The Discrete-Time Fourier Transform (DTFT) is a mathematical tool used in signal processing to analyze discrete-time signals that are not time-limited. In particular, it transforms a discrete-time signal, which is a sequence of samples taken at equally spaced intervals, into the frequency domain. The DTFT is defined as the sum of the input signal's samples multiplied by complex exponential functions at different frequencies.
More precisely, given a discrete-time signal x[n], the DTFT X(ω) is defined as the infinite sum from negative infinity to positive infinity of x[n] multiplied by the complex exponential e^(-jωn), where j denotes the imaginary unit and ω represents the continuous frequency variable.
The DTFT is a continuous function of frequency, allowing us to analyze the frequency content of a signal at any desired frequency. It provides a representation of the signal's spectrum, describing how much energy is contained at various frequencies.
The DTFT is often used in fields such as telecommunications, audio processing, image analysis, and control systems. It helps in understanding the behavior of signals in the frequency domain, enabling the design of filters, Fourier analysis, and other frequency-based signal processing techniques.
In summary, the DTFT is a mathematical technique used to convert discrete-time signals into the frequency domain, allowing for analysis of a signal's frequency content and enabling the application of various signal processing techniques.
