BER or SER for BPSK and QAM in Rayleigh fading channel

The probability of error can be calculated easily for a channel that has slow, flat fading with respect to the symbol period. Essentially, the signal-to-noise ratio (SNR) g  can be taken as fixed over the duration of the decision interval of one symbol. The average error probability can be computed by integrating over the fading distribution. So the Symbol-Error-Rate (SER) can be achieved by averaging the conditional error probability with respect to the random variable g as follows

                    .

where P(e |g) is the probability of error under AWGN, fg (g) is probability density of SNR g . For the special case of Rayleigh fading, SNR ³ has the exponential distribution

with local-mean SNR g_lm =E_s/N₀, which can be explained as the expected energy per symbol-to-noise power density ratio.

Bit-Error-Rate (BER) for BPSK and Symbol-Error-Rate for M-QAM can be calculated as follows.

BER for BPSK in Rayleigh fading

For BPSK in AWGN channel, the BER is

        .

For a specific channel SNR ³, the BER is

        .

Then the BER for BPSK in Rayleigh fading channel can be obtained by averaging,

        .

See also BER for Rice fading channel.

SER for M-QAM in Rayleigh fading

For M-QAM in AWGN channel, if log₂M is even integer, then the SER P=1-(1-p)² with

        .

If odd, the SER is

        .

Here we just calculate SER for M-QAM in Rayleigh fading channel, in the case that log₂M is even integer.

For a specific channel SNR ³ of M-QAM with log₂M an even integer, the probability of an error symbol is found as the probability that an error is caused by either one of the two directions, i.e.,

     P(e |g) = 1 - (1-p)² = 2p - p².

with p the probability of error from one direction,

     .

The SER of M-QAM in Rayleigh fading channel is

    P_s = 2E(p) - E(p²).

where the expectation E(p) and E(p²) can be calculated respectively, where

   .

and

    .

where we introduced t and for ease of notation,

     ,

     .

and assuming that g_lm is moderate, i.e., 0< g_lm < 2(M-1)/3. For large g_lm (i.e., g_lm > 2(M-1)/3), we are not aware of analytic solutions.

After executing the integration, the SER for M-QAM in Rayleigh fading channel can be obtained as

     .

See the Spreadsheet
Here you can see the figure for comparison of SER in AWGN and Rayleigh fading channel and Matlab code.

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