Unit 6 - Notes

MTH165

Unit 6: Fourier Series

1. Introduction and Euler's Formulae

1.1 Concept of Periodic Functions

A function $f(x)$ is said to be periodic with period $T$ if $f(x + T) = f(x)$ for all real $x$ . The Fourier series is an expansion of a periodic function into an infinite sum of sine and cosine terms. This tool is fundamental in engineering for signal analysis, vibration analysis, and solving partial differential equations.

1.2 Fourier Series Definition

If a function $f(x)$ is defined in the interval $(c, c+2\pi)$ , the Fourier series expansion is given by:

f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} a_n \cos(nx) + \sum_{n=1}^{\infty} b_n \sin(nx)

Where $a_0$ , $a_n$ , and $b_n$ are constants known as Fourier Coefficients.

1.3 Euler's Formulae for Coefficients

To determine the coefficients, we utilize the orthogonality properties of sine and cosine functions. For the interval $(c, c+2\pi)$ (often taken as $(-\pi, \pi)$ or $(0, 2\pi)$ ):

1. The Constant Term ( $a_0$ ):

a_0 = \frac{1}{\pi} \int_{c}^{c+2\pi} f(x) \, dx

(Note: The first term in the series is $\frac{a_0}{2}$ to make this formula consistent with the formula for $a_n$ when $n=0$ .)

2. The Cosine Coefficient ( $a_n$ ):

a_n = \frac{1}{\pi} \int_{c}^{c+2\pi} f(x) \cos(nx) \, dx

3. The Sine Coefficient ( $b_n$ ):

b_n = \frac{1}{\pi} \int_{c}^{c+2\pi} f(x) \sin(nx) \, dx

2. Conditions for a Fourier Expansion and Discontinuities

2.1 Dirichlet's Conditions

For a function $f(x)$ to be expanded as a Fourier series, it does not need to be continuous everywhere, but it must satisfy Dirichlet's Conditions. Sufficient conditions for the existence of a Fourier series in an interval $(c, c+2\pi)$ are:

Single-Valued and Finite: $f(x)$ is single-valued, finite, and periodic.
Piecewise Continuous: $f(x)$ has a finite number of finite discontinuities in any one period. Infinite discontinuities are not allowed.
Finite Extrema: $f(x)$ has a finite number of maxima and minima in any one period.

2.2 Points of Discontinuity

If $f(x)$ satisfies Dirichlet's conditions, the Fourier series converges:

To $f(x)$ at all points where $f(x)$ is continuous.
To the average of the left-hand and right-hand limits at points where $f(x)$ is discontinuous.

If $x_0$ is a point of finite discontinuity, the sum of the series at $x_0$ is:

S(x_0) = \frac{1}{2} \left[ \lim_{h \to 0} f(x_0 - h) + \lim_{h \to 0} f(x_0 + h) \right]

3. Change of Interval (Arbitrary Period)

In engineering problems, the period is rarely exactly $2\pi$ . Consider a function $f(x)$ with an arbitrary period $T = 2L$ , defined in the interval $(c, c+2L)$ (e.g., $[-L, L]$ or $[0, 2L]$ ).

The variable is transformed using the substitution $z = \frac{n\pi x}{L}$ .

3.1 General Formula

f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} a_n \cos\left(\frac{n\pi x}{L}\right) + \sum_{n=1}^{\infty} b_n \sin\left(\frac{n\pi x}{L}\right)

3.2 Generalized Euler Coefficients

The limits of integration change from $\pi$ to $L$ :

$a_0 = \frac{1}{L} \int_{c}^{c+2L} f(x) \, dx$
$a_n = \frac{1}{L} \int_{c}^{c+2L} f(x) \cos\left(\frac{n\pi x}{L}\right) \, dx$
$b_n = \frac{1}{L} \int_{c}^{c+2L} f(x) \sin\left(\frac{n\pi x}{L}\right) \, dx$

Common Intervals:

Interval $[-L, L]$ : Integration runs from $-L$ to $L$ .
Interval $[0, 2L]$ : Integration runs from $0$ to $2L$ .

4. Even and Odd Functions

Recognizing symmetry significantly reduces the workload by eliminating half of the coefficients.

