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. 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 is the original sampled data sequence. The inverse DFT is 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.
The discrete-time Fourier transform of a discrete set of real or complex numbers x[n], for all integersn, is a Fourier series, which produces a periodic function of a frequency variable. When the frequency variable, ω, has normalized units of radians/sample, the periodicity is 2π, and the Fourier series is:
The utility of this frequency domain function is rooted in the Poisson summation formula. Let X(f) be the Fourier transform of any function, x(t), whose samples at some interval T (seconds) are equal (or proportional) to the x[n] sequence, i.e. T⋅x(nT) = x[n]. Then the periodic function represented by the Fourier series is a periodic summation of X(f) in terms of frequency f in hertz (cycles/sec):
Fig 1. Depiction of a Fourier transform (upper left) and its periodic summation (DTFT) in the lower left corner. The lower right corner depicts samples of the DTFT that are computed by a discrete Fourier transform (DFT).
The integer k has units of cycles/sample, and 1/T is the sample-rate, fs (samples/sec). So X1/T(f) comprises exact copies of X(f) that are shifted by multiples of fs hertz and combined by addition. For sufficiently large fs the k = 0 term can be observed in the region [−fs/2, fs/2] with little or no distortion (aliasing) from the other terms. In Fig.1, the extremities of the distribution in the upper left corner are masked by aliasing in the periodic summation (lower left).
We also note that e−i2πfTn is the Fourier transform of δ(t − nT). Therefore, an alternative definition of DTFT is:[note 1]
The modulated Dirac comb function is a mathematical abstraction sometimes referred to as impulse sampling.
An operation that recovers the discrete data sequence from the DTFT function is called an inverse DTFT. For instance, the inverse continuous Fourier transform of both sides of Eq.3 produces the sequence in the form of a modulated Dirac comb function:
However, noting that X1/T(f) is periodic, all the necessary information is contained within any interval of length 1/T. In both Eq.1 and Eq.2, the summations over n are a Fourier series, with coefficients x[n]. The standard formulas for the Fourier coefficients are also the inverse transforms:
When the input data sequence x[n] is N-periodic, Eq.2 can be computationally reduced to a discrete Fourier transform (DFT), because:
All the available information is contained within N samples.
X1/T(f) converges to zero everywhere except at integer multiples of 1/(NT), known as harmonic frequencies.
The DTFT is periodic, so the maximum number of unique harmonic amplitudes is (1/T) / (1/(NT)) = N
The kernel x[n] e−i2πfTn is N-periodic at the harmonic frequencies, f = k/(NT). Introducing the notation to represent a sum over any n-sequence of length N, we can write:
Therefore, the DTFT diverges at the harmonic frequencies, but at different frequency-dependent rates. And those rates are given by the DFT of one cycle of the x[n] sequence. In terms of a Dirac comb function, this is represented by:
When the DTFT is continuous, a common practice is to compute an arbitrary number of samples (N) of one cycle of the periodic function X1/T:
where is a periodic summation:
The sequence is the inverse DFT. Thus, our sampling of the DTFT causes the inverse transform to become periodic. The array of |Xk|2 values is known as a periodogram, and the parameter N is called NFFT in the Matlab function of the same name.
In order to evaluate one cycle of numerically, we require a finite-length x[n] sequence. For instance, a long sequence might be truncated by a window function of length L resulting in three cases worthy of special mention. For notational simplicity, consider the x[n] values below to represent the values modified by the window function.
Case: Frequency decimation.L = N ⋅ I, for some integer I (typically 6 or 8)
A cycle of reduces to a summation of Iblocks of length N, or circular addition.[note 4] The DFT then goes by various names, such as:
Recall that decimation of sampled data in one domain (time or frequency) produces overlap (sometimes known as aliasing) in the other, and vice versa. Compared to an L-length DFT, the summation/overlap causes decimation in frequency, leaving only DTFT samples least affected by spectral leakage. That is usually a priority when implementing an FFT filter-bank (channelizer). With a conventional window function of length L, scalloping loss would be unacceptable. So multi-block windows are created using FIR filter design tools. Their frequency profile is flat at the highest point and falls off quickly at the midpoint between the remaining DTFT samples. The larger the value of parameter I, the better the potential performance.
