The Z-Transform and Fourier Transforms of DiscreteTime Signals
4.1 Definition of Z-Transform
1. Definition
The z-transform of a discrete-time signal x(n) is defined by
X ( z) 

 x ( n) z
n
n  
j
where z  re is a complex variable. The values of z for which the sum converges
define a region in z-plane referred to as the region of convergence (ROC).
2. Notation
If x(n) has a z-transform X(z), we write
Z
x(n) 
X ( z)
3. ROC
ROC is determined by the range of values of r for which

 x ( n) r
n

n  
Table 4-1 Common z-Transform Paris
Sequence
 (n )
z-Transform
1
1
1  z 1
1
1  z 1
z 1
(1  z 1 ) 2
 n u (n)
  n u (n  1)
n n u (n)
z 1
(1  z 1 ) 2
1  (cos  0 ) z 1
1  2(cos  0 ) z 1  z  2
1  (cos  0 ) z 1
1  2(cos  0 ) z 1  z  2
 n u(n  1)
n
cos( n 0 )u (n)
sin( n 0 )u(n)
4. Complex z-plane
z  Re( z )  j Im( z )  re j
Unit circle:
z 1
Region of Convergence
all z
z 
z 
z 
z 
z 1
z 1
Im(z)
Unit circle

Re(z)
4.2 Properties of Z-Transform
1. Linearity
If x(n) has a z-transform X(z) with a region of convergence Rx, and if y(n) has a ztransform Y(z) with a region of convergence Ry,
Z
w(n)  ax(n)  by(n) 
W ( z )  aX ( z )  bY ( z )
and the ROC of W(z) will include the intersection of Rx and Ry, that is,
Rw contains R x  R y .
2. Shifting property
If x(n) has a z-transform X(z),
Z
x(n  n0 ) 
z  n0 X ( z ) .
3. Time reversal
If x(n) has a z-transform X(z) with a region of convergence Rx that is the annulus
  z   , the z-transform of the time-reversed sequence x(-n) is
Z
x(n) 
X ( z 1 )
and has a region of convergence 1   z  1  , which is denoted by 1 Rx .
4. Multiplication by an exponential
If a sequence x(n) is multiplied by a complex exponential  n ,
Z
 n x(n) 
X ( 1 z ) .
5. Convolution theorm
If x(n) has a z-transform X(z) with a region of convergence Rx, and if h(n) has a ztransform H(z) with a region of convergence Rh,
Z
y(n)  x(n)  h(n) 
Y ( z)  X ( z) H ( z) .
The ROC of Y(z) will include the intersection of Rx and Rh, that is,
Ry contains R x  Rh .
With x(n), y(n), and h(n) denoting the input, output, and unit-sample response,
respectively, and X(z), Y(x), and H(z) their z-transforms. The z-transform of the unitsample response is often referred to as the system function.
6. Conjugation
If X(z) is the z-transform of x(n), the z-transform of the complex conjugate of x(n) is
Z
x  (n) 
X  (z ) .
7. Derivative
If X(z) is the z-transform of x(n), the z-transform of n k x(n) is
dX ( z )
Z
nx(n) 
z
.
dz
8. Initial value theorm
If X(z) is the z-transform of x(n) and x(n) is equal to zero for n<0, the initial value,
x(0), maybe be found from X(z) as follows:
x(0)  lim X ( z ) .
z 
4.3 Inverse Z-Transform
If X(z) is the z-transform of x(n), the inverse z-transform is given by the contour
integral
1
x ( n) 
X ( z ) z n1dz ,
2j C
where C is a counterclockwise closed contour in the region of convergence of X(z)
and encircling the origin of the z-plane.
4.4 Solution of Difference Equations Using Z-Transforms
1. The one-sided, or unilateral, z-transform is defined by

X 1 ( z )   x ( n) z  n
n 0
2. If x(n) has a one-sided z-transform X1(z), the one-sided z-transform of x(n-1) is
Z
x(n  1) 
z 1 X 1 ( z )  x(1) .
It is this property that makes the one-sided z-transform useful for solving difference
equations with initial conditions.