Saint-Petersburg State University

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Nov 29, 2013 (3 years and 8 months ago)

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AM
-
CP Faculty of SPbU




1

Saint
-
Petersburg State University




Applied

Mathematics



Control

Processes

Faculty




Professor

Evgeny

I
.
Veremey
,

Senior Lecturer Margarita V. Sotnikova




DIGITAL SYSTEMS

OF CONTROL

AND SIGNAL PROCES
S
ING


Lectures

Notes








Saint Peresburg

20
11

AM
-
CP Faculty of SPbU




2

I
n t r o d u c t i o n



1.
General preface



There are three main educational directions in our faculty
: A
pplied
mathematics and informatics,
A
pplied mathematics and physics
and I
nfo
r-
m
a
tion technologies. These directions are presented by the six teaching pr
o-
grams:
three four
-
year bachelor programs, two two
-
year master pr
o
grams and
one five
-
year specialist program.

I would like to propose to your attention some simplified variant of the
course “Introduction to digital systems” which is included to the master teac
h-
ing

program with direction Information technology.

This program

first of all

is oriented to the bachelors of information tec
h-
nologies

and the similar directions. We assume that they have got quite enough
training as a system and applied programmers and one o
f the main goals of
theirs master teaching is to
intensify their professional potential in the area of
formalized mathematical methods and numerical algorithms.



We are going to include the course Introduction to digital systems
also
to
the
joined master
program in the range of CBU
-
cooperation.



This course is intended to provide a modern treatment of digital signals
and systems at an introductory level. As such, it has the purpose to prepare st
u-
dents for upper
-
level courses in digital signal processing,

co
n
trol theory and
communication systems.


The goal of the course is to study fundamental ideas of the math
e
matical
modeling, analysis and synthesis of the information systems built on the base
of the modern digital devices.

In the center of attention is

the system models representation both in
time and frequency domain, the wide spectrum of modern approaches to its
stability and performance features estimation and the popular optim
i
zation
methods of its synthesis.

I am going to illustrate the base
points

of the course by the numerical e
x-
amples and to give possibility to do some exercises on the lessons with the help
of MATLAB package.




AM
-
CP Faculty of SPbU




3

2
.

The main concepts of digital system’s theory


The area of digital signals and systems
presents

one of the most powe
rful
technologies that will shape science and engineering in the twenty
-
first ce
n
t
u-
ry. Revolutionary changes have already been made in a
broad

range of fields:
communications, medical imaging, radar & sonar, high fidelity music repr
o-
duction, oil prospectin
g

and so on
.

Each of these areas has developed a
correspondent digital

technology,
with its own algorithms, mathematics, and specialized techniques.
To genera
l-
ize these achievements a theory of digital signals and systems was created.

This branch of know
ledge
is distinguished from other areas in co
m
puter
science by the unique type of data it uses:
signals
. In most cases, these signals
originate as sensory data from the real world: seismic vibr
a
tions, visual images,
sound waves, etc.

D
igital processing

on
the base of modern computers

starts

after
the signal
s

have been converted into a dig
i
tal form. This includes a wide
variety of
applications
, such as: enhancement of visual images, recogn
i
tion and
generation of speech, compression of data for storage and tr
ansmi
s
sion, etc.

One of the most significant application
s

of the digital theory
is oriented
to the various control processes and control systems due to an ou
t
standing fle
x-
ibility

and effectiveness

of digital technologies.

The presentation following below

is based on the fundamental conce
p-
tions of a signal and system.

A signal is formally defined as a function of one or more variables which
contains
some
information for the subject (person or system) using this si
g
nal.

A system is formally defined as an e
ntity that manipulates one or more
input signals to transform it to output signals extracting or injecting certain i
n-
form
a
tion.

{The wide using of the digital processing has been determined by the
personal computer revolution of the 1980s and 1990s.}

First
ly let

us consider one

practical example.

Let we have a cart with a mass
m
, moving on the rails (fig. 1) with the
friction coefficient
p
.
The cart is attached to the fixed ground by a spring with
the expansion
-
pr
essure coefficient
k
.

Let present the current position of the
cart by its displacement in relation to the fixed ground d
e
noting it as
)
(
t
y
.


Fig
.

1.
The car fixed by the spring
.

AM
-
CP Faculty of SPbU




4

Let
firstly
the cart is placed in the zero equilibrium position,
i.e.
0
,
0


dt
dy
y
.

Let we shift

the cart to right and then r
e
lease it. The cart
starts its motion at zero initial instant of time from the initial point
0
)
0
(
y
y


to return to eq
uilibrium position under the influence of the spring, and after
some oscillations the cart will stop its m
o
tion in the zero point.

The
dynamic

of the cart can be described by the linear differential equ
a-
tion in the form


0
2
2



ky
dt
dy
p
dt
y
d
m

(1)

on the
base of the mechanics lows. It is known that the particular solution of
this equation is presented by the expression




t
c
t
c
e
t
y
t






sin
cos
)
(
2
1
,

(2)

which corresponds to the mentioned motion. Here the numerical parameters

,

,
1
c

и

2
c

are uniquely

defined by the values of the coefficients
m
,
p
,
k

and by the initial displacement

0
y
.
The example of the curve
)
(
t
y

is presented
on the fig. 2
.

0
2
4
6
8
10
-1
-0.5
0
0.5
1

Fig
. 2.
Graph

of the equation (1) particular solution
.


Remark that it was supposed there are no external forces which influence
to
the cart’s

motion
.
This is reflected by the zero right part of the equation

(1).
Neverthel
ess the external forces should be always taken into account that leads
to the inhomogeneous equation



u
t
F
ky
dt
dy
p
dt
y
d
m




)
(
2
2
.

(3)

AM
-
CP Faculty of SPbU




5

This equation has two additional terms in the right part

which are dete
r-
mined by the external forces

(fig. 3). The first ter
m
)
(
t
F

is uncontrollable a
c-
tion of the external environment (maybe wind, defects of
the
rails etc.).

In co
n-
trary t
he second term
u

is

the control action
.



Fig
. 3.
The cart with external fo
rces
.


Let assume that the force
)
(
t
F

is negative for the process, because it d
e-
flects the cart from the desirable motion



zero equilibrium point
. In contrary,
the control force
u

is aimed to compensate undesirable

)
(
t
F

a
c
tion
.

The control force can be physically realized by the special engine
i
n-
stalled on the cart, but there are several ways
of the choice the value
u

as a
function of one o more variables.

This choice
has

p
rinciple

significance
because it determines the strategy
and tactics
of the control action to the cart.

Let consider tree admissible ways to form the control
u
:



on the base of the direct information about
external force
)
(
t
F
, for
example as
)
(
t
F
u


;



on the base of the information about the motion parameters, for exa
m-
ple as
ky
u


;



at last, on the base of the only information about the currant time as
)
(
t
f
u
u

.

