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					Lecturer: Dr. Peter Tsang
Room: G6505
Phone: 27887763
E-mail: eewmtsan@cityu.edu.hk


      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.
                        SIGNALS

Information expressed in different forms


                                $1.00, $1.20, $1.30, $1.30, …
  Stock Price



   Data File                    00001010 00001100 00001101



  Transmit
  Waveform                                      x(t)

                   Primary interest of Electronic Engineers
     SIGNALS PROCESSING AND ANALYSIS

Processing: Methods and system that modify signals



         x(t)            System               y(t)
     Input/Stimulus                         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)
       15
                                                0         0
       10
        5
                                                1         5
                                                2         8
x(t)




        0
        -5 0      5            10     15   20
                                                3         10
       -10
                                                4         8
       -15
                                t               5         5

       Detail but not informative
            TIME SIGNALS DESCRIPTION

1. Mathematical expression:              x(t)=Asin(wt+f)

                               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)         0
                              0   10        20   30   40
                         -5

                        -10
   Period = To          -15


                                       To

                        12

5. Aperiodic            10

                         8

                         6

                         4

                         2

                         0
                              0   10        20   30   40
                        -2
           TIME SIGNALS DESCRIPTION

                 xt )  x t )
                                                      15

6. Even signal                                        10

                                                       5

                                                       0
                                     -10    -5              0   5   10
                                                       -5

                                                      -10

                                                      -15




                                                      15


7. Odd signal    xt )   x t )                    10

                                                       5

                                                       0
                                     -10    -5              0   5   10
                                                       -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
                 t1 +TM

                    xt )dt
             1                  0

    xDC                        -5
                                     0     10     20     30      40


            TM     t1
                               -10

                               -15


                                                  TM
10. AC value

    x AC t )  xt )  xDC
                                     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

                 

                                            xt )
                                                    2

12. Instantaneous Power Pt )                          watts
                                              R
                                             t1 +TM

                                                Pt )dt
                                        1
13. Average Power                Pav 
                                       TM      t1


  Note: For periodic signal, TM is generally taken as To


Exercise:
Calculate the average power of x(t)=Acos(wt)
          TIME SIGNALS DESCRIPTION

                               P
14. Power Ratio   PR  10 log10 1    The unit is decibel (db)
                               P2

In Electronic Engineering and Telecommunication power is
usually resulted from applying voltage V to a resistive load
           V2
R, as   P
           R

Alternative expression for power ratio (same resistive load):

                      P            V12 / R
         PR  10 log10 1  10 log10 2
                      P2           V2 / R
                                    V1
                          20 log10
                                    V2
           TIME SIGNALS DESCRIPTION

15. Orthogonality

   Two signals are orthogonal over the interval   t1, t1 + TM 
   if
                         t1 +TM

                    r      x t )x t )dt  0
                           t1
                                  1   2




Exercise: Prove that sin(wt) and cos(wt) are orthogonal for
                                          2
                                  TM 
                                          w
            TIME SIGNALS DESCRIPTION

15. Orthogonality: Graphical illustration

x2(t)                           x2(t)




                       x1(t)                            x1(t)
       x1(t) and x2(t) are            x1(t) and x2(t) are
           correlated.                    orthogonal.
    When one is large, so is        Their values are totally
    the other and vice versa               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)




                                                          t
                               0
d t )  A    for   t 0
                            where A  
     0       otherwise

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                   
                                            t
                                0
  u t )  1    for   t0
        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 )      xt )e  jwt dt                w  2f
                  


Satisfied Parseval’s theorem
                               
      N        xt ) dt          X  f ) df
                       2                    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n )           X w )e jwn dw
                                                            1
     X w )     xn)e            jwn
                                                                
                n  
                                                           2   



Satisfied Parseval’s theorem
                                            2
               xn )  1 2 X  f ) df
                                         1
     N     
                              2

            n                         2
                     Deterministic Signals

Energy Density Spectrum (EDS)

     S xx  f )  X  f )
                             2




Equivalent expression for the (EDS)
                        
     S xx  f )      rxx m )e  jwm
                    m  



where
                    
     rxx m )     x* n )xn + m )     * Denotes complex conjugate
                  n  
     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
                  1   n0
         d n )  
                  0 otherwise

Given x(t) and x(n), we have
                                                     
                                         xn )     xk )d n  k )
         
 xt )   x )d t   )d    and
         
                                                   k  
    Two Elementary Deterministic Signals

Step function: A step response
                1   t0
       u t )  
                0 otherwise

Discrete Step function
                1   n0
       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  )  mx
                                                              2
           Properties of Random Signals

Variance of a random process                                2  cxx 0)   xx 0)  mx
                                                                                       2




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
                          

                          



For complex random process,                         xx   )   xx  )
                                                                   *



                                              
   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   xn pxn )dxn
             l
             n
                 l
                   


Autocorrelation                  xx m)  EX n EX k 

Autocovariance             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)   x
                                                   2
                                                                      x is the mean
    Properties of Discrete Random Signals

The variance of X(n) is given by

      2  cxx 0)   xx 0)   x
                                  2




Power Density Spectrum of a discrete random process
                      
      xx  f )      xx m )e  j 2fm
                    m  


                              xx m)   12 xx  f )e j 2fmdf
                                            1
Inverse relation:                           
                                                2




                                           0)   12 xx  f )df
                                                       1
                                  2
Average power:               EX   n         xx
                                                           2
                        Signal Modelling

Mathematical description of signal
            M
   xn )   ak n cosw k n + f k )
                 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

                                                     
                                        xn )      hk )wn  k )
Linear Random signal
model                                             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)

