Lecture-1_classifications-of-signals.ppt

SIGNALS Information expressed in different forms

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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

5

5

1

8

0

2

10

3

8

4

5

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

t

for 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)

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

