Page 487 - Applied Numerical Methods Using MATLAB

476 FOURIER TRANSFORM
Table E.2 Properties of DtFT (Discrete-Time Fourier Transform)
∞

(0) Definition X( ) = x[n]e −j n
n=−∞
(1) Linearity αx[n] + βx[n] → αX( ) + βY( )
∗
(2) Symmetry x[n] = x e [n] + x o [n]: real → X( ) ≡ X (− )
x e [n]: real and even → X e ( ) = Re{X( )}
x o [n]: real and odd → X o ( ) = jIm{X( )}
x[−n] → X(− )
(3) Time shifting x[n − n 1 ] → e −j n 1 X( )
(4) Frequency shifting e j 1 n x[n] → X( − 1 )
∞

(5) Real convolution g[n] ∗ x[n] = g[m]x[n − m] → G( )X( )
m=−∞
d
(6) Complex derivative nx[n] → j X( )
d
1
(7) Complex convolution x[n]y[n] → X( ) ∗ Y( ) (periodic/circular convolution)
2π

x[n/M] if n = mM(m : an integer)
(8) Scaling → X(M )
0, otherwise
∞
1
2
2
(9) Parseval’s relation |x[n]| = |x( )| d
2π
n=−∞ 2π