With infinite-precision arithmetic, the flow graphs shown in Figure P6.45 have the same...

With infinite-precision arithmetic, the flow graphs shown in Figure P6.45 have the same system function, but with quantized fixed-point arithmetic they behave differently. Assume that a and b are real numbers and 0 < a < 1.

(a) Determine x_max , the maximum amplitude of the input samples so that the maximum value of the output y[n] of either of the two systems is guaranteed to be less than one.

(b) Assume that the above systems are implemented with two’s-complement fixed-point arithmetic, and that in both cases all products are immediately rounded to B + 1 bits (before any additions are done). Insert round-off noise sources at appropriate locations in the above diagrams to model the rounding error. Assume that each of the noise sources inserted has average power equal to

(c) If the products are rounded as described in part (b), 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 f₁[n] and f₂[n] are the outputs due to the noise sources. Determine the power density spectra Φ_f1f1(e^jω) and Φ_f2f2(e^jω) of the output noise for both systems.

(d) Determine the total noise powers σ²_f1and σ²_f2 at the output for both systems.