Signals & Systems Test - 4 - PDF Flipbook
Signals & Systems Test - 4
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GATE
EEE
Signals
&
Systems
Test-04Solutions
SIGNALS & SYSTEMS
1. A continuous-time system is governed by the equation 3y3(t) +
2y2(t) + y(t) = x2(t) + x(t). (y(t) and x(t)) respectively are output
and input). The system is
a) linear and dynamic
b) linear and non-dynamic
c) non-linear and dynamic
d) non-linear and non-dynamic
Answer: (d)
Solution:
3Y3(t) + 2Y2(t) + Y(t) = X2(t) + x(t)
Since the output for each value of the independent variable at a
given time is dependent only on the input at the same time, the
system is memoryless or static or non-dynamic.
For a system to be linear, if x(t) = ax1(t) + bx2(t). then y(t) =
ay1(t) + by2(t). Since the given system does not satisfy the
condition of linearity, it is a non-linear system.
2. In what range should Re(s) remain so that the Laplace transform
of the function e(a + 2)t + 5exists?
a) Re(S) > a + 2
b) Re(s) > a + 7
c) Re(s) < 2
d) Re(s) > a + 5
Answer: (a)
1
Solution:
Remember the LTI pair:
eatu(t) → s−1a, Re(s) > a, a real
Therefore, e(a + 2)t + 5 = e5e(a + 2)t ⟶ e5 1
S−(a−2)
ROC: Re(s) > (a + 2)
3. Which one of the following statements is correct? A discrete
LTI system is non-causal if its impulse response is
a) anu(n − 2)
b) an − 2u(n)
c) an + 2u(n)
d) anu(n + 2)
Answer: (d)
Solution:
A system is non-causal if the present sample/ output depends on
future sample/input.
h(n) = an u(n + 2)
At n = 0,
h(0) = an u(2)
Thus sample at n =0, depends on future sample, u(2), at n = 2.
∴ sample is non-causal.
4. A 1.0 kHz signal is flat – top sampled at the rate of 1800
samples/sec and the samples are applied to an ideal rectangular
LPF with cut - off frequency of 1100 Hz, then the output of the
filter contains
2
a) only 800 Hz component.
b) 800 Hz and 900Hz components.
c) 800 Hz and 1000 Hz components.
d) 800 Hz, 900 Hz and 1000 Hz components.
Answer: (c)
Solution:
The input signal x(t) has spectrum with impulses at ± 1 KHZ.
Sampling frequency fS = 1.8 KHZ spectrum, XS(f) of the
sampled signal is given by XS (f) = fs ∑∞n=−∞ X(f − nfs)
∴ XS(f) will have impulses at frequencies (in KHZ) equal to ± 1,
± 0.8, ± 2.8, ± 2.6…... etc.
∴ The output of the low pass filter, with cutoff frequency, 1.1
KHz contains 0.8 KHZ and 1 KHZ components.
Note that in this question, the signal is under sampled because fS
< 2 fh. Therefore, in addition to 1 KHZ component a spurious
component 0.8 KHZ also, is present in the reconstructed signal.
5. Number of state variables of discrete time system, described by
y[n] − 3 y[n − 1] + 1 y[n − 2] = x[n] is
4 8
a) 2
b) 3
c) 4
d) 1
Answer: (a)
3
Solution:
Order of difference equation = 2.
∴ Number of state variables = 2.
6. The result of h(2t) *δ(t – t0) (“*” denotes convolution and "δ(∙)"
denotes the Dirac delta function) is
a) h (2t – 2t0)
b) h (2t0 – 2t)
c) h (– 2t – 2t0)
d) h (2t + 2t0)
Answer: (a)
Solution:
According to the convolution property
X(t) *δ (t – t0) = x (t – t0)
Hence, h(2t) * δ (t – t0) = h (2(t – t0)) = h (2t – 2t0)
7. X(t) = 1 + ∑NK=1 2 cos Kω0t, is the combined trigonometric
T0 T0
form of Fourier series for
a) Half rectified wave
b) Saw-tooth wave
c) Rectangular wave
d) Impulse train
Answer: (d)
Solution:
Given that,
X(t) = 1 + ∑KN = 1 2 cos Kω0t
T0 T0
4
a0 = 1 , aX = 2
T0 T0
As the Fourier series coefficient an is independent of ‘K’ signal
cannot be saw tooth, half rectified (or) rectangular. Hence,
impulse train.
(or)
The other way is evaluating Fourier series coefficients are
verifying.
8. Consider the following statements for continuous-time linear
time invariant (LTI) systems.
I. There is no bounded input bounded output (BIBO) stable
system with a pole in the right half of the complex plane.
II. There is no causal and BIBO stable system with a pole in the
right half of the complex plane. Which one among the
following is correct?
a) Both I and II are true
b) Both I and II are not true
c) Only I is true
d) Only II is true
Answer: (d)
Solution:
i. For example, consider a pole location at right half of
complex plane, if it is anti-causal, ROC is left sided, and
ROC includes jω axis, so it is a BIBO stable, so statement I
is false.
5
ii. If a causal system having a pole on right side of s-plane it is
compulsory unstable because ROC is not including jω-axis.
