Question Paper Details:

University: Rajasthan Technical University

Course: B.Tech Computer science & engineering

Subject: Digital Signal Processing

Course: B.Tech Computer science & engineering

Subject: Digital Signal Processing

Exam Year: Dec 2010/ Jan. 2011

Year or Semester: Third year/ Fifth SemesterPaper Code: 5E3257

Year or Semester: Third year/ Fifth SemesterPaper Code: 5E3257

Unit-I

1. Consider a system whose output y(n) is related to the input x(n) by y(n)= summision upto k=-infinity to + inifinity x(k) x(n+k).

Deter mine whether or not the system is

a) Linear,

b) Shift-inveriant,

c) Stable

d) Causal [Marks 16]

OR

2. Consider a system described by the difference equation y(n)=y(n-1)-y(n-2)+0.5x(n)+0.5x(n-1) Find the response of this system to the input x(n)=(0.5)^n u(n) with initial conditions y(-1)=0.75 and y(-2)=0.25 [Marks 16]

Unit-II

3. a) Prove the convolution theorem of DTFT. [Marks 8]

b) Find the inverse DTFT of X(e^be)= 1/ (1-1/3 e^-f10w) [Marks 8]

OR

4. a) Find the Z-transform of x(n)=|n| (1/2) ^|n| [Marks 8]

b) Find the inverse Z-transform of the second order system.

X(Z)=(1+1/4 Z^-1) / (1-1/2 Z^-1)^2, |Z|>2 [Marks 8]

Unit-III

5. Define and prove the sampling theorem. [Marks 16]

OR

6. Suppose that Xa(t) is band limited to 8KHz (that is, Xa(f)=0 for |f|>8000Hz).

i) What is the Nyquist rate for Xa(t)?

ii) What is the Nyquist rate for Xa(t) cos (2Pi 1000t)? [Marks 8+8=16]

Unit-IV

7. Explain overlap add and overlap save method for computing linear convolution using DFT for longer input sequence x(n). [Marks 16]

OR

8. Derive radix 2 decimation in fre3quency FFT algorithm for evaluating 8 point DFT. [Marks 16]

Unit-V

9. Consider the causal linear shift invariant filter with system function

H(Z)=(1+0.875Z^-1)/ (1+0.2Z^-1+0.9Z^-2)(1-0.7 Z^-1)

Draw a signal flow graph for this system using.

i) Direct form-I [Marks 2]

ii) Direct form-II [Marks 2]

iii) A cascade of first and second order system realized in direct form II [Marks 4]

iv) A cascade of first and second order system realized in transposed direct form II [Marks 4]

v) A parallel connection of first and second order system realized in direct form II [Marks 4]

OR

10. Use the window design method to design a linear phase FIR filter of order N=24 to approximate the following ideal frequency response magnitude:

|Hd(e hx)|= 1 |w|<=0.2pi

0 0.2pi<|w|<=Pi

Unit-I

Consider a system whose output y(n) is related to the input x(n) by y(n)= summision upto k=-infinity to + inifinity x(k) x(n+k).

Deter mine whether or not the system is

a) Linear,

b) Shift-inveriant,

c) Stable

d) Causal [Marks 16]

OR

2. Consider a system described by the difference equation y(n)=y(n-1)-y(n-2)+0.5x(n)+0.5x(n-1) Find the response of this system to the input x(n)=(0.5)^n u(n) with initial conditions y(-1)=0.75 and y(-2)=0.25 [Marks 16]

2. Consider a system described by the difference equation y(n)=y(n-1)-y(n-2)+0.5x(n)+0.5x(n-1) Find the response of this system to the input x(n)=(0.5)^n u(n) with initial conditions y(-1)=0.75 and y(-2)=0.25 [Marks 16]

Unit-II

a) Prove the convolution theorem of DTFT. [Marks 8]

a) Prove the convolution theorem of DTFT. [Marks 8]

b) Find the inverse DTFT of X(e^be)= 1/ (1-1/3 e^-f10w) [Marks 8]

OR

a) Find the Z-transform of x(n)=|n| (1/2) ^|n| [Marks 8]

a) Find the Z-transform of x(n)=|n| (1/2) ^|n| [Marks 8]

b) Find the inverse Z-transform of the second order system.

