2023-12-04 23:21:10 -07:00
#fourier
2023-11-26 18:14:37 -07:00
Remember the heat flow equation? We obtained that it's solution could be expressed in the form:
$$\sum_{n=1}^\infty c_{n}\sin\left( \frac{n\pi x}{L} \right)\quad\text{for}\quad0\leq x\leq L$$
2023-12-04 23:21:10 -07:00
But what is $c_{n}$? They are the coefficients of a Fourier transform. We want to develop a way to compute them.
Let's derive how to compute the coefficients of a Fourier transform. (feel free to skip to the end)
$f(x)=\sum_{n=1}^\infty b_{n}\sin\left( \frac{n\pi x}{L} \right)$ where $L$ is length of the rod
2023-11-26 18:14:37 -07:00
2023-12-04 23:21:10 -07:00
This is a Fourier series: it's a more general form of what we have above.
2023-11-26 18:14:37 -07:00
$f(x)=\frac{a_{0}}{2}+\sum_{n=1}^\infty\left( a_{n}\cos\left( \frac{n\pi x}{L} \right) + b_{n}\sin\left( \frac{n\pi x}{L}\right) \right)$
$x \in [-L,L]$
2023-12-04 23:21:10 -07:00
It converges to $f(x)$ almost everywhere (convergence will be discussed below)
Has a lot of benefits over Taylor series. $f(x)$ doesn't have to be infinitely differentiable (analytic)
$f(x)$ can even have jump discontinuities
Let's assume the equation is true when $x \in [-L,L]$
Integrate both sides, it will tell us the DC offset:
2023-11-26 18:14:37 -07:00
$\int _{-L} ^L f(x) \, dx=\int _ {-L}^L \frac{a_{0}}{2} \, dx+\int _{-L}^L (\text{put summation here}) \, dx$
$\int _{-L}^L \cos\left( \frac{n\pi x}{L} \right) \, dx=\frac{L}{n\pi}\sin\left( \frac{n\pi x}{L} \right)|_ {-L}^L=0$
same for $\int _{-L}^L \sin\left( \frac{n\pi x}{L} \right)\, dx=0$ (it equals 0)
so
$\int _{-L} ^L f(x) \, dx=\int _ {-L}^L \frac{a_{0}}{2} \, dx+\int _{-L}^L0 \, dx$
2023-12-04 23:21:10 -07:00
$\int _{-L} ^L f(x) \, dx=a_ {0}L$
$a_{0}=\frac{1}{L}\int _{-L}^{L} f(x) \, dx$
Now let's multiply both sides by $\cos\left( \frac{m\pi x}{L} \right)$ and integrate both sides, this will tell us the $\cos$ components:
$\int _{-L}^L f(x)\cos\left( \frac{m\pi x}{L} \right)\, dx=\frac{a_ {0}}{2}\cancelto{ 0 }{ \int _{-L}^L \cos\left( \frac{m\pi x}{L} \right) \, dx }+\sum_ {n=1}^\infty\left( a_{n}\int _{-L}^L\cos\left( \frac{n\pi x}{L} \right)\cos\left( \frac{m\pi x}{L} \right) \, dx +b_ {n}\int _{-L}^L \sin\left( \frac{n\pi x}{L} \right)\cos\left( \frac{m\pi x}{L} \right)\right) \, dx$
2023-11-26 18:14:37 -07:00
use trig identities (will be provided on exam):
$\cos(\alpha)\cos(\beta)=\frac{1}{2}(\cos(\alpha-\beta)+\cos(\alpha+\beta))$
$\sin(\alpha)\cos(\beta)=\frac{1}{2}(\sin(\alpha+\beta)+\sin(\alpha-\beta))$
$\sin(\alpha)\sin(\beta)=\frac{1}{2}(\cos(\alpha-\beta)-\cos(\alpha+\beta))$
2023-12-04 23:21:10 -07:00
$\int _{-L}^L \cos \frac{n\pi x}{L}\cos \frac{m\pi x}{L}\, dx=\frac{1}{2}(\int _ {-L}^L \left( \cos(\frac{(n-m)\pi x}{L} )+\cancelto{ 0 }{ \cos(\frac{(n+m)\pi x}{L} })\right)dx$
