Consider a stable and causal system with impulse response h(t) and system functi

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Home Aptitude

Question
Consider a stable and causal system with impulse response h(t) and system function H(S). Suppose H(S) is rational, contains a pole at S = - 2, and does not have a zero at the origin. The location of all other poles and zero is unknown for each of the following statements. Let us determine whether statement is true or false.
1. f[h(t) e^-3t] converges
3. h(t) has finite duration
4. H(s) = H(- s)
Choose correct option.
Options
A. 1 - True, 2 - False, 3 - True, 4 - False
B. 1 - False, 2 - False, 3 - False, 4 - True
C. 1 - False, 2 - False, 3 - False, 4 - False
D. 1 - True, 2 - can't say, 3 - True, 4 - can't say

Correct Answer

1 - False, 2 - False, 3 - False, 4 - False

Explanation

Statement 1 is false, since f{h(t)e^3t} corresponds to the value of the Laplace transform of h(t) at s = 3.

If this converges, it implies that s = - 3 is in the ROC.

A casual and stable system must always have its ROC to the right of all its poles. However, s = - 3 is not to the right of the pole at s = - 2.

Statement 2 is false, because it is equivalent to stating that H(0) = 0. This contradicts the fact that H(s) does not have a zero at the origin.

Statement 3 is false. If h(t) is of finite duration, then if its Laplace transform has any points in its ROC, ROC must be the entire s-plane.

However, this is not consistent with H(s) having a pole at s = - 2.

Statement 4 is false. If it were true, then H(s) has a pole at s = - 2, it must also have a pole at s = 2.

This is inconsistent with the fact that all the poles of a causal and stable system must be in the left half of the s-plane.

1. A transmission line has a characteristic impedance of 50 ? and a resistance of 0.1 ?/m, if the line is distortion less, the attenuation constant (in Np/m) is
Options
A. 500
B. 5
C. 0.014
D. 0.002

2. If the power spectral density of stationary random process is a sine-squared function of frequency, the shape of its autocorrelation is
Options
A.
B.
C.
D.

3. If x₁[n] x₁(z) with ROC = R₁ and x₂[n] x₂(z) with ROC = R₂ then ROC for x₁[n] + x₂[n] will be __________ (where R_i ? Region of convergence and x₁[n] and x₂[n] are causal).
Options
A. R₁ ? R₂
B. R₁ ? R₂
C. (R₁ ? R₂) ? (R₁)
D. (R₁ ? R₂) ? (R₁)

4. A MOSFET has a threshold voltage of 1 V and oxide thickness of 500 x 10^-8 [?_r = 3.9; ?₀ = 8.85 x 10^-14 F/cm, q = 1.6 x 10^-19 c]. The region under the gate is ion implanted for threshold voltage tailoring. The base and type of impant required to shift threshold voltage to - 1 V are __________ .
Options
A. 8.6 x 10¹¹/cm², p-type
B. 8.6 x 10¹¹/cm², n-type
C. 0.86 x 10⁹/cm², p-type
D. 1.02 x 10¹²/cm², n-type

5. Suppose that the modulating signal is m(t) = 2cos (2pf_mt) and the carrier signal is x_C(t) = A_C cos(2pf_ct), which one of the following is a conventional AM signal without over modulation?
Options
A. x(t) = A_cm(t)cos(2pf_ct)
B. x(t) = A_c [1 + m(t)]cos (2pf_ct)
C.
D. x(t) = A_ccos(2pf_mt)cos(2pf_ct) + A_csin(2pf_mt)sin(2pf_ct)

6. During transmission over a communicate channel, bit errors occur independently with probability . If a block of 3 bits are transmitted, the probability of at most one bit error is equal to
Options
A.
B.
C.
D. none

7. Denominator polynomial of a transfer function of certain network is:
s³ + s² + 2s + 24
Then the network is:
Options
A. stable
B. oscillatory
C. unstable
D. depends on numerator polynomial

8. In the circuit shown, in switch S is open for a long time and is closed at t = 0. The current i(t) for t ? 0⁺ is
Options
A. i(t) = 0.5 - 0.125e^-1000t A
B. i(t) = 1.5 - 0.125e^-1000t A
C. i(t) = 0.5 - 0.5e^-1000t A
D. i(t) = 0.375e^-1000t A