Discrete Fourier Transform — Study Notes

June 22, 2026 · DSP, FFT, signal processing

Summary

Reviewed the discrete Fourier transform (DFT) and how windowing reduces spectral leakage when analyzing non-periodic signals.

The DFT maps a finite time-domain sequence into frequency bins spaced by $f_s / N$ .
Multiplying by a window before the FFT trades main-lobe width for lower side lobes.
For embedded DSP, radix-2 FFT implementations favor lengths that are powers of two.

The DFT of a sequence $x[n]$ of length $N$ :

X[k] = \sum_{n=0}^{N-1} x[n]\, e^{-j 2\pi kn / N}, \quad k = 0, 1, \ldots, N-1

Parseval's relation (energy in time equals energy in frequency):

\sum_{n=0}^{N-1} |x[n]|^2 = \frac{1}{N} \sum_{k=0}^{N-1} |X[k]|^2

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Example code and experiments: GitHub — mookwonseo

Window choice (Hann, Hamming, Blackman) directly affects leakage and frequency resolution.
Next: implement a fixed-point radix-2 FFT and compare with CMSIS-DSP benchmarks.