Transcript +T - Department of Electronic Engineering

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

Week 6

Week 7-10

Week 11
:
:
:
:
Lecture
Test
Lecture
Test
Scores
Tests : 30% (15% for each test)
Exam : 70%
Tutorials

Group 01
Weeks

Group 02
Weeks

Group 03
Weeks
: Friday
: 2,3,4,7,8,9
: Monday
: 3,4,5,7,8,9
: Thursday
: 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.
SIGNALS
Information expressed in different forms
Stock Price
Data File
Transmit
Waveform
$1.00, $1.20, $1.30, $1.30, …
00001010 00001100 00001101
x(t)
Primary interest of Electronic Engineers
SIGNALS PROCESSING AND ANALYSIS
Processing: Methods and system that modify signals
x(t)
Input/Stimulus
System
y(t)
Output/Response
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)?
SIGNALS DESCRIPTION
To analyze signals, we must know how to describe or represent
them in the first place.
A time signal
t
x(t)
0
0
5
1
5
0
2
8
3
10
4
8
5
5
15
x(t)
10
-5 0
5
10
-10
-15
t
Detail but not informative
15
20
TIME SIGNALS DESCRIPTION
x(t)=Asin(wt+f)
1. Mathematical expression:
15
10
5
2. Continuous (Analogue)
0
0
5
10
15
20
-5
-10
-15
x[n]
n
3. Discrete (Digital)
TIME SIGNALS DESCRIPTION
15
4. Periodic
10
5
x(t)= x(t+To)
Period = To
0
0
10
20
30
40
20
30
40
-5
-10
-15
To
12
5. Aperiodic
10
8
6
4
2
0
-2
0
10
TIME SIGNALS DESCRIPTION
6. Even signal
xt )  x t )
15
10
5
0
-10
-5
0
5
10
0
5
10
-5
-10
-15
7. Odd signal
15
xt )   x t )
10
5
0
-10
-5
-5
-10
-15
T
Exercise: Calculate the integral
v   cos wt sin wtdt
T
TIME SIGNALS DESCRIPTION
8. Causality
Analogue signals: x(t) = 0
for t < 0
Digital signals: x[n] = 0
for n < 0
TIME SIGNALS DESCRIPTION
15
9. Average/Mean/DC value
10
5
xDC
1

TM
t1 +TM
 xt )dt
t1
0
0
10
20
30
40
-5
-10
-15
10. AC value
x AC t )  xt )  xDC
TM
DC: Direct Component
AC: Alternating Component
Exercise:
2
Calculate the AC & DC values of x(t)=Asin(wt) with TM 
w
TIME SIGNALS DESCRIPTION

11. Energy E 

xt ) dt
2

12. Instantaneous Power Pt ) 
13. Average Power
1
Pav 
TM
xt )
2
R
watts
t1 +TM
 Pt )dt
t1
Note: For periodic signal, TM is generally taken as To
Exercise:
Calculate the average power of x(t)=Acos(wt)
TIME SIGNALS DESCRIPTION
14. Power Ratio
P1
PR  10 log10
P2
The unit is decibel (db)
In Electronic Engineering and Telecommunication power is
usually resulted from applying voltage V to a resistive load
R, as
V2
P
R
Alternative expression for power ratio (same resistive load):
P1
V12 / R
PR  10 log10  10 log10 2
P2
V2 / R
V1
 20 log10
V2
TIME SIGNALS DESCRIPTION
15. Orthogonality
Two signals are orthogonal over the interval
if
r
t1, t1 + TM 
t1 +TM
 x t )x t )dt  0
1
2
t1
Exercise: Prove that sin(wt) and cos(wt) are orthogonal for
TM 
2
w
TIME SIGNALS DESCRIPTION
15. Orthogonality: Graphical illustration
x2(t)
x2(t)
x1(t)
x1(t)
x1(t) and x2(t) are
correlated.
When one is large, so is
the other and vice versa
x1(t) and x2(t) are
orthogonal.
Their values are totally
unrelated
TIME SIGNALS DESCRIPTION
16. Convolution between two signals
y t )  x1 t )  x2 t ) 


 x  )x t   )d   x  )x t   )d
1

2
2
1

Convolution is the resultant corresponding to the
interaction between two signals.
SOME INTERESTING SIGNALS
1. Dirac delta function (Impulse or Unit Response) d(t)
0
d t )  A
0
for
t 0
otherwise
t
where A  
Definition: A function that is zero in width and infinite in
amplitude with an overall area of unity.
SOME INTERESTING SIGNALS
2. Step function u(t)

