GATE EE Topic wise Questions SIGNALS & SYSTEMS

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1 GATE EE Topic wise Questions YEAR 010 ONE MARK Question. 1 For the system /( s + 1), the approximate time taken for a step response to reach 98% of the final value is (A) 1 s (B) s (C) 4 s (D) 8 s Question. The period of the signal xt ( ) = 8 sin 0.8t ` + 4 j is (A) 0.4 s (B) 0.8 s (C) 1.5 s (D).5 s Question. 3 The system represented by the input-output relationship 5t # yt () = x( τ) dτ, t > 0 3 (A) Linear and causal (B) Linear but not causal (C) Causal but not linear (D) Neither liner nor causal

2 EE Topic wise Question. 4 The second harmonic component of the periodic waveform given in the figure has an amplitude of (A) 0 (B) 1 (C) / (D) 5 YEAR 010 TWO MARKS Question. 5 xt () is a positive rectangular pulse from t = 1to t =+ 1 with unit height as shown in the figure. The value of X( ω) dω " where X( ω) -3 is the Fourier transform of xt ()} is. # 3 (A) (C) 4 (B) (D) 4 Question. 6 Given the finite length input xn [ ] and the corresponding finite length output yn [ ] of an LTI system as shown below, the impulse response hn [ ] of the system is Page

3 EE Topic wise (A) (C) hn [ ] = {1, 0, 0, 1} (B) - hn [ ] = {1, 1, 1, 1} (D) - Common Data Questions Q. 7 & 8 Given ft () and gt () as show below hn [ ] = {1, 0, 1} - hn [ ] = {1, 1, 1} - Question. 7 gt () can be expressed as (A) gt () = f( t 3) (B) gt () = f ` t 3j (C) gt () = ft 3 ` j (D) gt () = ft 3 ` j Question. 8 The Laplace transform of gt () is 1 3s 5s 1-5 s -3 s (A) ( ) s e e (B) ( ) s e e -3s (C) e -s (1 e ) s 1 5 s 3 s (D) ( ) s e e YEAR 009 ONE MARK Question. 9 A Linear Time Invariant system with an impulse response ht () produces output yt () when input xt () is applied. When the input xt ( τ) is applied to a system with impulse response ht ( τ), the output will be (A) y() τ (B) y( ( t τ)) (C) yt ( τ) (D) yt ( τ) Page 3

4 EE Topic wise YEAR 009 TWO MARKS Question. 10 A cascade of three Linear Time Invariant systems is causal and unstable. From this, we conclude that (A) each system in the cascade is individually causal and unstable (B) at least on system is unstable and at least one system is causal (C) at least one system is causal and all systems are unstable (D) the majority are unstable and the majority are causal Question. 11 The Fourier Series coefficients of a periodic signal xt () expressed 3 jkt/ T as xt () = / ae k are given by a- = j1, a 1 = j0., k =-3 a0 = j, a 1 = 05. j0., a = + j1 and ak = 0 for k > Which of the following is true? (A) xt () has finite energy because only finitely many coefficients are non-zero (B) xt () has zero average value because it is periodic (C) The imaginary part of xt () is constant (D) The real part of xt () is even Question The z-transform of a signal xn [ ] is given by 4z + 3z + 6z + z It is applied to a system, with a transfer function Hz () Let the output be yn. [ ] Which of the following is true? (A) yn [ ] is non causal with finite support (B) yn [ ] is causal with infinite support (C) yn [ ] = 0; n > 3 (D) Re[ Yz ( )] ji z e = Re[ Yz ( )] -ji = z= e Im[ Yz ( )] ji = Im[ Yz ( )] -j i; # θ < z= e z= e - 1 = 3z YEAR 008 ONE MARK Page 4 Question. 13 The impulse response of a causal linear time-invariant system is given

