Fourier analysis of signals and systems

Fourier Analysis of Signals
and
Systems
Babul Islam
Dept. of Electrical and Electronic Engineering
University of Rajshahi, Bangladesh
babul.apee@ru.ac.bd
1

Outline
• Response of LTI system in time domain
• Properties of LTI systems
• Fourier analysis of signals
• Frequency response of LTI system
2

• A system satisfying both the linearity and the time-
invariance 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.
Linear Time-Invariant (LTI) Systems
3

• Linear System:
+ T
)(1 nx
)(2 nx
1a
2a
 ][][)( 2211 nxanxany  T
   ][][)( 2211 nxanxany TT +
)(1 nx
)(2 nx
1a
2a
T
T
System, T is linear if and only if
i.e., T satisfies the superposition principle.
)()( nyny 
4

• Time-Invariant System:
A system T is time invariant if and only if
)(nx T )(ny
implies that
)( knx  T )(),( knykny 
Example: (a)
)1()()(
)1()(),(
)1()()(



knxknxkny
knxknxkny
nxnxny
Since )(),( knykny  , the system is time-invariant.
(b)
][)()(
][),(
][)(
knxknkny
knnxkny
nnxny



Since )(),( knykny  , the system is time-variant. 5

• Any input signal x(n) can be represented as follows:




k
knkxnx )()()( 
• Consider an LTI system T.
1






0for,0
0for,1
][
n
n
n
0 n1 2-1-2 ……
Graphical representation of unit impulse.
)( kn T ),( knh
)(n T )(nh
• Now, the response of T to the unit impulse is
)(nx T   ),()(][)( knhkxnxny
k



 T
• Applying linearity properties, we have
6

• 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.,
)()()()()( khkxknhkxny
k
 


)(nx T
(LTI)
)()(),()()( knhkxknhkxny
kk
 




• Applying the time-invariant property, we have
7

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]
8

• Convolution is Associative:
y[n] = h1[n]  [ h2[n]  x[n] ] = [ h1[n]  h2[n] ]  x[n]
h2x[n] y[n]
h1h2x[n] y[n]
h1
=
9

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.
10

Consider a continuous-time sinusoidal signal,
)cos()(   tAty
This signal is completely characterized by three parameters:
A = Amplitude of the sinusoid
 = Angular frequency in radians/sec = 2f
 = Phase in radians
• Fourier Series for Continuous-Time Periodic Signals:
A
Acos 
t
)cos()(   tAty
0
11

Complex representation of sinusoidal signals:
 ,
2
)cos()( )()( 
 
 tjtj
ee
A
tAty 
sincos je j


Fourier series of any periodic signal is given by:
 





1 1
000 cossin)(
n n
nn tnbtnaatx 
Fourier series of any periodic signal can also be expressed as:




n
tjn
nectx 0
)( 
where






T
n
T
n
T
tdtntx
T
b
tdtntx
T
a
dttx
T
a
0
0
0
cos)(
2
sin)(
2
)(
1


where 


T
tjn
n dtetx
T
c 0
)(
1 
12

Example:




T
n
T
tdtntx
T
a
dttx
T
a
0
0
0
0sin)(
2
0)(
1









 


11,7,,3for,
4
9,5,,1for,
4
2
sin
4
cos)(
2
0
n
n
n
nn
n
tdtntx
T
b
T
n




0
2
T
2
T

TT t
)(tx
1
 1






 ttttx 

5cos
5
1
3cos
3
1
cos
4
)(
13

• Power Density Spectrum of Continuous-Time Periodic Signal:




n
n
T
cdttx
T
P
22
)(
1
• This is Parseval’s relation.
• represents the power in the n-th harmonic component of the signal.
2
nc
2
nc
 2 323 0
Power spectrum of a CT periodic signal.
• If is real valued, then , i.e.,)(tx *
nn cc 
22
nn cc 
• Hence, the power spectrum is a symmetric function
of frequency.
14









2
22
)(
)(~
T
tperiodic
T
t
T
tx
tx
• Define as a periodic extension of x(t):)(~ tx




n
tjn
nectx 0
)(~ 



2/
2/
0
)(~1
T
T
tjn
n dtetx
T
c 






 dtetx
T
dtetx
T
c tjn
T
T
tjn
n
00
)(
1
)(
1
2/
2/

• Fourier Transform for Continuous-Time Aperiodic Signal:
• Assume x(t) has a finite duration.
• Therefore, the Fourier series for :)(~ tx
where
• Since for and outside this interval, then)()(~ txtx  22 TtT  0)( tx
15

.)(toapproaches)(~andvariable)s(continuou,0, 00 txtxnT  




 dtetxX tj
 )()(
• Now, defining the envelope of as)(X nTc
)(
1
0nX
T
cn 






n
tjn
n
tjn
enXenX
T
tx 000
00
)(
2
1
)(
1
)(~ 

 
• Therefore, can be expressed as)(~ tx
• As
• Therefore, we get



 


deXtx tj
)(
2
1
)(




 dtetxX tj
 )()(
16
Synthesis equation (inverse transform)
Analysis equation (direct transform)

