Dsp Unoversity Papers

Digital Signal Processing − November 2004 Time : 3 Hrs.] N.B.:

[Marks : 100

(1) Question No. 1 is compulsory. (2) Attempt any four out of remaining questions. (3) Assumptions made should be clearly stated. (4) Figures to the right indicate marks.

1.

State whether each of the following statement is true or false. Justify your answer in 4 or 5 sentences.

[20]

(a) Linear phase filters are always IIR (b) For a causal system | h(n) | tends to zero, as n tends to infinity, the system is stable. (c) A stable filter is always causal (d) A stable, causal FIR filter has its poles lying anywhere inside the unit circle in the z-plane (e) IIR filters have recursive realization always. 2.

(a) x (n) = δ(n) + δ(n– 1) – δ(n– 2) + δ(n– 3) + δ (n– 4) +δ (n– 5) h(n) = – δ(n) + 2δ(n– 1) – δ(n– 2) find y(n) using linear convolution (b) Obtain the plot of magnitude and phase response of a filter with the following impulse response h(n) = {– 1, 2, – 2, 1} Comment of the type of filter based on band magnitude and phase response.

[10]

[10]

3.

A causal system has the following transfer function. [20] 2 z H(z) = 2 (z − 0.1z − 0.12) ROC | z | > 0.4. (a) Plot the pole– zero locations in the z– plane (b) Find h(n) by (i) Partial fraction expansion (ii) Power series expansion by long division method (c) Find the difference equation and find h(n) from the difference equation (d) Show the direct from realization and canonic realization. Is there a difference in the two realizations.

4.

The impulse response of a system is given by h(n) = (1/2)n u(n) + (– 1/5)n u(n) (a) Find H(z) along with ROC

[20]

(b) Find H(ejw) directly and from z– transform (c) Comment on the system as causal, FIR, and BIBO stable (d) Find the energy in the sequence h(n) (e) Give a parallel realization of the system 5.

(a) If x(n) = δ(n) + δ(n–1) + δ(n– 2), (1) Find X(ejw) (2) Find X(k) 4– point DFT (do not use FFT) (3) Show that DFT is the sampled version of | X (ejw) |

[10]

(b) A sequence x(n) = {a, b, c, d} has DFT X(K) = {1, 2, 3, 2}. Find x(n) using 4 point DIF– FFT. If y(h) = [a, d, c, b], what is Y(K). Use DFT properties and state the property used. [10] 6.

(a) An ideal low pass filter has the response H(ejω) = 2 e– jωα | ω | < ωc = 0 ωc < | ω | < π Find h(n) ? For a transition width < π/32, calculate the window length and the values of α for (i) rectangular window (ii) hamming window.

[10]

(b) Design and realize a low pass filter using the bilinear transformation method to satisfy the following characteristics. [10] (i) monotonic stop band and pass band (ii) – 3dB cut off frequency of 0.5π rad (iii) stop band attenuation of 15dB at 0.75π rad 7.

(a) Write notes on the applications of DSP in (i) Image processing (ii) Speech Processing and (iii) Telecommunication

[12]

(b) Explain a DSP processor. State advantages of DSP processor over microprocessor for DSP applications. [8] 

− 13 −

Vidyalankar : BE − DSP

Digital Signal Processing − May 2005 Time : 3 Hrs.] N.B.:

[Marks : 100

(1) Question Nos. 1 is compulsory. (2) Attempt any four from rest. (3) Assume any suitable data if necessary with justification.

1.

(a) The impulse response of LTI system is h(n) = {1, 2, – 1, 3} [10] ↑ Determine the output response of the system to input x(n) = {1, 2, 3, 1} ↑ (b) Consider the analog signal – [10] xa(t) = 3 cos 4000 π t + 5 sin 6000 π t + 13 cos 2000πt (i) What is the Nyquist rate of sampling for this signal. (ii) If this signal is sampled at sampling frequency Fs = 5000 samples / sec. derive the expression for the discrete time signal after sampling. (iii) What is the analog signal if reconstruction is done from the above samples using ideal interpolation.

2.

If x(n) is input and y(n) is output sequence of a system then for every system given below determine whether the system is – [20] (i) static (ii) stable (iii) causal (iv) linear (v) shift invariant Justify your answer − (a) y(n) = ex(n) (b) y(n) = a x(n) + 6 (c) y(n) = x(n) + n x(n+1) (d) y(n) = x(– n+2)

3.

