CTFT

The continuous-time Fourier transform (CTFT) is a mathematical transform used in signal processing to analyze the frequency content of a continuous-time signal x(t). It represents x(t) as a complex-valued function X(jω) of angular frequency ω. The transform is defined for signals that satisfy certain conditions, such as absolute integrability, square integrability, or, in a broader sense, as a tempered distribution.

Definition: X(jω) = ∫ from −∞ to ∞ x(t) e^(−jωt) dt. Inverse transform: x(t) = (1/2π) ∫ from −∞ to ∞ X(jω) e^(jωt) dω. These formulas enable conversion between the time domain and the frequency domain. The pair allows the analysis of how different frequency components contribute to the signal.

Key properties include linearity, time shifting: if x(t − t0) then X(jω) multiplies by e^(−jωt0); frequency shifting: multiplying x(t) by e^(+jω0t) shifts X along ω by ω0; time scaling: x(at) corresponds to (1/|a|) X(jω/a); convolution in time corresponds to multiplication in frequency; Parseval’s theorem relates time-domain energy to frequency-domain energy. For real signals, X(jω) often satisfies symmetry properties.

Relation to the discrete-time Fourier transform (DTFT) and practical notes: the CTFT applies to continuous-time signals and yields a continuous spectrum; sampling a signal leads to a discrete-time spectrum and potential aliasing if not done carefully. For periodic signals, the CTFT consists of impulses at harmonic frequencies; for non-decaying signals, the CTFT exists in the sense of distributions.

Applications include spectral analysis, filtering, communications, modulation, and signal characterization. Common examples: a rectangular pulse has a sinc-shaped spectrum; a Gaussian signal is (up to scaling) its own Fourier transform. The CTFT provides a foundation for designing and analyzing systems in both time and frequency domains.