APP. B]
PROPERTIES OF LINEAR TIME-INVARIANT SYSTEMS TRANSFORMS
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
( h [ n ] ( d< t a:
Stability: n= -m
B.5 THE Z-TRANSFORM The Bilateral (or Two-sided) z-Transform: Dejnition:
Properties of the z-Transform:
Linearity: a l x l [ n ]+ a 2 x 2 [ nt]- , a , X 1 ( z )+ a 2 X 2 ( z ) ,R' 3 R , nR 2 Time shifting: x [ n - n o ]-2-"oX(z), R' 3 R n (0 < lzl < w) Multiplication by z:: z:x[n]
- -1, Z
x(%
R' = lzdR
Multiplication by ejR1tN:e ~ ~ o " ~ [ ~n( ]e - j n l ) z )R', 1
Time reversal: x[ - n ] t-,X
dX(z ) Multiplication by n: nx[n]o - z -, R' = R dz n 1 Accumulation: x[n] X ( z ) , R' 3 R k = - OC
-
-
Some Common z-Transforms Pairs:
6[n]
1 , all z
=R
1-2-'
n {lzl > 1)
449