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Digital Signal Processing Basics

The DFT of the 8-point real signal x[n] = {3, 1, 3, 1, 3, 1, 3, 1} is defined as X[k] = Σx[n]e^(−j2πkn/8). Which of the following is the value of X[4], the bin at the Nyquist frequency?

  1. A1
  2. B2
  3. C8
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Answer: C. 8

At k = N/2 = 4, e^(−jπn) = (−1)ⁿ, so X[4] = Σ(−1)ⁿx[n] = 3 − 1 + 3 − 1 + 3 − 1 + 3 − 1 = 8. The signal is a DC level of 2 plus a component (−1)ⁿ that alternates by ±1; the alternating component appears in the Nyquist bin and the DC level in X[0] = 16. For a real signal, X[0] and X[N/2] are always real.

  • A. This is the amplitude of the alternating component (X[4]/N = 8/8). With this DFT definition, the value is N times larger.
  • B. This is the mean of the signal (the DC component, X[0]/N = 16/8).
  • C. Correct. Multiplying by (−1)ⁿ and summing gives 8.
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More Digital Signal Processing Basics questions

  1. Suppose an N-point DFT takes about N² operations when computed from its definition and about N log₂N operations with the FFT (fast Fourier transform). For N = 1024, which of the following is closest to the reduction in the number of operations achieved by the FFT?
  2. A 256-point signal is followed by 768 zeros (zero padding) and a 1024-point DFT is computed. Compared with computing the DFT of the 256 points as they are, which of the following is correct?
  3. Which of the following is the main cause of spectral leakage in a DFT?
  4. Which of the following correctly compares the rectangular window (no window applied) and the Hann (Hanning) window applied before a DFT?
  5. Assume the number of points N is large enough. In a DFT with a rectangular window, a sine wave lies exactly halfway between two adjacent bins. Which of the following is closest to the drop in the amplitude read at the nearest bin, compared with a sine wave exactly at a bin center?