Fourier Analysis of Signals
and
Systems
Dr. Babul Islam
Dept. of Applied Physics and
Electronic Engineering
University of Rajshahi
1
Outline
• Response of LTI system in time domain
• Properties of LTI systems
• Fourier analysis of signals
• Frequency response of LTI system
2
Linear Time-Invariant (LTI) Systems
• A system satisfying both the linearity and the timeinvariance properties.
• LTI systems are mathematically easy to analyze and
characterize, and consequently, easy to design.
• Highly useful signal processing algorithms have been
developed utilizing this class of systems over the last
several decades.
• They possess superposition theorem.
3
• Linear System:
a1
x1 (n)
T
+
a2
x2 ( n )
x1 (n)
T
a1
+
x2 ( n )
y(n)  T a1x1[n]  a2 x2[n]
T
y(n)  a1T x1[n]  a2T x2[n]
a2
System, T is linear if and only if y(n)  y(n)
i.e., T satisfies the superposition principle.
4
• Time-Invariant System:
A system T is time invariant if and only if
x(n)
T
y (n)
implies that
x( n  k )
T
y(n, k )  y(n  k )
Example: (a) y (n)  x(n)  x(n  1)
y (n, k )  x(n  k )  x(n  k  1)
y (n  k )  x(n  k )  x(n  k  1)
Since y(n, k )  y(n  k ), the system is time-invariant.
(b)
y (n)  nx[n]
y (n, k )  nx[n  k ]
y (n  k )  (n  k ) x[n  k ]
Since y(n, k )  y(n  k ), the system is time-variant.
5
• Any input signal x(n) can be represented as follows:
x ( n) 
1,
0,

 [n]  
 x(k ) (n  k )
k  
for n  0
for n  0
1
• Consider an LTI system T.
• Now, the response of T to the unit impulse is
…
 (n)
T
-2 -1
0
1
2
…
n
h(n)
Graphical representation of unit impulse.
 (n  k )
T
h(n, k )
• Applying linearity properties, we have
x(n)
T
y(n)  T x[n] 

 x(k )h(n, k )
k  
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• Applying the time-invariant property, we have
x(n)
T
(LTI)
y(n) 


k  
k  
 x(k )h(n, k )   x(k )h(n  k )
• LTI system can be completely characterized by it’s impulse
response.
• Knowing the impulse response one can compute the output of
the system for any arbitrary input.
• Output of an LTI system in time domain is convolution of
impulse response and input signal, i.e.,
y(n) 

 x(k )h(n  k )  x(k )  h(k )
k  
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Properties of LTI systems
(Properties of convolution)
• Convolution is commutative
x[n]  h[n] = h[n]  x[n]
• Convolution is distributive
x[n]  (h1[n] + h2[n]) = x[n]  h1[n] + x[n]  h2[n]
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• Convolution is Associative:
y[n] = h1[n]  [ h2[n]  x[n] ] = [ h1[n]  h2[n] ]  x[n]
x[n]
h2
h1
y[n]
=
x[n]
h1h2
y[n]
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Frequency Analysis of Signals
• Fourier Series
• Fourier Transform
• Decomposition of signals in terms of sinusoidal or complex
exponential components.
• With such a decomposition a signal is said to be represented in the
frequency domain.
• For the class of periodic signals, such a decomposition is called a
Fourier series.
• For the class of finite energy signals (aperiodic), the decomposition
is called the Fourier transform.
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• Fourier Series for Continuous-Time Periodic Signals:
Consider a continuous-time sinusoidal signal,
y(t )  A cos(t   )
y(t )  A cos(t   )
A
Acos 
0
t
This signal is completely characterized by three parameters:
A = Amplitude of the sinusoid
 = Angular frequency in radians/sec = 2f
 = Phase in radians
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Complex representation of sinusoidal signals:
y (t )  A cos( t   ) 


A j (t  )
e
 e  j (t  ) ,
2
 e j  cos  j sin 
Fourier series of any periodic signal is given by:


n 1
n 1
x(t )  a0   an sin n0t   bn cos n0t
where
1
x(t )dt

T
T
2
an   x(t ) sin n0tdt
T T
2
bn   x(t ) cos n0tdt
T T
a0 
Fourier series of any periodic signal can also be expressed as:
x(t ) 

jn0t
c
e
n
n  
where
cn 
1
T

T
x(t )e  jn0t dt
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Example:
x(t )
1
T

T
2
0
T
2
T
t
 1
1 T
x (t ) dt  0

0
T
2 T
an   x (t ) sin ntdt  0
T 0
a0 
 4
, for n  1, 5, 9,

T
2
4
n  n
bn   x(t ) cos ntdt 
sin

0
T
n
2  4

, for n  3, 7,11,

 n
 x(t ) 
4
1
1

cos

t

cos
3

t

cos
5

t





3
5

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• Power Density Spectrum of Continuous-Time Periodic Signal:

