Communication systems often require conversion from time-division multiplexing (TDM) to...

Communication systems often require conversion from time-division multiplexing (TDM) to frequency-division multiplexing (FDM). In this problem, we examine a simple example of such a system. The block diagram of the system to be studied is shown in Figure P4.61-1. The TDM input is assumed to be the sequence of interleaved samples

Assume that the sequences x₁[n] = xc₁(nT ) and x₂[n] = xc₂(nT ) have been obtained by sampling the continuous-time signals xc₁(t) and xc₂(t), respectively, without aliasing.

Assume also that these two signals have the same highest frequency, ΩN, and that the sampling period is T = π/ΩN.

(a) Draw a block diagram of a system that produces x₁[n] and x₂[n] as outputs; i.e., obtain a system for demultiplexing a TDM signal using simple operations. State whether or not your system is linear, time invariant, causal, and stable.

The k^th modulator system (k = 1 or 2) is defined by the block diagram in Figure P4.61-2. The lowpass filter Hi(e^jω), which is the same for both channels, has gain L and cutoff frequency π/L, and the highpass filters H_k(e^jω) have unity gain and cutoff frequency ωk. The modulator frequencies are such that

Find ω₁ and L so that, after ideal D/C conversion with sampling period T/L, the Fourier transform of y_c(t) is zero, except in the band of frequencies

(c) Assume that the continuous-time Fourier transforms of the two original input signals are as sketched in Figure P4.61-3. Sketch the Fourier transforms at each point in the system.

(d) Based on your solution to parts (a)–(c), discuss how the system could be generalized to handle M equal-bandwidth channels.