Z-TRANSFORM In mathematics and signal processing, the Z-transform converts a discrete-time signal, which is a sequence of real or complex numbers, into a complex frequency domain representation.

It can be considered as a discrete-time equivalent of the Laplace transform.

The idea contained within the Z-transform is also known in mathematical literature as the method

of generating functions which can be traced back as early as 1730 when it was introduced by  de

Moivre in conjunction with probability theory. From a mathematical view the Z-transform can also be

viewed as a Laurent series where one views the sequence of numbers under consideration as the

(Laurent) expansion of an analytic function.

DefinitionThe Z-transform, like many integral transforms, can be defined as either a one-sided or two-

sided transform.

Bilateral Z-transform

The bilateral or two-sided Z-transform of a discrete-time signal x[n] is the formal power

series X(z) defined as

where n is an integer and z is, in general, a complex number:

where A is the magnitude of z, j is the imaginary unit, and ɸ is the complex argument (also

referred to as angle or phase) in radians.

Unilateral Z-transform

Alternatively, in cases where x[n] is defined only for n ≥ 0, the single-sided or unilateral Z-transform

is defined as

In signal processing, this definition can be used to evaluate the Z-transform of the unit impulse

response of a discrete-time causal system.

An important example of the unilateral Z-transform is the probability-generating function, where the

component x[n] is the probability that a discrete random variable takes the value n, and the

function X(z) is usually written as X(s), in terms of s = z−1. The properties of Z-transforms (below)

have useful interpretations in the context of probability theory.

Geophysical definitionIn geophysics, the usual definition for the Z-transform is a power series in z as opposed to z−1. This convention is used, for example, by Robinson and Treitel[7] and by Kanasewich.[8] The geophysical definition is:

The two definitions are equivalent; however, the difference results in a number of changes. For

example, the location of zeros and poles move from inside the unit HYPERLINK

"https://en.wikipedia.org/wiki/Unit_circle"circleusing one definition, to outside the unit circle using the

other definition.[7 HYPERLINK "https://en.wikipedia.org/wiki/Z-transform"] HYPERLINK "https://en.wikipedia.org/wiki/Z-transform"[8] Thus, care is required to note

which definition is being used by a particular author.

Properties

Properties of the z-transform

Time domain Z-domain Proof ROC

Notatio

n

Lineari

ty

Contains

ROC1 ∩

ROC2

Time

expans

ion r: integer

Decim

ationohio-state.edu  or  ee.ic.ac.uk

Time

shiftin

g

ROC,

except z =

0 if k> 0

and z = ∞

if k < 0

Scalin

g in

the z-

domai

n

Time

revers

al

Compl

ex

conjug

ation

Real

part

Imagin

ary

part

Differe

ntiatio

n

Convol

ution

Contains

ROC1 ∩

ROC2

Cross-

correla

tion

Contains

the

intersection

of ROC

of   

and 

First

differe

nce

Contains

the

intersection

of ROC

ofX1(z) and 

z ≠ 0

Accum

ulation

Multipl

ication-

Table of common Z-transform pairsHere:

is the unit (or Heaviside) step function and

is the discrete-time (or Dirac delta) unit impulse function. Both are usually not considered as

true functions but as distributions due to their discontinuity (their value on n= 0 usually does

not really matter, except when working in discrete time, in which case they become

degenerate discrete series ; in this section they are chosen to take the value 1 on n = 0,

both for the continuous and discrete time domains, otherwise the content of the ROC

column below would not apply). The two "functions" are chosen together so that the unit

step function is the integral of the unit impulse function (in the continuous time domain), or

the summation of the unit impulse function is the unit step function (in the discrete time

domain), hence the choice of making their value on n = 0 fixed here to 1.

Signal,  Z-transform,  ROC

1 1 all z

2

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Relationship to Fourier series and Fourier transformFor values of z in the region |z|=1, known as the unit circle, we can express the transform as a

function of a single, real variable, ω, by defining z=ejω.  And the bi-lateral transform reduces to

a Fourier series:

 

 

 

 

(Eq.

1)

which is also known as the discrete-time Fourier transform (DTFT) of the x[n] sequence. This 2π-

periodic function is the periodic summation of a Fourier transform, which makes it a widely used

analysis tool. To understand this, let X(f) be the Fourier transform of any function, x(t), whose

samples at some interval, T, equal the x[n] sequence. Then the DTFT of the x[n] sequence can be

written as:

When T has units of seconds,   has units of hertz. Comparison of the two series reveals that

  is a normalized frequency with units of radians per sample. The value ω=2π corresponds

to   Hz.  And now, with the substitution     Eq.1 can be expressed in terms of the Fourier

transform, X(•):

When sequence x(nT) represents the impulse response of an LTI system, these functions are also

known as its frequency response. When the x(nT) sequence is periodic, its DTFT is divergent at one

or more harmonic frequencies, and zero at all other frequencies. This is often represented by the

use of amplitude-variant Dirac delta functions at the harmonic frequencies. Due to periodicity, there

are only a finite number of unique amplitudes, which are readily computed by the much simpler

discrete Fourier transform.

