7.2 MCQs-Fourier Series

Fourier Series

Introduction and Basic Concepts

1. A Fourier series represents a periodic function as:

1. A sum of exponential functions

2. A sum of sines and cosines

3. A Taylor series expansion

4. A Laurent series expansion

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Answer: 2. A sum of sines and cosines

Explanation:

• A Fourier series decomposes a periodic function into an infinite sum of sine and cosine functions.

• The general form is: $f(x) = a_0 + \sum_{n=1}^{\infty} [a_n \cos(n\omega x) + b_n \sin(n\omega x)]$

• This representation is particularly useful for analyzing periodic signals in physics and engineering.

2. The Dirichlet conditions ensure:

1. The function is always continuous

2. The Fourier series converges to the function

3. The function is differentiable everywhere

4. The Fourier coefficients are zero

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Answer: 2. The Fourier series converges to the function

Explanation:

• Dirichlet conditions are sufficient conditions for a periodic function to have a convergent Fourier series.

• The three main conditions are:

  1. The function must be absolutely integrable over one period

  2. The function must have a finite number of maxima and minima in one period

  3. The function must have a finite number of discontinuities in one period

• If these conditions are satisfied, the Fourier series converges to the function at points of continuity and to the average of left and right limits at points of discontinuity.

3. The Fourier series of an even function contains:

1. Only sine terms

2. Only cosine terms

3. Both sine and cosine terms

4. Only constant term

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Answer: 2. Only cosine terms

Explanation:

• An even function satisfies $f(-x) = f(x)$ for all x in the domain.

• For even functions:

  • The sine terms vanish because sine is an odd function: $b_n = 0$ for all n

  • Only cosine terms (which are even functions) and the constant term remain

• This is known as a Fourier cosine series.

4. The Fourier series of an odd function contains:

1. Only sine terms

2. Only cosine terms

3. Both sine and cosine terms

4. Only constant term

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Answer: 1. Only sine terms

Explanation:

• An odd function satisfies $f(-x) = -f(x)$ for all x in the domain.

• For odd functions:

  • The cosine terms vanish because cosine is an even function: $a_n = 0$ for all n (including $a_0$)

  • Only sine terms (which are odd functions) remain

• This is known as a Fourier sine series.

Fourier Coefficients

5. The formula for $a_0$ in the Fourier series is:

1. $\frac{1}{T} \int_{-T/2}^{T/2} f(x) \cos(n\omega x) dx$

2. $\frac{1}{T} \int_{-T/2}^{T/2} f(x) \sin(n\omega x) dx$

3. $\frac{1}{T} \int_{-T/2}^{T/2} f(x) dx$

4. $\frac{2}{T} \int_{-T/2}^{T/2} f(x) dx$

Show me the answer

Answer: 3. $\frac{1}{T} \int_{-T/2}^{T/2} f(x) dx$

Explanation:

• $a_0$ represents the average value of the function over one period.

• The formula is: $a_0 = \frac{1}{T} \int_{-T/2}^{T/2} f(x) dx$

• Sometimes $a_0$ is written as $\frac{1}{2}a_0$ in the series, in which case the formula becomes $a_0 = \frac{2}{T} \int_{-T/2}^{T/2} f(x) dx$

• It's important to check the specific convention used in the Fourier series representation.

6. For a function with period $2\pi$, the Fourier coefficient $a_n$ is given by:

1. $\frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \cos(nx) dx$

2. $\frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \sin(nx) dx$

3. $\frac{1}{2\pi} \int_{-\pi}^{\pi} f(x) dx$

4. $\frac{2}{\pi} \int_{0}^{\pi} f(x) \cos(nx) dx$

Show me the answer

Answer: 1. $\frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \cos(nx) dx$

Explanation:

• For a function with period $2\pi$, the formulas simplify to: $a_0 = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(x) dx$ $a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \cos(nx) dx$ $b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \sin(nx) dx$

• These formulas come from the orthogonality properties of sine and cosine functions over $[-\pi, \pi]$.

