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QUESTION: 1

Value of b_{n} for the periodic function f with period 2π defined as follows :

Select one:

Solution:

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

QUESTION: 2

For the given periodic function The coefficient b_{1} of the continuous Fourier series associated with the given function f(t) can be computed as

Select one:

Solution:

The coefficient b_{1} of the continuous Fourier series associated with the above given function f(t) can be computed as

since and

Hence

b_{1} = –0.7468

The correct answer is: –0.7468

QUESTION: 3

Given the following periodic function, f(t).

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

The coefficient a_{0} of the continuous Fourier series associated with the above given function f(t) can be computed as

Select one:

Solution:

The coefficient a_{0} of the continuous Fourier series associated with the given function f(t) can be computed as

The correct answer is: 16/9

QUESTION: 4

For the given periodic function with a period T = 6. The Fourier coefficient a_{1} can be computed as

Select one:

Solution:

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

a_{1} = –0.9119

The correct answer is: –0.9119

QUESTION: 5

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

Select one:

Solution:

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

QUESTION: 6

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

Select one:

Solution:

Since if we replace “t” by “–t”, then the function value remains the same!

The correct answer is: t^{2}

QUESTION: 7

A “periodic function” is given by a function which

Select one:

Solution:

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)

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:

Solution:

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

The coefficient b_{1} of the continuous Fourier series associated with the above given function f(t) can be computed as

since

and

Hence

b_{1} = –0.7468

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

a_{1} = –0.9119

The correct answer is: –0.4560 + 0.3734i

QUESTION: 9

The function x^{2} is periodic with period 2l on the interval [–l, l]. The value of a_{n} is given by

Select one:

Solution:

The substitution transforms the function into a periodic function with period . Moreover it is an even function.

∴ b_{n} = 0, n = 1, 2, 3,.....

The correct answer is: for n even

QUESTION: 10

The function x^{2} extended as an odd function in [–l, l] by redefining it as

sum of series at x = l.

Select one:

Solution:

Substitution of transforms it into an odd periodic function on [-π, π],

so that the Fourier coefficients are

a_{n} = 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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