Editing Discrete Fourier transform (section)

==Generalized DFT (shifted and non-linear phase)==
It is possible to shift the transform sampling in time and/or frequency domain by some real shifts ''a'' and ''b'', respectively. This is sometimes known as a '''generalized DFT''' (or '''GDFT'''), also called the '''shifted DFT''' or '''offset DFT''', and has analogous properties to the ordinary DFT:

:<math>X_k = \sum_{n=0}^{N-1} x_n e^{-\frac{i 2 \pi}{N} (k+b) (n+a)} \quad \quad k = 0, \dots, N-1.</math>

Most often, shifts of <math>1/2</math> (half a sample) are used.
While the ordinary DFT corresponds to a periodic signal in both time and frequency domains, <math>a=1/2</math> produces a signal that is anti-periodic in frequency domain (<math>X_{k+N} = - X_k</math>) and vice versa for <math>b=1/2</math>.
Thus, the specific case of <math>a = b = 1/2</math> is known as an ''odd-time odd-frequency'' discrete Fourier transform (or O<sup>2</sup> DFT).
Such shifted transforms are most often used for symmetric data, to represent different boundary symmetries, and for real-symmetric data they correspond to different forms of the discrete [[discrete cosine transform|cosine]] and [[discrete sine transform|sine]] transforms.

Another interesting choice is <math>a=b=-(N-1)/2</math>, which is called the '''centered DFT''' (or '''CDFT''').  The centered DFT has the useful property that, when ''N'' is a multiple of four, all four of its eigenvalues (see above) have equal multiplicities (Rubio and Santhanam, 2005)<ref name=Santhanam/>

The term GDFT is also used for the non-linear phase extensions of DFT. Hence, GDFT method provides a generalization for constant amplitude orthogonal block transforms including linear and non-linear phase types. GDFT is a framework
to improve time and frequency domain properties of the traditional DFT, e.g. auto/cross-correlations, by the addition of the properly designed phase shaping function (non-linear, in general) to the original linear phase functions (Akansu and Agirman-Tosun, 2010).<ref name=Akansu/>

The discrete Fourier transform can be viewed as a special case of the [[z-transform]], evaluated on the unit circle in the complex plane; more general z-transforms correspond to ''complex'' shifts ''a'' and ''b'' above.

[[File:DirectAndFourierSpaceLocations.png|class=skin-invert-image|right|thumb|500px|Discrete transforms embedded in time & space.]]