Question: The diagram in Fig. depicts a cascade connection of two linear time-invariant 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.
(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).
(b) Suppose that System 1 is an FIR filter described by the difference equation y1[n] = x[n] + 5/6 x[n - 1], and System 2 is described by the system function H2(z) = 1 - 2z-1 + z -2. Determine the system function of the overall cascade system.
(c) For the systems in (b), obtain a single difference equation that relates y[n] to x[n] in Fig.
(d) For the systems in (b), plot the poles and zeros of 11(z) in the complex z-plane.
(e) Derive a condition on H(z) that guarantees that the output signal will always be equal to the input signal.
(f) If System I is the difference equation y1[n] x[n] + 5/6 x[n - 1]. Find a system function H2[z] so that output of the cascaded system will always be equal to its input. In other words, find H2(z), which will undo the filtering action of H1(z). This is called decowokiron and H2(z) is the inverse of H1(z).
(g) Suppose that H1(z) represents a general FIR filter. What conditions on H1(z) must hold if H2(z) is to be a stable and causal inverse filter for H1(z)?
