This problem explores the effect of interchanging the order of two operations on a signal, namely, sampling and performing a memoryless nonlinear operation.

(a) Consider the two signal-processing systems in Figure, where the C/D and D/C converters are ideal. The mapping g[x] = x² represents a memoryless nonlinear device.?

For the two systems in the figure, sketch the signal spectra at points 1, 2, and 3 when the sampling rate is selected to be 1/T = 2 f_m Hz and x_c(t) has the Fourier transform shown in Figure. Is y₁(t) = y₂(t)? If not, why not? Is y₁(t) = x²(t)? Explain your answer.

(b) Consider system 1, and let x(t) = A cos (30?t). Let the sampling rate be 1/T = 40 Hz. Is y₁(t) = x²_c(t)? Explain why or why not.?

(c) Consider the signal-processing system shown in Figure, where g[x] = x³ and g^?1[v] is the (unique) inverse, i.e., g^?1[g(x)] = x. Let x(t) = A cos (30?t) and 1/T = 40 Hz. Express v[n] in terms of x[n]. Is there spectral aliasing? Express y[n] in terms of x[n]. What conclusion can you reach from this example? You may find the following identity helpful:

(d) One practical problem is that of digitizing a signal having a large dynamic range. Suppose we compress the dynamic range by passing the signal through a memoryless nonlinear device prior to A/D conversion and then expand it back after A/D converter in our choice of the sampling rate?

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Part A System 1: C/D D/C XA1) System 2: w[n] (1)' C/D g|x] = x? D/C *(1) V(1) 2 X(j) -2nf.. 0) 2nf. Part C cos' 2o! cos 2ot + cos 32ot. g lv] g[x] = x' C/D v[n] y[r] x(t) velt)