Digital Signal Processing-Subha

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K.L.N.College of Engineering Department of Electronics and Instrumentation Compiled By.Prof.S.Nagammai HOD/EIE DIGITAL SIGNAL PROCESSING Unit 1 1)a) Explain the concept of energy and power signals. Also check whether the following signals are energy or power signal. 1) 2) b) Briefly explain Quantization. 2)Check whether the following systems for linearity, time invariance, causality and stability. i) ii) 3)Explain the basic signals in the study of DT signals and systems. 4)Determine the responses of the following systems to the input signal i) ii) iii) iv) 5)Clearly define the following with suitable examples. 1)Linear system 2)Time invariant system 3)Causal system. 6)Examine the system defined by with respect to the following properties: i)Linear or Non-linear.

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K.L.N.College of EngineeringDepartment of Electronics and InstrumentationCompiled By.Prof.S.Nagammai HOD/EIEDIGITAL SIGNAL PROCESSINGUnit 11)a) Explain the concept of energy and power signals. Also check whether thefollowing signals are energy or power signal.1){ }13( ) ( )nx n u n 2)( ) sin4x n nb) Briefly explain Quantization.2)Check whether the following systems for linearity, time invariance, causality and stability. i)( )( )x ny n e ii)( ) ( 2). y n x n +3)Explain the basic signals in the study of DT signals and systems.4)Determine the responses of the following systems to the input signal

| |, 3 3,( )0, .n nx notherwise 'i)( ) ( ) y n x n ii)( ) ( 1) y n x n iii)( ) ( 1) y n x n + iv) [ ]13( ) ( 1) ( ) ( 1) y n x n x n x n + + + 5)Clearly define the following with suitable examples. 1)Linear system2)Time invariant system3)Causal system.6)Examine the system defined by ( ) ( 2) y n x n + with respect to the following properties:i)Linear or Non-linear. ii)Static or dynamic.iii)Time invariant or time varying.iv)Causal or Non-causal. v)Stable or unstable.K.L.N.College of EngineeringDepartment of Electronics and InstrumentationCompiled By.Prof.S.Nagammai HOD/EIE7)Give the mathematical and graphical representation of the following sequences:1)Unit sample sequence. 2)Unit step sequence. 3)Unit ramp sequence. 4)Exponential sequence.8)Test the following properties: i)Linearity of ( )( ) ( )x ny n bx n ne + .ii)Time invariance property of 2( ) ( 1) ( ) . y n n x n c +iii)Causality and stability condition of

( ) ( ) ( 1) ( 2). y n x n x n x n + + +

( ) sin ( ). y n x n 9)i)Derive the convolution sum formula to determine the response y(n) of the discrete time LTI system whose impulse response is h(n) for an arbitrary input x(n).ii)Determine whether the following is causal, linear and time invariant.

( ) ( 2). y n x n 10)i)Explain the classifications of discrete time signals with suitable examples.ii)Check whether the unit step sequence is energy signal or power signal.11)For each of the following discrete time system determine whether or not the system is linear, time variant, causal and stable. 1)( ) ( 7) y n x n + 2)( ) ( ) y n nx n 3)3( ) ( ). y n x n 12)Check whether the system ( ) ( 1) ( ) y n ax n x n + is linear, casual, shift invariant and stable.13)Determine whether the following signals are Linear, Time variant, causal and stable.1)( ) cos[ ( )] y n x n 2)( ) ( 2) y n x n +3)( ) (2 ) y n x n 4)( ) ( ) ( 1) y n x n nx n + +.

