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SIGNALS AND SYSTEMS IMPORTA T 80
Q.1
CQ PDF WITH SOLUTION
The discretediscrete-time time signa signal x (n) = (-1) (A) 6 (C) 2
n
is periodic with fundamental period (B) 4 (D) 0
Ans: C Period = 2
Q.2
The frequency of a continuous time signal x (t) changes on transformation from x (t) to x ( α t), α > 0 by a factor 1 . (A) α . (B) α 2 (C) α . (D) a . Transfor x(t) x(at), a > 0 Ans: A a>1 com ression in t, expansion in f by a. a<1 expansion in t, compression in f by a.
Q.3
A useful useful prop propert erty y of the unit unit impulse impulse 6 (t) is that (A) 6 (at) = a 6 (t) . (B) 6 (at) = 6 (t) . (C) 6 (at) =
1
6 (t) .
a Ans: C Time-scaling property of 6(t): 6(at) = 1 6(t), a > 0 a Q.4
a
(D) 6(at) = [6(t)] .
The continuous time version of the the unit unit impulse impulse 6 (t) (t) is defined defined by the pai of relations 1
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{1 { [0
t = 0 (B) 6 (t) =1, t = 0 and t ≠ 0 . ∞ {1, t ≥ 0 . (C) 6 (t) = 0, t ≠ 0 nd ƒ 6 (t) dt = 1 . (D) 6(t ) = { -∞ [0, t < 0 (A) 6 (t) =
∞ ƒ 6 (t) dt =1. -∞
Ans: C 6(t) = 0, t Ç 0 → 6(t) Ç 0 at origin +œ
ƒ 6(t) dt = 1
Total area under the curve is unity.
-œ
[6(t [6(t)) is is als also o ca called Dirac-delta function] Q.5
.
Q.6
Two sequences x1 (n) a d x2 (n) are related by x 2 (n) = x1 (- n). In the z- domain, their ROC’s are (A) the same. (B) reciprocal of each other. (C) negative of ea h other. (D) complements of each other. z Ans: B x1(n) 1(z), RoC R x z Reciprocals x2(n) = x1(-n) X1(1/z), RoC 1/ R x jm t
The Fourie Fourierr transform transform o the exponential exponential signal e 0 is (A) a constant. (B) a rectangular gate. (C) an impulse. (D) a series of impulses. Ans: C
Q.7
Since the signal contains only a high frequency ωo its FT must be an impulse impulse at ω = ωo
If the Laplace transform of f (t) is (A) cannot be determined. (C) is unity.
ω
(s
2
, then the value of Lim f (t)
+ ω2 (B) is zero. (D) is infinity.
t→∞
L Ans: B f(t)
m s + m2
Lim f(t) = Lim t œ s 0 =
Lim s
Q.8
0
s F(s)
[Final value theorem]
sm s2 + m2
=0
The unit unit impu impuls lsee res respo pon nse of a linear time invariant system is the unit step function u(t) . For t > 0, the response of the system to an excitation e−at u(t), a > 0, will be (A) ae
−at
.
(B)
2
1− e−at . a
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(C) a 1−e
−at
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).
−at
(D) 1− e
.
Ans: B -at h(t) = u(t); x(t) = e u(t), a > 0 1J1
1 ]
System response y(t) = L− | . | [ s s + a ] 1 1 1 ] = L−1 |J − | a [ s
s + a]
-at
= 1 (1 - e ) a 0
Q.9 The z-transform of the function
) δ(n − k ) has the following region
f convergence
k =−∞
(A) z >1
(B) z = 1
(C) z < 1
(D) 0 < z < 1 0
x(n) =
Ans: C
) 6(n-k) k = -œ
0
x(z) =
) z-k
…..+ z3 + z2 + z + 1
(Sum of infinite geometric series)
k = -œ
= 1 , 1–z Q.10
|z| < 1
The auto-correlation function of a rectangular pulse of duration T is (A) a rectangular pulse of duration T. (B) a rectangular pulse of duration 2T. (C) a triangular p lse of duration T. (D) a triangular p lse of duration 2T. Ans: D T/2
R XX (ı) = 1 T
ƒ x(ı) x(t + ı) dı
triangular function of duration 2T.
