An allpass system is to be implemented with fixed-point arithmetic. Its system function...

An allpass system is to be implemented with fixed-point arithmetic. Its system function is

where a = re^jθ .

(a) Draw the signal flow graphs for both the direct form I and direct form II implementations of this system as a 2nd-order system using only real coefficients.

(b) Assuming that the products are each rounded before additions are performed, insert appropriate noise sources into the networks drawn in part (a), combining noise sources where possible, and indicating the power of each noise source in terms of , the power of a single rounding noise source.

(c) Circle the nodes in your network diagrams where overflow may occur.

(d) Specify whether or not the output noise power of the direct form II system is independent of r, while the output noise power for the direct form I system increases as r → 1. Give a convincing argument to support your answer. Try to answer the question without computing the output noise power of either system. Of course, such a computation would answer the question, but you should be able to see the answer without computing the noise power.

(e) Now determine the output noise power for both systems.