Due to the NN-sample periodicity of the complex exponential basis functions ej⁢2⁢π⁢n⁢kN 2 n k N in the DFT and IDFT, the resulting transforms are also periodic with NN samples.

X⁢k+N=X⁢k X k N X k x⁢n=x⁢n+N x n x n N

A shift in time corresponds to a phase shift that is linear in frequency. Because of the periodicity induced by the DFT and IDFT, the shift is circular, or modulo NN samples.

x⁢(n−m)modNX⁢k⁢e−(j⁢2⁢π⁢k⁢mN)  x n m N X k 2 k m N The modulus operator pmodN p N means the remainder of pp when divided by NN. For example, 9mod5=4 9 5 4 and -1mod5=4 -1 5 4

x⁢(−n)modN=x⁢(N−n)modNX⁢(N−k)modN=X⁢(−k)modN  x n N x N n N X N k N X k N Note: time-reversal maps 00  0 0 , 1N−1  1 N 1 , 2N−2  2 N 2 , etc. as illustrated in the figure below.

Figure 1: Illustration of circular time-reversal

(a) Original signal (b) Time-reversed

x⁢n*X⁢(−k)modN*  x n X k N

Circular convolution is defined as x⁢n*h⁢n≐∑ m =0N−1x⁢m⁢x⁢(n−m)modN ≐ x n h n m N 1 0 x m x n m N

Circular convolution of two discrete-time signals corresponds to multiplication of their DFTs: x⁢n*h⁢nX⁢k⁢H⁢k  x n h n X k H k

A similar property relates multiplication in time to circular convolution in frequency. x⁢n⁢h⁢n1N⁢X⁢k*H⁢k  x n h n 1 N X k H k

Parseval's theorem relates the energy of a length-NN discrete-time signal (or one period) to the energy of its DFT. ∑ n =0N−1|x⁢n|2=1N⁢∑ k =0N−1|X⁢k|2 n N 1 0 x n 2 1 N k N 1 0 X k 2

The continuous-time Fourier transform, the DTFT, and DFT are all defined as transforms of complex-valued data to complex-valued spectra. However, in practice signals are often real-valued. The DFT of a real-valued discrete-time signal has a special symmetry, in which the real part of the transform values are DFT even symmetric and the imaginary part is DFT odd symmetric, as illustrated in the equation and figure below.

x⁢n x n real  X⁢k=X⁢(N−k)modN* X k X N k N (This implies X⁢0 X 0 , X⁢N2 X N 2 are real-valued.)

Figure 2: DFT symmetry of real-valued signal

Real part of X(k) is even (a) Even-symmetry in DFT sense Imaginary part of X(k) is odd (b) Odd-symmetry in DFT sense