Linear Systems & Signals Signal Theory Zdzisław Papir Basic definitions Examples of signals and signal processing Classification of signal models Time-invariant

Linear Systems & Signals

„Signal Theory” Zdzisław Papir

•Basic definitions

•Examples of signals and signal processing

•Classification of signal models

•Time-invariant & Linear Systems (TILSs)

•TILS transfer function

•Components of a TILS response

•TILS response to a harmonic input

•Summary

Basic definitions

SYSTEM tx1

txi txk tym

ty j ty1

inputsignals

outputsignals

Signal – variation of some physical quantity in (t;x,y,z).Input signals – signals driving the system.Output signals – response of the system to input signals.Signal Theory is related to modeling of both:• signal properties,• signal processing in systems.Signal/system model – description of signal/systemusing functions or differential/integral equations „Signal Theory” Zdzisław Papir

Examples of signals &signal processingINFORMATION TRANSMISSION:

• radio and television signals,

• mobile and fixed telephony

• data transmission (data networks)

OBJECT IDENTIFICATION SIGNALS:

• ultrasound scanning,

• X-ray scanning,

• radar techniques,

• stock analysis,

• demographic trends.


Types of models of signals & signal processing

Analog modelsDiscret models

Time-invariant modelsTime-variant models

Linear modelsNonlinear models

Lumped modelsDistributed models

Deterministic modelsStochastic models

Static modelsDynamic models


Analog models


In analog models input and output signalsare continuous functions of time.

Seismogram recordedon an analog device

Electrocardiogram recordedon an analog device

Discret modelsIn discret models signals are changing stepwise.

Buffer Transmission channel

3

t0

1

2

4

5

6

7

1t 5t4t3t2t 6t

oT 5T4T3T2T1T 6T

nttN tN

Packet count is one of thepossible teletraffic models.


Static modelsStatic models do not depend on time.

1

,2,1,0,1

1Pr

L

L

j

jp jj

Packet buffering leads to multiplexingof traffic streams over a channel.

Buffer Channel

jL


Dynamic models

Buffer Channel t

tLDiffusion approximation

ttdttdL

tLt

tttttLttL

0Pr1


Dynamic models do depend on time.

Time – invariant modelsIn time-invariant models both signal parameters and system characteristics do not depend on time.

IN OUTLOGISTICITERATION

FEEDBACK

1n

1nn

x1

axx

„Teoria sygnałów” Zdzisław Papir

Time-invariant models 1n1nn x1axx

0 100 200 300 400 500 600 700 8000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Time-variant models

Frequency Modulation FM

Instantaneous frequency of the FM signaldepends on the modulating signal.

tkxt

dxktAt

tAtt

0

00FM

0

cos

cos


In time-variant models both signal parameters and system characteristics do depend on time.

Linear modelsIn linear models the system response to a composite input signal is combination of system responses to component signals.

txatxatxatxa 22112211 RRR

Preemphasis filter

21

21

1,

1

rCRC

Rr

R

C

rx1(t)

y1(t)

x2(t)

y2(t)


Linear models

100

101

102

103

104

10-2

10-1

100

f [dec]

H(f

) [d

B]

Preemphasis filter f2/f1 = 100log-log amplitude response


Nonlinear models

Weber-Fechner Law

The sensation change depends linearly ona relative stimulus change.

bw

bbw

bbw

ln

dd


In nonlinear models the system response to a composite input signal is not combination of system responses to component signals.

Nonlinear models

-compression

300100,10,1ln

1ln

xx

y

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

y


The aim of a nonlinear compression is to emphasizeweak signals while leaving strong signals almost unchanged.

Kompresja 0 0.5 1 1.5 2 2.5 3 3.5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Signal before compression

Signal after compression

-compression law is used in Northern America; Europeandigital telephony exploits the A-compression concept.

Nonlinear models


Lumped modelsIn lumped models energy is accumulated/disspated in isolated system points. Signals are transferred within the system without any delay.

R

C

r


Distributed parameter models

• power networks

• CATV coaxial network

• Digital Subscriber Lines

• Printed Circuit Boards (> 100 MHz)


In distributed models energy is accumulated/disspated in all system points. Signals are transferred within the system with some delay.

Deterministic models

Double-sidebandAmplitude modulation AM

tftfmAt cmc 2cos2cos1

tfmAte mc 2cos1AM


In deterministic models signal fluctuationsare described by functions or equations. The exact formula modeling the signal makes future signal values known.

Stochastic modelsStochastic models allow for a signal description exact to a probability distribution. The future signal values can be predicted with some accuracy only.