(a) State and prove the following properties of z– transform. (i) Find equivalent operation in time domain if differentiation is done in z– domain. (ii) What is the operation in z domain if two time signals are convalved in time domain. (b) Find the z– transform of the following sequences and specify ROC.

[10]

[10]

n

(i) x(n) =

⎛1⎞ n ⎜ ⎟ u(n) ⎝4⎠

(ii)

x(n) = n u(– n– 1)

4.

(a) Compute the response y(n) of the system if the input x(n) = n u(n). y(n) = 0.7 y(n– 1) – 0.12 y(n– 2) + x(n– 1) + x(n– 2) Comment on the stability of the system. (b) Using partial fraction expansion method, find inverse Z– transform of the following expression. z(z 2 − 4z + 5) X(z) = (z − 1)(z − 2)(z − 3) for following ROC (i) | z | > 3 (ii) 2 < | z | < 3 (iii) | z | < 2.

[10]

5.

(a) Determine discrete time Fourier Series representation for the signal – ⎡ nπ ⎤ x(n) = cos ⎢ ⎥ ⎣3⎦ plot spectrum of x(n).

[10]

(b) Obtain the power density spectrum of a periodic signal given by – x(n) = { ..... 0, 1, 2, 3, 0, 1, 2, 3, 0, ....} Also obtain total power.

[10]

(a) Sketch 8 – point signal flow graph for radix 2 DIT FFT.

[10]

(b) Using this diagram or otherwise determine DFT of the sequence – h(n) = { 1, 1, 1, 0, 0, 0, 1, 1 } Plot magnitude and phase response.

[10]

(a) Desired frequency response of linear phase FIR filter is given below – π ⎧ for | w | < ⎪⎪0 2 jw H (e ) = ⎨ π ⎪e− j2w for < | W | < π ⎪⎩ 2 Using Hamming window – ⎛ 2πn ⎞ w(n) = 0.54 – 0.46 cos ⎜ ⎟ 0≤n≤M– 1 ⎝ M ⎠ where M is filter length. (b) Plot magnitude and phase response of the above filter.

[20]

6.

7.



− 14 −

University Question Papers

Digital Signal Processing − December 2005 Time : 3 Hrs.] N.B.:

1.

(1) (2) (3) (4)

[Marks : 100

Question No. 1 is compulsory. Attempt any four questions out of remaining six questions. Figures to the right indicate full marks. Assume suitable data if necessary and state them clearly.

(a) Prove that the auto correlation sequence at zero lag has highest magnitude with respect to magnitude at any other lag. [4] (b) The z transform of x(n) is X(z)=1 + 2z−1. Find z transform of following and indicated their region of [4] convergence. n

⎛1⎞ (ii) x2(n) = ⎜ ⎟ x ( n − 2 ) ⎝2⎠ (c) Determine convolution of following pair of single using z transform. x1(n) = nu(n), x2(n) = 2nu(n − 1) (d) What is Discrete Hilbert Transform ? Why it is used ? (e) Write the relationship of Discrete Fourier transform to Fourier transforms and z transform.

(i) x1(n) = x(3 + .n)

[4] [4] [4]

2.

(a) Find the squared magnitude response of an filter, which has two poles P1, 2 = 0.8e± j π 2 and two zeros at origin. Plot it w.r.t. frequency and identify the filter type based on passband. [7] (b) A fourth order antisymmetric, linear phase filter has all real zeros. one of them is located at 0.5. What may be the location of remaining three zero ? Find the transfer function of this filter and identify the filer type [6] based an pass band. [7] (c) A certain discrete time LTI filter has the following data− Poles are at 0.2 and 0.6 Zeros are at 0.4 and origin Gain of filter is 5. Show (i) Direct from II and cascade realization (ii) If register length is 4 bits including sign bit. Calculate the effect on poles and zeros of above filter due to finite word length of filter.

3.

(a) If DFT of a sequence x(n) is given by x(k) = {1 2 3 2} What may be following signals ?

[8]

(ii) x2(n) = x ( n ) e jn π 2 .

(i) x1(n) = x(n + 2)

Use fast technique only once. (b) Let x(n) = {1 2 −3 2} [8] (i) find x(k) using DIF − FFT (ii) let y(n) = x(n/2) for n even. = 0 for n odd. Find 8 point y(k) using x (k) without performing DFT/FFT (c) x(n) and h(n) are two non − periodic sequences of length 7 and 5 respectively. The convolution of these sequences is to be find out using FFT technique. What sequence length is chosen for FFT? What are the [4] steps taken ? 4.

