1. The most general form of complex exponential function is:
a) eσt
b) eΩt
c) est
d) eat
Explanation: The general form of complex exponential function is: x(t) = est where s = σ + jΩ.
2. A complex exponential signal is a decaying exponential signal when
a) Ω = 0 and σ > 0
b) Ω = 0 and σ = 0
c) Ω ≠ 0 and σ < 0
d) Ω = 0 and σ < 0
Explanation: Let x(t) be the complex exponential signal
⇒ x(t) = est = e(σ+jΩ)t = eσt ejΩt
Now, when Ω = 0 ⇒ x(t) = eσt which will be an exponentially decaying signal if σ < 0.
3. When is a complex exponential signal sinusoidal?
a) σ =0 and Ω = 0
b) σ < 0 and Ω = 0
c) σ = 0 and Ω ≠ 0
d) σ ≠ 0 and Ω ≠ 0
Explanation: A signal is sinusoidal when σ = 0 and Ω ≠ 0
⇒ x(t) = est = e(σ+jΩ)t = eσt ejΩt = ejΩt = cosΩt + jsinΩt which is sinusoidal.
4. An exponentially growing sinusoidal signal is:
a) σ = 0 and Ω = 0
b) σ > 0 and Ω ≠ 0
c) σ < 0 and Ω ≠ 0
d) σ = 0 and Ω ≠ 0
Explanation: A complex exponential signal is sinusoidal when Ω has a definite value i.e., Ω ≠ 0. It can either be growing exponential or decaying exponential based on the value of σ.
∴ A signal is sinusoidal growing exponential when σ > 0 and Ω ≠ 0.
5. Determine the nature of the signal: x(t) = e-0.2t [cosΩt + jsinΩt].
a) Exponentially decaying sinusoidal signal
b) Exponentially growing sinusoidal signal
c) Sinusoidal signal
d) Exponential signal
Explanation: Clearly the signal has negative exponential ⇒ Decaying exponential signal.
The signal also has sinusoidal component.
The signal is exponentially decaying sinusoidal signal.
6. Is the function y[n] = cos(x[n]) periodic or not?
a) True
b) False
Explanation: ‘y’ will be periodic only if x attains the same value after some time, T. However, if x is a one-one discrete function, it may not be possible for some x[n].
7. If n tends to infinity, is the accumulator function an unstable one?
a) The function is marginally stable
b) The function is unstable
c) The function is stable
d) None of the mentioned
Explanation:The system would be unstable, as the output will grow out of bound at the maximally worst possible case.
8. Comment on the causality of the following discrete time system: y[n] = x[-n].
a) Causal
b) Non causal
c) Both Casual and Non casual
d) None of the mentioned
Explanation: For positive time, the output depends on the input at an earlier time, giving causality for this portion. However, at a negative time, the output depends on the input at a positive time, i.e. at a time in the future, rendering it non causal.
9. Comment on the causality of the discrete time system: y[n] = x[n+3].
a) Causal
b) Non Causal
c) Anti Causal
d) None of the mentioned
Explanation: The output always depends on the input at a time in the future, rendering it anti-causal.
10. Consider the system y[n] = 2x[n] + 5. Is the function linear?
a) Yes
b) No
Explanation: As we give two inputs, x1 and x2, and give an added input x1 and x2, we do not get the corresponding y1 and y2. Thus, additive rule is disturbed and hence the system is not linear.