5 EE Topic wise as ht. () Now consider the following two statements : Statement (I): Principle of superposition holds Statement (II): ht () = 0 for t < 0 Which one of the following statements is correct? (A) Statements (I) is correct and statement (II) is wrong (B) Statements (II) is correct and statement (I) is wrong (C) Both Statement (I) and Statement (II) are wrong (D) Both Statement (I) and Statement (II) are correct Question. 14 -αt A signal e sin( ωt) is the input to a real Linear Time Invariant system. Given K and φ are constants, the output of the system will -βt be of the form Ke sin( vt + φ) where (A) β need not be equal to α but v equal to ω (B) v need not be equal to ω but β equal to α (C) β equal to α and v equal to ω (D) β need not be equal to α and v need not be equal to ω YEAR 008 TWO MARKS Question. 15 A system with xt () and output yt () is defined by the input-output relation : # -t yt () = xtd () τ -3 The system will be (A) Casual, time-invariant and unstable (B) Casual, time-invariant and stable (C) non-casual, time-invariant and unstable (D) non-casual, time-variant and unstable Question. 16 sin( x) x A signal xt () = sinc( αt) where α is a real constant ^sinc() x h is the input to a Linear Time Invariant system whose impulse response ht () = sinc( βt), where β is a real constant. If min ( αβ, ) denotes the = Page 5

6 EE Topic wise minimum of α and β and similarly, max ( αβ, ) denotes the maximum of α and β, and K is a constant, which one of the following statements is true about the output of the system? (A) It will be of the form Ksinc( γ t) where γ = min( α, β) (B) It will be of the form Ksinc( γ t) where γ = max( α, β) (C) It will be of the form Ksinc( αt) (D) It can not be a sinc type of signal Question. 17 Let xt () be a periodic signal with time period T, Let yt ( ) = xt ( t0) + xt ( + t0) for some t 0. The Fourier Series coefficients of yt () are denoted by b k. If bk = 0 for all odd k, then t 0 can be equal to (A) T/ 8 (B) T/ 4 (C) T/ (D) T Question. 18 Hz () is a transfer function of a real system. When a signal xn [ ] = (1 + ) is the input to such a system, the output is zero. Further, the Region of convergence (ROC) of ^1 1 z -1 h H(z) is the entire Z-plane (except z = 0). It can then be inferred that Hz () can have a minimum of (A) one pole and one zero (B) one pole and two zeros (C) two poles and one zero D) two poles and two zeros j n Question. 19 Given Xz () = z with z > a, the residue of Xzz () n - 1 at z = a ( z a) for n $ 0 will be (A) a n - 1 (B) a n (C) na n (D) na n - 1 Page 6 Question. 0 Let xt () = rect^t h (where rect() x = 1 for # x # and sin( x) zero otherwise. If sinc() x = x, then the Fourier Transform of

7 EE Topic wise xt () + x( t) will be given by (A) sinc ` ω j (B) sinc ω ` j (C) sinc ω cos ω ` j ` j (D) sinc ω sin ω ` j ` j Question. 1 Given a sequence xn, [ ] to generate the sequence yn [ ] = x[ 3 4n], which one of the following procedures would be correct? (A) First delay xn ( ) by 3 samples to generate z1 [ n], then pick every 4 th sample of z1 [ n] to generate z [ n], and than finally time reverse z [ n] to obtain yn. [ ] (B) First advance xn [ ] by 3 samples to generate z1 [ n], then pick every 4 th sample of z1 [ n] to generate z [ n], and then finally time reverse z [ n] to obtain yn [ ] (C) First pick every fourth sample of xn [ ] to generate v1 [ n], timereverse v1 [ n] to obtain v [ n], and finally advance v [ n] by 3 samples to obtain yn [ ] (D) First pick every fourth sample of xn [ ] to generate v1 [ n], timereverse v1 [ n] to obtain v [ n], and finally delay v [ n] by 3 samples to obtain yn [ ] YEAR 007 ONE MARK Question. The frequency spectrum of a signal is shown in the figure. If this is ideally sampled at intervals of 1 ms, then the frequency spectrum of the sampled signal will be Page 7

8 EE Topic wise Question. 3 Let a signal a1sin( ω1t+ φ) be applied to a stable linear time variant system. Let the corresponding steady state output be represented as af( ω t+ φ). Then which of the following statement is true? (A) F is not necessarily a Sine or Cosine function but must be periodic with ω 1 = ω. (B) F must be a Sine or Cosine function with a = a 1 (C) F must be a Sine function with ω 1 = ω and φ 1 = φ Page 8 (D) F must be a Sine or Cosine function with ω 1 = ω

