Signals & Systems - Fourier Series for Continuous Time & Discrete Time Signals

1. • Early ideas of decomposing a
periodic function into the sum of
simple oscillating functions date
back to the 3rd century BC.
• The Fourier series is named in
honour of Jean-Baptiste Joseph
Fourier (1768–1830), who made
important contributions to the
study of trigonometric series.
• Fourier introduced the series for
the purpose of solving the heat
equation in a metal plate,
publishing his initial results in his
1807 and publishing Analytical
theory of heat in 1822.

2. • The heat equation is a partial differential equation. Prior to
Fourier's work, no solution to the heat equation was known in
the general case, although particular solutions were known if
the heat source behaved in a simple way, in particular, if the
heat source was a sine or cosine wave.
• Fourier's idea was to model a complicated heat source as a
superposition (or linear combination) of simple sine and
cosine. This superposition or linear combination is called the
Fourier series.
• Although the original motivation was to solve the heat
equation, it later became obvious that the same techniques
could be applied to a wide array of mathematical and physical
problems.

3. Determining the
Fourier Series Representation of a
Continuous Time Signal
tjntjk
k
k
tjn
eeaetx 000
)(  




 1)( )/2(0





tTjk
k
k
tjk
k
k
eaeatx 
A periodic CT signal can be expressed as a linear combination of harmonically related
complex exponentials of the form :-
Multiplying both sides with , we get :-
tjn
e 0
Now if we integrate both sides from 0 to T, we have :-
dteeadtetx
T
tjntjk
k
k
T
tjn






00
000
)( 

4. Interchanging the order of integration and summation :-
 2)(
00
000






 




T
tjntjk
k
k
T
tjn
dteeadtetx 
Applying Euler’s formula to the bracketed Integral :-
 
TTT
tnkj
tdtnkjtdtnkdte
0
0
0
0
0
)(
)sin()cos(0



T
tnkj
dte
0
)( 0 T, if k = n
0, if k n{
In the R.H.S. side of the integral, for ‘k’ not equal to ‘n’, both the integrals will be zero.
Fr k=n, the integrand equals ‘1’ and thus, the integral equals ‘T’.

5. 

T
tnkj
dte
0
)( 0 T, if k = n
0, if k n{

dte
T
tnkj 0)( 
Integrating from 0 to T is same as integrating over any interval of length T because we
are only concerned with integral number of periods of cosine and sine function in the
previous equation.
Now the R.H.S. of equation (2) reduces to Tan . Therefore :-
dtetx
T
a
T
tjn
n 


0
0
)(
1 
Consequently, we can write
dtetx
T
a
T
tjn
n 

 0
)(
1 

6. Hence the Fourier series of x(t) can be expressed as :-
 
 4)(
1
)(
1
3)(
)/2(
)/2(
0
0








dtetx
T
dtetx
T
a
eaeatx
T
tTjk
T
tjk
k
tTjk
k
k
tjk
k
k


Here equation (3) is the analysis equation and equation (4) is the synthesis equation.
Coefficients ak are called Fourier Series Coefficients or Spectral Coefficients of x(t).
Here the DC or constant component of x(t) is :-
dttx
T
a
T

0
0 )(
1

7. 7
Example :- Periodic square wave defined over one period as
 







2/tT0
t1
1
1
T
T
tx
1
 tx
-T -T/2 –T1 0 T1 T/2 T t
 
Defining
 10
1
sin
2
Tkc
T
T
ak 


k
Tk
ak
10sin
 When k 0
At k = 0  
T
T
dttx
T
a
T
T
1
2/
/2-
0
21
 

 
x
x
xc
sin
sin 

8. Fourier Series of some Common Signals

12. Conditions for convergence of
CT Fourier series
• Every function ƒ(x) of period 2п satisfying following conditions known as
DIRICHLET’S CONDITIONS, can be expressed in the form of Fourier series.
1. Over any period, x(t) must be absolutely integrable :-
it guarantees that each coefficient ak will be finite
2. In a single period, x(t) should have finite number of max and min
3. In any finite interval of time, there are only a finite number of discontinuities. Each
discontinuity should be finite.
 dttx
T
2
)(
ka