1
0
u t )  1
0
for
t
t0
otherwise
A more vigorous mathematical treatment on signals
Deterministic Signals
A continuous time signal x(t) with finite energy

N 

xt ) dt
2

Can be represented in the frequency domain
X w ) 

 jwt

)
x
t
e
dt

w  2f

Satisfied Parseval’s theorem

N 


xt ) dt 
2



X  f ) df
2
Deterministic Signals
A discrete time signal x(n) with finite energy


N 
xn )
2
n  
Can be represented in the frequency domain
Note: X w ) is periodic with period = 2rad / sec
X w ) 

 xn)e
xn ) 
 jwn
n  
Satisfied Parseval’s theorem
N 

2
2

)
x
n


1 X  f ) df
n  
2
1
2
1
2

jwn

)
X
w
e
dw


Deterministic Signals
Energy Density Spectrum (EDS)
S xx  f )  X  f )
2
Equivalent expression for the (EDS)
S xx  f ) 

 jwm

)
r
m
e
 xx
m  
where
rxx m ) 

*
x
 n)xn + m)
n  
* Denotes complex conjugate
Two Elementary Deterministic Signals
Impulse function: zero width and infinite amplitude

 d t )dt  1


 d t )g t )dt  g 0)

Discrete Impulse function
n0
1
d n )  
0 otherwise
Given x(t) and x(n), we have

xt )   x )d t   )d

and
xn ) 

 xk )d n  k )
k  
Two Elementary Deterministic Signals
Step function: A step response
t0
1
u t )  
0 otherwise
Discrete Step function
n0
1
u n )  
0 otherwise
Random 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 Signals
x(t,s0)
x(t,s1)
x(t,s2)
Deterministic Signals
Energy Density Spectrum (EDS)
S xx  f )  X  f )
2
Equivalent expression for the (EDS)

S xx  f )   rxx  )e  jw d

where
rxx  ) 

*
x
 t )xt +  )dt

* Denotes complex conjugate
Random 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
t1 , t2 ,....., t N 
the samples X t  X ti ) Is represented by:
i

p xt1 , xt2 ,....., xt N
)
A random process is known as stationary in the strict sense if

) 
p xt1 , xt2 ,....., xt N  p xt1 + , xt2 + ,....., xt N +
)
Properties of Random Signals
X ti ) is a sample at t=ti
The lth moment of X(ti) is given by the expected value
 

 )
E X   xtli p xti dxti
l
ti

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 Signals
X t1 ) and X t2 ) are samples at t=t1 and t=t2
The statistical correlation between the two samples are given
by the joint moment


E X t1 X t2  



 

)
xt1 xt2 p xt1 , xt2 dxt1 dxt2
This is known as autocorrelation function of the random
process, usually denoted by the symbol
 xx t1 , t2 )  EX t X t
1
2

For stationary process, the sampling instance t1 does not
affect the correlation, hence
 xx  )  EX t X t    xx   )
1
2
where   t1  t2
Properties of Random Signals
Average power of a random process
 xx 0)  EX t2 
1
Wide-sense stationary: mean value m(t1) of the process is
constant
Autocovariance function:



cxx t1 , t2 )  E X t1  mt1 ) X t2  mt2 )   xx t1 , t2 )  mt1 )mt2 )
For a wide-sense stationary process, we have
cxx t1 , t2 )  cxx  )   xx  )  mx2
Properties of Random Signals
 2  cxx 0)   xx 0)  mx2
Variance of a random process
Cross correlation between two random processes:
 xy t1 , t2 )  EX t Yt   
1
2



 

)
xt1 yt2 p xt1 , yt2 dxt1 dyt2
When the processes are jointly and individually
stationary,
 xy   )   yx  )  EX t Yt +   EX t  Yt
1
1
1
1