-T/2
Q.11 The Fourier transform (FT) of a function x (t) is X (f). The FT of dx(t)/ dt will be (A) dX(f ) / df .
(B) j2πf X(f ).
(C) jf X (f ) .
(D) X(f )/( jf ) . œ jmt
Ans: B (t) = 1 ƒ X( ) e 2n - œ
dm
œ
d x = 1 ƒ jm (f) e jmt dm dt 2n - œ ∴ d x ↔ j 2n f X( ) dt Q.12
The FT of a rectangular pulse existing between t = − T / 2 to t = T / 2 is a (A) sinc squared function. (B) sinc function. (C) sine squared unction. (D) sine function. 3
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Ans: B x(t) =
- Ç t ÇT 2 2 otherwise
1, 0, +œ
-jmt
X(jm) = ƒ x(t) e -œ
+T/2
dt =
ƒ e
-jmt
-T/2
-jmt
+T/2
dt = e jm
-T/2
= - 1 (e-jmT/2 - e jmT/2) = 2 jm m
e jmT/2 - e-jmT/2 2j
= 2 sin T = sin(mT/2) .T m 2 mT/2 Hence X(jm) is expressed in terms of a sinc function. Q.13
An analog signal has the spectrum shown in Fig. The minimum sampling rate needed to com letely represent this signal is (A) 3 KHz . (B) 2 KHz . (C) 1 KHz . (D) 0.5 KHz .
Ans: C
Q.14
For a band pass signal, the minimum sampling rate is twice th e bandwidth, which is 0.5 kHz here.
A given system is characterized by the differential equation: d2 y( t ) dy(t) − 2y(t ) x (t ). The system is: − 2 dt dt (A) linear and unstable. (B) linear and stable. (C) nonlinear an unstable. (D) nonlinear and stable. Ans:A
d2y(t) – dy(t) – 2y(t) = x(t), x(t) y(t) x(t) h(t) dt2 dt system The system is li near . Taking LT with zero initial conditions, we ge s2Y(s) – sY(s) 2Y(s) = X(s) or, H(s) = Y(s) = 1 = 2 X(s) s – s – 2
1 (s –2)(s + 1)
Because of the pole at s = +2, the system is unstable. Q.15
The system characterized by the equation y (t) = ax(t) + b is (A) linear for any value of b. (B) linear if b > 0. (C) linear if b < 0 . (D) non-linear. 4
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Ans: D The system s non-linear because x(t) = 0 does not lead to y (t) is a violation o the principle of homogeneity. Q.16
Inverse Fourier transf rm of u(ω) is 1 1 δ(t)+ . (A) 2 πt 1 (C) 2δ( t ) + . πt
(B)
0, which
1
δ( t ) . 2
(D) δ( t ) + sgn ( t ) .
FT Ans: A x(t) = u(t)
X(jm) = n 6(m) + Jm
Duality property: X(jt) u(m)
Q.17
1
2n x(-m)
1 6(t) + 1 2 nt
The impulse response f a system is h (n) = a n u( n) . The condition for th system to be BIBO stable is (A) a is real and ositive. (B) a is real and negative. (C) a > 1. (D) a < 1. +œ Ans: D Sum S =
+œ
) |h(n)| = n -œ +œ
n = -œ
) ||a| n
≤
) | an u(n) |
(
u(n) = 1 for n Ç 0 )
n= ≤
Q.18
1 1- |a|
if |a| < 1.
If R 1 is the region of c nvergence of x (n) and R 2 is the region of converg nce of y(n), then the region of convergence of x (n) convoluted y (n) is (A) R 1+R 2 . (B) R 1−R 2 . (C) R1 ∩ R2 . z (z), RoC R 1 Ans:C x(n) z y(n) Y z), RoC R 2
x(n) * y(n) Q.19
(D) R 1∪R 2 .
