(Solution) - This problem involves the use of crosscorrelation to detect a -(2025 Original AI-Free Solution)

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This problem involves the use of crosscorrelation to detect a signal in noise and estimate the time delay in the signal. A signal x(n) consists of a pulsed sinusoid corrupted by a stationary zero-mean white noise sequence. That is,
x(n) = y(n ? n0) + ?(n) 0 ? n ? N ? 1
where ?(n) is the noise with variance ?2? and the signal is
y(n) = A cos?0n, 0 ? n ? N ? 1
= 0, otherwise
The frequency ?0 is known but the delay n0, which is a positive integer, is unknown, and is to be determine by crosscorrelating x(n) with y(n). Assume that N > M + n0. Let
Denote the cross correlation sequence between x(n) and y(n). in the absence of noise this function exhibits a peak at delay m = n0. Thus n0 is determined with no error. The presence of noise can lead to errors in determining the unknown delay.
(a) For m = n0, determine E[rxy(n0)]. Also, determine the variance, var[rxy(n0)], due to the presence of the noise. In both calculations, assume that the double frequency term averages to zero. That is, M » 2?/?0.
(b) Determine the signal-to-noise ratio, defined as
SNR = {E[rxy(n0)]}2 / var[rxy(n0)]
(c) What is the effect of the pulse duration M on the SNR?