Consider the DSBAM demodulation system in Fig. P-12.8 whereFigure p.1We have shown that if...

Consider the DSBAM demodulation system in Fig. P-12.8 where

Figure p.1

We have shown that if x(t) has a bandlimited Fourier transform such that X(jω) = 0 for |ω| ≥ ωb and ωc > ωb and ϕ = 0 and ωb<ωco (2ωc – ωb), then the DSBAM signal y(t)= x(t)cos(ωct) can be demodulated by the system in Fig. P-1. That is, for perfect adjustment of the demodulator frequency and phase, υ(t)= x(t). In the following parts, assume that the input signal x(t) has a bandlimited Fourier transform represented by the following plot:

(a) Now suppose that ϕ ≠ 0. Use Euler’s formula for the cosine to show that

(b) From this eqoation obtain an equation for W(jω)in terms of X(jω)and use this equation to make a plot of W(jω)for the given X(jω).

(c) From this plot , assuming ωco = ωb, determine a plot of V(jω).

(d) From the plot of V(jω), obtain an equation for υ(t)in terms of x(t)and ϕ.