Inverse DFT

Fourier Analysis · Axiom Academy

LESSON Inverse Discrete Fourier Transform Reconstructing time-domain signals from frequency components The Inverse Discrete Fourier Transform reconstructs each time-domain sample x[n] by summing all frequency components with appropriate phase shifts: x[n] is the reconstructed time-domain sample at index n X[k] are the complex frequency coefficients from the DFT N is the total number of samples e^(i2πkn/N) rotates each frequency component to the correct phase 2. The Symmetry Between DFT and IDFT Compare the DFT and IDFT formulas side by side: Notice the remarkable symmetry: the formulas are nearly identical except for: The sign of the exponent : DFT uses -i, IDFT uses +i The 1/N normalization factor in the IDFT This symmetry means both transforms use the same mathematical machinery—just running in opposite directions! 3. Why the 1/N Normalization Factor? The 1/N factor ensures that applying DFT then IDFT returns the original signal. Without it, the reconstruction would be scaled by a factor of N. Some conventions split the normalization differently: 1/N in IDFT only (most common, shown here) 1/√N in both DFT and IDFT (symmetric normalization) All conventions are mathematically valid—just be consistent! 4. Verifying the Inverse Property The defining property of the IDFT is that it perfectly inverts the DFT: Let's verify this with a concrete example. Starting with signal x = [1, 2, 3, 4]:

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