[PDF] - Course Aims To introduce: 1. How signals can be represented and PDF Document

SLIDE 1

EE1: Introduction to Signals and Communications

Professor Kin K. Leung EEE and Computing Departments Imperial College kin.leung@imperial.ac.uk Lecture One

2

Course Aims

To introduce:

1. How signals can be represented and interpreted in

time and frequency domains

2. Basic principles of communication systems
3. Methods for modulating and demodulating signals to

carry information from an source to a destination

SLIDE 2

Recommended text book

B.P Lathi and Z. Ding, Modern Digital and Analog Communication Systems, Oxford University Press

Highly recommended
Well balanced book
It will be useful in the future
Slides based on this book, most of the figures are taken from this book

3

Handouts

Copies of the transparencies
Problem sheets and solutions
Everything is on the web

http://www.commsp.ee.ic.ac.uk/~kkleung/Intro_Signals_Comm

4

SLIDE 3

Communications

5

Input transducer Transmitter Channel Receiver Output transducer Distortion and noise Input message Input signal Transmitted signal Received signal Output signal Output message

Classifications of Signals

Continuous-time and discrete-time signals
Analog and digital signals
Periodic and aperiodic signals
Energy and power signals
Deterministic and probabilistic signals

6

SLIDE 4

Continuous-time and discrete-time signals

A signal that is specified for every value of time t is a continuous-time signal
A signal that is specified only at discrete values of t is a discrete-time signals

7

Periodic and aperiodic signals

A signal g(t) is said to be periodic if for some positive constant T0,

g(t) = g(t + T0) for all t

A signal is aperiodic if it is not periodic

Same famous periodic signals: sin ω0t, cos ω0t, ejω0t, where ω0 = 2π/T0 and T0 is the period of the function (Recall that ejω0t = cos ω0t + j sin ω0t)

8

SLIDE 5

Periodic Signal

A periodic signal g(t) can be generated by periodic extension of any segment

f g(t) of duration T0

9

Energy and power signal

First, define energy

The signal energy Eg of g(t) is defined (for a real signal) as
In the case of a complex valued signal g(t), the energy is given by
A signal g(t) is an energy signal if Eg < ∞

10

Eg  g2(t)

 



dt.

2

*( ) ( ) ( ) .

g

E g t g t dt g t dt

   

 

 

SLIDE 6

Power

A necessary condition for the energy to be finite is that the signal amplitude goes to zero as time tends to infinity. In case of signals with infinite energy (e.g., periodic signals), a more meaningful measure is the signal power. A signal is a power signal if A signal cannot be an energy and a power signal at the same time

11

2 2 2

1 lim ( )

T g T T

P g t dt T

 





2 2 2

1 0 lim ( )

T T T

g t dt T

 

  



Energy signal example

Signal Energy calculation

12

Eg  g2(t)

 



dt  (2)2dt

1



 4etdt





 4  4  8.

SLIDE 7

Power signal example

Assume g(t) = Acos(ω0t + θ), its power is given by

13

2 2 2 2 2 2 2 2 2 2 2 2 2 2

1 lim cos ( ) 1 lim 1 cos(2 2 ) 2 lim lim cos(2 2 ) 2 2 2

T g T T T T T T T T T T T

P A w t dt T A w t dt T A A dt w t dt T T A   

       

            

   

Power of Periodic Signals

Show that the power of a periodic signal g(t) with period T0 is Another important parameter of a signal is the time average:

14

2 2 2

1 ( )

T g T

P g t dt T







gaverage  lim

T 

1 T g(t)dt

T 2 T 2



.

SLIDE 8

Deterministic and probabilistic signals

A signal whose physical description is known completely is a deterministic

signal.

A signal known only in terms of probabilistic descriptions is a random

signal.

15

Useful Signals: Unit impulse function

The unit impulse function or Dirac function is defined as Multiplication of a function by an impulse:

16

( ) 0 ( ) 1 t t t d t  

  

  



( ) ( ) ( ) ( ) ( ) ( ) ( ). g t t T g T t T g t t T dt g T   

 

    



Area = 1

Δ→0

SLIDE 9

Useful Signals: Unit step function

Another useful signal is the unit step function u(t), defined by Observe that Therefore Use intuition to understand this relationship: The derivative of a ’unit step jump’ is an unit impulse function.

