DSP

Digital Signal Processing − November 2004Time : 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. [20] 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 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. 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 (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 (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) | [20] 4. 5. [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] − 15 − 0. (a) Determine discrete time Fourier Series representation for the signal – ⎡ nπ ⎤ x(n) = cos ⎢ ⎥ ⎣3⎦ plot spectrum of x(n). 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. 1 } Plot magnitude and phase response. 0. (b) Using partial fraction expansion method. 1.B..E.. (iii) What is the analog signal if reconstruction is done from the above samples using ideal interpolation.. find inverse Z– transform of the following expression. [10] ⎛1⎞ n ⎜ ⎟ u(n) ⎝4⎠ n (ii) x(n) = n u(– n– 1) [10] 4. 5. 6. − DSP Digital Signal Processing − May 2005 Time : 3 Hrs.Vidyalankar : B. [10] (a) Sketch 8 – point signal flow graph for radix 2 DIT FFT. 3.54 – 0. 0. (ii) If this signal is sampled at sampling frequency Fs = 5000 samples / sec. 2. y(n) = 0.. (a) The impulse response of LTI system is h(n) = {1. 3. (b) Using this diagram or otherwise determine DFT of the sequence – h(n) = { 1.. . 2. 0. 0. 1 is compulsory. 3. 2. 2.. derive the expression for the discrete time signal after sampling.. 1. 3. [20] − 16 − .] N.46 cos ⎜ ⎟ 0≤n≤M– 1 ⎝ M ⎠ where M is filter length. (b) Plot magnitude and phase response of the above filter.: (1) Question Nos.} Also obtain total power. 1.7 y(n– 1) – 0. (i) Find equivalent operation in time domain if differentiation is done in z– domain. [Marks : 100 1. 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. 3} [10] ↑ Determine the output response of the system to input x(n) = {1. (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. 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) (a) State and prove the following properties of z– transform. 0. (ii) What is the operation in z domain if two time signals are convalved in time domain. (2) Attempt any four from rest. – 1. (i) x(n) = [10] 2. (a) Compute the response y(n) of the system if the input x(n) = n u(n). (3) Assume any suitable data if necessary with justification. 1. (b) Find the z– transform of the following sequences and specify ROC. [10] [10] 7. 1. [10] (b) Obtain the power density spectrum of a periodic signal given by – x(n) = { .12 y(n– 2) + x(n– 1) + x(n– 2) Comment on the stability of the system. [5] (b) Determine the convolution of signals using z transforms.2 and 0. Attempt any four questions out of remaining six questions. Figures to the right indicate full marks. The convolution of these sequences is to be find out using FFT technique.B. then z1 = (1/r)ejwo is also zero.n) [4] [4] [4] 2. H1(z) H2(z) −1/3 −1/2 −1/2 (i) Find the transfer function of total system. n (i) x1(n) = x(3 + .4 and origin Gain of filter is 5. What sequence length is chosen for FFT? What are the steps taken ? [4] 4.8e± j π 2 and two zeros at origin.5. ⎝2⎠ (a) An FIR LTI system has an impulse response h(n) which is real valued. [4] ⎛1⎞ (ii) x2(n) = ⎜ ⎟ x ( n − 2 ) ⎝2⎠ (c) Determine convolution of following pair of single using z transform. 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. 