UNIT –VLINEAR SYSTEMS WITH RANDOM INPUTS

PART – A

1. Define a system. When is it called a linear system? A system is a functional relationship between the input and the output . The functional relationship is written as . If , then is called a linear system.

2. Write a note on linear system. If , then is called a linear system. If where , then is called a time – invariant system or and are said to form a time invariant system. If the output at a given time depends only on and not on any other past or future values of , then the system is called a memoryless system. If the value of the output at depends only on the past values of the input . (ie) , then the system is called a casual system.

3. State the properties of linear system. The properties of linear system are

(i) If a system is such that its input and its output are related by a convolution integral, then the system is a linear time invariant system.

(ii) If the input to a time-invariant, stable linear system is a WSS process, the output will also be a WSS process.

(iii) The power spectral densities of the input and output processes in the system are connected by the relation

, where is the Fourier transform of unit impulse response function .

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4. Define system weighting function. If the output of a system is expressed as the convolution of the input

and a function (ie) then is called the

system weighting function.

5. What is unit impulse response of a system? Why is it called so?

If a system is of the form then the

system weighting function is also called unit impulse response of the system. It is called so because the response (output) will be , when the input = the unit impulse function .

6. If the input of a linear system is a Gaussian random process, comment about the output random process. If the input of a linear system is a Gaussian random process, then the output will also be a Gaussian random process.

7. If the input of the system is the unit

impulse function, prove that .

Given

Put Therefore

[ by the property of

convolution ] .

8. If a system is defined as , find its unit

impulse function.

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Given

The unit impulse function .

9. If and in the system are WSS

processes, how are their autocorrelation functions related? , where * denotes convolution.

10. If the input and output of the system are

WSS processes, how are their power spectral densities related? , where is the Fourier transform of

11. Define the power transfer function or system function of the system. The power transfer function or system function of the system is the Fourier transform of the unit impulse function of the system.

12. If the system function of a convolution type of linear system is given by

. Find the relation between power spectral

density functions of the input and output processes.

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[since the first and second

integrand are even and odd functions]

.

13. What is thermal noise? By what type of random processes is it represented? Thermal noise is the noise because of the random motion of free electrons in conducting media such as a resistor. It is represented by Gaussian random processes.

14. Why is thermal noise called white noise? is a constant for all values of (ie) contains all frequencies in equal amount. White noise is called so in analogy to white light which consists of all colour.

15. Define white noise. Let be a sample function of a WSS noise process, then is called the white noise if the power density spectrum of

is constant at

all frequencies . (ie) where is a real positive

constant.

16. If the power spectral density of white noise is , find its

autocorrelation function.

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Therefore .

17. If the input to a linear time invariant system is white noise , what is power

spectral density function of the output? If the input to a linear time invariant system is white noise , then the power spectral density of the output is given by

.

where is the output process and is the power transfer function.

18. Find the average power or the mean square value of the white noise ? Average power =

19. A wide sense stationary noise process has an autocorrelation function with as a constant. Find its power density spectrum.

[since the first and

second integrand are even and odd functions]

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.

20. Define band limited white noise. Noise having a non-zero and constant spectral density over a finite frequency band and zero elsewhere is called band limited white noise. If is a band limited white noise, then

.

21. State a few properties of band limited white noise. Properties of band limited white noise are

(i)

(ii)

(iii) and are independent, where is a

non-zero integer.

22. Find the autocorrelation function of the band-limited white noise.

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[since the first and second integrand are even and odd functions]

.

23. Find the average power of the band – limited white noise. Average power

.

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PART – B

1. Show that if is a WSS process then the output is a WSS process.

Solution:

[ since is a WSS process ]

= a finite constant independent of

[ since is a

WSS ] Since the R.H.S of the above equation is a function of , so will be the L.H.S. Therefore, will be a function of . Therefore, is a WSS process.

2. For a linear system with random input , the impulse response and output , obtain the cross correlation function and

the output autocorrelation function .

Solution: (1) Let

Therefore,

Putting when when

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(2)

Assuming that and are jointly WSS

Therefore, .

3. For a linear system with random input , the impulse response and output , obtain the power spectrum and cross

power spectrum .Solution: By the previous problem, we have

----------------(1) -----------------(2) Taking Fourier transform of equations (1) and (2) , we have --------------(3) where is the Fourier transform of unit impulse response function and is the conjugate of . and ------------------(4) Inserting (3) in(4), we have Therefore, .

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4. Show that where and are the power spectral density functions of the input and the output

and is the system transfer function.Solution:

If is WSS, then

The above integral is a two fold convolution of autocorrelation with impulse response Therefore, Taking Fourier transform on both sides, we have --------------------------(1)

---------------------------------------

(2)

Putting when when

---------------------------------(3) Inserting (2) and (3) in (1), we have .

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5. A system has an impulse response function find the power spectral density of the output corresponding to the input . Solution: Given where

Therefore,

.

6. is the input voltage to a circuit and is the output voltage. is a stationary

random process with and . Find and if the

power transfer function is .

Solution:

[ since is given ]

Therefore, .

where

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[since the first and

second integrand are even and odd functions]

7. A wide sense stationary random process with autocorrelation function where and are real positive constants is applied to the input of an linearly time invariant system with impulse response where is a real positive constant. Find the spectral density of the output of the system.

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Solution:

.

.

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8. Establish the spectral representation theorem. Solution: Let be a signal and define

Represent in a Fourier series ( complex form )

where

[ taking the envelope of ]

Therefore,

As ,

Therefore, -----------------------(1)

and equivalently, ( by Fourier integral theorem )

-------------------------(2)

is usually called as spectrum of the aperiodic signal

9. Show that in an input – output system the energy of a signal is equal to the energy of the spectrum.

Solution: Energy of a signal

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[ by equation (1) of problem 8 above ]

[ by changing

the order of integration ]

[ by equation (2) of problem 8 above ]

= Energy of the spectrum.

10. If is a band limited process such that , when , prove that .

Solution:

[ since is an even function and therefore the first integrand is even and the second is an odd function ].

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[since is band

limited ]

-----------------------(1)

From trigonometry,

---------------------------(2)

Inserting (2) in (1) , we have

11. If , where is a constant, is a random variable with a uniform distribution in and is a band limited Gaussian white noise with a power spectral density

.

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Find the power spectral density of . Assume that and are independent. Solution: Given Since is uniformly distributed in , the density function is given by

( since and are independent )

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Taking Fourier transform on both sides , we have

where .

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