Chapter 4 The Fourier Series and Fourier Transform
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Transcript Chapter 4 The Fourier Series and Fourier Transform
Chapter 4
The Fourier Series and
Fourier Transform
Fourier Series Representation of
Periodic Signals
• Let x(t) be a CT periodic signal with period
T, i.e., x(t T ) x(t ), t R
• Example: the rectangular pulse train
The Fourier Series
• Then, x(t) can be expressed as
x(t )
ce
k
k
jk 0t
, t
where 0 2 / T is the fundamental
frequency (rad/sec) of the signal and
T /2
1
jk o t
ck
x(t )e
dt , k 0, 1, 2,
T T / 2
c0 is called the constant or dc component of x(t)
Dirichlet Conditions
•
A periodic signal x(t), has a Fourier series
if it satisfies the following conditions:
1. x(t) is absolutely integrable over any
period, namely
a T
| x(t ) | dt ,
a
a
2. x(t) has only a finite number of maxima
and minima over any period
3. x(t) has only a finite number of
discontinuities over any period
Example: The Rectangular Pulse Train
• From figure
T 2, so 0 2 / 2
• Clearly x(t) satisfies the Dirichlet conditions and
thus has a Fourier series representation
Example: The Rectangular Pulse
Train – Cont’d
k jk t
1
1
x(t)
sin e , t
2 k k
2
k 0
Trigonometric Fourier Series
• By using Euler’s formula, we can rewrite
x(t )
as
ce
k
k
jk 0t
, t
x(t ) c0 2 | ck |cos(k 0t ck ), t
k 1
dc component
k-th harmonic
as long as x(t) is real
• This expression is called the trigonometric
Fourier series of x(t)
Example: Trigonometric Fourier
Series of the Rectangular Pulse Train
• The expression
k jk t
1
1
x(t)
sin e , t
2 k k
2
k 0
can be rewritten as
1
x(t )
2
k 1
k odd
2
( k 1) / 2
cos k t (1)
1 , t
k
2
Gibbs Phenomenon
• Given an odd positive integer N, define the
N-th partial sum of the previous series
1
xN (t )
2
N
k 1
2
( k 1) / 2
cos k t (1)
1 , t
k
2
k odd
• According to Fourier’s theorem, it should be
lim | xN (t ) x(t ) | 0
N
Gibbs Phenomenon – Cont’d
x3 (t )
x9 (t )
Gibbs Phenomenon – Cont’d
x21 (t )
x45 (t )
overshoot: about 9 % of the signal magnitude
(present even if N )
Parseval’s Theorem
• Let x(t) be a periodic signal with period T
• The average power P of the signal is defined
as
T /2
1
2
P
x (t )dt
T T / 2
• Expressing the signal as x(t )
it is also
P
|c
k
k
2
|
k
ck e jk0t , t
Fourier Transform
• We have seen that periodic signals can be
represented with the Fourier series
• Can aperiodic signals be analyzed in terms of
frequency components?
• Yes, and the Fourier transform provides the
tool for this analysis
• The major difference w.r.t. the line spectra of
periodic signals is that the spectra of
aperiodic signals are defined for all real
values of the frequency variable not just
for a discrete set of values
Frequency Content of the
Rectangular Pulse
x(t )
xT (t )
x(t ) lim xT (t )
T
Frequency Content of the
Rectangular Pulse – Cont’d
• Since xT (t ) is periodic with period T, we
can write
xT (t )
ce
k
k
jk 0t
, t
where
T /2
1
jk o t
ck
x(t )e
dt , k 0, 1, 2,
T T / 2
Frequency Content of the
Rectangular Pulse – Cont’d
• What happens to the frequency components
of xT (t ) as T ?