The
first way is used very seldom: it needs to have very expensive sp
e-
cial sensor to measure the value of the force
)
(
t
F

for each instant of time. The
third way
is used more frequently, but it is not quite flexible and effective b
e-
cause of t
he force
)
(
t
F

is unknown in advance.

The second way is most popular and widely used in practice. It corr
e-
sponds to the feedback control principle and provides the best results in control
process. Naturally, it is necessary to have the se
nsor on the cart for its impl
e-
mentation to measure current deflection
)
(
t
y
.

So, the mentioned feedback control scheme is based on three steps
which
must be sequentially done during the process:



measuring of the current deflection
)
(
t
y

of the cart;

AM
-
CP Faculty of SPbU




6



processing of the obtained signal;



final
formation of the control force
)
(
t
u
.


The first step can be done by the sensor, the second


by the analog or
digital calculator and the third


by the engine
-
actuator.


Special
ly

remark, that usually the sensors and the actuators have an an
a-
log nature
, so they work in real continuous time.
If it is also used an analog
calculator for data processing, we have fully analog control system which is
not the
o
bject of our inte
rest.

Actually, the most advanced situation is connected with the digital a
p-
proach to data processing with the help of modern computer. Here we
have the
mixed analog
-
digital system

with digital processing of the mea
s
ured analog d
a-
ta. But digital processing

is fully based on the digital pre
s
entation of an input
and output information.

So it is necessary to expand the mentioned control scheme by the follo
w-
ing steps:



analog
-
digital
conversion

of a measured signal;



reverse digital
-
analog
convers
tion of the comp
uter’s output.


In general case if
we have any mixed information system (including co
n-
trol systems) with digital processing of analog signals
we

usually use the
sta
n
dard scheme of the signal conversion presented on the fig. 4.


This scheme
consists of

thre
e steps: coding of the initial analog sy
s
tem,
digital processing and encoding of its result.


The coding step includes two operations: preliminary filtering of the si
g-
nal and its discretization.
The filtering transforms initial analog signal
)
(
t
y

i
n-
to other analog signal
)
(
~
t
y

to
remove

the high
-
frequency component dete
r-
mined by the sensor noise.

Discretization or sampling allows to construct the
digital representation of the analog signal.


AM
-
CP Faculty of SPbU




7






)
(
t
y







)
(
~
t
y

Coding








]
[
n
y




Digital Processing



]
[
n
u







)
(
~
t
u

Encoding








)
(
t
u

Preliminary

Analog

Processing

(
Analog Filter
)

Analog
-
to
-
Digital

Conversion

(
А
/D
)

Digital


Transformation


(
Digital Processor
)

Digital
-
to
-
Analog

Conversion

(
D/A
)

Smoothing

of
Processed Signal

(
Analog Filter
)



Fig
. 4
. The general scheme of the analog signals digital processing

for the mixed information system.



After discretization the lattice function (numerical sequence)
]}
[
{
n
y
y


corresponds to the analog signal
)
(
~
t
y
. Every e
lement
]
[
n
y

of the
sequence is called
a sample
and the integer
n

is called a number of the sample
or a discrete instant of time.

AM
-
CP Faculty of SPbU




8

If discretization is done fore the equalyspaced instants of the continuous
time with t
he step
T
,
then the samples
]
[
n
y

are defined as


)
(
~
]
[
nT
n
y
y

,
,...
2
,
1
,
0

n


An example of the coding step for controlled cart is presented on the
fig.

5.

Here you can see three curves: initial

analog signal
)
(
t
y
, the result of
preliminary filtering
)
(
~
t
y

and the result of the sampling


digital signal
]}
[
{
n
y
y


presented by its samples.


-1.5
-1
-0.5
0
0.5
1
1.5
-1.5
-1
-0.5
0
0.5
1
1.5
0
5
10
15
20
25
30
-1.5
-1
-0.5
0
0.5
1
1.5
t(s)


Fig
. 5.
The curves of the cart’s displacement: initial
analog variant
)
(
t
y
, result
of filtering
)
(
~
t
y

and final discrete samples
)
(
~
]
[
nT
y
n
y

.

AM
-
CP Faculty of SPbU




9

-1.5
-1
-0.5
0
0.5
1
1.5
-1.5
-1
-0.5
0
0.5
1
1.5
0
5
10
15
20
25
30
-1.5
-1
-0.5
0
0.5
1
1.5
t(s)


Fig
. 6.
The curves for the control signal
:
initial digital samples
]
[
n
u
,

result of the

digital
-
analog

conversion

)
(
~
t
u

and final analog result

of smoothing

)
(
t
u
.


After that the digital representation
]}
[
{
n
y
y


of the analog signal

)
(
t
u

is constructed, it is accepted as input informatio
n for the correspondent
digital device to be
transformed



this is the main step of digital processing
.
The result of digital processing, which is done with fully defined purpose, is a
new numerical sequence
]}
[
{
n
u
u

. This sequence uniquely c
orresponds to
the sequence
]}
[
{
n
y
y

:

AM
-
CP Faculty of SPbU




10


]}
[
{
n
y
y


]}
[
{
n
u
u

.


d
igital processing



The last step of the considered scheme is
encoding
, i.e. the reverse
co
n-
version of the digital sequence

]}
[
{
n
u
u


to the analog signal
)
(
t
u
,
whi
ch is
an output of the system. As an intermediate step here the smoothing is used to
provide continuous differentiability of the output signal:


]}
[
{
n
u
u


)
(
t
u
u

.



encoding



On the fig. 6 three signals are presented as example of the encoding step.


3.

The
using of MATLAB package in the course


To increase the efficiency of this course we have decided to illustrate
some main positions with the help of appropriate algorithmic and programming
support. It has been chosen MATLAB


the main product of The MathWor
ks,
Inc.


as the most suitable and convenient instrumental environment.

The name MATLAB is acronym for MATrix LABoratory, since whose
basic data element is
matrix
. This system was originally written to provide easy
access to linear algebra software, crea
ted in New Mexico and Stanford unive
r-
sities. Today, MATLAB engines incorporate the LAPACK and BLAS libraries
as the most efficient packages of numerical analysis.

Now MATLAB system is one of the most popular and wide
-
spread

re
p-
resentatives of
science
-
consu
ming

modern computer technologies. This pac
k-
age of the program instruments is widely used both in industry and in univers
i-
ties for research work and as the basis element for teaching.


MATLAB

environment consists of the following main parts
:

1. Desktop Too
ls and Development Environment

This is the set of interactive tools for the system control. In most cases
these tools are presented by the graphical user interfaces. There are the
MA
T
LAB desktop, Command Window, an editor and debugger, various
browsers for

viewing help, objects and the workspace. The main tool is Co
m-
mand Wi
n
dow as a basis instrument which is constantly present on desktop.

