Let x (t) = 10 cos (4?t).
(a) Is x (t) bandlimited? Explain your answer. If it is bandlimited what is its Nyquist rate?
(b) If we multiply x (t) by rect (t) to form y (t) is y (t) bandlimited? Explain your answer. If it is bandlimited what is its Nyquist rate?
(c) If we multiply x (t) by sinc (t) to form y (t) is y (t) bandlimited? Explain your answer. If it is bandlimited what is its Nyquist rate?
(d) If we multiply x (t) by
to form y (t) is y (t) bandlimited? Explain your answer. If it is bandlimited what is its Nyquist rate?
(a) Is x (t) bandlimited? Explain your answer. If it is bandlimited what is its Nyquist rate?
The Fourier transform has impulses at ±2 in f and is zero everywhere else. The Nyquist rate is 4 Hz.
(b) If we multiply x (t) by rect (t) to form y (t) is y (t) bandlimited? Explain your answer. If it is bandlimited what is its Nyquist rate?
Multiplying it by a rectangle makes it time limited and it cannot therefore be bandlimited. It is not bandlimited.
(c) If we multiply x (t) by sinc (t) to form y (t) is y (t) bandlimited? Explain your answer. If it is bandlimited what is its Nyquist rate?
Fourier transform is two impulses convolved with a rectangle. Above a finite frequency magnitude the transform magnitude is zero.
(d) If we multiply x (t) by
to form y (t) is y (t) bandlimited? Explain your answer. If it is bandlimited what is its Nyquist rate?
The Fourier transform is two impulses convolved with a Gaussian of the form 
which approaches, but never reaches, zero at a finite frequency.
It is not bandlimited.
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