Flashcards on Spectral Analysis and Fourier Transforms
Spectral Analysis and Fourier Transforms: A Student's Guide
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Discrete Fourier Transform (DFT)
8 cards
Card 1
Question: What is the Discrete Fourier Transform (DFT) formula for X[k] given a finite-length signal x[n] of length N?
Answer: X[k] = Σ_{n=0}^{N-1} x[n] e^{-j(2π/N) k n}, for k = 0,1,…,N−1.
Card 2
Question: What is the inverse Discrete Fourier Transform (IDFT) formula for x[n] from X[k] for n in 0…N−1?
Answer: x[n] = (1/N) Σ_{k=0}^{N-1} X[k] e^{j(2π/N) k n}, for n = 0,1,…,N−1. (Note: some definitions place the 1/N in the forward transform instead.)
Card 3
Question: Why do we use the DFT instead of the continuous spectral density for a time-limited signal of length N?
Answer: Because the original spectral density is continuous, we sample it at N evenly spaced frequency points Ω_k to obtain N discrete spectral samples X[k],
Card 4
Question: What does the DFT implicitly assume about x[n] when computing the IDFT for indices outside 0…N−1?
Answer: Computing the IDFT outside 0…N−1 produces a periodic extension of x[n] with period N; the DFT therefore represents a periodic sequence.
Card 5
Question: How do DFT coefficients behave outside the base index range k = 0…N−1?
Answer: They repeat periodically: X[k + mN] = X[k] for any integer m; values outside 0…N−1 are periodic repetitions with no additional practical information.
Card 6
Question: State the symmetry property of DFT coefficients shown in the content.
Answer: X[N − k] = X*[k] (complex conjugate symmetry) when x[n] is real (implied by the provided relation X_{N−k} = X_k^*).
Card 7
Question: What is the number of DFT points and how is it denoted in the formulas?
Answer: The number of DFT points is N; it appears as the summation limit and as the normalization factor in the IDFT (1/N) when used.
Card 8
Question: How are the frequency sample points for the DFT defined in terms of Ω and k?
Answer: The DFT samples the continuous frequency at Ω_k = (2π/N) k, for k = 0,1,…,N−1.