2. The MATLAB Mathematical Function Library

This is a
rich

collection of computational algorithms ranging from el
e-
mentar
y functions, like sum, sine, cosine, and complex arithmetic, to more s
o-
AM
-
CP Faculty of SPbU




11

phisticated functions like matrix inverse, matrix eigenvalues, Bessel functions,
and fast Fourier transforms.

3. The MATLAB Language

This is a high
-
level matrix/array language with cont
rol flow statements,
functions, data structures, input/output, and object
-
oriented programming fe
a-
tures. It allows both “programming in the small” to rapidly create quick and
draft

programs, and “programming in the large” to create large and complex
applic
ation programs.

4. Graphics

MATLAB has extensive facilities for displaying vectors and matrices as
graphs, as well as annotating and printing these graphs. It includes high
-
level
functions for two
-
dimensional and three
-
dimensional data visualization, image

processing, animation, and presentation graphics. It also includes low
-
level
functions that allow you to fully customize the appearance of graphics as well
as to build complete graphical user interfaces on your MATLAB applications.

5. MATLAB External Inte
rfaces

This is a library that allows you to write C and Fortran programs that i
n-
teract with MATLAB. It includes facilities for calling routines from MATLAB
(dynamic linking), calling MATLAB as a computational engine, and for rea
d-
ing and writing MAT
-
files.

6. MATLAB Toolboxes

As it was said above, toolboxes are the problem
-
oriented collections of
the functions and user instrument which allows to expand the base MATLAB
possibilities. First of all we shall use Control System Toolbox
to do some e
x-
amples and tra
ining exercises.

7. Simulink Modelling Subsystem

This is a collection of
graphical uses instruments jointed to integrated
environment for computer modeling and simulation of nonlinear dynamical
systems.

MATLAB
can work both in command line and in programm
ing regime.

System is always ready to receive users commands after the
prompt

“>>” a
p-
pears in the command line. In this regime MATLAB can be treated as
e
x
trem
e-
ly large and powerful calculator with wide spectrum of possibilities.

To work in programming reg
ime you need to create your own folder and
to set it in
Current Directory

window.

After that it is necessary to start system editor with the help of “New M
-
file” button if you are going to create new program or with the help of menu
“File” (position “Open
”) if your program always exists. To run the program,
press F5 button. All the results are accessible in command window.


AM
-
CP Faculty of SPbU




12

PART 1

MATHEMATICAL MODELS of DIGITAL SYSTEMS


In this part of course we shall consider the main questions of the math
e-
matical repres
entation of the digital systems. It is obvious that mathematical
models in theirs various forms are the basis both for analysis, including co
m-
puter modeling, and for synthesis of these systems.

In further presentation we shall suppose that the digital and
discrete on
time systems are the same. Strictly speaking it is not quite so, because of the
digital systems are discrete not only on time but also on the levels of the pr
o-
c
essed signals. Nevertheless, we shall not take into account here the level di
s-
cretiz
ation supposing that the digits number for the used computer is suff
i
cien
t-
ly large.

Naturally, the main attention should be paid to the linear systems and
theirs models both in time and frequency domains.

1. Linear transformations of digital signals

Let co
nsider some system (fig. 1), which works in discrete time
1
N
n


and transforms input into output information in accordance with certain given
algorithm.


]}
[
{
n
u


{
]
[
n
y
}



H
n
{

}


Fig. 1. Digital system with discrete input and output sig
nals.


Let an input signal be presented by the
lattice

function or numerical s
e-
quence


]}
[
{
n
u
u

,

(1)

where
n

is the
sample number
, given by the integers from the range
)
,
(





let denote it by
1
N
n

.


The sequence (1) can be presented by the graph of the lattice function
with elements
]
[
n
u

as the samples.

Example 1.

For example fig. 2 represents mentioned graph for the sequence with e
l-
ements







.
12
,
0
,
12
,
]
[
2
.
0
n
if
n
if
e
n
u
n

AM
-
CP Faculty of SPbU




13


-15
-10
-5
0
5
10
15
0
0.2
0.4
0.6
0.8
1
n
u[n]


n


Fig
. 2.
An example of graphical representation of the sequence
]}
[
{
n
u
u

.


Exercise 1.

Use MATLAB to generate the graph of the sequence
]}
[
{
n
u
u


with the samples















.
8
0
,
6
.
0
sin
,
0
8
32
.
0
1
.
0
8
,
0
]
[
n
if
n
n
if
n
n
if
n
u



on the discrete time segmen
t
]
10
,
10
[


n
.

Let us also consider the signals
]}
[
{
n
y
y


of the same type which are
formed on the output of the system according to given algorithm. This alg
o-
rithm can be treated mathematically as an operator
n
H
, realized by the system,
that takes an input sequence
]}
[
{
n
u
u


and uniquely transforms it into output
sequence
]}
[
{
n
y
y

.

If an operator
n
H

is given, it means that it is known the rule, which a
l-
lows t
o calculate any element
]
[
n
y

of the sequence
y

by means of certain
(possibly


all) elements of the sequence
u
:




]
[
]}
[
{
n
u
H
n
y
n

.

(2)

It is naturally to say in this sense that output si
gnal
]}
[
{
n
y
y


is
sy
s-
tem response
to the input signal
]}
[
{
n
u
u

.


Let give some examples of an operator
n
H

definition:

AM
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CP Faculty of SPbU




14





].
1
[
2
]
2
[
4
]
[
5
.
0
]
[
)
;
]
2
[
5
])
1
[
(
3
])
[
(
2
]
[
)
;
])
[
(
2
]
[
)
2
3
2
.
0
2












n
y
n
u
n
u
n
y
c
n
u
n
u
n
u
e
n
y
b
n
u
n
y
a
n
;


Exercise 2.

Use MATLAB to construct on the time segment
]
15
,
15
[


n

the graph of the input signal
]}
[
{
n
u
u


with the samples









,
12
,
0
,
12
,
]
[
2
.
0
n
if
n
if
e
n
u
n


and the graph of the output signal with the samples


]
1
[
])
[
(
]
[
2



n
u
n
u
n
y
.

It is easy to see, that system operator can be both linear an
d nonlinear.


Definition 1.
Digital system is said to be
linear
, if an operator
n
H

is
linear operator
, i.e. if it has
homogeneous

and
additive

properties: for any
1
R



and for any input sequences
u
,
1
u
,
2
u

the following equalities

hols:






]
[
]
[
n
u
H
n
u
H
n
n



;








]
[
]
[
]
[
]
[
2
1
2
1
n
u
H
n
u
H
n
u
n
u
H
n
n
n





Definition 2.
Digital system is said to be
time invariant
, if an operator
n
H

is invariant with res
pect to a shift of time. By other words, if





]
[
]
[
n
u
H
n
y
n

,


then the system is called time invariant if and only if




]
[
]
[
p
n
y
p
n
u
H
n




for any
1
N
p

.

(3)

In this way any time shift for the input signal leads to the same
time shift
for its response, if the system is time invariant.