9 EE Topic wise YEAR 007 TWO MARKS Question. 4 A signal xt () is given by xt () = 1, T/ 4 < t # 3T/ 4 13, T/ 4< t # 7T/ 4 xt ( + T) * Which among the following gives the fundamental fourier term of xt ()? (A) 4 cos t ` T 4 j (B) cos t 4 ` + T 4j (C) 4 sin t ` T 4 j (D) sin t 4 ` + T 4j Statement for Linked Answer Question 5 & 6 : Question. 5 A signal is processed by a causal filter with transfer function Gs () For a distortion free output signal wave form, Gs () must (A) provides zero phase shift for all frequency (B) provides constant phase shift for all frequency (C) provides linear phase shift that is proportional to frequency (D) provides a phase shift that is inversely proportional to frequency Question. 6 Gz () = αz -1 + βz -3 is a low pass digital filter with a phase characteristics same as that of the above question if (A) α = β (B) α = β ( 1/ 3) (C) α = β (D) α ( 1/ 3) = β - Question. 7 Consider the discrete-time system shown in the figure where the impulse response of Gz () is g(0) = 0, g(1) = g() = 1, g(3) = g(4) = g = 0 Page 9

10 EE Topic wise This system is stable for range of values of K 1 (A) [ 1, ] (B) [ 1, 1] 1 (C) [, 1] (D) [ 1, ] Question. 8 If ut (), rt () denote the unit step and unit ramp functions respectively and ut ()* rt () their convolution, then the function ut ( + 1)* rt ( ) is given by (A) 1 ( t 1) u( t 1) (B) 1 ( t 1) u ( t ) 1 t (C) ( 1) u( t 1) (D) None of the above Question Xz () = 1 3 z, Yz () = 1+ z are Z transforms of two signals xn [ ], yn [ ] respectively. A linear time invariant system has the impulse response hn [ ] defined by these two signals as hn [ ] = xn [ 1]* yn [ ] where * denotes discrete time convolution. Then the output of the system for the input δ[ n 1] -1 (A) has Z-transform z X() z Y() z (B) equals δ[ n ] 3δ[ n 3] + δ[ n 4] 6δ[ n 5] (C) has Z-transform 1 3z + z 6z (D) does not satisfy any of the above three YEAR 006 ONE MARK Page 10 Question. 30 The following is true (A) A finite signal is always bounded (B) A bounded signal always possesses finite energy (C) A bounded signal is always zero outside the interval [ t0, t0] for some t 0 (D) A bounded signal is always finite

11 EE Topic wise Question. 31 xt () is a real valued function of a real variable with period T. Its trigonometric Fourier Series expansion contains no terms of frequency ω = ( k)/ T; k = 1,g Also, no sine terms are present. Then xt () satisfies the equation (A) xt () = xt ( T) (B) xt () = xt ( t) = x( t) (C) xt ( ) = xt ( t) = xt ( T / ) (D) xt ( ) = xt ( T) = xt ( T / ) Question. 3 A discrete real all pass system has a pole at z = + 30 % : it, therefore (A) also has a pole at % (B) has a constant phase response over the z -plane: arg Hz () = constant constant (C) is stable only if it is anti-causal (D) has a constant phase response over the unit circle: iω arg He ( ) = constant YEAR 006 TWO MARKS Question. 33 xn [ ] = 0; n< 1, n> 0, x[ 1] = 1, x[ 0] = is the input and yn [ ] = 0; n< 1, n>, y[ 1] = 1 = y[1], y[0] = 3, y[] = is the output of a discrete-time LTI system. The system impulse response hn [ ] will be (A) hn [ ] = 0; n< 0, n>, h[ 0] = 1, h[ 1] = h[ ] = 1 (B) hn [ ] = 0; n< 1, n> 1, h[ 1] = 1, h[ 0] = h[ 1] = (C) hn [ ] = 0; n< 0, n> 3, h[0] = 1, h[1] =, h[] = 1 (D) hn [ ] = 0; n<, n> 1, h[ ] = h[ 1] = h[ 1] = h[ 0] = 3 Question The discrete-time signal xn [ ] Xz ( ) 3 n n z n = /, where n = 0 + denotes a transform-pair relationship, is orthogonal to the signal Page 11

