The system in Fig. P-12.7(a) is called a quadrature modulation system. It is a method of s...

The system in Fig. P-12.7(a) is called a quadrature modulation system. It is a method of sending two bandlimited signals over the same channel. The demodulator in Fig. P-12.7(b) will recover one of the two input signals. Assume that both input signals are bandlimited such that the maximum frequency is ωm; i.e., X1(j ω) = 0 for |ω| ≥ ωm and X2(j ω) = 0 for |ω| ≥ ωm, where ωm ≪ ωc.

(a) The output of the quadrature modulator in Fig. P-12.7(a) is y(t) = x1(t) cos(ωct) + x2(t) sin(ωct). Determine an expression for the Fourier transform Y (j ω) in terms of X1(j ω) and X2(j ω). Make a sketch of Y(j ω). Assume simple (but different) shapes for the bandlimited Fourier transforms X1 (j ω) and X2 (j ω), and use them in making your sketch of Y(j ω).

(b) From the expression found in part (a) and the sketch that you drew, you should see that Y(jω) = 0 for |ω| ≤ ωa and for |ω| ≥ ωb. Determine ωa and ωb.

(c) Given the trigonometric identities 2 sin θ cos θ = sin 2θ, 2 sin2θ = (1 – cos2θ), and 2 cos2θ = (1 + cos 2θ), show that in the demodulator in Fig. P-12.7(b), the output of the mixer is

(d) The signal w(t) as determined in part (c) is the input to an LTI system. Determine the frequency response of that system so that its output is v (t) = x1(t). Give your answer as a carefully labeled plot of H(j ω).

(e) Draw a block diagram of a demodulator system whose output will be x2(t) when its input is y(t). This requires that you change the mixer.