FOURIER SERIES

 JeanBaptiste Joseph Fourier,

a French mathematician and a
physicist.

 He initialized Fourier series,

Fourier transforms and their
applications to problems of
heat transfer and vibrations.
 Torepresent any periodic signal x(t), Fourier developed an
expression called Fourier series.

 Thisis in terms of an infinite sum of sines and cosines or

exponentials. Fourier series uses orthogonality condition.
A Fourier series is a way of representing a periodic function
as a (possibly infinite) sum of sine and cosine functions. It is
analogous to a Taylor series, which represents functions as
possibly infinite sums of monomial terms.
Dirichlet conditions for the existence of
a Fourier Series of a periodic function
 There are sufficient conditions that are to be satisfied by a
function f(t) for its Fourier series representation with in the
interval (t1, t2). These conditions are called Direchlet’s
conditions:

1. The function f(t) is a single valued function of the variable t in

the interval (t1, t2).
2. The function f(t) has a finite number of discontinuities in the
interval (t1, t2).

3. The function f(t) has finite number of minima and maxima in the
interval (t1, t2).

4. The function is absolutely integral.

𝑡2
න 𝑓 𝑡 𝑑𝑡 < ∞
𝑡1
 Fourier Series Representation of
Continuous Time Periodic Signals
 A signal is said to be periodic if it satisfies the condition
x (t) = x (t + T) or x (n) = x (n + N).

Where T = fundamental time period,

ω0= fundamental frequency = 2π/T
 TRIGONOMETRIC FOURIER SERIES (TFS)

A set of harmonically related sine and cosine functions sin(nω0t)

and cos(nω0t) [n = 0, 1, 2, 3, …….. ] forms a complete orthogonal
set or closed orthogonal set over the interval (t0, t0+T), where 𝑇 =
2𝜋 2𝜋
𝑜𝑟 .
𝜔0 𝜔
 A functionf(t) can be represented by a Fourier series comprising
the following sine and cosine functions.

𝑓 𝑡
= 𝑎0 + 𝑎1 𝑐𝑜𝑠 𝜔0 𝑡 + 𝑎2 𝑐𝑜𝑠 2𝜔0 𝑡 + 𝑎3 𝑐𝑜𝑠 3𝜔0 𝑡
+ … … … . +𝑎𝑛 𝑐𝑜𝑠 𝑛𝜔0 𝑡 + … … … + 𝑏1 𝑠𝑖𝑛 𝜔0 𝑡 + 𝑏2 𝑠𝑖𝑛 2𝜔0 𝑡
+ ⋯ … … … . . +𝑏𝑛 𝑠𝑖𝑛 𝑛𝜔0 𝑡 + ⋯ … … . .
f t = a0 + σ∞
n=1[an cos nω0 t + bn sin nω0 t ]
for t 0 ≤ t ≤ t 0 + T
2π
Where, T =
ω0

The above equation is the Trigonometric Fourier Series

representation of f(t) over an interval (t0, t0+T).
 The constants are given by,
1 t0 +T
a0 = ‫׬‬t
f(t) dt
T 0

2 t0 +T
an = ‫׬‬t
f(t) cos(nω0 t) dt
T 0

2 t0 +T
bn = ‫׬‬t
f(t) sin(nω0 t) dt
T 0

Where, a0, an and bn are Fourier coefficients.

 The trigonometric series may also be represented in a simple form
as given below.

𝑓 𝑡 = 𝐶0 + σ∞
𝑛=1 𝐶𝑛 cos(𝑛𝜔0 𝑡 − 𝜑𝑛 )

Where, C0, Cn and φn are related to a0, an and bn by the equations

C0 = a0
𝐶𝑛 = (𝑎𝑛2 + 𝑏𝑛2 )
−1 𝑏𝑛
𝜑𝑛 = tan
𝑎𝑛
 EXPONENTIAL FOURIER SERIES (EFS)
 The set of complex exponential ൛ejnω0t , n = 0, ±1, ±2,
… … … . . ൟ forms a closed set of orthogonal functions over an
2π 2π
interval (t0, t0+T). where T = or .
ω0 ω

