Communication Fundamental s 1 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 • for any f(t) continuous at for finite t 0 . b a b t a t f dt t t t f elsewhe 0 ) ( ) ( ) ( 0 0 0 0 t t
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Communication Fundamentals1 Properties of Delta Function Delta function is a particular class of functions which plays a significant role in signal analysis.
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Communication Fundamentals 1
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
• for any f(t) continuous at for finite t0.
b
a
btatfdttttf
elsewhere0
)()()( 00
0
0tt
Communication Fundamentals 2
• 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.
• If f(t) = 1, then the above equation becomes
• Therefore (t) has unit area.
• Also
• The symmetry properties of delta function stipulates that
• Time scaling property --
1)( 0 b
adttt
00 allfor 0)( tttt
)()( tt )(
1)( t
aatt
Communication Fundamentals 3
• Multiplication by a time function
• Relationship with the Unit step function, which is given by
• For f(t)=1, we have
• Thus the derivative of the unit step function yields a delta function
)()()()( 000 tttftttf
0
00 0
1)(
tt
ttttu
t
b
a
t
ttudt
ttutt
ttdtdtf
)()(
or)(0
1)()()(
00
00
000
Communication Fundamentals 4
Communication Fundamentals 5
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 , 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
,)(tfT
T
Communication Fundamentals 6
• The new function fT(t) is a periodic function and consequently can be represented by an exponential Fourier Series
• we define Using these definitions we obtain
TdtetfT
FeFtfT
T
tjnTn
n
tjnnT /2;)(
1,)( 0
2/
2/
00
nnn TFFn )(;0
;)()(,)(1
)(2/
2/
T
T
tjTn
n
tjnT dtetfFeF
Ttf nn
Communication Fundamentals 7
• The spacing between adjacent lines in the line spectrum of fT(t) is Using this relation for T we obtain the alternate form
• Now as T becomes very large, becomes smaller and the spectrum becomes denser. As , the discrete lines in the spectrum merge and the frequency spectrum becomes continuous. Thus,
T/2
n
tjnT
neFtf
2)()(
T
n
tjn
TT
T
neFtf LimLim
)(2
1)(
Communication Fundamentals 8
• becomes
• similarly
• Symbolically
• The complex Fourier series coefficients can be evaluated in terms of the Fourier Transform
deFtf tj)(
2
1)(
dtetfF tj )()(
FtftfF 1)(;)(
0
1
n
n FT
F
Communication Fundamentals 9
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
Communication Fundamentals 10
• 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.
Communication Fundamentals 11
• e.g. Find the Fourier transform of a gate function defined as
• We have
2/0
2/1/
t
ttrect
2/
2/sin/
/)(
2/2/
2/
2/
jee
dtedtetrectF
jj
tjtj
Communication Fundamentals 12
Communication Fundamentals 13
Fourier Transform Involving Impulse Functions
• The Fourier transform of a unit impulse is
• The phase spectrum of the time-shifted impulse is linear with a slope that is proportional to the time shift.
0)()(
1)()(
00
0
tjtj
jtj
edtetttt
edtett
Communication Fundamentals 14
Complex Exponentials
• We would expect that the spectral density of
• as shown
0
01-
001
2
Thus, .2
1
have wesidesboth of ansformFourier tr Taking2
12
1
0
0
0
tj
tj
tj
tj
e
e
e
de
0at edconcentrat be will0 tje
Communication Fundamentals 15
Sinusoids
• The sinusoidal signals can be written in terms of the complex exponentials using Euler’s identities
tt 00 sin and cos
j
eet
eet
tjtj
tjtj
/
2
1
2
1sin
2
1
2
1cos
00
0
00
0
00
00
Communication Fundamentals 16
Communication Fundamentals 17
Signum Function and the Unit Step
• The Signum function, sgn(t), changes sign when its argument is zero
• The signum function has an average value of zero and is piecewise continuous, but not absolutely integrable. To make it absolutely integrable we multiply sgn(t) by
