Page 462 - Schaum's Outline of Theory and Problems of Signals and Systems

APP. B] PROPERTIES OF LINEAR TIME-INVARIANT SYSTEMS TRANSFORMS 449

B.4 DISCRETE-TIME LTI SYSTEMS
Unit sample response: h[n]
m
Convolution: y[n] = x[n] * h[n] = x[k]h[n - k]
k= -m
Causality: h[n] = 0, n < 0
m
Stability: (h[n](dt < a:
n= -m

B.5 THE Z-TRANSFORM
The Bilateral (or Two-sided) z-Transform:

Dejnition:

Properties of the z- Transform:
Linearity: alxl[n] + a2x2[n] t-,a,X1(z) +a2X2(z), R' 3 R, n R2

Time shifting: x[n - no] -2-"oX(z), R' 3 R n (0 < lzl < w)
Z
-
Multiplication by z:: z:x[n] - x(% 1, R' = lzdR
R'
Multiplication by ejR1tN: e~~o"~[n] ~(e-jnl)z), = R
1
Time reversal: x[ - n] t-, X

dX( z )
Multiplication by n: nx[n] o - z -, R' = R
n x[n] - 1 dz
Accumulation: X(z), R' 3 R n {lzl > 1)
1 -2-'
k= - OC

Some Common z-Transforms Pairs:
6[n] - all z
1,