Fourier Representation | Short Notes for Electrical Engineering - Electrical Engineering (EE) PDF Download

 Page 1


Fourier series representation of periodic signal 
Fourier series representation of continuous time periodic signals can be given as a
linear combination of harmonically related complex exponentials:-
A signal is periodic if x(t) = x(t + nT) for all t where T is a constant known as
period and n = 0, 1, 2, … and T = ?
0
/2p.
E.g. x(t) = cos?
0
t, x(t) = e
j?
0
t
Complex exponential
?
k
(t) = e
ik?0t
= e
jk(2p/T)t
k= 0, ±1, ±2…
8 8
i.e. x(t) = S a
k
. e
jk?0t
= S a
k
.e
jk(2p/T)t
-8 -8
if k = 0, x (t) is constant
k = ±1 fundamental frequency ?
0
is the fundamental components known as first
harmonic component.
k = ±2 second harmonic component.
k ± n is n
th
harmonic component.
This representation is known as Fourier series representation of periodic signal.
To determine a
k
multiply both with e
-jk?0t
8 8
i.e. x(t) e
-jk?0t
= S a
k
. e
jk?0t
e
-jk?0t
= S a
k
.e
jk(2p/T)t
e
-jk?0t
-8 -8
Integrating from 0 to T
T T 8 8 T
i.e. ?x(t) e
-jk?0t
= ? S a
k
. e
jk?0t
e
-jk?0t
= S ?a
k
[e
jk(k-n)?0t
dt]
0 0 -8 -8 0
Euler’s formula:
T T
?a
k
[e
jk(k-n)?0t
dt] = ? [cos(k-n)?
0
t + j sin(k-n)?
0
t]dt
0 0
For k ? n cos(k-n) and sin(k-n) are periodic and for k=n cos(k-n)=1 and ans is T.
There for
T
i.e. ?x(t) e
-j(k-n)?0t
= T, k=n
0
0, k?n
T T
Then for a
n
= 1/T ?x(t) e
-jn?0t
dt and for a
0
= 1/T ?x(t) dt
0 0
Page 2


Fourier series representation of periodic signal 
Fourier series representation of continuous time periodic signals can be given as a
linear combination of harmonically related complex exponentials:-
A signal is periodic if x(t) = x(t + nT) for all t where T is a constant known as
period and n = 0, 1, 2, … and T = ?
0
/2p.
E.g. x(t) = cos?
0
t, x(t) = e
j?
0
t
Complex exponential
?
k
(t) = e
ik?0t
= e
jk(2p/T)t
k= 0, ±1, ±2…
8 8
i.e. x(t) = S a
k
. e
jk?0t
= S a
k
.e
jk(2p/T)t
-8 -8
if k = 0, x (t) is constant
k = ±1 fundamental frequency ?
0
is the fundamental components known as first
harmonic component.
k = ±2 second harmonic component.
k ± n is n
th
harmonic component.
This representation is known as Fourier series representation of periodic signal.
To determine a
k
multiply both with e
-jk?0t
8 8
i.e. x(t) e
-jk?0t
= S a
k
. e
jk?0t
e
-jk?0t
= S a
k
.e
jk(2p/T)t
e
-jk?0t
-8 -8
Integrating from 0 to T
T T 8 8 T
i.e. ?x(t) e
-jk?0t
= ? S a
k
. e
jk?0t
e
-jk?0t
= S ?a
k
[e
jk(k-n)?0t
dt]
0 0 -8 -8 0
Euler’s formula:
T T
?a
k
[e
jk(k-n)?0t
dt] = ? [cos(k-n)?
0
t + j sin(k-n)?
0
t]dt
0 0
For k ? n cos(k-n) and sin(k-n) are periodic and for k=n cos(k-n)=1 and ans is T.
There for
T
i.e. ?x(t) e
-j(k-n)?0t
= T, k=n
0
0, k?n
T T
Then for a
n
= 1/T ?x(t) e
-jn?0t
dt and for a
0
= 1/T ?x(t) dt
0 0
Page 3


Fourier series representation of periodic signal 
Fourier series representation of continuous time periodic signals can be given as a
linear combination of harmonically related complex exponentials:-
A signal is periodic if x(t) = x(t + nT) for all t where T is a constant known as
period and n = 0, 1, 2, … and T = ?
0
/2p.
E.g. x(t) = cos?
0
t, x(t) = e
j?
0
t
Complex exponential
?
k
(t) = e
ik?0t
= e
jk(2p/T)t
k= 0, ±1, ±2…
8 8
i.e. x(t) = S a
k
. e
jk?0t
= S a
k
.e
jk(2p/T)t
-8 -8
if k = 0, x (t) is constant
k = ±1 fundamental frequency ?
