Figure P4.64-1 shows a system in which two continuous-time signals are multiplied and a...

Figure P4.64-1 shows a system in which two continuous-time signals are multiplied and a discrete-time signal is then obtained from the product by sampling the product at the Nyquist rate; i.e., y₁[n] is samples of y_c(t) taken at the Nyquist rate. The signal x₁(t) is bandlimited to 25 kHz (X1(j ) = 0 for | | ≥ 5π × 10⁴), and x2(t) is limited to 2.5 kHz (X2(j ) = 0 for | | ≥ (π/2) × 10⁴).

In some situations (digital transmission, for example), the continuous-time signals have already been sampled at their individual Nyquist rates, and the multiplication is to be carried out in the discrete-time domain, perhaps with some additional processing before and after multiplication, as indicated in Figure P4.64-2. Each of the systems A, B, and C either is an identity or can be implemented using one or more of the modules shown in Figure P4.64-3.

For each of the three systems A, B, and C, either specify that the system is an identity system or specify an appropriate interconnection of one or more of the modules shown in Figure P4.64-3. Also, specify all relevant parameters L, M, and ωc. The systems A, B, and C should be constructed such that y₂[n] is proportional to y₁[n], i.e.,

y ₂[n] = ky₁[n] = ky_c(nT ) = kx₁(nT) × x₂(nT),

and these samples are at the Nyquist rate, i.e., y₂[n] does not represent oversampling or

undersampling of y_c(t).