In this problem, we consider the implementation of a causal filter with system function

This system is to be implemented with (B + 1)-bit two's-complement rounding arithmetic with products rounded before additions are performed. The input to the system is a zeromean, white, wide-sense stationary random process, with values uniformly distributed between -xmax and +xmax.

(a) Draw the direct form flow graph implementation for the filter, with all coefficient multipliers rounded to the nearest tenth.

(b) Draw a flow graph implementation of this system as a cascade of two 1st-order systems, with all coefficient multipliers rounded to the nearest tenth.

(c) Only one of the implementations from parts (a) and (b) above is usable. Which one? Explain.

(d) To prevent overflow at the output node, we must carefully choose the parameter xmax. For the implementation selected in part (c), determine a value for xmax that guarantees the output will stay between -1 and 1. (Ignore any potential overflow at nodes other than the output.)

(e) Redraw the flow graph selected in part (c), this time including linearized noise models representing quantization round-off error.

(f) Whether you chose the direct form or cascade implementation for part (c), there is still at least one more design alternative:

(i) If you chose the direct form, you could also use a transposed direct form.

(ii) If you chose the cascade form, you could implement the smaller pole first or the larger pole first. For the system chosen in part (c), which alternative (if any) has lower output quantization noise power? Note you do not need to explicitly calculate the total output quantization noise power, but you must justify your answer with some analysis.