Fourier transforms 

In mathematics, the discretetime Fourier transform (DTFT), also called the finite Fourier transform, is a form of Fourier analysis that is applicable to a sequence of values.
The DTFT is often used to analyze samples of a continuous function. The term discretetime 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 discretetime Fourier transform of a discrete sequence of real or complex numbers x[n], for all integers n, is a Trigonometric 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 DTFT series is:^{[1]}^{: p.147 }

(Eq.1) 
The discretetime Fourier transform is analogous to a Fourier series, except instead of starting with a periodic function of time and producing discrete sequence over frequency, it starts with a discrete sequence in time and produces a periodic function in frequency. 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].^{[2]} 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):^{[a]}^{[A]}

(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 e^{−i2πfTn} is the Fourier transform of δ(t − nT). Therefore, an alternative definition of DTFT is:^{[B]}

(Eq.3) 
The modulated Dirac comb function is a mathematical abstraction sometimes referred to as impulse sampling.^{[4]}
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:

(Eq.4) 
When the input data sequence x[n] is Nperiodic, Eq.2 can be computationally reduced to a discrete Fourier transform (DFT), because:
The DFT coefficients are given by:
Substituting this expression into the inverse transform formula confirms:
as expected. The inverse DFT in the line above is sometimes referred to as a Discrete Fourier series (DFS).^{[1]}^{: p 542 }
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}: ^{[1]}^{: pp 557–559 & 703 }
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 X_{k}^{2} values is known as a periodogram, and the parameter N is called NFFT in the Matlab function of the same name.^{[5]}
In order to evaluate one cycle of 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 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 I segments of length N. 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 Llength DFT, the summation/overlap causes decimation in frequency,^{[1]}^{: p.558 } 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.^{[15]}^{[16]} 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.
When a symmetric, Llength window function () is truncated by 1 coefficient it is called periodic or DFTeven. The truncation affects the DTFT. A DFT of the truncated sequence samples the DTFT at frequency intervals of 1/N. To sample at the same frequencies, for comparison, the DFT is computed for one cycle of the periodic summation, ^{[E]}
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 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:
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).
Main article: Convolution theorem § Functions of a discrete variable (sequences) 
The convolution theorem for sequences is:
An important special case is the circular convolution of sequences x and y defined by where is a periodic summation. The discretefrequency nature of means that the product with the continuous function is also discrete, which results in considerable simplification of the inverse transform:
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 explained at Circular convolution and Fast convolution algorithms.
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 onetoone mapping between the four components of a complex time function and the four components of its complex frequency transform:^{[18]}^{: p.291 }
From this, various relationships are apparent, for example:
is a Fourier series that can also be expressed in terms of the bilateral Ztransform. I.e.:
where the notation distinguishes the Ztransform from the Fourier transform. Therefore, we can also express a portion of the Ztransform 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 X_{1/T}(f) remain a constant width and their separation 1/T scales up or down.
Some common transform pairs are shown in the table below. The following notation applies:
Time domain x[n] 
Frequency domain X_{2π}(ω) 
Remarks  Reference 

^{[18]}^{: p.305 }  
integer  
odd M 
integer  

The term must be interpreted as a distribution in the sense of a Cauchy principal value around its poles at .  
^{[18]}^{: p.305 }  
π < a < π

real number  

real number with  
real number with  
integer and odd integer  
real numbers with  
real number ,  
it works as a differentiator filter  
real numbers with  
Hilbert transform  
real numbers complex 
This table shows some mathematical operations in the time domain and the corresponding effects in the frequency domain.
Property  Time domain x[n] 
Frequency domain 
Remarks  Reference 

Linearity  complex numbers  ^{[18]}^{: p.294 }  
Time reversal / Frequency reversal  ^{[18]}^{: p.297 }  
Time conjugation  ^{[18]}^{: p.291 }  
Time reversal & conjugation  ^{[18]}^{: p.291 }  
Real part in time  ^{[18]}^{: p.291 }  
Imaginary part in time  ^{[18]}^{: p.291 }  
Real part in frequency  ^{[18]}^{: p.291 }  
Imaginary part in frequency  ^{[18]}^{: p.291 }  
Shift in time / Modulation in frequency  integer k  ^{[18]}^{: p.296 }  
Shift in frequency / Modulation in time  real number  ^{[18]}^{: p.300 }  
Decimation  ^{[F]}  integer  
Time Expansion  integer  ^{[1]}^{: p.172 }  
Derivative in frequency  ^{[18]}^{: p.303 }  
Integration in frequency  
Differencing in time  
Summation in time  
Convolution in time / Multiplication in frequency  ^{[18]}^{: p.297 }  
Multiplication in time / Convolution in frequency  Periodic convolution  ^{[18]}^{: p.302 }  
Cross correlation  
Parseval's theorem  ^{[18]}^{: p.302 } 