17

1 ( ) = 0

t

t d t   



    



1 ( ) t u t t      

du(t) dt (t).

1 t

Useful Signals: Sinusoids

Consider the sinusoid x(t) = C cos(2πf0t + θ) f0 (measured in Hertz) is the frequency of the sinusoid and T0 = 1/f0 is the period. Sometimes we use ω0 (radiant per second) to express 2πf0. Important identities with and

18

 

1 1 cos sin , cos , sin , 2 2 1 cos cos cos( ) cos( ) 2 cos sin cos( )

jx jx jx jx jx

e x j x x e e x e e j x y x y x y a x b x C x 

  

                    

C  a2  b2   tan1 b a

SLIDE 10

Signals and Vectors

Signals and vectors are closely related. For example,
A vector has components
A signal has also its components
Begin with some basic vector concepts
Apply those concepts to signals

19

Inner product in vector spaces

x is a certain vector. It is specified by its magnitude or length |x| and direction. Consider a second vector y . We define the inner or scalar product of two vectors as <y, x> = |x||y| cos θ. Therefore, |x|2 = <x, x>. When <y, x> = 0, we say that y and x are orthogonal (geometrically, θ = π/2).

20

x y θ

SLIDE 11

Signals as vectors

The same notion of inner product can be applied for signals. What is the useful part of this analogy? We can use some geometrical interpretation of vectors to understand signals! Consider two (energy) signals x(t) and y(t). The inner product – correlation integral - is defined as For complex signals where y*(t) denotes the complex conjugate of y(t). Two signals are orthogonal if <x(t), y(t)> = 0.

21

x(t), y(t)  x(t)y(t)

 



dt x(t), y(t)  x(t)y*(t)

 



dt

Energy of orthogonal signals

If vectors x and y are orthogonal, and if z = x + y |z|2 = |x|2 + |y|2 (Pythagorean Theorem). If signals x(t) and y(t) are orthogonal and if z(t) = x(t) + y(t) then Ez = Ex + Ey . Proof for real x(t) and y(t) : since

22

2 2 2

( ( ) ( )) ( ) ( ) 2 ( ) ( ) 2 ( ) ( )

z x y x y

E x t y t dt x t dt y t dt x t y t dt E E x t y t dt E E

         

         

    

x(t)y(t)

 



dt  0.

z y x

SLIDE 12

Power of orthogonal signals

The same concepts of orthogonality and inner product extend to power signals. For example, g(t) = x(t) + y(t) = C1 cos(ω1t + θ1) + C2 cos(ω2t + θ2) and ω1 ≠ ω2. The signal x(t) and y(t) are orthogonal: <x(t), y(t)> = 0. Therefore,

23

P

x  C1 2

2 , P

y  C2 2

2 . P

g  P x  P y  C1 2

2  C2

2

2 .

Signal comparison: Correlation

If vectors x and y are given, we have the correlation measure as Clearly, −1 ≤ cn ≤ 1. In the case of energy signals: again −1 ≤ cn ≤ 1.

24

cn  cos  y,x x y 1 ( ) ( )

n y x

c y t x t dt E E

 





SLIDE 13

Best friends, worst enemies and complete strangers

cn = 1. Best friends. This happens when g(t) = Kx(t) and K is positive. The

signals are aligned, maximum similarity.

cn = −1. Worst Enemies. This happens when g(t) = Kx(t) and K is
negative. The signals are again aligned, but in opposite directions. The

signals understand each others, but they do not like each others.

cn = 0. Complete Strangers The two signals are orthogonal. We may

view orthogonal signals as unrelated signals.

