In Problem 1.45, we introduced and examined some of the basic properties of relation fun...

In Problem 1.45, we introduced and examined some of the basic properties of relation functions for continuous-time signals. The discrete-time counterpart of the correlation function has essentially the same properties as those in continuous time and both are extremely important in numerous applications (as is discussed in problems 2.66 and 2.67). In this problem, we introduce the discrete-time correlation function and examine several more of its properties.

Let x[n] and y[n] be two real-valued discrete-time signals. The autocorrelation functions respectively, are defined by the expressions

and

and the cross-correlation functions are given by

and

As in continuous time, these functions possess certain symmetry properties. Specifically, .

(a), Compute the autocorrelation sequences for the signals and x₄[n] depicted in Figure P2.65.

(b) Compute the cross-correlation sequences

for x_i [n], i = 1, 2, 3, 4, as shown in Figure P2.65.

(c) Let x[n] be the input to an LTI system with unit sample response h[n], and let the corresponding output be y[n]. Find expressions for can be viewed as the output

of LTI systems with as the input. (Do this by explicitly specifying the impulse response of each of the two systems.)

(d) Let h[n) = x_t [n] in Figure P2.65, and let y[n] be the output of the LTI system with impulse response h[n] when the input x[n] also equals x₁ (n]. Calculate using the results of part (c).