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Test: Fourier Series- 2


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10 Questions MCQ Test Signals and Systems | Test: Fourier Series- 2

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Test: Fourier Series- 2 - Question 1

Value of bn for the periodic function f with period 2π defined as follows :

Select one:

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

The function is bounded, integrable and piece wise monotonic on  
Let us determine the Fourier coefficients



The correct answer is: - 1/n, for n even

Test: Fourier Series- 2 - Question 2

For the given periodic function  The coefficient b1 of the continuous Fourier series associated with the given function f(t) can be computed as
Select one:

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

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 correct answer is: –0.7468

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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