25

Correlation

Why do we bother poor undergraduate students with correlation? Correlation is widely used in engineering. For instance

To design receivers in many communication systems
To identify signals in radar systems
For classifications

26

SLIDE 14

Correlation examples

Find the correlation coefficients between:

x(t) = A0 cos(ω0t) and y(t) = A1 sin(ω1t) cx,y = 0.
x(t) = A0 cos(ω0t) and y(t) = A1 cos(ω1t) and ω0 ≠ ω1 cx,y = 0.
x(t) = A0 cos(ω0t) and y(t) = A1 cos(ω0t) cx,y = 1.
x(t) = A0 sin(ω0t) and y(t) = A1 sin(ω1t) and ω0 ≠ ω1

cx,y = 0.

x(t) = A0 sin(ω0t) and y(t) = A1 sin(ω0t) cx,y = 1.
x(t) = A0 sin(ω0t) and y(t) = −A1 sin(ω0t) cx,y = -1.

27

Signal representation by orthogonal signal sets

Examine a way of representing a signal as a sum of orthogonal signals
We know that a vector can be represented as the sum of orthogonal

vectors

The results for signals are parallel to those for vectors
Review the case of vectors and extend to signals

28

SLIDE 15

Orthogonal vector space

Consider a three-dimensional Cartesian vector space described by three mutually orthogonal vectors, x1, x2 and x3. Any three-dimensional vector can be expressed as a linear combination of those three vectors: g = a1x1+ a2x2+ a3x3 where In this case, we say that this set of vectors is complete. Such vectors are known as a basis vector.

29

 xm,xn  m  n | xm |

2

m  n     

ai  g,xi |xi|2

Orthogonal signal space

Same notions of completeness extend to signals. A set of mutually orthogonal signals x1(t), x2(t), ..., xN(t) is complete if it can represent any signal belonging to a certain space. For example: If the approximation error is zero for any g(t) then the set of signals x1(t), x2(t), ..., xN(t) is complete. In general, the set is complete when N → ∞. Infinite dimensional space (this will be more clear in the next lecture).

30

g(t) ~ a1x1(t) a2x2(t)... aNxN (t)

SLIDE 16

EE1: Introduction to Signals and Communications

Professor Kin K. Leung EEE and Computing Departments Imperial College kin.leung@imperial.ac.uk Lecture Two

Trigonometric Fourier series

Consider a signal set

{1, cos ω0t, cos 2ω0t, ..., cos nω0t, ..., sin ω0t, sin 2ω0t, ..., sin nω0t, ...}

A sinusoid of frequency nω0t is called the nth harmonic of the sinusoid,

where n is an integer.

The sinusoid of frequency ω0 is called the fundamental harmonic.
This set is orthogonal over an interval of duration T0 = 2π/ω0, which is the

period of the fundamental harmonic.

32

SLIDE 17

Trigonometric Fourier series

The components of the set {1, cos ω0t, cos 2ω0t, ..., cos nω0t, ..., sin ω0t, sin 2ω0t, ..., sin nω0t, ...} are orthogonal as for all m and n means integral over an interval from t = t1 to t = t1 + T0 for any value of t1.

33

cos cos 2 sin sin 2 sin cos

T T T

m n n t m tdt T m n m n n t m tdt T m n n t m tdt                         

  

T0



Trigonometric Fourier series

This set is also complete in T0. That is, any signal in an interval t1 ≤ t ≤ t1 + T0 can be written as the sum of sinusoids. Or Series coefficients

34

g(t)  a0  a1cos0t  a2 cos20t  ... b

1sin0t  b2 sin20t  ...

 a0  an cosn0t

n1 



 bn sinn0t ( ),cos ( ),sin cos ,cos sin ,sin

n n

g t n t g t n t a b n t n t n t n t        

SLIDE 18

Trigonometric Fourier Coefficients

Therefore As We get

35

1 1 1 1

2

( )cos cos

t T t n t T t

g t n tdt a n tdt  

 

 



1 1 1 1 1 1

1 ( ) 2 ( )cos 1,2,3,... 2 ( )sin 1,2,3,...

t T t t T n t t T n t

a g t dt T a g t n tdt n T b g t n tdt n T  

  

    

  

1 1 1 1

2 2

cos 2, sin 2.

t T t T t t

n tdt T n tdt T  

 

 

 

Compact Fourier series

Using the identity where The trigonometric Fourier series can be expressed in compact form as For consistency, we have denoted a0 by C0.