12 EE Topic wise / n (A) y1[ n] ) Y1( z) = 3 z n = 0 `3j 3 n = 0 -n (B) y [ n] ) Y ( z) = ( 5 n) z / (C) y [ n] ) Y ( z) = z 3 3 / n 3 - n n =-3 -n (D) y [ n] ) Y ( z) = z + 3z ( n + 1) Question. 35 A continuous-time system is described by yt () = e - xt (), where yt () is the output and xt () is the input. yt () is bounded (A) only when xt () is bounded (B) only when xt () is non-negative (C) only for t # 0 if xt () is bounded for t $ 0 (D) even when xt () is not bounded Question. 36 The running integration, given by yt () = xtdt () (A) has no finite singularities in its double sided Laplace Transform Ys () (B) produces a bounded output for every causal bounded input (C) produces a bounded output for every anticausal bounded input (D) has no finite zeroes in its double sided Laplace Transform Ys () t -# 3 ' ' YEAR 005 TWO MARKS Question. 37 For the triangular wave from shown in the figure, the RMS value of the voltage is equal to Page 1

13 EE Topic wise (A) 1 6 (B) 1 3 (C) 3 1 (D) 3 Question. 38 The Laplace transform of a function ft () is Fs () = t " 3, f( t) approaches (A) 3 (B) 5 (C) 17 (D) 3 5s + 3s+ 6 as ss ( + s+ ) Question. 39 The Fourier series for the function fx () = sinxis (A) sin x+ sin x (B) 1 cos x (C) sin x+ cos x (D) cosx Question. 40 If ut () is the unit step and δ () t is the unit impulse function, the inverse z -transform of Fz () = z for k > 0 is (A) ( 1) k δ( k) (B) δ() k ( ) 1 k (C) ( 1) k uk ( ) (D) uk () ( ) 1 k YEAR 004 TWO MARKS Question. 41 The rms value of the periodic waveform given in figure is (A) 6 A (B) 6 A (C) 43A / (D) 1.5 A Page 13

14 EE Topic wise Question. 4 The rms value of the resultant current in a wire which carries a dc current of 10 A and a sinusoidal alternating current of peak value 0 is (A) 14.1 A (B) 17.3 A (C).4 A (D) 30.0 A YEAR 00 ONE MARK Question. 43 Fourier Series for the waveform, ft () shown in Figure is (A) 8 sin( t) 1 sin( 3 t) 1 sin( 5 t) B (B) 8 sin( t) 1 cos(3 t) 1 sin(5 t) B (C) 8 cos( t) 1 cos( 3 t) 1 cos( 5 t) B (D) 8 cos( t) 1 sin( 3 t) 1 sin( 5 t) B Question. 44 Let st () be the step response of a linear system with zero intial conditions; then the response of this system to an an input ut () is t t (A) # st ( τ) u( τ) dτ (B) d st ( τ) u( τ) dτ dt ; # E 0 t t 1 # # (D) [ ( )] ( ) (C) st ( τ) ; u( τ1) dτ1edτ # 0 st τ uτ dτ Page 14

15 EE Topic wise Question. 45 Let Ys () be the Laplace transformation of the function yt, () then the final value of the function is (A) LimY() s (B) LimY() s s " 0 s " 3 (C) Lim sy() s s " 0 (D) Lim sy() s s " 3 Question. 46 What is the rms value of the voltage waveform shown in Figure? (A) ( 00/ ) V (B) ( 100/ ) V (C) 00 V (D) 100 V YEAR 001 ONE MARK Question. 47 Given the relationship between the input ut () and the output yt () to be t -3( t -τ) yt () = # ( + t τ) e u( τ) dτ, 0 The transfer function Ys ()/ Us () is -s (A) e s + 3 (C) s + 5 s + 3 (B) s + ( s + 3 ) (D) s + 7 ( s + 3 ) Common data Questions Q Consider the voltage waveform v as shown in figure Page 15

16 EE Topic wise Question. 48 The DC component of v is (A) 0.4 (B) 0. (C) 0.8 (D) 0.1 Question. 49 The amplitude of fundamental component of v is (A) 1.0 V (B).40 V (C) V (D) 1 V *********** Page 16

LTI Systems (Continuous & Discrete) - Basics

LTI Systems (Continuous & Discrete) - Basics 1. A system with an input x(t) and output y(t) is described by the relation: y(t) = t. x(t). This system is (a) linear and time-invariant (b) linear and time-varying