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Test: Fourier Series - Civil Engineering (CE) MCQ


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10 Questions MCQ Test - Test: Fourier Series

Test: Fourier Series for Civil Engineering (CE) 2024 is part of Civil Engineering (CE) preparation. The Test: Fourier Series questions and answers have been prepared according to the Civil Engineering (CE) exam syllabus.The Test: Fourier Series MCQs are made for Civil Engineering (CE) 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Test: Fourier Series below.
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Test: Fourier Series - Question 1

If we use the Fourier transform  to solve the partial differential equation   in the half-plane {(x, y) : -∞ < x < ∞, 0 < y < ∞} the Fourier modes ϕk(y) depend on y as yα and yβ. The values of α and β are  

Detailed Solution for Test: Fourier Series - Question 1

The Fourier transform, 
CALCULATION:
We have;

Now take the derivative of ϕ(x, y) we have;

Again derivative we have;

Now putting these values in equation 1) we have;

Now, by applying Cauchy's differential equation we have;

Again solving we have;
The roots are written as;

Hence option 3) is the correct answer.

Test: Fourier Series - Question 2

What is Fourier series?

Detailed Solution for Test: Fourier Series - Question 2

The Fourier series is the representation of non periodic signals in terms of complex exponentials, or equivalently in terms of sine and cosine waveform leads to Fourier series. In other words, Fourier series is a mathematical tool that allows representation of any periodic wave as a sum of harmonically related sinusoids.

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Test: Fourier Series - Question 3

 Consider  where t ∈ R. Here [t] represents the largest integer less than or equal to t and [t] denotes the samllest integer greater than or equal to t. The coefficient of the second harmonic component of the fourier series representing g(t) is

Test: Fourier Series - Question 4

The Fourier series expansion of x3 in the interval −1 ≤ x < 1 with periodic continuation has

Detailed Solution for Test: Fourier Series - Question 4

f(x) = x3
find f(x) is even or odd
put x = -x
f(-x) = - x3
f(x) = -f(-x) hence it is odd function
for odd function, ao = an = 0 
Fourier Series for odd function has only bn term

Hence only sine terms are left in Fourier expansion of x3
Additional Information
Fourier Series

Test: Fourier Series - Question 5

Who discovered Fourier series?

Detailed Solution for Test: Fourier Series - Question 5

The Fourier series is the representation of non periodic signals in terms of complex exponentials or sine or cosine waveform. This was discovered by Jean Baptiste Joseph Fourier in 18th century.

Test: Fourier Series - Question 6

Let g: [0,∞) → [0,∞) be a function defined by g(x) = x – [x], where [x] represents the integer part of x. (i.e., it is the largest integer which is less than or equal to x). The value of the constant term in the Fourier series expansion of g(x) is

Test: Fourier Series - Question 7

F(t) is a periodic square wave function as shown. It takes only two values, 4 and 0, and stays at each of these values for 1 second before changing. What is the constant term in the Fourier series expansion of F(t)?

Detailed Solution for Test: Fourier Series - Question 7

Fourier Series is defined as

Calculation:
Given:
f(t) is an even periodic function, Since
f(-t) = f(t)
The constant term in the Fourier series is: ao/2

Since the function is not continious, we need to break the integral according to interval
from -1 to 0, f(t) = 0,
from 0 to 1, f(t) = 4

Mistake Points
Avoid using the integral property of even function because the given function f(t) is not continuous. Though the function given is an even periodic function but it is not continuous

Test: Fourier Series - Question 8

What are the conditions called which are required for a signal to fulfil to be represented as Fourier series?

Detailed Solution for Test: Fourier Series - Question 8

When the Dirichlet’s conditions are satisfied, then only for a signal, the fourier series exist. Fourier series is of two types- trigonometric series and exponential series.

Test: Fourier Series - Question 9

Let x(t) be a periodic signal with time period T, Let y(t) = x(t – to) + x(t + to) for some to. The fourier series coefficients of y(t) are denoted by bk. If bk = 0 for all odd K. Then to can be equal to

Test: Fourier Series - Question 10

Choose the condition from below that is not a part of Dirichlet’s conditions?

Detailed Solution for Test: Fourier Series - Question 10

Even if the Fourier series demands periodicity as the major necessity for its formation still it is not a part of Dirichlet’s condition. It is the basic necessity for Fourier series.

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