frequencydomain

Frequency domain is a representation of signals and systems based on frequency components rather than time. In this view, a signal is described by the amplitudes and phases of sinusoidal components across frequencies. The frequency-domain representation is typically obtained by applying Fourier transforms to time-domain signals.

For a continuous-time signal x(t), the Fourier transform is X(f) = ∫ x(t) e^{-j2π f t} dt, with the inverse x(t) = ∫ X(f) e^{j2π f t} df. For discrete-time signals, the discrete-time Fourier transform or the discrete Fourier transform (DFT) is used: X[k] = ∑_{n=0}^{N-1} x[n] e^{-j 2π k n / N}, with the inverse transform reconstructing x[n]. The spectrum comprises the magnitude |X(f)| and the phase ∠X(f); the power spectrum is |X(f)|^2 and relates to signal energy via Parseval’s theorem.

Key properties include that linear time-invariant systems become simple multiplications in the frequency domain: convolution in time corresponds to multiplication in frequency, and multiplication in time corresponds to convolution in frequency. This makes many analyses and designs more convenient.

Applications span spectrum analysis, filter design, communications, audio and image processing, and system identification. Practical considerations include spectral leakage and resolution, which motivate windowing and zero-padding; the sampling rate imposes a Nyquist limit to prevent aliasing. Time-localized analyses (such as the short-time Fourier transform) extend frequency-domain methods to non-stationary signals.