In signal processing, phase describes the position of a sinusoidal waveform relative to a reference point in time. It is measured in radians or degrees, and it determines how much a sine wave is shifted along the time axis.
A pure sinusoid can be written as:
$x(t) = A \cdot \sin(2\pi f t + \phi)$
Here, $\phi$ (phi) is the phase. If $\phi = 0$, the wave starts at zero. If $\phi = \pi/2$ (90°), it starts at its maximum value — which is exactly a cosine wave. This simple shift changes the shape of the signal in the time domain, even though the frequency and amplitude remain the same.
The plot above shows three sine waves with identical frequency (5 Hz) and amplitude, but different phases. The blue wave starts at zero. The orange wave (phase = 45°) is shifted to the left. The green wave (phase = 90°) is shifted so that it starts at its maximum — it is a cosine wave.
In the frequency domain, phase is the second component of the Fourier transform, alongside magnitude. Every frequency bin $X[k]$ is a complex number:
$X[k] = |X[k]| \cdot e^{j \cdot \angle X[k]}$
The magnitude $|X[k]|$ tells us how strong a frequency is. The phase $\angle X[k]$ tells us where in time that frequency component is located relative to the start of the window.
Phase is often overlooked, but it carries essential information:
- Signal reconstruction: Without phase, the inverse Fourier transform cannot recover the original signal. Phase determines how different frequency components align in time.
- Time delay: A pure time delay manifests as a linear phase shift across frequencies. This is why phase is used in radar and sonar to measure distance.
- Waveform shape: Two signals with identical magnitude spectra but different phase spectra can have completely different shapes in the time domain.
- Filtering: Phase distortion can alter the shape of a signal even when magnitude is preserved. Linear-phase filters are designed to avoid this.
In practice, phase is often ignored when analysing the frequency content of a signal (e.g., audio spectrum analysis). But in applications such as communications, radar, and audio processing, phase is just as important as magnitude — it carries timing, distance, and structural information.