Fourier Properties and Transform Pairs

The one property that wears four hats

Compress a signal in time and its spectrum expands, reciprocally. That single fact explains four things that look unrelated: a short pulse must occupy wide bandwidth; a sharp filter cutoff must have a long impulse response; fine frequency resolution requires long observation; and a signal cannot be arbitrarily concentrated in both domains at once. The last is the uncertainty principle, and it is mathematically the same statement as the one in quantum mechanics.

Shifting a signal in time leaves the magnitude spectrum completely unchanged and adds a phase shift proportional to frequency. So a pure delay is a straight sloping line on a phase plot, and “linear phase” means every frequency is delayed equally — which is exactly what preserves waveform shape.

Pairs worth knowing by heart

Multiplication in time is convolution in frequency. Multiplying by a sinusoid convolves with a pair of spikes, producing shifted copies of the spectrum — which is amplitude modulation, and why frequency translation requires a nonlinear operation. It also explains windowing retroactively: applying a window multiplies in time, so it convolves your spectrum with the window’s own spectrum. Leakage is that window spectrum, smeared across every line.

Gibbs, harmonics, and chirps

Reconstruct a square wave from finitely many components and there is about 9% overshoot at each edge. Adding more terms narrows it but never shrinks it — it is sinc side lobes from truncating the spectrum, and the fix is to taper the truncation rather than extend it. The faint halos near sharp edges in an over-compressed JPEG are exactly this.

Any periodic non-sinusoidal signal has energy at integer multiples of its fundamental. A square wave has odd harmonics falling as one over the harmonic number. Since a linear system cannot create frequencies, harmonics appearing at an output are proof of nonlinearity — which is what total harmonic distortion measures.

A chirp sweeps frequency over time, giving it long duration and wide bandwidth. Duration supplies energy, bandwidth supplies timing precision, and correlating against a copy compresses it to a sharp peak. That is radar pulse compression, fully explained — the puzzle posed back in episode one.

Key points

1. The transform is linear: scaling scales the spectrum, and adding signals adds their spectra. But magnitudes do not add, because relative phase determines whether components reinforce or cancel.
2. Shifting a signal in time leaves the magnitude spectrum completely unchanged and adds a phase shift proportional to frequency. On a phase plot, a pure delay is a straight sloping line.
3. Linear phase means every frequency is delayed by the same time, so waveform shape is preserved. Nonlinear phase delays components differently and distorts the shape even when every magnitude is untouched. This is why Bessel filters preserve shape and Chebyshev filters ring.
4. A symmetric signal has a purely real transform and zero phase; an antisymmetric one is purely imaginary with ninety degree phase. Symmetric filter kernels give linear phase for exactly this reason.
5. The spectrum is periodic with period N, which is the mathematical statement of aliasing. It is also symmetric about zero, giving negative frequencies — meaningless for real signals but not optional in the accounting, and the source of persistent factor-of-two calibration confusion.
6. Compressing a signal in time expands its spectrum, and vice versa, reciprocally. A short pulse must have wide bandwidth; a sharp filter cutoff must have a long impulse response; fine frequency resolution requires long observation. These are one fact wearing four hats.
7. Multiplication in time is convolution in frequency. Multiplying by a sinusoid convolves with a pair of spikes, producing shifted copies of the spectrum — which is amplitude modulation, and why frequency translation needs a nonlinear operation.
8. Windowing is multiplication in time, so it convolves your spectrum with the window's spectrum. Leakage is the window's own spectrum smeared across every spectral line. Not windowing convolves with a sinc, so there is no neutral option.
9. Parseval's relation says total energy is the same computed in either domain. Useful as a check on scaling, and for computing the energy within a frequency band.
10. A delta function transforms to a flat spectrum, and a constant transforms to a delta at zero frequency. White noise being spectrally flat and temporally uncorrelated are the same statement.
11. A rectangular pulse transforms to a sinc, and by duality a sinc transforms to a rectangle. Since an ideal low pass filter is a rectangle in frequency, its impulse response must be a sinc — which extends infinitely and therefore cannot be built. All filter design is about how you truncate it.
12. The Gaussian is its own transform, and it is the only shape that is. It has no ripples in either domain, so Gaussian filters produce no ringing at all, which is why they dominate image processing.
13. Exponential decay transforms to a Lorentzian, which is the line shape of every resonance. Faster decay gives a broader line — reciprocity again.
14. The Gibbs effect gives about nine percent overshoot at a reconstructed discontinuity, and adding more terms narrows it without reducing its height. It is sinc side lobes from truncating the spectrum, and the fix is to taper the truncation rather than extend it. J P E G ringing is exactly this.
15. A periodic non-sinusoidal signal has energy at integer multiples of its fundamental. A square wave has odd harmonics with amplitudes falling as one over the harmonic number. Harmonic patterns are what make instruments distinguishable, and harmonics appearing at a system's output are proof of nonlinearity.
16. A chirp sweeps frequency over time, giving it both long duration and wide bandwidth. Duration supplies energy, bandwidth supplies timing precision, and correlating against a copy compresses it to a sharp peak. That is radar pulse compression, fully explained.