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UNIT IV
CORRELATION AND SPECTRAL DENSITIESPART – A
1. Define correlation of the process .Answer:
If the process is stationary either in the strict sense or in the wide sense, then is a function of , denoted by or
. This function is called the autocorrelation function of the process .
2. State any two properties of an autocorrelation function.Answer:
i. is an even function of .ii. If is the autocorrelation function of a stationary
process with no periodic component, then
, provided the limit exists.3. Prove that for a WSS process , .
Answer:
Therefore is an even function of .4. Show that the autocorrelation function is maximum at .
Answer:is maximum at
Cauchy-Schwarz inequality isPut and , then
[Since and are constants for a stationary process]
Taking square root on both sides,. [Since is positive]
5. Statistically independent zero mean random processes andhave autocorrelation functions andrespectively. Find the autocorrelation function of the sum
.Answer:
Given ,
If , then
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[Since the processes are independent]
Similarly
6. The autocorrelation function of a stationary process is
. Find the mean and variance of the process.
Answer:
Given
Mean
Variance
7. If the autocorrelation function of a stationary processes is
. Find the mean and variance of the process.
Answer:
Given
Mean
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Variance
.8. The random process has an autocorrelation function
. Find and .
Answer:
Given
Removing the periodic components of , then
Mean
9. Define cross correlation function and state any two of its properties.Answer:
If the process and are jointly wide sense stationary, then is a function of , denoted by . This function
is called the cross correlation function of the process and.
Properties of cross correlation function are:i. .ii. If the process and are orthogonal, then .iii. If the process and are independent, then
.
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10. Define power spectral density function of stationary random processes .Answer:
If is a stationary process with autocorrelation function , then the Fourier transform of is called the power spectral density function of and denoted as or .
.
11. Define cross spectral density.Answer:
If and are two jointly stationary random processes with cross correlation function , then the Fourier transform of
is called the cross spectral density of and and denoted as .
.
12. State and prove any one of the properties of the cross spectral density function.Answer:
Cross spectral density function is not an even function of , but it has a symmetry relationship.
Proof:
Putting whenwhen
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13. Find the power spectral density of a random signal with autocorrelation function .Answer:
Given
(Since the first integrand is even and
the second integral is odd)
14. The autocorrelation function of the random telegraph signal process is given by . Determine the power density spectrum of the random telegraph signal.Answer:
Given
(Since the first integrand is even and
the second integral is odd)
15. Find the power spectral density of a WSS process with autocorrelation function .
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Answer:Given
Put
WhenWhen
16. IF the power spectral density of a WSS process is give by
.
Answer:
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.
17. The power spectral density function of a zero mean wide sense stationary process is given by . Find
.Answer:
[ The 1st integrand is
even and the 2nd is odd]
.
18. The power spectral density of the random process is given by . Find its autocorrelation function.Answer:
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[ The 1st integrand is
even and the 2nd is odd]
.
19. Given the power spectral density , find the
average power of the process.Answer:
[Where is the closed contour consisting
of the real axis from to and the upper half of the circle ]
Average power
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PART – B
1. If is a WSS process with autocorrelation function and if. Show that .
Answer:Given
aatXatXE
aatXatXE
atXatXEatXatXE
2
2
aRaRRR XXXXXXXX 22 .
2. A stationary random process has autocorrelation function given
by . Find the mean and variance of .
Answer:
Given
Mean
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Variance
.3. State the properties of autocorrelation function for a WSS process.
Answer:i. is an even function of .
ii. is maximum at . iii. If the autocorrelation function of a real stationary
function is continuous at , it is continuous at every other point.
iv. If is the autocorrelation function of a stationary
process with no periodic component, then
, provided the limit exists.4. Given a stationary random process where
followed uniform distribution. Find the autocorrelation function of the process.Answer:
Given is uniformly distributed in , the PDF of is
.
.
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5. Consider two random processes and
where and is uniformly distributed random variable over
. Verify whether .Answer:
Given is uniformly distributed in , the PDF of is
.
and
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Since the function takes the maximum and minimum values to be 1 to -1, .
Hence .
6. The autocorrelation function for a stationary process is given by. Find the mean value of the random variable
and variance of .
Answer:Given .
Mean of
Variance of
.
