GATE QUESTIONS ON SIGNALS & SYSTEMS
1. The Fourier Series of an odd periodic function, contains only a. Odd harmonics
b. even harmonics
[
c. cosine terms
] d. sine terms
Ans: D
GATE 1994
2. The RMS value of a rectangular wave o f period T, having a value of +V for a duration, T1(
b. T1-T2 V T
c. V/√2
[
]
d. T1 V T
Ans: A
GATE 1995
3. The Trigonometric Fourier Series of a periodic time function can have only a. cosine terms
b. sine terms c. cosine & sine terms
[
d. d.c. and cosine terms
Ans: C
GATE 1998
4. Which of the following cannot be the Fourier Series expansion of a periodic signal? [ a. x(t) = 2 cos t +3 cos 3t
c. x(t) = cos t +0.5
b. x(t) = 2 cos πt +7 cos t
d. x(t) = 2 cos 1.5πt + sin 3.5 πt
Ans: B
a. 3 sin (25t) b. 4 cos(20t+3) +3 sin (10t)
c. exp(-|t|)sin (25t)
Ans: C
GATE 2005
b. cosine terms
c. sine terms
Ans: C
[
]
d. odd harmonic terms GATE 1996, GATE 2011
7. The Trigonometric Fourier Series of a real periodic function has only
[
P. cosine terms if it is even Q. sine terms if it is even R. cosine terms if it is odd S. sine terms if it is odd Which of the above statements are correct? b. P and R
c. Q and S
Ans: A
8. A function is given by f(t) = sin 2t + cos 2t. which w hich of the following is true? a. f has frequency components at 0 and 1/2π Hz b. f has frequency components at 0 and 1/π Hz c.
]
d. 1
6. The Trigonometric Fourier Series of an even function of time does not have
a. P and S
]
GATE 2002
5. choose the function f(t); -∞ - ∞ < t < +∞, for which a Fourier Series cannot be defined [ defined [
a. d.c. terms
]
f has frequency components at 1/2π Hz and and 1/π Hz
d. f has frequency components at 0, 1/2π Hz and 1/π Hz
d. Q and R GATE 2009
]
Ans: B
GATE 2009
9. A half – wave rectified sinusoidal waveform has a peak voltage of 10 V. Its average value and the peak value of the fundamental component are respectively given by:
[
]
a. 20 V, 10 V π √2 b. 10 V, 10 V π √2 c.
10 V, 5 V π
d. 20 V, 5 V π Ans: C
GATE 1987
10. Which of the following signals is/are periodic? a. s(t) = cos 2t + cos 3t + cos 5t
[
b. s(t) = exp(j8πt)
]
c. s(t) = exp(-7t)sin 10πt
d. s(t) = cos 2t cos 4t Ans: A, B &D
GATE 1992
11. The PSD and the power of a signal g(t), are respectively, Sg(w), Pg. The PSD and the power of the signal a g(t) are, respectively, a. a2Sg(w), a2 Pg
b. a2Sg(w), a Pg c. aSg(w), a2 Pg d. aSg(w), a Pg
Ans: a
GATE 2001
12. The Fourier Transform of a real valued time signal has a. Odd symmetry
b. Even symmetry
c. conjugate symmetry d. no symmetry
Ans: C
GATE 1996
13. The amplitude spectrum of a Gaussian pulse is a. Uniform
b. a sine function
c. Gaussian
d. an impulse function
Ans: C
GATE 1998
14. If a signal f(t) has energy E, the energy of the signal f(2t) is e qual to a. E
b. E/2
c. 2E
d. 4E
Ans: B
GATE 2001
15. The Fourier Transform F{e-t u(t)} is equal to 1/(1+j2πf). Therefore F{1/(1+j2πt)} is [ f
a. e u(f) Ans: C
-f
b. e u(f)
f
c. e u(-f)
-f
d. e u(-f) GATE 2002
]