0
is the fundamental components known as first
harmonic component.
k = ±2 second harmonic component.
k ± n is n
th
harmonic component.
This representation is known as Fourier series representation of periodic signal.
To determine a
k
multiply both with e
-jk?0t
8 8
i.e. x(t) e
-jk?0t
= S a
k
. e
jk?0t
e
-jk?0t
= S a
k
.e
jk(2p/T)t
e
-jk?0t
-8 -8
Integrating from 0 to T
T T 8 8 T
i.e. ?x(t) e
-jk?0t
= ? S a
k
. e
jk?0t
e
-jk?0t
= S ?a
k
[e
jk(k-n)?0t
dt]
0 0 -8 -8 0
Euler’s formula:
T T
?a
k
[e
jk(k-n)?0t
dt] = ? [cos(k-n)?
0
t + j sin(k-n)?
0
t]dt
0 0
For k ? n cos(k-n) and sin(k-n) are periodic and for k=n cos(k-n)=1 and ans is T.
There for
T
i.e. ?x(t) e
-j(k-n)?0t
= T, k=n
0
0, k?n
T T
Then for a
n
= 1/T ?x(t) e
-jn?0t
dt and for a
0
= 1/T ?x(t) dt
0 0
10. Integration: ?x(t)dt (finite value) (its periodic only if a
0
= 0) has the Fourier series
coefficient a
k
/jkw
0
.
11. Conjugate symmetry for real signals: i.e. if x(t) is real then a
k
= *a
k
,
re{ a
k
}=re{ a
-k
},im{ a
k
}= -im{ a
-k
}, | a
k
|=| a
-k
| and
12. Real and even: If x(t) is real and even then their coefficient is also real and even.
13. Real and odd: If x(t) is real and odd then the coefficient is purely imaginary and
odd.
14. Decomposition of real signal: x
e
(t) = Ev{x(t)} [x(t) is real] then coefficient
Re{ a
k
}and x
o
(t) = Od{x(t)} [x(t) is real] then coefficient is Im{ a
k
}
Fourier series representation of discrete time periodic signals 
Linear combination of harmonically related complex exponentials:-
A signal is periodic if x[n] = x(n + mN) for all n where N is a constant known as
period and m = 0, 1, 2, … and N = ?
0
/2p. E.g. x[n] = cos?
0
t, x[n] = e
j?
0
n
Complex exponential
?
k
[n] = e
ik?0n
= e
jk(2p/N)n
k= 0, ±1, ±2…
?
k
[n] = ?
k+N
[n]
i.e. ?
0
[n] = ?
N
[n], ?
1
[n] = ?
N+1
[n]
x[0] = S a
k
., x[1] = S a
k
. e
jk(2p/N)
k=<N> k=<N>
i.e. x[n] = S a
k
. e
jk?0n
= S a
k
.e
jk(2p/N)n
k=<N> k=<N>
if k = 0, 1, …. N -1 or K=3, 4, …, N+2 etc…
K= N is N
th
harmonic component or N successive integers.
This representation is known as Fourier series representation of periodic signal of
discrete type.
To determine a
k
multiply both with e
-jr?0n
i.e. x[n] e
-jr?0n
= S a
k
. e
jk?0n
e
-jr?0n
= S a
k
.e
jk(k-r)(2p/N)n
k<n> k=<n>
There for inner most sum if k = r is N and if K ? r is 0.
a
r
= 1/N S x[n] e
-jr?0n
where ?
0
= 2p/N
k=<N>
a
k
= 1/N S x[n] e
-jk?0n
where ?
0
= 2p/N
k=<N>
Page 4


Fourier series representation of periodic signal 
Fourier series representation of continuous time periodic signals can be given as a
linear combination of harmonically related complex exponentials:-
A signal is periodic if x(t) = x(t + nT) for all t where T is a constant known as
period and n = 0, 1, 2, … and T = ?
0
/2p.
E.g. x(t) = cos?
0
t, x(t) = e
j?
0
t
Complex exponential
?
k
(t) = e
ik?0t
= e
jk(2p/T)t
k= 0, ±1, ±2…
8 8
i.e. x(t) = S a
k
. e
jk?0t
= S a
k
.e
jk(2p/T)t
-8 -8
if k = 0, x (t) is constant
k = ±1 fundamental frequency ?
0
is the fundamental components known as first
harmonic component.
k = ±2 second harmonic component.
k ± n is n
th
harmonic component.
