user at k^{th} chip instant, is given by.

where γ_{i}^{g} is the g^{th} user symbol at i^{th} time instant, x_{k}^{g} is k^{th} chip of the g^{th} user chaotic spreading sequence or chip within an information bit (i.e. k = 1, ··· , 2β ) and 2β is the spreading factor. Superscript p is used to denote the terms related to pilot instead of g.

For L multipath channels, the k^{th} chip of received signal r_{k}^{mg} at m^{th} antenna of g^{th} user at time i is given by.

where ξ_{k,m} denotes the k^{th} chip of complex additive white Gaussian noise for m^{th} receiving antenna with power spectral density equal to N_{}. τ_{l} is the delay for the l^{th} path with respect to the first path. Delay of the first path is assumed to be zero, i.e. τ_{} = 0, and other delays are more than one chip duration. h_{i,m,l} is the complex channel for the l^{th} path to m^{th} receiver at i^{th} time instant. This complex channel model includes amplitude variations (a_{i,m}) and phase variations (φ_{i,m}) caused by the channel, i.e.

Above equation represents the generalized channel, i.e. the further derivation is not depen-dent on the channel model.

**Receiver diversity performance**

In this section theoretical BER expression for antenna diversity in a frequency selective fading channel is evaluated. The pilot signal and g^{th} user symbol are extracted from the received signal by multiplying corresponding chaotic sequences. Received pilot symbols are used by the channel estimator to calculate the channel coefficients (aˆ_{i,m,l} e jφ^{i,m,l} ). LMS algorithm is discussed in Appendix C.1. Complex conjugates of estimated coefficients are multiplied with extracted user symbols to generate the intermediate decision variable Z_{i}^{mg}. This process is done at every branch and the final decision variable Z_{i}^{g}

**Simulation results**

First order Markov process is used to model the fading process of channel [94], which is described as where v_{i,l,m} is the complex Gaussian process for antenna m at time i. In the simulation the variance of the Gaussian process is set to 1. α_{m} is the correlation coefficient that depends on maximum Doppler frequency f_{d} and is defined as where J_{}(·) is the Bessel function of first kind and zeroth order, and T_{s} is the signaling rate.

f_{d} T_{s} is set to 0.1 in the simulation.

In LMS algorithm, the order of the filter is one and the step size µ is set to 0.9. Following Chebyshev polynomial function is used to generate the chaotic sequence [90].

where x_{k}^{j} denotes the k^{th} chip value of j^{th} user and pilot.

Two simulation results are shown to demonstrate the performance of the system with four users and pilot signal. LMS BER and true channel BER represents the bit error rate performance of the system using LMS estimator and perfect channel estimation cases. SISO and SIMO represents the one and two receiving antennas respectively. In Fig. 4.2 perfor-mance degradation of the system is shown with an increase in number of paths. We can see that the performance of the system improves with two receiving antennas. Performance of the system can be further improved by choosing the high value of spreading factor as shown in Fig. 4.3.

**Conclusion**

In this chapter the BER performance degradation of chaos-based CDMA systems with an-tenna diversity due to frequency selective channels is evaluated. Fading coefficients are complex and the effect of phase distortion by channel is studied. Analytical BER equation shows that the performance of the system depends on the performance of the channel esti-mator. We have shown analytically that the performance of the system degrades an increase in multipath components. Performance of LMS estimator in the proposed system is com-pared with the perfect channel estimator. Performance of the system is greatly reduced due to intersymbol interference created by the multipath channel. Since the chaotic sequences are uncorrelated with their shifted version, therefore the signals delayed by more than one chip can be separated, and performance of this system can be further improved by using RAKE receiver techniques as shown in the next chapter.

**Rake receiver performance**

**Introduction**

In many cases the received multipath components are far beyond the symbol period, there-fore there is an overlapping of the current received symbol with previously received sym-bols. Such channels are known as frequency selective channels, and the overlapping is called intersymbol interference [75]. Frequency selective channels such as indoor channels, wireless mobile communication channels and underwater acoustic channels are often en-countered in communication systems. Conventional systems require an equalizer to combat intersymbol interference caused by frequency selective fading [74]. Since the spreading codes in DS-CDMA systems have very low correlation value for delayed chips, therefore delayed multipath components can be seen as uncorrelated noise in CDMA systems. Hence the CDMA system can mitigate the frequency selective fading effects. Further RAKE re-ceivers are the most commonly used method to achieve path diversity gain in multipath fading environments [4, 95].

RAKE receivers utilize the multiple correlator known as fingers to accomplish diversity gain. These fingers are delayed at least by one chip duration. Each finger extracts one of the separable multipath components and generates a decision variable. The decision variables of each finger are then combined to generate the final decision variable using maximal ratio combining technique which requires the knowledge of fading coefficients. Analytical per-formance of RAKE receiver in frequency selective channel with known multipath channel coefficients is shown in [96–98]. Performance of the adaptive RAKE receiver is analyzed in [30, 99, 100]. This performance can be further improved using a higher number of fingers than multipath components. Such RAKE receivers are called generalized RAKE receiver Generalized RAKE receivers with adaptive finger placement and weight computation is studied in [101]

Since the channels are time varying in wireless communication systems, an adaptive channel estimator is required to estimate the fading coefficients. For estimating the channel coefficients a predefined sequence, known as pilot, is transmitted along with the actual mes-sage. The channel estimation is performed at each finger to remove the fading effect from that particular finger.