K.L.N.College of EngineeringDepartment of Electronics and InstrumentationCompiled By.Prof.S.Nagammai HOD/EIEUnit 21)i)Determine the Z-transform of 0( ) cos ( ) x n n u n . (6)ii)State and prove the following properties of Z-transforms: 1)Time shifting2)Time reversal3)Differentiation in Z-domain4)Scaling in Z-domain. (10)2)i)Determine the inverse Z transform of 11 21 3( )1 3 2zX zz z ++ + for |z|>2. (6)ii)Compute the response of the system and check for stability

( ) 0.7 ( 1) 0.12 ( 2) ( 1) ( 2) y n y n y n x n x n + + to input ( ) ( ) x n nu n .(10)3)The impulse response of a system is

1; 0 ( 1)( )0;n Nh notherwise ' Find the transfer function and frequency response. (16)4)i)Response of a system is { } 1, 2, 3, 4. Find Impulse response of the system if the input is { } 1, 4, 3, 2. (8) ii)Determine the convolution sum of two sequences{ } ( ) 4, 8, 2,1, 0, 5 x n and { } ( ) 3, 2,1, 5, 6 h n . (8)5)Find the inverse Z transform for the following signal 2( 3)( 5)zz z +. When the signal is right sided?(8)6)i)State and prove convolution property of discrete Fourier transform.(8)ii)State and prove the shifting property of DFT. (8)7)i)Determine the Fourier transform of the signal | |( ) ;nx n a -1 K.L.N.College of EngineeringDepartment of Electronics and InstrumentationCompiled By.Prof.S.Nagammai HOD/EIE and 2)13| | z >.(8)18)Compute the convolution of the following signals by means of the z transform.(16) ( )( )13112, 0( ), 0nnnx nn '< ( )12 2( ) ( )nx n u n .19)Determine the solution of the difference equation 5 16 6( ) ( 1) ( 2) ( ) y n y n y n x n + for( ) 2 ( )nx n u n .(16)20)Determine the causal signal x(n) having the Z transform

1 1 21( )(1 2 )(1 )x zz z + +.21)Use the convolution to find x(n) if X(Z) is given by 1 11 12 41(1 )(1 ) z z + for ROC 12| | z >.22) Determine the signalx(n) for the Fourier transform 4( )jjx e efor

.23)Find the magnitude and phase of the system ( ) 0.6 ( 1) ( ) y n y n x n at 4 .24)Find the convolution of ( ) ( ) x n h n through z-transform method.

( )( )1214( ) ( )( ) ( )nnx n u nh n u n25) i) Obtain and sketch the impulse response of shift invariant system described by

( ) 0.4 ( ) ( 1) 0.6 ( 2) ( 3) 0.4 ( 4). y n x n x n x n x n x n + + + + (8) ii) An LTI system is described by the equation ( ) ( ) 0.81 ( 1) 0.81 ( 2) 0.45 ( 2). y x x n x n x n y n + + Determine the transfer function of the system. Sketch the poles and K.L.N.College of EngineeringDepartment of Electronics and InstrumentationCompiled By.Prof.S.Nagammai HOD/EIE zeroes on the z-plane. (8)26)Sketch the block diagram representation of the DT system described by

1 1 14 2 2( ) ( 1) ( ) ( 1). y n y n x n x n + + 27) Consider a causal and stable LTI system whose input x(n) and output y(n) are related through the second order differential equation

1 16 6( ) ( 1) ( 2) ( ) y n y n y n x n . Determine the impulse response h(n) for the system.28)Determine the impulse response and the unit step response of the systems described by the difference equation.