X(z).Y(z), RoC at least R 1 fi R 2
( )
The continuous time system described by y(t ) = x t 2 is (A) causal, linear and time varying. (B) causal, non- inear and time varying. (C) non causal, on-linear and time-invariant. (D) non causal, linear and time-invariant.
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Ans: D y(t) = x(t2)
y(t) depends on x(t2) i.e., future values of input if t > 1. System is ant icipative or non-causal α x1(t) → y1(t) = α x1(t2) β x2(t) → y2(t) = β x2(t2) α x1(t) + β x2(t) → y(t) = αx1(t2) +β x2(t2) = y1(t) + y2(t) System is Linea System is time varying. Check with x(t) = u(t) – u(t-z) → y(t) and x1(t) = x(t – 1) Q.20
y1(t) and find that y1(t) ≠ y (t –1).
If G(f) represents the ourier Transform of a signal g (t) which is real and odd symmetric in time, then G (f) is (A) complex. (B) imaginary. (C) real. (D) real and non-negative. FT Ans: B g(t)
G(f)
g(t) real, odd s mmetric in time G*(jm) = - G(j ); G(jm) purely imaginary. Q.21
For a random variable x having the PDF shown in the Fig., the mean and t e variance are, respecti ely, (A) 1 and 2 . 2 3 (B) 1 and 4 . 3 (C) 1 and 2 . 3 (D) 2 and 4 . 3 +œ
Ans:B Mean = �x(t) = ƒ x f x(t) (x) dx -œ 3
= ƒ x 1 x = 1 x2 3 = 9 – 1 1 = 1 -1 4 4 2 -1 2 2 4 +œ
Variance = ƒ (x - �x)2 f x (x) dx -œ 3
=
ƒ (x - 1) 2 1 d(x-1) -1 4 6
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= 1 x - 1)3 3 = 1 [8 + 8] = 4 -1 4 3 12 3 Q.22
If white noise is input to an RC integrator the ACF at the output is proport onal {| − τ [| {−τ[ (A) exp (B) exp| |. . | | [ RC J [ RC J (C) exp (τ RC . (D) exp(- τ RC).
to
Ans: A
R N(ı) = N0 e p - | ı | 4RC RC Q.23
n
x(n) = a , a < 1 is (A) (B) (C) (D)
an energy signal. a power sig al. neither an e ergy nor a power signal. an energy a well as a power signal.
Ans: A œ +œ 2 2 Energy = ) x (n = ) a n=-œ
n=-œ
œ
œ 2
= 1+ 2 ) a2
= ) (a ) n=-œ
n=1
= finite since |a| < 1 ∴This is an energy signal. Q.24
The spectrum of x (n) extends from − mo to +mo , while that of h(n) exte nds ∞
from − 2mo to + 2mo . The spectrum of y(n) =
) h(k ) x(n − k ) extends
k =−∞
from (A)
− 4mo to + 4mo .
(B) − 3mo to + 3mo .
(C)
− 2mo to + 2mo .
(D) − mo to + mo
. jm Ans: D Spectrum dep nds on H( e ) Q.25
jm
X( e ) Smaller of the two ran es.
The signals x1 (t) and x 2 (t) are both bandlimited to (− m1, + m1 ) and
(− m2 , + m2 ) respe
tively. The Nyquist sampling rate for the signal x1 t) x 2 (t)
will be (A) 2m1 if m1> m2 .
(B) 2m2 if m1< m2 .
(C) 2 (m1+m2 ) .
(D)
Ans: C Q.26
(m1+m 2 )
2
.
Nyquist sampling rate = 2(Bandwidth) = 2(m1 – (-m2)) = 2(
(
If a periodic function (t) of period T satisfies f (t) = −f t + T series expansion, 7
2
1
+ m 2)
), then in its Fourier
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(A) the constant term will be zero. (B) there will be no cosine terms. (C) there will be no sine terms. (D) there will be no even harmonics. Ans: T
T/2
T
T/2
1 ƒ f(t) dt = 1 ƒ f(t) dt + ƒf(t) dt T/2 T 0 T 0
Q.27
= 1 T
T/2
ƒ f(t) dt + ƒ f(ı + T/2 dı = 0 0
0
A band pass signal ext nds from 1 KHz to 2 KHz. The minimum sampling frequency needed to retain all inf rmation in the sampled signal is (A) 1 KHz. (B) 2 KHz. (C) 3 KHz. (D) 4 KHz. Ans: B Minimum sampling frequency = 2(Bandwidth) = 2(1) = 2 kHz
Q.28
The region of convergence of the z-transform of the signal 2n u(n) − 3n u(− n − 1) (A) is z > 1.