36

an cosn0t  bn sinn0t  Cn cos(n0t  n) Cn  an

2  bn 2 n  tan1(bn an).

g(t)  C0  Cn cos(n0t  n)

n1 



t1  t  t1  T0.

SLIDE 19

Periodicity of the Trigonometric series

We have seen that an arbitrary signal g(t) may be expressed as a trigonometric Fourier series

ver any interval of T0 seconds.

What happens to the Trigonometric Fourier series outside this interval? Answer: The Fourier series is periodic of period T0 (the period of the fundamental harmonic). Proof: for all t and for all t

37

(t)  C0  Cn cos(n0t  n)

n1 



     

1 1 1

( ) cos cos 2 cos ( )

n n n n n n n n n

t T C C n t T C C n t n C C n t t         

     

                

  

Properties of trigonometric series

The trigonometric Fourier series is a periodic function of period T0 = 2π/ω0.
If the function g(t) is periodic with period T0, then a Fourier series

representing g(t) over an interval T0 will also represent g(t) for all t.

38

SLIDE 20

Example

39

Example

ω0 = 2π / T0 = 2 rad / s. We can plot

the amplitude Cn versus ω this gives us the amplitude spectrum
the phase θn versus ω (phase spectrum).

This two plots together are the frequency spectra of g(t).

40

n

1 2 3 4 Cn 0.504 0.244 0.125 0.084 0.063 θn

75.96
82.87
85.84
86.42

g(t)  C0  Cn cos(2nt  n)

n1 



SLIDE 21

Amplitude and phase spectra

41

Exponential Fourier Series

Consider a set of exponentials The components of this set are orthogonal. A signal g(t) can be expressed as an exponential series over an interval T0:

42

e jn0t n  0,1,2,...

1 ( ) ( )

jn t jn t n n T n

g t D e D g t e dt T

    

 

 

SLIDE 22

Trigonometric and exponential Fourier series

Trigonometric and exponential Fourier series are related. In fact, a sinusoid in the trigonometric series can be expressed as a sum of two exponentials using Euler’s formula.

43

( ) ( )

cos( ) 2 2 2

n n n n

j n t j n t n n n j jn t j jn t n n jn t jn t n n

C C n t e e C C e e e e D e D e

         

 

      

                       1 1 2 2

n n

j j n n n n

D C e D C e

   

 

Amplitude and phase spectra. Exponential case

44

SLIDE 23

Parseval’s Theorem

Trigonometric Fourier series representation The power is given by Exponential Fourier series representation Power for the exponential representation

45

P

g  C0 2  1

2 Cn

2 n1 



. g(t)  C0  Cn cos(n0t  n).

n1 



g(t)  Dne jn 0t.

n 



2 g n n

P D

 

 

Transition from Fourier Series to Fourier Transforms

Fourier series is used to represent a periodic signal or any signal over a fixed

period of time T0.

Fourier transform is used to represent a periodic or aperiodic signal over the

whole time horizon (subject to some mathematical requirements).

SLIDE 24

Aperiodic signal representation

We have an aperiodic signal g(t) and we consider a periodic version gT0(t)

f such

signal obtained by repeating g(t) every T0 seconds.

47

The periodic signal gT0(t)

The periodic signal gT0(t) can be expressed in terms of g(t) as follows: Notice that, if we let T0 → ∞, we have

48

( ) ( ).

T n

g t g t nT

 

 



lim ( ) ( ).