3. [10] ⎛1⎞ x1(n) = ⎜ ⎟ u ( n ) . x1(n) = nu(n). 2 = 0. even and has finite duration of [8] (2N + 1). which is described by a convolution simulation is LTI and relaxed. 1. x2(n) = cos πn u(n).t. frequency and identify the filter type based on passband. linear phase filter has all real zeros. (a) Find the squared magnitude response of an filter. 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. 1 is compulsory. Show (i) Direct from II and cascade realization (ii) If register length is 4 bits including sign bit. [5] x(n) δ(n −no) = x(no). (ii) Show that a discrete time system.] N. 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. (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 n 5.University Question Papers Digital Signal Processing − December 2005 Time : 3 Hrs. Assume suitable data if necessary and state them clearly.: [Marks : 100 (1) (2) (3) (4) Question No. which has two poles P1. [4] (b) The z transform of x(n) is X(z)=1 + 2z−1. [3] [2] − 17 − . (a) If DFT of a sequence x(n) is given by x(k) = {1 2 3 2} What may be following signals ? (i) x1(n) = x(n + 2) (ii) x2(n) = x ( n ) e jn π 2 . δ(n−no) and x(n) * δ(n−no) = x(n−no).r. one of them is located at 0. [7] (c) A certain discrete time LTI filter has the following data− Poles are at 0. (a) Prove that the auto correlation sequence at zero lag has highest magnitude with respect to magnitude at any other lag. Show that if z1 = yejwo is a zero of the system. = 0 for n odd. Plot it w. [8] 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. Find z transform of following and indicated their region of convergence.6 Zeros are at 0. (ii) Find difference equation of system. [7] (b) A fourth order antisymmetric. Calculate the effect on poles and zeros of above filter due to finite word length of filter. (a) (i) Prove and explain graphically the difference between the relation. if input data sequence is long and impulse response has a few [4] number of samples. 0.1z −1 [6] [6] [4] 7.4 y(n − 1) − 0. − DSP ⎛ −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.: [Marks : 100 (1) Question Nos. (a) Determine the frequency response. 2.E. (b) Show pole−zero locations of the normalized Fourth order Butterworth IIR filter. 3.B. (a) A digital LPF is required to meet the following specifications Pass band ripple ≤ 1db Pass band edge = 4KHz Stop band attenuation ≥ 40db Stop band edge = 6KHz Sampling frequency = 24KHz.Vidyalankar : B. [4] (d) Prove the BIBO stability condition in time domain. (i) Determine the order of Butterworth filter which meets above specifications. 3.25 y(n − 2) what is ROC of this system ? (c) Explain overlap and save method of filtering.01 πn) (c) Find x(n) using convolution for X(z) = [5] 2. x(n) = {1. 0}. Find X1 (K) and X2 (K) by using result of X (K) [12] − 18 − . (4) Assume suitable data if necessary . 0. x2(n) = {1. (b) Find the Energy of the signal x(n) = (1/4)n = 2 n [5] [5] n≥0 n<0 [5] 1 ⎛ 1 −1 ⎞ ⎛ 1 −1 ⎞ ⎜1 − z ⎟ ⎜1 + z ⎟ ⎝ 2 ⎠⎝ 4 ⎠ (d) Determine whether following signals are periodic (i) cos (0. n [3] [4] [8] 6. 1.] N. 1} (ii) Determine the causal signal x(n) having the z − transform 1 X(z) = −1 (1 + z )(1 − z −1 ) 2 [5] [5] 3. 2. 1}. (ii) Determine cut off frequency of filter. 2. 3. magnitude response. 0. 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.3 π n + π/6) (ii) x(n) = sin (0. 1. 0. h(n) = {4. 2. magnitude response. 