Transcript ppt
Fourier Analysis of Signals
and
Systems
Dr. Babul Islam
Dept. of Applied Physics and
Electronic Engineering
University of Rajshahi
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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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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.
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• 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.
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• 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.
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• 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]
h1h2
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 = 2f
= 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 n0t bn cos n0t
where
1
x(t )dt
T
T
2
an x(t ) sin n0tdt
T T
2
bn x(t ) cos n0tdt
T T
a0
Fourier series of any periodic signal can also be expressed as:
x(t )
jn0t
c
e
n
n
where
cn
1
T
T
x(t )e jn0t 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 ntdt 0
T 0
a0
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, for n 1, 5, 9,
T
2
4
n n
bn x(t ) cos ntdt
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.
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• c n represents the power in the n-th harmonic component of the signal.
• If x(t ) is real valued, then
cn 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
jn0t
n
T /2
1 ~
jn0t
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
jn0t
jn0t
cn
x
(
t
)
e
dt
x
(
t
)
e
dt
T T / 2
T
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• Now, defining the envelope X ( ) of Tcn as
1
X () x(t ) e jt dt
T
1
X ( n 0 )
T
~
• Therefore, x (t ) can be expressed as
cn
~
x (t )
1
1
jn0t
X
(
n
)
e
0
T
2
n
X (n )e
n
0
jn0t
0
• As T , 0 0, n0 (continuou
s variable)and ~
x (t ) approachesto x(t ).
• Therefore, we get
1
x(t )
2
X ( )e jt d
1
X () x(t ) e jt 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 jt d
2
1
*
X ( )d x(t )e jt 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.
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• 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 2kn / N
k 0
where
1
ck
N
• Since ck N
N 1
j 2kn / 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 2kn / 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 ( )
jn
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 ( 2k )
x ( n)e
j ( 2k ) n
n
x ( n)e
n
j ( 2k ) n
x
(
n
)
e
n
jn j 2kn
e
x(n)e jn 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 jm d x(n)e jn e jm d
n
2x(m), m n
j ( m n )
x ( n) e
d
mn
n
0,
1
x ( n)
2
X ( )e jn 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 jt
H e j t
h(t )
cos t
H cos t H
• For discrete-time LTI system
e
H e j n
jn
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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