Properties

Properties of DFT

All of these properties of the discrete Fourier transform (DFT) are applicable for discrete-time signals that have a DFT. Meaning these properties of DFT apply to any generic signal x(n) for which an X(k) exists.

(x(n)   X(k))

where X(k)= ∑_n=0^N-1 x(n) e^-2πjkn/N.

Property	Mathematical Representation
Linearity	a₁x₁(n)+a₂x₂(n) a₁X₁(k) + a₂X₂(k)
Periodicity	if x(n+N) = x(n) for all n then x(k+N) = X(k) for all k
Time reversal	x(N-n) X(N-k)
Duality	x(n) Nx[((-k))_N]
Circular convolution
Circular correlation	For x(n) and y(n), circular correlation r_xy(l) is r_xy(l) R_xy(k) = X(k).Y*(k)
Circular frequency shift	x(n)e^2πjln/N X(k+l) x(n)e^-2πjln/N X(k-l)
Circular time shift	x((n-l))N = x(n-l) X(k)e^-2πjlk/N or X(k)W^kl_Nwhere W is the twiddle factor.
Circular symmetries of a sequence	If the circular shift is in anti-clockwise direction (positive): Delayed discrete-time signal clockwise direction (negative): Advanced discrete-time signal Time reversal: Obtained by reversing samples of the discrete-time sequence about zero axis/locating x(n) in a clockwise direction.
Multiplication
Complex conjugate	x(n) X(N-k)
Symmetry	For even sequences: X(k) = ∑_n=0^N-1 x(n)Cos(2πnk/N) For odd sequences: X(k) = ∑_n=0^N-1 x(n)Sin(2πnk/N)
Parseval’s theorem	∑_n=0^N-1 x(n).y(n) = (1/N)∑_n=0^N-1 X(k).Y(k)