negativefrequency

Negative frequency is a term used in Fourier analysis and signal processing to describe frequency components at negative values in the frequency domain. In continuous-time theory, the Fourier transform represents a signal x(t) as a continuum of complex exponentials e^{j2πft} for all real f. The negative frequency f = -F is the mathematical counterpart of the positive frequency F, and together they form a complete basis for real-valued signals.

For real-valued signals, the spectrum is conjugate symmetric: X(-f) = X*(f). This symmetry means that a real signal can be constructed from pairs of positive and negative frequencies. In particular, a real cosine at frequency f results from equal contributions at ±f. Negative frequencies are not separate physical oscillations; they are part of the representation that ensures the time-domain signal remains real.

In contrast, complex-valued signals can have independent information at negative and positive frequencies, so their negative frequency components may carry distinct content. In discrete contexts, the discrete Fourier transform of a real sequence exhibits similar symmetry, making negative-frequency information redundant for the real signal itself.

Analytic signal concepts use this idea by suppressing negative frequencies. Forming an analytic signal x_a(t) = x(t) + j Hilbert{x(t)} yields a spectrum that is zero for f < 0, facilitating instantaneous amplitude and phase estimation.

Overall, negative frequency is a conventional label for part of the frequency-domain decomposition. Its interpretation depends on whether the signal is real or complex, and on the chosen representation or processing approach.