7. The orthogonality relation for cosine functions is:

1. $\int_{-\pi}^{\pi} \cos(mx) \cos(nx) dx = \pi \delta_{mn}$

2. $\int_{-\pi}^{\pi} \cos(mx) \cos(nx) dx = 0$

3. $\int_{-\pi}^{\pi} \cos(mx) \cos(nx) dx = 2\pi \delta_{mn}$

4. $\int_{-\pi}^{\pi} \cos(mx) \cos(nx) dx = \begin{cases} 0 & \text{if } m \neq n \\ \pi & \text{if } m = n \neq 0 \\ 2\pi & \text{if } m = n = 0 \end{cases}$

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Answer: 4. $\int_{-\pi}^{\pi} \cos(mx) \cos(nx) dx = \begin{cases} 0 & \text{if } m \neq n \\ \pi & \text{if } m = n \neq 0 \\ 2\pi & \text{if } m = n = 0 \end{cases}$

Explanation:

• This orthogonality relation is crucial for deriving the Fourier coefficients.

• The complete set of orthogonality relations over $[-\pi, \pi]$ are:

  • $\int_{-\pi}^{\pi} \cos(mx) \cos(nx) dx = \begin{cases} 0 & m \neq n \\ \pi & m = n \neq 0 \\ 2\pi & m = n = 0 \end{cases}$

  • $\int_{-\pi}^{\pi} \sin(mx) \sin(nx) dx = \begin{cases} 0 & m \neq n \\ \pi & m = n \neq 0 \\ 0 & m = n = 0 \end{cases}$

  • $\int_{-\pi}^{\pi} \sin(mx) \cos(nx) dx = 0$ for all m, n

Complex Fourier Series

8. The complex form of Fourier series uses:

1. Sine and cosine functions

2. Exponential functions with imaginary exponents

3. Hyperbolic functions

4. Bessel functions

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Answer: 2. Exponential functions with imaginary exponents

Explanation:

• The complex Fourier series uses Euler's formula: $e^{i\theta} = \cos\theta + i\sin\theta$

• The series is written as: $f(x) = \sum_{n=-\infty}^{\infty} c_n e^{in\omega x}$

• The coefficients are given by: $c_n = \frac{1}{T} \int_{-T/2}^{T/2} f(x) e^{-in\omega x} dx$

• This form is often more compact and easier to manipulate mathematically.

9. The relationship between complex coefficients $c_n$ and real coefficients $a_n$, $b_n$ is:

1. $c_n = a_n + ib_n$

2. $c_n = \frac{1}{2}(a_n - ib_n)$

3. $c_n = a_n - ib_n$

4. $c_0 = a_0, \ c_n = \frac{1}{2}(a_n - ib_n), \ c_{-n} = \frac{1}{2}(a_n + ib_n)$

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Answer: 4. $c_0 = a_0, \ c_n = \frac{1}{2}(a_n - ib_n), \ c_{-n} = \frac{1}{2}(a_n + ib_n)$

Explanation:

• The relationships come from Euler's formulas: $\cos\theta = \frac{e^{i\theta} + e^{-i\theta}}{2}$ $\sin\theta = \frac{e^{i\theta} - e^{-i\theta}}{2i}$

• Specifically:

  • $c_0 = a_0$

  • For n > 0: $c_n = \frac{1}{2}(a_n - ib_n)$

  • $c_{-n} = \frac{1}{2}(a_n + ib_n)$

• Conversely: $a_n = c_n + c_{-n}$ and $b_n = i(c_n - c_{-n})$

Parseval's Theorem

10. Parseval's theorem for Fourier series states:

1. The integral of the function equals the sum of Fourier coefficients

2. The average power equals the sum of squares of Fourier coefficients

3. The function can be reconstructed from its Fourier coefficients

4. The Fourier series converges uniformly

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Answer: 2. The average power equals the sum of squares of Fourier coefficients

Explanation:

• Parseval's theorem relates the power (or energy) of a periodic signal in the time domain to the power in the frequency domain.

• For real Fourier series: $\frac{1}{T} \int_{-T/2}^{T/2} |f(x)|^2 dx = a_0^2 + \frac{1}{2} \sum_{n=1}^{\infty} (a_n^2 + b_n^2)$

• For complex Fourier series: $\frac{1}{T} \int_{-T/2}^{T/2} |f(x)|^2 dx = \sum_{n=-\infty}^{\infty} |c_n|^2$

• This theorem is important in signal processing and physics.

