Question 1

What does the frequency spectrum of a signal reveal?

Accepted Answer

The frequency spectrum shows how much energy is present at each frequency in a signal. A pure sine wave at 440 Hz produces a single spike at 440 Hz in the spectrum. A complex musical chord shows spikes at multiple harmonically related frequencies. A square wave shows energy at the fundamental and all odd harmonics with amplitude falling as 1/n. Noise has energy spread across all frequencies (white noise) or with a specific spectral shape. The spectrum reveals periodicities, harmonic relationships, and frequency components that are not obvious in the raw time-domain signal.

Question 2

What is spectral leakage and how do window functions reduce it?

Accepted Answer

Spectral leakage occurs when the DFT implicitly assumes the signal repeats periodically, but the signal within the analysis window does not begin and end at the same value. The discontinuity at the window boundary spreads energy from a pure tone across neighboring frequency bins, obscuring nearby spectral features. Window functions (Hann, Hamming, Blackman) taper the signal smoothly to zero at both ends, eliminating the discontinuity. This trades frequency resolution (slightly broadened peaks) for reduced leakage (suppressed sidelobes), with different windows optimizing different trade-offs.

Question 3

Why is the FFT so much faster than the DFT?

Accepted Answer

The DFT computes each of N frequency components as a sum of N time-domain samples, requiring N squared multiplications total. The FFT exploits the DFT's symmetry and periodicity properties to factor the computation recursively: an N-point DFT is broken into two N/2-point DFTs, each broken into N/4-point DFTs, and so on for log2(N) levels. Total operations become N*log2(N)—a 50,000-fold speedup for N=1,000,000 that made real-time signal processing feasible.

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