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Signal Analysis. Lecturer: Dr. Peter Tsang Room: G6505 Phone: 27887763 E-mail: [email protected]. Website: www.ee.cityu.edu.hk/~csl/sigana/ Files: SIG01.ppt, SIG02.ppt, SIG03.ppt, SIG04.ppt Restrict access to students taking this course. Signal Analysis. Suggested reference books - PowerPoint PPT Presentation
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Lecturer: Dr. Peter Tsang
Room: G6505
Phone: 27887763
E-mail: [email protected]
Website: www.ee.cityu.edu.hk/~csl/sigana/Files: SIG01.ppt, SIG02.ppt, SIG03.ppt, SIG04.ppt
Restrict access to students taking this course.
Suggested reference books
1. M.L. Meade and C.R. Dillon, “Signals and Systems”, Van Nostrand Reinhold (UK).
2. N.Levan, “Systems and Signals”, Optimization Software, Inc.
3. F.R. Connor, “Signals”, Edward Arnold.
4*. A. Oppenheim, “Digital Signal Processing”, Prentice Hall.
Note: Students are encouraged to select reference books in the library.
* Supporting reference
Course outline Week 2-4 : Lecture Week 6 : Test Week 7-10 : Lecture Week 11 : Test
Scores
Tests : 30% (15% for each test)
Exam : 70%
Tutorials Group 01 : Friday
Weeks : 2,3,4,7,8,9 Group 02 : Monday
Weeks : 3,4,5,7,8,9 Group 03 : Thursday
Weeks : 2,3,4,7,8,9
Course outline
1. Time Signal Representation.
2. Continuous signals.
3. Fourier, Laplace and z Transform.
4. Interaction of signals and systems.
5. Sampling Theorem.
6. Digital Signals.
7. Fundamentals of Digital System.
8. Interaction of digital signals and systems.
Coursework Tests on week 6 and 11: 30% of total score.
Notes in Powerpoint Presented during lectures and very useful for studying
the course.
Study Guide A set of questions to build up concepts.
Discussions Strengthen concepts in tutorial sessions.
Reference books Supplementary materials to aid study.
Expectation from students Attend all lectures and tutorials. Study all the notes. Participate in discussions during tutorials. Work out all the questions in the study guide at least
once. Attend the test and take it seriously. Work out the questions in the test for at least one more
time afterwards.
SIGNALSSIGNALS
Information expressed in different forms
Stock Price
Transmit Waveform
$1.00, $1.20, $1.30, $1.30, …
Data File
x(t)
00001010 00001100 00001101
Primary interest of Electronic Engineers
SIGNALS PROCESSING AND ANALYSISSIGNALS PROCESSING AND ANALYSIS
Processing: Methods and system that modify signals
System y(t)x(t)
Analysis:• What information is contained in the input signal x(t)?• What changes do the System imposed on the input?• What is the output signal y(t)?
Input/Stimulus Output/Response
SIGNALS DESCRIPTIONSIGNALS DESCRIPTION
To analyze signals, we must know how to describe or represent them in the first place.
