2.3 Fourier Transform:
2.3.1 From Fourier Series to Fourier Transforms


First, the signal x(t) must satisfy the following condition:
1.
x(t) is absolutely integrable on the real line; that is,
2.
x(t) is an energy type signal



x(t ) dt  
Then the Fourier transform (or Fourier integral) of x(t), defined by

X ( f )   x(t )e  j 2ft dt


The original signal can be obtained from its Fourier transform by

x(t )   X ( f )e j 2ft df

Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea:
http://dasan.sejong.ac.kr/~ojkwon/
1
FOURIER TRANSFORM

The following observations concerning the Fourier transform

X(f) is generally a complex function

Its magnitude | X(f) | and phase  X(f) represent the amplitude and phase of
various frequency components in x(t)



X(f), | X(f) |, | X(f) |2 is sometimes referred to as the spectrum of the signal x(t)
Shorthand for both FT and iFT relations: x(t )  X ( f )

X(f) : Fourier transform of x(t), the notation: X ( f )  x(t )

x(t) : Inverse Fourier transform of X(f), the notation: x(t )  1X ( f )
If the variable in the Fourier transform is ω rather than f, then

1 
jt
X ( )   x(t )e jt dt
x
(
t
)

X
(

)
e
d




2
Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea:
http://dasan.sejong.ac.kr/~ojkwon/
2
FOURIER TRANSFORM

Proof of FT


 xn e j 2nf t
x(t ) 
From FS:
0
n  



n  

Let f 0 
[
1
T0
T0 / 2
T / 2 x( )e
j
2n
T0
d ]e
j
2nt
T0
0
1
, and T0  , i.e. f 0  0, then
T0
x(t )  lim
f 0 0


 j 2nf 0 
j 2nf 0t
[
x
(

)
e
d

]
e
f 0

n  





  [  x( )e  j 2f d ]e j 2ft df
X( f )
Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea:
http://dasan.sejong.ac.kr/~ojkwon/
3
Example 2.3.1:

[(t )]  sinc( f )
Example 2.3.2:

[ (t )]  1
Example 2.3.3:
  nt
e , t 0
 nt
 xn (t )   e , t  0
 0,
t 0


Xn( f )  ?
Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea:
Lettingn  , [sgn(t )] 
http://dasan.sejong.ac.kr/~ojkwon/
1
jf
4
Fourier Transform of Real, Even, and Odd Signals



X ( f )   x(t )e

 j 2ft




dt  x(t ) cos(2ft )dt  j  x(t ) sin(2ft )dt
For real x(t), the transform X(f) is a Hermitian function: X ( f )  X * ( f )
ReX ( f )  ReX ( f ) ImX ( f )   ImX ( f )
X ( f )  X ( f )
X ( f )  X ( f )
Figure 2.36 Magnitude and phase of the spectrum of a real signal.


If, in addition to being real, x(t) is an even signal : the Fourier
transform X(f) will be real and even.
If x(t) is real and odd : the real part of its Fourier transform vanishes
and X( f ) will be imaginary and odd.
Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea:
http://dasan.sejong.ac.kr/~ojkwon/
5
Signal Bandwidth




The bandwidth of a signal represents the range of frequencies
present in the signal
The higher the bandwidth, the larger the variations in the
frequencies present
In general, we define the bandwidth of a real signal x(t) as the
range of positive frequencies present in the signal
In order to find the bandwidth of x(t)



we first find X(f), which is the Fourier transform of x(t)
we find the range of positive frequencies that X(f) occupies
The bandwidth is BW = Wmax – Wmin


Wmax : Highest positive frequency present in X(f)
Wmin : Lowest positive frequency present in X(f)
Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea:
http://dasan.sejong.ac.kr/~ojkwon/
6
2.3.2 Basic Properties of the Fourier Transform

Theorem : Linearity
x(t )  X ( f )
and
y(t )  Y ( f )
ax(t )  by(t )  aX ( f )  bY ( f )

Theorem : Duality
x(t )  X ( f )
X (t )  x( f ) and X (t )  x( f )
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7
Example 2.3.5:

Ex 2.3.1에서 [(t )]  sinc( f ) 이므로 [sinc(t )]  ( f )  ( f )
Example:

Ex 2.3.2에서 [ (t )]  1 이므로 [1]   ( f )   ( f )
Example 2.3.6:

Ex 2.3.3에서 [sgn(t )]  1 / jf 이므로
[1 / jt ]  sgn( f )   sgn( f ) [1 / t ]   j sgn( f )
Example 2.3.4:

u1 (t ) 
1 1
1
1
 sgn(t )  [u1 (t )]   ( f ) 
2 2
2
j 2f
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8
Basic Properties of the Fourier Transform

Theorem : Shift in Time Domain
x(t )  X ( f )

Theorem : Scaling
x(t )  X ( f )


1 f
x(at)  X  
a a
If a > 1, then x(at) is a contracted form of x(t)
 If a < 1, x(at) is an expanded version of x(t)
If we expand a signal in the time domain, its frequency-domain
representation (Fourier transform) contracts
If we contract a signal in the time domain, its frequency domain
representation expands
This is exactly what we expect since contracting a signal in the time
domain makes the changes in the signal more abrupt, thus increasing its
frequency content.


x(t  t0 )  e j 2ft0 X ( f )
Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea:
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9
Basic Properties of the Fourier Transform

Theorem : Convolution
x(t )  X ( f )
and
y(t )  Y ( f )
x(t ) * y(t )  X ( f )Y ( f )

This theorem is very important and is a direct result of the fact that the
complex exponentials are eigenfunctions of LTI systems (or,
equivalently, eigenfunctions of the convolution operation).

Finding the response of an LTI system to a given input is much easier
in the frequency domain than it is the time domain.

This theorem is the basis of the frequency-domain analysis of LTI
systems.
Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea:
http://dasan.sejong.ac.kr/~ojkwon/
10
Example 2.3.9:
3 0  t  4
x(t )  
0 otherwise.
X( f ) ?
Example 2.3.10:
x(t )  (t ).
X( f ) ?
Example 2.3.11:
t
t
x(t )      .
4
2
Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea:
X( f ) ?
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11
Basic Properties of the Fourier Transform

Theorem : Modulation

 x(t )e
j 2f 0t
 


x(t )e j 2f 0t e  j 2ft df

  x(t )e  j 2t ( f  f 0 ) df

 X ( f  f0 )

This theorem is the dual of the time-shift theorem.

The time-shift theorem says that a shift in the time domain results in a
multiplication by a complex exponential in the frequency domain

The modulation theorem states that a multiplication in the time domain by
a complex exponential results in a shift in the frequency domain

A shift in the frequency domain is usually called modulation
Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea:
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12
Example 2.3.14


Determine the Fourier transform of the signal x(t ) cos(2f 0t )
Solution
1
1
1
 1
x(t ) cos(2f 0t )   x(t )e j 2f t  x(t )e  j 2f t   X ( f  f 0 )  X ( f  f 0 )
0

0
2
2
2
 2
In Chapter 3, we will see that this relation is the basis of the operation of
amplitude modulation systems.
Figure 2.38 Effect of
modulation in both the
time and frequency
domain.
Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea:
http://dasan.sejong.ac.kr/~ojkwon/
13
Basic Properties of the Fourier Transform

Theorem : Parseval’s Relation
 If
the Fourier transforms of the signals x(t) and y(t) are
denoted by X(f) and Y(f) respectively, then



 Note
x(t ) y (t )dt   X ( f )Y * ( f )df

that if we let y(t) = x(t) , we obtain





*
x(t ) dt  
2


2
X ( f ) df
This is known as Rayleigh's theorem and is similar to Parseval's
relation for periodic signals
Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea:
http://dasan.sejong.ac.kr/~ojkwon/
14
Example 2.3.16:






sinc4 (t )dt  ?
sinc3 (t )dt  ?
Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea:
http://dasan.sejong.ac.kr/~ojkwon/
15
Basic Properties of the Fourier Transform

Theorem : Time Autocorrelation

The time autocorrelation function of the signal x(t) is
denoted by Rx() and is defined by

Rx ( )   x(t ) x* (t   )dt

 The autocorrelation theorem states that
Rx ( )  X ( f ) 2
Rx ( )  x( )  x* ( )