( ) 0.7 ( 1) 0.1 ( 2) 2 ( ) ( 2) y n y n y n x n x n + . (16)29)Determine the impulse response for the cascade of two linear time invariant systems having impulse responses: ( )11 2( ) ( )nh n u n and( )12 4( ) ( )nh n u n .30)Determine the transfer function, magnitude and phase response, impulse response for the system 3 1 14 8 3( ) ( 1) ( 2) ( ) ( 1) y n y n y n x n x n + + . (10)31)Determine the impulse response and the unit step response of thesystems described by the difference equation. ( ) 0.6 ( 1) 0.08 ( 2) ( ) y n y n y n x n +.32)Describe the stability of the system described by 11 2( )1zH zz z .Unit 31)Derive and draw the flow graph of the Radix-2 DIFFFT algorithm for the computation of 8-point DFT.2)What are the differences and the similarities between DIT and DIFFFT algorithms?3)Compute the 8-point DFT of the sequence{ } ( ) 1, 2, 3, 4, 4, 3, 2,1 x n .4)Illustrate the concept of circular convolution property of DFT.K.L.N.College of EngineeringDepartment of Electronics and InstrumentationCompiled By.Prof.S.Nagammai HOD/EIE5)Obtain the cascade and parallel realization of the system described by ( ) 0.1 ( 1) 0.2 ( 2) 3 ( ) 3.6 ( 1) 0.6 ( 2) y n y n y n x n x n x n + + + + .6)A signal x(n) has the following Fourier transform 1( )1jXae. Determine the Fourier transforms of the following signals 2) (2 1)) ( ) ( 1)) ( 2)) ( ) cos(5 )ni x nii x n x niii e x niv x n n++ +7)An LTI system with impulse response 1( ) ( )4nh n u n _

,. Determine and sketch the magnitude and phase response of input and output for the following signals. { }3) ( ) cos ,10) ( ) ...1, 0,1,1,1, 0,1,1,1, 0,1...ni x n nii x n < < 8)Develop and draw 16 point FFT(DIT) algorithm using 2 point butterfly structure.9)Find the circular convolution of ( ) ( ) x n h n . ( )12( ) 1; 0 10( ) ; 0 10nx n nh n n 10)State and prove Parsavals theorem.11)Draw the FFT flowchart for radix-2, DIF algorithm.Assume N=8.12)Compute the DFT of x(n) 0, 0 4( )1, 5 7.nx nn ' 13)Find X(k) of the following signals using DIT-FFT algorithm: { } ) ( ) 0,1, 2, 3 i x n K.L.N.College of EngineeringDepartment of Electronics and InstrumentationCompiled By.Prof.S.Nagammai HOD/EIE) ( ) 2nii x n where 0,1, 2, 3, 4, 5, 6, 7 n .14)Given{ } ( ) 1, 2, 3, 4, 4, 3, 2,1 x n , find X(k) using DIF-FFT algorithm.15)Compute the DFT of the sequence ( ) cos2nx n, where 0,1, 2, 3; n using DIF algorithm.16)Using the raix-2 DITFFT algorithm, compute the 8 point DFT of the sequence{ } ( ) 1, 2, 3, 4, 4, 3, 2,1 x n . Draw the flow graph and show all the intermediate results. 17)Draw the flow graph of 16-point DITFFT.18)Compute the IDFT of the sequence { } ( ) 7, 0.707 0.707, , 0.707 0.707, 0.707 0.707, , 0.707 0.707 X K j j j j j j + + using DIT algorithm.19)Derive the signalflow graph for the N=4 point, radix-4, decimation in time FFT algorithm in which input sequence is in the normal order and the computations are done in peace. 20)Compute the 16-point DFT sequence ( ) cos ; 0 152nx n n _ , using radix- 2 decimation in time algorithm.21)Compute linear and circular convolution of the 2 sequences{ }1( ) 1,1, 2, 2 x n and{ }2( ) 1, 2, 3, 4 x n .22)Find out the output of the filter whose impulse response is{ } ( ) 1, 2 h n and the input signal{ } ( ) 1, 2, 1, 2, 3, 2, 3, 1,1,1, 2, 1 x n , using overlap save method.23)Derive the key equations of radix-2 decimation-in-time FFT algorithm and draw the relevant flow graph taking the computation of an 8-point DFT for your illustration. Using the above flow graph compute the DFT K.L.N.College of EngineeringDepartment of Electronics and InstrumentationCompiled By.Prof.S.Nagammai HOD/EIEsequence x(n) defined by { }1 1 1 12 2 2 2, , , , 0, 0, 0, 0.Show the intermediate results on the flow graph itself.24)Explain how FFT algorithm is used to compute the IDFT.25)Find the IDFT of the sequence { } ( ) 36, 4 9.656, 4 4, 4 1.656, 4, 4 1.656, 4 4, 4 9.656 X k j j j j j j + + + using DIF FFT algorithm.26)Find the inverse z-transform of 11 21 3( ) ;| | 21 3 2ZX Z ZZ Z + >+ +27)Compute the response of the system and check for the stability ( ) 0.7 ( 1) 0.12 ( 2) ( 1) ( 2) Y n y n y n x n x n + + and ( ) . ( ) x n n u n .28)Perform the circular convolution of two sequences: { }{ }12( ) 2, 4, 2,1( ) 1, 2, 1, 2XnX n 29)Determine the response of the system whose input and impulse responses are