Convergence and Applications

11. The Gibbs phenomenon refers to:

1. Slow convergence of Fourier series

2. Overshoot near discontinuities in Fourier series approximation

3. Divergence of Fourier series at discontinuities

4. Aliasing in Fourier reconstruction

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Answer: 2. Overshoot near discontinuities in Fourier series approximation

Explanation:

• The Gibbs phenomenon occurs when approximating a function with a jump discontinuity using a finite number of terms in its Fourier series.

• The partial sums overshoot the function value near the discontinuity by about 9% of the jump height.

• This overshoot does not disappear as more terms are added; instead, it moves closer to the discontinuity.

• It was first observed by Henry Wilbraham and later analyzed by Josiah Willard Gibbs.

12. Fourier series are particularly useful for solving:

1. Algebraic equations

2. Ordinary differential equations with constant coefficients

3. Partial differential equations with boundary conditions

4. Integral equations of the first kind

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Answer: 3. Partial differential equations with boundary conditions

Explanation:

• Fourier series are essential tools for solving partial differential equations (PDEs) such as:

  • Heat equation (diffusion equation)

  • Wave equation

  • Laplace's equation

• They allow separation of variables and transformation of PDEs into simpler ordinary differential equations.

• Boundary conditions determine whether to use sine series, cosine series, or full Fourier series.

Half-Range Expansions

13. A half-range Fourier sine expansion is used when:

1. The function is defined on [0, L] and extended as an odd function

2. The function is defined on [0, L] and extended as an even function

3. The function is periodic with period L

4. The function has symmetry about x = L/2

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Answer: 1. The function is defined on [0, L] and extended as an odd function

Explanation:

• Half-range expansions are used when a function is defined only on half the interval [0, L] instead of the full period [-L, L].

• For half-range sine expansion:

  • Extend f(x) as an odd function: $f(-x) = -f(x)$

  • The Fourier series contains only sine terms: $f(x) = \sum_{n=1}^{\infty} b_n \sin\left(\frac{n\pi x}{L}\right)$

  • Coefficients: $b_n = \frac{2}{L} \int_{0}^{L} f(x) \sin\left(\frac{n\pi x}{L}\right) dx$

14. For a function f(x) defined on [0, π], the half-range cosine expansion coefficients are:

1. $a_0 = \frac{1}{\pi} \int_{0}^{\pi} f(x) dx, \ a_n = \frac{2}{\pi} \int_{0}^{\pi} f(x) \cos(nx) dx$

2. $a_0 = \frac{2}{\pi} \int_{0}^{\pi} f(x) dx, \ a_n = \frac{2}{\pi} \int_{0}^{\pi} f(x) \cos(nx) dx$

3. $a_0 = \frac{1}{\pi} \int_{0}^{\pi} f(x) dx, \ a_n = \frac{1}{\pi} \int_{0}^{\pi} f(x) \cos(nx) dx$

4. $b_n = \frac{2}{\pi} \int_{0}^{\pi} f(x) \sin(nx) dx$

Show me the answer

Answer: 2. $a_0 = \frac{2}{\pi} \int_{0}^{\pi} f(x) dx, \ a_n = \frac{2}{\pi} \int_{0}^{\pi} f(x) \cos(nx) dx$

Explanation:

• For half-range cosine expansion on [0, π]:

  • Extend f(x) as an even function about x = 0

  • The series is: $f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} a_n \cos(nx)$

  • Coefficients: $a_0 = \frac{2}{\pi} \int_{0}^{\pi} f(x) dx$

  • $a_n = \frac{2}{\pi} \int_{0}^{\pi} f(x) \cos(nx) dx$

• Note the factor of 2 in the formulas compared to full-range expansions.

Fourier Series of Common Functions

15. The Fourier series of f(x) = x on (-π, π) contains:

1. Only cosine terms

2. Only sine terms

3. Both sine and cosine terms

4. Only constant term

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Answer: 2. Only sine terms

Explanation:

• f(x) = x is an odd function: f(-x) = -x = -f(x)

• Therefore, its Fourier series contains only sine terms.

• The Fourier series is: $x = 2 \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} \sin(nx)$

• This series converges to x for -π < x < π, and to 0 at x = ±π (by Dirichlet conditions).