Properties of Random Signals
Cross covariance between two random processes:
cxy t1 , t2 )   xy t1 , t2 )  mx t1 )my t2 )
When the processes are jointly and individually
stationary,
 xy   )   yx  )  EX t Yt +   EX t  Yt
1
1
1
1
Two processes are uncorrelated if
  
cxy t1 , t2 ) or  xy t1 , t2 )  E X t1 E Yt2

Properties of Random Signals
Power Spectral Density: Wiener-Khinchin theorem

xx  f )    xx  )e  j 2f d

An inverse relation is also available,

 xx  )   xx  f )e j 2f df

Average power of a random process
 xx 0)   xx  f )df  EX t2   0


Properties of Random Signals
Average power of a random process
 xx 0)   xx  f )df  EX t2   0


 xx   )   xx*  )
For complex random process,

xx  f )    xx  )e
*

*
j 2f

d    xx   )e j 2f d  xx  f )


Cross Power Spectral Density:
xy  f )    xy  )e  j 2f d
For complex random process,
xy*  f )  xy  f )

Properties of Discrete Random Signals
X n , or X n )
is a sample at instance n.
The lth moment of X(n) is given by the expected value
 

E X   xnl pxn )dxn
l
n

Autocorrelation
Autocovariance
 xx m)  EX n EX k 
cxx n, k )   xx n, k )  EX n EX k 
For stationary process, let
m  nk
cxx m)   xx m)  EX n EX k    xx m)   x2
 x is the mean
Properties of Discrete Random Signals
The variance of X(n) is given by
 2  cxx 0)   xx 0)   x2
Power Density Spectrum of a discrete random process
xx  f ) 

 j 2fm

)

m
e
 xx
m  
 xx m)   12 xx  f )e j 2fmdf
1
Inverse relation:
Average power:

  
EX
2
n
2
2

)
0

xx
 1 xx  f )df
1
2
Signal Modelling
Mathematical description of signal
M
xn )   ak nk cosw k n + f k )
k  1 or 0  k  1
k 1
ak , k ,w k ,fk 1k M
are the model parameters.
M
Harmonic Process model
xn )   ak cosw k n + f k )
k 1
Linear Random signal
model
xn ) 

 hk )wn  k )
k  
Signal Modelling
Rational or Pole-Zero model
xn)  axn  1) + wn)
Autoregressive (AR) model
p
xn ) +  ak xn  k )  wn )
k 1
Moving Average (MA) model
q
xn )   bk wn  k )
k 0
SYSTEM DESCRIPTION
1. Linearity
x1(t)
System
y1(t)
x2(t)
System
y2(t)
x2(t) + x2(t)
System
y1(t) + y2(t)
IF
THEN
SYSTEM DESCRIPTION
2. Homogeneity
IF
x1(t)
System
y1(t)
THEN
ax1(t)
System
ay1(t)
Where a is a constant
SYSTEM DESCRIPTION
3. Time-invariance: System does not change with time
IF
x1(t)
System
y1(t)
THEN
x1(t)
System
y1(t)
x1(t)
y1(t)
t
x1(t)
t
y1(t)

t

t
SYSTEM DESCRIPTION
3. Time-invariance: Discrete signals
IF
x1[n]
System
y1 [n]
THEN
x1[n - m
System
y1[n - m
x1[n]
y1 [n]
t
t
y1[n - m
x1[n - m
m
t
m
t
SYSTEM 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 PARTS
Signal Representation and Analysis
System Representation and Implementation
Output Response
Signal 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 Series
Ground Rule: All periodic signals are formed by sum of
sinusoidal waveforms


xt )  ao +  an cos nwt + bn sin nwt
1
(1)
1
T/2
2
an 
xt ) cos nwtdt

T T / 2
T/2
1
ao 
xt )dt

T T / 2
(2)
T/2
2
bn 
xt ) sin nwtdt

T T / 2
(3)
Fourier Series – Parseval’s Identity
Energy is preserved after Fourier Transform


1 T/2
1
2
2
2
2

)


x
t
dt

a
+
a
+
b

o
n
T T / 2
2 1 n


1
1
)
(4)
xt )  ao +  an cos nwt + bn sin nwt
 xt ) dt
T/2
2
T / 2
T/2
 ao 
T / 2

xt )dt +  an 
1
T/2
T / 2

xt ) cos nwtdt + bn 
1
T/2
T / 2
xt ) sin nwtdt
Fourier Series – Parseval’s Identity
2

