1. 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.
2. 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.
3. 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.
4. 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.
5.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.
\(P_∞ = lim_{T→∞} \frac{1}{2T} \int_{-T}^T |x(t)|^2 \,dt = ∞.\)
6. 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.
\(P_∞ = lim_{T→∞} \frac{1}{2T} \int_{-T}^T |x(t)|^2 \,dt = 1.\)
7. 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.
8. 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.
9. 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
Explanation: 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.
10. How is the discrete time impulse function defined in terms of the step function?
a) d[n] = u[n+1] – u[n].
b) d[n] = u[n] – u[n-2].
c) d[n] = u[n] – u[n-1].
d) d[n] = u[n+1] – u[n-1].
Explanation: Using the definition of the Heaviside function, we can come to this conclusion.