Texts: All questions are related.. Please help.
1. Find the Fourier transform (spectrum) of the following signals:
(a) s1(t) = 1.5 cos(2000Ï€rt) cos(7000Ï€t)
(b) s2(t) = 2(sin(8000Ï€t))
(c) s3(t) = 0.5s1(t+0.7s2t)
Hints:
(i) You may want to use the identities 2 cos(α) cos(β) = cos(α + β) + cos(α - β)
(ii) Use Euler's formula: e^(jθ) = cos(θ) + j sin(θ)
2. The signal s3(t) of Problem 1(c) above is passed through an ideal bandpass filter (passband gain = 1) with lower and upper cutoff frequencies 1.9 kHz and 5.5 kHz, respectively, to obtain the output y(t). Find y(t).
Hint: First find Y(f), the Fourier transform of y(t), then take its inverse Fourier transform to get y(t).
3. A signal m(t) = s1(t) of Problem 1(a) is sampled at the rate of fs = 8000 samples/sec. The sampled signal is input to an ideal lowpass filter (passband gain = 1/8000) with a cutoff frequency of 4.0 kHz.
(a) Sketch the spectrum of the sampled signal m(t) (you may draw two different sketches).
(b) What is the output of the lowpass filter? Provide a mathematical expression in time.
4. Repeat Problem 3 when the sampling rate is 10,000 samples/sec, and the lowpass filter cutoff frequency and passband gain are 5.0 kHz and 1/10,000, respectively.
