Table 3: Properties of the z-Transform
Property
Sequence
Transform
ROC
x[n]
x1 [n]
x2 [n]
X(z)
X1 (z)
X2 (z)
R
R1
R2
Linearity
ax1 [n] + bx2 [n]
aX1 (z) + bX2 (z)
At least the intersection
of R1 and R2
Time shifting
x[n − n0 ]
z −n0 X(z)
R except for the
possible addition or
deletion of the origin
Scaling in the
ejω0 n x[n]
R
z-Domain
z0n x[n]
n
a x[n]
−jω0
X(e
z)
X zz0
X(a−1 z)
x[−n]
X(z −1 )
Inverted R (i.e., R−1
= the set of points
z −1 where z is in R)
X(z k )
R1/k
Time reversal
Time expansion
x(k) [n] =
x[r], n = rk
0,
n 6= rk
for some integer r
z0 R
Scaled version of R
(i.e., |a|R = the
set of points {|a|z}
for z in R)
(i.e., the set of points z 1/k
where z is in R)
Conjugation
x∗ [n]
X ∗ (z ∗ )
R
Convolution
x1 [n] ∗ x2 [n]
X1 (z)X2 (z)
At least the intersection
of R1 and R2
First difference
x[n] − x[n − 1]
(1 − z −1 )X(z)
At least the
intersection of R and |z| > 0
Accumulation
Pn
1
X(z)
1−z −1
At least the
intersection of R and |z| > 1
Differentiation
in the z-Domain
nx[n]
−z dX(z)
dz
R
k=−∞ x[k]
Initial Value Theorem
If x[n] = 0 for n < 0, then
x[0] = limz→∞ X(z)
Table 4: Some Common z-Transform Pairs
Signal
Transform
ROC
1. δ[n]
1
All z
2. u[n]
1
1−z −1
|z| > 1
3. −u[−n − 1]
1
1−z −1
|z| < 1
4. δ[n − m]
z −m
All z except
0 (if m > 0) or
∞ (if m < 0)
5. αn u[n]
1
1−αz −1
|z| > |α|
6. −αn u[−n − 1]
1
1−αz −1
|z| < |α|
7. nαn u[n]
αz −1
(1−αz −1 )2
|z| > |α|
8. −nαn u[−n − 1]
αz −1
(1−αz −1 )2
|z| < |α|
9. [cos ω0 n]u[n]
1−[cos ω0 ]z −1
1−[2 cos ω0 ]z −1 +z −2
|z| > 1
10. [sin ω0 n]u[n]
[sin ω0 ]z −1
1−[2 cos ω0 ]z −1 +z −2
|z| > 1
11. [r n cos ω0 n]u[n]
1−[r cos ω0 ]z −1
1−[2r cos ω0 ]z −1 +r 2 z −2
|z| > r
12. [r n sin ω0 n]u[n]
[r sin ω0 ]z −1
1−[2r cos ω0 ]z −1 +r 2 z −2
|z| > r
Table of Laplace and Z-transforms
X(s)
x(t)
1.
–
–
2.
–
–
3.
4.
1
s
1
s+a
x(kT) or x(k)
1(t)
1(k)
e-at
e-akT
5.
1
s2
t
kT
6.
2
s3
t2
(kT)2
7.
6
s4
t3
(kT)3
8.
a
s (s + a )
1 – e-at
1 – e-akT
9.
b−a
(s + a )(s + b )
e-at – e-bt
e-akT – e-bkT
te-at
kTe-akT
(1 – at)e-at
(1 – akT)e-akT
t2e-at
(kT)2e-akT
10.
11.
12.
(s + a )
2
(s + a )2
Tz −1
(1 − z )
T z (1 + z )
(1 − z )
T z (1 + 4 z + z )
(1 − z )
(1 − e )z
(1 − z )(1 − e z )
(e − e )z
(1 − e z )(1 − e z )
−1 2
−1
(s + a )
3
−1
−1
akT – 1 + e-akT
14.
ω
s +ω 2
sin ωt
sin ωkT
15.
s
s +ω 2
cos ωt
cos ωkT
2
e-at sin ωt
e-akT sin ωkT
2
e-at cos ωt
e-akT cos ωkT
2
2
ω
s+a
17.
(s + a )
18.
–
–
ak
19.
–
–
ak-1
k = 1, 2, 3, …
20.
–
–
kak-1
21.