(B) is z < 1.
(C) is 2 < z < 3 .
(D) does not exist.
Ans: n
2 u(n) n
3 u(-n-1)
1 , |z| > 2 -1 1 –2 z 1 , |z| < 3 -1 1 – 3z
ROC is 2 < | | < 3. Q.29
The number of possi le regions of convergence of the function (A) 1. (C) 3.
(e −2− − 2)z is )( ) (z − e z − 2
(B) 2. (D) 4.
Ans: C -2
Possible ROC ’s are |z| > e -2 , |z| < 2 and e < |z| < 2 Q.30
The Laplace transform of u(t) is A(s) and the Fourier transform of u(t) is B( jω). Then 1 1 (A) B( jω) = (s) s = jω . (B) A(s) = but B( jω) ≠ . s jω 1 1 1 1 (C) A(s) ≠ (D) A(s) ≠ but B( jω) ≠ . ut B( jω) = . s jω s jω L Ans: B u(t) A(s) = 1 s 8
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F.T u(t)
B(jm) = 1 + n 6(m) jm
A(s) = 1 but B(j m) Ç 1 s jm Q.31
Given a unit step function u(t), its time-derivative is: (A) a unit impulse . (B) another step function. (C) a unit ramp fu nction. (D) a sine function. Ans: A
Q.32
The impulse respo se of d2 y + y(t) = x(t) will be dt 2 (A) a constant. (C) a sinusoid.
a
system
described
by
the
differ ntial
equation
(B) an impulse function.. (D) an exponentially decaying f unction.
Ans: C Q.33
The function
sin(πu)
(πu) (A) sin c(nu). (C) signum.
is denoted by: (B) sin c(u). (D) none of these.
Ans: C Q.34
The frequency respon e of a system with h(n) = 6(n) - 6(n-1) is given by jm (A) 6(m) - 6(m - 1 . (B) 1 - e . -jm (C) u(m) – u(m -1 . (D) 1 – e . Ans: D
Q.35
The order of a linear onstant-coefficient differential equation representing a system refers to the number of (A) active devices. (B) elements including sources. (C) passive devi es. (D) none of those. Ans: D
Q.36
z-transform converts convolution of time-signals to (A) addition. (B) subtraction. (C) multiplicatio . (D) division. Ans: C
Q.37
Region of convergence of a causal LTI system (A) is the entire -plane. (B) is the right-half of s-plane. (C) is the left-half of s-plane. (D) does not exist. 9
WWW.ALLEXAMREVIEW.COM Ans: B Q.38
The DFT of a signal x(n) of length N is X(k). When X(k) is given and x(n) is computed from it, the length of x(n) (A) is increased to infinity (B) remains N 2 (C) becomes 2N – 1 (D) becomes N Ans: A
Q.39
The Fourier transform of u(t) is 1 . (A) j2πf 1 . (C) 1+ j2πf
(B) j2nf. (D) none of these.
Ans: D Q.40
For
the
probability
density
function
of
a
random
variable
X
given
−Kx
f x (x) = 5e 1 (A) 5 (C) 25
u(x) , where u(x) is the unit step function, the value of K is 1 (B) 25 (D) 5
Ans: D Q.41
The system having input x(n) related to output y(n) as y(n) = log10 x(n) is: (A) nonlinear, causal, stable. (C) nonlinear, causal, not stable.
(B) linear, noncausal, stable. (D) linear, noncausal, not stable.