2023-11-26 18:14:37 -07:00
$= \begin{cases}0 & n\ne m \\L & n=m\end{cases}$
2023-12-04 23:21:10 -07:00
$\int _{-L}^L \sin \frac{n\pi x}{L}\cos \frac{m\pi x}{L} \, dx=\int _ {-L}^L \text{odd}\, dx=0$
so:
$\int _{-L} ^L f(x)\cos\left( \frac{m\pi x}{L} \right)\, dx=a_ {m}L$
Similarly can be done for when multiplying both sides by $\sin\left( \frac{m\pi x}{L} \right)$ and integrating both sides to find the $\sin$ coefficients:
$\int _{-L}^L f(x)\sin\left( \frac{m\pi x}{L} \right)\, dx=\frac{a_ {0}}{2}\cancelto{ 0 }{ \int _{-L}^L \sin\left( \frac{m\pi x}{L} \right) \, dx }+\sum_ {n=1}^\infty\left( a_{n}\cancelto{ \text{odd} }{ \int _{-L}^L\cos\left( \frac{n\pi x}{L} \right)\sin\left( \frac{m\pi x}{L} \right) } \, dx +b_ {n}\int _{-L}^L \sin\left( \frac{n\pi x}{L} \right)\sin\left( \frac{m\pi x}{L} \right)\right) \, dx$
$\int _{-L} ^L \sin\left( \frac{n\pi x}{L} \right)\sin\left( \frac{m\pi x}{L} \right) \, dx=\frac{1}{2}\int_ {-L}^L \cos\left(\frac{(n-m)\pi x}{L} \right)-\cos\left( \frac{(n+m)\pi x}{L} \right)dx$
$=\begin{cases}0, & n\ne m \\L, & n=m\end{cases}$
so:
$\int _{-L} ^L f(x)\sin\left( \frac{m\pi x}{L} \right)\, dx=b_ {m}L$
In conclusion:
2023-11-26 18:14:37 -07:00
$$a_{m}=\frac{1}{L}\int _{-L}^L f(x)\cos \frac{m\pi x}{L} \, dx \quad\text{valid for all }m=0,1,2,\dots$$
$$b_{m}=\frac{1}{L}\int _{-L}^L f(x)\sin \frac{m\pi x}{L} \, dx=b_ {m} \quad m=1,2,\dots$$
2023-12-04 23:21:10 -07:00
Now we know how to compute the coefficients for Fourier series!
2023-11-26 18:14:37 -07:00
properties:
for functions $f$, $g$, If $\int _{-L}^Lf(x)g(x) \, dx=\begin{cases}0 & f\ne g \\L & f=g \end{cases}$
then $f, g$ are orthogonal
2023-12-04 23:21:10 -07:00
the Fourier expansion is called an ortho normal expansion, Taylor is not orthonormal.
2023-11-26 18:14:37 -07:00
#end of lec 28
#start of lec 29
2023-12-06 22:48:05 -07:00
Last lecture we derived how to find the coefficients of a Fourier series.
2023-11-26 18:14:37 -07:00
$f(x)=\frac{a_{0}}{2}+\sum_{n=1}^\infty\left( a_{n}\cos\left( \frac{n\pi x}{L} \right) + b_{n}\sin\left( \frac{n\pi x}{L}\right) \right)$
$x \in [-L,L]$
2023-12-04 23:21:10 -07:00
### 1st convergence theorem:
If $f$ and $f'$ are piecewise continuous on $[-L,L]$, then the Fourier series converges to:
2023-11-26 18:14:37 -07:00
$\frac{1}{2}(f(x^-)+f(x^+))$ for all $x \in (-L,L)$
2023-12-04 23:21:10 -07:00
and on $x=\pm L$ the Fourier series converges to $\frac{1}{2}(f(-L^+)+f(L^-))$
2023-12-03 17:47:59 -07:00
2023-12-04 23:21:10 -07:00
Recall the definition of piecewise continuous: $f(t)$ is piecewise continuous on an interval $I$ if $f(t)$ is continuous on $I$, except possibly at a < u > finite< / u > number of points of < u > jump< / u > discontinuity (horizontal asymptotes not allowed).
