This problem involves the use of crosscorrelation to detect a signal in noise and estima...

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,

where w(n) is the noise with variance and the signal is

The frequency is known but the delay n_o, which is a positive integer, is unknown, and is to be determined by crosscorrelating x (n) with y (n) . Assume that N > M +n_o.

Let

denote the crosscorrelation sequence between x (n) and y(n). In the absence of noise this function exhibits a peak at delay m = n_o. Thus n_o is determined with no error. The presence of noise can lead to errors in determining the unknown delay.

(a) For m = n_o, determine E[r_xy(n_o)]. Also, determine the variance, var[r_xy (n_o)] due to the presence of the noise. In both calculations, assume that the double frequency term averages to zero. That is, M >>

(b) Determine the signal-to-noise ratio, defined as

(c) What is the effect of the pulse duration M on the SNR?