Discrete-time Fourier transform

In mathematics, the discrete-time Fourier transform (DTFT) is a form of Fourier analysis that is applicable to a sequence of discrete values.

The DTFT is often used to analyze samples of a continuous function. The term discrete-time refers to the fact that the transform operates on discrete data, often samples whose interval has units of time. From uniformly spaced samples it produces a function of frequency that is a periodic summation of the continuous Fourier transform of the original continuous function. In simpler terms, when you take the DTFT of regularly-spaced samples of a continuous signal, you get repeating (and possibly overlapping) copies of the signal's frequency spectrum, spaced at intervals corresponding to the sampling frequency. Under certain theoretical conditions, described by the sampling theorem, the original continuous function can be recovered perfectly from the DTFT and thus from the original discrete samples. The DTFT itself is a continuous function of frequency, but discrete samples of it can be readily calculated via the discrete Fourier transform (DFT) (see § Sampling the DTFT), which is by far the most common method of modern Fourier analysis.

Both transforms are invertible. The inverse DTFT reconstructs the original sampled data sequence, while the inverse DFT produces a periodic summation of the original sequence. The fast Fourier transform (FFT) is an algorithm for computing one cycle of the DFT, and its inverse produces one cycle of the inverse DFT.

Let ${\displaystyle s(t)}$ be a continuous function in the time domain. We begin with a common definition of the continuous Fourier transform, where ${\displaystyle f}$ represents frequency in hertz and ${\displaystyle t}$ represents time in seconds:

We can reduce the integral into a summation by sampling ${\displaystyle s(t)}$ at intervals of ${\displaystyle T}$ seconds (see Fourier transform § Numerical integration of a series of ordered pairs). Specifically, we can replace ${\displaystyle s(t)}$ with a discrete sequence of its samples, ${\displaystyle s(nT)}$, for integer values of ${\displaystyle n}$, and replace the differential element ${\displaystyle dt}$ with the sampling period ${\displaystyle T}$. Thus, we obtain one formulation for the discrete-time Fourier transform (DTFT):

This Fourier series (in frequency) is a continuous periodic function, whose periodicity is the sampling frequency ${\displaystyle 1/T}$. The subscript ${\displaystyle 1/T}$ distinguishes it from the continuous Fourier transform ${\displaystyle S(f)}$, and from the angular frequency form of the DTFT. The latter is obtained by defining an angular frequency variable, ${\displaystyle \omega \triangleq 2\pi fT}$ (which has normalized units of radians/sample), giving us a periodic function of angular frequency, with periodicity ${\displaystyle 2\pi }$:

The utility of the DTFT is rooted in the Poisson summation formula, which tells us that the periodic function represented by the Fourier series is a periodic summation of the continuous Fourier transform:

The components of the periodic summation are centered at integer values (denoted by ${\displaystyle k}$) of a normalized frequency (cycles per sample). Ordinary/physical frequency (cycles per second) is the product of ${\displaystyle k}$ and the sample-rate, ${\displaystyle f_{s}=1/T.}$   For sufficiently large ${\displaystyle f_{s},}$ the ${\displaystyle k=0}$ term can be observed in the region ${\displaystyle [-f_{s}/2,f_{s}/2]}$ with little or no distortion (aliasing) from the other terms.  Fig.1 depicts an example where ${\displaystyle 1/T}$ is not large enough to prevent aliasing.