16. The Fourier Transform of a conjugate symmetric function is always a. Imaginary
b. conjugate anti-symmetric
[
c. real d. conjugate symmetry
Ans: C
GATE 2004
17. Hilbert Transform of [cosw1t + sinw2t] is a.
]
sinw1t- cosw2t
b. cosw1t + sinw2t
[ c. cosw1t – sinw2t
Ans: A
]
d. sinw1t + sinw2t GATE 2000
18. If G(f) represents the Fourier Transform of a signal g(t) which is real and odd symmetric in time, then a. G(f) is complex
b. G(f) is imaginary
c. G(f) is real
d. G(f) is real and non-negative
Ans: B
GATE 1992
19. The Fourier Transform of a signal h(t) is H(jw) = (2 cos w)(sin 2w)/w. The value of h(0) is a. ¼
b. ½
c. 1
d. 2
[
Ans: C
]
GATE 2012
20. The Laplace Transforms of the functions t u(t) and u(t) sin(t) are respectively a. 1/s2 , s/( s2+1)
b. 1/s , 1/( s2+1)
c. 1/s2 , 1/( s2+1)
Ans: C
[
]
d. s , s/( s2+1) GATE 1987
21. The Laplace Transforms of the function f(t)u(t),where f(t) is periodic with period T, is A(s) times the L.T. of its first period. Then a. A(s) = s
[
b. A(s) = 1/(1-exp(-Ts)) c. A(s) = 1/(1+exp(-Ts)) d. A(s)= exp(Ts)
Ans: B
GATE 1988
22. In what range should Re(s) remain so that the L.T. of the function e(a+2)t+5 exists? [ a. Re(s) > a+2
b. Re(s) > a+7
c. Re(s) <2
Ans: A
GATE 2005
Ans: B
a. 1
]
d. Re(s) > a+5
23. If F(s) = L[f(t)] = 2(s+1) then the initial and final values are respectively S2+4s+7 a. 0,2 b. 2,0 c. 0, 2/7 d. 2/7,0
24. Given f(t) = L-1
]
[
]
GATE 2011 3+
. +4 +(−3) b. 2
If
lim () 1 , then the value of K is [
→∞
c. 3
]
d. 4
Ans: D
GATE 2010
25. Given that F(s) is a one sided L.T. of f(t), the L.T. of ∫0 () is
[
]
a. sF(s)-f(0) b. 1/s F(s)
c. ∫0 ()
d. 1/s[F(s)-f(0)]
Ans: B
GATE 2009
26. consider the function f(t) having the L.T. F(s) =
w0 ( s + w0 2) b. 1
[
]
Re[s]>0 the final value of f(t) would be
2
a. 0
c. -1≤ f(∞) ≤ 1
d. ∞
Ans: C
GATE 2006
27. The first six points of the 8 – point DFT of a real valued sequence are 5, 1-j3, 0, 3-j4, 0, and 3+j4. The last two points of the DFT are respectively [ ] a. 0, 1-j3 b. 0, 1+j3 c. 1+j3, 5 d. 1-j3,5 Ans: B
GATE 2011
28. The z- Transform of the function ∑∞ () is =0 a. (z-1)/z
b. z/(z-1)2
c. z/(z-1)
[
d. (z-1)2/z
Ans: C
GATE 1998
29. The z Transform of the sequence x[n] is given by X[z] =
0.5 . it is given that the region of −2
convergence of X[z] includes the unit circle. The value of x[0] is a. -0.5
b. 0
c. 0.25
[
]
d. 0.5
Ans: B
GATE 2007
30. The Z-transform F(z) of the function f(nT)=a^(nT) is
(a) z/(z-a^T)
]
(b) z/(z+a^T) (c) z/(z-a^(-T))
[GATE-1999]
(d) z/(z+a^T)
Answer: a 30. A signal x(t) has a Fourier Transform X(w). If x(t) is a real and odd function of t, Then X(w) is
[GATE-1999]
(a) a real and even function of w (b) an imaginary and odd function of w (c) an imaginary and even function of w (d) a real and odd function of w Answer: b 31. A system with an input x(t) and output y(t) is described by the relation: y(t)=t x(t).
This system is
[GATE-2002]
(a) linear and time-invariant (b) linear and time varying (c) non-linear and time-invariant (d) non-linear and time-varying Answer: b 32. The region of convergence of the z-transform of a unit step function is (a) IzI > 1
[GATE-2002]
(b) IzI < 1 (c) (Real part of z) > 0 (d) (Real part of z) < 0 Answer: a 33. If a signal f(t) has energy E, the energy of the signal f(2t) is equal to (a) E
(b) E/2
(c) 2E
(d) 4E
[GATE-2002]
Answer: b
34. Let x(t) be the input to a linear, time-invariant system. The required output is 4x(t-2). The transfer function of the system should be (a) 4exp(j4*pi*f)
(b) 2exp(-j8*pi*f)
(c) 4exp(-j4*pi*f)
[GATE-2003] (d) 2exp(j8*pi*f)
Answer: c 35. The impulse response h[n] of a linear time-invariant system is given by h[n]=u[n+3] + u[n+2] – 2u[n-7] where u[n] is the unit step response sequence. The above system is (a) Stable but not causal (b) Stable and causal (c) Causal but unstable (d) Unstable and not causal Answer: a 36. The Fourier transform of a conjugate symmetric function is always
[GATE-2004]
(a) Imaginary
[GATE-2004]
(b) Conjugate antisymmetric (c) Real (d) Conjugate symmetric Answer: c 37. The z-transform of a system is H(z)= z/(z-0.2)
[GATE-2004]
If the ROC is IzI<0.2, then the impulse response of the system is (a) (0.2)^n u[n]
