magnitudephase

Magnitude and phase describe the size and angle of a complex number or a complex-valued signal. If z = a + ib, its magnitude (modulus) is |z| = sqrt(a^2 + b^2). Its phase (argument) is theta = atan2(b, a), the angle the complex number forms with the positive real axis.

These two quantities allow a polar representation z = |z| e^{i theta} or z = |z| (cos theta + i sin theta). The phase is often taken modulo 2pi, with common ranges such as (-pi, pi] or [0, 2pi).

In signal processing, a complex spectrum is described by magnitude and phase for each frequency component. The magnitude indicates how strong the component is; the phase indicates its timing relative to a reference. Real signals have conjugate-symmetric spectra, which constrains phase relationships between positive and negative frequencies.

Computationally, magnitude is computed as sqrt(re^2 + im^2) and phase as arctan2(im, re). Phase unwrapping removes artificial jumps of 2pi in sequences of phase values, yielding a continuous phase function.

Applications span phasor representations in AC circuits, optics, and communications. In impedance analysis, magnitude and phase describe the amplitude and lag between voltage and current. In Fourier analysis and related signal processing tasks, magnitude and phase together determine the reconstructed signal.