• For k 0 :
c0 1/ T
• For k 1, 2, :
k0 1
k0
ck
sin
sin
k0T
2 k
2
2
0 2 / T
Frequency Content of the
Rectangular Pulse – Cont’d
plots of T | ck |
vs. k 0
for T 2,5,10
Frequency Content of the
Rectangular Pulse – Cont’d
• It can be easily shown that
lim Tck sinc
T
2
,
where
sinc( )
sin( )
Fourier Transform of the
Rectangular Pulse
• The Fourier transform of the rectangular
pulse x(t) is defined to be the limit of Tck
as T , i.e.,
X ( ) lim Tck sinc
T
2
| X ( ) |
,
arg( X ( ))
The Fourier Transform in the
General Case
• Given a signal x(t), its Fourier transform
X ( ) is defined as
X ( )
x(t )e
j t
dt ,
• A signal x(t) is said to have a Fourier
transform in the ordinary sense if the above
integral converges
The Fourier Transform in the
General Case – Cont’d
•
The integral does converge if
1. the signal x(t) is “well-behaved”
2. and x(t) is absolutely integrable, namely,
| x(t ) | dt
•
Note: well behaved means that the signal
has a finite number of discontinuities,
maxima, and minima within any finite time
interval
Example: The DC or Constant Signal
• Consider the signal x(t ) 1, t
• Clearly x(t) does not satisfy the first
requirement since
| x(t ) | dt dt
• Therefore, the constant signal does not have
a Fourier transform in the ordinary sense
• Later on, we’ll see that it has however a
Fourier transform in a generalized sense
Example: The Exponential Signal
bt
• Consider the signal x(t ) e u (t ), b
• Its Fourier transform is given by
X ( )
e
bt
u (t )e
j t
dt
e
0
t
( b j ) t
1
( b j ) t
dt
e
b j
t 0
Example: The Exponential Signal –
Cont’d
• If b 0 , X ( ) does not exist
• If b 0 , x(t ) u (t ) and X ( ) does not
exist either in the ordinary sense
• If b 0 , it is
1
X ( )
b j
amplitude spectrum
1
| X ( ) |
b2 2
phase spectrum
arg( X ( )) arctan
b
Example: Amplitude and Phase
Spectra of the Exponential Signal
x(t ) e 10t u (t )
Rectangular Form of the Fourier
Transform
• Consider
X ( )
x(t )e
j t
dt ,
• Since X ( ) in general is a complex
function, by using Euler’s formula
X ( ) x(t ) cos( t )dt j x(t )sin( t )dt
R ( )
X ( ) R( ) jI ( )
I ( )
Polar Form of the Fourier Transform
• X ( ) R( ) jI ( ) can be expressed in
a polar form as
X ( ) | X ( ) | exp( j arg( X ( )))
where
| X ( ) | R ( ) I ( )
2
2
I ( )
arg( X ( )) arctan
R ( )
Fourier Transform of
Real-Valued Signals
• If x(t) is real-valued, it is
X ( ) X ( )
• Moreover
Hermitian
symmetry
X ( ) | X ( ) | exp( j arg( X ( )))
whence
| X ( ) || X ( ) | and
arg( X ( )) arg( X ( ))
Example: Fourier Transform of the
Rectangular Pulse
• Consider the even signal
• It is / 2
2
2
t / 2
X ( ) 2 (1) cos( t ) dt sin( t ) t 0 sin
2
0
sinc
2
Example: Fourier Transform of the
Rectangular Pulse – Cont’d
X ( ) sinc
2
Example: Fourier Transform of the
Rectangular Pulse – Cont’d
amplitude
spectrum
phase
spectrum
Bandlimited Signals
• A signal x(t) is said to be bandlimited if its
Fourier transform X ( ) is zero for all B
where B is some positive number, called
the bandwidth of the signal
• It turns out that any bandlimited signal must
have an infinite duration in time, i.e.,
bandlimited signals cannot be time limited
Bandlimited Signals – Cont’d
• If a signal x(t) is not bandlimited, it is said
