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Digital Signal Processing Basics

A 64-point DFT of a real signal x[n] gives X[10] = 3 + 4j. Which of the following is X[54]?

  1. A−3 + 4j
  2. B3 + 4j
  3. C3 − 4j
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Answer: C. 3 − 4j

The DFT of a real signal is conjugate symmetric: X[N − k] = X[k]* (the complex conjugate of X[k]). With N = 64 and k = 10, N − k = 54, so X[54] = (3 + 4j)* = 3 − 4j. The magnitude |X[k]| = 5 is the same at k and N − k, and the phases have opposite signs. For a real signal, therefore, the bins k = 0 to N/2 contain all the information in the spectrum.

  • A. This flips the sign of the real part. The complex conjugate flips the sign of the imaginary part.
  • B. This is the same value as X[10]. The magnitude is the same, but the phase has the opposite sign.
  • C. Correct. X[N − k] is the complex conjugate of X[k].
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  1. 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?
  2. 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?
  3. 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?
  4. Which of the following is the main cause of spectral leakage in a DFT?
  5. Which of the following correctly compares the rectangular window (no window applied) and the Hann (Hanning) window applied before a DFT?