Properties of Exponential Form of Fourier Series Coefficients
Note : c, and d, are exponential form of Fourier series coeflicients of x(t) and y()
n
Property
Continuous time
Fourier series
periodic signal
coefficients
Linearity
A x(t) + B y(t)
Time shifting
x(t -,)
e
Conjugation
x*(t)
Time reversal
x(-t)
x(at) ; a>0
jm20t
x(t)
(x(0) is period with period T/a)
Multiplication
x(t) y(t)
d
Differentiation
-x(t)
d
x(t) dt
Integration
Ac, + B d,
jns 20to
Frequency shifting
Time scaling
respectively.
n- m
C
C,
No change in Fourier coefficient)
too
-
m =-0
jn2,Cn
1
jns2,
(Finite valued and periodic only if a, 0)
Periodic convolution
x(t) y(t -) dt
Te, d
Cn = C-n
Symmetry of real signals
x(t) is real
Re{cn) = Re{c-n
Im{cn}
-Im{c-n)
Real and even
x(t) is real and even
C, are real and even
Real and odd
x(t) is real and odd
c, are imaginary and odd
Parseval's relation
Average power, P of x(t) is defined as,
P =
The average power, P in terms
of Fourier series coefficients is,
P=
n=-0
Note : l. The term lc. respresent the power in n" harmonic component ofx(t). The
total average
power in aperiodic signal is equal to the sum of power in all of its harmonics.
2. The term lc. for n= 0, l, 2, ...is the distribution of poweras a
function of frequency and
so itis called power density spectrum or power spectral density of the periodic signal
Gibbs Phenomenon
The exponential form of Fourier series of a continuous time periodic signal x(t) is given by
+00
x(t)=c, emag!
n=-0
The above equation is frequency domain representation of the signal x(t) as a sum of infinite serio
with each term in the series representing a harmonic frequency component. When the signal x() i
reconstructed or synthesised with only N number of terms of the infinite seires, the reconstructed sional
exhibits oscillations (or overshoot or ripples), especially in signals with discontinuities.
x(t)
A
Fig 4.17a.
0
2
n=1
Fig 4.17b.
t
x(t)4
n=3
A
Fig 4.17c.
x(t)A
n=5
A
Fig 4.17à
,(t)4
n=59
A
Fig 4.17e.
T
0
2
Fig 4.17 : Successively closer
to a square pulse signal.
Consider a periodic square pulse signalapproximations
shown in fig 4.17a. The reconstructed signal using
of Fourier series are shown in fig
N-tems
4.17b, c, d and e. (Refer MATLAB program
4.5
in
be observed that the
reconstructed signal exhibits oscillations and the oscillations are section 4.18). It can
points of discontinuities with increasing value of N.
compressed towards
Also it can be observed that, at the points ot
discontinuity, the Fourier series converges to average value of the
signal on either side of discontinuy
This phenomenon was named after a
famous
mathematician, Josiah Gibbs, as Gibbs phenomenon and
the oscillations are called Gibbs
oscillations. (Josiah Gibbs is a famous mathematician who first
mathematical explanation for this phe
x(t)+
Example
Determine the trigonometric form of Fourier series of the
waveform shown in fig4.1.1.
t
Solution
The waveform shown in fig 4.1.1 has even symmetry,
half wave symmetry and quarter wave symmetry.
b, = 0 and a, =
The mathema
x(t) cosnslt dt
Fig 4.1. 1.
iation of the square wave is,
x(t) =
r
T
t = 0 to
T
t =
A
T
to
4
2
Evaluation ofa,
T/4
A
COsnsz,t dt =
T/2
Acosns2,t dt + ;|-A) cosn2,t dt
T/4
4A
qT/4
2n. ]T/4
4A sinn2gt qTi2
sinng,!