16. For the square wave function defined by f(x) = 1 for 0 < x < π and f(x) = -1 for -π < x < 0, the Fourier series is:

1. $\frac{4}{\pi} \sum_{n=1}^{\infty} \frac{1}{n} \sin(nx)$

2. $\frac{4}{\pi} \sum_{n=1}^{\infty} \frac{1}{2n-1} \sin((2n-1)x)$

3. $\frac{2}{\pi} \sum_{n=1}^{\infty} \frac{1}{n} \sin(nx)$

4. $\frac{2}{\pi} \sum_{n=1}^{\infty} \frac{1}{2n-1} \sin((2n-1)x)$

Show me the answer

Answer: 2. $\frac{4}{\pi} \sum_{n=1}^{\infty} \frac{1}{2n-1} \sin((2n-1)x)$

Explanation:

• This square wave is an odd function, so only sine terms appear.

• The Fourier coefficients are: $b_n = \frac{2}{\pi} \int_{0}^{\pi} \sin(nx) dx = \frac{2}{n\pi} [1 - (-1)^n]$

• When n is even, $b_n = 0$

• When n is odd (n = 2k-1), $b_{2k-1} = \frac{4}{(2k-1)\pi}$

• Thus: $f(x) = \frac{4}{\pi} \sum_{k=1}^{\infty} \frac{1}{2k-1} \sin((2k-1)x)$

Differentiation and Integration

17. If a Fourier series converges uniformly, then:

1. The derivative of the series equals the series of derivatives

2. The integral of the series equals the series of integrals

3. Both 1 and 2

4. Neither 1 nor 2

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Answer: 3. Both 1 and 2

Explanation:

• For uniformly convergent Fourier series:

  • Term-by-term differentiation is valid if the differentiated series also converges uniformly

  • Term-by-term integration is always valid for Fourier series, even without uniform convergence

• However, differentiation is more restrictive than integration:

  • The original function must be continuous and have a piecewise continuous derivative

  • Integration can be performed term-by-term on any Fourier series, regardless of convergence type

Frequency Domain Concepts

18. The fundamental frequency in a Fourier series with period T is:

1. $\frac{1}{T}$

2. $\frac{2\pi}{T}$

3. $T$

4. $\frac{T}{2\pi}$

Show me the answer

Answer: 2. $\frac{2\pi}{T}$

Explanation:

• The fundamental angular frequency is $\omega = \frac{2\pi}{T}$

• The fundamental frequency in Hz is $f = \frac{1}{T}$

• Higher harmonics have frequencies that are integer multiples of the fundamental frequency: $n\omega$ or $nf$

• In the Fourier series: $f(x) = a_0 + \sum_{n=1}^{\infty} [a_n \cos(n\omega x) + b_n \sin(n\omega x)]$

Symmetry Properties

19. If f(x) has half-wave symmetry, meaning f(x + T/2) = -f(x), then:

1. Only even harmonics are present

2. Only odd harmonics are present

3. All harmonics are present

4. No harmonics are present

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Answer: 2. Only odd harmonics are present

Explanation:

• Half-wave symmetry: $f(x + T/2) = -f(x)$

• For such functions:

  • All even harmonics (n = 2, 4, 6, ...) vanish

  • Only odd harmonics (n = 1, 3, 5, ...) are present

• Square waves and triangular waves typically exhibit half-wave symmetry

• This property can simplify Fourier coefficient calculations

Fourier Series vs Fourier Transform

20. The main difference between Fourier series and Fourier transform is:

1. Fourier series is for periodic functions, Fourier transform is for aperiodic functions

2. Fourier series uses discrete frequencies, Fourier transform uses continuous frequencies

3. Both 1 and 2

4. Fourier series is more accurate than Fourier transform

Show me the answer

Answer: 3. Both 1 and 2

Explanation:

• Fourier series:

  • For periodic functions

  • Discrete frequency spectrum (harmonics at nω)

  • Time-domain representation is periodic

• Fourier transform:

  • For aperiodic (or periodic treated over infinite interval) functions

  • Continuous frequency spectrum

  • Time-domain representation is non-periodic

• Fourier series can be seen as a special case of Fourier transform for periodic functions, where the transform becomes a series of delta functions at harmonic frequencies.