X(Z)=(1+1/4 Z^-1) / (1-1/2 Z^-1)^2, |Z|>2 [Marks 8]

Unit-III

Define and prove the sampling theorem. [Marks 16]

Define and prove the sampling theorem. [Marks 16]

OR

Suppose that Xa(t) is band limited to 8KHz (that is, Xa(f)=0 for |f|>8000Hz).

i) What is the Nyquist rate for Xa(t)?

ii) What is the Nyquist rate for Xa(t) cos (2Pi 1000t)? [Marks 8+8=16]

Unit-IV

7. Explain overlap add and overlap save method for computing linear convolution using DFT for longer input sequence x(n). [Marks 16]

OR

8. Derive radix 2 decimation in fre3quency FFT algorithm for evaluating 8 point DFT. [Marks 16]

Unit-V

9. Consider the causal linear shift invariant filter with system function

H(Z)=(1+0.875Z^-1)/ (1+0.2Z^-1+0.9Z^-2)(1-0.7 Z^-1)

Draw a signal flow graph for this system using.

i) Direct form-I [Marks 2]

ii) Direct form-II [Marks 2]

iii) A cascade of first and second order system realized in direct form II [Marks 4]

iv) A cascade of first and second order system realized in transposed direct form II [Marks 4]

v) A parallel connection of first and second order system realized in direct form II [Marks 4]

OR

10. Use the window design method to design a linear phase FIR filter of order N=24 to approximate the following ideal frequency response magnitude:

|Hd(e hx)|= 1 |w|<=0.2pi

0 0.2pi<|w|<=Pi

Unit-I

Consider a system whose output y(n) is related to the input x(n) by y(n)= summision upto k=-infinity to + inifinity x(k) x(n+k).

Deter mine whether or not the system is

a) Linear,

b) Shift-inveriant,

c) Stable

d) Causal [Marks 16]

OR

2. Consider a system described by the difference equation y(n)=y(n-1)-y(n-2)+0.5x(n)+0.5x(n-1) Find the response of this system to the input x(n)=(0.5)^n u(n) with initial conditions y(-1)=0.75 and y(-2)=0.25 [Marks 16]

2. Consider a system described by the difference equation y(n)=y(n-1)-y(n-2)+0.5x(n)+0.5x(n-1) Find the response of this system to the input x(n)=(0.5)^n u(n) with initial conditions y(-1)=0.75 and y(-2)=0.25 [Marks 16]

Unit-II

a) Prove the convolution theorem of DTFT. [Marks 8]

a) Prove the convolution theorem of DTFT. [Marks 8]

b) Find the inverse DTFT of X(e^be)= 1/ (1-1/3 e^-f10w) [Marks 8]

OR

a) Find the Z-transform of x(n)=|n| (1/2) ^|n| [Marks 8]

a) Find the Z-transform of x(n)=|n| (1/2) ^|n| [Marks 8]

b) Find the inverse Z-transform of the second order system.

X(Z)=(1+1/4 Z^-1) / (1-1/2 Z^-1)^2, |Z|>2 [Marks 8]

Unit-III

Define and prove the sampling theorem. [Marks 16]

Define and prove the sampling theorem. [Marks 16]

OR

Suppose that Xa(t) is band limited to 8KHz (that is, Xa(f)=0 for |f|>8000Hz).

i) What is the Nyquist rate for Xa(t)?

ii) What is the Nyquist rate for Xa(t) cos (2Pi 1000t)? [Marks 8+8=16]

Unit-IV

7. Explain overlap add and overlap save method for computing linear convolution using DFT for longer input sequence x(n). [Marks 16]

OR

8. Derive radix 2 decimation in fre3quency FFT algorithm for evaluating 8 point DFT. [Marks 16]

Unit-V

9. Consider the causal linear shift invariant filter with system function

H(Z)=(1+0.875Z^-1)/ (1+0.2Z^-1+0.9Z^-2)(1-0.7 Z^-1)

Draw a signal flow graph for this system using.

i) Direct form-I [Marks 2]

ii) Direct form-II [Marks 2]

iii) A cascade of first and second order system realized in direct form II [Marks 4]

iv) A cascade of first and second order system realized in transposed direct form II [Marks 4]

v) A parallel connection of first and second order system realized in direct form II [Marks 4]

OR

10. Use the window design method to design a linear phase FIR filter of order N=24 to approximate the following ideal frequency response magnitude:

|Hd(e hx)|= 1 |w|<=0.2pi

0 0.2pi<|w|<=Pi

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