T T

g t g t





SLIDE 25

The Fourier representation of gT0(t)

The signal gT0(t) is periodic, so it can be represented in terms of its Fourier series. The basic intuition here is that the Fourier series of gT0(t) will also represent g(t) in the limit for T0 → ∞. The exponential Fourier series of gT0(t) is where and

49

0 ( )

,

jn t T n n

g t D e

  

 

Dn  1 T0 gT0(t)

T0 2 T0 2



e

 jn0tdt

2 . T    The Fourier representation of gT0(t)

Integrating gT0(t) over ( −T0/2, T0/2) is the same as integrating g(t) over ( −∞, ∞). So we can write If we define a function then we can write the Fourier coefficients Dn as follows:

50

1 ( ) .

jn t n

D g t e dt T

   





( ) ( )

j t

G g t e dt



  

 

1 ( ).

n

D G n T  

SLIDE 26

Computing the lim T0→∞ gT0(t)

Thus gT0(t) can be expressed as: Assuming (i.e., replace notation by ∆ω), we get In the limit for T0 → ∞, ∆ω → 0 and gT0(t) → g(t). We thus get:

51

1 T0   2

gT0(t)  Dne

jn0t n 



 G(n0) T0 e

jn0t n 



( )

( ) ( ) . 2

j n t T n

G n g t e



  

  

   

g(t)  limT0 gT0(t)  lim0 G(n) 2

n 



e j(n)t  1 2 G()e jt d.

 



0 where 0  2 T0 . Fourier Transform and Inverse Fourier Transform

What we have just learned is that, from the spectral representation G(ω) of g(t), that is, from we can obtain g(t) back by computing Fourier transform of g(t): Inverse Fourier transform: Fourier transform relationship:

52

( ) ( ) ,

j t

G g t e dt



  

  1 ( ) ( ) . 2

j t

g t G e d



  

 





( ) ( ) .

j t

G g t e dt



  

 

1 ( ) ( ) . 2

j t

g t G e d



  

 





( ) ( ). g t G  

SLIDE 27

Example

Find the Fourier transform of g(t) = e-atu(t).

Since , we have that Therefore:

53 ( ) ( )

1 ( ) ( ) .

at j t a j t a j t

G e u t e dt e dt e a j

  

 

         

    

 

1 2 2

1 1 ( ) , ( ) , ( ) tan ( ).

g

G G a j a a       



      1

j t

e

 

 limt  eate jt.

Some useful functions

The Unit Gate Function:

The unit gate function rect(x) is defined as:

54

rect(x)  0 x 1 2 1 x 1 2     

SLIDE 28

Some useful functions

The function sin(x)/x ‘sine over argument’ function is denoted by sinc(x):

sinc(x) is an even function of x.
sinc(x) = 0 when sin(x) = 0 and x ≠ 0.
Using L’Hospital’s rule, we find that sinc(0) = 1
sinc(x) is the product of an oscillating signal sin(x) and a monotonically decreasing

function 1/x.

55

Fourier transform of rectangular signal

Find the Fourier transform of g(t) = rect( t/τ ). Therefore

56

2 2 2 2

( ) 1 2sin( 2) ( ) sin( 2) sin ( 2). ( 2)

j t j t j j

t G rect e dt e dt e e j c

     

         

     

             

 

( ) sin ( 2) rect t c    

SLIDE 29

Fourier transforms of unit impulse and dc signals

Find the Fourier transform of the unit impulse δ(t): Therefore Find the inverse Fourier transform of δ(ω): Therefore

57

( ) 1.

j t j t t

t e dt e

 



    

 



1 1 ( ) . 2 2

j t

e d



    

 





( ) 1 t  

1 2 ( )   

Four transform of complex exponential signal

Find the inverse Fourier transform of δ( – 0): Therefore and

58

1 1 ( ) . 2 2

j t j t

e d e 



     

 

 



2 ( )

j t

e       2 ( )

j t

e



  



 

SLIDE 30

Fourier transform of cosine signal

Find the Fourier transform of the everlasting sinusoid cos(ω0t). Since and using the fact that and we discover that

59

 

1 cos( ) 2

j t j t

t e e

 





 

   

cos( ) . t                

 

2

j t

e      

 

2 ,

j t

e



  



  Summary of Fourier Transforms

Fourier transform of g(t): Inverse Fourier transform: Fourier transform relationship: Important Fourier transforms:

60

   

cos( ) . t                

( ) ( ) ,

j t

G g t e dt



  

  1 ( ) ( ) . 2

j t

g t G e d



  

 





