(b) Use the result in Equation 1 and the modulation property of the Fourier Transform to obtain W(jω) in terms of X(jω). Then plot W(jω).
(c) Assuming that ωco = ωb + Δ, find V(jω) in terms of X(jω). Relate your result to the plot of W(jω).
(d) From the equation of V(jω) and modulation property of the Fourier transform, obtain an equation for v(t) in terms of x(t) and Δ.
Problem 6.2 [30 Points] (Ch 12: Sinewave Amplitude Modulation)
Consider the double sidebanded AM demodulation system in Figure 2 below:
wt XX
LTI System
v(t)
>vt
ht, H(jω)
cos((ωt)
Figure 2: A DSBAM Demodulation System
where the filter's frequency response is given as:
2ω0H(jω) = 0 for |ω| > ω0
and the input signal is equal to:
yt = x(t)cos(ωct)
where t has a Fourier transform shown in Figure 3.
We have shown that if t has a bandlimited Fourier transform such that X(jω) = 0 for |ω| > ωb and if ωb = 0, and ω < ωco < 2ωc, then the DSB AM signal y(t) = x(t)cos(ωct) can be demodulated by the system in Figure 2. That is, for the perfect adjustment of the demodulator frequency and phase, v(t) = x(t).
X(jω)
-ωb
Figure 3: Fourier transform of x(t)
In the following parts, assume that ωe and the input signal x(t) has a bandlimited Fourier transform represented by the following plot:
(a) Suppose that ω = 0. Use Euler's formula for the cosine function to show that:
wt = ycos(ωc + Δt) + x(t)cos(ωc - Δt)
(1)