13. Properties of CT Fourier Series
1. Linearity
z(t) = Ax(t) + By(t) ck = Aak + Bbk
2. Time Shifting
z(t) = x(t-t0) ck =
• In time shifting magnitude of Fourier series coefficient remains the same
|ck| = |ak|
3. Time Reversal
z(t) = x(-t) ck = a-k
• If x(t) is even, ak = a-k
• If x(t) is odd, ak = -a-k
x(t) & y(t) are two periodic signals with period T and
Fourier coefficients ak & bk respectively
00tjk
k ea 

14. 4. Time Scaling
z(t) = x(αt) ck = ak
• But, the fundamental period is (T/α)
5. Multiplication
z(t) = x(t)y(t) ck =
• (DT convolution between coefficients)
6. Conjugation & Conjugate Symmetry
• Real x(t) a-k = a*k (conjugate symmetric)
• Real & Even x(t) ak = a*k (real & even ak)
• Real & Odd x(t) ak = -a*k (purely imaginary & odd ak ; a0 = 0)
• z(t) = Even part of x(t) ck = Real{ak}
• z(t) = Odd part of x(t) ck = jImaginary{ak}




l
lklba

15. 7. Periodic Convolution
8. Parseval’s Relation
• Total average power = sum of average power in all harmonic components
• Energy in time domain equals to energy in frequency domain
 dtyxtytx
T
)()()(*)(   Takbk

 








k
k
T
tjk
k
k
T k
tjk
k
T
adtea
T
dtea
T
dttx
T
222
2
2
0
0
1
1
)(
1



16. Determining the
Fourier Series Representation of a
Discrete Time Signal
 1][ )/2(0
   Nk
nNjk
k
Nk
njk
k eaeanx 
A periodic DT signal can be expressed as set of N linear equations for N unknown
coeffficients ak as k ranges over a set of N successive integers :-
Nn
nNjk
e )/2(  N, if k = 0,+N,+2N…
0, otherwise
{
According to the fact that :-

17. 


Nk
nNrkj
k
nNjr
eaenx )/2)(()/2(
][ 
Multiplying both sides with , we get :-
nNjr
e )/2( 
Now if summing over N terms, we have :-
   




Nn Nk
nNrkj
k
Nn
nNjr
eaenx )/2)(()/2(
][ 
Interchanging the order of summation, we have :-
   




Nk Nn
nNrkj
k
Nn
nNjr
eaenx )/2)(()/2(
][ 

18. 

Nn
nNrkj
e )/2)((  N, if k = 0,+N,+2N…
0, otherwise
{
According to the fact stated before, we can conclude that :-
So the R.H.S. of the equation reduces to Nar



Nn
nNjr
r enx
N
a )/2(
][
1 
Hence the Fourier series of x[n] can be expressed as below where equation (3) is the
analysis equation and equation (4) is the synthesis equation
 
 4][
1
][
1
3][
)/2(
)/2(
0
0









Nn
nNjk
Nn
njk
k
Nk
nNjk
k
Nk
njk
k
enx
N
enx
N
a
eaeanx



19. Properties of DT Fourier Series
1. Linearity
z[n] = Ax[n] + By[t] ck = Aak + Bbk
2. Time Shifting
z[t] = x[n-n0] ck =
• In time shifting magnitude of Fourier series coefficient remains the same
|ck| = |ak|
3. Time Reversal
z[t] = x[-n] ck = a-k
• If x[n] is even, ak = a-k
• If x[n] is odd, ak = -a-k
x[n] & y[n] are two periodic signals with period N and
Fourier coefficients ak & bk respectively periodic with period N
0)/2( nNjk
k ea 