This representation is known as Fourier series representation of periodic signal.
To determine a
k
multiply both with e
-jk?0t
8 8
i.e. x(t) e
-jk?0t
= S a
k
. e
jk?0t
e
-jk?0t
= S a
k
.e
jk(2p/T)t
e
-jk?0t
-8 -8
Integrating from 0 to T
T T 8 8 T
i.e. ?x(t) e
-jk?0t
= ? S a
k
. e
jk?0t
e
-jk?0t
= S ?a
k
[e
jk(k-n)?0t
dt]
0 0 -8 -8 0
Euler’s formula:
T T
?a
k
[e
jk(k-n)?0t
dt] = ? [cos(k-n)?
0
t + j sin(k-n)?
0
t]dt
0 0
For k ? n cos(k-n) and sin(k-n) are periodic and for k=n cos(k-n)=1 and ans is T.
There for
T
i.e. ?x(t) e
-j(k-n)?0t
= T, k=n
0
0, k?n
T T
Then for a
n
= 1/T ?x(t) e
-jn?0t
dt and for a
0
= 1/T ?x(t) dt
0 0
10. Integration: ?x(t)dt (finite value) (its periodic only if a
0
= 0) has the Fourier series
coefficient a
k
/jkw
0
.
11. Conjugate symmetry for real signals: i.e. if x(t) is real then a
k
= *a
k
,
re{ a
k
}=re{ a
-k
},im{ a
k
}= -im{ a
-k
}, | a
k
|=| a
-k
| and
12. Real and even: If x(t) is real and even then their coefficient is also real and even.
13. Real and odd: If x(t) is real and odd then the coefficient is purely imaginary and
odd.
14. Decomposition of real signal: x
e
(t) = Ev{x(t)} [x(t) is real] then coefficient
Re{ a
k
}and x
o
(t) = Od{x(t)} [x(t) is real] then coefficient is Im{ a
k
}
Fourier series representation of discrete time periodic signals 
Linear combination of harmonically related complex exponentials:-
A signal is periodic if x[n] = x(n + mN) for all n where N is a constant known as
period and m = 0, 1, 2, … and N = ?
0
/2p. E.g. x[n] = cos?
0
t, x[n] = e
j?
0
n
Complex exponential
?
k
[n] = e
ik?0n
= e
jk(2p/N)n
k= 0, ±1, ±2…
?
k
[n] = ?
k+N
[n]
i.e. ?
0
[n] = ?
N
[n], ?
1
[n] = ?
N+1
[n]
x[0] = S a
k
., x[1] = S a
k
. e
jk(2p/N)
k=<N> k=<N>
i.e. x[n] = S a
k
. e
jk?0n
= S a
k
.e
jk(2p/N)n
k=<N> k=<N>
if k = 0, 1, …. N -1 or K=3, 4, …, N+2 etc…
K= N is N
th
harmonic component or N successive integers.
This representation is known as Fourier series representation of periodic signal of
discrete type.
To determine a
k
multiply both with e
-jr?0n
i.e. x[n] e
-jr?0n
= S a
k
. e
jk?0n
e
-jr?0n
= S a
k
.e
jk(k-r)(2p/N)n
k<n> k=<n>
There for inner most sum if k = r is N and if K ? r is 0.
a
r
= 1/N S x[n] e
-jr?0n
where ?
0
= 2p/N
k=<N>
a
k
= 1/N S x[n] e
-jk?0n
where ?
0
= 2p/N
k=<N>
Page 5


Fourier series representation of periodic signal 
Fourier series representation of continuous time periodic signals can be given as a
linear combination of harmonically related complex exponentials:-
A signal is periodic if x(t) = x(t + nT) for all t where T is a constant known as
period and n = 0, 1, 2, … and T = ?
0
/2p.
E.g. x(t) = cos?
0
t, x(t) = e
j?
0
t
Complex exponential
?
k
(t) = e
ik?0t
= e
jk(2p/T)t
k= 0, ±1, ±2…
8 8
i.e. x(t) = S a
k
. e
jk?0t
= S a
k
.e
jk(2p/T)t
-8 -8
if k = 0, x (t) is constant
k = ±1 fundamental frequency ?
0
is the fundamental components known as first
harmonic component.
k = ±2 second harmonic component.
k ± n is n
th
harmonic component.
This representation is known as Fourier series representation of periodic signal.