{ }{ }( ) 1, 2, 3, 1, 2, 3, 4, 5, 6( ) 2,1, 1x nh n using overlap save method.30)Compute the 8-point DFT of the sequence using DIT and DIF algorithm.{ } ( ) 0.5, 0, 0.5, 3, 2,1, 0.5,1 x n 31)Compute the IDFT of the sequence using DIT and DIF algorithm. { } ( ) 8,1 2,1 , 0,1, 0,1 ,1 2 X K j j j j + + 32)Compute the DFT of the sequence{ } ( ) 1, 2, 3, 4 x n .33)Determine all possible signals x(n) associated with the z-transformK.L.N.College of EngineeringDepartment of Electronics and InstrumentationCompiled By.Prof.S.Nagammai HOD/EIE ( ) ( )1 21( )1 112 4X zz z +.34)Perform the linear convolution of the two sequences{ } ( ) 1, 2, 3, 4 x n and { } ( ) 4, 3, 2,1 h n . Use DFT.35)Compute linear and circular convolution of the two sequences { }1( ) 1, 2, 2, 2 x n and{ }2( ) 1, 2, 3, 4 x n .36)Compute the FFT using DIT algorithm for the sequence { } ( ) 1, 2, 3, 4, 4, 3, 2,1 x n and draw the corresponding flow diagram.37)Prove the multiplication of the DFTs of 2 sequences is equivalent to the DFT of the convolution of the 2 sequences in time domain.38)Discuss in detail the use of FFT algorithms in linear filtering.39)Draw the flow group of an 8-point DIF,FFT and explain.40)Given{ } ( ) 0,1, 2, 3, 4, 5, 6, 7 x n find X(k) using DIT FFT algorithm.41)Using decimation-in-time draw the butterfly line diagram for 8-point FFT calculation.42)Given( ) 2nx n and N=8, find X(k) using DIF FFT algorithm.43)What is DIT FFT algorithm.44)What is DIF FFT algorithm.45)Find DFT for { } 1,1, 2, 0,1, 2, 0,1 using FFT DIT butterfly algorithm and plot the spectrum.46)Find IDFT for { } 1, 4, 3,1 using FFT-DIF method.47)Find DFT for { } 1, 2, 3, 4,1.K.L.N.College of EngineeringDepartment of Electronics and InstrumentationCompiled By.Prof.S.Nagammai HOD/EIE48)Realize the following filter using cascade and parallel form with direct form-I structure.

