Consider an all-pass system with system function A flow graph for an implementa...

Consider an all-pass system with system function

A flow graph for an implementation of this system is shown in Figure P6.42.

(a) Determine the coefficients b, c, and d such that the flow graph in Figure P6.42 is a direct realization of H(z).

(b) In a practical implementation of the network in Figure P6.42, the coefficients b, c, and d might be quantized by rounding the exact value to the nearest tenth (e.g., 0.54 will be rounded to 0.5 and 1/0.54 = 1.8518 . . . will be rounded to 1.9).Would the resulting system still be an all-pass system?

(c) Show that the difference equation relating the input and output of the all-pass system with system function H(z) can be expressed as

y[n] = 0.54(y[n − 1] − x[n]) + x[n − 1].

Draw the flow graph of a network that implements this difference equation with two delay elements, but only one multiplication by a constant other than ±1.

(d) With quantized coefficients, would the flow graph of part (c) be an all-pass system? The primary disadvantage of the implementation in part (c) compared with the implementation in part (a) is that it requires two delay elements. However, for higher-order systems, it is necessary to implement a cascade of all-pass systems. For N all-pass sections in cascade, it is possible to use all-pass sections in the form determined in part (c) while requiring only (N + 1) delay elements. This is accomplished by sharing a delay element between sections.

(e) Consider the all-pass system with system function

Draw the flow graph of a “cascade” realization composed of two sections of the form obtained in part (c) with one delay element shared between the sections. The resulting flow graph should have only three delay elements.

( f ) With quantized coefficients a and b, would the flow graph in part (e) be an all-pass system?