(b) (0.2)^n u[-n-1]
(c) -(0.2)^n u[n]
(d) - (0.2)^n u[-n-1]
Answer: d 38. A 1kHz sinusoidal signal is ideally sampled at 1500 samples/sec and the sampled signal is passed through an ideal low pass filter with cut-off frequency 800 Hz. The output signal has the frequency (a) zero Hz
[GATE-2004]
(b) 0.75 Hz
(c) 0.5 Hz
(d) 0.25 Hz
Answer: c 38. Let x(n)=(1/2)^n u(n), y(n)=(x(n))^2 and Y(exp(jw)) be the Fourier transform of y(n). Then Y(exp(j0)) is (a) 1/4
[GATE-2005] (b) 2
(c) 4
(d) 4/3
Answer: d 39. The power in the signal s(t)= 8cos(20*pi*t-pi/2) + 4sin(15*pi*t) is (a) 40
(b) 41
(c) 42
(d) 82
[GATE-2005]
Answer: a 39. Match the following and choose the correct combination. GROUP-1
E. Continuous and aperiodic signal
[GATE-2005]
F. Continuous and periodic signal G. Discrete and aperiodic signal H. Discrete and periodic signal GROUP-2
1. Fourier representation is continuous and aperiodic 2. Fourier representation is discrete and aperiodic 3. Fourier representation is continuous and periodic 4. Fourier representation is discrete and periodic
(a) (b) (c) (d)
E-3,F-2,G-4,H-1 E-1,F-3,G-2,H-4 E-1,F-2,G-3,H-4 E-2,F-1,G-4,H-3 Answer: c
40. For a signal x(t) the Fourier transform is X(f). Then the inverse Fourier transform of X(3f+2) is given by [GATE-2005] (a) (1/2) x(t/2) exp(j3*pi*t)
(b) (1/3) x(t/3) exp(-j4*pi*t/3)
(c) 3 x(3t) exp(-j4*pi*t)
(d) x(3t+2)
Answer: b 40. A system with input x[n] and output y[n] is given as y[n]= (sin(5/6)*pi*n) x(n) . The system is (a) (b) (c) (d)
[GATE-2006]
linear, stable and invertible non-linear, stable and non-invertible linear, stable and non-invertible linear, unstable and invertible Answer: c
41. If the Laplace transform of a signal y(t) is Y(s)= 1/(s(s-1)), then its final value is (a) -1
(b) 0
(c) 1
(d) Unbounded
[GATE-2007]
Answer: d 42. The z-transform X[z] of a sequence x[n] is given by X[z]= 0.5/(1-2z^(-1)) . It is given that the region of convergence of X[z] includes a unit circle. The value of X(0) is (a) -0.5
(b) 0
(c) 0.25
(d) 0.5
[GATE-2007]
Answer: b 43. The input and output of a continuous time system are respectively denoted by x(t) and y(t). Which of the following descriptions is corresponds to a causal system? [GATE-2008] (a) (b) (c) (d)
y(t) = x(t-2) + x(t+4) y(t) = (t-4) x(t+1) y(t) = (t+4) x(t-1) y(t) = (t+5) x(t+5) Answer: c
44. A Discrete time linear shift-invariant system has an impulse response h[n] with h[0]=1, h[1]= -1, h[2]= 2, and zero otherwise. The system is given an input sequence x[n] with x[0]=x[2]= 1 and zero otherwise. The number of nonzero samples in the output sequence y[n], and the value of y[2] are, respectively [GATE-2008] (a) 5,2
(b) 6,2
(c) 6,1
(d) 5,3
Answer: d
Statement for Linked Answer Questions 20 and 21:
[GATE-2008]
45. The frequency response H(w) of this system in terms of angular frequency w,is given by, H(w)= (a) 1/(1+j2w)
(b) (sinw)/w
(c) 1/(2+jw)
(d) jw/(2+jw)
Answer: c 46. The output of this system, to the sinusoidal input x(t)=2cos(2t) for all the time t, is (a) (b) (c) (d)
0 2^(0.25) cos(2t-0.125*pi) 2^(0.5) cos(2t-0.125*pi) 2^(0.5) cos(2t-0.25*pi) Answer: d
47. The Fourier series of a real periodic function has only
[GATE-2009]
P. Cosine terms if it is even Q. Sine terms if it is even R. Cosine terms if it is odd S. Sine terms if it is odd Which of the above statements are correct? (a) P and S
(b) P and R
(c) Q and S
(d) Q and R
Answer: a 48. A system with transfer function H(z) has impulse response h(.) defined as h(2)=1, h(3)= -1 and h(k)=0 otherwise. Consider the following statements. [GATE-2009] S1: H(z) is a low pass filter S2: H(z) is an FIR filter Which of the following is correct? (a) (b) (c) (d)
Only S2 is true Both S1 and S2 are false Both S1 and S2 are true, and S2 is a reason for S1 Both S1 and S2 are true, but S2 is not a reason for S1. Answer: a
49. The 4-point discrete Fourier transform (DFT) of a discrete time sequence {1, 0, 2, 3) is (a) (b) (c) (d)
[0, -2+2j, 2, -2-2j] [2, 2+2j, 6, 2-2j] [6, 1-3j, 2, 1+3j] [6, -1+3j, 0, -1-3j] Answer: d
[GATE-2009]
50. The trigonometric Fourier series of an even function does not even have the (a) (b) (c) (d)
Dc term Cosine terms Sine terms Odd terms Answer: c
[GATE-2011]