Ans: A Q.42
To obtain x(4 – 2n) from the given signal x(n), the following precedence (or priority) rule is used for operations on the independent variable n: (A) Time scaling ‹ Time shifting ‹ Reflection. (B) Reflection ‹ Time scaling ‹ Time shifting. (C) Time scaling ‹ Reflection ‹ Time shifting. (D) Time shifting ‹ Time scaling ‹ Reflection. Ans: D
Q.43
The unit step-response of a system with impulse response h(n) = 6(n) – 6(n – 1) is: (A) 6(n – 1). (B) 6(n). (C) u(n – 1). (D) u(n). Ans: B
10
by
WWW.ALLEXAMREVIEW.COM Q.44
If φ(ω) is the phase-response of a communication channel and ωc is the channel φω d ( ) frequency, then − ω=ωc represents: dω (A) Phase delay (B) Carrier delay (C) Group delay (D) None of these. Ans: C
Q.45
Zero-order hold used in practical reconstruction of continuous-time signals is mathematically represented as a weighted-sum of rectangular pulses shifted by: (A) Any multiples of the sampling interval. (B) Integer multiples of the sampling interval. (C) One sampling interval. (D) 1 second intervals. Ans: B ℑ
Q.46
If x ( t ) ↔ X(s), then ℑ (A)
dX(s)
.
J dx(t)]
| | is given by: [ dt ]
−1
(B) X(s) x (0) − . s s
ds −
(C) sX(s) − x(0 ) .
(D) sX(s) – sX(0).
Ans: C Q.47
The region of convergence of the z-transform of the signal x(n) ={2, 1, 1, 2} ‡ is n=0 (A) all z, except z = 0 and z = œ (B) all z, except z = 0. (C) all z, except z = œ. (D) all z. Ans: A
Q.48
When two honest coins are simultaneously tossed, the probability of two heads on any given trial is: 3 (A) 1 (B) 4 1 1 (C) (D) 2 4 Ans: D
Q.49
Let u[n] be a {1, (A) x[n] = { [0, {1, (C) x[n] = 0,
unit step sequence. The sequence u[ N − n] can {1, n < N (B) x[n] = { otherwise [0, n > N {1, (D) x[n] = { otherwise [0, 11
be described as n ≤ N otherwise n ≥ N otherwise
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{1, n ≤ N Ans (B) x[n] = { [0, otherwise Here the function u(-n) is delayed by N units. Q.50
A continuous-time periodic signal x(t ) , having a period T, is convolved with itself. The resulting signal is (B) periodic having a period T (A) not periodic (C) periodic having a period 2T (D) periodic having a period T /2 Ans (B) periodic having a period T Convolution of a periodic signal (period T ) with itself will give the same period T .
Q.51
If the Fourier series coefficients of a signal are periodic then the signal must be (A) continuous-time, periodic (C) continuous-time, non-periodic
(B) discrete-time, periodic (D) discrete-time, non-periodic
Ans B) discrete-time, periodic This is the property of the discrete-time periodic signal. Q.52
The Fourier transform of a signal x(t ) = e 2t u(−t ) is given by 1 2 (A) (B) . 2 − jω 1− jω 1 2 (C) (D) j2 − ω j2 − ω 1 Ans (A) 2 − jω 1 1 . Therefore, FT of u(-t ) = . If a function x(t ) is multiplied FT u(t ) = − jω jω by e2t , then its FT will be F ( jω )
ω → ω −2
. Hence the answer.