( ) ( ). g t G  

( ) sin ( 2) rect t c    

( ) 1 t  

SLIDE 31

Some properties of Fourier transform

61

Some properties of Fourier transform

62

SLIDE 32

Some properties of Fourier transform

63

Some properties of Fourier transform

64

SLIDE 33

Fourier transform pair

65

Symmetry Property

Consider the Fourier transform pair
Then
Example

66

g(t)  G() G(t)  2g()

SLIDE 34

Scaling Property

Consider the Fourier transform pair
Then
Example

67

g(t)  G()

1 ( ) ( ) g at G a a   Time-Shifting Property

Consider the Fourier transform pair
Time shifting introduces phase shift
Example

68

g(t)  G()

g(t  t0)  G()e jt0

t g(t) g(t-to) to

SLIDE 35

Frequency-Shifting Property

Consider the Fourier transform pair
Exponential multiplication introduces frequency shift
Cosine multiplication leads to

69

g(t)  G()

g(t)e j0t  G( 0) g(t)e j0t  G(  0)

 

1 ( )cos ( ) ( ) 2 1 ( )cos ( ) ( ) 2

j t j t

g t t g t e g t e g t t G G

 

     



         

Frequency-Shifting Property

70

SLIDE 36

Frequency-Shifting Property

71

Fourier transform of periodic functions

Find the Fourier transform of a general periodic signal g(t) of period T0
A periodic signal g(t) can be expressed as an exponential Fourier series as

72

 

2 ( ) ( ) ( ) 2

jn t n n jn t n n n n

g t D e T g t F D e g t D n

 

     

     

        

  

SLIDE 37

Fourier transform of periodic functions

Consider a periodic waveform given by
where

73

non-zero 2 ( ) ( ) ( ) 0 otherwise

n n

T t g t w t nT w t



        



g(t)  Dn e

jn0t n n



w(t) W() g(t)  G()  2 Dn( n0)

n n



Dn  1 T0 g(t)e

 jn0t dt  T0



W(n0) T0

|D-4| |D-3| |D-2| |D-1| |D0| |D1| |D2| |D3| |D4| G() 2

40
30
20
0

0 0 20 30 40 ……  ……

Fourier transform of periodic functions

Find the Fourier transform of a unit impulse train δ(t) of period T0

74

w(t) (t)  W()  F((t)) 1 Dn  W(n0) T0  1 T0 g(t)  2 T0 (  n0)

n n



SLIDE 38

Convolution

The convolution of two functions g(t) and w(t),

Consider two waveforms
Convolution in time domain
Convolution in the frequency domain

75

( )* ( ) ( ) ( ) g t w t g w t d   

 

 



1 1 2 2

( ) ( ) ( ) ( ) g t G g t G    

1 2 1 2

( ) * ( ) ( ) ( ) g t g t G G   

1 2 1 2

1 ( ) ( ) ( )* ( ) 2 g t g t G G     Time Differentiation and Time Integration

Consider the Fourier transform relationship
The following relationship exists for integration
The following relationship exists differentiation

76

( ) ( ) g t G  

( ) ( ) (0) ( )

t

G g d G j       



 



dg(t) dt  jG() d ng(t) dt n  ( j)nG()

SLIDE 39

Important Fourier Transform Operations

77

Linear Systems

78

Linear Time Invariant System h(t) g(t) y(t)

SLIDE 40

Linear Systems (continued)

A system converts an input signal g(t) in an output signal y(t).
Assume the output for an input signal g1(t) is y1(t) and the output for an

input g2(t) is y2(t). The system is linear if the output for input g1(t) + g2(t) is y1(t) + y2(t).

A system is time invariant if its properties do not change with the time. That

is, if the response to g(t) is y(t), then the response to g(t - t0) is going to be y(t - t0)

79

Linear System g(t) y(t) Linear System g1(t) + g2(t) y1(t) + y2(t) Linear System g(t-t0) y(t-t0)

Unit impulse response of a LTI system

Consider a linear time invariant (LTI) system. Assume the input signal is a Dirac function δ(t). Call the observed output h(t).

h(t) is called the unit impulse response function.
With h(t), we can relate the input signal to its output signal through the

convolution formula:

80

y(t)  h(t)* g(t)  h()g(t  )d.