The Fourier transform of the autocorrelation of a signal
is always a real-valued, positive function
Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea:
http://dasan.sejong.ac.kr/~ojkwon/
16
Basic Properties of the Fourier Transform


Theorem : Differentiation
x(t ) is differentiable and x(t )  X ( f )
x(t )  X ( f )
 dx(t ) 

 j 2fX ( f )

 dt 
 d n x(t ) 
n




j
2

f
X( f )
n 
 dt 
Theorem : Differentiation in the Frequency Domain
j d
tx(t ) 
X( f )
2 df
Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea:


n
n
j
d


 t n x(t )   
X( f )
n
 2  df
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17
Example 2.3.17:
x(t ) 
d
 (t ).
dt
X(f ) ?
Example 2.3.18:
x(t )  t.
X( f ) ?
기출문제:
y(t )  x((t  2)) 그림. Y ( f )를 X ( f )로 표현하시오.
Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea:
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18
Basic Properties of the Fourier Transform

Theorem : Integration
t
X( f ) 1


  x( )d 
 X (0) ( f )
 
 j 2f 2

Theorem : Moments


n
n
j
d


n
 t x(t )  
X( f )

n
 2  df

For the special case of n = 0, we obtain this simple
relation for finding the area under a signal, i.e.,



x(t )dt  X (0)
Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea:
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19
Fourier Transform Pairs
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20
Fourier Transform Properties
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21
2.3.4 Transmission over LTI Systems


The output of an LTI system is equal to the convolution of the input and
the impulse response of the system
If we translate this relationship in the frequency domain using the
convolution theorem, then X(f), H(f), and Y(f) are the Fourier transforms of
the input, system impulse response, and the output, respectively.
Y ( f )  X ( f )H ( f )




The input-output relation for an LTI system in the frequency domain is
much simpler than the corresponding relation in the time domain
In the time domain, we have the convolution integral; however, in the
frequency domain we have simple multiplication
To find the output of an LTI system for a given input, we must find the
Fourier transform of the input and the Fourier transform of the system
impulse response.
Then, we must multiply them to obtain the Fourier transform of the output.
To get the time-domain representation of the output, we find the inverse
Fourier transform of the result
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http://dasan.sejong.ac.kr/~ojkwon/
22
Transmission over LTI Systems

Lowpass signals


Signals with a frequency domain representation that contains
frequencies around the zero frequency and does not contain
frequencies beyond some Wl
Ideal lowpass filter

An LTI system that can pass all frequencies less than some W and
rejects all frequencies beyond W

An ideal lowpass filter will have a frequency response that is 1 for
all frequencies -W  f W and is 0 outside this interval

W is the bandwidth of the filter.
Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea:
http://dasan.sejong.ac.kr/~ojkwon/
23
Transmission over LTI Systems

Ideal highpass filter


There is unity outside the interval -W  f W and zero inside
Ideal bandpass filters


have a frequency response that is unity in some interval W1  |f| W2
and zero otherwise
The bandwidth of the filter is W2 – W1
Figure 2.45 Various filter types.
Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea:
http://dasan.sejong.ac.kr/~ojkwon/
24
2.6 HILBERT TRANSFORM AND ITS PROPERTIES



In this section, we will introduce the Hilbert transform of a signal and explore some
of its properties
The Hilbert transform is unlike many other transforms because it does not involve a
change of domain
 In contrast, Fourier, Laplace, and z-transforms start from the time-domain
representation of a signal and introduce the transform as an equivalent
frequency-domain (or more precisely, transform-domain) representation of the
signal
 The resulting two signals are equivalent representations of the same signal
in terms of two different arguments, time and frequency
Strictly speaking, the Hilbert transform is not a transform in this sense
 First, the result of a Hilbert transform is not equivalent to the original signal,
rather it is a completely different signal
 Second, the Hilbert transform does not involve a domain change, i.e., the
Hilbert transform of a signal x(t) is another signal denoted by xˆ(t ) in the same
domain (i.e., time domain with the same argument t)
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25
HILBERT TRANSFORM

The Hilbert transform of a signal x(t) is a signal xˆ (t ) whose absolute
frequency components lag the frequency components of x(t) by 90
ˆ (t ) has exactly the same frequency components present in x(t) with the same
 x
amplitude–except there is a 90 phase delay


