Assume that a in the flow graphs shown in Figure P6.47 is a real number and 0 < a &lt...

Assume that a in the flow graphs shown in Figure P6.47 is a real number and 0 < a < 1. Note that under infinite-precision arithmetic, the two systems are equivalent.

(a) Assume that the two systems are implemented with two’s-complement fixed-point arithmetic, and that in both cases all products are immediately rounded (before any additions are done). Insert round-off noise sources at appropriate locations in both flow graphs to model the rounding error (multiplications by unity do not introduce noise). Assume that each of the noise sources inserted has average power equal to = 2^−2B/12.

(b) If the products are rounded as described in part (a), the outputs of the two systems will differ; i.e., the output of the first system will be y₁[n] = y[n] + f₁[n] and the output of the second system will be y₂[n] = y[n] + f₂[n], where y[n] is the output owing to x[n] acting alone, and f₁[n] and f₂[n] are the outputs owing to the noise sources. Determine the power density spectrum of the output noise Φf₁f₁ (e^jω). Also determine the total noise power of the output of flow graph #1; i.e., determine .

(c) Without actually computing the output noise power for flow graph #2, you should be able to determine which system has the largest total noise power at the output. Give a brief explanation of your answer.