time-frequency representation

A time-frequency representation plots a signal as it unfolds across both axes simultaneously: horizontal for time, vertical for frequency, with color or brightness showing amplitude at each point. Unlike a simple oscilloscope trace that shows only amplitude against time, or a spectrum analyzer that collapses everything into frequency bins, a time-frequency plot reveals how the frequency content of a signal changes moment by moment. This matters because most real signals, from speech to radar returns to vibration data, are not stationary. They contain tones that appear and disappear, frequencies that sweep up or down, and transients that last only milliseconds.

The most common time-frequency representation in practice is the spectrogram, built from the short-time Fourier transform (STFT). The STFT divides the signal into overlapping windows of a few milliseconds each, applies a window function such as Hamming or Hann to reduce edge artifacts, and computes the discrete Fourier transform on each window. The magnitude of each frequency bin becomes a pixel in a 2D grid. The choice of window length is a tradeoff: shorter windows give good time resolution but poor frequency resolution; longer windows do the opposite. A 10 ms window on audio sampled at 44.1 kHz resolves frequencies to roughly 100 Hz but freezes any tone changes faster than 100 milliseconds.

Variants and alternative approaches

Wavelet transforms offer a different path. Unlike STFT, which uses the same window width across all frequencies, a continuous wavelet transform uses narrow windows at high frequencies and wide windows at low frequencies. This matches how the ear hears: we distinguish pitch changes in the bass over longer timescales but notice tiny variations in treble. Wavelets scale naturally and require no arbitrary choice of window width. They produce coefficients rather than a simple magnitude grid, and the Morlet wavelet is standard in audio and vibration analysis. Reassignment and synchrosqueezing are post-processing tricks that sharpen spectrograms by moving energy to its true instantaneous frequency rather than a bin center.

Time-frequency representations are essential wherever a signal's frequency content shifts during measurement. In condition monitoring, a bearing defect produces impulses at a specific frequency; watching these grow over days or weeks requires the ability to see that frequency isolated from the noise floor as it develops. In radio and radar, a chirp signal sweeps through frequencies; only a time-frequency plot shows the slope and duration clearly. In music and speech processing, formants drift continuously; a static spectrum at one moment tells you nothing about the next. Power quality engineers use spectrograms to spot interharmonics that appear only when a particular load switches on.

The term reflects the signal processing world's older division of labor: you either worked in the time domain (oscilloscopes, waveform capture) or the frequency domain (spectrum analyzers). A time-frequency representation refuses this choice. The mathematics traces back to work on signal uncertainty in the 1940s, but the practical tools came after personal computers made it feasible to compute and display thousands of FFTs. Today every modern measurement instrument, from oscilloscopes to hand-held vibration meters, includes spectrogram plotting as standard.