  IF
               x2(t)           System   y2(t)




  THEN         x2(t) + x2(t)   System   y1(t) + y2(t)
                 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                              t

x1(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

x1[n - m                       y1[n - m

                            t                               t
            m                               m
                 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                            T/2

            / xt )cos nwtdt             / xt )dt
        2                             1
   an                           ao                   (2)
        T T 2                        T T 2

                     T/2

                   / xt ) sin nwtdt
               2
          bn                                          (3)
               T T 2
                      Fourier Series – Parseval’s Identity

       Energy is preserved after Fourier Transform

               1 T/2
               T T / 2
                                             1  2 2
                                             2 1
                                                    
                         xt )2 dt  ao 2 +  a n + bn    )                 (4)


                                          
 xt )  ao +  an cos nwt +  bn sin nwt
                      1                    1



 xt ) dt
 T/2              2
T / 2
                                                           
                  xt )dt +  an          xt ) cos nwtdt + bn          xt ) sin nwtdt
         T/2                         T/2                             T/2
 ao 
         T / 2                  T / 2                          T / 2
                           1                                1
                        Fourier Series – Parseval’s Identity

        xt )2 dt
 T/2
T / 2
                                                             
                    xt )dt +  an          xt ) cos nwtdt + bn          xt ) sin nwtdt
           T/2                         T/2                             T/2
 ao 
           T / 2                  T / 2                          T / 2
                              1                               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
 T T / 2
                     1  2 2
  xt ) dt  ao +  a n + bn
           2       2

                     2 1
                                                         )
       Periodic Signal Representation – Fourier Series
            -T/2        x(t) T/2
                    1                                       t            x(t)
-t                                                t    -T/2 to –T/4      -1
                                                       -T/4 to +T/4      +1
                                         -1
                -T/4     T/4                           +T/4 to +T/2      -1

             T/2
                                           2
              / xt )cos nwtdt
          2
     an                                w
          T T 2                           T
         2                                           
            T /4            T /4         T /2
            cos nwtdt +  cos nwtdt   cos nwtdt 
         T T / 2         T / 4        T /4         
         2  sin nwt  T / 4  sin nwt  T / 4  sin nwt  T / 2 
                                                                     
                            +                         
         T  nw  T / 2  nw  T / 4  nw  T / 4 
                                                                    
       Periodic Signal Representation – Fourier Series

                         x(t)
                                  1                         t            x(t)
-t                                                t    -T/2 to –T/4      -1
                                                       -T/4 to +T/4      +1
                                         -1
                -T/4     T/4                           +T/4 to +T/2      -1

             T/2
                                           2
              / xt )cos nwtdt
          2
     an                                w
          T T 2                           T

         2  sin nwt  T / 4  sin nwt  T / 4  sin nwt  T / 2 
                                                                     
                            +                         
         T  nw  T / 2  nw  T / 4  nw  T / 4 
                                                                    
            8     nwT   4      nwT 
            sin         sin      
          nwT  4  nwT  2 
       Periodic Signal Representation – Fourier Series

           2          x(t)
        w                    1                    t          x(t)
           T
-t                                        t   -T/2 to –T/4    -1
                                              -T/4 to +T/4    +1
                                  -1
                -T/4   T/4                    +T/4 to +T/2    -1


            8     nwT   4      nwT 
     an     sin         sin      
          nwT  4  nwT  2 
                                                    zero for all n
               n  2
         sin    sin n )
          4
         n  2  n

                              4           4
     We have, ao  0, a1  , a2  0, a3     ,.......
                                         3
      Periodic Signal Representation – Fourier Series

          2           x(t)
       w                     1                         t          x(t)
          T
-t                                             t   -T/2 to –T/4    -1
                                                   -T/4 to +T/4    +1
                                      -1
               -T/4    T/4                         +T/4 to +T/2    -1


     It can be easily shown that bn = 0 for all values of n. Hence,

             4                                        
      xt )   coswt  cos3wt + cos5wt  cos7wt + ....
                       1        1        1
                     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        x(t) T/2
                  A                                t          x(t)
-t                                       t    -/2 to –/2     A
                 -/2 /2                    -T/2 to -  /2    0

                                             +  /2 to +T/2    0

                            /2
                                    A
        T/2

         / xt )dt  T / 2Adt  T
     1                1
ao 
     T T 2
                         2 2                          2
 an   T xt )cosnwtdt   TAcosnwtdt
     2 T2
                                                    w
     T 2                T  2                        T
                       
       2 A  sin nw  2 4A   nw
           nw    nwT sin 2
       T            
                        2
       Periodic Signal Representation – Fourier Series
           -T/2        x(t) T/2
                   A                                    t           x(t)
-t                                            t    -/2 to –/2      A
                  -/2 /2                        -T/2 to -  /2     0

                                                  +  /2 to +T/2     0
                         
          2 A  sin nw  2 4A   nw                           2
     an       nw    nwT sin 2                         w
          T                                                  T
                             2


It can be easily shown that bn = 0 for all values of n. Hence, we have


                   A 2 A        
                                    sin nw / 2)
           xt )    +             nw / 2) cosnw
                   T   T          1
     Periodic Signal Representation – Fourier Series

                     A 2 A     
                                   sin nw / 2)
             xt )    +          nw / 2) cosnw
                     T   T       1


     Note:   sin  y ) y  0     for        y  nw 2  k
                                  nw             2k
     Hence: an  0         for         k  nw             k 1,2 ,3 ,...
                                   2               

A
     T
         0                                                             w

                      2               4
                                           