T
Jo
no
sinn
4A
T
JTI4
n
4A
2r
=
2T T
sin
sin 0
T
2T
4A
T
2n
T
2A
4A
T
n
n
sin
n
4A
n
sin
2
sin n
4A Sinr
2
2
For even values of n, sin= 0
For odd values of n, sin= 1
2
a = 0
a, =
.. a, =
; for even values of n
4A
sin
4A
1x T
4A
3 x TU
ag =
a7 =
4A
5 X T
4A
7 x T
; for odd values of n
2
4A
T
sin 2
sin
sin
sin
TU
3n
4A
2
3T
5t
4A
2
5
7n
4A
2
7n
JTI4
2T T
T2
2
T
2nT
2A
2
T
Jo
4A
2 7T/2
sinnt
T
and so on.
2nT
sin
sin 0 = 0
sin n =0
for integer n
Fourier Series
The trigonometric form of Fourier series of x(t) is,
x(t) =
b, sinn2,t
Xa, cosnl,t +
+
2
n=1
n = 1
Here, a, =0,b =0 and a exists only for oddvalues of n.
n
a, cosns2,t
.. x(t) =
n= odd
= a, cos,t + a, COs 32,t + a, cos 52,t + a, cos7lot +
4A
=
4A
T
cOs2t
cos2,t
4A
3T
cos312%t +
4A
5t
cos502,t
cos 32,t
cos 502,!
3
5
4A
7
cos72t +
cos792,t ....
7
+
Fourier Transform
Development of Fourier Transform From Fourier Series
The exponential form of Fourier series representation of a periodic signal is given by,
x(t) = )c,eintlg
+00
D=-0
T/2
where, c, =
-jns20t
T
dt
-T/2
In the Fourier representation using equation (4.29), the c for various values of n are the spectral
components of the signal x(), located at intervals of fundamental frequency B Therefore the frequency
spectrumn is discrete in nature.
The Fourier representation of a signal using cquation (4.29) is applicable for periodic signals. For
rourier representation of non-periodic signals, let us consider that the fundamental period tends to
ninity. When the fundamental period tends to infinity, the tundamental frequency 2, tends to zero or
becomes very small. Since fundamental frequency 9is very small, the spectral components will lie very
ach other andso the frequency spectrum becomes continuous.
In order to obtain the Fourier representation of a non-periodic signal let us consider that the
fundamental frequency S2, is very small.
Let. 2,
On replacing S2, by A2 in cquation (4.29) we get.
N() = ) c,cn
n
On substituting for c, in the abOve cquationfrom cquation (4.30)(by taking t as dummy variable
for integration) we get.
T
dte'inAt
jnAs2r
x(t) =
..431)
-T/2
2r
We know that, S2, = 2nE, = :
T
Since S2, ’ AQ.
T
2T
..4.32)
2T
On substituting for
1
from equation (4.32) in equation (4.31) we get,
T/2
+00
X(t) enAlr drejnA2t
x(t)= )
2T
2
-T/2
T/2
x(t) ejn
dten
AN
n=--T/2
For non-periodic signals, the fundamental period T tends to infinity. On letting limit T
tends to
infinity in the above equation we get,
x(t)= T’
Lt
1
21
+
T/2
-jnA2r drejnat A
n=- -T/2
When T ’ ;
-jns2r do
.. x(t) =
eJn do
e'
1
2T
where, Xj2) = Xx(t) e h dr =
The equation (4.34) is Fourier
.(4.33)
Since t is a dummy variable, Let r =t.
dt
..(4.34)
transtorm of x() and equation (4.33) is nverse Fourier
transtom
Since the equalion (4.34) extracts the
frequency components of the signal,
usng
equation (4.34) is also called analysis of the signal
x(). Since the cquation (4,33) transtomation
combines the frequency
components of the signal, ihe nverse transtormalion using equation
is also called synthesis of the
(4.33)
signal x().
of x(t).
Definition of Fourier Transform
Let,
x(t) = Continuous tme signal
X(i) =Fourier transform of x()
The Fourier transform of continuous time signal, x(t) is defined as,
X(G2) = x()e
t
Also, X(iS) isdenoted as F{x(t)} where "F" is the symbol used todenote the Fourier transform
operation.
F{x()} = X(j2) = x(t) e Mdt
..(4.35)
Note : Sometimes the Fouiertransformis erpressed as afunction of cyclic frequency F, rather than
radian frequency S2. The Fourier transform as afunction of cyclic frequency F, is defined as,
+00
XGF) = x() e
dt
Condition for Existence of Fourier Transform
The Fourier transform of x() exists if it satisfies the following Dirichlet condition.