We will use the acronym
DLTI
bellow for
Digital Linear Time Invar
i-
ant
s
ystems. It is necessary to remark, that vast majority of discrete systems
AM
-
CP Faculty of SPbU




15

used in practice are DLTI systems. Therefore we
will pay central attention to
this class of systems in our course.

From the formal point of view, to specify DLTI system it is necessary to
define an operator
n
H
, which is realized by the system. So
this operator
, gi
v-
en in any mathemat
ical form,
can be treated as mathematical model

of DLTI
system.

There are a lot of various forms of models to define an operator
n
H
. One
of widely used forms of mathematical representation is so
-
called
impulse r
e-
sponse
.


To introduce th
is model let at first consider one of the elementary di
s-
crete signals
]}
[
{
n




known as
unit discrete impulse.

This

is

the

sequence

with

elements









.
0
,
0
,
0
1
]
[
n
if
n
if
n

(4)

Accordingly it is possible to define
unit shifted impulse
]
[
k
n


,
1
N
k

:










.
,
0
,
1
]
[
k
n
if
k
n
if
k
n


It is easy to see that any sequence
]}
[
{
n
u
u


can be represented as a
sum of unit shifted impulses:









k
k
n
k
u
n
u
]
[
]
[
]
[
,

(5)

that is very convenient for the analysis o
f linear system output with an arbitrary
input signal
u
.

Let consider the linear system response to discrete signal
]}
[
{
n
u
u


with the samples presented in the form (5). With this purpose firstly introduce
special notati
on


]
[
n
h
h
k
k


for the response to unit shifted i
m-
pulse
]
[
k
n


:




]
[
]}
[
{
k
n
H
n
h
n
k



.

(6)

Remark that sequence
k
h

depends on instant
k

of the unit impulse appearance,
i.e. for every

addend

in the sum (5) system has its own response in the form of
correspondent sequence. So, taking into account an operator
k
H

linearity
property, obtain

AM
-
CP Faculty of SPbU




16



























k
n
k
n
n
k
n
H
k
u
k
n
k
u
H
n
u
H
n
y
]
[
]
[
]
[
]
[
]
[
]}
[
{
,


from where it follows on the base of (6)


.
]
[
]
[
]
[





k
k
n
h
k
u
n
y

(7)

It is necessary to say that formula (7) is not practically convenient to find
response
]}
[
{
n
y
y


because it requires the sequences


]
[
n
h
h
k
k


for every
instant
k
. So in general case equali
ty (7) is used only in theoretical researches.

Nevertheless, if mentioned system is not only linear, but is also time i
n-
variant, we have essentially different situation.

Suppose that
n
H

is an operator of DLTI system. Let find response


]
[
n
h
h


of this system to non
-
shifted unit impulse


]
[
n



:




]
[
]}
[
{
n
H
n
h
n


.

(8)

The sequence


]
[
n
h
h


is said to be
impulse (weight, transient) response
of
linear time
-
invariant system.

Note that it
is possible that there exists an integer
0

N

such that
:
n


N
n


0
]
[

n
h
. Than we say that DLTI system has
Finite Impulse R
e-
sponse

and it is called
FIR

system.

On the contrary, if i
mpulse response has infinite number of non
-
zero e
l-
ements we say that DLTI system has
Infinite Impulse Response

and it is called
IIR

system.

Now, let find a response
k
h

of DLTI system to unit shifted impulse
:





]}
[
{
]
[
]}
[
{
k
n
h
k
n
H
n
h
n
k





,

(9)

using (8) and taking into account time invariant property (3). Thus it is very
easy to obtain system reaction to every shifted impulse: it is sufficient to shift
impulse response to
k

samples along the time axis.

Using (9), it is
possible to get system response
]}
[
{
n
y
y


to any input
signal
]}
[
{
n
u
u


for DLTI systems on the base of the formula (7):








k
k
n
h
k
u
n
y
]
[
]
[
]
[

(10)

AM
-
CP Faculty of SPbU




17



this is very important expression for the theory and practice of digital si
gnal
processing and control theory.

Remark, that we can obtain response
y

in accordance with (10) if we
have only one (in contrast with (7)) additional sequence


]
[
n
h
h




system
impulse response.

In this sense, equat
ion (10) can be treated as
mathematical model
on the
base of
impulse response
for DLTI system.

This equation can be written in
other widely used equivalent form.

Definition

3.
If the sequences
]}
[
{
n
u
u


and
]}
[
{
n
y
y


are co
n
nec
t-
ed by the formula (10), then sequence
y

is known as
discrete convolution

of
the sequences
u

and
h



it is denoted as follows


]
[
]
[
]
[
n
h
n
u
n
y


.


One of the most important properties o
f the convolution is
commutativity
, r
e-
flected by the equality


]
[
]
[
]
[
]
[
n
u
n
h
n
h
n
u





or


]
[
]
[
]
[
]
[
k
n
u
k
h
k
n
h
k
u
k
k











.

(11)

This

equality

easy

to

prove
:
let

denote
k
n
p


,
i
.
e
.
p
n
k


,

replacing
k

in (11
),

we obtain


]
[
]
[
]
[
]
[
p
h
p
n
u
k
n
h
k
u
p
k











.


Then replacing
p

by
k

in the right part, we get (11).

Thus, on the base of
commutativity
, formula (10) can be presented as


]
[
]
[
]
[
]
[
]
[
n
u
n
h
n
h
n
u
n
y




.

(12)

In conclusion
,

let

us consider a practical algorithm to compute the
n
-
th
sample of the output sequence with the help of the formula (10). It is easy to
see that to find the sample
]
[
n
y

it is firstly necessary to multiply correspo
n
d-
en
t elements of the sequences
]}
[
{
k
u
u


and


]
[
k
n
h
h


, where
1
N
k

,
and
n

is considered as a fixed parameter. Then

the

products

]
[
]
[
k
n
h
k
u


are

summarized

for

all

1
N
k

.

AM
-
CP Faculty of SPbU




18

Remark, that to find the element
]
[
n
y

all the samples of the mentioned
sequences are used. The first of them is fixed, but the second is changed for
every
n
. Nevertheless it is very easy to find these chan
ged sequences


]
[
k
n
h
h



on the base of impulse response


]
[
k
h
h

, which is given.

Really, note that
]
[
]
[
n
k
h
k
n
h




, so if we have the auxiliary s
e-
quence


]
[
1
k
h
h


, than the resulting sequence


]
[
k
n
h
h



can be
formed by the shift of
1
h

to
n

samples along the time axis. But the s
e
quence
1
h

is the simple reflection of the response


]
[
k
h
h


in relation to origin.

So
it is necessary to do only two steps to find the sequence


]
[
k
n
h
h



for the fixed value
n
: reflection in discrete time of the r
e-
sponse
h

and shifting the result to
n

samples al
ong the
k

axis.


