DTFTiin

The Discrete-Time Fourier Transform (DTFT) is a mathematical tool used in signal processing to analyze the frequency content of discrete-time signals. Unlike the Fourier Transform for continuous signals, the DTFT operates on sequences of numbers that represent samples of a signal taken at regular intervals. The DTFT transforms a discrete-time signal into a continuous-frequency domain representation, showing how the signal's energy is distributed across different frequencies.

The DTFT is defined by a summation. For a discrete-time signal x[n], its DTFT, denoted as X(e^jω), is given by the infinite sum of x[n] multiplied by e^(-jωn) over all integer values of n. Here, ω represents the normalized angular frequency, typically ranging from -π to π. The output of the DTFT is a periodic function of ω with a period of 2π, reflecting the fact that discrete-time systems inherently have aliasing due to sampling.

The DTFT is particularly useful for understanding the spectral characteristics of signals in digital systems. It allows engineers to identify dominant frequencies, analyze bandwidth, and design filters. While the DTFT produces a continuous frequency spectrum, in practical digital signal processing, it is often approximated using the Discrete Fourier Transform (DFT), which computes the transform at a finite number of discrete frequencies. The DTFT itself is a theoretical concept that underpins many digital signal processing techniques.