A time signal
-15
-10
-5
0
5
10
15
0 5 10 15 20
t
x(t)
t x(t)
0 0
1 5
2 8
3 10
4 8
5 5
Detail but not informative
TIME SIGNALS DESCRIPTIONTIME SIGNALS DESCRIPTION
1. Mathematical expression: x(t)=Asin(t
2. Continuous (Analogue)
-15
-10
-5
0
5
10
15
0 5 10 15 20
3. Discrete (Digital)
x[n]
n
TIME SIGNALS DESCRIPTIONTIME SIGNALS DESCRIPTION
4. Periodic
-15
-10
-5
0
5
10
15
0 10 20 30 40x(t)= x(t+To)
To
Period = To
5. Aperiodic
-2
0
2
4
6
8
10
12
0 10 20 30 40
TIME SIGNALS DESCRIPTIONTIME SIGNALS DESCRIPTION
6. Even signal txtx
Exercise: Calculate the integral
7. Odd signal txtx
-15
-10
-5
0
5
10
15
-10 -5 0 5 10
-15
-10
-5
0
5
10
15
-10 -5 0 5 10
T
T
tdttv sincos
TIME SIGNALS DESCRIPTIONTIME SIGNALS DESCRIPTION
8. Causality
Analogue signals: x(t) = 0 for t < 0
Digital signals: x[n] = 0 for n < 0
TIME SIGNALS DESCRIPTIONTIME SIGNALS DESCRIPTION
9. Average/Mean/DC value
MTt
tMDC dttx
Tx
1
1
1
Exercise: Calculate the AC & DC values of x(t)=Asin(twith
2
MT
TM
-15
-10
-5
0
5
10
15
0 10 20 30 40
10. AC value
DCAC xtxtx DC: Direct ComponentAC: Alternating Component
TIME SIGNALS DESCRIPTIONTIME SIGNALS DESCRIPTION
11. Energy
dttxE2
Exercise: Calculate the average power of x(t)=Acos(t
12. Instantaneous Power watts
R
txtP
2
13. Average Power
MTt
tMav dttP
TP
1
1
1
Note: For periodic signal, TM is generally taken as To
TIME SIGNALS DESCRIPTIONTIME SIGNALS DESCRIPTION
14. Power Ratio2
11010
P
PlogPR
In Electronic Engineering and Telecommunication power is usually resulted from applying voltage V to a resistive load
R, as
The unit is decibel (db)
R
VP
2
Alternative expression for power ratio (same resistive load):
R/V
R/Vlog
P
PlogPR 2
2
21
102
110 1010
2
11020
V
Vlog
TIME SIGNALS DESCRIPTIONTIME SIGNALS DESCRIPTION
15. Orthogonality
Exercise: Prove that sin(tand cos(tare orthogonal for
Two signals are orthogonal over the interval if
021
1
1
dttxtxrMTt
t
MTtt 11,
2
MT
TIME SIGNALS DESCRIPTIONTIME SIGNALS DESCRIPTION
15. Orthogonality: Graphical illustration
x1(t)
x2(t)
x1(t) and x2(t) are correlated.
When one is large, so is the other and vice versa
x1(t)
x2(t)
x1(t) and x2(t) are orthogonal.
Their values are totally unrelated
TIME SIGNALS DESCRIPTIONTIME SIGNALS DESCRIPTION
16. Convolution between two signals
dtxxdtxxtxtxty
122121
Convolution is the resultant corresponding to the interaction between two signals.
1. Dirac delta function (Impulse or Unit Response) (t)
0t
otherwise
tAt
0
0for
where A
Definition: A function that is zero in width and infinite in amplitude with an overall area of unity.
SOME INTERESTING SIGNALSSOME INTERESTING SIGNALS
2. Step function u(t)
0t
otherwise
ttu
0
0for 1
SOME INTERESTING SIGNALSSOME INTERESTING SIGNALS
1
A more vigorous mathematical treatment on signals
Deterministic SignalsDeterministic SignalsDeterministic SignalsDeterministic Signals
A continuous time signal x(t) with finite energy
dttxN
2
Can be represented in the frequency domain
dtetxX tj
Satisfied Parseval’s theorem
dffXdttxN
22
f 2
Deterministic SignalsDeterministic SignalsDeterministic SignalsDeterministic Signals
A discrete time signal x(n) with finite energy
n
N nx2
Can be represented in the frequency domain
n
njenxX
Satisfied Parseval’s theorem
dffXnxn
N
22
1
21
2
deXnx nj
2
1
Note: X is periodic with period = sec/2 rad
Deterministic SignalsDeterministic SignalsDeterministic SignalsDeterministic Signals
Energy Density Spectrum (EDS)
2fXfSxx
Equivalent expression for the (EDS)
where
mj
mxxxx emrfS
n
xx mnxnxmr ** Denotes complex conjugate
Two Elementary Deterministic SignalsTwo Elementary Deterministic SignalsTwo Elementary Deterministic SignalsTwo Elementary Deterministic Signals
Impulse function: zero width and infinite amplitude
1
dtt
Discrete Impulse function
otherwise
nn
0
01
dtxtx
0gdttgt
Given x(t) and x(n), we have
knkxnxk
and
Two Elementary Deterministic SignalsTwo Elementary Deterministic SignalsTwo Elementary Deterministic SignalsTwo Elementary Deterministic Signals
Step function: A step response
Discrete Step function
otherwise
nnu
0
01
otherwise
ttu
0
01
Random SignalsRandom SignalsRandom SignalsRandom Signals
Infinite duration and infinite energy signals
e.g. temperature variations in different places, each have its own waveforms.