)


x
t
dt
T / 2
T/2
T/2
 ao 
T / 2

xt )dt +  an 
1
T/2
T / 2

xt ) cos nwtdt + bn 
1
T  T
 ao T +  an +  bn
2 1
2
1
2

T  T
 ao T +  an +  bn
2 1
2
1
2


1 T/2
1  2 2
2
2
  xt ) dt  ao +  a n + bn
T T / 2
2 1
)
T/2
T / 2
xt ) sin nwtdt
Periodic Signal Representation – Fourier Series
-T/2
1
x(t) T/2
-t
t
-T/4
T/2
T/4
2
an 
xt ) cos nwtdt

T T / 2
-1
t
x(t)
-T/2 to –T/4
-1
-T/4 to +T/4
+1
+T/4 to +T/2
-1
2
w
T
T /4
T /4
T /2

2
    cos nwtdt +  cos nwtdt   cos nwtdt 
T T / 2
T / 4
T /4

T / 4
T /4
T /2

2   sin nwt 
 sin nwt 
 sin nwt  
 
+





T  nw  T / 2  nw  T / 4  nw  T / 4 
Periodic Signal Representation – Fourier Series
x(t)
1
-t
t
-T/4
T/2
T/4
2
an 
xt ) cos nwtdt

T T / 2
-1
t
x(t)
-T/2 to –T/4
-1
-T/4 to +T/4
+1
+T/4 to +T/2
-1
2
w
T
T / 4
T /4
T /2

2   sin nwt 
 sin nwt 
 sin nwt  
 
+





T  nw  T / 2  nw  T / 4  nw  T / 4 
4
 nwT 
 nwT 

sin 
sin 


nwT  4  nwT  2 
8
Periodic Signal Representation – Fourier Series
-t
2
w
T
x(t)
1
t
-T/4
-1
T/4
4
 nwT 
 nwT 
an 
sin 
sin 


nwT  4  nwT  2 
t
x(t)
-T/2 to –T/4
-1
-T/4 to +T/4
+1
+T/4 to +T/2
-1
8
zero for all n
4
 n  2
 sin    sin n )
n  2  n
4
We have, ao  0, a1  , a2  0, a3 
,.......

3
4
Periodic Signal Representation – Fourier Series
-t
2
w
T
x(t)
1
t
-T/4
T/4
-1
t
x(t)
-T/2 to –T/4
-1
-T/4 to +T/4
+1
+T/4 to +T/2
-1
It can be easily shown that bn = 0 for all values of n. Hence,
4
1
1
1

xt )   coswt  cos3wt + cos5wt  cos7wt + ....

3
5
7

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 Series
-T/2
A
x(t) T/2
-t
t
-/2 /2
t
x(t)
-/2 to –/2
A
-T/2 to -  /2
0
+  /2 to +T/2
0
/2
1
1
A
ao 
xt )dt 
Adt 


T T / 2
T  / 2
T
T/2
2 T2
2 2
an   T xt )cosnwtdt   TAcosnwtdt
T 2
T  2

2 A  sin nw  2
4A
nw


sin


T  nw   nwT
2
2
2
w
T
Periodic Signal Representation – Fourier Series
-T/2
A
x(t) T/2
-t
t
-/2 /2

2 A  sin nw  2
4A
nw
an 

sin


T  nw   nwT
2
t
x(t)
-/2 to –/2
A
-T/2 to -  /2
0
+  /2 to +T/2
0
2
w
T
2
It can be easily shown that bn = 0 for all values of n. Hence, we have
A 2 A
xt ) 
+
T
T
sin nw / 2)
1 nw / 2) cosnw

Periodic Signal Representation – Fourier Series
A 2 A
xt ) 
+
T
T
Note:
sin  y ) y  0
Hence: an  0
A
for
sin nw / 2)
1 nw / 2) cosnw

y  nw 2  k
for
nw
2k
 k  nw 
2

k 1,2 ,3 ,...
T
w
0
2

4