 



SLIDE 41

Physical interpretation of linear system response

δ(t)

t

81

Input Output δ(t-to)

t

to

t

???

t

h(t) : unit-impulse response

Physical interpretation of linear system response

δ(t)

t

82

Input Output δ(t-to)

t

to

t

to δ(t-to)

t

to δ(t)

t

???

t

h(t): unit-impulse response h(t-to) Time invariant

SLIDE 42

Physical interpretation of linear system response

δ(t)

t

83

Input Output δ(t-to)

t

to

t

to δ(t-to)

t

to δ(t)

t

h(t) h(t-to)

t

h(t)+h(t-to) to Linearity

Physical interpretation of linear system response

84

Input Output b δ(t-to)

t

to a δ(t)

t

a h(t)+b h(t-to) to Linearity g(nΔτ)

…

Δτ

t t

???

n t

…

input g(nΔτ): output g(nΔτ)Δτ h(t-nΔτ) g(t) y(t) = ∑ g(nΔτ)Δτ h(t-nΔτ) 

  

      d t h g t g t h t y ) ( ) ( ) ( * ) ( ) (

SLIDE 43

Intuitive explanation of the convolution formula

g(t) can be approximated as g(t) ≅ Σng(n∆τ)∆τδ(t−n∆τ).
In the limit as ∆τ→0 this approximation approaches

the true function g(t).

The response ŷ(t) of the LTI system to the input as

Σng(n∆τ)∆τδ(t−n∆τ) is going to be Σng(n∆τ)h(t−n∆τ)∆τ .

Thus, y(t) = lim∆→0Σng(n∆τ)h(t−n∆τ)∆τ=

85

g()h(t  )d.

 



Graphical Interpretation of Convolution (1)



  

 

u

du u t g u f t g t f ) ( ) ( ) ( * ) (

t

g(t)

t

f(t)

b

a

86

SLIDE 44

Graphical Interpretation of Convolution (2)



  

 

u

du u t g u f t g t f ) ( ) ( ) ( * ) (

u

g(-u)

u

g(u)

u

g(t-u) t<0

u

g(t-u) t>0

u

f(u)

b

a

a, b: positive

87

right shift by t left shift by t

Graphical Interpretation of Convolution (3)



  

 

u

du u t g u f t g t f ) ( ) ( ) ( * ) (

g(t-u)

<t<-a

u

f(u)

b

a

g(t-u) t>b

u

f(u)

b

a

g(t-u)

a<t<b

u

f(u)

b

a

Depending on t, the convolution integral is the area under f(u)g(t-u). Search “Convolution” on the Wikipedia site for an animation of convolution.

88

SLIDE 45

Convolution in the frequency domain

The convolution of two functions g(t) and h(t), denoted by g(t) ∗ h(t), is defined by the integral If g(t) ⇔ G(ω) and h(t) ⇔ H(ω) then the convolution reduces to a product in the Fourier domain H(ω) is called the system transfer function or the system frequency response or the spectral response. Notice that, for symmetry, a product in the time domain corresponds to a convolution in frequency domain. That is

89

( ) ( )* ( ) ( ) ( ) . y t h t g t h x g t x dx

 

  



( ) ( )* ( ) ( ) ( ) ( ). y t h t g t Y w H G     

1 2 1 2

1 ( ) ( ) ( )* ( ). 2 g t g t G G    

Ideal Low-Pass Filter

Ideal low-pass filter response Ideal low-pass filter impulse response

90

h(t)  W  sinc W t td

 

    H()  rect  2W

 e

 j td

SLIDE 46

Ideal High-Pass and Band-pass filters

91

Figure 1: Ideal high-pass filter Figure 2: Ideal band-pass filter

Practical filters

The filters in the previous examples are ideal filters.
They are not realizable since their unit impulse responses are everlasting

(Think of the sinc function).

Physically realizable filter impulse response h(t) = 0 for t < 0.
Therefore, we can only obtain approximated version of the ideal low-pass,

high-pass and band-pass filters.

92