PN sequences, e.g. maximum length (ML) sequences and Gold sequences, are well known periodic spreading sequences and have very good correlation properties [87, 102]. But they can be generated using linear shift registers [103], hence security is an issue with such sequences. Also, only a limited number of such sequences can be generated from these registers. On the other hand chaotic sequences are very sensitive to initial conditions and aperiodic in nature. Therefore infinite numbers of chaotic sequences can be generated from the same chaotic map with different initial conditions. Due to these properties chaotic sequences have very low probability of intercept (LPI) [104]. Hence, the chaotic sequences gain significant interest among researchers [39, 48, 49, 52, 92, 105].

The majority of work in chaos based CDMA fields is investigated for flat fading channel with real and positive coefficients in [10, 49–51]. Theoretical analysis for complex flat fading channel coefficients is done in [52]. In this analysis perfect channel estimation is assumed. Further the performance improvement in chaos based CDMA with a multiple input multiple output system using Alamouti space time code scheme [57] with perfect channel estimation is investigated in [58]. Performance of these systems can degrade if complex channel coefficients are not known at the receiver.

The objective of this chapter is to investigate the performance of the RAKE receiver in chaos based CDMA systems for slowly time varying frequency selective fading channels. BER equation in closed form is derived for BPSK system with adaptive RAKE receiver. BER expression shows that the probability of error depends on cosine of the estimated phase error. Various estimation algorithms are shown in [32] but we have used LMS filter for estimating the complex fading coefficients because of its simplicity. When fading is very slow, i.e. variation of channel is almost constant, the LMS filter can estimate the channel variation within the data block. Performance degradation of the system in frequency selec-tive channels is compared with flat fading cases. Further, BER performance improvement using RAKE receiver is shown. Finally we have shown that BER equations used in [10, 49] are a special case of the BER equation derived in this chapter.

This chapter is organized as follows. Section 5.2 shows the proposed chaos based adap-tive CDMA system with RAKE receiver. Analytical performance of the RAKE receiver is shown in Section 5.3. Section 5.4 presents the simulation results. Concluding remarks are given in Section 5.5.

**System model**

Fig. 5.1 shows the baseband representation of the proposed transceiver scheme. The base-band signal s^{g}_{k} for g^{th} user at k^{th} chip instant, is given by.

where γ_{i}^{g} is the g^{th} user symbol at i^{th} time instant, x_{k}^{g} is k^{th} chip of the g^{th} user chaotic spreading sequence or chip within an information bit (i.e. k = 1, ··· , 2β ) and 2β is the spreading factor. Superscript p is used to represent the signal related to pilot instead of g. Pilot symbols are usually the series of ones, i.e. γ_{i}^{p} = 1. Here we assume that the spreading sequence of pilot and data have the same chip power P_{c}.

For L separable multipath channels, the k^{th} chip of received signal r_{k}^{g} at g^{th} user can be written.

where τ_{l} is the delay for l^{th} path with respect to first path which has the zero delay. Minimum delay between τ_{l} is assumed to be more than one chip duration. h_{i,l} is the complex channel coefficient for l^{th} path at i^{th} time instant which is defined.

where a_{i,l} and φ_{i,l} represent the amplitude and phase variations of the channel respectively.

**Table of contents**

**Table of contents **

**List of figures **

**List of tables **

**Nomenclature **

**List of publications **

**1 Introduction **

1.1 Wireless communication channel

1.2 Channel estimators

1.3 Chaos-based communication

1.4 Diversity

1.5 Problem statement

1.6 Thesis contribution

1.7 Thesis framework

**2 Imperfect channel estimation in receiver antenna diversity **

2.1 Introduction

2.2 System model

2.3 Performance analysis

2.4 Simulation results

2.5 Conclusion

**3 Adaptive chaos-based CDMA systems with diversity **

3.1 Introduction

3.2 System model

3.3 Correlation filter and decision variable

3.4 Performance analysis of MRC technique

3.5 Simulation Results

3.6 Conclusion

**4 Diversity analysis in frequency selective fading channels **

4.1 Introduction

4.2 System model

4.3 Receiver diversity performance

4.4 Simulation results

4.5 Conclusion

**5 Rake receiver performance **

5.1 Introduction

5.2 System model

5.3 Performance analysis of multiuser RAKE receiver

5.4 Simulation results

5.5 Conclusion

**6 Anti-jamming performance under pulse noise jammer **

6.1 Introduction

6.2 System model

6.3 Receiver performance

6.4 Simulation results

6.5 Conclusion

**7 Anti-jamming performance under pulse noise jammer with diversity **

7.1 Introduction

7.2 System model

7.3 Receiver performance

7.4 Simulation results

7.5 Conclusion

**8 Multicarrier CDMA systems **

8.1 Introduction

8.2 System model

8.3 Performance analysis of MC-CDMA system

8.4 Simulation results

8.5 Conclusion

**9 Conclusions and future works **

9.1 Conclusions

9.2 Future works

References

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Performance analysis of adaptive chaos-based DS-CDMA system with imperfect channel estimation