1
2
2
P   x(t ) dt   cn
T T
n  
• This is Parseval’s relation.
2
• c n represents the power in the n-th harmonic component of the signal.
• If x(t ) is real valued, then
cn  cn* , i.e., cn  c n
2
2
cn
2
• Hence, the power spectrum is a symmetric function
of frequency.
 3  2  
0

2
3
Power spectrum of a CT periodic signal.
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• Fourier Transform for Continuous-Time Aperiodic Signal:
• Assume x(t) has a finite duration.
• Define ~
x (t ) as a periodic extension of x(t):
T
T

x
(
t
)


t


~
2
2
x (t )  
T
 periodic
t 
2

x (t ) :
• Therefore, the Fourier series for ~
~
x (t ) 

c e
n  
jn0t
n
T /2
1 ~
 jn0t
c

x
(
t
)
e
dt
where n

T T / 2
x (t )  x(t ) for  T 2  t  T 2 and x(t )  0 outside this interval, then
• Since ~
T /2

1
1
 jn0t
 jn0t
cn 
x
(
t
)
e
dt

x
(
t
)
e
dt


T T / 2
T 
15
• Now, defining the envelope X ( ) of Tcn as

1
X ()   x(t ) e jt dt
T 
1
X ( n 0 )
T
~
• Therefore, x (t ) can be expressed as
 cn 
~
x (t ) 

1
1
jn0t
X
(
n

)
e


0
T
2
n  

 X (n )e
n  
0
jn0t
0
• As T  , 0  0, n0   (continuou
s variable)and ~
x (t ) approachesto x(t ).
• Therefore, we get
1
x(t ) 
2



X ( )e jt d

1
X ()   x(t ) e jt dt
T 
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• Energy Density Spectrum of Continuous-Time Aperiodic Signal:

E   x(t ) dt  

2


X ( ) d
2
• This is Parseval’s relation which agrees
the principle of conservation of energy in
time and frequency domains.

E   x(t ) x* (t )dt


 1 

  x(t )dt   X * ( )e  jt d 

 2 


 1 

*
  X ( )d   x(t )e  jt dt

 2 

  X * ( )d  X ( )  




X ( ) d
2
• X ( ) represents the distribution of
2
energy in the signal as a function of
frequency, i.e., the energy density
spectrum.
17
• Fourier Series for Discrete-Time Periodic Signals:
• Consider a discrete-time periodic signal x(n) with period N.
 x(n  N )  x(n) for all n
• Now, the Fourier series representation for this signal is given by
N 1
 x(n)   ck e j 2kn / N
k 0
where
1
ck 
N
• Since ck  N
N 1
 j 2kn / N
x
(
n
)
e

n 0
1 N 1
1 N 1
 j 2 ( k  N ) n / N
  x ( n) e
  x(n)e  j 2kn / N  ck
N n 0
N n 0
• Thus the spectrum of x(n) is also periodic with period N.
• Consequently, any N consecutive samples of the signal or its
spectrum provide a complete description of the signal in the time
or frequency domains.
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• Power Density Spectrum of Discrete-Time Periodic Signal:

1 
2
P   x(n)   ck
N n 0
k  
2
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• Fourier Transform for Discrete-Time Aperiodic Signals:
• The Fourier transform of a discrete-time aperiodic signal is given by
 X ( ) 

 jn
x
(
n
)
e

n  
• Two basic differences between the Fourier transforms of a DT and
CT aperiodic signals.
• First, for a CT signal, the spectrum has a frequency range of  ,  .
In contrast, the frequency range for a DT signal is unique over the
range   ,  , i.e., 0, 2 , since
X (  2k ) 

 x ( n)e
 j (  2k ) n

n  


 x ( n)e
n  

 j (  2k ) n
x
(
n
)
e

n  
 jn  j 2kn
e

  x(n)e  jn  X ( )
n  
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• Second, since the signal is discrete in time, the Fourier transform
involves a summation of terms instead of an integral as in the case
of CT signals.
• Now x(n) can be expressed in terms of X ( ) as follows:






X ( )e jm d     x(n)e  jn  e jm d

 n  



2x(m), m  n
j ( m  n )
  x ( n)  e
d  

mn
n  
0,
1
 x ( n) 
2




X ( )e jn d
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• Energy Density Spectrum of Discrete-Time Aperiodic Signal:

1
E   x ( n) 
2
n  
2



X ( ) d
2
• X ( ) represents the distribution of energy in the signal as a function of
2
frequency, i.e., the energy density spectrum.
• If x(n) is real, then X * ()  X () .
 X ()  X ()
(even symmetry)
• Therefore, the frequency range of a real DT signal can be limited further to
the range 0     .
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Frequency Response of an LTI System
• For continuous-time LTI system
e jt
H   e j  t
h(t )
cos t 
H  cos t  H 
• For discrete-time LTI system
e
H  e j  n
jn
cos n 
h[n]
H  cos n  H 
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Conclusion
• The response of LTI systems in time domain has been examined.
• The properties of convolution has been studied.
• The response of LTI systems in frequency domain has been analyzed.
• Frequency analysis of signals has been introduced.
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