• and then take the limit as
01
00
01
)sgn(
t
t
t
t
tt
tae
0a teLimt ta
asgn)sgn(
0
Communication Fundamentals 18
• Interchanging the operations of taking the limit and integrating we have
• The unit step function can be expressed as
• Thus
ja
j
dtedte
dtetet
a
tjatja
a
tjta
a
22lim
lim
sgnlimsgn
220
0
00
0
ttu sgn2
1
2
1
j
tu1
Communication Fundamentals 19
Periodic Functions
• A periodic function, of period T, can be expressed as
• Taking the Fourier transform, we find
TeFtfn
tjnnT /2 where 0
0
nn
tjn
nn
n
tjnnT
nF
eFeFtf
02
00
Communication Fundamentals 20
Communication Fundamentals 21
Communication Fundamentals 22
Properties of Fourier TransformLinearity
• This follows directly from the integral definition of Fourier transform
Complex Conjugate• For any complex signal we have
• If f(t) is real, then
22112211 FaFatfatfa
**
**
** toDue .
Fdtetfdtetftf
Ftf
tjtj
FFtftf ** and
Communication Fundamentals 23
Symmetry
• Any signal can be expressed as a sum of an even function and an odd function
Duality• Duality exists between time and frequency domain as
shown below
0cos2
sincos
:Proof .imaginary) (and and (real)
tdttf
tdttfjtdttfdtetftf
FtfFtf
e
eetj
ee
ooee
ftFFtf 2 then ,)(
Communication Fundamentals 24
• The proof can be done by interchanging t and in the Fourier transform integral.
tFdtetFf
dtetFfdeftF
deFtfdtetfF
tj
tjtj
tjtj
-
-
-
2 Thus
2
1 ;
get we and t inginterchang2
1 ;
Communication Fundamentals 25
• e.g. It is given that find
• Let
•
2/Satrect 2/tSa
rectrecttF
tSatFSaF
22
2/2/
Communication Fundamentals 26
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),
Ftf
FF
dxexfxftftx
dtetftfFtf
xj
tj
1simply or
0for 1
and;0for 1
/ have wefor
and 1
/
Communication Fundamentals 27
• 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.
Communication Fundamentals 28
Time Shifting
Frequency Shifting
dxexfedxexfxf
ttxdtettfttf
eFttf
xjtjtxj
tj
tj
00
0
get wethenlet ;
:Proof
000
0
0
)(
0
000
0
:Proof
;
Fdtetfdteetfetf
Fetf
tjtjtjtj
tj
Communication Fundamentals 29
Differentiation and Integration• If df/dt is absolutely integrable, then
• The corresponding integration property is
deFjdeFdt
dtf
dt
d
deFtf
Fjtfdt
d
tjtj
tj
2
1
2
1
;2
1 :Proof
;
Communication Fundamentals 30
• Consider the function g(t) defined as
• Let g(t) have Fourier transform G(). Now
• However, for g(t) to have a transform G() must exist. One condition is that This means
• 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
''
tdttftg
FGjtfdt
tdg thatsee weabove From.
0lim
tgt
0dttf
00 F
01
'' FFj
dttft
Communication Fundamentals 31
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 , the system impulse response is defined as where is the delay or age variable. If the system is time-invariant, h(t,) takes the special form . The input signal f(t) may be expressed in terms of impulse functions by
• If we define
ttf ,tht T
thtT
dtfdtftf
dττtδτftg T
Communication Fundamentals 32
• From integration theory, we can rewrite this as
• 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
• Therefore we have
• This is a key result in signal analysis for it links the input to the output by means of an integral operation.
nnn tfTtg
0lim
nn tftg T0
lim dthftg ,
Communication Fundamentals 33
• The equation reduces to
• 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.
• Proof
thtfdthftg
HFhf
dtedthfhf tj
Communication Fundamentals 34
• Changing the order of integration and integrating with respect to t first yields
HFdHefhf
Heth
ddtethfhf
j
j
tj
have weThus.
have weproperty,delay timeusing
Communication Fundamentals 35
Frequency Convolution
• A dual to the preceding property can be established
2121
2211
2Then
,
ffFF
FtfFtf
Communication Fundamentals 36
Some Convolution Relationships
• The convolution integral
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 --
• Distributive Law --
• Associative Law --
dthftg
1221 ffff 3121321 fffffff
321321 ffffff
Communication Fundamentals 37
Convolution involving Singularity Functions
• The unit step response is the indefinite integral of the unit impulse response as shown
• This provides a technique for determining the impulse response of a system in the laboratory.
• Convolution with the unit impulse function gives
t
dxxhhu
txdthdthuthtu
then
Let . 0
000 ttfdttfttf
Communication Fundamentals 38
• Example: Find as shown:
•
thtfhf , for the,
20
20sin
00
22sinsin
2sin
2,sin
t
ttA
t
tg
tutAtutA
dttuAhftg
ttthttuAtf
Communication Fundamentals 39
Graphical Interpretation of Convolution
• The graphical interpretation of convolution permits to understand visually the results of the more abstract mathematical operations. For instance
• 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
dtffff
221
Communication Fundamentals 40
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
0
21021tt
tftftff
Communication Fundamentals 41
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.
Communication Fundamentals 42
• Example:
• Find the convolution of a rectangular pulse and a triangular pulse