HW7.3 (Frequency Mismatch in Amplitude Modulation) A bandlimited continuous- time signal x has the triangular spectrum shown below: X(w) -A +A 3 The following diagram shows an amplitude modulation-demodulation scheme to communicate the signal x to a receiver. In this problem, you'll explore the effects r(t) r(t) x(t) ✗ y(t) qR(t) = cos(wit) qr(t) = cos(wot) of frequency mismatch between the transmitter and receiver carriers qr and QR, respectively. In particular, assume that 0<< A < 0 = W₁ + € (a) Determine reasonably simple expressions for the signals r and y. You may find the following trigonometric identity useful: cos a cos = [cos(a + B) + cos(a - ẞ)] (b) Provide well-labeled plots of R(w) and Y(w), the spectra of the signals r and y, respectively. (c) Explain why the information-bearing signal x is irrecoverable, even if we send the signal y through an ideal low-pass filter.

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HW7.3 (Frequency Mismatch in Amplitude Modulation) A bandlimited continuous-
time signal x has the triangular spectrum shown below:
-A
↑x X (w)
+A
ω
The following diagram shows an amplitude modulation-demodulation scheme to
communicate the signal x to a receiver. In this problem, you'll explore the effects
r(t)
Y
r(t)
x(t)
y(t)
IR(t) = cos(wit)
qr(t) = cos(wot)
of frequency mismatch between the transmitter and receiver carriers qT and QR,
respectively. In particular, assume that
0<< A < W₁ = W₁ + €
(a) Determine reasonably simple expressions for the signals r and Y. You may
find the following trigonometric identity useful:
1
cos a cosẞ
=
[cos(a + B) + cos(a – ẞ)]
-
2
(b) Provide well-labeled plots of R(w) and Y(w), the spectra of the signals r and
y, respectively.
(c) Explain why the information-bearing signal x is irrecoverable, even if we
send the signal y through an ideal low-pass filter.
Transcribed Image Text:HW7.3 (Frequency Mismatch in Amplitude Modulation) A bandlimited continuous- time signal x has the triangular spectrum shown below: -A ↑x X (w) +A ω The following diagram shows an amplitude modulation-demodulation scheme to communicate the signal x to a receiver. In this problem, you'll explore the effects r(t) Y r(t) x(t) y(t) IR(t) = cos(wit) qr(t) = cos(wot) of frequency mismatch between the transmitter and receiver carriers qT and QR, respectively. In particular, assume that 0<< A < W₁ = W₁ + € (a) Determine reasonably simple expressions for the signals r and Y. You may find the following trigonometric identity useful: 1 cos a cosẞ = [cos(a + B) + cos(a – ẞ)] - 2 (b) Provide well-labeled plots of R(w) and Y(w), the spectra of the signals r and y, respectively. (c) Explain why the information-bearing signal x is irrecoverable, even if we send the signal y through an ideal low-pass filter.
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