Discretetime Fourier transform
In mathematics, the discretetime Fourier transform (DTFT) is a form of Fourier analysis that is applicable to the uniformlyspaced samples of a continuous function. The term discretetime refers to the fact that the transform operates on discrete data (samples) whose interval often has units of time. From only the 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.
Fourier transforms 

Continuous Fourier transform 
Fourier series 
Discretetime Fourier transform 
Discrete Fourier transform 
Discrete Fourier transform over a ring 
Fourier analysis 
Related transforms 
Contents
Definition
The discretetime Fourier transform of a discrete set of real or complex numbers x[n], for all integers n, 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:

(Eq.1)
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. . Then the periodic function represented by the Fourier series is a periodic summation of X(f). In terms of frequency in hertz (cycles/sec):

(Eq.2)
The integer k has units of cycles/sample, and 1/T is the samplerate, f_{s} (samples/sec). So X_{1/T}(f) comprises exact copies of X(f) that are shifted by multiples of f_{s} hertz and combined by addition. For sufficiently large f_{s} the k=0 term can be observed in the region [−f_{s}/2, f_{s}/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 is the Fourier transform of Therefore, an alternative definition of DTFT is:^{[note 1]}

(Eq.3)
The modulated Dirac comb function is a mathematical abstraction sometimes referred to as impulse sampling.^{[1]}
Inverse transform
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 X_{1/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:
Periodic data
When the input data sequence x[n] is Nperiodic, Eq.2 can be computationally reduced to a discrete Fourier transform (DFT), because:
 All the available information is contained within N samples.
 converges to zero everywhere except integer multiples of known as harmonic frequencies.
 The DTFT is periodic, so the maximum number of unique harmonic amplitudes is
The kernel is Nperiodic at the harmonic frequencies, Introducing the notation to represent a sum over any nsequence of length N, we can write:
Therefore, the DTFT diverges at the harmonic frequencies, but at different frequencydependent 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:
 ^{[note 2]}^{[note 3]}
Sampling the DTFT
When the DTFT is continuous, a common practice is to compute an arbitrary number of samples (N) of one cycle of the periodic function X_{1/T}:
where x_{N} is a periodic summation:
The x_{N} sequence is the inverse DFT. Thus, our sampling of the DTFT causes the inverse transform to become periodic. The array of is known as a periodogram, where parameter N is known to Matlab users as NFFT.^{[2]}
In order to evaluate one cycle of x_{N} numerically, we require a finitelength x[n] sequence. For instance, a long sequence might be truncated by a window function of length L resulting in two cases worthy of special mention: L ≤ N and L = I•N, for some integer I (typically 6 or 8). For notational simplicity, consider the x[n] values below to represent the modified values.
When L = I•N a cycle of x_{N} reduces to a summation of I blocks of length N. This goes by various names, such as:^{[3]}^{[4]}^{[5]}^{[6]}^{[7]}
 windowpresum FFT
 Weight, OverLap, Add (WOLA)^{[note 4]}
 polyphase FFT
 multiple block windowing
 timealiasing.
An interesting way to understand/motivate the technique is to recall that decimation of sampled data in one domain (time or frequency) produces aliasing in the other, and vice versa. The x_{N} summation is mathematically equivalent to aliasing, leading to decimation in frequency, leaving only DTFT samples least affected by spectral leakage. That is usually a priority when implementing an FFT filterbank (channelizer). With a conventional window function of length L, scalloping loss would be unacceptable. So multiblock windows are created using FIR filter design tools.^{[8]}^{[9]} 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. We note that the same results can be obtained by computing and decimating an Llength DFT, but that is not computationally efficient.
When L ≤ N the DFT is usually written in this 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 NL of them are zeros. Therefore, the case L < N is often referred to as "zeropadding".
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 zeropadding to graphically display and compare the detailed leakage patterns of window functions. To illustrate that for a rectangular window, consider the sequence:
 and
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 zerocrossings. Rather than the DTFT of a finitelength 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 DFTeven Hann window).
Convolution
The convolution theorem for sequences is:
An important special case is the circular convolution of sequences x and y defined by x_{N} * y where x_{N} is a periodic summation. The discretefrequency nature of DTFT{x_{N}} "selects" only discrete values from the continuous function DTFT{y}, which results in considerable simplification of the inverse transform. As shown at Convolution theorem#Functions of discrete variable sequences:
For x and y sequences whose nonzero duration is less than or equal to N, a final simplification is:
The significance of this result is expounded at Circular convolution and Fast convolution algorithms.
DTFT of real signals
A complex discretetime signal is a real signal (i.e if ) if and only if its DTFT is conjugate symmetric.
Spectrum of symmetric signals
If is an even symmetric signal (i.e. if ), then the DTFT is real for all .
If is an odd symmetric signal (i.e. if ), then the DTFT is purely imaginary for all .
Relationship to the Ztransform
The bilateral Ztransform is defined by:
 where z is a complex variable.
We denote this function as to avoid confusion with the Fourier transform. For values of z in the region z=1, known as the unit circle, we can express the transform as a function of a single, real variable, ω, by defining z=e^{iω}. And the bilateral transform reduces to a Fourier series:
Note that when parameter T changes, the terms of remain a constant separation (2π) apart, and their width scales up or down. The terms of remain a constant width and their separation (1/T) scales up or down.
Table of discretetime Fourier transforms
Some common transform pairs are shown in the table below. The following notation applies:
 ω = 2πfT is a real number representing continuous angular frequency (in radians per sample). (f is in cycles/sec, and T is in sec/sample.) In all cases in the table, the DTFT is 2πperiodic (in ω).
 X_{2π}(ω) designates a function defined on −∞ < ω < ∞.
 X_{o}(ω) designates a function defined on −π < ω ≤ π, and zero elsewhere. Then:
 δ(ω) is the Dirac delta function
 sinc(t) is the normalized sinc function
 rect(t) is the rectangle function
 tri(t) is the triangle function
 n is an integer representing the discretetime domain (in samples)
 u[n] is the discretetime unit step function
 δ[n] is the Kronecker delta δ_{n, 0}
Time domain x[n] 
Frequency domain X_{2π}(ω) 
Remarks 

