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Digital Signal Processing Basics

The DFT of the 4-point signal x[n] = {1, −1, 2, 0} is defined as X[k] = Σx[n]e^(−j2πkn/4). Which of the following is Σ|X[k]|² (k = 0 to 3)?

  1. A1.5
  2. B6
  3. C24
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Answer: C. 24

By Parseval's theorem, Σ|X[k]|² = N·Σ|x[n]|² holds for a DFT with this definition. Since Σ|x[n]|² = 1 + 1 + 4 + 0 = 6, Σ|X[k]|² = 4 × 6 = 24. Computing X[k] = 2, −1 + j, 4, −1 − j directly gives |X[k]|² = 4, 2, 16, 2, which also sum to 24. The theorem states that energy is conserved (apart from the factor N) between the time side and the frequency side.

  • A. This is Σ|x[n]|² = 6 divided by N = 4 (the average power per sample).
  • B. This is Σ|x[n]|² on the time side itself. With this DFT definition, the result is N = 4 times larger.
  • C. Correct. 4 × 6 = 24, which matches the X[k] computed directly.
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