 Any arbitrary function f(t) can be represented by a complex

exponential series over an interval (t0, t0+T).
f t = F0 + F1 ejω0t + F2 ej2ω0t + F3 ej3ω0t + ⋯ … … +
Fn ejnω0 t … … . . +F−1 e−jω0t + F−2 e−j2ω0t … . + F−n e−jnω0t +
…………..
∞

f t = ෍ Fn ejnω0t for t 0 ≤ t ≤ t 0 + T
n=−∞
2π
Where, T = and Fn is the Fourier coefficient and it is given by
ω0
t0 +T
1
Fn = න f t e−jnω0t dt
T
t0
 Note:
𝑗𝑥
𝑒 = cos 𝑥 + 𝑗 sin(𝑥)
and
𝑒 −𝑗𝑥 = 𝑐𝑜𝑠 𝑥 − 𝑗 𝑠𝑖𝑛(𝑥)
𝑒 𝑗𝑥 +𝑒 −𝑗𝑥
cos 𝑥 =
2
and
𝑒 𝑗𝑥 −𝑒 −𝑗𝑥
sin 𝑥 =
2𝑗
 DERIVATION OF TFS FROM CEFS
 We know that, the CEFS,
f t = F0 + F1 ejω0t + F2 ej2ω0t + F3 ej3ω0t + ⋯ … +
Fn ejnω0t … … . . +F−1 e−jω0t + F−2 e−j2ω0t … . + F−n e−jnω0t +
…………..

f t
= F0 + F1 cos ω0 t + j sin ω0 t + F2 cos 2ω0 t + j sin 2ω0 t
+F3 cos 3ω0 t + j sin 3ω0 t + ⋯ …
+ Fn cos nω0 t + j sin nω0 t … … . . +F−1 [cos ω0 t − j sin ω0 t ]
+ F−2 [cos 2ω0 t − j sin 2ω0 t ] … . + F−n [cos nω0 t − j sin nω0 t ]
+ …………..
𝑓 𝑡 = 𝐹0 + 𝐹1 + 𝐹−1 𝑐𝑜𝑠 𝜔0 𝑡 + 𝐹2 + 𝐹−2 𝑐𝑜𝑠 2𝜔0 𝑡 +
⋯ … . . + 𝐹𝑛 + 𝐹−𝑛 𝑐𝑜𝑠 𝑛𝜔0 𝑡 + … … + 𝑗 𝐹1 − 𝐹−1 sin 𝜔0 𝑡 +
𝑗 𝐹2 − 𝐹−2 𝑠𝑖𝑛 2𝜔0 𝑡 + … … + 𝑗 𝐹𝑛 − 𝐹−𝑛 𝑠𝑖𝑛 𝑛𝜔0 𝑡 + … …

 The trigonometric Fourier series is given by

𝑓 𝑡
= 𝑎0 + 𝑎1 𝑐𝑜𝑠 𝜔0 𝑡 + 𝑎2 𝑐𝑜𝑠 2𝜔0 𝑡 + 𝑎3 𝑐𝑜𝑠 3𝜔0 𝑡
+ … … … . +𝑎𝑛 𝑐𝑜𝑠 𝑛𝜔0 𝑡 + … … … + 𝑏1 𝑠𝑖𝑛 𝜔0 𝑡 + 𝑏2 𝑠𝑖𝑛 2𝜔0 𝑡
+ ⋯ … … … . . +𝑏𝑛 𝑠𝑖𝑛 𝑛𝜔0 𝑡 + ⋯ … …

Comparing above two equations, we get,

𝐹0 = 𝑎0 , 𝑎𝑛 = 𝐹𝑛 + 𝐹−𝑛
and 𝑏𝑛 = 𝑗 𝐹𝑛 − 𝐹−𝑛
Derive the polar Fourier Series. Prove that,
Dn = 2|Cn| or Dn = 2|Fn|
 The Trigonometric FS is given by
∞

f t = a0 + ෍ [an cos nω0 t + bn sin nω0 t ]