1. The x(t) be absolutely integrable.
+0
i.e.,
x(t) dt < o
Thex(t) should have a finitenumber of maxima and minima within any finite interval.
3The x() can have a finite number of discontinuities within any interval.
Definition of Inverse Fourier Transform
The inverse Fourier transform of X(2) is defined as,
+00
x(t) = F{X(j2)} =
1
2
X(j2) e dQ
....(4.36)
The signals x(t) and X(/2) are called Fourier transform pair and can be expressed as shown
below,
x(t)
F
X(j2)
Note : When Fourier transform is expressed as afunction ofcyclic frequency F he inverse Fourier
transform is defined as,
Properties of Fourier Transform
1. Linearity
Let, F{x, (t)} - X,(G2): F{x,()} - X,((2)
The linearity property of Fourier transform says that.,
Fla, x,() +a, x,(t)} =a, X,(2) ta, X,(j2)
Proof:
By definition of Fourier transform,
X,j2) =
x,(t) e- dt
and X,() =
x, t) e t
..4.40)
Consider the linear combination a x(t)+a, x,(t). On taking Fourier transform of this signalwe get,
T{o, x,(t) + 4, x,(1} = | a, x,t) + a, X,t)] e dt = , x, (t) e dt +
a,
x,t) e dt + a,x, t) e
2. Time shifting
If F{x()}= Xj2) then
F{x(t-t,)} = e ig! Xi2)
dt
Using equation (4.40)|
= 0, Xj2) + a, X,i2)
The time shifing property of Fourier transform says
dt
that,
Proof:
By definition of Fourier transform,
F{x()} = Xlij2) = | x(t) e t
....14.4)
Let, -
: F{t - t,)} =(t - t,)e" dt =x) e
..t=t+f
tol dr
On differentiating
dt = de
4
|xt) e
xe odr =e Mo xe)e de
Since t is a dummy
variable for ingetration
we can change to t.
-
+00
= eMox(t) e
t, =t
dt = eMo X(j2)
Usingequation (4.41)
3. Time scaling
The time scaling property of Fourier transform says that,
If F{x(t)}
=X(j2) then
1
F{x(a)}
x2)
Proof:
By definition of Fourier transform,
+00
xt) e dt
F{x) = Xlj2) =
F{xlat)) =
xlat) e
xt)e
dt =
|xt:)e)dr
|Put, at=t
dt =
: .. t=
The term
-;dt =
x\c) e
dt
d is
The above transform is applicable for positive values of "a".
similar to the form of Fourier
If "a" happens to be negative then it can be proved that,
transform except that
replaced by ().
dr = X)
Fklat} =
Hence in general,
Fxlat)) =
G
) for both positive and negative values of"a'
4. Time reversal
The time reversal property of Fourier transform says that,
If F{x(t)} =X(j2) then
F {x(-t)} - X(-j2)
Proof:
From time scaling property we know that,
FIat)) = 2)
Let, a =-1.
.:.
FIx(-)) = X(H)
is
S. Conjugation
The conjugation property ot Fourier transforn says that.
If F{x(t)} = X(ÇQ) then
Fx(0) = X'(-j2)
Proof:
By definition of Fourier transform,
F{xtM} = Xi2) =xlt) e dt
The term.xlt)e
dt
is similar to the form of Fourier transform
except that 2is replaced by -S2
| x(t)e-'dt = X(- i2)
- [X-i2)] = X*(-2)
6. Frequency shifting
says that,
The frequency shifting property of Fourier transform
If F{x(t)} = X(j2) then
F{ePal x(1)} - XG(2-2,)
Proof:
By definition of Fourier transform,
F{xt)} = Xij2) =
x(t) e" dt
The term | xt) e il0-20" dt is similar to
FfePo' xtn}
[|eo' xt] e" t=
xnefo' e dhe formof Fourier transformexcept that
2is replaced by 2-S
[x(0) e- -00" dt = x(i(2 - lo))
7. Time differentiation
The differentiation property of Fourier transform says that,
If F{x()}
X(j9) then
= j2 X(j2)
Proof:
Consider the definition of Fourier transform of xt).