(a) (i) Prove and explain graphically the difference between the relation. [5] x(n) δ(n −no) = x(no). δ(n−no) and x(n) * δ(n−no) = x(n−no). (ii) Show that a discrete time system. which is described by a convolution simulation is LTI and relaxed. [5] (b) Determine the convolution of signals using z transforms. [10] n

5.

⎛1⎞ x1(n) = ⎜ ⎟ u ( n ) , x2(n) = cos πn u(n). ⎝2⎠ (a) An FIR LTI system has an impulse response h(n) which is real valued, even and has finite duration of [8] (2N + 1). Show that if z1 = yejwo is a zero of the system, then z1 = (1/r)ejwo is also zero. (b) A causal DT system has transfer function H(z) such that H(z) = H1(z)H2(z). The pole−zero diagram of H1(z) and H2(z) is as follows

H1(z)

−1/3

H2(z)

−1/2

−1/2

(i) Find the transfer function of total system. (ii) Find difference equation of system.

[3] [2]

− 15 −

Vidyalankar : BE − DSP

n

⎛ −1 ⎞ (iii) Find the response of the system to the input x (n) = ⎜ ⎟ u ( n ) ⎝ 2⎠ (iv) What is magnitude and phase response of the system at w = 0 and w = π.

6.

7.

(a) A digital LPF is required to meet the following specifications Pass band ripple ≤ 1db Pass band edge = 4KHz ≥ 40db Stop band attenuation Stop band edge = 6KHz Sampling frequency = 24KHz. (i) Determine the order of Butterworth filter which meets above specifications. (ii) Determine cut off frequency of filter. (b) Show pole−zero locations of the normalized Fourth order Butterworth IIR filter. (c) Let x(n) = {1 2 3 2 4 3 2 1} Find DFT of x(n) using DIT − FFT algorithm only. (a) Find the values x(0) and x(∞) for a function x(n), if x ( z ) =

1 1 − 0.1z

−1

− 0.2z −2

[3] [4] [8]

[6] [6] [4]

(b) A causal DT system has a difference equation [4] y(n) = x(n) − 0.4 y(n − 1) − 0.25 y(n − 2) what is ROC of this system ? (c) Explain overlap and save method of filtering, if input data sequence is long and impulse response has a few [4] number of samples. [4] (d) Prove the BIBO stability condition in time domain. [4] (e) Find the energy of the signal. n

⎛1⎞ x ( n ) = ⎜ ⎟ u ( n ) + 3n u ( − n − 1) ⎝ 2⎠



Digital Signal Processing − May 2006 Time : 3 Hrs.] N.B.:

[Marks : 100

(1) Question Nos. 1 is compulsory. (2) Attempt any four questions from remaining six questions (3) Figures to right indicate full marks. (4) Assume suitable data if necessary . ∞

1.

∑

(a) Show that

x 2 (n) =

n = −∞

∞

∑

n = −∞

x e2 (n) +

∞

∑

n =−∞

x 02 (n)

where xe(n) and x0(n) are even and odd parts of x(n). (b) Find the Energy of the signal x(n) = (1/4)n = 2

n

[5]

n≥0 n<0

1 ⎛ 1 −1 ⎞ ⎛ 1 −1 ⎞ ⎜1 − z ⎟ ⎜1 + z ⎟ ⎝ 2 ⎠⎝ 4 ⎠ (d) Determine whether following signals are periodic (i) cos (0.3 π n + π/6) (ii) x(n) = sin (0.01 πn)

(c) Find x(n) using convolution for X(z) =

2.

[5]

[5]

(a) Determine the frequency response, magnitude response, magnitude response, phase response of the system [10] given by 1 y(n) − y (n − 1) = x(n) − x(n −1) 2 (b) (i) Determine cross correlation of the following sequence. x(n) = {1, 0, 0, 1}, h(n) = {4, 3, 2, 1} (ii) Determine the causal signal x(n) having the z − transform 1 X(z) = −1 (1 + z )(1 − z −1 ) 2

3.