So statement II is true.
9. Which one of the following is the correct relation?
a) F(at) ↔ a F(ω/a)
b) F(at) ↔ a F(aω)
c) F(t/a) ↔ aF(ω/a)
d) F(at) ↔ (1/a) F(ω/a)
Answer: (d)
10. Which one of the following transfer functions does correspond
to a non-minimum phase system?
a) S2 + S + 1
2S
b) S2 S +1 1
+ 2S +
c) S2 S +1 1
+ 2S −
d) S2 S −1 1
+ 2S +
Answer: (d)
Solution:
When transfer function has at least one pole or zero in the RHS
of s-plane, it is called non-minimum phase transfer function.
Transfer function S2 s−1 1 has one zero (i.e., s = 1) at RHS of s-
+ 2s +
plane, hence it is a non-minimum phase system.
6
11. A voltage having the Laplace transform 4S2 + 3S + 2 is applied
7S2 + 6S + 5
across a 2H inductor having zero initial current. What is the
current in the inductor at t = ∞?
a) Zero
b) (1/5)A
c) (2/7)A
d) (2/5)A
Answer: (b)
Solution:
i (∞) = Sli→m0 SI(s)
∵ V(s) = L.SI(s)
∴ i (∞) = Sli→m0 L. SI(s)
= Sli→m0 1 �47SS22 + 3S + 52�
2 + 6S +
= (1/5) A
12. A continuous-time function x(t) is periodic with period T. The
function is sampled uniformly with a sampling period TS. In
which one of the following cases is the sampled signal periodic?
a) T = √2 TS
b) T = 1.2 TS
c) Always
d) Never
Answer: (b)
7
Solution:
A discrete time signal X(n) = cos (ω0n) is said to be periodic if
ω0 is a rational number.
2π
13. The unit step response y(t) of a linear system is y(t) = (1 – 3e–t
+ 3e–2t) u(t) For the system function, the frequency at which the
forced response becomes zero is
a) 1 rad⁄s
√2
b) 1 rad⁄s
2
c) √2 rad⁄s
d) 2 rad/s
Answer: (c)
Solution:
Y(t) = 1 – 3e – t + 3 e – 2t
Taking Laplace transform of both sides
Y(s) = 1 − (S 3 1) + (S 3 2)
S + +
Y(s) = S(S S2 + 2 2)
+ 1)(S +
So, the forced response become zero when S2 + 2 = 0
∴ S = ± j √2
i.e., ω = √2 rad/sec
14. The Fourier transform of unit step sequence is
a) π δ(Ω)
b) 1 1
− e−jΩ
8
c) π δ(Ω) + 1
1 − e−jΩ
d) 1 – e–jΩ
Answer: (c)
Solution:
15. A system can be represented in the form of state equations as
S (n + 1) = AS(n) + Bx(n)
Y(n) = CS(n) + Dx(n)
Where A, B, C and D are matrices, S(n) is the state vector. x(n)
is the input and y(n) is the output. The transfer function of the
system H(z) = Y(z)/x(z) is given by
a) (ZI – B)– 1C + D
b) (ZI – C)–1 D + A
c) (ZI – A)–1 B + D
d) (ZI – A)–1 C + B
Answer: (c)
16. For an all-pass system H (Z) = �Z−1 − b� , where �H�e−jω�� = 1,
(1 − aZ−1)
for all ω. If Re (a) ≠ 0, Im (a) ≠ 0 then b equals
a) a
b) a*
c) 1/a*
d) 1
a
Answer: (b)
9
Solution:
H(Z) = − b + Z−1
1 − aZ−1
For an APF, pole is conjugate reciprocal of zero.
17. If X(Z) is 1 with |Z| > 1, then what is the corresponding
1 − Z−1
x(n)?
a) e–n
b) en
c) u(n)
d) δ(n)
Answer: (c)
Solution:
anu[n]↔Z 1 − 1aZ−1, |Z| > a
Putting a = 1
u[n]↔Z 1 −1Z−1, |Z| > 1
18. For the discrete signal x[n] = an u[n] the z-transform is
a) Z
Z+a
b) Z− a
Z
c) Z
a
d) Z a
Z−
Answer: (d)
10
Solution:
X(Z) = ∑n∞= − ∞ x[n]Z−n
= ∑∞n = − ∞ anZ−n = 1 + (aZ−1) + (aZ−1)2 + …
= 1− 1 = Z Z a
aZ−1 −
19. Suppose x[n] is an absolutely assumable discrete time signal.
Its z-transform is a rational function with two poles and two
zeroes. The poles are at z = ± 2j. Which one of the following
statements is TRUE for the signal x[n]?
a) It is a finite duration signal.
b) It is a causal signal.
c) It is a non-causal signal.
d) It is a periodic signal.
Answer: (c)
Solution:
Poles are + 2j, – 2j
Assume zeros are Z1, Z2
X (Z) = (Z − Z1)(Z − Z2)
(Z + 2j)( − 2 )
X (Z) = ( − 1)( − 2)
2 + 4
X(n) is absolutely summable discrete time signal so ROC
includes unit circle. It is possible only when the ROC is | |
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