Shown in Fig. P9.37 is the coupled-form implementation of a two-pole filter with poles a...

Shown in Fig. P9.37 is the coupled-form implementation of a two-pole filter with poles at There are four real multiplications per output point. Let i = 1, 2, 3, 4 represent the round-off noise in a fixed-point implementation of the filter. Assume that the noise sources are zero-mean mutually uncorrelated stationary white noise sequences. For each n the probability density function p(e) is uniform in the range

(a) Write the two coupled difference equations for y(n) and v(n), including the noise sources and the input sequence x(n).

(b) From these two difference equations, show that the filter system functions and between the input noise terms and the output y(n) are:

We know that

Determine

(c) Determine a closed-form expression for the variance of the total noise from at the output of the filter.