Dsp U Lec05 The Z Transform

EC533: Digital Signal Processing
5 l l

Lecture 5
The Z-Transform

5.1 - Introduction

• The Laplace Transform (s domain) is a valuable tool for
representing, analyzing & designing continuos-time signals &
l d l
systems.
• The z transform is convenient yet invaluable tool for representing,
z-transform
analyzing & designing discrete-time signals & systems.
• The resulting transformation from s-domain to z-domain is called
z-transform.
z-transform

• The relation between s-plane and z-plane is described below :
z = esT
• The z-transform maps any point s = σ + jω in the s-plane to z-
plane (r θ).

5.2 – The Z-Transform
For continuous-time signal,
Time Domain
Ti D i S‐Domain

For discrete-time signal,

Ƶ Z‐Domain
Time Domain
Ƶ-1

Causal
System

where,

5.2.1 – Z-Transform Definition
• The z-transform of sequence x(n) is defined by
∞
−n
X ( z ) = ∑ x ( n) z Two sided z transform
Bilateral z transform
n = −∞

For causal system
∞
−n
X (z) = ∑ x(n)z One sided z transform
Unilateral z transform
n=0
• The z transform reduces to the Discrete Time Fourier transform
(DTFT) if r=1; z = e−jω.
DTFT
∞
X ( e jω ) = ∑
n = −∞
x ( n ) e − jω n

5.2.2 – Geometrical interpretation of
z transform
z-transform
• The point z = rejω is a
p
vector of length r from Im z
origin and an angle ω with j
respect to real axis. z = rejω

r
ω
Re
R z
• Unit circle : The contour -1 1

|z| = 1 is a circle on the z-
plane with unity radius
l ith it di -j

DTFT is to evaluate z-transform on a unit circle.

5.2.3 – Pole-zero Plot

• A graphical representation
Im z
of z-transform on z-plane j
– Poles denote by “x” and
– zeros denote by “o”
Re z
-1 1

-j

Example
Find the z-transform of,

Solution:

It’s a geometric sequence
Recall: Sum of a Geometric Sequence

where, a: first term, r: common ratio,
n: number of terms

5.3 – Region Of Convergence (ROC)
• ROC of X(z) is the set of all values of z for which X(z) attains a
finite value.
• Give a sequence, the set of values of z for which the z-transform
converges, i.e., |X(z)|<∞, is called the region of convergence.
∞ ∞
−n
| X ( z ) |= ∑ x(n) z = ∑ | x(n) || z |− n < ∞
n = −∞ n = −∞
∞
−n
Im ∑ | x(n)r |< ∞
n = −∞
ROC is an annual ring centered on
r the origin.

Re Rx − <| z |< Rx +

ROC = {z = re jω | Rx − < r < Rx + }

Ex. 1 Find the z-transform of the following sequence
x = {2 -3, 7 4 0 0 ……..}
{2, 3 7, 4, 0, 0, }
∞
X ( z) = ∑ x[n]z − n = 2 − 3 z −1 + 7 z − 2 + 4 z −3
n = −∞

2 z 3 − 3z 2 + 7 z + 4
= , |z|>0
z 3

The ROC is the entire complex z - plane except the origin.

Ex. 2 Find the z-transform of δ [n]
∞
X ( z) = ∑ δ [ n] z − n = 1
n = −∞

with an ROC consisting of the entire z - plane.

Ex 3 Find the z transform of δ [n -1]
Ex. z-transform 1]
∞
1
X ( z) =
n = −∞
∑ δ [n − 1] z − n = z −1 =
z
with an ROC consisting of the entire z - plane except z = 0 .

Ex. 4 Find the z-transform of δ [n +1]
∞
X ( z) = ∑ δ [n + 1] z − n = z
n = −∞

with an ROC consisting of the entire z - plane except z = ∞,
i.e., there is a pole at infinity.