• Energy Density Spectrum of Continuous-Time Aperiodic Signal:
• This is Parseval’s relation which agrees
the principle of conservation of energy in
time and frequency domains.
• represents the distribution of
energy in the signal as a function of
frequency, i.e., the energy density
spectrum.
2
)(X
17





 

dXdttxE
22
)(
2
1
)(
 


 
 










































dX
XdX
dtetxdX
deXdttx
dttxtxE
tj
tj
2
*
*
*
*
)(
2
1
)()(
2
1
)(
2
1
)(
)(
2
1
)(
)()(

• Fourier Series for Discrete-Time Periodic Signals:
• Consider a discrete-time periodic signal with period N.)(nx
nnxNnx allfor)()( 




1
0
/2
)(
N
k
Nknj
kecnx 
18
• The Fourier series representation for consists of N
harmonically related exponential functions
)(nx
1,,1,0,/2
 Nke Nknj

and is expressed as
• Again, we have


 


 otherwise
NNkN
e
N
n
Nknj
,0
,2,,0,1
0
/2 











 1,
1
1
1,1
0 a
a
a
aN
a N
N
n
n

19
• Since k
N
n
Nknj
N
n
NnNkj
Nk cenx
N
enx
N
c  







1
0
/2
1
0
/)(2
)(
1
)(
1 
• Thus the spectrum of is also periodic with period N.)(nx
• Consequently, any N consecutive samples of the signal or its
spectrum provide a complete description of the signal in the time
or frequency domains.





1
0
/2
)(
1 N
n
Nknj
k enx
N
c 
110
1
0
1
0
2
1
0
2
 








N,,,l,Ncece)n(x l
N
n
N
k
N/n)lk(j
k
N
n
Nlnj

• Now,
• Therefore,

20
• Power Density Spectrum of Discrete-Time Periodic Signal:






1
0
2
1
0
2
)(
1 N
k
k
N
n
cnx
N
P

• Fourier Transform for Discrete-Time Aperiodic Signals:
• The Fourier transform of a discrete-time aperiodic signal is given by





n
nj
enxX 
 )()(
• 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 since
 ., 
   ,2,0i.e.,,, 
)()()(
)()()2(
2
)2()2(




Xenxeenx
enxenxkX
n
nj
n
knjnj
n
nkj
n
nkj
















21

• 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 can be expressed in terms of as follows:)(nx )(X














 









nm
nmmx
denx
deenxdeX
nmj
n
mj
n
njmj
,0
),(2
)(
)()(
)( 



















deXnx nj
)(
2
1
)(
22

• Energy Density Spectrum of Discrete-Time Aperiodic Signal:




dXnxE
n
 



22
)(
2
1
)(
• represents the distribution of energy in the signal as a function of
frequency, i.e., the energy density spectrum.
2
)(X
• If is real, then)(nx .)()(*
  XX
)()(   XX (even symmetry)
• Therefore, the frequency range of a real DT signal can be limited further to
the range .0  
23

24
Frequency Response of an LTI System
• For continuous-time LTI system
• For discrete-time LTI system
][nh
nj
e    nj
eH 

 ncos       HnH cos
)(th
tj
e 
  tj
eH 

     HtH cos tcos 

25
The z-Transform
• The z-transform of a discrete-time signal x(n) is defined as the power series





n
n
znxzX )()(
where z is a complex variable.
• Since the z-transform is an infinite power series, it exists only for those
values of z for which series converges.
• The set of all values of z for which X(z) attains a finite value is known as
Region of Convergence (ROC).
• From a mathematical point of view the z-transform is simply an alternative
representation of a signal.
• The coefficient of z-n is the value of the signal at time n.

26
• Rational z-transform: Poles and Zeros
• The zeros of a z-transform are the values of z for which .0)( zX
• The poles of a z-transform are the values of z for which .)(zX
Example:
Determine the z-transform and find its pole, zero and ROC for the following
signal:
1),()(  anuanx n
az
z
az
znuaznxzX
n n
nnn



  






0
1
1
1
)()()(
azROC :
Thus has one zero at and one pole at .)(zX 01 z ap 1

27
j
rez 
   n
n z
njnnjj
rnxernxrenxreX 





   )()())(()( 
• In polar form z can be expressed as
where r is the magnitude of z and  is the angle of z.
• If r = 1, the z-transform reduces to the Fourier transform, that is,
)()(  XzX j
ez

• Relationship between the z-transform and the Fourier transform:

28
)(n T )(nh
• Now, for an LTI system, T :





n
n
znhzHnh )()()(
)(nx T
(LTI)
)(*)()()(),()()( khkxknhkxknhkxny
kk
 




)(
)(
)(
)()()(
zX
zY
zH
zHzXzY


which is known as system response

Fourier analysis of signals and systems

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