2. (ii) If x1(n) = {1. if x ( z ) = 1 1 − 0.2z −2 (b) A causal DT system has a difference equation [4] y(n) = x(n) − 0. − 0. (c) Let x(n) = {1 2 3 2 4 3 2 1} Find DFT of x(n) using DIT − FFT algorithm only. 2} (i) Find X[K] using DIT − FFT. 2. 3. 2. 1 is compulsory. (a) If x(n) = {1. ⎛1⎞ x ( n ) = ⎜ ⎟ u ( n ) + 3n u ( − n − 1) ⎝ 2⎠ n Digital Signal Processing − May 2006 Time : 3 Hrs. 2}. 3. (2) Attempt any four questions from remaining six questions (3) Figures to right indicate full marks. [4] (e) Find the energy of the signal. (a) Show that n = −∞ ∑ ∞ x 2 (n) = n = −∞ ∑ ∞ 2 x e (n) + n =−∞ ∑ ∞ 2 x 0 (n) where xe(n) and x0(n) are even and odd parts of x(n). 0. (a) Find the values x(0) and x(∞) for a function x(n). (a) (i) Write a short note on Chirp Algorithm. 4. (a) Design a digital Butterworth Filter satisfies the following constraint using bilinear Transformation. 1. (ii) Determine the z− Transform of the x(n) = −n an u(n − 1) (b) Explain in brief Hillbert Transform. (c) Realize the filter using cascade form using first order modules. (4 + j2)} (i) Find X(K) by DFT Equation (ii) Let P [n] = {1. 1 is compulsory. (2 + j2). (g) State whether the filter is stable? Why or why not? (h) Find magnitude squared response at w = 0 and W = Π x (n) [3] [2] [3] [4] [2] [2] [2] [2] ∑ 5/8 Z −1 ∑ 3/4 y (n) Z−1 −1/16 1/8 − 19 − . (3 + j3).9 ≤ 2 π jw H(e ) ≤ 0. 3.25z −2 (ii) Find whether the Linear and Time variant y(n) = 2x(n) + 3 [6] [4] [5] [5] [10] 7.University Question Papers (b) Using DIF − FFT. 3} 4. (a) Find the transfer function of the filter. 2} Find P[K] and Q(K) using X(K) and not otherwise. (b) (i) Find the initial and final value of the function 1 + z −1 X(z) = 1 − 0. [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. −2. π H (e jw ) ≤ 1 0≤w≤ 0.] N. 4. (2) Attempt any four questions out of remaining six questions. 2.2 3 ≤w≤π 4 (b) A low pass filter has desired response as given below [4] [3] [3] [4] [6] 5. Assume T = 1s. (e) Find the impulse response function of the filter.B. 3. (a) A sequence x(n) = {(1 + j). (f) Show pole−zero pattern of filter. (b) Find the corresponding difference equation.: [Marks : 100 (1) Question No. Figure shows the direct form − II realization of IIR filter. −2. Find DFT of the following sequence X(n) = {1. (d) Realize the filter using parallel form using first order modules. 1. 3. 2. 4} and q (n) = {1. [10] [10] e − j3w H d (e jw ) = 0 0≤w≤ π 2 π ≤w≤π 2 [10] Determine the filter co−efficient h(n) for M = 7 using frequency sampling technique. Digital Signal Processing − November 2006 Time : 3 Hrs. 6. − DSP 2. 1/2}. 4) [8] [12] 3. (i) x1 ( n ) = x ( 3 + n ) [10] (i) What is Nyquist rate for this signal ? (ii) Suppose. 1.2 0 <= w <= Π/2 3Π/4 <= w <= Π [5] (b) Determine the energy of the signal given by x (n) = (1/4) n n >= 0 n<0 = 2n 5. 6.9 <= H e jw <= 1 jw ( ) H ( e ) <= 0. ⎛1⎞ (ii) x 2 ( n ) = ⎜ ⎟ × ( n − 2 ) .54−0. 2. otherwise find the DFT of (0. 2) (b) Find the DFT of x (n) = (1. 1. (a) Find z−transform of an (cosw0n) u(n). 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 (a) Determine the DFT of a sequence using DIT−FFT Algorithm x(n) = (1.46Cos 2Πn / (M−1) 0<n<(M−1) =0 otherwise (b) The impulse response of a LTI system h (n) = {1. Justify the same. the signal is sampled at sampling rate fs = 5000 sample/sec. 1. 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. (3) Assume suitable data wherever necessary. (ii) Tabulation technique. 