Example 2.
Figure 3 illustrates the steps for DLTI with the FIR.

0
0.5
1) impulse responce h(k)
0
0.5
2) impulse responce h(-k)
-10
-8
-6
-4
-2
0
2
4
6
8
10
0
0.5
3) impulse responce h(-k+n), n=3
k

Fig
. 3.
Two main steps to compute the convolution (10) for
3

n
.


In conclusion let remark that an impulse response has a special
signif
i-
cance since it can be easy find experimentally. Really, it is sufficient to give
the unit impulse to system’s input, and the correspondent output signal presents

the sequence
]}
[
{
n
h
h

.


AM
-
CP Faculty of SPbU




19

Exercise 3.

Calculate system response to the inp
ut signal
]}
[
{
n
u
u


on
the time segment
]
22
,
0
[

n

with the samples











.
8
,
0
,
0
,
8
0
,
]
[
2
.
0
n
n
if
n
if
e
n
u
n


Use MATLAB function
conv

to find convolution of two finite s
e
quences
]}
[
{
n
u
u


and
]}
[
{
n
h
h

. Impulse respo
nse for the system is given as


]}
[
{
n
h
h

={1
-
1 2 3
-
2
-
1
-
0.5 2 3
-
1}


for the instants of digital time


n
={0 1 2 3 4 5 6 7 8 9}.


2. Mathematical Models of DLTI systems

on the base of difference equations


If mathematical model i
s formed by the theoretical way, it often is repr
e-
sented by the systems of difference equations. In this case difference equations
are considered as primary model, but impulse response


as secondary model
of DLTI system.


Definition 1.
The sequence
]}
[
{
n
y
y




with elements


]
1
[
]
[
]
[




n
y
n
y
n
y
,
1
N
n


(1)

is said to be the
first backward finite difference

with
respect

to the sequence
]}
[
{
n
y
y

.

Finite difference
]}
[
{
n
y
y




is an analogue of the

first derivative for
the function
)
(
t
y

of continuous time.

In accordance with definition it is possible to introduce the second finite
difference and so on:


]
2
[
]
1
[
]
1
[
]
[
]
1
[
]
[
]
[
2













n
y
n
y
n
y
n
y
n
y
n
y
n
y


or


]
2
[
]
1
[
2
]
[
]
[
2






n
y
n
y
n
y
n
y
;


]
1
[
]
[
]
[
1
1








n
y
n
y
n
y
N
N
N
, …

AM
-
CP Faculty of SPbU




20

It is easy to see, that
n
-
th sample for difference
y
N


is uniquely d
e-
fines by the samples of the sequence
]}
[
{
n
y
y


with the numbers
n
,
1

n
,
… ,
N
n

. So it depends only on the current and previous instants of discrete
time and does not depend on the future time.


Definition

2.
The

equality




0
]
[
...,
],
[
],
[
],
[
,
2




n
y
n
y
n
y
n
y
n
F
N
,

(2)

is called to be the
ordinary difference equation of the
N
-
th order
. It connects
n
-
th samples of the sequence
y

and its finite differences including difference
of the
N
-
th order. Any sequence (lattice function)
]}
[
{
n
y
y

, s
atisfying (2)
for all
1
N
n


is called a solution of
difference equation
.


Note, that difference equation for continuous time can be treated as an
analogue of differential equation for discrete time, but it has essential uniqu
e-
ness: there

is no problem to obtain numerical solution of this equation.

Really, if all the finite differences in (2) are presented on the base of cu
r-
rent and previous samples of
y
, then difference equation can be written as




0
]
[
...,
],
2
[
],
1
[
],
[
,
1




N
n
y
n
y
n
y
n
y
n
F
.

(3)

It is said to be the
regular case

if it is possible to solve equation (3) with r
e-
gard to the sample
]
[
n
y
:




]
[
...,
],
2
[
],
1
[
,
]
[
2
N
n
y
n
y
n
y
n
F
n
y




.

(4)

Thus equality (4) can be treated as a
recurrence formula
, which allows to find
the samples
]
[
n
y
,
]
1
[

n
y
,
]
2
[

n
y
,…, if the previous
N

samples are
known. Note, that the first
N
samples for this recurrence process should be
given as initial conditions.

It
is obvious, that linear difference equations can be used to model DLTI
systems. The initial form of linear difference equation linearly connects the
current end previous samples of input and output signals and their finite diffe
r-
ences. If reduce linear equ
ation to the form (4), we get


].
[
...
]
1
[
]
[
]
[
...
]
1
[
]
[
1
0
1
M
n
u
b
n
u
b
n
u
b
N
n
y
a
n
y
a
n
y
M
N












(5)

The difference equation in the form (5) is widely used in theory and
practice as mathematical model of SISO DLTI system in time domain.

Often equation (5) is called a
model of digital filter
, ta
king into account,
that the formula

AM
-
CP Faculty of SPbU




21











M
k
k
N
k
k
k
n
u
b
k
n
y
a
n
y
0
1
]
[
]
[
]
[
,


follows from (5), unequally determines digital algorithm of input and output
signals processing on the base of their current and previous samples.

By analogy with the random processes theory, t
he equation (5) is said to
be the model of
autoregression with moving a
v
erage (ARMA model)
.

If signal processing algorithm does not use the samples of input on the
previous instants, i.e. if
0

M

and equation has the form



],
[
]
[
...
]
1
[
]
[
0
1
n
u
b
N
n
y
a
n
y
a
n
y
N







(6)

the system is said to be presented by the
autoregressive
or

AR model
.

If, vise versa, previous output samples are not used, i.e. if
0

N
, then
few say that filter equ
a
tion



]
[
...
]
1
[
]
[
]
[
1
0
M
n
u
b
n
u
b
n
u
b
n
y
M







(7)

is
moving av
erage
or
MA model

of digital system.

Besides that the filters with AR or ARMA model are called
recursive fi
l-
ters
, underlining dependence of the current value of the output from the prev
i-
ous samples of input and output. Unlike that, MA filters are said to b
e
non
-
recursive

ones.

If we have a mathematical model in the form of difference equation (5),
it is very easy to calculate an impulse response of the DLTI system. Actually,
let us consider the unit impulse as an input signal
]}
[
{
]}
[
{
n
n
u
u



, i.e.
1
]
0
[

u
,
0
]
[

m
u

for

any

0

m
.
Also

let

the

initial

conditions

be

zero
:
0
]
[

m
y

for any

0

m
.

Then, in accordance with (5), we have the signal
h
y


wi
th the sa
m
ples


0
]
[

n
h
,
if

0

n
,


0
]
0
[
b
h

,


1
1
]
0
[
]
1
[
b
h
a
h





]
0
[
]
1
[
1
1
h
a
b
h


,


2
2
1
]
0
[
]
1
[
]
2
[
b
h
a
h
a
h






]
0
[
]
1
[
]
2
[
2
1
2
h
a
h
a
b
h



, …


M
N
b
N
M
h
a
M
h
a
M
h






]
[
...
]
1
[
]
[
1









}
,
min{
1
]
[
]
[
M
N
k
k
M
k
M
h
a
b
M
h
,

AM
-
CP Faculty of SPbU




22







}
,
min(
1
]
[
]
[
N
n
k
k
k
n
h
a
n
h
,
if

M
n

.