to have infinite bandwidth or an infinite
spectrum
• Time-limited signals cannot be
bandlimited and thus all time-limited
signals have infinite bandwidth
• However, for any well-behaved signal x(t)
it can be proven that lim X ( ) 0
whence it can be assumed that
| X ( ) | 0 B
B being a convenient large number
Inverse Fourier Transform
• Given a signal x(t) with Fourier transform
X ( ) , x(t) can be recomputed from X ( )
by applying the inverse Fourier transform
given by
1
x(t )
2
X ( )e
j t
d , t
• Transform pair
x(t ) X ( )
Properties of the Fourier Transform
x(t ) X ( )
y (t ) Y ( )
• Linearity:
x(t ) y (t ) X ( ) Y ( )
• Left or Right Shift in Time:
x(t t0 ) X ( )e
• Time Scaling:
j t0
1
x(at ) X
a a
Properties of the Fourier Transform
• Time Reversal:
x(t ) X ( )
• Multiplication by a Power of t:
n
d
t x(t ) ( j )
X ( )
n
d
n
n
• Multiplication by a Complex Exponential:
x(t )e
j0t
X ( 0 )
Properties of the Fourier Transform
• Multiplication by a Sinusoid (Modulation):
j
x(t )sin( 0t ) X ( 0 ) X ( 0 )
2
1
x(t ) cos( 0t ) X ( 0 ) X ( 0 )
2
• Differentiation in the Time Domain:
n
d
n
x
(
t
)
(
j
)
X ( )
n
dt
Properties of the Fourier Transform
• Integration in the Time Domain:
t
1
x( )d j X ( ) X (0) ( )
• Convolution in the Time Domain:
x(t ) y (t ) X ( )Y ( )
• Multiplication in the Time Domain:
x(t ) y (t ) X ( ) Y ( )
Properties of the Fourier Transform
• Parseval’s Theorem:
1
x(t ) y(t )dt 2
X
( )Y ( )d
1
2
if y (t ) x(t ) | x(t ) | dt
| X ( ) | d
2
2
• Duality:
X (t ) 2 x( )
Properties of the Fourier Transform Summary
Example: Linearity
x(t ) p4 (t ) p2 (t )
2
X ( ) 4sinc
2sinc
Example: Time Shift
x(t ) p2 (t 1)
j
X ( ) 2sinc e
Example: Time Scaling
p2 (t )
p2 (2t )
2sinc
sinc
2
a 1 time compression frequency expansion
0 a 1 time expansion frequency compression
Example: Multiplication in Time
x(t ) tp2 (t )
d
d sin
cos sin
X ( ) j
2sinc j 2
j2
d
d
2
Example: Multiplication in Time –
Cont’d
cos sin
X ( ) j 2
2
Example: Multiplication by a Sinusoid
x(t ) p (t ) cos( 0t )
sinusoidal
burst
1
( 0 )
( 0 )
X ( ) sinc
sinc
2
2
2
Example: Multiplication by a
Sinusoid – Cont’d
1
( 0 )
( 0 )
X ( ) sinc
sinc
2
2
2
0 60 rad / sec
0.5
Example: Integration in the Time
Domain
2|t |
v(t ) 1
p (t )
dv (t )
x(t )
dt
Example: Integration in the Time
Domain – Cont’d
• The Fourier transform of x(t) can be easily
found to be
X ( ) sinc
j 2sin
4
4
• Now, by using the integration property, it is
1
2
V ( )
X ( ) X (0) ( ) sinc
j
2
4
Example: Integration in the Time
Domain – Cont’d
V ( ) sinc
2
4
2
Generalized Fourier Transform
• Fourier transform of (t )
j t
(
t
)
e
dt
1
(t ) 1
• Applying the duality property
x(t ) 1, t
2 ( )
generalized Fourier transform
of the constant signal x(t ) 1, t
Generalized Fourier Transform of
Sinusoidal Signals
cos(0t ) ( 0 ) ( 0 )
sin(0t ) j ( 0 ) ( 0 )
Fourier Transform of Periodic Signals
• Let x(t) be a periodic signal with period T;
as such, it can be represented with its
Fourier transform
x(t )
ce
jk 0 t
k
k
0 2 / T
• Since e j0t 2 ( 0 ), it is
X ( )
2 c ( k
k
k
0
)
Fourier Transform of
the Unit-Step Function
• Since
t
u (t )
( )d
using the integration property, it is
t
1
u (t ) ( )d
( )
j
Common Fourier Transform Pairs