Lecture 09: Exact BER Expression for Rayleigh Fading Wireless Channel

Summary

Key Points

• The average BER is initially represented as an integral involving the Q function, the signal-to-noise ratio (SNR), and the Rayleigh density function of the fading channel amplitude. 0:45
• The module's primary objective is to derive and evaluate the exact bit error rate (BER) expression for a wireless communication system. 1:33
• The derivation begins by substituting SNR with 'mu' and replacing the Q function with its integral definition, resulting in a double integral. 2:57
• To simplify the calculation, the order of integration for the double integral is reversed, allowing the inner integral with respect to 'a' to be evaluated first. 7:42
• The inner integral is solved by comparing it to a known result for the variance of a Gaussian distribution, leading to a simplified expression in terms of 'mu' and 'u'. 8:47
• The outer integral, now simplified, is evaluated using a trigonometric substitution (u = square root of 2/mu tan(theta)), which transforms the integral limits and integrand. 15:13
• After simplification and integration of the trigonometric expression, the result is expressed in terms of sine of an inverse tangent function. 15:47
• A trigonometric identity is applied to simplify the sine of the inverse tangent term, leading to an expression involving the square root of mu and (2 + mu). 15:47
• Finally, substituting 'mu' back with SNR, the exact average bit error rate for BPSK transmission in a Rayleigh fading channel is derived as half times (1 minus square root of SNR divided by (2 plus SNR)). 18:40