Transcript Corporate Communication Network

Properties of Delta Function
• Delta function is a particular class of functions which plays
a significant role in signal analysis.
• They have simple mathematical form but they are either
not finite everywhere or they do not have finite derivatives
of all orders everywhere. They are also known as
singularity functions.
• Unit impulse function or Dirac delta function is a
singularity function of great importance. This function has
the property b
 f (t0 ) a  t0  b
a f (t ) (t  t0 )dt  0
elsewhere
• for any f(t) continuous at t  t 0 for finite t0.
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• The impulse function selects or sifts out a particular value
of the function f(t), namely, the value at t=t0, in the
integration process.
b
• If f(t) = 1, then the above equation becomes a  (t  t0 )dt  1
• Therefore (t) has unit area.
• Also  (t  t0 )  0 for all t  t0
• The symmetry properties of delta function stipulates that
 (t )   (t )
1
• Time scaling property -- t (at )   (t )
a
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• Multiplication by a time function f (t ) (t  t0 )  f (t0 ) (t  t0 )
• Relationship with the Unit step function, which is given by
t  t0
1
u (t  t0 )  
0
t  t0
• For f(t)=1, we have

b

t
a

1
f ( ) (  t0 )d    (  t0 )d  

0
t
t  t0 
  u (t  t0 ) or
t  t0 
 (  t0 )d  u (t  t0 )
• Thus the derivative of the unit step function yields a delta
function
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Fourier Transform
• We consider an aperiodic function f(t) as shown
• We wish to represent this function as a sum o fexponential
functions over the entire interval  ,  . For this
purpose, we construct a new periodic function fT (t ) , with
period T so that the function f(t) is forced to repeat itself
completely every T seconds. The original function can be
obtained by letting T  
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• The new function fT(t) is a periodic function and
consequently can be represented by an exponential Fourier
Series
fT (t ) 

F e
n  
n
jn 0t
1 T /2
, Fn  
fT (t )e  jn 0t dt ;  0  2 / T
T T / 2
• we define  n  n 0 ; F ( n )  TFn Using these
definitions we obtain

T /2
1
j n t
fT (t )   F ( n )e , F ( n )  
fT (t )e  j nt dt ;
T / 2
n   T
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• The spacing between adjacent lines in the line spectrum of
fT(t) is   2 / T Using this relation for T we obtain
the alternate
form

j n t 
fT (t )   F ( n )e
2
n  
• Now as T becomes very large,  becomes smaller and
the spectrum becomes denser. As T   , the discrete
lines in the spectrum merge and the frequency spectrum
becomes continuous. Thus,
Lim
T 
1
fT (t )  Lim
T  2

j n t
F
(

)
e
 n 
n  
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1 
jt
f (t ) 
F
(

)
e
d

• becomes
2 

 jt
• similarly F ( ) 
f
(
t
)
e
dt


1




F  
F
(

)


f
t
;
f
(
t
)


• Symbolically
• The complex Fourier series coefficients can be evaluated
in terms of the Fourier Transform
1
Fn  F  
T
  n 0
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Spectral Density Function
• The area under the spectral density function F() gas the
dimensions voltage. Each point on the F() curve
contributes nothing to the representation of f(t). It is the
area that contributes. But each point does indicate the
relative weighting of each frequency component. The
contribution of a given frequency band to the
representation of f(t) may be found by integrating to find
the desired area.
• A periodic waveform has its amplitude components at
discrete frequencies. At each of these discrete frequencies
there is some definite contribution. To portray the
amplitude components of a periodic waveform on a
spectral-density graph requires a representation with area
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• equal to the respective amplitude components yet
occupying zero frequency width. This can be done by
representing each amplitude component of the periodic
function by an impulse function. The area of the impulse
is equal to the amplitude component and the position of the
impulse is determined by the particular discrete frequency.
• Summarizing, a signal of finite energy can be described by
a continuous spectral density function. This spectral
density function is found by taking the Fourier transform
of the signal.
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• e.g. Find the Fourier transform of a gate function defined

t  /2
as
1
rect t /    
t  /2

0
• We have

F ( )   rect t /  e
 jt

 e
e
 j / 2
dt  
e  jt dt
 / 2

j / 2
 /2

sin  / 2
/  j   
 / 2
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Fourier Transform Involving Impulse Functions
• The Fourier transform of a unit impulse is

 (t )    (t )e  jt dt  e j 0  1


 (t  t0 )    (t  t0 )e

 j t
dt  e
j t 0
• The phase spectrum of the time-shifted impulse is linear
with a slope that is proportional to the time shift.
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Complex Exponentials
• We would expect that the spectral density of
 j 0 t
e
will be concentrat ed at   0 as shown
•
1 
jt


