magnitudespectrum

The magnitude spectrum of a signal is the function that describes the amplitude of its frequency components. It is obtained from the Fourier transform X(f) = ∫ x(t) e^{-j2πft} dt for continuous signals, or from the discrete Fourier transform X[k] = Σ x[n] e^{-j2πkn/N} for discrete-time signals. The magnitude spectrum is M(f) = |X(f)| or M[k] = |X[k]|.

For real-valued signals, the magnitude spectrum is typically symmetric about zero frequency because X(-f) is the complex conjugate of X(f), so |X(-f)| = |X(f)|. The phase spectrum, Φ(f) = arg(X(f)), contains information about the signal’s timing and structure, and the magnitude together with phase fully determine the original signal.

The power spectrum is P(f) = |X(f)|^2, and is often visualized in decibels using 20 log10|X(f)| for amplitude or 10 log10 P(f) for power. In practice, magnitude spectra are used to assess dominant frequencies, bandwidth, and spectral envelope of signals such as audio, speech, and vibration data.

Computation and interpretation considerations include windowing, sampling rate, and spectral resolution. In discrete analysis, zero-padding increases the number of frequency samples without adding true resolution, while leakage can affect peak accuracy. To study time-varying spectra, the short-time Fourier transform (STFT) yields a sequence of magnitude spectra, enabling spectrogram representations.

In image processing, the two-dimensional magnitude spectrum is obtained from a 2D Fourier transform and reveals spatial frequency content, useful for filtering, texture analysis, and feature extraction.