1 2 31 1 21 5(1 )(1 2 4 )z z zz z z + + ++ + +49)Find H(s) for the3rd order lowpass butter worth filter.50)Find DFT of a sequence{ } ( ) 1,1, 0, 0 x n and find the IDFT of { } ( ) 1, 0,1, 0 Y k .51)Compute the 4-point DFT of sequence{ } ( ) 0,1, 2, 3 x n using DIT and DIF algorithm.52)By means of DFT IDFT, determine the sequence 3( ) x n corresponding to the circular convolution of the sequences{ } { }1 2( ) 2,1, 2,1 , ( ) 1, 2, 3, 4 x n x n .53)State the difference between Overlap save method and Overlap add method.54)Compute the FFT for the sequence ( ) 1 x n n + where N=8 using the in place radix-2 decimation in frequency algorithm.Unit 41)Discuss about any three window functions used in the design of FIR filters.2)Deign a digital Butterworth filter satisfying the following constraints withT=1sec. Using Bilinear transformation.Realize the same in Direct form II. ( )0.707 | | 1; 02jH e ( )| | 0.2;34jH e 3)List the three well known methods of design techniques for FIR filters and explain any one.K.L.N.College of EngineeringDepartment of Electronics and InstrumentationCompiled By.Prof.S.Nagammai HOD/EIE4)Explain in detail triangular window method of FIR filter design 2 | | ( 1) 11 ;( ) 1 2 20;Tx N Nfor nWn Notherwise '.5)Explain with procedural steps the design of low pass digital Butterworth filter and list its properties.6)List the design procedure for low pass digital Chebyshev IIR filter.7)Convert analog filter with system function ( )aHs into digital filter using bilinear transformation. 20.3( )( 0.3) 16asHss++ +.8)Design a digital filter using with impulse invariance method

21( )9 18H sS S+ + with T=1sec.9)Implement the following filter using direct form I and parallel structures. 1 11(1 0.5 )(1 0.25 ) Z Z .10)An IIR filter has transfer function of 0.875ZZ assuming input signal

(0) 0.75 x computed results are rounded to 4 bit binary number (1 bit allotted for sign) check for dead band effect. 11)Design a second order digital LPF using bilinear transformation with cutoff frequency 1rad/sec and 1 secsT m .12)Design a linear phase FIR filter approximately the ideal response

1;| |6( )0; | |6H ' Using Hamming window with 9points.13)The desired frequency response of a digital filter is K.L.N.College of EngineeringDepartment of Electronics and InstrumentationCompiled By.Prof.S.Nagammai HOD/EIE

3;4 4( )0; | |4jdeH '< Determine the filter coefficients if the window function is defined as

1; 0 5( )0;xxotherwise '.14)List the various steps in designing FIR filters.15)Derive the frequency response of linear phase FIR filter when impulse response is symmetrical and N is odd.16)How is the design of linear phase FIR filter done by frequency sampling method? Explain.17)Apply the bilinear transformation of 2( )( 1)( 2)aHss s+ + with T=1sec and find H(z).18)The normalized function of an analog filter is given by

21( )1.414 1a nn nHss s+ + Convert the analog filter to a digital filter with a cutoff frequency of 0.4using bilinear transformation.19)Enumerate the various steps involved in the design of low pass digital Butterworth IIR filter.20)The specification of the desired low pass filter is

0.8 | ( ) | 1.0; 0 0.2| ( ) | 0.2; 0.32HH Design a Butterworth digital filter using impulse invariant transformation.21)Explain the various design methods of IIR filters through analog filters.22)Describe in detail, the design of FIR filters using window function method.K.L.N.College of EngineeringDepartment of Electronics and InstrumentationCompiled By.Prof.S.Nagammai HOD/EIE23)Convert the analog filter with system function 20.1( )( 0.1) 9asHss++ + into a digital IIR filter by means of the impulse invariance method.24)Convert the analog filter with the system function 20.1( )( 0.1) 16asHss++ +into a digital IIR filter by means of the bilinear transformation. The digital filter is to have a resonant frequency of 2r 25)Design a digital Butterworth filter that satisfies the following constraint using bilinear transformation. Assume T=ls.

