# Signals & Systems Questions and Answers – Periodic and Non-Periodic Signals

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This set of Signals & Systems Multiple Choice Questions & Answers (MCQs) focuses on “Periodic and Non-Periodic Signals”.

1. Given the signal
X (t) = cos t, if t<0
Sin t, if t≥0
The correct statement among the following is?
a) Periodic with fundamental period 2π
b) Periodic but with no fundamental period
c) Non-periodic and discontinuous
d) Non-periodic but continuous

Explanation: From the graphs of cos and sin, we can infer that at t=0, the function becomes discontinuous.
Since, cos 0 = 1, but sin 0 = 0
As 1 ≠ 0, so, the function X (t) is discontinuous and therefore Non-periodic.

2. The fundamental period of the signal X (t) = 10 cos2(10 πt) is __________
a) 0.2
b) 0.1
c) 0.5
d) No fundamental period exists

Explanation: X (t) = 10 cos2 (10 πt)
Since, cos 2t = 2cos2 t – 1
Or, cos2 t = $$\frac{1+cos⁡2t}{2}$$
∴ X (t) = 5 + 5 cos 20πt
Now, Y (t) = cos 20πt
Fundamental period of the signal is = $$\frac{2π}{20π} = \frac{1}{10}$$ = 0.1.

3. The even component of the signal X (t) = ejt is _________________
a) Sin t
b) Cos t
c) Sinh t
d) Cosh t

Explanation: Let Xe (t) represents the even component of X (t)
Now, Xe (t) = $$\frac{1}{2}$$[X (t) + X (-t)]
= $$\frac{1}{2}$$[ejt + e-jt]
= cos t.

4. The odd component of the signal X (t) = ejt is _______________
a) Sin t
b) Cos t
c) Sinh t
d) Cosh t

Explanation: Let Xo (t) represents the odd component of X (t)
Now, Xo (t) = $$\frac{1}{2}$$[X (t) – X (-t)]
= $$\frac{1}{2}$$[ejt + e-jt]
= sin t.

5. The period of the signal X (t) = 24 + 50 cos 60πt is _______________
a) $$\frac{1}{30}$$ s
b) 60 π s
c) $$\frac{1}{60π}$$ s
d) Non-periodic

Explanation: Period of cos t = 2π
Period of cos at = $$\frac{2π}{a}$$
Here, a = 60π
So, period of cos 60πt = $$\frac{2π}{60π}$$
= $$\frac{1}{30}$$ s.

6. The period of the signal X (t) = 10 sin 5t – 4 cos 9t is _______________
a) $$\frac{24π}{35}$$
b) $$\frac{4π}{35}$$
c) 2π
d) Non-periodic

Explanation: Period of cos t = 2π
Period of cos at = $$\frac{2π}{a}$$
Here, a = 9
So, period of cos 9t = $$\frac{2π}{9}$$
Again, Period of sin t = 2π
Period of sin at = $$\frac{2π}{a}$$
Here, a = 5
So, period of sin 5t = $$\frac{2π}{5}$$
∴ Period of X (t) = LCM [Period of X1 (t), Period of X2 (t)]
∴ Period of X (t) = LCM ($$\frac{2π}{5}, \frac{2π}{9}$$) = 2π.

7. The period of the signal X (t) = 5t – 2 cos 6000 πt is ________________
a) 0.96 ms
b) 1.4 ms
c) 0.4 ms
d) Non-periodic

Explanation: Period of cos t = 2π
Period of cos at = $$\frac{2π}{a}$$
Here, a = 6000π
So, period of cos 6000πt = $$\frac{2π}{6000π}$$
= $$\frac{1}{3000}$$
Again, Period of t = indefinite
∴ Period of X (t) = LCM [Period of X1 (t), Period of X2 (t)]
∴ Period of X (t) = LCM ($$\frac{1}{3000}$$, ∞) = Indefinite.

8. The period of the signal X (t) = 4 sin 6t + 3 sin $$\sqrt{3}$$t is ________________
a) $$\frac{2π}{3}$$ s
b) $$\frac{2π}{\sqrt{3}}$$ s
c) 2π s
d) Non-periodic

Explanation: Period of sin t = 2π
Period of sin at = $$\frac{2π}{a}$$
Here, a = 6
So, period of sin 6t = $$\frac{2π}{6}$$
Again, a = $$\sqrt{3}$$
So, period of sin $$\sqrt{3}$$t = $$\frac{2π}{\sqrt{3}}$$
∴ Period of X (t) = LCM [Period of X1 (t), Period of X2 (t)]
∴ Period of X (t) = LCM ($$\frac{π}{3}, \frac{2π}{\sqrt{3}}$$) = Indefinite.