1 Q.53
For the function H ( jω ) = (A) 1 (C) 2
(B) 1/2 (D) 3
2 + 2 jω + ( jω )
2
, maximum value of group delay is
Ans None of the given answers is correct. Q.54
A continuous-time signal x(t ) is sampled using an impulse train. If X ( j ω ) is the Fourier transform of x(t ) , the spectrum of the sampled signal can be expressed as ∞
(A)
∞
) X ( jω + k ω )δ (ω )
(B)
) X ( jω )* δ (ω + k ω )
(D)
s
k =−∞ ∞
(C)
) X ( jk ω ) * δ (ω + k ω ) s
k =−∞ ∞
s
k =−∞
) X ( jω )δ (ω + k ω ) s
k =−∞
12
WWW.ALLEXAMREVIEW.COM ∞
Ans (A)
) X ( jω + k ω )δ (ω ) s
k =−∞
Since the spectrum consists of various harmonics k = −∞ to ∞ and discretely spread at an interval of fundamental frequency f s. Hence the answer. Q.55
The region of convergence of a causal finite duration discrete-time signal is (A) (B) (C) (D)
the entire z -plane except z = 0 the entire z -plane except z = ∞ the entire z -plane a strip in z-plane enclosing jω -axis
Ans (A) The entire z -plane except z = 0 n2
X ( z ) =
)
x[n] z −n . This sum should converge provided each term in the sum is
n=n1
finite. However, if there is a non-zero causal component for n2>0, then X ( z ) will have a term involving z -1 and thus ROC cannot include z = 0. Q.56
j Let H (e
) ( ) be the frequency response of a discrete-time be the frequency response of its inverse. Then, e ω
jω
H I
( ) ( )= 1 H (e )* H (e ) = 1 ω
jω
ω
jω
jω
( ) H (e ) = 1 Since H(e ) and H (e ) are jω
ω
j j (B) H e HI e
jω
I
I
Ans (A) H e
and
( ) ( )= δ (ω ) (D) H (e )* H (e )= δ (ω )
ω
j j (A) H e H I e
(C)
LTI system,
jω
I
jω
jω
I
the inverse of each other, their product should
equal 1. 1 Q.57
1
The transfer function of a stable system is H ( z ) = + . 1 − 0.5 z −1 1 − 2 z −1 Its impulse response will be (A) (C)
(0.5)n u[n] + (2)n u[n] (0.5)n u[n] − (2)n u[−n −1] n
n
n
n
n
(B) − (0.5) u[−n −1] + (2) u[n] (D) − (0.5) u[−n − 1] − (2) u[−n − 1]
n
Ans (C) (0.5) u[n] − (2) u[−n − 1] (A) and (C) are the possible IFTs of the given system function. However, the system is stable; therefore (C) is the only correct answer. Q.58
The probability cumulative distribution function must be monotone and (A) increasing (B) decreasing (C) non-increasing (D) non-decreasing Ans (D) non-decreasing The probability cumulative distribution function increases to 1 monotonically and there after remains constant.
Q.59
The average power of the following signal is
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A2
(B) A2
2 2
2 (D) A T1
(C) AT1
Ans: (D) T 1 / 2
W =
2
2
x(t ) dt = A T 1
−T / 2 1
Q.60
Convolution is us d to find: (A) The impulse response of an LTI System (B) Frequency r sponse of a System (C) The time res onse of a LTI system (D) The phase response of a LTI system Ans: (C) k =∞
Time response
y(n) = x(n) * h(n) =
) x(k )h(n − k )
k =−∞
Q.61
The Fourier Transform of a rectangular pulse is (A) Another rectangular pulse (B) Triangular pulse (C) Sinc functio (D) Impulse. Ans: (C) This can be seen y putting the value of pulse function in the definition of Fourier transform.
Q.62
The property of F urier Transform which states that the compression in time domain is equival nt to expansion in the frequency domain is (A) Duality. (B) Scaling. (C) Time Scaling. (D) Frequency Shif ting. Ans: (B) Substituting the s uare pulse function f (t ) in ∞
F ( jω ) = ƒ f (t )e
dt
−∞
gives the sinc fun tion. Q.63
jω t
What is the Nyquist Frequency for the signal x(t ) =3 cos 50nt +10 sin 300nt – cos100nt ? (A) 50 Hz (C) 200 Hz
(B) 100 Hz (D) 300 Hz
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Ans:
(D) Here the highest frequency present in the signal is ω m = 300π or
f m = 150 Hz. Therefore the Nyquist frequency f s = 2 f m = 300 Hz. Q.64
The step response of a LTI system when the impulse response h(n) is unit step u(n) is (B) n (A) n+1 (C) n-1 (D) n 2 Ans: (A) ∞
y(n) = x(n) * h(n) = u(n) * u(n) =
6
)u(k )u(n − k ) = )u(k )u(n − k )
k =−∞
k =0
y(0) = 1, y(1) = 2, y(2) = 3, ...., y(n) = (n + 1) y(n) = (n + 1) .