Relationship to Laplace transformBilinear transfom

The bilinear transform can be used to convert continuous-time filters (represented in the Laplace

domain) into discrete-time filters (represented in the Z-domain), and vice versa. The following

substitution is used:

to convert some function   in the Laplace domain to a function   in the Z-domain (Tustin

transformation), or

from the Z-domain to the Laplace domain. Through the bilinear transformation, the complex s-plane

(of the Laplace transform) is mapped to the complex z-plane (of the z-transform). While this mapping

is (necessarily) nonlinear, it is useful in that it maps the entire jΩ axis of the s-plane onto the unit

circle in the z-plane. As such, the Fourier transform (which is the Laplace transform evaluated on

the jΩ axis) becomes the discrete-time Fourier transform. This assumes that the Fourier transform

exists; i.e., that the jΩ axis is in the region of convergence of the Laplace transform.

Linear constant-coefficient difference equationThe linear constant-coefficient difference (LCCD) equation is a representation for a linear system

based on the autoregressive moving-average equation.

Both sides of the above equation can be divided by α0, if it is not zero, normalizing α0 = 1 and

the LCCD equation can be written

This form of the LCCD equation is favorable to make it more explicit that the "current"

output y[n] is a function of past outputs y[n−p], current input x[n], and previous inputs x[n−q].

Transfer function

Taking the Z-transform of the above equation (using linearity and time-shifting laws) yields

and rearranging results in

Zeros and poles

From the fundamental theorem of algebra the numerator has M roots (corresponding to zeros of H)

and the denominator has N roots (corresponding to poles). Rewriting the transfer function in terms of

poles and zeros

where qk is the k-th zero and pk is the k-th pole. The zeros and poles are commonly complex and

when plotted on the complex plane (z-plane) it is called the pole–zero plot.

In addition, there may also exist zeros and poles at z = 0 and z = ∞. If we take these poles and zeros

as well as multiple-order zeros and poles into consideration, the number of zeros and poles are

always equal.

By factoring the denominator, partial fraction decomposition can be used, which can then be

transformed back to the time domain. Doing so would result in the impulse response and the linear

constant coefficient difference equation of the system.

Output response

If such a system H(z) is driven by a signal X(z) then the output is Y(z) = H(z)X(z). By

performing partial fraction decomposition on Y(z) and then taking the inverse Z-transform the

output y[n] can be found. In practice, it is often useful to fractionally decompose before multiplying

that quantity by z to generate a form of Y(z)which has terms with easily computable inverse Z-

transforms.

APPLICATIONS OF Z TRANSFORM

Z transform is used to convert discrete time Domain into a complex frequency domain where, discrete time domain

represents an order of complex or real Numbers. It is generalize form of Fourier Transform, which we get when we

generalize Fourier transform and get z transform. The reason behind this is that Fourier transform is not sufficient to

converge on all sequence and when we do this thing then we get the power of complex variable theory that we deal

with noncontiguous time systems and signals.

This transform is used in many applications of mathematics and signal processing. The lists of applications of z

transform are:-

-Uses to analysis of digital filters.-Used to simulate the continuous systems.-Analyze the linear discrete system.-Used to finding frequency response.-Analysis of discrete signal.-Helps in system design and analysis and also checks the systems stability.-For automatic controls in telecommunication.-Enhance the electrical and mechanical energy to provide dynamic nature

of the system.

If we see the main applications of z transform than we find that it is analysis tool that analyze the whole discrete time

signals and systems and their related issues. If we talk the application areas of

This transform wherever it is used, they are:--Digital signal processing.-Population science.-Control theory.

Z-transforms represent the system according to their location of poles and zeros of the system during transfer

function that happens only in complex plane. It is closely related toLaplace HYPERLINK

"http://math.tutorcircle.com/calculus/laplace-transform.html" Transform. Main functionality of this transform is to

provide access to transient behavior (transient behavior means changeable) that monitors all states stability of a

system or all behavior either static or dynamic. This transform is a generalize form of Fourier transform from a

discrete time signals and Laplace transform is also a generalize form of Fourier transform but from continuous time

signals.