–
–
k2ak-1
22.
–
–
k3ak-1
− aT
−1
–
–
k4ak-1
24.
–
–
ak cos kπ
x(t) = 0
for t < 0
x(kT) = x(k) = 0 for k < 0
Unless otherwise noted, k = 0, 1, 2, 3, …
−1
− bT
−1
−1
−bT
Te − aT z −1
(1 − e
− aT
z −1
−1
)
2
1 − (1 + aT )e − aT z −1
(1 − e z )
T e (1 + e z )z
(1 − e z )
[(aT − 1 + e )+ (1 − e − aTe )z ]z
(1 − z ) (1 − e z )
−1 2
− aT
− aT
− aT
−1
−1
−1 3
− aT
− aT
−1 2
− aT
− aT
−1
−1
z −1 sin ωT
1 − 2 z −1 cos ωT + z − 2
1 − z −1 cos ωT
1 − 2 z −1 cos ωT + z − 2
e − aT z −1 sin ωT
1 − 2e − aT z −1 cos ωT + e − 2 aT z − 2
1 − e − aT z −1 cos ωT
1 − 2e z −1 cos ωT + e − 2 aT z − 2
1
1 − az −1
z −1
1 − az −1
− aT
z −1
(1 − az )
z (1 + az )
(1 − az )
−1 2
−1
23.
−2
− aT
− aT
at – 1 + e-at
+ω
−1
−1 4
− aT
a2
2
s (s + a )
+ω
−1
−1 3
2
2
2
1
1 − z −1
1
1 − e − aT z −1
− aT
s
(s + a )
z-k
3
1
2
1
2
13.
16.
X(z)
Kronecker delta δ0(k)
1
k=0
0
k≠0
δ0(n-k)
1
n=k
0
n≠k
−1
−1 3
(
z −1 1 + 4az −1 + a 2 z −2
(1 − az )
)
−1 4
(
z −1 1 + 11az −1 + 11a 2 z −2 + a 3 z −3
(1 − az )
−1 5
1
1 + az −1
)
−1
Definition of the Z-transform
Z{x(k)} = X ( z ) =
∞
∑ x(k ) z − k
k =0
Important properties and theorems of the Z-transform
x(t) or x(k)
Z{x(t)} or Z {x(k)}
1.
ax(t )
aX (z )
2.
ax1( t ) + bx2 ( t )
aX 1 ( z ) + bX 2 ( z )
3.
x( t + T ) or x( k + 1 )
zX ( z ) − zx( 0 )
4.
x( t + 2T )
z X ( z ) − z 2 x( 0 ) − zx( T )
5.
x( k + 2 )
z 2 X ( z ) − z 2 x( 0 ) − zx( 1 )
6.
x( t + kT )
z k X ( z ) − z k x( 0 ) − z k −1 x( T ) − K − zx( kT − T )
7.
x( t − kT )
z −k X ( z )
8.
x( n + k )
z k X ( z ) − z k x( 0 ) − z k −1 x( 1 ) − K − zx( k1 − 1 )
9.
x( n − k )
z −k X ( z )
10.
tx( t )
− Tz
d
X( z )
dz
11.
kx( k )
−z
d
X( z )
dz
12.
e − at x( t )
X ( zeaT )
13.
e − ak x( k )
X ( ze a )
14.
a k x( k )
⎛z⎞
X⎜ ⎟
⎝a⎠
15.
ka k x( k )
16.
x( 0 )
17.
x( ∞ )
lim 1 − z −1 X ( z ) if 1 − z −1 X ( z ) is analytic on and outside the unit circle
18.
∇x( k ) = x( k ) − x( k − 1 )
(1 − z )X ( z )
19.
∆x( k ) = x( k + 1 ) − x( k )
(z − 1)X ( z ) − zx( 0 )
20.
∑ x( k )
1
X( z )
1 − z −1
21.
∂
x( t , a )
∂a
∂
X ( z,a )
∂a
22.
k m x( k )
d ⎞
⎛
⎜− z ⎟ X( z )
dz
⎠
⎝
23.
∑ x( kT ) y( nT − kT )
X ( z )Y ( z )
n
k =0
2
−z
d ⎛z⎞
X⎜ ⎟
dz ⎝ a ⎠
lim X ( z ) if the limit exists
z →∞
[(
z →1
) ] (
)
−1
m
n
k =0
∞
24.
∑ x( k )
k =0
X (1)