Ensemble of time functions (random process): The set of all possible waveforms
Ensemble of all possible sample waveforms of a random process: X(t,S), or simply X(t).t denotes time index and S denotes the set of all possible sample functions
A single waveform in the ensemble: x(t,s), or simply x(t).
Random SignalsRandom SignalsRandom SignalsRandom Signals
x(t,s0)
x(t,s1)
x(t,s2)
Deterministic SignalsDeterministic SignalsDeterministic SignalsDeterministic Signals
Energy Density Spectrum (EDS)
2fXfSxx
Equivalent expression for the (EDS)
where
derfS jxxxx
dttxtxrxx
** Denotes complex conjugate
Random SignalsRandom SignalsRandom SignalsRandom Signals
Each ensemble sample may be different from other.
Not possible to describe properties (e.g. amplitude) at a given time instance.
Only joint probability density function (pdf) can be defined. Given a sequence of time instants
Nttt ,.....,, 21 the samples it tXXi Is represented by:
A random process is known as stationary in the strict sense if
NN tttttt xxxpxxxp ,.....,,,.....,,
2121
Nttt xxxp ,.....,,
21
Properties of Random SignalsProperties of Random SignalsProperties of Random SignalsProperties of Random Signals
is a sample at t=ti itX
The lth moment of X(ti) is given by the expected value
iiii tt
lt
lt dxxpxXE
The lth moment is independent of time for a stationary process.
Measures the statistical properties (e.g. mean) of a single sample.
In signal processing, often need to measure relation between two or more samples.
Properties of Random SignalsProperties of Random SignalsProperties of Random SignalsProperties of Random Signals
are samples at t=t1 and t=t2 21 tXandtX
The statistical correlation between the two samples are given by the joint moment
21212121, tttttttt dxdxxxpxxXXE
This is known as autocorrelation function of the random process, usually denoted by the symbol
2121, ttxx XXEtt
For stationary process, the sampling instance t1 does not affect the correlation, hence
xxttxx XXE21 21 where tt
Properties of Random SignalsProperties of Random SignalsProperties of Random SignalsProperties of Random Signals
2
10 txx XEAverage power of a random process
Wide-sense stationary: mean value m(t1) of the process is constant
Autocovariance function:
21212121 ,,21
tmtmtttmXtmXEttc xxttxx
For a wide-sense stationary process, we have
221, xxxxxxx mcttc
Properties of Random SignalsProperties of Random SignalsProperties of Random SignalsProperties of Random Signals
22 00 xxxxx mc Variance of a random process
Cross correlation between two random processes:
21212121,, 21 ttttttttxy dydxyxpyxYXEtt
When the processes are jointly and individually stationary,
1111 ttttyxxy YXEYXE
Properties of Random SignalsProperties of Random SignalsProperties of Random SignalsProperties of Random Signals
Cross covariance between two random processes:
212121 ,, tmtmttttc yxxyxy
When the processes are jointly and individually stationary,
1111 ttttyxxy YXEYXE
Two processes are uncorrelated if
212121 ,or , ttxyxy YEXEttttc
Properties of Random SignalsProperties of Random SignalsProperties of Random SignalsProperties of Random Signals