Remark that if an impulse response is obtained experimentally, then
mentioned expressions can be treated as the system of linear algebr
aic equation
in relation to unknown coefficients
i
a
,
j
b
,
N
i
,
1

,
M
j
,
1


for (5).

Let us pay special attention to the MA model (7) and calculate its output,
as a re
sponse
to the un
it impulse (i.e.
1
]
0
[

u
,
0
]
[

m
u

for any
0

m
):


0
]
[

n
h
,

if

0

n
,


0
]
0
[
b
h

,
1
]
1
[
b
h

,
2
]
2
[
b
h

, … ,
M
b
M
h

]
[
,
0
]
[

n
h
,
if


M
n

.


From here it follows very important conclusion: an impulse r
e
sponse of
MA digital filter is the finite sequence
0
]
0
[
b
h

,
1
]
1
[
b
h

, … ,
M
b
M
h

]
[
.

The r
everse statement is also correct:
if an impulse response of DLTI
system is presented by the finite sequence

with M samples,
then this system
has MA model (7)

with coefficients

]
0
[
0
h
b

,
]
1
[
1
h
b

, …,
]
[
M
h
b
M

.

O
n the base of the
last
statements
DLTI systems with MA models are
said to be FIR digital filters

or filters with
Finite Impulse R
e
sponse
.

On the contrary, the
systems with AR or ARMA models are called IIR
digital filters

or filters with
Infinite Impulse Re
sponse
.

Let us consider now very significant question about a conversion of the
N
-
th order difference equation (3) to the system of first order equations solved
for the first derivatives.

As it is known, if this equation can be presented in the recurrent f
orm
(4), it is possible to realize such a conversion.

At first let us illustrate this statement for the AR model equation (6),
from where it follows


]
[
]
[
]
1
[
...
]
2
[
]
1
[
]
[
0
1
2
1
n
u
b
N
n
y
a
N
n
y
a
n
y
a
n
y
a
n
y
N
N













.

(8)

In accordance with (8) let introduce
N

auxiliary variab
les
N
x
x
x
,...,
,
2
1

by the formulas


],
1
[
]
[
],
2
[
]
[
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
],
1
[
]
[
],
[
]
[
1
2
1










n
y
n
x
n
y
n
x
N
n
y
n
x
N
n
y
n
x
N
N

(9)

AM
-
CP Faculty of SPbU




23

therefore we have


].
[
]
1
[
],
1
[
]
1
[
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
],
1
[
]
1
[
1
1
n
y
n
x
n
y
n
x
N
n
y
n
x
N
N












Substituting

here

(9)
and
(8),
we obtain


].
[
]
[
...
]
[
]
1
[
],
[
]
1
[
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
],
[
]
1
[
0
1
1
1
2
1
n
u
b
N
x
a
n
x
a
n
x
n
x
n
x
n
x
n
x
N
N
N
N
N












(10)

Now
let

introduce the vector


T
2
1
]
[
...
]
[
]
[
]
[
n
x
n
x
n
x
n
N

x



the state
ve
ctor of DLTI system
, and

four matrices

























1
2
1
...
1
...
0
0
0
...
...
...
...
...
0
...
1
0
0
0
...
0
1
0
a
a
a
a
N
N
N
A
,

















0
0
...
0
0
b
b
,




1
2
1
...
a
a
a
a
N
N
N







c
,
0
b

d
.



This allows us to present the equations (10) in matrix form


].
[
]
[
]
[
],
[
]
[
]
1
[
n
u
n
n
n
u
n
n
d
cx
y
b
Ax
x







(11)

The equalities (11)
are said to be
state
-
space AR model of DLTI
.

Remark

that

in

general

case

for

the

model

(5)
if

N
M

it

is

possible

to

obtain

the

following

state
-
space

presentation


],
[
]
[
]
[
],
[
]
[
]
1
[
n
n
n
n
n
n
Du
Cx
y
Bu
Ax
x






(16)

where (for any fixed
n
)

N
E
n

]
[
x

is state vector,
M
E
n

]
[
u



input vector,
K
E
n

]
[
y



output vector.

AM
-
CP Faculty of SPbU




24

3.
Z
-
transformation

(
Laurent

transformation
)


It is convenient to use the analog of Laplace transformation for modeling
a
nd analysi
s of DLTI systems. This analog is known as the z
-
transformation.


Let consider the sequence of the complex numbers






n
f
n
f
f


]
[
,
,...
2
,
1
,
0



n
.

(1)

Definition

1
.
Z
-
transformation of the sequence

]}
[
{
n
f
f


is called
the following map








n
n
z
n
f
z
F
]
[
)
(
.

(2)

So the function
)
(
z
F

of the complex variable
1
C
j
z






uniquely co
r-
responds to the given sequence
f

and is called
z
-
transform

of this sequence,
that is denoted as


]}
[
{
)
(
n
f
Z
z
F

.

In accordance with the given definition function
)
(
z
F

is the sum of the
Laurent series with coefficients
]
[
n
f
. The region of convergence of thi
s series
is, as known, the ring
R
z
r



with the center in origin. The inner and outer
radius are determined by the features of the transformed sequence
f
. The
function
)
(
z
F

is an analytical function

in the region of convergence.

Let us find the
z
-
transforms for some significant sequences.



1.

Unit impulse
. This is the sequence with the samples









.
0
,
0
,
0
1
]
[
n
if
n
if
n


In

this

case

we

have


1
1
]
[
]}
[
{
)
(
0












z
z
n
n
Z
z
F
n
n
.

2.

Unit

ste
p
.
The elements of this sequence are given by the formula

AM
-
CP Faculty of SPbU




25








,
0
,
0
,
0
1
]
[
n
if
n
if
n
u
e


so we get















0
1
)
(
]
[
]}
[
{
)
(
n
n
n
n
e
z
z
n
n
u
Z
z
F
,


i
.
e
.
z
-
transform of the unit step is the sum of the geometric progression with
denominator
1


z
q
. If the condition
1
1



z
q

is hold, we get


1
1
1
1
1
)
(
)
(
1
0
1













z
z
z
q
z
z
F
n
n
.

So the region of convergence for this transform is determined by the inequality
1

z



it is the outer region of the unit disc on the complex plane.

3
.

Exponential sequence
or

geometric progression


]
[
]
[
n
u
A
n
f
e
n


,


where
A

and




are any complex numbers. In the case of
e


,
A



real
number, we have
real expo
nential sequence

with the transform





















0
1
0
)
(
]
[
]}
[
{
)
(
n
n
n
n
n
n
n
z
A
z
A
z
n
f
n
f
Z
z
F
.