2. Find z−transform of following and indicate their region of [5] convergence. shift multiply and sum concept.E. ⎝ 2⎠ (c) Consider the analog signal x a ( t ) = 3cos 2000πt + 5sin 6000πt + 10 cos1200πt. The Hamming Window function is W (n) = 0. (a) Design a digital Butterworth filter that satisfies the following constraint using bilinear transformation. 1. 0. 1. (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. − 20 − [10] . 1 is compulsory.B. 2. [20] Digital Signal Processing − May 2007 Time : 3 Hrs. 3. (v) stable or unstable (1) y (n) = cos [x (n)] (2) y (n) = x (n) cos w0n Write short note on (i) Fetal ECG monitoring (iii) Hillbert transforms. [15] 0.: [Marks : 100 (1) Question No. (a) Show that if x (n) is an odd signal then n =−∞ ∑ x (n) = 0 . 2) Using the same result. 2. 2. (2) Attempt any four out of remaining six questions.Vidyalankar : B. (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. Find the response of the system when input x(n) = {1. (a) The desired response of a low pass filter is Hd e ( )=e jw [10] −3 jw −3Π/4 <= w <= 3Π/4 3Π/4 < w < Π =0 Determine H(ejw) for M = 7 using Hamming Window. [10] [10] 7. 2. [15] [5] 4. [10] (a) Prove the following properties of DFT with example. 2. (b) The discrete time system is represented by following equation. 2. 3} By (i) fold. n ∞ [5] (b) The z−transform of x (n) is X (z) = 1 + 2z−1. (ii) DSP processor − TMS 320 C5X. Assume T = 1s.] N. 1. y (n) = 3/2 [y (n−1)] − 1/2 [y (n−2)] + x (n) with initial conditions y (−1) = 0. 3. −2 j} find x ( n ) (b) If x ( n ) = {5. ⎧ ⎪ x ( n ) for 0 ≤ n ≤ 3 (iii)Let Y1 ( n ) = ⎨ for 4 ≤ n ≤ 7 ⎪0 ⎩ Find Y1(k) without performing DFT/FFT. −2 + 2 j. − 21 − . poles are at 0. 1 − 0. Show (i) Direct Form−I. (a) If DFT of a sequence X (n) is given by X ( k ) = {10. (iii) A stable filter is always causal (iv) A linear phase FIR filter have antisymmetric co−efficient can not be “High Pass Filter”. (b) Write the relationship of DFT and z − transform. ( ) 5. (b) Design a digital Butterworth filter satisfies the following constraint using bilinear transformation.3z −1 + 0. − 2. causal FIR filter has its poles lying anywhere inside the unit circle in z−plane.4 and origin [10] Gain of filter is 5. (v) If H ( z ) = 1 − 0. (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). [4] [16] 4. 6.25z −2 (iii) Test linearity.3z −3 − 1 . [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. (iv) Let b(n) = x (n −1) find B(K) using X(K) and not otherwise. (ii) If register length is 4 bits including sign bit.2 and 0. time variance and causality of the following system : x n (ii) y n = e ( ) (i) y(n) = 2x n + 3 1 + z −1 ( ) ( ) (iv) State and prove the Parservals Energy relation.6. ⎝ 2⎠ (iv) Find the step response of the system. Justify your answer in 4 or 5 sentences. [20] (i) Linear phase filter are always IIR (ii) A stable.9 ≤ H e jw ≤ 1 0≤w≤ 2 3π H e jw ≤ 0. 8} (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. 4 ( ) ( ) ( ) 7. (a) State whether each of the following statement is true or false. (v) Identify the filter type based on passband if H ( z ) = 1 + 5z −1 1 + 0. (iii) Show that h(n) is the convolution of the following signal : ⎛1⎞ h2 (n ) = ⎜ ⎟ u (n ) h1 ( n ) = δ ( n ) + δ ( n − 1) . Assume [10] T = 15 π 0. 