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History

The z-transform was introduced, under this name,

by Ragazzini and Zadeh in 1952. The modified or advanced Z-transform was later developed by E. I. Jury, and presented in his

bookSampled-Data Control Systems (John Wiley & Sons 1958). The

idea contained within the Z-transform was previously known as the

"generating function method."

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Why z-Transform?

y A generalization of Fourier transformy Why generalize it?

y FT does not converge on all sequencey Notation good for analysisy Bring the power of complex variable theory deal with the

discrete-time signals and systems

In mathematics and signal processing, the Z-transform converts a discrete time-domain signal, which is

a sequence of real or complex numbers, into a complex frequency-domain representation.

It can be considered as a discrete equivalent of the Laplace transform. This similarity is explored in the

theory of time scale calculus.

The z-transform is an extension of the discrete time. Fourier transform to a function that is defined on

regions of the complex plane. Some sequences that do not have Fourier transforms will have z-transforms, but the reverse is true as well. For our purposes, the special significance of z transform is as

a tool to facilitate designing systems with the desired attributes that can be practically implemented.

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X(z) = 7 x[n] z n

linear

differenceequation

time

domainsolution

z transformedequation

z transformsolution

summing

algebra

z transforminverse z transform

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Relation btw various

transforms

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y The z-transform is a function of the complex z variable

y Convenient to describe on the complex z-planey If we plot z=e j[ for[=0 to 2T we get the unit circle

z -plane

Re

Im

X (z )

Re

Im

z = e j [

[

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g!! §§g

g!

g

g! n

n

n

n z n x z n x z X |||)(|)(|)(|

The region of convergence (ROC) is

the set of points in the complex plane

for which the Z-transform summationconverges.

Region of Convergence

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Right Sided Sequence

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Left Sided Sequence

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Two Sided Sequence

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Poles and Zeroes

)(

)()(

z Q

z P z X !

where P (z ) and Q(z ) are

polynomials in z .

Zeros: The values of z ¶s such that X (z ) = 0

Poles: The values of z ¶s such that X (z ) = g

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Properties of ROC

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Most real signals are analog and in order to utilise the processing power of modern digital processors it is necessary to convert these analog signalsinto some form which can be stored and processed by digital devices. The

standard method is to sample the signal periodically and digitize it with an A to D converter using a standard number of bits 8, 16 etc. Digital signalprocessing is primarily concerned with the processing of these sampledsignals.

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A mathematical representation of the sampled signal is shown below.

This is equivalent to modulating a train of delta functions by theanalog signal. The delta function effectively "filters" out the values of the signal at times corresponding to the zeros in the argument of thedelta function. This process is also referred to as "ideal" sampling sinceit results in sampled signals of "zero" width and whose spectrum isperfectly periodic.

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Digital Filter y In electronics, computer science and mathematics, a digital filter is a

system that performs mathematical operations on a sampled, discrete-time signal to reduce or enhance certain aspects of that signal.

y A digital filter system usually consists of an analog-to-digital converterto sample the input signal, followed by a microprocessor and someperipheral components such as memory to store data and filtercoefficients etc. Finally a digital-to-analog converter to complete theoutput stage. Program Instructions (software) running on themicroprocessor implement the digital filter by performing thenecessary mathematical operations on the numbers received from the

ADC.

y Digital filters are commonplace and an essential element of everyday electronics such as radios, cellphones, and stereo receivers.

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aracterization o igita i ters

y A digital filter is characterized by its transfer function, or equivalently, its

difference equation.Mathematical analysis of the transfer function candescribe how it will respond to specifications any input. As such, designinga filter consists of developing appropriate to the problem (for example, asecond-order low pass filter with a specific cut-off frequency), and thenproducing a transfer function which meets the specifications.

y The transfer function for a linear, time-invariant, digital filter can be

expressed as a transfer function in the Z -domain; if it is causal, then it hasthe form:

y where the order of the filter is the greater of N or M .

y This is the form for a recursive filter with both the inputs (Numerator) and outputs(Denominator), which typically leads to an IIR infinite impulse response behaviour,but if the denominator is made equal to unity i.e. no feedback, then this becomesan FIR or finite impulse response filter.

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Difference equation y In discrete-time systems, the digital filter is often implemented by converting

the transfer function to a linear constant-coefficient difference equation

(LCCD) via the Z-transform. The discrete frequency-domain transfer functionis written as the ratio of two polynomials. For example:

y This is expanded:

and divided by the highest order of z :

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y The coefficients of the denominator, ak , are the 'feed-backward' coefficients

and the coefficients of the numerator are the 'feed-forward' coefficients, bk . Theresultant linear difference equation is:

or, for the example above:

rearranging terms

then by taking the inverse z -transform:

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y This equation shows how to compute the next output sample, y[n], interms of the past outputs, y[n p], the present input, x [n], and the pastinputs, x [n p]. Applying the filter to an input in this form isequivalent to a Direct Form I or II realization, depending on the exactorder of evaluation.

and finally, by solving for y [n]:

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Implementatio n

Direct Fo rm 1y The most straightforward implementation is the Direct Form 1, which

has the following difference equation:

y Here the b0, b1 and b2 coefficients determine zeros, and a1, a2 determine theposition of the poles.

y Flow graph of biquad filter in Direct Form 1:

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Digital signal processing (DSP) is concerned with

the representation of signals by a sequence of numbers

or symbols and the processing of these signals.GOA LS OF DSP:

1. to measure, filter and/or compress continuous real- world analog signals

y convert the signal from an analog to a digital form using ananalog-to-digital converter

y the required output signal is another analog output signal, which requires a digital-to-analog converter (DAC)

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2.other advantages:

error detection and correction in transmission data &

compression of data.

DSP logirithm standard computers using digital signalprocessors or purpose built hardwares.

Some more powerful general purpose

1. microprocessors

2. field-programmable gate arrays (FPGAs)

3. digital signal controllers (mostly for industrial appssuch as motor control)

4.stream processors

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DSP DOM AIN:

engineers usually study digital signals in one of the

following domains: time domain (one-dimensionalsignals), spatial domain (multidimensional signals),frequency domain and wavelet domains.

TIME AND SPACE DOM AIN:

The most common processing approach in the time orspace domain is enhancement of the input signalthrough a method called filtering. Digital filteringgenerally consists of some linear transformation of a

number of surrounding samples around the currentsample of the input or output signal

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FR EQUENCY DOM AIN:

Signals are converted from time or space domain to the

frequency domain usually through the Fouriertransform. The Fourier transform converts the signalinformation to a magnitude and phase component of each frequency.

Z-DOM AIN ANA L Y SIS:y analog filters are usually analysed on the s-plane;

digital filters are analysed on the z-plane or z-domainin terms of z-transforms

y Most filters can be described in Z-domain (a complexnumber superset of the frequency domain) by theirtransfer functions. A filter may be analysed in the z-domain by its characteristic collection of zeroes and

poles.

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V ARIOUS SUBFIELDS:

1. audio and speech signal processing

2. sonar and radar signal processing3. sensor array processing

4. digital image processing

5. signal processing for communications

6. control of systems

7. seismic data processing

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