in Partial Fulfillment of the Requirement for the Degree of

The effect of diversity order and fading parameters on performance measures is studied by means of numerical evaluation of the obtained expressions. This effect is also attributed to the time-varying impulse response of the wireless channel.

Diversity Combining

Polarization diversity: Independent fading path can also be realized by simultaneous transmission of horizontal and vertical polarizations of signals [8,9]. Angle diversity: The angle of arrival in the case of beamforming antennas can also be used to generate independent fading channels [8,9].

Multipath Fading Models

Also, the Nakagami mdistribution is widely used to model the urban environment, and the PDF of the envelope is given by [2]. +1+2κκ)2 denotes the number of multipath clusters and Iv(·)is the modified Bessel function of the first kind and fifth order.

Literature Survey

Recently, [38] also presented closed-form expressions for the ASER of a dual-MRC receiver via correlated Hoyt fading channels for a number of coherent and incoherent modulations using a novel decorrelation transform technique. Recently, [48] also presented closed-form expressions for the MRC receiver outage probability due to η−µfading channels for a number of coherent and incoherent modulations.

Motivation

Problem Formulation

In [46], ASER expressions for QAM modulation are presented and in [47] it is given for BPSK modulation. Recently, [48] also presented closed-form expressions for the outage probability of an MRC receiver overshooting η−µ channels for a number of coherent and incoherent modulations. and receiver SC for arbitrary number of branches. a) SC receiver performance over independent and interconnected fading channels with arbitrary number of branches.

Thesis Contributions

Organization

Selection Combining

Assuming identical noise power in all the received branches, the output of the combiner can be given by With uncorrelated fading at the input receive antennas, the cumulative distribution function (CDF) of the combiner output Pγ(γsc) can be expressed as.

Maximal Ratio Combining

Equal Gain Combining

The output of the combiner is fed as input to the detector to decide which of the symbols from the transmitter's signal set has been sent. Therefore, the detection rule is the threshold value in the middle of the signal constellation for the ML detector [2].

Correlated Fading Models

For example, for a BPSK modulation scheme which has only two symbols in its signal set, the detector must decide whether a '10 or '00 was transmitted. In an exponential correlation model, the correlation coefficient between ite and jte received fading signals is defined as [1,2,51].

Analytical Methods for System Performance Evaluation

Probability Density Function Based Approach
Moment Generating Function Based Approach
Characteristic Function Based Approach
PDF of Combiner Output Signal-to-Noise Ratio
Moments of Combiner Output Signal-to-Noise Ratio
Outage Probability of Combiner
Average Bit Error Rate
Results and Discussion

Performance measurements using the MGF of the combiner output SNR can be obtained as below. Thus, an expression for the ABER for binary non-coherent modulations can be obtained as.

Table 2.1: Conditional BER of coherent and noncoherent modulation schemes with instantaneous SNR γ [3]

Maximal Ratio Combining in Independent and Identical Fading Channels

Average Output Signal-to-Noise Ratio
Outage Probability of Combiner Output Signal-to-Noise Ratio
Amount of Fading of Combiner Output Signal
Upper Bound on Truncation Error

For incoherent binary modulations, a simplified expression for the conditional BER from Table 2.1 is given in equation 2.25. Substituting pe,ncoh(ε|γ) and fγmrc(γmrc) into equation 2.17, an expression for the incoherent ABER can be given as.

Figure 3.5: Outage probability vs. ¯ γ N as a function of L and q .

Dual Diversity in Correlated Hoyt Fading channels

Maximal Ratio Combining Receiver

The ABER for a diversity receiver is defined in Section 2.3 and an expression for the ABER is given in Equation 2.17. Solving the above integral using [1, A-(6)] (reproduced in equation B.13), an expression for ABER can be obtained. A combined expression for conditional BER for incoherent DPSK and BFSK modulations obtained from Table 2.1 is given in equation 2.25.

Solving the integral in the above expression using reproduced in equation B.10), an expression for non-coherent ABER can be obtained as.

