# PERFORMANCE ANALYSIS OF DIFFERENT M-ARY MODULATION TECHNIQUES IN FADING CHANNELS USING DIFFERENT DIVERSITY

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3 5.1 M-ary Phase Shift Keying An M-ary phase-shift-keyed (M-PSK) signal occurs when Ө(t) takes on equiprobable values, i =1, 2...M, in each symbol interval Ts. As such, Ө(t) is modeled as a random pulse stream, that is, Where Ө n is the information phase in the n th symbol interval nts t (n+1) Ts ranging over the set of M possible values β i as above and p t is again a unit amplitude rectangular pulse of duration T s seconds. The conditional probability of symbol error for coherent MPSK is given as follows [4] So the probability of symbol error for MPSK over Rician fading channels with Rician parameter K and diversity N as follows Figure 1: SEP for MPSK over Rician, Rayleigh and AWGN fading channel for different values of K and M=16; solid line for N=1, dashed line for N= M-ary Differential Phase Shift Keying MDPSK is the non-coherent version of the MPSK. It eliminates the need for a coherent reference signal at the receiver by combining two basic operations at the transmitter: (1) differential encoding of the input signal and (2) phase shift keying hence, the name, M-ary differential phaseshift keying (MDPSK). The conditional probability of symbol error is given by [4] The probability of symbol error for coherent MPSK for Rayleigh fading with diversity N is obtainable from (5) by the substitution K=0 in (5), which is given by Now using the following relation given in [3] when K approaches infinity, (5) reduces to (4). The probability of symbol error for MDPSK over Rician fading channels with Rician parameter K and diversity N is shown in (9) [4] 25

4 The error probability as a function of K, and N of the system can be calculated by averaging the conditional probability of error over the pdf of γ, i.e. Where is the mean symbol SNR. Where, PS (E /γ) is the conditional probability of symbol error. It is shown easily as K goes to zero that a substitution of K = 0 in (9) yields the probability of symbol error for MDPSK over a Rayleigh fading channel with diversity N, i.e. The probability of symbol error for QAM over a Gaussian channel is given as [1] After the substitution of (13) into (12) and putting On the other hand, it is interesting to note, when K approaches infinity, that (9) reduces to (7). This means that the system is equivalent to a non diversity reception in AWGN. When N = 1 is substituted into (9); a single link analysis is obtained. [4] we get Using the following relation given in [3] Fig 2: SEP for MDPSK over Rician, Rayleigh and AWGN fading channel for different values of K and M=16; solid line for N=1, dashed line for N= M-ary Quadrature Amplitude Modulation In M-QAM modulation scheme, the in-phase and quadrature components are both in-dependently PAM Modulated. The signal constellation for M- QAM consists of a square lattice of message points. The probability of symbol error for M-QAM over i.i.d. Rician fading channels with Rician parameter K, diversity N, and mean symbol SNR,, is given as follows 26

5 6. PERFORMANCE COMPARISON OF MDPSK, MPSK AND M-QAM On comparing the SEP expressions for MDPSK and MPSK, available in the literature [4] with our SEP expression (17), obtained using mathematical analysis for M-QAM over slow, flat, i.i.d Rician fading channels when MRC is applied at the receiver, it can be easily seen that SEP expressions for MDPSK and MPSK are in the integral form whereas our SEP expression for M-QAM is in simple closed form and contains only exponential functions. On substituting, N = 1 in equation (17), we found The Comparison curves of three modulation techniques using equation (5), (9) and (17) (MPSK, MDPSK and M-QAM) are given below: Moreover, it can also be shown that as K goes to zero that a substitution of K=0 in (17) yields the probability of symbol error for M-QAM over a Rayleigh fading channel with diversity N Fig 4: SEP for MDPSK, MPSK, M-QAM for M=8, N=1, K=0. It can be easily shown that as K goes to infinity, that a substitution of K= in (17) yields the probability of symbol error for M-QAM over a Gaussian channel as Fig 5: SEP for MDPSK, MPSK, M-QAM for M=8, N=1, K=6. Fig 3: SEP for M-QAM over Rician, Rayleigh and AWGN fading channel for different values of K and M=16; solid line for N=1, dashed line for N=4. Fig 6: SEP for MDPSK, MPSK, M-QAM for M=8, N=1, K=12. 27

### Keywords - Maximal-Ratio-Combining (MRC), M-ary Phase Shift Keying (MPSK), Symbol Error Probability (SEP), Signal-to-Noise Ratio (SNR).

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