4.1 Even Functions

A function is Even if $f(-x) = f(x)$ (Symmetric about the y-axis).

Property: $\int_{-L}^{L} f(x) \, dx = 2 \int_{0}^{L} f(x) \, dx$ .
Fourier Series: Contains only cosine terms (and constant).
Coefficients:
- $b_n = 0$ (Sine terms vanish).
- $a_0 = \frac{2}{L} \int_{0}^{L} f(x) \, dx$ .
- $a_n = \frac{2}{L} \int_{0}^{L} f(x) \cos\left(\frac{n\pi x}{L}\right) \, dx$ .

4.2 Odd Functions

A function is Odd if $f(-x) = -f(x)$ (Symmetric about the origin).

Property: $\int_{-L}^{L} f(x) \, dx = 0$ .
Fourier Series: Contains only sine terms.
Coefficients:
- $a_0 = 0$ .
- $a_n = 0$ (Cosine terms vanish).
- $b_n = \frac{2}{L} \int_{0}^{L} f(x) \sin\left(\frac{n\pi x}{L}\right) \, dx$ .

5. Half Range Series

Sometimes a function is defined only over a range $[0, L]$ , which represents half the physical period. We can extend this function to the range $[-L, L]$ to create a full Fourier series. The manner of extension determines the type of series.

5.1 Half Range Cosine Series

We extend the function as an Even function into $[-L, 0]$ .

Because it is even, $b_n = 0$ .
Expansion:
$f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} a_n \cos\left(\frac{n\pi x}{L}\right)$
Coefficients (integrated over $0$ to $L$ ):
$a_0 = \frac{2}{L} \int_{0}^{L} f(x) \, dx$
$a_n = \frac{2}{L} \int_{0}^{L} f(x) \cos\left(\frac{n\pi x}{L}\right) \, dx$

5.2 Half Range Sine Series

We extend the function as an Odd function into $[-L, 0]$ .

Because it is odd, $a_0 = 0$ and $a_n = 0$ .
Expansion:
$f(x) = \sum_{n=1}^{\infty} b_n \sin\left(\frac{n\pi x}{L}\right)$
Coefficients (integrated over $0$ to $L$ ):
$b_n = \frac{2}{L} \int_{0}^{L} f(x) \sin\left(\frac{n\pi x}{L}\right) \, dx$

6. Parseval's Formula

Parseval's identity relates the average energy (or power) of the function in the time/space domain to the sum of the squares of its Fourier coefficients in the frequency domain.

For a function $f(x)$ defined in $(-L, L)$ :

\frac{1}{L} \int_{-L}^{L} [f(x)]^2 \, dx = \frac{a_0^2}{2} + \sum_{n=1}^{\infty} (a_n^2 + b_n^2)

Significance

Root Mean Square (RMS): The LHS relates to the square of the RMS value of the function.
Summation of Series: This formula is frequently used to find the sum of infinite numerical series (e.g., $\sum \frac{1}{n^2}$ ) by evaluating the integral of the squared function.

7. Complex Form of Fourier Series

The sine and cosine form is convenient for real calculus, but the complex exponential form is standard in electrical engineering and signal processing (e.g., FFT).

Using Euler's identity $e^{i\theta} = \cos\theta + i\sin\theta$ , we can rewrite the series.

7.1 The Expansion

For a function $f(x)$ defined in $(-L, L)$ with period $2L$ :

f(x) = \sum_{n=-\infty}^{\infty} c_n e^{i \frac{n\pi x}{L}}

Where $i = \sqrt{-1}$ . Note that the summation runs from $-\infty$ to $+\infty$ .

7.2 Complex Coefficient ( $c_n$ )

c_n = \frac{1}{2L} \int_{-L}^{L} f(x) e^{-i \frac{n\pi x}{L}} \, dx

7.3 Relationship with Real Coefficients

$c_0 = \frac{a_0}{2}$
$c_n = \frac{a_n - i b_n}{2}$ (for $n > 0$ )
$c_{-n} = \frac{a_n + i b_n}{2}$ (Complex conjugate of $c_n$ )
Amplitude Spectrum: $|c_n| = \frac{1}{2}\sqrt{a_n^2 + b_n^2}$