[5]

[5] [5]

(a) If x(n) = {1, 2, 3, 2} (i) Find X[K] using DIT − FFT. (ii) If x1(n) = {1, 0, 2, 0, 3, 0, 2, 0}, x2(n) = {1, 2, 3, 2, 1, 2, 3, 2}, Find X1 (K) and X2 (K) by using result of X (K)

− 16 −

[12]

University Question Papers

(b) Using DIF − FFT. Find DFT of the following sequence X(n) = {1, 3, −2, 4, 1, 4, −2, 3} 4.

5.

[8]

(a) The difference equation of the system is given by − 3 1 y(n) + y(n − 1) + y(n − 2) = x(n) + x(n − 1) 4 8 (i) Determine transfer function (ii) Plot poles and zero diagram (iii) Find impulse response of the system (iv) Find step response of the system (v) Show Direct form−I and Direct Form II realization.

[4] [3] [3] [4] [6]

(a) Design a digital Butterworth Filter satisfies the following constraint using bilinear Transformation. Assume T = 1s. π H (e jw ) ≤ 1 0≤w≤ 0.9 ≤ 2 π jw H(e ) ≤ 0.2 3 ≤w≤π 4 (b) A low pass filter has desired response as given below

e − j3w H d (e jw ) = 0

0≤w≤

[10]

[10]

π 2

π ≤w≤π 2

Determine the filter co−efficient h(n) for M = 7 using frequency sampling technique. 6.

(a) A sequence x(n) = {(1 + j), (2 + j2), (3 + j3), (4 + j2)} (i) Find X(K) by DFT Equation (ii) Let P [n] = {1, 2, 3, 4} and q (n) = {1, 2, 3, 2} Find P[K] and Q(K) using X(K) and not otherwise.

[10]

(b) (i) Find the initial and final value of the function 1 + z −1 X(z) = 1 − 0.25z −2 (ii) Find whether the Linear and Time variant y(n) = 2x(n) + 3 7.

[6]

[4]

(a) (i) Write a short note on Chirp Algorithm. (ii) Determine the z− Transform of the x(n) = −n an u(n − 1) (b) Explain in brief Hillbert Transform.

[5] [5] [10]



Digital Signal Processing − November 2006 Time : 3 Hrs.] N.B.:

1.

[Marks : 100

(1) Question No. 1 is compulsory. (2) Attempt any four questions out of remaining six questions.

Figure shows the direct form − II realization of IIR filter. (a) Find the transfer function of the filter. (b) Find the corresponding difference equation. (c) Realize the filter using cascade form using first order modules. (d) Realize the filter using parallel form using first order modules. (e) Find the impulse response function of the filter. (f) Show pole−zero pattern of filter. (g) State whether the filter is stable? Why or why not? (h) Find magnitude squared response at w = 0 and W = Π x (n)

∑

∑ −1

5/8

Z

3/4

Z−1

−1/16

1/8

− 17 −

[3] [2] [3] [4] [2] [2] [2] [2]

y (n)

Vidyalankar : BE − DSP

2.

(a) Find z−transform of an (cosw0n) u(n). (b) The discrete time system is represented by following equation. y (n) = 3/2 [y (n−1)] − 1/2 [y (n−2)] + x (n) with initial conditions y (−1) = 0, y(−2) = −2 and x (n) = (1/4)n u(n) Determine : (i) zero input response (ii) zero state response (iii) total response of the system

[8] [12]

3.

(a) Determine the DFT of a sequence using DIT−FFT Algorithm x(n) = (1, 2, 1, 2, 0, 2, 1, 2) Using the same result, otherwise find the DFT of (0, 2, 1, 2, 1, 2, 1, 2) (b) Find the DFT of x (n) = (1, 2, 3, 4)

[15]

4.

(a) Design a digital Butterworth filter that satisfies the following constraint using bilinear transformation. [15] Assume T = 1s.

( ) H ( e ) <= 0.2

0.9 <= H e jw <= 1 jw

5.

0 <= w <= Π/2 3Π/4 <= w <= Π

(b) Determine the energy of the signal given by x (n) = (1/4) n n >= 0 n<0 = 2n

[5]

(a) The desired response of a low pass filter is

[10]

( )=e

Hd e

jw

−3 jw

−3Π/4 <= w <= 3Π/4 3Π/4 < w < Π

=0

Determine H(ejw) for M = 7 using Hamming Window. The Hamming Window function is 0
7.

[5]

[10]

(a) Prove the following properties of DFT with example. (i) periodicity property (ii) convolution property (b) Check whether the following systems are (i) static or dynamic (ii) linear or nonlinear (iii) shift variant or shift invariant (iv)causal or non causal. (v) stable or unstable (1) y (n) = cos [x (n)] (2) y (n) = x (n) cos w0n

[10]

Write short note on (i) Fetal ECG monitoring (iii) Hillbert transforms.