So under the condition
1
1



z
, which determines a region of convergence


z
, we can use the formula for the sum of geometric progression:















z
Az
z
A
z
A
z
F
n
n
1
0
1
1
)
(
)
(
.


Let us point to the basic properties, which are directly follows from the
definition 1 of Laurent transformation.

A. Linearity.

If
]}
[
{
)
(
1
1
n
f
Z
z
F


and
]}
[
{
)
(
2
2
n
f
Z
z
F

, then




)
(
)
(
]
[
]
[
2
2
1
1
2
2
1
1
z
F
z
F
n
f
n
f
Z







,
1
2
1
,
C




,

i.e.
the Laurent transformation is
a linear operator
.


AM
-
CP Faculty of SPbU




26

B. Time Shifting.

Let
]}
[
{
)
(
n
f
Z
z
F


and let any integer number
m

is given. Then the equality


)
(
]}
[
{
z
F
z
m
n
f
Z
m




is holds, that can be easy proved directly on the base o
f the formula (2):



















m
p
n
m
n
p
z
m
n
f
m
n
f
Z
n
n
]
[
]}
[
{



)
(
]
[
]
[
z
F
z
z
p
f
z
z
p
f
m
p
p
m
p
m
p
















.


The z
-
transforms and the properties of the z
-
transformation obtained
above are summarized in Table 1.

Table

1.















Remark,

that

introduced definition 1 is referred to as a definition of the
bilateral z
-
transformation

to distinguish it from
unilateral (or one
-
side) z
-
transformation
, which in some cases is more convenient to use.

It is possible also to use u
nilate
ral transformation

for only positive
n
.

Along with the direct z
-
transformation it is widely used also its inverse
variant.

Definition

2
.
The

map







dz
z
z
z
F
j
n
f
n
)
(
2
1
]
[

is called the inverse z
-
transformation, which allows to calculate a numerical
seq
uence


)
(
]}
[
{
1
z
F
Z
n
f
f



, such that
]}
[
{
)
(
n
f
Z
z
F

.
Here


is


]
[
n
f

)
(
z
F

1.

]
[
n


1

2.

]
[
n
u
e

1

z
z

3.

]
[
n
u
A
e
n




z
Az


]
[
1
n
f
,
]
[
2
n
f

)
(
1
z
F
,
)
(
2
z
F

4.

]
[
]
[
2
2
1
1
n
f
n
f




)
(
)
(
2
2
1
1
z
F
z
F




5.

]
[
m
n
f


)
(
z
F
z
m


6.

]
[
]
[
2
1
n
f
n
f


)
(
)
(
2
1
z
F
z
F

AM
-
CP Faculty of SPbU




27

any closed contour in the region of the Laurent series convergence enclosing
the origin.

In general case the residue theorem is used to calculate the inverse

z
-
transform:















i
n
k
i
n
z
z
z
z
F
dz
z
z
z
F
j
n
f
,
)
(
Res
)
(
2
1
]
[
1
,


where
i
z

are the singular points of the function
)
(
z
F
,
k
i
,
1

.

It is necessary to note that the Laurent transformation is very convenient
instrument for the
solving and analysis of the linear difference equations with
the constant coefficients.

Let consider the equation of the mention type in the form


].
[
...
]
1
[
]
[
]
[
...
]
1
[
]
[
1
0
1
M
n
u
b
n
u
b
n
u
b
N
n
y
a
n
y
a
n
y
M
N












(
3
)

Transform both the left and the right parts of (
3
) by the bilateral z
-
transfo
rmation:








]
[
...
]
1
[
]
[
]
[
...
]
1
[
]
[
1
0
1
M
n
u
b
n
u
b
n
u
b
Z
N
n
y
a
n
y
a
n
y
Z
M
N














or, on the base of the linearity and shifting properties,











.
]
[
...
]
[
...
1
2
2
1
1
0
2
2
1
1
n
u
Z
z
b
z
b
z
b
b
n
y
Z
z
a
z
a
z
a
M
M
N
N

















(
4
)

Denote the z
-
transforms as





]
[
)
(
n
y
Z
z
Y

,


]
[
)
(
n
u
Z
z
U



for the sequences
y

and
u

correspondently, we have from (
4
):



)
(
...
1
...
)
(
2
2
1
1
2
2
1
1
0
z
U
z
a
z
a
z
a
z
b
z
b
z
b
b
z
Y
N
N
M
M

















(
5
)



this is z
-
transform of the equation (
3
) solution.

Let consider the system


]
[
]
[
]
1
[
n
n
n
Bu
Ax
x




(
6
)

AM
-
CP Faculty of SPbU




28

of the difference equations with the constant coeffici
ents, where
N
E
n

]
[
x

is
unknown for any
n

vector of solution,
M
E
n

]
[
u

is the given vector, which
determines no homogeneity in the system.

It is possible to use bilateral z
-
transformation to calculate the

solution:








]
[
]
[
]
1
[
n
Z
n
Z
n
Z
Bu
Ax
x



,


from

where

it

follows








]
[
]
[
]
[
n
Z
n
Z
n
zZ
u
B
x
A
x


.

(
7
)

Introducing the denotes





]
[
)
(
n
Z
z
x
X

,


]
[
)
(
n
Z
z
u
U



for the vectors of z
-
transforms of the sequences
x

and
u

respectively, on the
base of (
7
) we have


)
(
)
(
)
(
1
z
z
z
BU
A
E
X




(
8
)



z
-
transform of the difference system (
6
) solution.


4. Mathematical models of DLTI systems in z
-
dom
ain


The Laurent transformation discussed above allows us to introduce

a
special form of the DLTI systems mathematical models, which are widely used
side by side with its continuous analog on the base of Laplace transformation.

Let accept the difference equations in standard ss
-
form as an initial mo
d-
el of DLTI system:


].
[
]
[
]
[
],
[
]
[
]
1
[
n
n
n
n
n
n
Du
Cx
y
Bu
Ax
x






(1)

Here (for any fixed value of
n
)
s
n
E
n

]
[
x
is the state vector,
s
m
E
n

]
[
u
and
s
k
E
n

]
[
y
are the vector of input and output variables correspondently. Su
p-
pose initial conditi
ons on state are zero; then applying Laurent transformation
for the first equation in (1), we have


)
(
)
(
)
(
1
z
z
z
BU
A
E
X



,


AM
-
CP Faculty of SPbU




29

where
E

is the identity matrix with dimension
s
s
n
n

. Substituting this e
x-
pression to the
second equation, we get




)
(
)
(
)
(
1
z
z
z
U
D
B
A
E
C
Y




.