### 2nd Convergence theorem (uniform convergence):
If $f(x)$ is continuous on $(-\infty,\infty)$ and $2L$ periodic and if $f'$ is piecewise continuous on $[-L,L]$, then its Fourier series converges to $f(x)$ everywhere (i.e., the Fourier series converges uniformly).
2023-12-03 17:47:59 -07:00
2023-12-04 23:21:10 -07:00
#ex #fourier
Let's compute the Fourier transform of:
$$f(x)=\begin{cases}1, & -\pi\leq x\leq 0 \\x, & 0< x \leq \pi \end { cases }$$
2023-11-30 10:41:13 -07:00
$L$ here is $\pi$ clearly.
2023-12-04 23:21:10 -07:00
Let's find the coefficients $a_{n}$ and $b_{n}$
Use the formula we derived earlier:
$a_{n}=\frac{1}{\pi}\left( \int _{-\pi}^0 1\cos\left( \frac{n\pi x}{\pi} \right)\, dx +\int _ {0}^\pi x\cos(nx)\, dx\right)$
Using integration by parts (for the second integral):
$=\frac{1}{\pi}\left( \frac{1}{n}\cancelto{ 0 }{ \sin(nx) |_{-\pi}^0} + \frac{1}{n}x\cancelto{ 0 }{ \sin(nx) |_{0}^\pi}-\frac{1}{n}\int _{0}^\pi \sin(nx) \, dx \right)$
$a_{n}=\frac{1}{n^2\pi}(\cos(nx)|_{0}^\pi)=\frac{1}{n^2\pi}(\underbrace{ \cos(n\pi) }_{ (-1)^n }-1)$
$a_{n}=\frac{1}{n^2\pi}((-1)^n-1)\quad n=1,2,\dots$
Now let's find $b_{n}$
2023-11-26 18:14:37 -07:00
$b_{n}=\frac{1}{\pi}\left( \int _{-\pi}^0\sin(nx) \, dx+\int _ {0}^\pi x\sin(nx) \, dx \right)$
2023-12-04 23:21:10 -07:00
$=\frac{1}{\pi}[ \frac{-1}{n}\underbrace{ \cos(nx)|_{-\pi}^0 }_{ 1-(-1)^n }-\frac{1}{n}( \underbrace{ x\cos(nx)|_{0}^\pi }_{ \pi(-1)^n-0 }-\underbrace{ \int _{0}^\pi \cos(nx)\, dx }_ { 0 } ) ]$
$b_{n}=\frac{1}{n\pi}((-1)^n-1-\pi(-1)^n)$
2023-11-30 10:41:13 -07:00
$b_{n}=\frac{1}{n\pi}((-1)^n(1-\pi)-1) \quad n=1,2,\dots$
2023-12-04 23:21:10 -07:00
we find that
$$a_{2n}=0 \qquad n=1,2,3,\dots$$
$$a_{2k-1}=-\frac{2}{n^2\pi} \qquad k=1,2,3\dots$$
2023-11-26 18:14:37 -07:00
what about when $n=0$?
$a_{0}=\frac{1}{\pi}\left( \int _{-\pi}^0 \, dx+\int _ {0}^\pi x \, dx \right)$
2023-12-04 23:21:10 -07:00
$a_{0}=\frac{1}{\pi}\left( x|_{-\pi}^0+\frac{x^2}{2}|_{0}^\pi \right)$
$a_{0}=\frac{1}{\pi}\left( 0+\pi+\frac{\pi^2}{2} \right)$
$$a_{0}=\frac{\pi}{2}+1$$
2023-11-26 18:14:37 -07:00
2023-12-04 23:21:10 -07:00
#ex Let's compute the Fourier transform of:
$$f(x)=x \qquad -\pi\leq x\leq \pi$$
We have to take a windowed form of $f$ to make this possible, $L=\pi$
At the left and right edge of the interval, the Fourier series is equal to 0. (1st convergence theorem.)