Power Spectral Density: Wiener-Khinchin theorem
def fjxxxx
2
An inverse relation is also available,
Average power of a random process
dfef fjxxxx
2
00 2
txxxx XEdff
Properties of Random SignalsProperties of Random SignalsProperties of Random SignalsProperties of Random Signals
Cross Power Spectral Density: def fjxyxy
2
Average power of a random process
00 2
txxxx XEdff
For complex random process,
fdedefxxxxxxxx
fjfj
22**
*xxxx
For complex random process, ffxyxy*
is a sample at instance n. nXX n or ,
The lth moment of X(n) is given by the expected value
nnln
ln dxxpxXE
Properties of Discrete Random SignalsProperties of Discrete Random SignalsProperties of Discrete Random SignalsProperties of Discrete Random Signals
Autocorrelation
mean theis x
Autocovariance knxxxx XEXEknknc ,,
For stationary process, let knm
2xxxknxxxx mXEXEmmc
knxx XEXEm
The variance of X(n) is given by
22 00 xxxxxc
Properties of Discrete Random SignalsProperties of Discrete Random SignalsProperties of Discrete Random SignalsProperties of Discrete Random Signals
Power Density Spectrum of a discrete random process
fmj
mxxxx emf 2
Inverse relation: 21
21
2 dfefm fmjxxxx
Average power: dffXE xxxxn 21
21
2 0
Mathematical description of signal
kknk
M
kk nanx
cos1
Signal ModellingSignal ModellingSignal ModellingSignal Modelling
are the model parameters. Mkkkkka 1,,,
Harmonic Process model
10or 1 kk
kk
M
kk nanx
cos1
Linear Random signal model
k
knwkhnx
Rational or Pole-Zero model
Signal ModellingSignal ModellingSignal ModellingSignal Modelling
nwnaxnx 1
nwknxanxp
kk
1
Autoregressive (AR) model
q
kk knwbnx
0
Moving Average (MA) model
SYSTEM DESCRIPTIONSYSTEM DESCRIPTION
1. Linearity
System y1(t)x1(t)
System y2(t)x2(t)
IF
System y1(t) + y2(t)x2(t) + x2(t)THEN
SYSTEM DESCRIPTIONSYSTEM DESCRIPTION
2. Homogeneity
System y1(t)x1(t)
System ay1(t)ax1(t)
IF
THEN
Where a is a constant
SYSTEM DESCRIPTIONSYSTEM DESCRIPTION
3. Time-invariance: System does not change with time
System y1(t)x1(t)
System y1(tx1(t
IF
THEN
t
x1(t)
t
y1(t)
t
x1(t
t
y1(t
SYSTEM DESCRIPTIONSYSTEM DESCRIPTION
3. Time-invariance: Discrete signals
System y1 [n]x1[n]
System y1[n - mx1[n - m
IF
THEN
t t
t t
x1[n]
x1[n - m
y1 [n]
y1[n - m
mm
SYSTEM DESCRIPTIONSYSTEM DESCRIPTION
4. Stability
The output of a stable system settles back to the quiescent state (e.g., zero) when the input is removed
The output of an unstable system continues, often with exponential growth, for an indefinite period when the input is removed
5. Causality
Response (output) cannot occur before input is applied, ie.,
y(t) = 0 for t <0
THREE MAJOR PARTSTHREE MAJOR PARTS
Signal Representation and Analysis
System Representation and Implementation
Output Response
Signal Representation and AnalysisSignal Representation and Analysis
An analogy: How to describe people?
(A) Cell by cell description – Detail but not useful and impossible to make comparison
(B) Identify common features of different people and compare them. For example shape and dimension of eyes, nose, ears, face, etc..
