Discrete-time Fourier transform

The Discrete-Time Fourier Transform (DTFT) is an essential mathematical tool for analyzing discrete-time signals, converting them from the time domain to the frequency domain. This transformation allows for examining the frequency components of discrete signals, providing insights into their spectral characteristics. In the DTFT, the continuous integral used in the continuous-time Fourier transform is replaced by a summation to accommodate the discrete nature of the signal.

One of the notable properties of the DTFT is its periodicity. The Fourier spectrum X(Ω) is periodic with a period of 2π. This periodicity implies that X(Ω) can be represented as a Fourier series, facilitating various analytical and computational techniques. The periodic nature of the DTFT also enables the computation of its inverse, the Inverse Discrete-Time Fourier Transform (IDTFT), which reconstructs the original discrete-time signal from its frequency spectrum.

Here, Ω represents the frequency variable, differentiated from the continuous frequency variable typically denoted by ω. The result, X(Ω), is the Fourier spectrum of the discrete signal. The existence and convergence of X(Ω) depend on the summability of the discrete-time signal x[n]. If x[n] is absolutely summable, then X(Ω) exists and converges.

Despite the discrete nature of the original signal, X(Ω) is a continuous function of the frequency variable Ω, highlighting the DTFT's role as a bridge between discrete and continuous domains. This characteristic is pivotal in various practical applications, particularly in the design and analysis of digital filters used in audio and video processing, communication systems, and biomedical signal processing.

In summary, the DTFT is a foundational tool in signal processing, enabling the analysis and manipulation of discrete-time signals in the frequency domain. Its properties and applications underscore its importance in both theoretical and practical aspects of modern engineering and technology.