Question

Determine the periodicity of the following CT signal
X(t) = 2 cos 3t + 3 sin 7t​

Answer 1

Answer:

The fundamental period is the LCM of individual time periods

Step-by-step explanation:

$2\pi \: f = 3$

$f = \frac{3}{2\pi}$

or

$t = \frac{2\pi}{3}$

Similarly time period of second signal is

$t = \frac{2\pi}{7}$

LCM is

$\frac{2\pi}{21}$

Hence fundamental frequency is

$\frac{21}{2\pi}$

Answer 2

The periodicity of X(t) is 2π.

Given:

X(t) = 2 cos 3t + 3 sin 7t​

To Find:

The periodicity of the following CT signal.

Solution:

We are required to find the periodicity of the following CT signal.

X(t) = 2 cos 3t + 3 sin 7t​

where, X₁(t) = 2 cos 3t and X₂(t) = 3 sin 7t​

For X₁(t) = 2 cos 3t  

Period of cos t = 2π

Period of cos at = 2π/a

Here, a = 3

So, the period of cos 3t = 2π/3

For X₂(t) = 3 sin 7t​

The Period of sin t = 2π

Period of sin at = 2π/a

Here, a = 7

So, the period of sin t = 2π/7

∴ Period of X (t) = LCM [Period of X₁ (t), Period of X₂ (t)]

Period of X (t) = LCM[2π/3,2π/7]

Period of X (t) = 2π

Therefore, The periodicity of X(t) is 2π.

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