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Test: Fourier Series- 2 - Electronics and Communication Engineering (ECE) MCQ


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10 Questions MCQ Test - Test: Fourier Series- 2

Test: Fourier Series- 2 for Electronics and Communication Engineering (ECE) 2025 is part of Electronics and Communication Engineering (ECE) preparation. The Test: Fourier Series- 2 questions and answers have been prepared according to the Electronics and Communication Engineering (ECE) exam syllabus.The Test: Fourier Series- 2 MCQs are made for Electronics and Communication Engineering (ECE) 2025 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Test: Fourier Series- 2 below.
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Test: Fourier Series- 2 - Question 1

Detailed Solution for Test: Fourier Series- 2 - Question 1

Given the periodic function f(t) with a period of 2π, defined as:

The Fourier series is given by:

The Fourier coefficients aₙ and bₙ are calculated as follows:

For a₁ (cosine coefficients):

Since f(t) = 0 for π < t < 2π, only the first integral needs to be considered.

For b₁ (sine coefficients):

Again, f(t) = 0 for π < t < 2π, so we only calculate the first integral.

Let's compute these coefficients.

Step 1: Calculate a₁

Using the identity sin(t) cos(t) = (1/2) sin(2t), we can simplify the integral and compute.

Step 2: Calculate b₁

Using the identity sin²(t) = (1 - cos(2t))/2, we can compute the integral for b₁.

The Fourier series coefficients are:

a₁ = 0
b₁ = A/2

Thus, the correct option is:

d) a₁ = 0 ; b₁ = A/2

Test: Fourier Series- 2 - Question 2

Detailed Solution for Test: Fourier Series- 2 - Question 2




Test: Fourier Series- 2 - Question 3

Given the following periodic function, f(t).

f (t) = { t2   for 0 ≤ t ≤ 2  ; 
          -t + 6   for 2 ≤ t ≤ 6



The coefficient a0 of the continuous Fourier series associated with the above given function f(t) can be computed as
Select one:

Detailed Solution for Test: Fourier Series- 2 - Question 3

The coefficient a0 of the continuous Fourier series associated with the given function f(t) can be computed as


The correct answer is: 16/9

Test: Fourier Series- 2 - Question 4

For the given periodic function  with a period T = 6. The Fourier coefficient a1 can be computed as
Select one:

Detailed Solution for Test: Fourier Series- 2 - Question 4

The coefficient a1 of the continuous Fourier series associated with the above given function f(t) can be computed with k = 1 and T = 6 as following :


a1 = –0.9119
The correct answer is: –0.9119

Test: Fourier Series- 2 - Question 5

Sum of the series at  for the periodic function f with period 2π is defined as

Select one:

Detailed Solution for Test: Fourier Series- 2 - Question 5

The function is piece wise monotonic, bounded and integrable on [-π, π]  Let us compute its Fourier coefficients

The function is continuous at all points of [-π, π] except 

which holds at all points with the exception of all discontinuities, 

At  the sum of the series

The correct answer is: 0

Test: Fourier Series- 2 - Question 6

Which of the following is an “even” function of t?
Select one:

Detailed Solution for Test: Fourier Series- 2 - Question 6

Since if we replace “t” by “–t”, then the function value remains the same!
The correct answer is: t2

Test: Fourier Series- 2 - Question 7

A “periodic function” is given by a function which
Select one:

Detailed Solution for Test: Fourier Series- 2 - Question 7

Since the function’s value remains the same value after a period (or multiple periods) has passed!
The correct answer is: satisfies f(t + T) = f(t)

Test: Fourier Series- 2 - Question 8

For the given periodic function  with a period T = 6. The complex form of the Fourier series can be expressed as   The complex coefficient  can be expressed as
Select one:

Detailed Solution for Test: Fourier Series- 2 - Question 8

The coefficient  (corresponding to k = 1) can be expressed  as :

The coefficient b1 of the continuous Fourier series associated with the above given function f(t) can be computed as

since  
and 
Hence 

b1 = –0.7468
The coefficient a1 of the continuous Fourier series associated with the above given function f(t) can be computed with k = 1 and T = 6 as following :


a1 = –0.9119

The correct answer is: –0.4560 + 0.3734i

Test: Fourier Series- 2 - Question 9

The function x2 is periodic with period 2l on the interval [–l, l]. The value of an is given by
Select one:

Detailed Solution for Test: Fourier Series- 2 - Question 9

The substitution transforms the function into a periodic function with period . Moreover it is an even function.
∴ bn = 0, n = 1, 2, 3,.....



The correct answer is:  for n even

Test: Fourier Series- 2 - Question 10

The function x2 extended as an odd function in [–l, l] by redefining it as

sum of series at x = l.
Select one:

Detailed Solution for Test: Fourier Series- 2 - Question 10

Substitution of  transforms it into an odd periodic function on [-π, π],
so that the Fourier coefficients are
an = 0 for n = 0, 1, 2, 3


At x = 0, a point of continuity of the function, the sum of the series is zero, a fact which may be verified directly from series.
At x = l, the sum of series = 
The correct answer is: 0

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