7. Direct Form −II and Cascade realization. The filter can not be a low−pass filter. ⎪0 ⎩ jw Determine H (e ) for M = 7 using Hamming window. Attempt any four of the following : 1 (i) Find x ( n ) using convolution x ( z ) = ⎛ 1 −1 ⎞ ⎛ 1 −1 ⎞ ⎜1 − z ⎟ ⎜1 + z ⎟ ⎝ 2 ⎠⎝ 4 ⎠ [20] (ii) Find the initial and final value of the function x ( z ) = .2z −1 . zeros are at 0. −2.University Question Papers (b) A certain discrete time LTI filter has the following data.2 ≤ w ≤ π. (c) What is discrete Hilbert transform ? Why it is used ? 6. Calculate the effect on poles and zeros of above filter due to finite word length of filter. n [2] (b) (i) Show the condition on impulse response for a system to be BIBO stable. 3. [10] [3] [3] [4] 5.5 and one zero at z = [3] [2] [3] ⎛ 1⎞ x(n) = ⎜ − ⎟ u(n). (2) Attempt any four questions out of the remaining six questions. 0.] N. (3) Assume suitable data if necessary. DTFT and DFT. [7] [6] [7] − 22 − . − DSP Digital Signal Processing − November 2007 Time : 3 Hrs.05 y(n − 2) (b) A causal FIR system has three cascaded block. H2(z) 1 . −2. 1 is compulsory. (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. h[n] = {1. (a) Find the step response of a system having difference equation Y(n) = x(n) − 0. 2. first two of them have individual impulse responses. h1(n) = δ(n) + 2δ (n − 1) + 2δ (n − 2) h2(n) = u(n) − u(n − 2) Find the impulse response h3(n) of a third block. Answer the following : (a) Derive the relationship between z−transform. 2 (i) Find transfer function of system. 2. −1} (a) A causal DT system has transfer function H(z) such that H(z) = H1(z) . 3 1 H2(z) has one pole at z = 0 and one zero at z = − . (iii) Find response of system to i/p.: [Marks : 100 (1) Question No. [5] (ii) State the convolution theorem of z−transform. (c) Show that the group delay of a linear phase FIR filter N −1 τq = 2 [6] [8] [6] 4.4 y(n − 1) + 0. 1] (a) Let x(n) = {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. 3] x2(n) = [−2. [5] [5] [5] [5] 2. (ii) Find difference equation of system. 2] (c) Find the convolution and correlation of two sequences x1(n) and x2(n) x1(n) = [2. Using this property determine the convolution of [5] following pair of signal = n u (n) h1(n) h2(n) = 2n u(n − 1) 3. 3. 1. −1. 3. 4} (i) Find DFT of x(n) using DITFFT (ii) Let h(n) = {1. 2. 2.Vidyalankar : B. ⎝ 2⎠ (iv) Draw pole−zero plot of the overall system and hence comment on the stability of system.E. z . | z | > 1. 1. if an overall impulse response is h(n) = [2. 5. H1(z) has one pole at z = 0. 6.B. 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). Prove your answer. 4 pt DFT (iii) Show that DFT is sampled version of | X(ejw) |. (f) Show pole zero pattern of filter. Write short notes on any two : (a) Fast convolution method (b) DSP processor TMS320C40 (c) Goertzel algorithm. (a) (i) A certain DT system is stable and has transfer function as given below.3 π Sampling frequency = 5 kHz [12] Use (i) Impulse invariance technique (ii) Bilinear Transformation technique. (a) Design a linear phase FIR filter of seventh order.: [Marks : 100 (1) Question No. 1. (g) State whether the filter is stable or not. (b) Determine the order and cut−off frequency of a butterworth low pass digital filter for following specifications : Pass−band attenuation ≤ 1 dB Stop−band attenuation ≥ 20 dB Pass−band frequency = 0.……. causal FIR filter has its poles lying any where inside the unit circle in the z−plane. Justify your answer in 4 or 5 sentences. 