0.9 | ( ) | 1; 02| ( ) | 0.2;34iiH eH e .26)A LPF is to be designed with the following desired frequency response 2,4 4( )0, | |4iideH e ' Determine the filter coefficients ( )dh n if the window function is defined as1; 0 4( )0;nnotherwise '. Also determine the determine the frequency responseof the designed filter.27)Determine the coefficients {h(n)} of a linear phase FIR filter of lengthM=15 which has a symmetric unit sample response of frequency response that satisfies the condition 1, 0,1, 2, 320, 4, 5, 6, 7 15rkkHk _ ' ,28)Determine the system function H(z) of the lowest order Chebyshev digital filter that meets the following specifications: i)1 db ripple in the passband 0 0.3 ii)Atleast 60 db attenuation in the stop band 0.35 . Use the bilinear transformation method.K.L.N.College of EngineeringDepartment of Electronics and InstrumentationCompiled By.Prof.S.Nagammai HOD/EIE29)Derive the conditions that are to be satisfied by the impulse response of an FIR system to have a linear phase. 30)The derived response of a low pass filter is 33, 3 34 4( )0, 34jjdeH e ' Determine( )jH e for N-7 using a Hamming window.31)Design a Chebyshev low pass filter such that the passband magnitude is constant to within 1 db for frequencies below0.2and stop band filter attenuation > 15 db for frequencies between0.3and . Design digital from analog filter using bilinear transformation.32)Design a digital filter equivalent to 2( )( 1)( 2)H ss s+ + using impulse invariant method.33)Find H(s) using impulse invariant technique for the analog system function 21( )( 0.5)( 0.5 2)H ss s s+ + + use T=1s34)Obtain the direct form II, cascade form, parallel form structures for the system 3 228 4 11 2( )( 0.25)( 0.5)z z zH zz z z + +.35)Explain the type 1 and type 2 design of FIR filters using frequency sampling technique. 36)Obtain the transformation formula for the bilinear transformation.37)Compare the frequency domain characteristics of the different types of window functions.38)Explain the procedure for designing an FIR filter using Kaiser window.K.L.N.College of EngineeringDepartment of Electronics and InstrumentationCompiled By.Prof.S.Nagammai HOD/EIE39)Design a digital filter with 21( )7 12H ss s+ + using T=1 sec.40)Design a digital filter using bilinear transform for 2( )( 1)( 2)H ss s+ + with cutoff frequency as 100 rad/sec and sampling time=1.2 ms.41)Design a linear HPF using Hamming window with N=9 1;( )0;cHotherwise < < '42)Realize the following FIR systems in i)Direct formii)Cascade form

1 2 3 41 21 1 11) ( ) 1 22 2 22) ( ) 1 3 2H z z z z zH z z z + + + +43)Consider the transfer function 1 21.0( )1 0.94 0.64H zz z +. Find the pole location and effect due to rounding to 3-bit (excluding sign bit).44)For the desired response 3,8 8( )0,8jjdeH e ' Determine( )jH e for N=7 and compare the response for Hammingwindow and rectangular window.45)Design a digital filter corresponding to an analog filter 0.5( 4)( )( 1)( 4)sH ss s++ +using the impulse invariant method to work at a sampling frequency ofsamples/sec.

46)Determine the Direct Form I, Direct form II, Cascade and parallel structure for the system ( ) 0.1 ( 1) 0.72 ( 2) 0.7 ( ) 0.25 ( 2) y n y n y n x n x n + + .K.L.N.College of EngineeringDepartment of Electronics and InstrumentationCompiled By.Prof.S.Nagammai HOD/EIE47)What is the main drawback of impulse invariant method? How is this overcome by bilinear transformation? 48)Compare the frequency domain characteristics of various window functions.Explain how a linear phase FIR can be used using window method.49)Explain the type I frequency sampling method of designing an FIR filter.50)Obtain direct and cascade form realizations for the transfer function of the system given by 1 2 1 21 3 1 1( ) 1 14 8 8 2H z z z z z _ _ + +

, ,.51)Design a FIR linear phase digital filter approximating the ideal frequency response 1,6( )0,6dH '< .Determine the coefficient of a 11 tap based window method using a rectangular window and Hamming window.Unit 51)Describe the function of onchip peripherals of TMS 320 C 54 DSP processor. What are the different buses of TMS 320 C 54 DSP processor?2)Discuss in detail the various quantization effects in the design of digital filters.3)Draw the circuit of a sample and hold circuit and explain its operation. K.L.N.College of EngineeringDepartment of Electronics and InstrumentationCompiled By.Prof.S.Nagammai HOD/EIE