Table 3.1: Number of terms (N) required for an accuracy at 7 th place of decimal digit in the numer- numer-ical evaluation of Equation 3.50.

Maximal Ratio Combining for Unequal Fading Parameters

An expression for the failure probability of the dual MRC receiver in correlated Hoyt fading channels can be obtained by substituting fγmrc(γmrc) from equation 3.58 into equation 2.16. The failure probability in equation 3.60 is given as a function ofγth and not as a function of ¯γN. Thus, by putting pe,coh(ε|γ) from equation 2.24 and fγmrc(γmrc) from equation 3.58 into equation 2.17, an expression for binary coherent ABER can be given as.

By placing pe,ncoh(ε|γ) from equation 2.25 and fγmrc(γmrc) from equation 3.58 into equation 2.17, an expression for non-coherent ABER can be given as.

Equal Gain Combining Receiver

The ABER performance of a diversity communication system is defined in Section 2.3, and a general mathematical expression is given in Equation 2.17. To obtain an expression for the ABER of a correlated EGC receiver, the obtained expression for the PDF of the output SNR γegc in equation 3.69 and an expression for the conditional BER pe,coh(ε|γ) corresponding to the used modulation scheme are required . A simplified combined expression for both conditional BERs is obtained and is given in Equation 2.24.

Substituting pe,ncoh(ε|γ) and fγεgc(γegc) from Equations 2.25 and 3.69 into Equation 2.17, an expression for the incoherent ABER can be given as.

Selection Combining Receiver

An expression for the PDF ofα can be obtained by differentiating F(α) in equation 3.77 w.r.t. By inserting equation 3.78 into equation 3.9, an expression for the Nth moment of γsc can be given as By substituting equation 3.78 into equation 2.16, an expression for the outcome probability for correlated dual-SC combiner can be given as

Expressing the hypergeometric function in infinite series, the integrations in equation 3.82 can be solved by applying the reproduction in equation B.6).

Figure 3.22: Outage probability of correlated dual-SC receiver.

Maximal Ratio Combining with Arbitrary Order Diversity

Equal Correlation Model

For the special case ρ=0 (independent channels with fading) it can be shown that equation 3.96 reduces to ¯γmrc=L¯γ as expected. ABER is given by Equation 2.17, which requires an expression for the PDF of γmrc and the conditional bit error rate pe,coh(ε|γ). For binary coherent modulations (ie, BPSK and BFSK), the expression for the conditional BER is given in Equation 2.24.

Putting pe,coh(ε|γ) and fγmrc(γmrc) from equations 2.24 and 3.93 into equation 2.17, the ABER expression can be given as

Figure 3.26: Outage probability of the L -MRC receiver with equal correlation.

Exponential Correlation Model

By substituting fγmrc(γmrc) from Equation 3.108 into Equation 2.16, the expression for the cutoff probability can be expressed for a threshold γthas. For coherent modulations, the expression for conditional BER can be given as pe,coh(ε|γ) =aQp. Putting pe,coh(ε|γ) and fγmrc(γmrc) into equation 3.114, the ASER for coherent modulation can be expressed as.

Putting pe,ncoh(ε|γ) and fγmrc(γmrc) into equation 3.114, an expression for the ASER for non-coherent modulations can be given as.

Table 3.5: Values of a and b for some coherent and noncoherent modulations.

Performance Comparison Among the Diversity Schemes

Summary

For binary coherent modulations (BPSK and BFSK), an expression for the conditional BER is given in Equation 2.24. an expression for ABER can be written as. 4.8) The integral in Equation 4.8 cannot be solved in this form. The integral in Equation 4.9 can be solved after expressing the Q(·) function in incomplete gamma function using [1, A-(6)] (represented in Equations B.13 ) as below. By substituting pe,ncoh(ε|γ) and fγsc(γsc) from Equations 2.25 and 4.4 into Equation 2.17, an expression for the incoherent ABER can be given as.

