Questions and Answers
What is the DFT of the sequence x(n)={1,2,3,4} with no aliasing? What are the magnitude and phase spectra? How can you verify it?
The DFT of the sequence x(n)={1,2,3,4} with no aliasing is X(k)={10,-2+2j,-2,-2-2j}. The magnitude spectrum is |X(k)|={10,2√2,2,2√2} and the phase spectrum is θ(k)={0,3π/4,π,-3π/4}. You can verify it by finding the IDFT.
If x(t)=4cos(100πt) is sampled at twice the Nyquist rate for one period, what is its DFT? What is the frequency resolution in terms of analog frequency? What happens if it's sampled at the Nyquist rate?
If x(t)=4cos(100πt) is sampled at twice the Nyquist rate for one period, its DFT is X(k)={0,8,0,8} and the frequency resolution in terms of analog frequency is ∆F=50Hz. If it's sampled at the Nyquist rate, the DFT is X(k)={0,8} and the frequency resolution is still ∆F=50Hz.
What is the DTFT of the sequence x(n)={0,1,2,3,4,5}? If it's sampled at four equally spaced points to obtain a 4-point DFT, what is the IDFT? What can you comment on the result?
The DTFT of the sequence x(n)={0,1,2,3,4,5} is X(e^jω)=1+e^jω+e^2jω+e^3jω+e^4jω+e^5jω. If it's sampled at four equally spaced points to obtain a 4-point DFT, the IDFT is x(n)=1.5, 0.366, -0.5, 0.366. This result might not accurately represent the original sequence due to the limited number of samples used in the DFT.
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