Case: L = N+1, where N is even-valued
This case arises in the context of Window function design, out of a desire for real-valued DFT coefficients. When a symmetric sequence is associated with the indices [-M ≤ n ≤ M], known as a finite Fourier transform data window, its DTFT, a continuous function of frequency is real-valued. When the sequence is shifted into a DFT data window, [0 ≤ n ≤ 2M], the DTFT is multiplied by a complex-valued phase function: . But when sampled at frequencies for integer values of the samples are all real-valued. To achieve that goal, we can perform a -length DFT on a periodic summation with 1-sample of overlap. Specifically, the last sample of a data sequence is deleted and its value added to the first sample. Then a window function, shortened by 1 sample, is applied, and the DFT is performed. The shortened, even-length window function is sometimes called DFT-even. In actual practice, people commonly use DFT-even windows without overlapping the data, because the detrimental effects on spectral leakage are negligible for long sequences (typically hundreds of samples).[note 6]
Fig 2. DFT of ei2πn/8 for L = 64 and N = 256
Fig 3. DFT of ei2πn/8 for L = 64 and N = 64
Case: Frequency interpolation.L ≤ N
In this case,the DFT simplifies to a more familiar form:
In order to take advantage of a fast Fourier transform algorithm for computing the DFT, the summation is usually performed over all N terms, even though N − L of them are zeros. Therefore, the case L < N is often referred to as "zero-padding".
Spectral leakage, which increases as L decreases, is detrimental to certain important performance metrics, such as resolution of multiple frequency components and the amount of noise measured by each DTFT sample. But those things don't always matter, for instance when the x[n] sequence is a noiseless sinusoid (or a constant), shaped by a window function. Then it is a common practice to use zero-padding to graphically display and compare the detailed leakage patterns of window functions. To illustrate that for a rectangular window, consider the sequence:
Figures 2 and 3 are plots of the magnitude of two different sized DFTs, as indicated in their labels. In both cases, the dominant component is at the signal frequency: f = 1/8 = 0.125. Also visible in Fig 2 is the spectral leakage pattern of the L = 64 rectangular window. The illusion in Fig 3 is a result of sampling the DTFT at just its zero-crossings. Rather than the DTFT of a finite-length sequence, it gives the impression of an infinitely long sinusoidal sequence. Contributing factors to the illusion are the use of a rectangular window, and the choice of a frequency (1/8 = 8/64) with exactly 8 (an integer) cycles per 64 samples. A Hann window would produce a similar result, except the peak would be widened to 3 samples (see DFT-even Hann window).
When the real and imaginary parts of a complex function are decomposed into their even and odd parts, there are four components, denoted below by the subscripts RE, RO, IE, and IO. And there is a one-to-one mapping between the four components of a complex time function and the four components of its complex frequency transform::p. 291
From this, various relationships are apparent, for example:
The transform of a real-valued function (xRE+ xRO) is the even symmetric function XRE+ i XIO. Conversely, an even-symmetric transform implies a real-valued time-domain.
The transform of an imaginary-valued function (i xIE+ i xIO) is the odd symmetric function XRO+ i XIE, and the converse is true.
The transform of a even-symmetric function (xRE+ i xIO) is the real-valued function XRE+ XRO, and the converse is true.
The transform of a odd-symmetric function (xRO+ i xIE) is the imaginary-valued function i XIE+ i XIO, and the converse is true.
where the notation distinguishes the Z-transform from the Fourier transform. Therefore, we can also express a portion of the Z-transform in terms of the Fourier transform:
Note that when parameter T changes, the terms of remain a constant separation apart, and their width scales up or down. The terms of X1/T(f) remain a constant width and their separation 1/T scales up or down.