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Fourier Analysis of Signals and
Systems
Dr. Babul Islam
Dept. of Applied Physics and Electronic
Engineering
University of Rajshahi1
Outline
• Response of LTI system in time domain
• Properties of LTI systems
• Fourier analysis of signals
• Frequency response of LTI system
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• 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
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• Linear System:
+ T
)(1 nx
)(2 nx
1a
2a
][][)( 2211 nxanxany T
][][)( 2211 nxanxany TT +
)(1 nx
)(2 nx
1a
2aT
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
nn
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 ),()(][)( knhkxnxnyk
T
• Applying linearity properties, we have
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• 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.,
)()()()()( khkxknhkxnyk
)(nx T(LTI)
)()(),()()( knhkxknhkxnykk
• Applying the time-invariant property, we have
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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]
h2x[n] y[n]
h1h2x[n] y[n]
h1
=
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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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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 = 2f
= Phase in radians
• Fourier Series for Continuous-Time Periodic Signals:
AAcos
t
)cos()( tAty
0
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Complex representation of sinusoidal signals:
,2
)cos()( )()( tjtj eeA
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
tjnnectx 0)(
where
Tn
Tn
T
tdtntxT
b
tdtntxT
a
dttxT
a
0
0
0
cos)(2
sin)(2
)(1
where T
tjnn dtetxT
c 0)(1
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Example:
T
n
T
tdtntxT
a
dttxT
a
0
00
0sin)(2
0)(1
11, 7, ,3for ,4
9, 5, ,1for ,4
2sin
4cos)(
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nn
nnn
ntdtntx
Tb
T
n
02
T
2
T TT t
)(tx1
1
ttttx
5cos
5
13cos
3
1cos
4)(
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• Power Density Spectrum of Continuous-Time Periodic Signal:
n
nTcdttx
TP
22)(
1
• This is Parseval’s relation.
• represents the power in the n-th harmonic component of the signal.2
nc
2
nc
2 323 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.
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2
22)(
)(~T
tperiodic
Tt
Ttx
tx
• Define as a periodic extension of x(t):)(~ tx
n
tjnnectx 0)(~
2/
2/
0)(~1 T
T
tjnn dtetxT
c
dtetxT
dtetxT
c tjnT
T
tjnn
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
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.)( toapproaches )(~ and variable)s(continuou ,0, 00 txtxnT
dtetxT
X tj )(1
)(
• Now, defining the envelope of as)(X nTc
)(1
0nXT
cn
n
tjn
n
tjn enXenXT
tx 00000 )(
2
1)(
1)(~
• Therefore, can be expressed as)(~ tx
• As
• Therefore, we get
deXtx tj)(
2
1)(
dtetxT
X tj )(1
)(
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• Energy Density Spectrum of Continuous-Time Aperiodic Signal:
dXdttxE
22)()(
dXXdX
dtetxdX
deXdttx
dttxtxE
tj
tj
2*
*
*
*
)()()(
)(2
1)(
)(2
1)(
)()(
• 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
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• Fourier Series for Discrete-Time Periodic Signals:• Consider a discrete-time periodic signal with period N. )(nx
nnxNnx allfor )()(
• Now, the Fourier series representation for this signal is given by
1
0
/2)(N
k
Nknjkecnx
where
1
0
/2)(1 N
n
Nknjk enxN
c
• Since k
N
n
NknjN
n
NnNkjNk cenx
Nenx
Nc
1
0
/21
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. 18
• Power Density Spectrum of Discrete-Time Periodic Signal:
k
kn
cnxN
P2
0
2)(
1
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• Fourier Transform for Discrete-Time Aperiodic Signals:• The Fourier transform of a discrete-time aperiodic signal is given by
n
njenxX )()(
• 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
.,
,2,0 i.e., ,,
)()()(
)()()2(
2
)2()2(
Xenxeenx
enxenxkX
n
nj
n
knjnj
n
nkj
n
nkj
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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 can be expressed in terms of as follows:)(nx )(X
nm
nmmxdenx
deenxdeX
nmj
n
mj
n
njmj
,0
),(2)(
)()(
)(
deXnx nj)(2
1)(
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• 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
22
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Frequency Response of an LTI System
• For continuous-time LTI system
• For discrete-time LTI system
][nhnje njeH
n cos HnH cos
)(th
tje tjeH
HtH cos t cos
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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