The diagram in Fig. 1(a) depicts a cascade connection of two LTI systems; i.e., the output...

The diagram in Fig. 1(a) depicts a cascade connection of two LTI systems; i.e., the output of the first system is the input to the second system and the overall output is the output of the second system.

Figure 1

(a) Use z-transforms to show that the system function for the overall system (from x[n] to y[n]) is H(z) = H2(z)H1(z), where Y(z) = H(z)X(z).

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(b) Use the result of (a) to show that the order of the systems is not important; i.e., show that for the same input x[n] into the systems of Figs. 1(a) and 1(b), the overall outputs are the same (w[n] = y[n]).

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(c) Suppose that System 1 is a 3-point averager described by the difference equation and System 2 is described by the system function . Determine the system function of the overall cascade system.

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(d) Obtain a single difference equation that relates y[n] to x[n] in Fig. 1(a). Is the cascade of two 3-point averagers the same as a 6-point averager? Why would a better term be “weighted averager”?

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(e) Plot the poles and zeros of H(z) in the complex z-plane.

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(f) From H(z) obtain an expression for the frequency response  and sketch the magnitude of the frequency response of the overall cascade system as a function of frequency for .