MyQuestionIcon

1

You visited us 1 times! Enjoying our articles? Unlock Full Access!

The fourier s...

Question

The fourier series for a periodic signal is given as
$x (t) = cos (1.2 π t) + cos (2 π t) + cos (2.8 π t) .$
The fundamental frequency of the signal is

Open in App

Solution

$Given signal is$
$x (t) = cos (1.2 π t) + cos (2 π t) + cos (2.8 π t) .$
Step-1:
$ω_{1} = 1.2 π$

$\Rightarrow \frac{2 π}{T_{1}} = 1.2 π$

$\Rightarrow T_{1} = \frac{2 π}{1.2 π}$

$= \frac{20}{12} = \frac{5}{3}; ω_{2} = 2 π$

$\Rightarrow \frac{2 π}{T_{2}} = 2 π$

$\Rightarrow T_{2} = 1$

$ω_{3} = 2.8 π$

$\Rightarrow \frac{2 π}{T_{3}} = 2.8 π$

$\Rightarrow T_{3} = \frac{2}{2.8} = \frac{5}{7}$

Step - 2:
$\frac{T_{1}}{T_{2}} = \frac{5}{3}, \frac{T_{1}}{T_{3}} = \frac{7}{3}$

Step-3:
L.C.M. of denominators of step 2 is
L.C.M. = 3.

Step-4:
$T = (L.C.M.) \times T_{1}$

$= 3 \times \frac{5}{3} = 5 sec$

Fundamental period = 5 sec
Fundamental frequency of the signal
$= \frac{1}{T} = \frac{1}{5} = 0.2 H z$

Suggest Corrections

thumbs-up

1

Similar questions

Q. What is the expression for Fourier transform of a periodic signal x(t) with exponential Fourier series coefficient as

C_{n}

?

Q. The exponential Fourier series representation of a continuous - time periodic signal x(t) is defined as

x (t) = \sum_{k = - \infty}^{\infty} a_{k} e^{j k ω_{o}} t

ω_{o}

is the fundamental angular frequency of x(t) and the coefficient of the series are

a_{k}

. The following information is given about x(t) and

a_{k}

.
I. x(t) is real and even, having a fundamental period of 6.
II. The average value of x(t) is 2.
III.

a_{k} = {\begin{matrix} 2, & 1 \leq k \leq 3 0, & k > 3 \end{matrix}

The average power of the signal x(t) (rounded off to one decimal place) is

Q. The Fourier series expansion of a real periodic signal with fundamental frequency

f_{0}

g_{p} (t) = \sum_{n = - \infty}^{\infty} c_{n} e^{j 2 π n f_{0} t}

it is given that

c_{3} = 3 + j 5.

C_{- 3}

Q. Let x(t) be a continuos time periodic signal with fundamental period = 1 seconds. Let

a_{k}

be the complex Fourier series coefficients of x(t), where k is integer valued. Consider the following statements about x(3t) :

I. The complex Fourier series coefficients of x(3t) are

a_{k}

where k is integer valued.

II. The complex Fourier series coefficients of X(3t) are 3

a_{k}

where k is integer valued.

III. The fundamental angular frequency of x(3t) is

6 π

rad/s.

For the three statements above, which one of the following correct ?

Q. Consider a periodic signal x(t) as shown below:

It has a Fourier series representation

x (t) = \infty \sum k = - \infty a_{k} e^{j (2 π / T) k t}

Which one of the following statements is TRUE?

View More

Join BYJU'S Learning Program

similar_icon

Related Videos

lock

Monopolistic Competition

Watch in App

Join BYJU'S Learning Program

CrossIcon