     0  




e
d
0

2 
1  j 0 t

e
2
Taking Fourier tr ansform of both sides we have
1


1
     0  
 e  j 0t . Thus,
2
 e  j 0t  2    0 
-1


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Sinusoids
• The sinusoidal signals cos  0t and sin  0t can be written
in terms of the complex exponentials using Euler’s
identities
 1 j 0 t 1  j  0 t 
cos  0t   e  e

2
2

     0       0 
 1 j 0 t 1  j 0 t 
sin  0t   e  e

2
2

     0       0  / j
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Signum Function and the Unit Step
• The Signum function, sgn(t), changes sign when its
argument is zero
1
t 
sgn( t )   0
t 
 1
t 0
t 0
t0
• The signum function has an average value of zero and is
piecewise continuous, but not absolutely integrable. To
a t
e
make it absolutely integrable we multiply sgn(t) by
• and then take the limit as a  0 sgn( t )   Lim e a t sgn t 
 

a 0
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• Interchanging the operations of taking the limit and
integrating we have
 e
 lim   e
sgn t   lim
a 0

a t

0
a 0

sgn t e  jt dt
 a  j t


dt   e a  j t dt
0

  2 j  2
 lim  2


2
a 0 a  

 j
1 1


• The unit step function can be expressed as u t   sgn t 
2 2
1
• Thus u t      
j
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Periodic Functions
• A periodic function, of period T, can be expressed as
fT t  

jn 0t
F
e
where  0  2 / T
 n
n  
• Taking the Fourier transform, we find


 

 fT t     Fn e jn 0t    Fn  e jn 0t
n  
 n 
 2


 F    n 
n  
n
0
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Properties of Fourier Transform
Linearity
a1 f1 t   a2 f 2 t   a1F1    a2 F2  
• This follows directly from the integral definition of Fourier
transform
Complex Conjugate
• For any complex signal we have
 f * t   F *   . Due to
 
f t   
*
f * t e jt dt    f t e jt dt   F *   
 


• If f(t) is real, then f * t   f t  and F *     F  
*


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Symmetry
• Any signal can be expressed as a sum of an even function
and an odd function
 f e t   Fe   (real) and  f o t   Fo   (and imaginary) . Proof :
 f e t   


f e t e
 j t
dt  


f e t  cos tdt  j 


f e t sin tdt

 2  f e t  cos tdt
0
Duality
• Duality exists between time and frequency domain as
shown below
 f (t )  F  , then F t   2f   
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• The proof can be done by interchanging t and  in the
Fourier transform integral.

1
F     f t e dt ; f t  

2
interchang ing t and  we get

F t    f  e

 jt
 jt
d ;


-
1
f   
2
F  e  jt d


-
F t e jt dt

Thus 2f      F t e  jt dt  F t 
-
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• e.g. It is given that rect t   Sa / 2 find Sat / 2
• Let F    Sa / 2  F t   Sat / 2 
F t   2rect     2rect  
•
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Coordinate Scaling
• The expansion or compression of a time waveform affects
the spectral density of the waveform. For a real-valued
scaling constant  and any pulse signal f(t),

 
 f t   F   and  f t    f t e  jt dt

  
1

for x  t we have  f t    f  x    f x e  jx /  dx / 

 
 F   for   0;
  
 
and   F   for   0
  
1
1
 
or simply  f t   F  
  
1
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• If  is positive and greater than unity, f(t) is compressed,
and its spectral density is expanded in frequency by 1/  .
The magnitude of the spectral density also changes -- an
effect necessary to maintain energy balance between the
two domains. If  > 0 but less than unity, f(t) is an
expanded version of f(t) and its spectral density is
compressed. When  < 0, f(t) is reversed in time
compared to f(t) and is expanded or compressed depending
whether | | is greater than or less than unity.
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Time Shifting
 f t  t0   F  e  jt0 Proof :

 f t  t0    f t  t0 e  jt dt ; let x  t  t0 then we get


 f  x    f  x e
 j  x  t 0 

dx  e
 j t 0



f  x e
 j x
dx
Frequency Shifting


Proof : f t e    f t e
 f t e j0t  F    0 ;
j 0t


j 0 t  j t
e

dt   f t e  j ( 0 )t dt  F    0 

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Differentiation and Integration
• If df/dt is absolutely integrable, then
d
f t   jF  ;
dt
1 
jt
Proof : f t  
F  e d ;

