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Unit 6 - Practice Quiz

MTH165

1 Which of the following represents the trigonometric Fourier series of a periodic function defined in the interval ?

A.
B.
C.
D.

2 For a function defined in , Euler's formula for the coefficient () is given by:

A.
B.
C.
D.

3 For a function defined in , Euler's formula for the coefficient is given by:

A.
B.
C.
D.

4 If the Fourier series is expressed as , what is the formula for in the interval ?

A.
B.
C.
D.

5 Which of the following is NOT one of Dirichlet's conditions for the existence of a Fourier series for ?

A. must be periodic, single-valued, and finite.
B. must have a finite number of discontinuities in any one period.
C. must be differentiable everywhere in the interval.
D. must have a finite number of maxima and minima in any one period.

6 If a function satisfies Dirichlet's conditions, at a point of finite discontinuity , the Fourier series converges to:

A.
B.
C.
D.

7 The value of the integral (where are integers) is:

A. if
B. $0$ for all integer
C. if
D. $1$

8 For an even function defined in , which Fourier coefficients are always zero?

A.
B.
C.
D. None of the coefficients

9 If is an odd function in , then is given by:

A.
B.
C.
D.

10 Which of the following functions is an Even function?

A.
B.
C.
D.

11 In the Fourier expansion of in , the value of is:

A.
B.
C. $0$
D.

12 For a function defined in the interval , the coefficient is calculated as:

A.
B.
C.
D.

13 What is the Fourier Series expansion of in the Half Range Cosine Series for ?

A.
B.
C.
D.

14 To expand as a Half Range Sine Series in , we extend the function to the interval as:

A. An even function
B. An odd function
C. A periodic function
D. A constant function

15 The coefficient in a Half Range Sine Series for in is:

A.
B.
C.
D.

16 Parseval's formula for a function with period in states that:

A.
B.
C.
D.

17 In the Complex form of Fourier Series defined in , the coefficient is given by:

A.
B.
C.
D.

18 The relationship between the complex coefficient and the real coefficients (for ) is:

A.
B.
C.
D.

19 For a real-valued function , the complex coefficients satisfy the property:

A. (Complex Conjugate)
B.
C.
D.

20 The value of where is an integer is:

A.
B.
C.
D.

21 The value of where is an integer is:

A.
B.
C.
D.

22 If in , which coefficients are zero?

A. and
B. only
C. only
D. No coefficients are zero

23 What is the period of the function ?

A.
B.
C.
D.

24 In the Fourier series of the square wave defined by for and for , the value of is:

A.
B.
C.
D.

25 Which term represents the 'fundamental harmonic' in a Fourier series?

A. The term with
B. The term
C. The term with
D. The term with

26 Parseval's identity is useful for:

A. Finding the sum of infinite series involving squares of integers.
B. Finding the phase angle of the harmonics.
C. Determining if a function is even or odd.
D. Calculating the derivative of .

27 If in , the value of is:

A.
B.
C.
D.

28 The complex coefficient corresponds to:

A. The average value of the function over the period.
B. The amplitude of the fundamental frequency.
C. Zero always.
D. The value of the function at .

29 If the interval of a function is changed from to , the basis functions change from to:

A.
B.
C.
D.

30 Which of the following conditions ensures that the Dirichlet's integral converges?

A. The integral of over the period is finite (Absolute Integrability).
B. The function is continuous everywhere.
C. The function is differentiable everywhere.
D. The function is strictly increasing.

31 For a function expanded in a Fourier series, the term represents:

A. The amplitude of the -th harmonic.
B. The phase angle of the -th harmonic.
C. The power of the -th harmonic.
D. The frequency of the -th harmonic.

32 In the Fourier series of in , the coefficients are:

A.
B.
C.
D.

33 What happens to the Fourier series of a discontinuous function near the discontinuity?

A. Gibbs Phenomenon (overshoot)
B. It becomes infinite.
C. It becomes zero.
D. It becomes a straight line.

34 For the Half Range Sine Series of in , the value of is:

A.
B.
C.
D.

35 The constant term in the Fourier Series of in is:

A.
B.
C.
D.

36 If is periodic with period , then equals:

A.
B.
C.
D.

37 The complex form of the Fourier series is often preferred in signal processing because:

A. It simplifies algebraic manipulations involving derivatives and integrals.
B. It uses only real numbers.
C. It eliminates the DC component.
D. It only works for even functions.

38 When expanding as a half-range cosine series in , the period of the extended function is:

A.
B.
C.
D.

39 The value of for is:

A.
B.
C.
D.

40 For a function defined in , the coefficient is:

A.
B.
C.
D.

41 If is even, then is:

A. Odd
B. Even
C. Periodic but neither even nor odd
D. Constant

42 Identify the relation used to derive the Fourier coefficients:

A. Orthogonality of trigonometric functions.
B. Pythagorean theorem.
C. Taylor series expansion.
D. Laplace transform.

43 Which series contains only terms with odd harmonics ()?

A. Functions with Half-wave symmetry: .
B. Even functions.
C. Odd functions.
D. Functions with period .

44 The RMS (Root Mean Square) value of a function over is related to Parseval's theorem by:

A.
B.
C.
D.

45 In the Fourier series of in , the function is:

A. Neither even nor odd
B. Even
C. Odd
D. Zero

46 If is the complex Fourier coefficient, then represents:

A. The power spectrum at frequency .
B. The phase spectrum.
C. The time domain signal.
D. The derivative of the signal.

47 For a half-range cosine series of (constant) in :

A. , all other .
B. for all .
C. for all .
D. All coefficients are non-zero.

48 The sum of the series derived from the Fourier series of a square wave is:

A.
B.
C.
D.

49 If is continuous and piecewise smooth, the Fourier series converges:

A. Uniformly to .
B. Only at .
C. To infinity.
D. Pointwise but not uniformly.

50 In the expansion of in , the value of is:

A.
B.
C.
D.