7. If and where is a constant
and is a random variable uniformly distributed in , find ,
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, and . Verify two properties of autocorrelation function and cross correlation function.Answer:
Given is uniformly distributed in , the PDF of is
.
and
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[ ]
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(1)
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(2)
8. Define power spectral density and cross spectral density of a random process. State their properties.Answer:
Power spectral density: If is a stationary process with the autocorrelation function , then the Fourier transform of is called the power spectral density function of and denoted as
.
.
Properties of power spectral density function:i. The value of the spectral density function at zero
frequency is equal to the total area under the graph of the autocorrelation function.
ii. The mean square value of a wide sense stationary process is equal to the total area under the graph of the spectral density.
iii. The spectral density function of a real random process is an even function.
iv. The spectral density of a process , real or complex is a real function of and nonnegative.
v. The spectral density and the autocorrelation function of a real WSS process form a Fourier cosine transform pair.
Cross spectral density: If and are two jointly stationary random processes with cross correlation function , then the Fourier transform of is called the cross spectral density of and and denoted as .
Properties of cross spectral density function:i.
ii. and are even functions of
iii. and are odd functions of
iv. and if and are orthogonal.v. If and are uncorrelated, then
.
9. State and prove Wiener-Khinchine theorem.Answer:
Statement: If is the Fourier transform of the truncated random process defined as where is a
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real WSS process with power spectral density function , then
.
Proof:
Given
[ is real]
[ is WSS]
(say) (1)
The double integral (1) is evaluated over the area of the square bounded by and as shown in the figure.
We divide the area of the square in to a number of strips like, where is given by and is given by .When is at the initial position , . When it is at
the final position , . Hence when varies from to , the area is covered.
Now Elemental area of the plane.Area of (2)
and
andWhenArea of
[Omitting ] (3)From (2) and (3), we have (4)Using (4) in (1), we have
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[Assuming that is
bounded]
.
10. Find the power spectral density of the random process, if its autocorrelation function is given by .Answer:
[Since the first integrand
is even and the second is odd]
11. Autocorrelation function of an ergodic process is. Obtain the spectral density of .
Answer:
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[Since the first integrand
is even and the second is odd]
.
12. The autocorrelation function of a signal is , where
is a constant. Find the power spectral density and average power.Answer:
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Put When
When
Average power13. The autocorrelation function of the random process is given
by . Find the power spectrum of the process
.Answer:
[Since the first integrand
is even and the second is odd]
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.
14. The autocorrelation function of the Poisson increment process is given by . Prove the spectral density is given by
.
Answer:
Where is the Fourier transform of
The inverse Fourier transform of is given by
.
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Now let us find corresponding to where is the unit impulse function.
[ ]
.
15. A random process is given by whereand are independent random variables such that
and . Find the power spectral density of the process.Answer:
Consider
Therefore
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16. Show that if whereis week sense stationary.Answer:
GivenThen [Proved in the 1st
problem.]Taking Fourier transform on both sides, we have
Put when Put whenwhen when
.
17. If and are uncorrelated random processes, then find the power spectral density of if . Also find the cross spectral density and .Answer:
Given
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Given and are uncorrelated random variables, we have.
.18. Obtain the spectral density of , when is an
intendment of such that .
Answer:
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.
19. The power spectral density function of a zero mean WSS process is given by . Find and show also
that and are uncorrelated.
Answer:
To show that and are uncorrelated, we have to
show that .
But
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and are uncorrelated.
20. The cross power spectrum of real random process and is given by . Find the cross correlation function.Answer:
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.
21. If the cross spectral density of and is given by
where and are constants, find the
cross correlation function.Answer:
22. Given the power spectral density of a continuous process as
, find the mean square value of the process.
Answer:
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Consider
(1)
The integral (1) is evaluated by contour integration technique.
Consider , where is the closed contour
consisting of the real axis from to and the upper half of the circle.The singularities of the integrand lying within are and .
Using Cauchy’s residue theorem, taking limits as and using Jordan’s lemma, we get
(2)
Using (2) in (1), we have
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Mean square value of the process
.
23. The power spectrum of the WSS process is given by
. Find its autocorrelation and average power.
Answer:
(1)
The integral (1) is evaluated by contour integration technique.
Consider , where is the closed contour consisting of
the real axis from to and the upper half of the circle .The only pole that lies inside the circle is of order 2.
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Using Cauchy’s residue theorem, taking limits as and using Jordan’s lemma, we get
(2)
Substitute (2) in (1), we have
Average power of
.
24. Given that a process has the autocorrelation function, find the power spectrum.
Answer:
(1)
Note that
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(2)
Equation (1)
.
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