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
Representation of Signals in Terms
of Frequency Components
• Consider the CT signal defined by
N
x(t ) Ak cos( k t k ), t
k 1
• The frequencies `present in the signal’ are the
frequency k of the component sinusoids
• The signal x(t) is completely characterized by
the set of frequencies k , the set of amplitudes
Ak , and the set of phases k
Example: Sum of Sinusoids
• Consider the CT signal given by
x(t ) A1 cos(t ) A2 cos(4t / 3) A3 cos(8t / 2),
t
• The signal has only three frequency
components at 1,4, and 8 rad/sec, amplitudes
A1 , A2 , A3 and phases 0, / 3, / 2
• The shape of the signal x(t) depends on the
relative magnitudes of the frequency
components, specified in terms of the
amplitudes A1 , A2 , A3
Example: Sum of Sinusoids –Cont’d
A1 0.5
A2 1
A 0
3
A1 1
A2 0.5
A 0
3
A1 1
A2 1
A 0
3
Example: Sum of Sinusoids –Cont’d
A1 0.5
A2 1
A 0.5
3
A1 1
A2 0.5
A 0.5
3
A1 1
A2 1
A 1
3
Amplitude Spectrum
• Plot of the amplitudes Ak of the sinusoids
making up x(t) vs.
• Example:
Phase Spectrum
• Plot of the phases k of the sinusoids
making up x(t) vs.
• Example:
Complex Exponential Form
• Euler formula: e
• Thus
j
cos( ) j sin( )
Ak cos( k t k ) Ak e j (k t k )
real part
whence
N
x(t ) Ak e
k 1
j ( k t k )
, t
Complex Exponential Form – Cont’d
• And, recalling that ( z ) ( z z ) / 2 where
z a jb, we can also write
N
1
j ( k t k )
j ( k t k )
, t
x(t ) Ak e
Ak e
2
k 1
• This signal contains both positive and
negative frequencies
• The negative frequencies k stem from
writing the cosine in terms of complex
exponentials and have no physical meaning
Complex Exponential Form – Cont’d
• By defining
Ak j k
ck
e
2
c k
Ak j k
e
2
it is also
N
x(t ) ck e jk t c k e jk t
k 1
N
k N
k 0
ck e jk t , t
complex exponential form
of the signal x(t)
Line Spectra
• The amplitude spectrum of x(t) is defined as
the plot of the magnitudes | ck | versus
• The phase spectrum of x(t) is defined as the
plot of the angles ck arg(ck ) versus
• This results in line spectra which are defined
for both positive and negative frequencies
• Notice: for k 1, 2,
| ck || c k |
ck c k
arg(ck ) arg(c k )
Example: Line Spectra
x(t ) cos(t ) 0.5cos(4t / 3) cos(8t / 2)
0.
0.
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
jk0t
, t
where 0 2 / T is the fundamental
frequency (rad/sec) of the signal and
T /2
1
jko t
ck
x(t )e
dt , k 0, 1, 2,
T T / 2
c0 is called the constant or dc component of x(t)
The Fourier Series – Cont’d
• The frequencies k 0 present in x(t) are
integer multiples of the fundamental
frequency 0
• Notice that, if the dc term c0 is added to
x(t )
N
ce
k N
k 0
j k t
k
and we set N , the Fourier series is a
special case of the above equation where all
the frequencies are integer multiples of 0
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 whence 0 2 / 2
• Clearly x(t) satisfies the Dirichlet conditions and
thus has a Fourier series representation
Example: The Rectangular Pulse
Train – Cont’d
1
1
x(t )
(1)|( k 1) / 2| e jk t , t
2 k k
k odd
Trigonometric Fourier Series
• By using Euler’s formula, we can rewrite
x(t )
as
ce
k
k
jk0t
, t
x(t ) c0 2 | ck |cos(k 0t ck ), t
k 1
dc component
k-th harmonic
• This expression is called the trigonometric
Fourier series of x(t)
Example: Trigonometric Fourier
Series of the Rectangular Pulse Train
• The expression
1
1
|( k 1) / 2| jk t
x(t )
(1)
e , t
2 k k
k odd
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
k odd
2
( k 1) / 2
cos k t (1)
1 , t
k
2
• 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
jk0t
, t
where
T /2
1
jko 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
1
c0
T
• For k 0
k 0 1
k 0
ck
sin
sin
, k 1, 2,
k 0T
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 ( ))
Fourier Transform of the Rectangular
Pulse – Cont’d
• The Fourier transform X ( ) of the
rectangular pulse x(t) can be expressed in
terms of x(t) as follows:
1
jko t
ck x(t )e
dt , k 0, 1, 2,
T
x(t ) 0 for t T / 2 and t T / 2
whence
Tck
x(t )e
jko t
dt , k 0, 1, 2,
Fourier Transform of the Rectangular
Pulse – Cont’d
• Now, by definition X ( ) lim Tck and,
T
since k 0 as T
X ( )
x(t )e
j t
dt ,
• The inverse Fourier transform of X ( ) is
1
x(t )
2
X ( )e
j t
d , t
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 ) e10t 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 ( ))
Fourier Transforms of
Signals with Even or Odd Symmetry
• Even signal: x(t ) x(t )
X ( ) 2 x(t ) cos(t )dt
0
• Odd signal: x(t ) x(t )
X ( ) j 2 x(t )sin(t )dt
0
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