Q.65
The Laplace transform of u(t ) is 1
(A)
(B) s
2
s 1
(C) s
(D) s
2
Ans: (A) ∞
Substituting f (t ) = u(t ) in the relation F ( s) =
ƒ f (t )e
− st
dt gives the answer.
o
Q.66
The function which has its Fourier transform, Laplace transform, and Z transform unity is (A) Gausian (B) impulse (C) Sinc (D) pulse Ans: (B) Substituting f (t ) = δ (t ) in the definitions of Fourier, Laplace and Z-transform, we get the transforms in each case as 1.
Q.67
The Z transform of δ (n − m) is (A) z −n
(B) z −m
1 (C)
(D)
z−n
1 z − m
Ans: (B) -m The Z-transform of a delayed f unction f (n-m) is z times the Z-transform of the function f (n). Q.68
1
If the joint probability pdf of f ( x, y) = , 0 ≤ x, y ≤ 2, P ( x + y ≤ 1) is 4
(A) 1 8 1 (C) 4
(B) (D)
1 16 1 2
15
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Ans: (A) 1 1− y 1
ƒ ƒ
P ( x + y) =
0 0
Q.69
1 dxdy =
ƒ
4
4
1 0
1 − y 1 x dy = 0 4
1
1
0
8
ƒ (1− y)dy = .
The period of the signal x(t ) = 10 sin 12 π t + 4 cos18π t is π 1 (B) (A) 4 6 1 1 (D) (C) 9 3 Ans: (D) There are two waveforms of frequencies 6 and 9, respectively. Hence the combined frequency is the highest common factor between 6 and 9,i.e., 3. Hence period is 1/3.
Q.70
The autocorrelation of a rectangular pulse is (A) another rectangle pulse (C) Triangular pulse
(B) Square pulse (D) Sinc pulse
Ans: (C) Autocorrelation involves the integration of a constant which gives a ramp function. Hence the triangular pulse. Q.71
If the Fourier series coefficients of a signal are periodic then the signal must be (A) continuous-time, periodic (B) discrete-time, periodic (C) continuous-time, non periodic (D) discrete-time, non perodic Ans: (B) It is the property of the discrete-time periodic signal. ∞
Q.72
The area under the curve (A) ∞ (C) 0
δ (t ) dt is ƒ −∞
(B) unity (D) undefined
Ans: (B) ∞
By definition of delta function, Q.73
ƒ δ (t ) = 1 −∞
A transmission is said to be replica of the input signal.
if the response of the system is exact
(A) LTI (C) Distortionless
(B) Distorted (D) Causal
Ans: (C) Since y(n) = x(n) . 16
WWW.ALLEXAMREVIEW.COM Q.74
n
Laplace Transform of t is always equal to n (A) (B) n! sn sn n! All (C) (D) n+1 s Ans: (C) n t =
=
ƒ0
£ Q.75
n − st ∞ t e dt
n! n+1
s
For a stable system (A)
z < 1
(B)
z = 1
(C)
z > 1
(D)
z ≠ 1
Ans: (A) For the system to be stable, the ROC should include the unit circle. Q.76
The region of convergence of a causal finite duration discrete time signal is (A) The entire ‘z’ plane except z = 0 (B) The entire ‘z’ plane except z = ∞ (C) The entire ‘z’ plane (D) A strip in z-plane Ans: (A) The ROC of the causal finite duration will have negative power of z. The ROC is the entire z-plane except z = 0.
