Page 464 - Schaum's Outline of Theory and Problems of Signals and Systems

APP. B] PROPERTIES OF LINEAR TIME-INVARIANT SYSTEMS TRANSFORMS 45 1

B.6 THE DISCRETE-TIME FOURIER TRANSFORM
Definition:

m
X(R) = C x [n] eeJf'"
n= -0c

Properties of the Discrete-Time Fourier Transform:
Periodicity: x[n] - X(R) = X(R + 2rr)
Linearity: a,x,[n] + a2x2[n] ++ a, XJR) + a2X2(R)
Time shifting: x[n - no] ++ e-~~"llX(R)
Frequency shifting: e~"~"x[n] - X(R - R,)
Conjugation: x*[n] - X*( -R)
Time Reversal: x[ - n] - X( - R)
ifn=km
Time Scaling: x(,,,[n] = -X(mR)
ifnzkm
dX(R)
Frequency differentiation: m[n] - j------
dR
First difference: x[n] - x[n - 11 ++ (1 - e-jo)x(fl)
n 1
Accumulation: C x[k ] - rrX(0) S(R) + A x(n)
1 - e -jn
k= -CC
Convolution: x,[n] * x2[n] ++ X,(R)X,(R)

1
Multiplication: x,[n]x2[n] - -X,(R) @ X2(R)
2rr
Real sequence: x[n] =x,[n] + x,[n] ++ X(R) = A(R) + jB(R)
X(-R) =X*(R)
Even component: x,[n] - Re{X(R)) =A(R)
Odd component: x,[n] - j 1m{X(R)) = jB(R)

Parseval's Relations: