x[n] and y[n] are two real-valued, positive, finite-length sequences of length 256;...

x[n] and y[n] are two real-valued, positive, finite-length sequences of length 256; i.e.,

x[n] > 0, 0 ≤ n ≤ 255,

y[n] > 0, 0 ≤ n ≤ 255,

x[n] = y[n] = 0, otherwise.

r[n] denotes the linear convolution of x[n] and y[n]. R(e^jω) denotes the Fourier transform of r[n]. R_s[k] denotes 128 equally spaced samples of R(e^jω); i.e.,

Given x[n] and y[n], we want to obtain Rs [k] as efficiently as possible. The only modules available are those shown in Figure P8.62. The costs associated with each module are as follows:

Modules I and II are free.

Module III costs 10 units.

Module IV costs 50 units.

Module V costs 100 units.

By appropriately connecting one or several of each module, construct a system for which the inputs are x[n] and y[n] and the output is R_s [k]. The important considerations are (a) whether the system works and (b) how efficient it is. The lower the total cost, the more efficient the system is.