250+ TOP MCQs on Periodic and Non-Periodic Signals and Answers

Signals & Systems Multiple Choice Questions 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

Answer: C
Clarification: 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

Answer: B
Clarification: 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) = e^jt is _________________
A. Sin t
B. Cos t
C. Sinh t
D. Cosh t

Answer: B
Clarification: Let X_e (t) represents the even component of X (t)
Now, X_e (t) = (frac{1}{2})[X (t) + X (-t)]
= (frac{1}{2})[e^jt + e^-jt]
= cos t.

4. The odd component of the signal X (t) = e^jt is _______________
A. Sin t
B. Cos t
C. Sinh t
D. Cosh t

Answer: A
Clarification: Let X_o (t) represents the odd component of X (t)
Now, X_o (t) = (frac{1}{2})[X (t) – X (-t)]
= (frac{1}{2})[e^jt + 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

Answer: A
Clarification: 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

Answer: C
Clarification: 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

Answer: D
Clarification: 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

Answer: D
Clarification: 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

Answer: B
Clarification: 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

Answer: C
Clarification: 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

Answer: A
Clarification: 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})

Answer: D
Clarification: E = 2(int_0^5 x^2 (t) ,dt)
= 2 (int_0^4 1^1 ,dt + 2int_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

Answer: A
Clarification: 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

Answer: D
Clarification: 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

Answer: A
Clarification: Given that, N₁ = 4, N₂ = 16, N₃ = 8
We know that period of X [n] (say N) = LCM (N₁, N₂, N₃)
∴ Period of X [n] = LCM (4, 16, 8) = 16.