To determine a
k
multiply both with e
-jk?0t
8 8
i.e. x(t) e
-jk?0t
= S a
k
. e
jk?0t
e
-jk?0t
= S a
k
.e
jk(2p/T)t
e
-jk?0t
-8 -8
Integrating from 0 to T
T T 8 8 T
i.e. ?x(t) e
-jk?0t
= ? S a
k
. e
jk?0t
e
-jk?0t
= S ?a
k
[e
jk(k-n)?0t
dt]
0 0 -8 -8 0
Euler’s formula:
T T
?a
k
[e
jk(k-n)?0t
dt] = ? [cos(k-n)?
0
t + j sin(k-n)?
0
t]dt
0 0
For k ? n cos(k-n) and sin(k-n) are periodic and for k=n cos(k-n)=1 and ans is T.
There for
T
i.e. ?x(t) e
-j(k-n)?0t
= T, k=n
0
0, k?n
T T
Then for a
n
= 1/T ?x(t) e
-jn?0t
dt and for a
0
= 1/T ?x(t) dt
0 0
10. Integration: ?x(t)dt (finite value) (its periodic only if a
0
= 0) has the Fourier series
coefficient a
k
/jkw
0
.
11. Conjugate symmetry for real signals: i.e. if x(t) is real then a
k
= *a
k
,
re{ a
k
}=re{ a
-k
},im{ a
k
}= -im{ a
-k
}, | a
k
|=| a
-k
| and
12. Real and even: If x(t) is real and even then their coefficient is also real and even.
13. Real and odd: If x(t) is real and odd then the coefficient is purely imaginary and
odd.
14. Decomposition of real signal: x
e
(t) = Ev{x(t)} [x(t) is real] then coefficient
Re{ a
k
}and x
o
(t) = Od{x(t)} [x(t) is real] then coefficient is Im{ a
k
}
Fourier series representation of discrete time periodic signals 
Linear combination of harmonically related complex exponentials:-
A signal is periodic if x[n] = x(n + mN) for all n where N is a constant known as
period and m = 0, 1, 2, … and N = ?
0
/2p. E.g. x[n] = cos?
0
t, x[n] = e
j?
0
n
Complex exponential
?
k
[n] = e
ik?0n
= e
jk(2p/N)n
k= 0, ±1, ±2…
?
k
[n] = ?
k+N
[n]
i.e. ?
0
[n] = ?
N
[n], ?
1
[n] = ?
N+1
[n]
x[0] = S a
k
., x[1] = S a
k
. e
jk(2p/N)
k=<N> k=<N>
i.e. x[n] = S a
k
. e
jk?0n
= S a
k
.e
jk(2p/N)n
k=<N> k=<N>
if k = 0, 1, …. N -1 or K=3, 4, …, N+2 etc…
K= N is N
th
harmonic component or N successive integers.
This representation is known as Fourier series representation of periodic signal of
discrete type.
To determine a
k
multiply both with e
-jr?0n
i.e. x[n] e
-jr?0n
= S a
k
. e
jk?0n
e
-jr?0n
= S a
k
.e
jk(k-r)(2p/N)n
k<n> k=<n>
There for inner most sum if k = r is N and if K ? r is 0.
a
r
= 1/N S x[n] e
-jr?0n
where ?
0
= 2p/N
k=<N>
a
k
= 1/N S x[n] e
-jk?0n
where ?
0
= 2p/N
k=<N>
13. Real and odd: If x[n] is real and odd then the coefficient is purely imaginary and
odd.
14. Decomposition of real signal: x
e
[n] = Ev{x[n]} [x[n] is real] then coefficient
Re{ a
k
}and x
o
[n] = Od{x[n]} [x[n] is real] then coefficient is Im{ a
k
}
Paseval’s relation:
1. Continuous time periodic signal:
8
1/T ? |x(t}|
2
dt = S | a
k
|
2
where a
k
is Fourier series coefficient and T is the time
T k=-8
period of the signal.
I.e. average power or energy per unit time in one period.
Total paseval’s relation = sum of the average power in all harmonic components.
2. Discrete time periodic signal:
1/N S |x[n]|
2
= S | a
k
|
2
where a
k
is Fourier series coefficient and N is the time
n=<N> k=<N>
period of the signal.
I.e. | a
k
| = average power or energy per unit time in one period.
Total paseval’s relation = sum of the average power in all harmonic components.
Fourier transforms representation of periodic signal 
Representation of a-periodic signal:
x(t) = 1 |t| <T
0, T
1
< |t| < T/2 period T
Formula for Fourier transform
if x(t) is a signal:
8
X(j?) = ? x(t). e
-j?t
dt
-8
Formula for inverse Fourier transform
if X(j?) is the transformed signal:
8
x(t) = 1/2p ? X(j?). e
-j?t
d?
-8