Page 460 - Schaum's Outline of Theory and Problems of Signals and Systems

APP. B] PROPERTIES OF LINEAR TIME-INVARIANT SYSTEMS TRANSFORMS 447

Integration in the Time Domain:

Initial value theorem: ~(0') = lirn sX,(s)
S-m
Final value theorem: lim x(t) = lim sX,(s)
t-oc s-0

B.3 THE FOURIER TRANSFORM
DeJinition:

Properties of the Fourier Transform:
Linearity: alxl(t) + a2x2(t) c*alX,(o) + a2X2(o)
Time shifting: x(t - to) c* e-~"'o~(w)
Time scaling: x(at) -
Frequency shifting: eJw~~'x(t) c*X(o - oo)

Time reversal: x( - t ) c* X( - o)
Duality: X(t) c* 2lrx( -o)
Wt)
Time differentiation: - c*jwX(w)
dt
dX(4
Frequency differentiation: ( -jt )x(t) c* -
do

Convolution: x,(t)* x2(t) -X,(w)X2(w)
1
Multiplication: x,(t )x2(t) - -X,(o) * X2( W)
2lr
Real signal: x(t) =xe(t) +xo(t) -X(o) =A(o) + jB(o)
X( -0) = X*(o)
Even component: xe(t) - Re{X(o)) = A(w)
Odd component: xo(t) c* j Im(X(o)) = jB(o)