Ex.5 Find the z-transform of the following right-sided sequence
(causal)
x [ n] = a u [ n]
n

∞ ∞
−n
X ( z ) = ∑ a u[n]z
n
= ∑ (az −1 ) n
n = −∞ n =0

This f
Thi form to find inverse
fi d i
ZT using PFE

Ex.6 Find the z-transform of the following left-sided sequence

Ex. 7 Find the z-transform of

Rewriting x[n] as a sum of left-sided and right sided sequences
left sided right-sided
and finding the corresponding z-transforms,

where

Notice from the ROC that the z-transform
doesn’t exist for b > 1

5.3.1 – Characteristic Families of Signals with Their
Corresponding ROC

5.3.2 – Properties of ROC

• A ring or disk in the z-plane centered at the origin.
g p g
• The Fourier Transform of x(n) is converge absolutely iff the
ROC includes the unit circle.
• The ROC cannot include any poles
• Finite Duration Sequences: The ROC is the entire z-plane
except possibly z=0 or z=∞.
• Right sided sequences (causal seq.): The ROC extends
outward from the outermost finite pole in X(z) to z=∞.
• Left sided sequences: The ROC extends inward from the
innermost nonzero pole in X(z) to z=0.
• Two-sided sequence: The ROC is a ring bounded by two
circles passing through two pole with no poles inside the ring
circles passing through two pole with no poles inside the ring

5.4 - Properties of z-Transform
(1) Linearity : a x[n] + b y[n] ←→ a X ( z ) + b Y ( z )

⎛z⎞
(4) Z - scale Property : a x[n] ←→ X ⎜ ⎟
n

⎝a⎠
1
(5) Time Reversal : x [−n] ←→ X ( )
l
z
(6) Convolution : h [n] ∗ x [n] ←→ H ( z ) X ( z )
Transfer
Function

5.5 - Rational z-Transform

For most practical signals, the z-transform can be expressed
as a ratio of two polynomials
f l l

N ( z) ( z − z1 )( z − z 2 ) L ( z − z M )
X ( z) = =G
D( z ) ( z − p1 )( z − p2 ) L ( z − p N )
where
G is scalar gain,
z1 , z 2 , L, z M are the zeroes of X(z), i.e., the roots
of the numerator polynomial
and p1 , p2 , L , p N are the poles of X(z), i.e., the roots
of the denominator polynomial.

5.6 - Commonly used z-Transform pairs

Sequence z‐Transform ROC
δ[n] 1 All values of z
All values of z

u[n] 1 |z| > 1
1 − z −1

1
αnu[n] |z| > |α|
1 − αz −1
αz −1
nαnu[n] (1 − αz −1 ) 2 |z| > |α|
|z| > |α|

1
(n+1) αnu[n] |z| > |α|
(1 − αz −1 ) 2

1 − (r cos ω0 ) z −1
(rn cos ω on) u[n] |z| > |r|
1 − (2r cos ω0 ) z −1 + r 2 z −2

1 − (r sin ω0 ) z −1
(rn sin ωon) [n] |z| > |r|
1 − (2r cos ω0 ) z −1 + r 2 z −2

5.7 - Z-Transform & pole-zero distribution &
Stability considerations
y

Thus,
unstable
z
stable R.H.S.
Mapping between S-plane & Z-plane is done as follows: L.H.S.

1) Mapping of Poles on the jω‐axis of the s‐domain to the z‐domain
1) Mapping of Poles on the jω axis of the s domain to the z domain ωs/4

Maps to a unit circle & represents Marginally stable terms 1

ωs/2 ω=0
ω=ωs
3ωs/4

5.7 - Z-Transform & pole-zero distribution &
Stability considerations – cont.
y
2) Mapping of Poles in the L.H.S. of the s‐plane to the z‐plane

Maps to inside the unit circle & represents stable terms & the
system is stable.

3) Mapping of Poles in the R.H.S. of the s‐plane to the z‐plane

Outside the unit circle & represents unstable terms.

Discrete Systems Stability Testing Steps

1) Find the pole positions of the z-transform.
2) If any pole is on or outside the unit circle. (Unless coincides with zero on the unit
circle) The system is unstable.

5.7.1 - Pole Location and Time-domain
Behavior of Causal Signals

5.7.2 - Stable and Causal Systems
Causal Systems : ROC extends outward from the outermost pole.
C lS t t d t df th t t l
Im

R
Re

Stable Systems : ROC includes the unit circle. Im

A stable system requires that its Fourier transform is 1
uniformly convergent.
Re

Dsp U Lec05 The Z Transform

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