Fourier Series | Sahithyan's Notes

Used to decompose periodic non-sinusoidal waveforms into a sum of sinusoidal ones.

Suppose $f(x)$ is a $2\pi$ -periodic function.

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

\int_{\tau}^{\tau + T} \cos(nx)\, \text{d}x = \int_{\tau}^{\tau + T} \sin(nx)\, \text{d}x = 0

\int_{\tau}^{\tau + T} \sin(nx) \cos(mx) \, \text{d}t = 0

\int_{\tau}^{\tau + T} \sin(nx) \sin(mx) \, \text{d}x = \begin{cases} \frac{T}{2} & \text{if } n = m \\ 0 & \text{otherwise} \end{cases}

\int_{\tau}^{\tau + T} \cos(nx) \cos(mx) \, \text{d}x = \begin{cases} \frac{T}{2} & \text{if } n = m \\ 0 & \text{otherwise} \end{cases}

Unknown Coefficients

$\forall n \geq 0$ .

a_n = \frac{2}{T} \int_{\tau}^{\tau + T} f(x)\cos(nx) \, \text{d}x

b_n = \frac{2}{T} \int_{\tau}^{\tau + T} f(x)\sin(nx)\, \text{d}x

$\cfrac{a_0}{2}$ is the DC offset.

For any waveform, subtracting the DC offset results in a symmetrical waveform.

When a wave is symmetric about the y-axis.

f(x) = f(-t)

The fourier series of an even waveform contains only cosine terms.

a_n = \frac{4}{T} \int_{\tau}^{\tau + T/2} f(x)\cos(nx) \, \text{d}x \;\;\text{and}\;\; b_n = 0

When a wave is symmetric about the origin.

f(x) = -f(-t)

The fourier series of an odd waveform contains only sine terms.

a_0 = a_n = 0 \;\;\text{and}\;\; b_n = \frac{4}{T} \int_{\tau}^{\tau + T/2} f(x)\sin(nx) \, \text{d}x

When a wave repeats itself with a reversal of sign after half a period.

f(x) = -f\left(t + \frac{T}{2}\right) = -f\left(t - \frac{T}{2}\right)

The coefficients can be found by:

a_n = \begin{cases} \frac{4}{T} \int_{\tau}^{\tau + T/2} f(x)\cos(nx) \, \text{d}x & \text{if } n \text{ is odd} \\ 0 & \text{if } n \text{ is even} \end{cases}

b_n = \begin{cases} \frac{4}{T} \int_{\tau}^{\tau + T/2} f(x)\sin(nx) \, \text{d}x & \text{if } n \text{ is odd} \\ 0 & \text{if } n \text{ is even} \end{cases}

Half-wave symmetry can co-exist with odd symmetry or even symmetry. In that case:

a_n = \begin{cases} \frac{8}{T} \int_{\tau}^{\tau + T/4} f(x)\cos(nx) \, \text{d}x & \text{if } n \text{ is odd} \\ 0 & \text{if } n \text{ is even} \end{cases}

b_n = \begin{cases} \frac{8}{T} \int_{\tau}^{\tau + T/4} f(x)\sin(nx) \, \text{d}x & \text{if } n \text{ is odd} \\ 0 & \text{if } n \text{ is even} \end{cases}

Sufficient conditions for a real-valued, periodic function $f$ to be equal to its Fourier series at a point of continuity.

Suppose $f$ is a periodic function with period $2L$ . If:

Then:

\frac{f(c^-)+f(c^+)}{2}

Suppose:

f(x) = \begin{cases} \rho(x) & 0\lt x \lt c \\ \theta(x) & c \lt x \lt 2\pi \end{cases}

then compute coefficients by splitting integrals:

\begin{equation} \nonumber \begin{split} a_0 & =\frac{1}{\pi}\left[\int_0^cρ(x)\,dx+\int_c^{2\pi}θ(x)\,dx\right] \\ a_n & =\frac{1}{\pi}\left[\int_0^cρ(x)\cos nx\,dx+\int_c^{2\pi}θ(x)\cos nx\,dx\right] \\ b_n & =\frac{1}{\pi}\left[\int_0^cρ(x)\sin nx\,dx+\int_c^{2\pi}θ(x)\sin nx\,dx\right] \end{split} \end{equation}

Used when a function is defined only on (0,π). Extend the function to (-π,π) artificially.

The function is extended as an even function.

\begin{equation} \nonumber \begin{split} a_0 & =\frac{2}{\pi}\int_0^\pi f(x)\,dx \\ a_n & =\frac{2}{\pi}\int_0^\pi f(x)\cos nx\,dx \end{split} \end{equation}

And $b_n = 0$ .

The function is extended as an odd function.

b_n=\frac{2}{\pi}\int_0^\pi f(x)\sin nx\,dx

And $a_0 = a_n = 0$ .

For period $2l$ :

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

Coefficients:

a_0=\frac{1}{l}\int_{-l}^{l}f(x)\,dx

a_n=\frac{1}{l}\int_{-l}^{l}f(x)\cos\left(\frac{n\pi x}{l}\right)\,dx

b_n=\frac{1}{l}\int_{-l}^{l}f(x)\sin\left(\frac{n\pi x}{l}\right)\,dx

A method to find higher order algebraic series, that is, for $n>2$ :

\sum_{i=1}^\infty \frac{1}{i^n}

For period $2c$ :

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

For period 2π:

f(x)=\sum_{n=-\infty}^{\infty} c_n e^{-inx}

where

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

Here:

General period $2l$ also included.