Q.77
The CDF for a certain random variable is given as {0,
| F ( x) = {kx 2 , | 100k , [
− ∞ < x ≤ 0 0 < x ≤ 10 10 < x < ∞
The value of k is (A) 100 (B) 50 (C) 1/50 (D) 1/100 Ans: (D) From the given F(x), we get dF ( x) dx
= 0 + 2kx + 0 = 2kx
10
∴ 2kxdx = 1 ƒ 0
or 100k = 1 → k = 1/100
Q.78
The group delay function τ(ω) is related to phase function φ(ω) as −d d (A) τ (ω ) = φ (ω ) (B) τ ( ω ) = 2 φ ( ω ) d ω d ω 17
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d
(C) τ (ω ) =
)
2
d2 (D) τ (ω ) = φ (ω ) d ω
φ (ω
d ω 2
Ans: (A): By definition. Q.79
Two sequences x1(n) and x2 (n)are related by x2 (n) = x1(− n ) . In the Z-domain, their ROCs are (A) same (B) reciprocal of each other (C) negative of each other (D) complement of each other Ans: (B) ROC of Z [ x2 (n)] is outside the circle of radius r2 while ROC of Z [ x1 (−n)] is
inside the circle of radius r1 such that r2 = 1/ r1. Q.80
The autocorrelation of a sinusoid is (A) Sinc pulse (C) Rectangular pulse
(B) another sinusoid (D) Triangular pulse
Ans: (B) ∞
φ XX (t ) = ƒ x(τ ) x(τ − t )d τ −∞∞
= ƒ A sin ωτ × A sin ω (τ − t ) d τ −∞2 A ∞ = [(cos t − cos 2ωτ .cos t − sin 2ωτ .sin t ]d τ 2 −∞ A 2 J π [(cos t − cos 2ωτ .cos t − sin 2ωτ .sin t ]d τ ] = K
ƒ
=
2 A 2 2
K
|[ ƒ−π J π
ƒ
|]
]
[cos t ]d τ = K ' cos t |[ −π |]
.
Thus the autocorrelation is a sinusoid. Q.81
Which of the following is true for the system represented by y(n) = x(− n) (A) Linear (C) Causal
(B) Time invariant (D) Non Linear
Ans.: (A) The given function is of the form y = mx . Hence linear. Q.82
The Fourier transform of impulse function is (B) 2πω (A) δ(ω) (C) 1 (D) sinc f Ans: (C) ∞
FT of
ω δ (t ) = ƒ δ (t )e − j t dt = 1 −∞
18
WWW.ALLEXAMREVIEW.COM Q.83 Convolution is used to find (A) amount of similarity between the signals
19
WWW.ALLEXAMREVIEW.COM (B) response of the system (C) multiplication of the signals (D) Fourier transform
Ans: (B) Convolution of the input signal x(n) and the impulse response h(n) is given by ∞
y(n) = x(n) * h(n) =
) x(k )h(n − k)
, where y(n) is the response of the system.
k =−∞
Q.84
[
]
The final value of x(t ) = 2 + e−3t u(t ) is (A) 2 (C) e
(B) 3 −3t
(D) 0
Ans: (A)
[
]
−3t u(t ) = 2 . Final value = Lt t →∞ x(t ) = Lt t →∞ 2 + e
Q.85
Discrete time system is stable if the poles are (A) within unit circle (C) on the unit circle
(B) outside unit circle (D) None
Ans: (A) The ROC should include the unit circle. Q.86
The z transform of − u(− n − 1) is 1 (A) 1 − z 1 (C) 1 − z −1
(B) (D)
z 1 − z z 1 − z −1
Ans: (C) −∞
z [− u(−n −1)] = − ) [u(−n −1) ]z
−n
2
3
= −[ z + z + z + ...] =
∞
The area under Gaussian pulse
=
z − 1
n=−1
Q.87
z
ƒ e
−π
t2
dt is
−∞
(A) Unity (C) Pulse
Ans: (A) ∞ − π t 2
ƒ ∞e −
Q.88
(B) Infinity (D) Zero
∞
dt = ƒ e −∞
− x
x 2π π
dx = 2 π ƒ
∞
−∞
The spectral density of white noise is (A) Exponential (C) Poisson
20
x e
− x
dx =1.
(B) Uniform (D) Gaussian
1
1 − z−1
WWW.ALLEXAMREVIEW.COM
Ans: (B) The distribution of White noise is homogeneous over all frequencies. Power spectrum is the Fourier transform of the autocorrelation function. Therefore, power spectral density of white noise is uniform.
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