F{x(n} = Xij2) =xt) e t dt
...4.42)
dt
d
= exloo) - e*" x-o) + j2
= j9
x)=0
x(t) e " t
e0
x(th e dt = 2Xi)
|Using equation (442)
8. Time integration
The integration property of Fourier transform says that,
IfF{x()} -X(j2) and X(0) =0then
j2
X(j2)
Proof:
Considera continuoUs time signal x(t). Let Xlj) be Fourier transform of xt). Since integration and differentiafion
are inverse operations, xt) can be expressed as shown below.
= x(t)
dt
On taking Fourier transform of the above equation we get,
Using time differentiation
=F{at1}
1
property of Fourier transform.
F{xt} =X(2)
X(j2)
9. Frequency differentiation
The frequency differentiation property of Fourier transform says that,
If F{x(t)} =X(2), then
d
F{t x(t)} = jdS2 X(j2)
Progf:
By definition of Fourier transform,
X(js2) = Fit) =
xlt) e
dt
On differentiating the above equation with respect to 2 we get,
d
+
-X(j2) =
dQ
+
x(t) e dt
d
do
e
dt
|Interchanging the order of
integration and differentiation
da
Using definition of
Fourier transform.
d
10. Convolution theorem
The convolution theorem of Fourier transform says that, Fourier transform of convolution of two
signals is given by the product of the Fourier transform of the individual signals.
i.e., if F{x,()} = X, (i2) and F{x,()}
X,(j2) then,
...4.43)
F{x,() * x,()} = X,(2) X,(S)
The equation (4.43) is also known a convolution property of Fourier transforrn.
With reference to chapter-2, section -2.9 we get,
x,() x,(t - ) dr
x(t) * x, (t) =
.(4.44)
where t is a dummy variable used for integration.
Proof:
Let x{t) and x,() be two time domain signals. Now, by definition of Fourier transform,
X(2) = Fo,t)} = x,t) e
dt
X,(i2) = Fh,tt} =
dt
x,(t) e
..44.45)
...4.46)
Using definition of Fourier transform we can write,
e- dt
=
Let, e
e
x e
xe
:
e
xe-d = e-t
where, M=t- and so, dM = dt
...4.47)
....4.48)
....4.49)
Using equations (4.48) and (4.49), the equation (4.47) can be written as,
+0
-
x) e dt xX, (M) ekM dM
...4.50)
In equation (4.50), tand Mare dummy variables used for integration,and so they can be
changed to t.
Therefore equation (4.50) can be writen as,
F{*,#) " x, }=x,() e dt xX t) e dt
= X(i2) X, (i2)
Using equations
|(4.45) and (4.46)
|1. Frequecy convolution
Let. F{x,(tO} = X,((2): F{x,(t)} - X,(G2).
The frequency convolution property of Fourier transform says that.
F{ (0) x,))
X,() x,(0 - 2) d
2
Proof:
By definition of Fourier transtorm,
F(xth} = X-) = xt e t
F{*,)x,)} = x4)x,t) e d
4.5)
By the definition of inverse Fourier transform we get,
1
|X,2) e dQ = 2r
x,t) =FX,(2)} = 2
(4.521
On substituting for X,(t) from equation (4.52) in equation (4.5l] we get,
|Here 2is the vairable
USed for ingetrafion.
Let us change 2 to À
)
2
x,(t) e e dt da
1
X,t) e - dt dA
X,à)
2
-
|Interchanging the
order of integration.
x,it) et- dt
|The term,
is similar to the fon of Fourier transforn
except thgt 2is reploced by S2
2
12. Parseval's relation
The Parseval's relation says that,
IfF{x(0)} =X(j2) then
x(t)}' dt =
Proof:
2
Let x(t) be a continuous time signal and x") be conjugate of x().