Solving the integral in Equation 4.12 using [3, (C.1)] (given in Equation B.15) expression for the incoherent ABER can be obtained as.

Figure 4.1: ABER vs. ¯ γ for SC receiver with CPSK and CFSK modulations.

Selection Combining in Exponentially Correlated Channels

Joint PDF of Exponentially Correlated η − µ Random Variables

The XL}s obtained in equation 4.14 cannot be used in this format since it is not possible to derive Xl2+Yl2, to estimate the joint PDF of the exponentially correlated η−µ envelopes (Zls). 4.15). An expression for the joint PDF of exponentially correlated RVsYl2s can be obtained by putting equation 4.16 in the exine place. It can be given as 4.17) Since Xl2andYl2 are independent RVs, applying the convolution operation an expression for the joint PDF of exponentially correlated RVsZl2=Xl2+Yl2 can be derived as.

Outage Probability

4.17) Since Xl2andYl2 are independent RVs, applying the convolution operation an expression for the joint PDF of exponentially correlated RVsZl2=Xl2+Yl2 can be derived as. After these substitutions and some algebraic manipulations, the expression for the exponentially integrated relation of the PDF ofγls can be given as For the SC receiver, an expression for thePout(γth) can be obtained by evaluating Equation 4.21 atγi=γth,∀i.

Results and Discussion

Therefore, increasing η means a weak channel and as expected it can be observed that the outage probability is high for the highest value of η in Figure 4.3. We observed that the probability for fixed κα and µ of the break is small for L=2 and ρ=0 compared to L=3 and ρ=0.8 for lower ¯γN. Therefore, an important conclusion that can be drawn from these observations is that it is better to go for lower diversity with sufficient antenna space rather than higher order diversity in limited space and scenario. of power.

It can be seen that the numerical results are in close agreement with the simulation results.

Figure 4.3: Outage probability of SC receiver in exponentially correlated η − µ fading channels.

Maximal Ratio Combining Receiver in Equally Correlated Fading Channels

PDF of Combiner Output SNR
Moments of the Combiner Output SNR
Outage Probability

The PDF of γmrc can be obtained by scaling equation 4.27 (applying the concept of transformation of RV) by the multiplication factor (Eb/N0). The average value of the combiner output SNR can be obtained by putting N = 1 in equation 4.31, as. Using [1, A-(8a)] (shown in equation B.14), the above integration can be solved and an expression for ABER can be given as.

By introducing pe,ncoh(ε|γ) and fγmrc(γmrc) from equations 2.25 and 4.28, an expression for non-coherent ABER can be given in equation 2.17 as.

Figure 4.4: Outage probability of the L -MRC receiver with equal correlation.

Summary

Results and discussion

By expressing the Bessel function involved in equation 5.2 in infinite series using [54] (reproduced in equation B.17) we can rewrite the CDF ofγlas. A general expression to obtain the ABER is given in equation 2.17 which needs the PDF ofγsc and the conditional bit error rate pe,coh(ε|γ), for evaluation. For coherent binary BPSK and BFSK modulations, the expression for conditional BER is given in equation 2.24.

Using [1, A-(6)] (represented in equation B.13 ) we can write the Q(·) function in incomplete gamma function, and equation 5.11 can thus be rewritten as.

Summary

The performance of a variety of SC, EGC, and MRC receivers is analyzed on Hoyt, η−µ, and κ−µ fading channels. By focusing on the analytical approach, mathematical expressions for various performance measures such as ASNR, outage probability and ABER/ASER of various receivers were obtained. It is emphasized that it is necessary to analyze diversity receivers with arbitrary order of diversity with correlated fading channels, as these cases are often encountered in field deployment of diversity receivers.

Receiver systems with the specific configurations analyzed are summarized below. i) Expression of PDF of output SNR for independent fading channels:.

