In this computer experiment we study the statistical characteristics of a random process X (t) defined by X (t) = A cos (2??_ct + ?) + W (t). Where the phase ? of the sinusoidal component is a uniformly distributed random variable over the interval [-?, ?], and W (t) is a white Guassian noise component of zero mean and power spectral density N₀/2. The two components of X (t) are statistically independent; hence the autocorrelation function of X (t) is RX (?) = A²/2 cos (2??_c?) + N₀/2 ? (?). This equation shows that foe | ? | > 0 the autocorrelation function R_X (?) has the same sinusoidal waveform as the signal component of X (t). The purpose of this computer experiment is to perform the computation of R_X (?) using two different methods.

(a) Ensemble averaging. Generate M = 50 randomly picked realizations of the process X (t). Hence compute the product x(t + ?) x (t) fir the M realizations of X (t), and there by compute the average of these computations over M. Repeat this sequence of computations for different values of ?.

(b) Time averaging. Compute the time-averaged autocorrelation function, where x (t) is a particular realization of X (t), and 2T is the total observation interval. For this computation, use the Fourier-transform pair, where | XT (?) | ² /2T is the period-gram of the process X (t). Specifically, compute the Fourier transform X_T (?) of the time-windowed function. Hence compute the inverse Fourier transform | X_T (?) | ² / 2T. Compare the result of your computation of R_X (?) using two approaches.

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