All of these properties of the discrete Fourier transform (DFT) are applicable for discretetime 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 .
Property  Mathematical Representation 
Linearity  a_{1}x_{1}(n)+a_{2}x_{2}(n) a_{1}X_{1}(k) + a_{2}X_{2}(k) 
Periodicity  if x(n+N) = x(n) for all n
then x(k+N) = X(k) for all k 
Time reversal  x(Nn) X(Nk) 
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(kl) 
Circular time shift  x((nl))N = x(nl) X(k)e^{2πjlk/N}
or X(k)W^{kl}_{N }where W is the twiddle factor. 
Circular symmetries of a sequence  If the circular shift is in

Multiplication  
Complex conjugate  x*(n) X*(Nk) 
Symmetry  For even sequences:
X(k) = x(n)Cos(2πnk/N) For odd sequences: X(k) = x(n)Sin(2πnk/N) 
Parseval’s theorem  x(n).y*(n) = X(k).Y*(k) 
Proofs of the properties of the discrete Fourier transform