[20]

[10]

(ii) DSP processor − TMS 320 C5X. 

Digital Signal Processing − May 2007 Time : 3 Hrs.] N.B.:

[Marks : 100

(1) Question No. 1 is compulsory. (2) Attempt any four out of remaining six questions. (3) Assume suitable data wherever necessary, Justify the same. ∞

1.

(a) Show that if x (n) is an odd signal then

∑ x (n) = 0 .

[5]

n =−∞

(b) The z−transform of x (n) is X (z) = 1 + 2z−1. Find z−transform of following and indicate their region of [5] convergence. n

⎛1⎞ (ii) x 2 ( n ) = ⎜ ⎟ × ( n − 2 ) . ⎝ 2⎠ (c) Consider the analog signal x a ( t ) = 3cos 2000πt + 5sin 6000πt + 10 cos1200πt.

(i)

x1 ( n ) = x ( 3 + n )

[10]

(i) What is Nyquist rate for this signal ? (ii) Suppose, the signal is sampled at sampling rate fs = 5000 sample/sec. What is the discrete time signal obtained after sampling ? (iii) What is the analog signal if the reconstruction is done from the above samples using ideal interpolation? 2.

(a) If x ( n ) = δ ( n ) + δ ( n − 1) − δ ( n − 2 ) + δ ( n − 3) + δ ( n − 4 ) + δ ( n − 5 ) h ( n ) = δ ( n ) + 2δ ( n − 2 ) (i) Find y(n) by linear convolution

(ii) Find y(n) by circular convolution. − 18 −

[10]

University Question Papers

(b) A certain discrete time LTI filter has the following data, poles are at 0.2 and 0.6, zeros are at 0.4 and origin [10] Gain of filter is 5. Show (i) Direct Form−I, Direct Form −II and Cascade realization. (ii) If register length is 4 bits including sign bit. Calculate the effect on poles and zeros of above filter due to finite word length of filter. 3.

(a) If DFT of a sequence X (n) is given by X ( k ) = {10, −2 + 2 j, − 2, −2, −2 j} find x ( n )

[4]

(b) If x ( n ) = {5, 6, 7, 8}

[16]

(i) Find X(K) using DIT−FFT ⎪⎧ x ( n / 2 ) for n even (ii) If y ( n ) = ⎨ for n odd ⎪⎩0 Find 8−point Y(K) using X(K) without performing DFT/FFT. ⎪⎧ x ( n ) for 0 ≤ n ≤ 3 (iii)Let Y1 ( n ) = ⎨ for 4 ≤ n ≤ 7 ⎪⎩0 Find Y1(k) without performing DFT/FFT. (iv) Let b(n) = x (n −1) find B(K) using X(K) and not otherwise. 4.

(a) State whether each of the following statement is true or false. Justify your answer in 4 or 5 sentences. [20] (i) Linear phase filter are always IIR (ii) A stable, causal FIR filter has its poles lying anywhere inside the unit circle in z−plane. (iii) A stable filter is always causal (iv) A linear phase FIR filter have antisymmetric co−efficient can not be “High Pass Filter”.

(

)

(v) If H ( z ) = 1 − 0.3z −1 + 0.3z −3 − 1 . The filter can not be a low−pass filter. 5.

(a) Consider the system shown below : x(n) Z−1

y(n)

[12]

1 2 Z−1 (i) Find Difference equation (ii) Find impulse response of the system h(n). (iii) Show that h(n) is the convolution of the following signal : ⎛1⎞ h2 (n ) = ⎜ ⎟ u (n ) h1 ( n ) = δ ( n ) + δ ( n − 1) , ⎝ 2⎠ (iv) Find the step response of the system. (b) Write the relationship of DFT and z − transform. (c) What is discrete Hilbert transform ? Why it is used ? 6.

[4] [4]

(a) The desired response of low pass filter is : [10] − j3w −3π / 4 ≤ w ≤ 3π / 4 ⎪⎧e H d e jw = ⎨ 3π / 4 < w ≤ π. ⎪⎩0 jw Determine H (e ) for M = 7 using Hamming window. (b) Design a digital Butterworth filter satisfies the following constraint using bilinear transformation. Assume [10] T = 15 π 0.9 ≤ H e jw ≤ 1 0≤w≤ 2 3 π H e jw ≤ 0.2 ≤ w ≤ π. 4

( )

( )

( )

7.