Find the coefficients:
2023-11-26 18:14:37 -07:00
$a_{n}=\frac{1}{\pi}\int _{-\pi}^\pi x\cos(nx) \, dx=0$
2023-12-04 23:21:10 -07:00
$$a_{n}=0$$
Why is it zero? because the integrand is an odd function. (odd times even is odd.) and because we are integrating from $-\pi$ to $\pi$ (a symmetric interval)
2023-11-26 18:14:37 -07:00
definition of odd: $f(x)=-f(-x)$
definition of even: $f(x)=f(-x)$
odd times even is odd.
odd times odd is even.
even times even is even.
huge exam time saving technique.
2023-12-04 23:21:10 -07:00
Find $b_{n}$:
$b_{n}=\frac{1}{\pi}\int _{-\pi}^\pi x\sin(nx) \, dx=\frac{2}{\pi}\int _ {0}^\pi x\sin(nx) \, dx$ < - that ' s even , don ' t be a silly goose and say it ' s $ 0 $
Using integration by parts:
$b_{n}=\frac{2}{\pi}\left( x\left( -\frac{1}{n}\cos(nx)|_{0}^\pi \right)-\int_{0}^\pi -\frac{1}{n}\cos(nx) \, dx \right)$
$b_{n}=\frac{2}{\pi}\left( -\frac{\pi}{n}(-1)^n+\frac{1}{n^2}\cancelto{ 0 }{ \sin(nx)|_{0}^\pi } \right)$
$$b_{n}=\frac{2}{n}(-1)^{n+1}$$
another take away: if $f$ is odd, the $\cos$ terms are $0$
if $f$ is even, the $\sin$ terms are $0$.
2023-11-26 18:14:37 -07:00
2023-12-04 23:21:10 -07:00
if $f$ is only defined between $0$ and $L$:
you can create an odd extension: $\bar{f}(x)=\begin{cases}f(x) & 0\leq x\leq L \\-f(-x), & -L\leq x< 0 & & \end { cases }$
this will contain only $\sin$ terms.
You also have a choice to extend it as an even function, symmetrically across the $y$ axis.
$\bar{f}(x)=\begin{cases}f(x) & 0\leq x\leq L \\f(-x), & -L\leq x< 0 & & \end { cases }$
This will contain only $\cos$ terms.
#end of lec 29
2023-11-26 18:14:37 -07:00
#start of lec 30
2023-12-04 23:21:10 -07:00
From last lecture:
2023-11-26 18:14:37 -07:00
$f(x)$ is defined on $[0,L]$
2023-12-04 23:21:10 -07:00
odd extension:
2023-11-26 18:14:37 -07:00
$\bar{f}(x)=\begin{cases}f(x), & 0\leq x\leq L \\-f(-x,) & -L\leq x< 0 \end { cases }$
2023-12-04 23:21:10 -07:00
and the $a$ coefficients ($\cos$ terms) are zero.
not only that, but the $b$ coefficients are:
$b_{n}=\frac{1}{L}\int _{-L}^L\bar{f}(x) \sin\left( \frac{n\pi x}{L} \right) \, dx$
2023-11-26 18:14:37 -07:00
$$b_{n}=\frac{2}{L}\int _{0}^L f(x)\sin\left( \frac{n\pi x}{L} \right)\, dx$$
2023-12-04 23:21:10 -07:00
and $\bar{f}(x)$ is:
$$\bar{f}(x)=\sum_{n=1}^\infty b_{n}\sin\left( \frac{n\pi x}{L} \right)$$
"How about that, this is called a Fourier sine series."
