The document discusses the z-transform, which is useful for analyzing discrete-time signals and systems. It defines the z-transform as the sum of a discrete-time signal multiplied by z-n, where z is a complex variable. The region of convergence for a z-transform consists of the values of z where this sum converges. The z-transform has properties like linearity, time shifting, and convolution that allow discrete-time signals and systems to be analyzed in the complex z-domain. Key topics covered include the definition of the z-transform, properties like region of convergence and poles/zeros, and properties like linearity, time shifting, and the convolution theorem.

1. THE Z-TRANSFORM
Presented By: Karansinh Parmar

2. THE Z-TRANSFORM:
INTRODUCTION
 Why z-Transform?
1. Many of signals (such as x(n)=u(n), x(n) = (0.5)n
u(-n),
x(n) = sin(nω) etc. ) do not have a DTFT.
2. Advantages like Fourier transform provided:
 Solution process reduces to a simple algebraic procedures
 The temporal domain sequence output y(n) = x(n)*h(n) can
be represent as Y(z)= X(z)H(z)
 Properties of systems can easily be studied and
characterized in z – domain (such as stability..)
 Topics:
 Definition of z –Transform
 Properties of z- Transform
 Inverse z- Transform

3. DEFINITION OFTHE Z-TRANSFORM
1. Definition:The z-transform of a discrete-time signal x(n) is defined by
where z = rejw
is a complex variable.The values of z for which the sum
converges define a region in the z-plane referred to as the region of
convergence (ROC).
2. Notationally, if x(n) has a z-transform X(z), we write
3. The z-transform may be viewed as the DTFT or an exponentially weighted
sequence. Specifically, note that with z = rejw
, X(z) can be looked as the DTFT
of the sequence r--n
x(n) and ROC is determined by the range of values of r of
the following right inequation.

4. ROC & Z-PLANE
 Complex z-plane
z = Re(z)+jIm(z) = rejw
 Zeros and poles of X(z)
Many signals have z-transforms that are rational
function of z:
Factorizing it will give:
The roots of the numerator polynomial, βk,are referred to
as the zeros (o) and αkare referred to as poles (x). ROC
of X(z) will not contain poles.

5. ROC PROPERTIES
 ROC is an annulus or disc in the z-plane centred at the origin.
i.e.
 A finite-length sequence has a z-transform with a region of
convergence that includes the entire z-plane except, possibly,
z = 0 and z = .The point z = will be included if x(n) = 0 for n < 0,
and the point z = 0 will be included if x(n) = 0 for n > 0.
 A right-sided sequence has a z-transform with a region of
convergence that is the exterior of a circle:
ROC: |z|>α
 A left-sided sequence has a z-transform with a region of
convergence that is the interior of a circle:
ROC: |z|<β
 The FourierTransform of x(n) converges absolutely if and only
if ROC of z-transform includes the unit circle

13. PROPERTIES OF Z-TRANSFORM
 Linearity
If x(n) has a z-transform X(z) with a region of convergence Rx, and if
y(n) has a z-transform Y(z) with a region of convergence Ry,
and the ROC of W(z) will include the intersection of Rx and Ry, that
is, Rw contains .
 Shifting property
If x(n) has a z-transform X(z),
 Time reversal
If x(n) has a z-transform X(z) with a region of convergence Rx that is
the annulus , the z-transform of the time-reversed sequence
x(-n) is
and has a region of convergence , which is denoted by
)()()()()()( zbYzaXzWnbynaxnw Z
+=→←+=
yx RR 
)()( 0
0 zXznnx nZ −
→←−
βα << z
)()( 1−
→←− zXnx Z
αβ 11 << z
xR1

14. PROPERTIES OF Z-TRANSFORM
 Multiplication by an exponential
 If a sequence x(n) is multiplied by a complex exponential αn
.
 Convolution theorm
If x(n) has a z-transform X(z) with a region of convergence Rx, and if h(n)
has a z-transform H(z) with a region of convergence Rh,
The ROC of Y(z) will include the intersection of Rx and Rh, that is,
Ry contains Rx R∩ h .
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 unit-sample response is often referred to as the system
function.
 Conjugation
If X(z) is the z-transform of x(n), the z-transform of the complex conjugate
of x(n) is
)()( 1
zXnx Zn −
→← αα
)()()()()()( zHzXzYnhnxny Z
=→←∗=
)()( ∗∗∗
→← zXnx Z

15. PROPERTIES OF Z-TRANSFORM
 Derivative
 If X(z) is the z-transform of x(n), the z-transform of is
 Initial value theorem
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:
dz
zdX
znnx Z )(
)( −→←
)(lim)0( zXx
z ∞→
=