2 
d
1 d 
1
jt
f t  
F  e d  j

dt
2 dt 
2



F  e jt d
• The corresponding integration property is
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• Consider the function g(t) defined as g t    f t 'dt '
• Let g(t) have Fourier transform G(). Now
dg t 
 f t . From above we see that jG    F  
dt
t
• However, for g(t) to have a transform G() must exist.
g t   0 This means   
One condition is that lim
f t dt  0
t 




F
0

0
• which is equivalent to F(0) = 0. If
then g(t) is no
longer an energy function and the transform will include an
impulse function

t

1
f t 'dt ' 
F    F 0  
j
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Time Convolution
• There are two ways of characterizing a system -- frequency
response and impulse response. The two can be related
using the principle of convolution.
• For the test signal f t    t    , the system impulse
response is defined as T t     ht ,  where  is the
delay or age variable. If the system is time-invariant, h(t,)
takes the special form T t     ht    . The input
signal f(t) may be expressed in terms of impulse functions


by f t  
f     t d 
f   t   d



• If we define g t   T




f τ δ t τ dτ
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• From integration theory, we can rewrite this as



g t   T  lim  f  n  t   n  
 0 n 

• Using the principle of superposition, we move the system
operator inside the summation. Also, the f(n) are the
weights (areas) of the impulse functions and are constants
for each impulse. Therefore we have
g t   lim  f  n T t   n 
 0

• Therefore we have g t    f  ht , d

• This is a key result in signal analysis for it links the input
to the output by means of an integral operation.
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• The equation reduces to

g t    f  ht   d  f t   ht 

• This is known as the convolution integral.
• An important property of the Fourier transform is that it
reduces the convolution integral to an algebraic product.
 f  h  F  H  
• Proof


 f  h    f  ht   d e  jt dt

 
 

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• Changing the order of integration and integrating with
respect to t first yields


 f  h   f    ht   e  jt dt  d using time delay property, we have
 


ht     e  j H  . Thus we have

 f  h  


f  e  j H  d  F  H  
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Frequency Convolution
• A dual to the preceding property can be established
 f1 t   F1  ,
 f 2 t   F2  
Then F1  F2  2 f1 f 2 
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Some Convolution Relationships
 

 

• The convolution integral g t   f  h t   d
holds as long as the system is linear, time-invariant, and
causal. Thus h(t) = 0 for all t < 0 and there is no
contribution to the integration for (t-) < 0.
• Often the input, f(t), also satisfies f(t) = 0 for t < 0.
• Properties of Convolution
• Commutative Law -- f1  f 2  f 2  f1
• Distributive Law --
f1   f 2  f 3   f1  f 2  f1  f 3
• Associative Law --
f1   f 2  f 3    f1  f 2  f 3
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Convolution involving Singularity Functions
• The unit step response is the indefinite integral of the unit
impulse response as shown



0
u t   ht    u  ht   d   ht   d . Let x  t  
then u  h   hx dx
t

• This provides a technique for determining the impulse
response of a system in the laboratory.
• Convolution with the unit impulse function gives

f   t  t0    f   t  t0   d  f t  t0 

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• Example: Find f  h, for the f t , ht 
as shown:
f t   A sin tut , ht    t    t  2
•

A sin u   t      t  2   d

 A sin t u t   A sin  t  2 u t  2
g t   f  h  
0

g t    A sin t
0


t0
0t 2
t2
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Graphical Interpretation of Convolution
• The graphical interpretation of convolution permits to
understand visually the results of the more abstract
mathematical operations. For instance

f1  f 2   f   f 2 t   d

• The required operations are as listed below:
 Replace t by  in f1(t) giving f1()
 Replace t by (- ) in f2(). This folds the function f2() about
the vertical axis passing through the origin of the  axis.
 Translate the entire frame of reference of f2(- ) by an amount
t. Thus the amount of translation, t, is the difference
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between the moving frame of reference and the fixed frame
of reference and the fixed frame of reference. The origin
in the moving is at  = t, the origin in the fixed frame is at
 = 0. The function in the moving frame represents f2(t- ).
The function in the fixed frame represents f1(t).
 At any given relative shift between the frames of reference,
e.g. t0, we must find the area under the product of the two
functions
 f   f t
1
2
0
     f1 t   f 2 t 
t t 0
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 This procedure is to be repeated for different values of t=t0
by successively progressing the movable frame and finding
the values of the convolution integral at those values of t.
 If the amount of shift of the movable frame is along the
negative  axis, t is negative. If the shift is along the
positive  axis, t is positive.
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• Example:
• Find the convolution of a rectangular pulse and a triangular
pulse
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