2. find the impulse response of system if − [5] 3 z H(z) = (z − 0. (c) Realise the filter using cascade form. (b) Using partial fraction Expansion method. Digital Signal Processing − May 2008 Time : 3 Hrs. − 23 − [3] [2] [3] [3] [3] [3] [3] . (a) Linear phase filters are always IIR. (e) Find the impulse response function of filter.2 π Stop−band frequency = 0. [20] 7.46 cos ⎢ ⎥. (2) Attempt any four questions from remaining six questions. [20] 1. (a) Find the transfer function of the filter. Use following [8] window function. 1 is compulsory. N − 1. 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. (d) Realise the filter using parallel form. (3) Assume suitable data if necessary. (d) A stable. (b) Find the corresponding difference equation. (b) For a causal system | h(n) | tends to zero. x(n) ∑ Z−1 ∑ y(n) −1/2 Z−1 −3/16 Figure shows the direct Form−II realization of IIR filter. the system is stable.5) (z − 2) 1 − 0.] N. 2. ⎡ 2πn ⎤ w(n) = 0. as n tends to infinity.707 z 1 + 2 z −1 If this filter is known to be unstable.54 − 0. State whether each of the following statement is true or false.2 z −1 −1 − 1 [5] [10] 3.2) (z − 0. Justify the same.B. ⎣ (N − 1) ⎦ n = 0. (e) IIR filter has always recursive realization. with cut−off frequency 1 radian/sec. (i) | z | > 3 (ii) 2 < | z | < 3 (iii) | z | < 2 (ii) If H(z) = 1 − 0.University Question Papers 6. (c) A stable filter is always causal. find all possible impulse response of filter. 1.E. −2. 3. [5] = (a) Using DIF−FFT find DFT of the following sequence x(n) = {1. : 1. Assume [10] T = 1s 0. x 2 (n) = ⎢1 + ⎜ ⎟ ⎥ u(n) x1(n) ⎝ 4⎠ ⎣ ⎝ 2⎠ ⎦ [10] [5] (ii) X(z) = z 2 [5] 6. 1} x1(n) = {4. [Marks : 100 (1) Question No. 4. for a transition width < π . (2) Attempt any four questions from remaining five questions. 3. −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. 2. 3} (b) Find the z transform of the following sequence and specify the ROC : (i) x(n) = ⎛1⎞ n ⎜ ⎟ u(n) ⎝ 4⎠ n [10] [10] (ii) x(n) = n u (−n − 1) [5] 3. −2.Vidyalankar : B. 4. DTFT and DFT.B. (a) An ideal low pass filter has the response − H (e jw )= [10] 2e = 0 − jwα Find h(n). 0.5 z − 0. Sequence p(n) is related to x(n) has DFT [5] P(K) = {6. 1.01 πn) (d) Determine whether following signals are periodic (i) cos (0. −2. 3. −2. (a) A four point sequence x(n) has DFT X(K) = {6.9 ≤ | H(ejw) | ≤ 1 0 ≤ w ≤ π/2 | H(ejw) | ≤ 0. 3. 2} (i) Determine x(n) (ii) Plot x1(n) if x1(k) = x(k) e 2 . 1 is compulsory. (b) Design a digital Butterworth filter satisfies the following constraint using bilinear transformation.4 y (n − 1) + 0.5 z + z −1 (c) Determine the convolution of following pair of signal by z−transform − n ⎡ ⎛ 1 ⎞n ⎤ ⎛1⎞ = ⎜ ⎟ u(n − 1). − iπk [5] [20] 7. 2}. 4. 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. −2. 1.5 z + z −1 − 24 − . 1. −2}. calculate the window length and the values of α for (i) rectangular 32 | w | < wc wc < | w | < π window (ii) Hamming window. 2} x2(n) [5] (c) DFT of a sequence x(n) is given by x(k) = {6.2 3π/4 < w ≤ π 5. − DSP 4. (b) Find the step response of a system having difference equation y(n) = x(n) − 0.] N. Write short notes on : (a) Application of DSP in Telecommunication (b) DSP processor (c) Effect of finite word length (d) Chirp z−algorithm Digital Signal Processing − November 2008 Time : 3 Hrs. [5] [5] [5] (a) Derive the relationship between Z transform.5z − 0. 