time-frequency analysis

Time-frequency analysis is a method for breaking down a signal into its component frequencies while tracking how those frequencies change over time. Traditional Fourier analysis shows you what frequencies exist in a signal, but loses all timing information; time-frequency analysis recovers both. This is essential in electrical work because real-world signals are not stationary: a 60 Hz hum may spike during motor startup, a fault may produce a burst of high-frequency energy for milliseconds, or a bearing may emit changing vibration signatures as it degrades. You need to know not just what is present, but when it happens.

The most common time-frequency representation is the spectrogram, which divides the signal into short, overlapping time windows and computes the frequency spectrum within each window. A spectrogram displays time on the horizontal axis, frequency on the vertical axis, and power as color intensity. Other representations include the Wigner-Ville distribution, which offers higher resolution but can produce artifacts, and wavelet transforms, which adapt the window size to the frequency being analyzed: wider windows for low frequencies, narrower windows for high frequencies. This adaptability makes wavelets powerful for detecting transients and faults that occur at specific frequencies.

Where it matters in practice

Electrical engineers use time-frequency analysis for condition monitoring, power quality assessment, and fault diagnosis. In rotating machinery, bearing defects produce repeating impacts that show up as energy bursts at specific frequencies and intervals; a spectrogram reveals both the impact frequency and its evolution over weeks or months. In power systems, disturbances like capacitor bank switching or arc faults create transient energy spread across a wide band and concentrated in time; time-frequency tools pinpoint the event and its spectral signature. Equipment manufacturers embed spectrogram analysis in vibration analyzers and power analyzers sold for industrial maintenance.

The mathematics involves a fundamental trade-off: high time resolution means poor frequency resolution, and vice versa. This is not a limitation of the tool but a property of the uncertainty principle in signal processing. A window 10 milliseconds wide gives you precise timing but blurs frequencies closer than about 100 Hz apart. A window 1 second wide separates frequencies well but smears events to a one-second blur. Wavelet methods partially escape this trap by using different window widths at different frequencies, which is why they are favored when you must catch a brief, high-frequency transient.

Practical time-frequency analysis requires choosing parameters carefully: window type (Hann, Hamming, or Blackman windows are common), window length, overlap, and whether to use standard Fourier-based spectrograms or wavelet analysis. Many engineers rely on software tools with presets for their application rather than setting parameters manually. The quality of your result depends on sampling rate, signal length, and how noise is handled; undersampling will alias high frequencies, too short a signal will blur low frequencies, and noise without proper filtering will create false features across the entire time-frequency plane.