The Hilbert transform of x(t) = Acos(2f0t + ) is Acos(2f0t +  - 90) =
Asin(2f0t + )
A lag of /2 at all absolute frequencies
j ( 2f 0t  2 )
j2

f
t
0
 e
will become e
  je j 2f0t
 j ( 2f 0t  2 )
 e-j2f0t will become e
 je j 2f0t


At positive frequencies, the spectrum of the signal is multiplied by -j

At negative frequencies, it is multiplied by +j
This is equivalent to saying that the spectrum (Fourier transform) of the signal
is multiplied by -jsgn(f).
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http://dasan.sejong.ac.kr/~ojkwon/
26
HILBERT TRANSFORM

In this section, assume that x(t) is real
xˆ(t )   j sgn( f ) X ( f )
1  j sgn( f ) 
1
1
xˆ (t )   x(t ) 
t


1
t
x( )
 t   d

The operation of the Hilbert transform is equivalent to a convolution, i.e., filtering
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27
Example 2.6.1


Determine the Hilbert transform of the signal x(t) = 2sinc(2t)
Solution

We use the frequency-domain approach to solve this problem. Using the
scaling property of the Fourier transform, we have
1 f
1
1
f


x(t )  2         f     f  
2 2
2
2
2




In this expression, the first term contains all the negative frequencies and
the second term contains all the positive frequencies
To obtain the frequency-domain representation of the Hilbert transform of
x(t), we use the relation xˆ(t )   j sgn( f )[ x(t )], which results in
1
1


xˆ (t )  j f    j f  
2
2



Taking the inverse Fourier transform, we have
xˆ (t )  je jt sinc(t )  je jt sinc(t )   j (e jt  e jt )sinc(t )
  j  2 j sin(t )sinc(t )  2 sin(t )sinc(t )
Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea:
http://dasan.sejong.ac.kr/~ojkwon/
28
HILBERT TRANSFORM AND ITS PROPERTIES

Evenness and Oddness


Proof







The Hilbert transform of an even signal is odd, and the Hilbert transform
of an odd signal is even
If x(t) is even, then X(f) is a real and even function
Therefore, -jsgn(f)X(f) is an imaginary and odd function
Hence, its inverse Fourier transform xˆ (t ) will be odd
If x(t) is odd, then X(f) is imaginary and odd
Thus -jsgn(f)X(f) is real and even
Therefore, xˆ (t ) is even
Sign Reversal


Applying the Hilbert-transform operation to a signal twice causes a sign
reversal of the signal, i.e., xˆˆ(t )   x(t )
Proof
2
[ xˆˆ(t )]   j sgn( f ) X ( f )
Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea:
[ xˆˆ(t )]   X ( f )
http://dasan.sejong.ac.kr/~ojkwon/
29
HILBERT TRANSFORM AND ITS PROPERTIES

Energy


The energy content of a signal is equal to the energy content of its Hilbert
transform
Proof

Using Rayleigh's theorem of the Fourier transform, we have

Ex   x(t ) dt  
2




2
X ( f ) df


Exˆ   xˆ (t ) dt    j sgn( f ) X ( f ) df
2

2

Orthogonality


The signal x(t) and its Hilbert transform are orthogonal
Proof




Using Parseval's theorem of the Fourier transform, we obtain

0


x(t ) xˆ * (t )dt   X ( f )[ j sgn( f ) X ( f )]* df   j 


X ( f ) df  j  X ( f ) df  0
2
2
0
In the last step, we have used the fact that X(f) is Hermitian; | X(f)|2 is even
Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea:
http://dasan.sejong.ac.kr/~ojkwon/
30
Recommended Problems











Textbook Problems from p113
2.39.3, 2.39.5, 2.39.9, 2.39.11
2.41.1-4
2.43.(b)
2.46.1, 2.46.4, 2.46.6, 2.46.7, 2.46.9
2.47
2.49.(a), 2.49.(d), 2.49.(e)
2.50
2.53.4
2.59.3, 2.59.6
2.60, 2.62, 2.63, 2.65
Oh-Jin Kwon, EE dept., Sejong Univ., Seoul, Korea:
http://dasan.sejong.ac.kr/~ojkwon/
31