integer M  
odd M 
integer M > 0  

The term must be interpreted as a distribution in the sense of a Cauchy principal value around its poles at ω = 2πk.  
π < a < π

real number a  

real number a; –π < a < π  
real number a; –π < a < π  
integer M  
real number a  
real number W 0 < W < 1 

real number W 0 < W < 0.5 

it works as a differentiator filter  
real numbers W, a 0 < W = 1 

Hilbert transform  
real numbers A, B complex C 
Properties
This table shows some mathematical operations in the time domain and the corresponding effects in the frequency domain.
 is the discrete convolution of two sequences
 x[n]* is the complex conjugate of x[n]
Property  Time domain 
Frequency domain 
Remarks 

Linearity  
Shift in time / Modulation in frequency  integer k  
Shift in frequency / Modulation in time  real number a  
Downsampling  ^{[note 5]}  integer M  
Expansion  integer M  
Time reversal / Frequency reversal  
Time conjugation  
Time reversal & conjugation  
Derivative in frequency  
Integration in frequency  
Differencing in time  
Convolution in time / Multiplication in frequency  
Multiplication in time / Convolution in frequency  Periodic convolution  
Cross correlation  
Parseval's theorem 
See also
Notes

^ In fact Eq.2 is often justified as follows:
 ^ Substituting this expression into formula produces the correctly scaled inverse DFT for the x(nT) sequence.
 ^ The generalized function is not unitless. It has the same units as T.
 ^ WOLA is not to be confused with the Overlapadd method of piecewise convolution.

^ This expression is derived as follows:
References
 ^ Rao, R. Signals and Systems. PrenticeHall Of India Pvt. Limited. ISBN 9788120338593.
 ^ "Periodogram power spectral density estimate  MATLAB periodogram".
 ^ Gumas, Charles Constantine (July 1997). "Windowpresum FFT achieves highdynamic range, resolution". Personal Engineering & Instrumentation News: 58–64. Archived from the original on 20010210.
 ^ Lyons, Richard G. (June 2008). "DSP Tricks: Building a practical spectrum analyzer". EE Times. Note however, that it contains a link labeled weighted overlapadd structure which incorrectly goes to Overlapadd method.
 ^ Lillington, John. "Comparison of Wideband Channelisation Architectures". RF Engines Ltd. Retrieved 20161030.
 ^ Chennamangalam, Jayanth (20161018). "The Polyphase Filter Bank Technique". CASPER Group. Retrieved 20161030.
 ^ Dahl, Jason F. (20030206). Time Aliasing Methods of Spectrum Estimation (Ph.D.). Brigham Young University. Retrieved 20161031.
 ^ Lin, YuanPei; Vaidyanathan, P.P. (June 1998). "A Kaiser Window Approach for the Design of Prototype Filters of Cosine Modulated Filterbanks" (PDF). IEEE Signal Processing Letters. 5 (6): 132–134. Retrieved 20170316.
 ^ cmfb.m, Caltech, retrieved 20170316
Further reading
 Crochiere, R.E.; Rabiner, L.R. (1983). Multirate Digital Signal Processing. Englewood Cliffs, NJ: Prentice Hall. pp. 313–326. ISBN 0136051626.
 Oppenheim, Alan V.; Schafer, Ronald W. (1999). DiscreteTime Signal Processing (2nd ed.). Prentice Hall Signal Processing Series. ISBN 0137549202.
 Porat, Boaz (1996). A Course in Digital Signal Processing. John Wiley and Sons. pp. 27–29 and 104–105. ISBN 0471149616.
 Siebert, William M. (1986). Circuits, Signals, and Systems. MIT Electrical Engineering and Computer Science Series. Cambridge, MA: MIT Press. ISBN 0262690950.
 Lyons, Richard G. (2010). Understanding Digital Signal Processing (3rd ed.). Prentice Hall. ISBN 9780137027415.