(2)

Definition 1.
A

transfer matrix
of the DLTI system (1
) from the input
u

to the output
y

is
s
s
m
k


matrix




D
C
B
A
D
B
A
E
C
F
,
,
,
:
)
(
)
(
1





z
z

(3)

with st
rictly proper rational (as the functions of
z
) components
.
In addition
we say that the equation


,
)
(
u
F
y
z


(4)

with the sense of (2) is the
mathematical model of DLTI system in z
-
domain
or its tf
-
model
.

Let us remark th
at a transition from ss
-

to tf
-
model of DLTI system is
made uniquely. Nevertheless a reverse transmission is not unique: it is defined
by the linear transformation of the state vector even for a minimal realization
of the transfer matrix
)
(
z
F
.

The DLTI system is said to be
SISO system

(Single Input, Single Ou
t-
put) if it has the only input and the only output. In this case we use the term
transfer function
(not matrix)

and denote it as
)
(
z
F
.

It is obvious, that the digi
tal filters mentioned above with the equations


]
[
...
]
1
[
]
[
]
[
...
]
1
[
]
[
1
0
1
M
n
u
b
n
u
b
n
u
b
N
n
y
a
n
y
a
n
y
M
N












(5)

are the typical examples of the SISO systems.

It is easy to see, that a transfer function of a digital filter can be directly
calculated without intermediate transformation of the

difference equation (5) to
the normal form (1).

Really, let apply z
-
transformation to the both pats of (5). As it was
shown above, this gives us the z
-
transform of the solution:



)
(
...
1
...
)
(
2
2
1
1
2
2
1
1
0
z
U
z
a
z
a
z
a
z
b
z
b
z
b
b
z
Y
N
N
M
M















.

(6)

Definition 2.
A

transfer function
of the
digital filter (5
) from its input
u

to the output
y

is the rational function of
1

z

AM
-
CP Faculty of SPbU




30



N
N
M
M
z
a
z
a
z
a
z
b
z
b
z
b
b
z
U
z
Y
z
F
















...
1
...
)
(
)
(
)
(
2
2
1
1
2
2
1
1
0


(7)



the ratio of the z
-
transforms
Y
and
.
U

In this connection the equation


u
z
F
y
)
(

,

(8)

is said to be the mathematical model of the digital filter in z
-
domain or tf mo
d-
el “input
-
output” with the sense

(6).

It is obvious, that the expression (7) can be converted to equiv
alent r
a-
tional function of
z
. Actually
,
if
N
M

after

multiplying

to

N
z

we have
:


N
N
N
N
N
M
N
M
N
N
N
a
z
a
z
a
z
a
z
z
b
z
b
z
b
z
b
z
F
















1
2
2
1
1
2
2
1
1
0
...
...
)
(
.

(9)

O
n contrary, if
N
M


it should be done multiplying to
M
z
, so that


N
M
N
M
M
M
M
M
M
M
M
z
a
z
a
z
a
z
b
z
b
z
b
z
b
z
b
z
F
















...
...
)
(
2
2
1
1
1
2
2
1
1
0
.

(10)

Remark that in both cases a transfer function
)
(
z
F

is the proper fraction
that allows converting the model (8) to the equations (1) in ss form, so defin
i-
tions 1 and 2 are equivalent in the

sense mentioned above.

It is easy to see, that
for
conversion of tf model (8) to ss model (1) we
obtain the state vector with dimension



M
N
n
s
,
max

.

For example it is useful to consider ss
-
model of the FIR filter, presented
in the section 2. R
eally, its difference equation is


]
[
...
]
1
[
]
[
]
[
1
0
M
n
u
b
n
u
b
n
u
b
n
y
M






,

(11)

so the transfer matrix of the filter can be presented by the polynomial of
1

z


M
M
z
b
z
b
b
z
F






...
)
(
1
1
0


or by the proper fraction


M
M
M
M
z
b
z
b
z
b
z
F





...
)
(
1
1
0

(12)

of the vari
able
z
. So, tf model of the filter is


u
z
F
y
)
(

,

(13)

AM
-
CP Faculty of SPbU




31

where the transfer function
)
(
z
F

is defined by the formula (12).

But it was shown earlier that the equation (11) can be presented in
equivalent s
s
-
form


],
[
]
[
]
[
],
[
]
[
]
1
[
n
du
n
n
y
n
u
n
n





cx
b
Ax
x

(14)

where the state vector
x

has dimension
M
, and matrices
c
b
A
,
,
,
d

are d
e-
fined by the expressions



















0
1
...
0
0
...
...
...
...
...
0
0
...
1
0
0
0
...
0
1
0
0
...
0
0
A
,


















1
2
1
...
b
b
b
b
M
M
b
,


1
...
0
0
0

c
,
0
b
d

.

It is easy to calculate the transfer function of the system (14) from input
u

to output
y

in accordance with (3):


d
z
z
d
z
z
F
M








)
(
)
(
)
(
)
(
1
b
A
E
c
.

(15)

Here

we

have

M
z
z
z




)
det(
)
(
A
E
,


M
M
M
M
M
M
M
b
z
b
z
b
b
b
b
z
b
z
z





























...
1
...
0
0
...
...
...
...
...
0
...
1
0
0
...
1
0
...
0
det
)
(
2
2
1
1
1
2
1
,


(the Kramer’s formulas) and after substitution to (15) we get


M
M
M
M
M
z
b
z
b
z
b
b
z
z
z
F









...
)
(
)
(
)
(
1
1
0
0
.


Naturally, this expression is the same as (12).

AM
-
CP Faculty of SPbU




32

PATH 2

ANALYSIS OF DIGITAL SIGNAL
S and SYSTEMS


This part of the course is devoted to the mathematical methods of the
signals and system investigations. First of all the frequency features are di
s-
cussed on the base of various forms of Fourier transformation. B
e
sides that the
pro
b
lems conn
ected with the stability of the discrete systems are considered.
And, at last, essential attention is given to the various forms of performance
indexes introducing for the digital signals and sy
s
tems.

All the mentioned points now are widely used in practic
e and in theoret
i-
cal researches to give a full complex of digital system characteristics which r
e-
flect various sides of its dynamical properties.

A special attention is paid to the spectral analysis because it is used as
the basis for the frequency approac
h to digital systems research. The spectral
analysis of the digital systems is mostly based on the Fourier tran
s
formation.
Hear its application is used by the following ways. First it allows us to calc
u-
late a signal's
frequency spectrum

to analyze informa
tion which is hidden in the

frequency, phase or magnitude of the separate components. For example, h
u-
man speech and hearing can be analyzed by the mentioned way.

Second, the Fourier transformation allows to find a system's frequency
response from the syst
em's impulse response, and vice versa. This is system
analysis in the
frequency domain
, just as convolution allows systems to be an
a-
lyzed in the
time domain
.

Third, this transformation can be used as an intermediate step in more
advanced analysis and synth
esis techniques. For example, it is the optimal sy
s-
tem synth
e
sis in the sense of the spaces
2
H

and

H

norms.




AM
-
CP Faculty of SPbU




33