Attempt any four of the following :

[20]

1 (i) Find x ( n ) using convolution x ( z ) = ⎛ 1 −1 ⎞ ⎛ 1 −1 ⎞ ⎜1 − z ⎟ ⎜1 + z ⎟ ⎝ 2 ⎠⎝ 4 ⎠

(ii) Find the initial and final value of the function x ( z ) =

1 + z −1

. 1 − 0.25z −2 (iii) Test linearity, time variance and causality of the following system : x n (ii) y n = e ( ) (i) y(n) = 2x n + 3

( )

( )

(iv) State and prove the Parservals Energy relation. (v) Identify the filter type based on passband if H ( z ) =

1 + 5z −1 1 + 0.2z −1

 − 19 −

.

Vidyalankar : BE − DSP

Digital Signal Processing − November 2007 Time : 3 Hrs.] N.B.:

1.

2.

[Marks : 100

(1) Question No. 1 is compulsory. (2) Attempt any four questions out of the remaining six questions. (3) Assume suitable data if necessary.

Answer the following : (a) Derive the relationship between z−transform, DTFT and DFT. z , | z | > 1. (b) Determine the initial and final value of x(n) if X(z) = 2 2z − 3z + 1 (c) Sketch pole−zero plot for the system with transfer function z 6 − 26 H(z) = 5 z (z − 2) is this system stable ? (d) Find the given system is linear phase or not. Prove your answer. h[n] = {1, −2, 0, 2, −1}

[5] [5] [5]

[5]

(a) A causal DT system has transfer function H(z) such that H(z) = H1(z) . H2(z) 1 . 3 1 H2(z) has one pole at z = 0 and one zero at z = − . 2 (i) Find transfer function of system. (ii) Find difference equation of system. (iii) Find response of system to i/p.

H1(z) has one pole at z = 0.5 and one zero at z =

[3] [2] [3]

n

⎛ 1⎞ x(n) = ⎜ − ⎟ u(n). ⎝ 2⎠ (iv) Draw pole−zero plot of the overall system and hence comment on the stability of system.

[2]

(b) (i) Show the condition on impulse response for a system to be BIBO stable. [5] (ii) State the convolution theorem of z−transform. Using this property determine the convolution of [5] following pair of signal = n u (n) h1(n) h2(n) = 2n u(n − 1) 3.

4.

5.

(a) Find the step response of a system having difference equation Y(n) = x(n) − 0.4 y(n − 1) + 0.05 y(n − 2) (b) A causal FIR system has three cascaded block, first two of them have individual impulse responses. = δ(n) + 2δ (n − 1) + 2δ (n − 2) h1(n) h2(n) = u(n) − u(n − 2) Find the impulse response h3(n) of a third block, if an overall impulse response is h(n) = [2, 5, 6, 3, 2, 2] (c) Find the convolution and correlation of two sequences x1(n) and x2(n) = [2, 1, 3] x1(n) = [−2, −1, 1] x2(n) (a) Let x(n) = {1, 2, 3, 4} (i) Find DFT of x(n) using DITFFT (ii) Let h(n) = {1, 2, 3, 2} Find y(n) = x(n) * h(n) Using FFT/IFFT (b) If x(n) = δ(n) + δ (n − 1) + δ(n − 2) (i) Find X(ejw) (ii) Find X(k), 4 pt DFT (iii) Show that DFT is sampled version of | X(ejw) |. (a) Determine impulse response h(n) of a linear phase FIR filter of length 4 for which | H(w) |w = 0 = 1 and | H(w) |w = π/2 = 1/2 (b) Show that the zeros of a linear phase FIR filter occur at reciprocal location. (c) Show that the group delay of a linear phase FIR filter N −1 τq = 2

− 20 −

[6] [8]

[6]

[10]

[3] [3] [4]

[7] [6] [7]

University Question Papers

6.

(a) Design a linear phase FIR filter of seventh order, with cut−off frequency 1 radian/sec. Use following [8] window function. ⎡ 2πn ⎤ w(n) = 0.54 − 0.46 cos ⎢ ⎥, ⎣ (N − 1) ⎦ n = 0, 1, 2,……, N − 1. (b) Determine the order and cut−off frequency of a butterworth low pass digital filter for following specifications : ≤ 1 dB Pass−band attenuation Stop−band attenuation ≥ 20 dB Pass−band frequency = 0.2 π Stop−band frequency = 0.3 π [12] Sampling frequency = 5 kHz Use (i) Impulse invariance technique (ii) Bilinear Transformation technique.