Characteristic Function of Sum of Hoyt Square RVs

Joint Characteristic Function of Dual Correlated Hoyt RVs

From the model given in equation A.12, α2 can be written in terms of a quadratic Gaussian distribution as Since RVXl2 and Yl2 are independent from equation A.12, the total CF of RVs α21 and α22 can be obtained by multiplying the total CF of Xl2 and Yl2as. The joint density function of the correlated Gaussian distribution X1 and X2 with variance σ2x is given in [59] as

Carrying out the RV transformation operation in Equation A.21 the joint PDF of X12 and X22 can be written as

Joint PDF of Dual correlated Hoyt RV

Characteristic Function of Hoyt RV with Unequal q

Correlation Coefficient of Hoyt RV with Unequal q

PDF of Sum of Exponentially Correlated Gamma RVs

Joint PDF of Generalized Rayleigh RV

Upper Bound on Truncation Error

Equation 3.6
Equation 3.28
Equation 3.34
Equation 3.78
Equation 3.113
Equation 3.118
Equation 4.28

Block diagram of a diversity receiver system
Block diagram showing principle of operation of the Selection Combiner
Block diagram showing principle of operation of a Maximal Ratio combiner
Outage probability of L independent SC receiver
ABER of L-independent SC receiver for CPSK and CFSK modulations
ABER of L-independent SC receiver for NCFSK and DPSK modulations
SNR gain per branch vs. L for CPSK and DPSK modulations as a function of q
Outage probability vs. ¯ γ N as a function of L and q
Amount of fading vs. L as a function of q
ABER vs. ¯ γ for CPSK ( a = 1) and BFSK ( a = 0.5) modulations as a function of L , q . 43
Diversity gain vs. L for average output SNR, CPSK and DPSK modulations
Truncation error for ABER in Equation 3.34 as a function of K
Outage probability vs. ¯ γ N for correlated dual-MRC receiver
ABER vs. ¯ γ for correlated dual-MRC receiver with coherent modulation
ABER vs. ¯ γ for correlated dual-MRC receiver with noncoherent modulation
Excess SNR vs. correlation coefficient for CPSK and DPSK modulations
Probability of outage for an ABER of 10 −3 and average fading power decay factor
ABER vs. ¯ γ for correlated dual-MRC receiver with binary coherent modulations
ABER vs. ¯ γ for correlated dual-MRC receiver with binary noncoherent modulations. 63
ABER vs. ¯ γ for EGC receiver with noncoherent modulations
Excess SNR vs. correlation coefficient for EGC receiver for CPSK and DPSK mod-
Outage probability of correlated dual-SC receiver
ABER of correlated dual-SC receiver for CPSK and CFSK modulations
ABER of correlated dual-SC receiver for NCFSK and DPSK modulations
Excess SNR required for an ABER of 10 −3 for dual-SC receiver
Outage probability of the L-MRC receiver with equal correlation
ABER vs. ¯ γ for L-MRC receiver with binary coherent modulations
ABER vs. ¯ γ for L -MRC receiver with binary noncoherent modulations
Outage probability vs. ¯ γ N as a function of L and q
ASER vs. ¯ γ for some coherent modulation scheme as a function of L ,ρ and q
ASER vs. ¯ γ for some noncoherent modulation scheme as a function of L,ρ and q
E K ASER vs. K for 8PSK modulation scheme as a function of L,ρ and ¯ γ with q = 0.5. 98
ABER vs. ¯ γ for SC receiver with DPSK and NCFSK modulations
Outage probability of SC receiver in exponentially correlated η − µ fading channels. 115

The hypergeometric function involved in equation A.54 can be shown to be monotonically decreasing over all values ofk, so the error can be bounded above as. Finally, by regularizing the series as a function of the hypergeometric function, an upper bound expression on the truncation error can be given. The hypergeometric function involved in equation A.62 can be shown to be monotonically decreasing over all positive values of (2Lτ+t).

It can be shown that the hypergeometric function involved in equation 3.118 is monotonically decreasing, and therefore the error can be upper bounded. A.67).