Linear Systems & Signals Signal Theory Zdzisław Papir Basic definitions Examples of signals and...
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Transcript of Linear Systems & Signals Signal Theory Zdzisław Papir Basic definitions Examples of signals and...
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.
„Signal Theory” Zdzisław Papir
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
„Signal Theory” Zdzisław Papir
Analog models
„Signal Theory” Zdzisław Papir
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.
„Signal Theory” Zdzisław Papir
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
„Signal Theory” Zdzisław Papir
Dynamic models
Buffer Channel t
tLDiffusion approximation
ttdttdL
tLt
tttttLttL
0Pr1
„Signal Theory” Zdzisław Papir
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
„Signal Theory” Zdzisław Papir
Time-variant models
Frequency Modulation FM
Instantaneous frequency of the FM signaldepends on the modulating signal.
tkxt
dxktAt
tAtt
0
00FM
0
cos
cos
„Signal Theory” Zdzisław Papir
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)
„Signal Theory” Zdzisław Papir
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
„Signal Theory” Zdzisław Papir
Nonlinear models
Weber-Fechner Law
The sensation change depends linearly ona relative stimulus change.
bw
bbw
bbw
ln
dd
„Signal Theory” Zdzisław Papir
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
„Signal Theory” Zdzisław Papir
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
„Teoria sygnałów” Zdzisław Papir
Lumped modelsIn lumped models energy is accumulated/disspated in isolated system points. Signals are transferred within the system without any delay.
R
C
r
„Signal Theory” Zdzisław Papir
Distributed parameter models
• power networks
• CATV coaxial network
• Digital Subscriber Lines
• Printed Circuit Boards (> 100 MHz)
„Signal Theory” Zdzisław Papir
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
„Signal Theory” Zdzisław Papir
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
„Signal Theory” Zdzisław Papir
+
–
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
„Teoria sygnałów” Zdzisław Papir
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
„Signal Theory” Zdzisław Papir
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).
„Signal Theory” Zdzisław Papir
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
„Signal Theory” Zdzisław Papir
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
„Signal Theory” Zdzisław Papir
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”.
„Signal Theory” Zdzisław Papir
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
„Signal Theory” Zdzisław Papir
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
„Signal Theory” Zdzisław Papir
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.
„Signal Theory” Zdzisław Papir
Preemphasis filter
21
21
1,
1
rCRC
RrR
1/Cs
r
x(t) y(t)
„Signal Theory” Zdzisław Papir
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:
„Signal Theory” Zdzisław Papir
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
„Signal Theory” Zdzisław Papir
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, () = - (-)
„Signal Theory” Zdzisław Papir
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
„Signal Theory” Zdzisław Papir
Preemphasis filter
100
101
102
103
104
10-2
10-1
100
f [dek]
H(f
) [d
B]
„Signal Theory” Zdzisław Papir
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
„Signal Theory” Zdzisław Papir
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.
„Signal Theory” Zdzisław Papir
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).
„Signal Theory” Zdzisław Papir
Chebyshev filter
gnTjHA
22
22
1
1
22 1
n = 6g
„Signal Theory” Zdzisław Papir
ButterworthChebyshev
Summary
„Signal Theory” Zdzisław Papir
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
„Signal Theory” Zdzisław Papir
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.