20. 4. Multiplication
z[n] = x[n]y[n] ck =
5. Conjugation & Conjugate Symmetry
• Real x[n] a-k = a*k (conjugate symmetric)
• Real & Even x[n] ak = a*k (real & even ak)
• Real & Odd x[n] ak = -a*k (purely imaginary & odd ak )
• z[n] = Even part of x[n] ck = Real{ak}
• z[n] = Odd part of x[n] ck = jImaginary{ak}
6. Periodic Convolution


Nl
lklba


Nr
rnyrxnynx ][][][*][ Nakbk

21. 7. Parseval’s Relation
• Total average power = sum of average power in all harmonic components
• Energy in time domain equals to energy in frequency domain
22
][
1
 

Nk
k
Nn
anx
N

22. FOURIER SERIES, which is an infinite series representation of such
functions in terms of ‘sine’ and ‘cosine’ terms, is useful here. Thus,
FOURIER SERIES, are in certain sense, more UNIVERSAL than TAYLOR’s
SERIES as it applies to all continuous, periodic functions and also to the
functions which are discontinuous in their values and derivatives.
FOURIER SERIES a very powerful method to solve ordinary and partial
differential equation..
As we know that TAYLOR SERIES representation of functions are valid
only for those functions which are continuous and differentiable. But
there are many discontinuous periodic function which requires to
express in terms of an infinite series containing ‘sine’ and ‘cosine’
terms.
Advantages of using Fourier Series

23. Consider a mass-spring system as before, where we have a mass m on a
spring with spring
constant k, with damping c, and a force F(t) applied to the mass.
Suppose the forcing function F(t) is 2L-periodic for some
L > 0.
The equation that governs this
particular setup is
The general solution consists of the
complementary solution xc, which
solves the associated
homogeneous equation mx” + cx’ + kx = 0, and a particular solution of (1)
we call xp.
mx”(t) + cx’(t) + kx(t) = F(t)
Applications of using Fourier Series
1. Forced Oscillation

24. For c > 0,
the complementary solution xc will decay as time goes by. Therefore, we
are mostly interested in a
particular solution xp that does not decay and is periodic with the same
period as F(t). We call this
particular solution the steady periodic solution and we write it as xsp as
before. What will be new in
this section is that we consider an arbitrary forcing function F(t) instead of
a simple cosine.
For simplicity, let us suppose that c = 0. The problem with c > 0 is very
similar. The equation
mx” + kx = 0
has the general solution,
x(t) = A cos(ωt) + B sin(ωt);
Where,

25. Any solution to mx”(t) + kx(t) = F(t) is of the form
A cos(ωt) + B sin(ωt) + xsp.
The steady periodic solution xsp has the same period as F(t).
In the spirit of the last section and the idea of undetermined
coefficients we first write,
Then we write a proposed steady periodic solution x as,
where an and bn are unknowns. We plug x into the deferential
equation and solve for an and bn in terms of cn and dn.

26. • It turns out that (almost) any kind of a wave can be written as a sum of sines and
cosines. So for example, if a voice is recorded for one second saying something, I
can find its Fourier series which may look something like this for example
• and this interactive module shows you how when you add sines and/or cosines the
graph of cosines and sines becomes closer and closer to the original graph we are
trying to approximate.
• The really cool thing about Fourier series is that first, almost any kind of a wave
can be approximated. Second, when Fourier series converge, they converge very
fast.
• So one of many applications is compression. Everyone's favorite MP3 format uses
this for audio compression. You take a sound, expand its Fourier series. It'll most
likely be an infinite series BUT it converges so fast that taking the first few terms is
enough to reproduce the original sound. The rest of the terms can be ignored
because they add so little that a human ear can likely tell no difference. So I just
save the first few terms and then use them to reproduce the sound whenever I
want to listen to it and it takes much less memory.
• JPEG for pictures is the same idea.
2. Speech/Music Recognition

27. 3. Approximation Theory :- We use Fourier series to write a
function as a trigonometric polynomial.
4. Control Theory :- The Fourier series of functions in the
differential equation often gives some prediction about the
behavior of the solution of differential equation. They are
useful to find out the dynamics of the solution.
5. Partial Differential equation :- We use it to solve higher order
partial differential equations by the method of separation of
variables.