and Interpretation
The DFT formula, frequency bins, the DFT as a matrix operation, and the circular nature that drives zero-padding and overlap-add filtering.
N time samples go in. N complex frequency values X[k] come out. Each bin encodes amplitude and phase at a discrete frequency.
Reconstruction
The IDFT recovers x[n] exactly. The only change: the exponent sign flips to +j and a 1/N normalization factor appears.
Bin k → frequency f = k · fs/N Hz. Frequency resolution Δf = fs/N.
X = WN x — the DFT is a change of basis. Each row of WN is a complex sinusoid. Direct computation: O(N²). The FFT exploits symmetry to reach O(N log N).
The DFT treats x[n] as periodic with period N. Index arithmetic is mod N. Consequence: multiplying spectra gives circular convolution, not linear convolution.
Not Sharper
Zero-padding interpolates the spectrum — the plot looks smoother. But true resolution requires more data, not more zeros. To resolve Δf Hz you need at least 1/Δf seconds of signal.
- More zeros → finer sampling of DTFT → smoother curve
- More samples → genuinely more frequency information
- Two tones at Δf Hz apart need T ≥ 1/Δf seconds of data
- DFT: X[k] = Σ x[n] e−j2πkn/N; IDFT has +j and 1/N factor
- Bin k ↔ frequency k·f_s/N Hz; resolution Δf = f_s/N
- |X[k]| = amplitude, ∠X[k] = phase; real signals have conjugate symmetry
- DFT = matrix multiplication X = WNx; O(N²) direct, O(N log N) via FFT
- DFT is periodic (circular) — index arithmetic mod N
- Spectral multiplication → circular convolution; zero-pad for linear convolution
- Zero-padding smooths the spectrum plot but doesn't improve true resolution