Signals can be described by similar concepts: “Decompose into common set of components”
Periodic Signal Representation – Fourier SeriesPeriodic Signal Representation – Fourier Series
Ground Rule: All periodic signals are formed by sum of sinusoidal waveforms
(1)
(2)
11
tnsinbtncosaatx nno
tdtncostxT
a/T
/T
n
2
2
2
(3)
tdtnsintxT
b/T
/T
n
2
2
2
dttxT
a/T
/T
o
2
2
1
Fourier Series – Parseval’s IdentityFourier Series – Parseval’s Identity
Energy is preserved after Fourier Transform
(4)
1
2222
2
2
2
11no
/T
/Tbaadttx
T n
11
tnsinbtncosaatx nno
dttnsintxbtdtncostxadttxa
dttx
/T
/Tn
/T
/Tn
/T
/To
/T
/T
1
2
21
2
2
2
2
2
2
2
Fourier Series – Parseval’s IdentityFourier Series – Parseval’s Identity
dttnsintxbtdtncostxadttxa
dttx
/T
/Tn
/T
/Tn
/T
/To
/T
/T
1
2
21
2
2
2
2
2
2
2
22 11
2 Tb
TaTa nno
22 11
2 Tb
TaTa nno
1
2222
2
2
2
11no
/T
/Tbaadttx
T n
Periodic Signal Representation – Fourier SeriesPeriodic Signal Representation – Fourier Series
t
x(t)
-t
1
-1T/4-T/4
t x(t)
-T/2 to –T/4 -1
-T/4 to +T/4 +1
+T/4 to +T/2 -1
tdtncostxT
a/T
/T
n
2
2
2
2/
4/
4/
4/
4/
2/
coscoscos2 T
T
T
T
T
T
tdtntdtntdtnT
2/
4/
4/
4/
4/
2/
sinsinsin2T
T
T
T
T
T n
tn
n
tn
n
tn
T
T
2
-T/2 T/2
Periodic Signal Representation – Fourier SeriesPeriodic Signal Representation – Fourier Series
t
x(t)
-t1
-1T/4-T/4
t x(t)
-T/2 to –T/4 -1
-T/4 to +T/4 +1
+T/4 to +T/2 -1
tdtncostxT
a/T
/T
n
2
2
2
2/
4/
4/
4/
4/
2/
sinsinsin2T
T
T
T
T
T n
tn
n
tn
n
tn
T
2sin
4
4sin
8 Tn
Tn
Tn
Tn
T
2
Periodic Signal Representation – Fourier SeriesPeriodic Signal Representation – Fourier Series
t
x(t)
-t1
-1T/4-T/4
t x(t)
-T/2 to –T/4 -1
-T/4 to +T/4 +1
+T/4 to +T/2 -1
2
4
4
8 Tnsin
Tn
Tnsin
Tnan
T
2
nsinn
nsin
n
2
2
4
zero for all n
We have, ,ao 0 ,a4
1 ,a 02 ,.......a34
3
Periodic Signal Representation – Fourier SeriesPeriodic Signal Representation – Fourier Series
t
x(t)
-t1
-1T/4-T/4
t x(t)
-T/2 to –T/4 -1
-T/4 to +T/4 +1
+T/4 to +T/2 -1
T
2
It can be easily shown that bn = 0 for all values of n. Hence,
....tttttx
cos7
7
1cos5
5
1cos3
3
1cos
4
Only odd harmonics are present and the DC value is zero
The transformed space (domain) is discrete, i.e., frequency components are present only at regular spaced slots.
Periodic Signal Representation – Fourier SeriesPeriodic Signal Representation – Fourier Series
t
x(t)
-t
At x(t)
-/2 to –/2 A
-T/2 to - /2 0
+ /2 to +T/2 0
-/2 /2
-T/2 T/2
T
AdtA
Tdttx
Ta
/
/
/T
/T
o
2
2
2
2
11
2
2
2
2
cos2
cos2
T
T
TtdtnA
Ttdtntx
Tan T
2
2sin
4sin2 2
2
n
Tn
A
n
n
T
A
Periodic Signal Representation – Fourier SeriesPeriodic Signal Representation – Fourier Series
t
x(t)
-t
At x(t)
-/2 to –/2 A
-T/2 to - /2 0
+ /2 to +T/2 0
-/2 /2
-T/2 T/2
T
2
2sin
4sin2 2
2
n
Tn
A
n
n
T
Aan
It can be easily shown that bn = 0 for all values of n. Hence, we have
1
cos2
2sin2
n/n
/n
T
A
T
Atx
Periodic Signal Representation – Fourier SeriesPeriodic Signal Representation – Fourier Series
1
cos2
2sin2
n/n
/n
T
A
T
Atx
Note: knyyy 2for 0sin
Hence: ,...,,kn
knk
na 321
2
2for 0
2
4
0
TA