Page 458 - Schaum's Outline of Theory and Problems of Signals and Systems

Appendix B

Properties of Linear Time-Invariant Systems
and Various Transforms

B.l CONTINUOUS-TIME LTI SYSTEMS
Unit impulse response: h(t)
w
Convolution: y(t) =x(t)* h(t) =
Causality: h( t ) = 0, t < 0
w
Stability: / Ih(t)l dt < m
- ?D

B.2 THE LAPLACE TRANSFORM
The Bilateral (or Two-sided) Laplace Transform

Definition:

Properties of the Bilateral Laplace Transform:
Linearity: a,x,(t) + a,x,(t) -a,X,(s) + a,X,(s), R'3R, nR,
Time shifting: x( t - t,) H e-"oX(s), R' = R
Shifting in s: e"llx(t) - X(s - so), R' = R + Re(s,)
1
Time scaling: x(at) H -X(S), R' = aR
la l
Time reversal: x( - t) - X( -s), R' = -R
Wt)
Differentiation in t: - -sX(s), R' 3R
dt
fl(s)
Differentiation in s: - tx(t - - , R'=R
ds
I 1
Integration: x(r) dr - -X(s), R' >R n {Re(s) > 0)
1- w S
Convolution: x,( t) * x,( t) t, X,(s)X,(s), R' 3 R, n R,