Field Guide · algorithm

Also known as: Fourier transform, FT

The Fourier transform decomposes a signal into the frequencies that compose it, converting a description in terms of time into an equivalent description in terms of frequency.1 It is the mathematical foundation of spectrum analysis — the reason a waveform that looks like noise in the time domain can reveal sharp carriers, tones, and sidebands once viewed as a spectrum. It is named for Joseph Fourier, who showed that arbitrary functions can be written as sums of sinusoids.

time domain FT frequency domain
The Fourier transform expresses a signal as a sum of frequencies — converting a time waveform into a spectrum.

How it works

The transform works by correlating the signal against sinusoids of every candidate frequency. Where the signal contains energy at a given frequency, its correlation with a sinusoid at that frequency is large; where it does not, the correlation averages to zero. The result is a spectrum: a function that reports, for each frequency, both the amplitude and the phase of the sinusoid present there. Because it captures phase as well as magnitude, the transform is invertible — the inverse Fourier transform reassembles the original waveform exactly, so no information is lost in the round trip.

For a real-valued signal the spectrum is symmetric, but SDR works with complex IQ data, whose spectrum is one-sided and can distinguish frequencies above and below the tuned centre. This is why quadrature sampling matters: it lets the transform place a signal on the correct side of the local oscillator.

Variants

The Fourier family is really several related transforms that differ in whether time and frequency are continuous or discrete:

  • Continuous Fourier transform — the pure mathematical form, defined by an integral over all time, mapping a continuous signal to a continuous spectrum.
  • Fourier series — for periodic signals, giving a spectrum of discrete harmonics.
  • Discrete-time / discrete Fourier transform — the discrete Fourier transform (DFT) operates on a finite block of samples and produces a finite set of frequency bins. This is the only form a computer can evaluate directly, and it is what every SDR spectrum display actually computes.
  • Fast Fourier transform — the FFT is not a different transform but an efficient algorithm for the DFT, cutting the cost from O(N²) to O(N log N) and making real-time waterfalls possible.

A closely related operation is the Hilbert transform, which shifts every frequency component by 90° to build the analytic signal used in single-sideband and envelope detection.

Relevance to SDR

Turning IQ samples into a spectrum — and thus a waterfall — is a Fourier transform, the reason you can see signals on an SDR. GopherTrunk uses FFTs to survey a band, to locate a steady control channel, and inside multi-channel channelizers that split a wide capture into per-channel streams. Beyond scanning, the transform underpins OFDM demodulation, matched filtering, and fast convolution throughout modern radio, so it is fair to call it the single most-used idea in digital signal processing.

Sources

  1. Fourier transform — Wikipedia, for the mathematical definition and time/frequency-domain duality. See also Frequency domain for the spectrum concept. ↩

See also

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