4. (d) Write relationship of DFT to Fourier transform and z−transform. (a) Find initial and final value of x(n) for following causal system : 2z 2 + 1 z (i) x(z) = (ii) x(z) = 2 2 z − 0.05 y (n − 2) (c) Show that if x(n) is odd signal then n = −∞ ∑ ∞ x(n) = 0 sin (0. (a) Using DIF−FFT find DFT of the following sequence − x(n) = {1. 2.3 πn + π/6) (ii) x (n) 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) ∈r1 Find y(n) by linear convolution. [5] [10] [10] (a) Design a digital Butterworth filter satisfying the following constraints using bilinear transformation. 3. (a) Derive the relationship between Z-Transform. 4. (b) Impulse response of LT1 system is h(n) = {1 2 −1 3} ↑ Determine the output response of the system to input x(n) = {1 2. 3. magnitude response. (a) Design a digital butterworth filter satisfying.9 ≤ | H(e jw ) | ≤ 1 0 ≤ w≤ 2 3π jw | H(e ) | ≤ 0. (3) Make the necessary assumption wherever required.2 ≤ w≤π 4 (b) What is discrete hilbert transform ? Why it is used. −2. 3} (b) Determine the frequency response. [10] [10] 4. −2.B. 3 1} ↑ Write short notes on : (a) Application of DSP in Telecommunication (b) DSP processors (c) Fetal ECG monitoring (d) Chirp 2 Algorithm [5] [5] [10] 5. (b) Prove the following properties of DFT with example : (i) Periodically property (ii) Convolution property 2. 1 is compulsory. (a) A low pass filter has desired response as given below π e − j3w 0≤w≤ 2 H d (e jw ) = π 0 ≤w≤π 2 Determine the filter coefficient h(n) for M = 7 using the sampling technique. 0. (b) What are the conditions to be satisfied by LTZ system? [10] 5. (c) What are the conditions to be satisfied by LT1 system. phase response of the system given by y(n) − ½ y(n −1) = x(n) − x(n − 1) (a) A low pass filter has desired response is given below: 0≤ w <π / 2 e − j3w Hd(e w)= π≤ W ≤ π 0 Determine the filter coefficient h(n) for M = T using frequency sampling technique. Assume T = 1 sec.4 y(n − 1) + 0. The following constraints using billinear transformation. 2 −1} (a) Using DIF-FFT find DFT of the following sequence : x(n) = {1.University Question Papers (b) Sketch the pole−zero plot for the system with transfer function : z 6 − 26 H(z) = is the system stable. DTFT and DFT. [20] Digital Signal Processing − May 2009 Time : 3 Hrs. 3] x2(n) = [−2. 1.] N. [5] [5] [5] [5] [10] [10] [10] 1. [10] 6.3 πn + π/6) (ii) x(n) = sin (0. 1] 4. (2) Attempt any four questions from remaining five. −2. : [Marks : 100 (1) Question No. 4. 1. [10] 0. −1. x1(n) = [2.2 3π/4 ≤ w ≤ π − 25 − . z5 (z − 2) (c) Find the convolution and correlation of two sequences x1(n) and x2(n).05 y(n − 2) (d) Find the given system is linear phase or not prove your answer : h(n) = {1. (b) Determine whether following signals are periodic (i) cos (0.9 < | H (jw) | ≤ 1 0 ≤ w ≤ π/2 | H (jw) | ≤ 0. Assume T = 1S π 0.01 πn) (c) Find the step response of a system having difference equation y(n) = x(n) − 0. Find y(n) by circular convolution. Vidyalankar : B. [20] − 26 − .E. (d) Pole zero pattern of filter (e) State whether filter is stable or not. − DSP (b) x(n) Σ Σ y(n) [10] Z−1 Z−1 −3/16 −1/2 For the direct form−II realization of IIR filter find : (a) Transfer function of filter (b) Corresponding difference equation (c) Impulse response fraction of filter 6. Write short notes on any two: (a) Fetal ECG monitoring (b) DSP processors (c) Chirp−2 Algorithm. 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