9. The period of the signal Z (t) = sin3t + cos 4t is _______________
a) periodic without a definite period
b) periodic with a definite period
c) non- periodic over an interval
d) non-periodic throughout

Explanation: Period of cos t = 2π
Period of cos at = $$\frac{2π}{a}$$
Here, a = 4
So, period of cos 4t = $$\frac{2π}{4}$$
= $$\frac{π}{2}$$
Again, Period of sin t = 2π
Period of sin at = $$\frac{2π}{a}$$
Here, a = 3
So, period of sin 3t = $$\frac{2π}{3}$$
∴ Period of X (t) = LCM [Period of X1 (t), Period of X2 (t)]
∴ Period of X (t) = LCM ($$\frac{2π}{5}, \frac{2π}{4}$$) = definite
Hence Z (t) is periodic with a definite period.

10. The signal X (t) = e-4t u (t) is _______________
a) Power signal with P = $$\frac{1}{4}$$
b) Power signal with P = 0
c) Energy signal with E = $$\frac{1}{4}$$
d) Energy signal with E = 0

Explanation: If a signal has E∞ as ∞ and P∞ as a finite value, then the signal is a power signal. If a signal has E∞ as a finite value and P as ∞, then the signal is an energy signal.
|x (t)| < ∞, E = $$\int_{-∞}^∞ |x(t)|^2 \,dt$$
= $$\int_∞^∞ e^{-4t} u(t) \,dt$$
= $$\in_∞^∞ e^{-4t} \,dt = \frac{1}{4}$$
So, this is not a power signal but an energy signal.
$$P_∞ = lim_{T→∞} \frac{1}{2T} \int_{-T}^T |x(t)|^2 \,dt = ∞.$$

11. The signal X (t) = $$e^{j(2t + \frac{π}{6})}$$ is ________________
a) Power signal with P = 1
b) Power signal with P = 2
c) Energy signal with E = 2
d) Energy signal with E = 1

Explanation: If a signal has E as ∞ and P as a finite value, then the signal is a power signal. If a signal has E as a finite value and P as ∞, then the signal is an energy signal.
|x (t)| = 1, E = $$\int_{-∞}^∞ |x(t)|^2 \,dt = ∞$$
So, this is a power signal not an energy signal.
$$P_∞ = lim_{T→∞} \frac{1}{2T} \int_{-T}^T |x(t)|^2 \,dt = 1.$$.

12. Signal X (t) is as shown in the figure below.

The total energy of X (t) is _______________
a) 0
b) 13
c) $$\frac{13}{3}$$
d) $$\frac{26}{3}$$

Explanation: E = 2$$\int_0^5 x^2 (t) \,dt$$
= 2 $$\int_0^4 1^1 \,dt + 2\int_4^5 (5 – t^2) \,dt$$
= 8 + $$\frac{2}{3} = \frac{26}{3}$$.

13. A discrete time signal is as given below
$$X [n] = cos \frac{πn}{9} + sin (\frac{πn}{7} + \frac{1}{2})$$
The period of the signal X [n] is ______________
a) 126
b) 32
c) 252
d) Non-periodic

Explanation: Given that, N1 = 18, N2 = 14
We know that period of X [n] (say N) = LCM (N1, N2)
∴ Period of X [n] = LCM (18, 14) = 126.

14. A discrete time signal is as given below
$$X [n] = cos (\frac{n}{8}) cos (\frac{πn}{8})$$
The period of the signal X [n] is _____________
a) 16 π
b) 16(π+1)
c) 8
d) Non-periodic

Explanation: We know that for X [n] = X1 [n] × X2 [n] to be periodic, both X1 [n] and X2 [n] should be periodic with finite periods.
Here X2 [n] = cos ($$\frac{πn}{8}$$), is periodic with fundamental period as $$\frac{8}{n}$$
But X1 [n] = cos ($$\frac{n}{8}$$) is non periodic.
∴ X [n] is a non-periodic signal.

15. A discrete time signal is as given below
$$X [n] = cos (\frac{πn}{2}) – sin (\frac{πn}{8}) + 3 cos (\frac{πn}{4} + \frac{π}{3})$$
The period of the signal X [n] is _____________
a) 16
b) 4
c) 2
d) Non-periodic