7.

Write short notes on any two : (a) Fast convolution method (b) DSP processor TMS320C40 (c) Goertzel algorithm.

[20]



Digital Signal Processing − May 2008 Time : 3 Hrs.] N.B.:

[Marks : 100

(1) Question No. 1 is compulsory. (2) Attempt any four questions from remaining six questions. (3) Assume suitable data if necessary.

1.

State whether each of the following statement is true or false. Justify your answer in 4 or 5 sentences. (a) Linear phase filters are always IIR. (b) For a causal system | h(n) | tends to zero, as n tends to infinity, the system is stable. (c) A stable filter is always causal. (d) A stable, causal FIR filter has its poles lying any where inside the unit circle in the z−plane. (e) IIR filter has always recursive realization.

2.

(a) (i) A certain DT system is stable and has transfer function as given below, find the impulse response of [5] system if − 3 z H(z) = (z − 0.2) (z − 0.5) (z − 2) (ii) If H(z) =

1 − 0.2 z −1

−

1

1 − 0.707 z 1 + 2 z −1 If this filter is known to be unstable, find all possible impulse response of filter. (b) Using partial fraction Expansion method, find inverse z−transform of the following expression − z(z 2 − 4z + 5) x(z) = (z − 1) (z − 2) (z − 3) for the following Roc. (i) | z | > 3 (ii) 2 < | z | < 3 (iii) | z | < 2

3.

x(n)

−1

[20]

∑

∑

[5] [10]

y(n)

Z−1

−1/2 Z−1

−3/16 Figure shows the direct Form−II realization of IIR filter. (a) Find the transfer function of the filter. (b) Find the corresponding difference equation. (c) Realise the filter using cascade form. (d) Realise the filter using parallel form. (e) Find the impulse response function of filter. (f) Show pole zero pattern of filter. (g) State whether the filter is stable or not. Justify the same. − 21 −

[3] [2] [3] [3] [3] [3] [3]

Vidyalankar : BE − DSP

4.

(a) An ideal low pass filter has the response − H (e

jw

)=

− jwα

[10]

| w | < wc wc < | w | < π

2e = 0

Find h(n), for a transition width <

π , calculate the window length and the values of α for (i) rectangular 32

window (ii) Hamming window. (b) Design a digital Butterworth filter satisfies the following constraint using bilinear transformation. Assume [10] T = 1s 0 ≤ w ≤ π/2 0.9 ≤ | H(ejw) | ≤ 1 | H(ejw) | ≤ 0.2 3π/4 < w ≤ π 5.

(a) Using DIF−FFT find DFT of the following sequence − x(n) = {1, 3, −2, 4, 1, 4, −2, 3} (b) Find the initial and final values of x(n) for the following causal system − 2z 2 + 1 (i) X(z) = z 2 − 0.5 z − 0.5 (ii) X(z)

=

z

[5]

(a) A four point sequence x(n) has DFT X(K) = {6, 2, 1, 2}. Sequence p(n) is related to x(n) has DFT [5] P(K) = {6, −2, 1, −2}. What is the relation between x[n] and p[n] ? (b) Find circular convolution of the following sequences using DFT/IDFT any − [5] = {1, 2, 3, 1} x1(n) = {4, 3, 2, 2} x2(n) [5] (c) DFT of a sequence x(n) is given by x(k) = {6, 0, −2, 2} (i) Determine x(n) −

iπk

(ii) Plot x1(n) if x1(k) = x(k) e 2 . (d) Write relationship of DFT to Fourier transform and z−transform. 7.

[5]

2

z + z −1 (c) Determine the convolution of following pair of signal by z−transform − n ⎡ ⎛ 1 ⎞n ⎤ ⎛1⎞ = ⎜ ⎟ u(n − 1), x 2 (n) = ⎢1 + ⎜ ⎟ ⎥ u(n) x1(n) ⎝ 4⎠ ⎣ ⎝ 2⎠ ⎦

6.

[10]

Write short notes on : (a) Application of DSP in Telecommunication (b) DSP processor (c) Effect of finite word length (d) Chirp z−algorithm

[5] [20]



− 22 −