This problem explores the effect of interchanging the order of two operations on a signa...

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 P4.55-1, 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 = 2fm H_z and x_c(t) has the Fourier transform shown in Figure P4.55-2. 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 (30πt). 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 P4.55-3, where g[x] = x³ and g ⁻¹[v] is the (unique) inverse, i.e., g ⁻¹[g(x)] = x. Let x(t) = A cos (30πt) 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 conversion. What is the impact of the nonlinear operation prior to the A/D converter in our choice of the sampling rate?