n=1

∞
[an cos nω0 t + bn sin nω0 t ]
f t = a0 + ෍ [a2n + b2n ]
n=1 [a2n + b2n ]
 ∞
an cos nω0 t bn sin nω0 t
f t = a0 + ෍ [a2n + b2n ] +
n=1 a2n + b2n a2n + b2n

an bn
Setting, = cos φn and = sin(φn )
a2n +b2n a2n +b2n

sin(φn ) bn −1
bn
= tan φn = =⇒ ϕn = tan
cos(φn ) an an
Therefore,
∞

f t = a0 + ෍ [a2n + b2n ] [cos φn cos nω0 t + sin(φn ) sin nω0 t ]

n=1

f t = a0 + σ∞
n=1 [a2
n + b2
n ] [cos nω0 t − ϕn ]
 f t = a0 + σ∞
n=1 [a 2
n + b 2
n ] [cos nω0 t − ϕn ]

 The polar form of TFS is given by

f t = D0 + σ∞
n=1 Dn cos(nω0 t − φn )

Comparing above two equations, we get,

−1 bn
D0 = a0 ,Dn = [a2n + b2n ] and ϕn = tan
an

We know that,
F0 = a0 , an = Fn + F−n and bn = j Fn − F−n
 Now, Dn = [a2n + b2n ] = (Fn + F−n )2 +j2 (Fn − F−n )2

 Dn = Fn2 + F−n
2
+ 2Fn F−n − (Fn2 + F−n
2
− 2Fn F−n )

 Dn = Fn2 + F−n
2
+ 2Fn F−n − Fn2 − F−n
2
+ 2Fn F−n

 Dn = 4Fn F−n

 We know that F−n = Fn∗ and Fn × Fn∗ = |Fn |2

 Therefore, Dn = 4Fn Fn∗ = 4|Fn |2 = 2|Fn |

 FOURIER SERIES REPRESENTATION OF
PERIODIC SIGNALS
𝟐𝛑
The periodic condition, f(t) = f(t+T) for all t and Where, 𝐓 =
𝛚𝟎

The TFS is given by, 𝑓 𝑡 = 𝑎0 + σ∞

𝑛=1[𝒂𝒏 𝒄𝒐𝒔 𝒏𝝎𝟎 𝒕 + 𝒃𝒏 𝒔𝒊𝒏 𝒏𝝎𝟎 𝒕 ]

Now, f t + T = a0 + σ∞
n=1[an cos nω0 (t + T) + bn sin nω0 (t + T) ]

f t + T = a0 + ෍ [an cos nω0 t + nω0 T + bn sin nω0 t + nω0 T ]

n=1
 ∞

f t + T = a0 + ෍ [an cos nω0 t + n2π + bn sin nω0 t + n2π ]

n=1

For different integer values of ‘n’,

cos nω0 t + 2nπ = cos nω0 t and sin nω0 t + 2nπ = sin nω0 t

Therefore, 𝑓 𝑡 + 𝑇 = 𝑎0 + σ∞
𝑛=1[𝑎𝑛 𝑐𝑜𝑠 𝑛𝜔0 𝑡 + 𝑏𝑛 𝑠𝑖𝑛 𝑛𝜔0 𝑡 ]

f t + T = f(t)

Hence, The TFS is a periodic signal with period T

Similarly, for a CEFS is also a periodic signal with period T
 COMPLEX FOURIER SPECTRUM
 The Fourier spectrum of a periodic signal x(t) is a plot of its
Fourier coefficients versus frequency ω.

 It is in two parts : (a) Amplitude spectrum and (b) phase spectrum.

 The plot of the amplitude of Fourier coefficients verses frequency

is known as the amplitude spectra.
The plot of the phase of Fourier coefficients verses
frequency is known as phase spectra.
 The two plots together are known as Fourier frequency spectra of
x(t). This type of representation is also called frequency domain
representation.

 The Fourier spectrum exists only at discrete frequencies nωo,

where n = 0, 1, 2, ….. Hence it is known as discrete spectrum or
line spectrum.

 The envelope of the spectrum depends only upon the pulse shape,
but not upon the period of repetition.
 The trigonometric representation of a periodic signal x(t) contains both sine
and cosine terms with positive and negative amplitude coefficients (an and bn)
but with no phase angles.