Transition graph for the Miller’s code 0Pr1,1Pr pp „Signal Theory” Zdzisław Papir

+

–

0

1

1

10

Miller’s code

0Pr1

1Pr

p

p

1 0 001 1 00 00 1 1 1


+

–

0

1

1

10

Miller’s code +

–

p

p1

p

pp1 0Pr1

1Pr

p

p

S()

Spectral density function

Bipolar code

Miller’s code

Biphase code


Time-invariant Linear Systems (TILS)

TILS tx ty

Linear System

tyatya

txatxa

tytx

tytx

2211

2211

22

11

tytx

tytx

tytx

Time-invariantSystem

„Signal Theory” Zdzisław PapirTILS

Exponential input

Linear System

TILS ste etx ?tye

? tyetx est

e

Time-invariant System

tyeee

tye

essts

est

tyeee

tye

etssts

est

see etyty


Exponential input

TILS ste etx s

ee etyty

see etyty

The single and nontrivial solution to an equation:

is an exponential signal:

ste esHty

The amplitude H(s) depends on some constant s C.The exponential signal is an invariant to LinearTime-invariant Systems (TILS).


Exponential input ste etx ste esHty

Let’s assume that an extra solution does exist:

setvtv Let’s substracte the identity side by side:

ststs esHeesH

sstts

sststs

etvesHtvesH

etvesHetvesH


TILS

s

sstts

etztz

etvesHtvesH

The conclusion is:

st

st

esH

tvtvesH

We state that:

We do not receive a new solution:

stesHtv

2

10


Exponential input

TILS transfer function

TILS ste etx st

e esHty

ste

e

tysH

The transfer function of any TILS:

is defined as a ratio of the system response tothe exponential driving function.The transfer function can be interpreted asa TILS „amplification”.


TILS (R, L, C) impedance

ste

e

e

e

tu

ti

tusZ

TILS impedance(voltage/current transfer function):

TILS ste eti st

e esZtu R

C

L

1Cs

Ls

R

sZ


TILS (R, L, C) admittance

st

e

e

e

e

ti

tu

tisY

Admittance(current/voltage transfer function):

ULS ste etu st

e esYti R

C

L

Cs

Ls

R

sY 1

1


TILS (R, L, C) transfer functionDerivation of the TILS (R, L, C) transfer function is supported by various theorems:

• serial/parallel combination of impedances,

• Kirchoff’s current law,

• Kirchoff’s voltage law,

• Thevenin/Norton theorems,

• transformation of current/voltage sources.


Preemphasis filter

21

21

1,

1

rCRC

RrR

1/Cs

r

x(t) y(t)


TILS response toa sinusoidal input

TILSste stesH

TILStje tjejH

TILS

tjtj eet 2

1cos tjtj ejHejH

2

1

TILS response to a sinusoidal (harmonic) input:


Harmonic excitationTILS

tjtj eettx 2

1cos tjtj ejHejHty

2

1

The transfer function H(j) is a rational function soit follows the Hermite symmetry:

jHjH *

Using the exponential representation we get:

jj

j

eAeA

eAjH


Harmonic excitationTILS

tjtj eettx 2

1cos tjtj ejHejHty

2

1

TILS response to the harmonic excitation:

tAty cos

A() - amplitude-frequency characteristic() - phase -frequency characteristic

A-f function A() is an even function, A() = A(-)P-f function () is an odd function, () = - (-)


Preemphasis filter

21

21

1,

1

rCRC

Rr

R

1/Cs

r

x(t) y(t)

2

212

1

22

21

2

1

,1

,

,

1

1

1

1

1

1

Rr

R

rjHA

j

j

R

rjH

rCs

RCs

R

rsH


Preemphasis filter

100

101

102

103

104

10-2

10-1

100

f [dek]

H(f

) [d

B]


Preemphasis filter f2/f1 = 100Log-log amplitude response

Butterworth filter

10-2

100

102

104

106

10-4

10-2

100

10-2

100

102

104

106

-200

-150

-100

-50

0

A-f functionn = 2, fg = 1

kHz

P-f functionn = 2, fg = 1

kHz

ng

jHA2

22

1

1


Butterworth filter

12,,2,1

00 22

nk

d

Ad

d

Adk

k

k

k

ng

jHA2

22

1

1

Butterworth filters have a maximaly flat a-f function in both passband and stopband.


Chebyshev filter

gnTjHA

22

22

1

1

,4,3,2

12,

21

221

nvTvvTvT

vvTvvT

nnn

Chebyshevpolynomials:

121,1

Oscillation level of A2()in the passband:

The Chebyshev a-f function decreases faster thanthe Butterworth a-f function (for the same order).


Chebyshev filter

gnTjHA

22

22

1

1

22 1

n = 6g


ButterworthChebyshev

Summary


In time-invariant models both signal parameters and system characteristics do not depend on time.

In linear models the system response to a composite input signal is combination of system responses to component signals.The exponential signal is an invariant to LinearTime-invariant Systems (TILS).

The transfer function of any TILS is defined as a ratioof the system response to the exponential driving function.

Signal Theory is related to modeling of both:• signal properties,• signal processing in systems.

Summary


The TILS response to a harmonic excitation is a harmonic signal as well. The frequency remains unchanged. Amplitude and phase can be derived fromamplitude and phase functions.

The transfer function of the TILS = (R, L, C) can bederived from a differential equation or using theoremsof the circuit theory.

Documents

Linear Systems & Signals Signal Theory Zdzisław Papir Basic definitions Examples of signals and signal processing Classification of signal models Time-invariant