Now, lxtI' = x) x*)
On integrating the above equation with respect to twe get,
(4.53)
By definiticn of inverse Fourier transform, we can write,
xt) =F{Xje} =
21
transform
Fourier
definitionof |Using
.:X(i2)=2rx(12)
xfi).
x-2)= function
x,(-i2)
2rx,l-j2)
even For Note:
:.X,lj2)
2n =
= d
dn x,t)e :x2)= 2
(4.55)equation |Using
interchanging
andt| is2
Replacingt
-tby
x1-)-X2)e
+3
2 x,=(t)
transform,
Fourier inverse definition
of
T
By
4.55)
form.
X,i2) F{x,))=
similar
in
(-j2)
functions similar
functions
below.
are X,j2)
and x,I)
and X,(2) F{x,l1}=
Let,
let,
Proof:
jS2) X,( then
2rx,
(j2)
X, ’(t) x,
If
propertyis duality
Alternatively
shown expressed
as
are j2) rx,(2are (j2) X,
2rx,(-j2),
i.c.,
similar
are (j2) X,and x,(t) i.e., (G2),
X,(j2) (t)}-
(j) = X,
then
X,x,(t)=
ifand
and X,(j2)
F{x,
F{x,(0}=
If
Duality
called
isit soand
x(). signal
function
s2 of
energy
term
The
density
X(j2)
energy spectrum
or
epresents
the
the density
of spectral
energy
as distribution
of
13.
Not:e
2n
S2-0
=
2r
d2
XijQ) |X*i2)
Xlj X*{j)= X(jS)
transform.
definition
of
1
2n
Fourier
Using
tegration.
erchanging
the
1
2n
order
of
writlen
as,
beca(4.53)
n equation
the(4.54)equation
dn )e
.4.54)
gei. equation
we
above
x|
Using
2n
thconjugate
e of
x*(t)
=
faking
On
14. Area under a time domain signal
Area under x(t) =x() dt
If x(t) and X(i2) are Fourier transfom pair,
then,x(t) dt= X(0)
where, X(0)= j2Lt’0 X((2)
Proof:
By definition of Fourier transform,
Xi2) =xl|) e d
:.X(0) =
Xj2) = Lt
x(0) e- dt
= (x() e d=) dt
i.x() dt = X{0)
15. Area under a frequency domain signal
+o0
Area under X(j2) =X((2) d2
If x(t) and X(() are Fourier transform pair,
then,XG2) dQ =2r x(0)
where, x(0)= t ’ Lt0 x(t)
Proof:
By definition of inverse Fourier transform,
x[t) =
1
2r
.. x0)= Lt x(t) = LI
1-0 2
2
| X(jS2) d = 2r x(0)
Summary of Properties of Fourier Transform
Let, F{x()} = X((): Fx.)} = X.G); F{(t)} = ,0s Frequency domain signal
Time domain signal
Property
a, X,() +a, X,G2)
Linearity
a, x,() +a,x,(0)
Time shi
x(t - )
Time scaling
x(at)
Time reversal
x()
Conjugation
x()
Frequency shifting
X(9)
X(j)
X(-jN)
XG(2-))
eo! x(t)
x(t)
Time differentiation
jS2 XGL)
Time integration
x(t) dr
Frequency differentiation
t x(t)
Xj) =
X(0) S(9)
400
Time convolution
(t)*x,(t)= x() X, (t- t)dt
X,G2) X,i2)
0
1
Frequency convolution
(or Multiplication)
2
x,() x,()
X(j2)) - X(2)
LXj2) = -LX-j2)
Symmetry of real signals
x(t) is real
Re(X(j2))
Re(X(-j2))
Im(X((2))
- Im{X(-j))
Real and even
x(t) is real and even
X(i2) are real and even
Real and odd
x(t) is real and odd
X(i2) are imaginary and odd
Duality
[ie., x,() and X,(2) are similar functions]
then X,(j)= 2rx(jS2) [i.e.,X,() and 2rx,(j) are similar functions
If
x,() = X,G)
+00
= 2r x(0)
Area under a
domain signalfrequency
Area under a time
domain signal
x() t =X(o)
Energy in frequency domain is,
Energy in time domain is,
+00
2T
Parseval's relation
1
2T