 The exponential representation of a periodic signal x(t) contains amplitude

coefficients Fn or Cn Which are complex. Hence, they can be represented by
magnitude and phase.

 Therefore, we can plot two spectra, the magnitude spectrum(⎹Cn⎸versus ω)

and phase spectrum ( 𝐶𝑛 versus ω).
 The spectra can be plotted for both positive and negative frequencies. Henceit
is called two-sided spectra.
 The magnitude spectrum is symmetrical about the vertical axis
passing through the origin, and the phase spectrum is anti
symmetrical about the vertical axis passing through the origin.

 Sothe magnitude spectrum exhibits even symmetry and phase

spectrum exhibits odd symmetry.

 When x(t) is real , then C-n = 𝐶𝑛*, the complex conjugate of Cn.
Fourier composition of a square wave.
 Properties of Fourier series
 x1(t)and x2(t) are two periodic signals with period T and with
Fourier series coefficients Cn and Dn respectively.
That is, FS[x1(t)] = Cn and FS[x2(t)] = Dn

 Let us use the following notation to represent the Fourier

coefficients FS[f(t)] = Cn (or) FS[x(t)] = Fn
f t = FS−1 Cn or x t = FS −1 Fn
 The other notation that can be used to represent Fourier series
F.S
coefficients are f t Cn
 Linearity Property
F.S
Statement: The linearity property states that, if x1 (t) Cn and
F.S F.S
x2 (t) Dn Then Ax1 t + Bx2 t ACn + BDn

Proof: 1 t0+T
FS x1 t = Cn = න x1 t e−jnω0t dt
T t0
1 t0+T
FS x2 t = Dn = න x2 t e−jnω0t dt
T t0

1 t0 +T
FS Ax1 t + Bx2 t = න Ax1 t + Bx2 t e−jnω0 t dt
T t0
 1 t0 +T 1 t0 +T
FS Ax1 t + Bx2 t = ‫׬‬ Ax1 t e−jnω0 t dt + ‫׬‬t Bx2 t e−jnω0 t dt
T t0 T 0

1 t0 +T 1 t0 +T
FS Ax1 t + Bx2 t =A ‫׬‬t
x1 t e−jnω0 t dt +B ‫׬‬t x2 t e−jnω0 t dt
T 0 T 0

FS Ax1 t + Bx2 t = A 𝐶𝑛 + 𝐵𝐷𝑛

 Time shifting property
Statement: The time shifting property states that,
𝐹.𝑆 𝐹.𝑆
if 𝑥(𝑡) 𝐶𝑛 Then 𝑥 𝑡 − 𝑡0 𝑒 −𝑗𝑛𝜔0𝑡0 𝐶𝑛

Proof:
x t = σ∞ C
n=−∞ n ejnω0 t
= FS −1
[Cn ]

and

x t − 𝑡0 = σ∞ C
n=−∞ n ejnω0 (𝑡−𝑡0 )
x t − 𝑡0 = σ∞ C
n=−∞ n ejnω0 𝑡 −jnω0 𝑡0
e

= σ∞ [C
n=−∞ n e−jnω0 𝑡0 jnω0 𝑡
]e = FS −1
[Cn e−jnω0 𝑡0
]

FS[x t − 𝑡0 ] = Cn e−jnω0𝑡0
 Time reversal property
 Statement: The time reversal property states that,
𝐹.𝑆 𝐹.𝑆
if 𝑥(𝑡) 𝐶𝑛 Then 𝑥 −𝑡 𝐶−𝑛

 Proof: x t = FS −1 Cn = σ∞ C
n=−∞ n ejnω0 t

t is replaced by –t, we get,

∞

x −t = ෍ Cn e−jnω0t
n=−∞
 substituting n = -p in the right hand side, we get
−∞

x −t = ෍ C−p e−j(−p)ω0t
p=∞
∞

==⇒ x −t = ෍ C−p ejpω0t

p=−∞
substituting p = n , we get
∞

x −t = ෍ C−n ejnω0t
n=−∞
=⇒ x −t = FS−1 C−n =⇒ FS x −t = C−n