2023-11-26 18:14:37 -07:00
For even extension:
$\bar{f}(x)=\begin{cases}f(x), & 0\leq x\leq L \\f(-x,) & -L\leq x< 0 \end { cases }$
2023-12-04 23:21:10 -07:00
and the $b$ coefficients ($\sin$ terms) are zero.
not only that but the $a$ coefficients are:
$a_{n}=\frac{1}{L}\int _{-L}^L\bar{f}(x) \cos\left( \frac{n\pi x}{L} \right) \, dx$
$$a_{n}=\frac{2}{L}\int _{0}^L f(x)\cos\left( \frac{n\pi x}{L} \right)\, dx$$
and $\bar{f}$ is:
$$\bar{f}(x)=\frac{a_{0}}{2}+\sum_{n=1}^\infty a_{n}\cos\left( \frac{n\pi x}{L} \right)$$
Remember that $\sum_{n=1}^\infty b_{n}\sin\left( \frac{n\pi x}{L} \right)$ was the expansion of the eigen value function from the heat equation?
then $\frac{a_{0}}{2}+\sum_{n=1}^\infty a_{n}\cos\left( \frac{n\pi x}{L} \right)$ is also an expansion of some related eigen value function. It's interesting to note.
2023-11-26 18:14:37 -07:00
2023-12-04 23:21:10 -07:00
#ex #fourier Fourier sine series for:
$$f(x)=x^2 \qquad 0\leq x\leq \pi$$
Well that means we want the odd extension:
2023-12-03 17:47:59 -07:00
2023-12-04 23:21:10 -07:00
the $a_{n}$ (cosine) terms are zero.
the $b_{n}$ terms are:
2023-11-26 18:14:37 -07:00
$b_{n}=\frac{2}{\pi}\int _{0}^\pi x^2\sin(nx) \, dx$
2023-12-04 23:21:10 -07:00
$=-\frac{2}{n\pi}\left[ x^2\cos(nx)|_{0}^\pi-2\int _{0}^\pi x\cos(nx)\, dx \right]$
2023-11-26 18:14:37 -07:00
$=-\frac{2}{n\pi}\left[ \pi^2(-1)^n-\frac{2}{n}\left( x\cancelto{ 0 }{ \sin(nx) }|_{0}^\pi-\int _{0}^\pi \sin(nx)\, dx \right) \right]$
2023-12-04 23:21:10 -07:00
$b_{n}=-\frac{2}{n\pi}[ \pi^2(-1)^n-\frac{2}{n^2}\underbrace{ \cos(nx)|_{0}^\pi }_{ (-1)^n-1 }]=\frac{2\pi}{n}(-1)^{n+1}+\frac{4}{n^3\pi}((-1)^n-1)$ for $n=1,2,3,\dots$
Note no $n=0$ so no division by zero problems here.
2023-11-26 18:14:37 -07:00
2023-12-04 23:21:10 -07:00
#ex #fourier Fourier cosine series of $f(x)=\sin(x)$ for $0\leq x\leq \pi$
2023-12-03 17:47:59 -07:00
2023-12-04 23:21:10 -07:00
$b_{n}$ (sine) terms are all zero obviously, as it's asking for a Fourier cosine series, i.e., we are doing an even extension.
$a_{n}=\frac{2}{\pi}\int _{0}^\pi \sin(x)\cos(nx)\, dx$ for $n=0,1,2,\dots$ < -Odd , but not symmetrical bounds . We can ' t rule out that it ' s zero .
>Don't be a silly goose and try changing the bounds by removing that 2 in the front. If you did, you'd also have to change $\sin(x)$ to $\bar{f}$ which is $abs(\sin(x))$ and then you're integrating $a_{n}=\frac{1}{\pi}\int _{-\pi}^\pi |\sin(x)|\cos(nx)\, dx$ which is even$\times$even.
2023-11-26 18:14:37 -07:00
2023-12-04 23:21:10 -07:00
Use trig identity: (by the way the identities will be provided in the final exam.)
$=\frac{2}{\pi} \frac{1}{2}\int _{0}^\pi \left[\sin((1-n)x)+\sin((n+1)x)\right]\, dx$
Integrating gives you:
$\frac{1}{\pi}( \frac{-1}{1-n}\underbrace{ \cos((1-n)x)|_{0}^\pi }_{ (-1)^{n+1}-1 } +\frac{-1}{n+1}\underbrace{ \cos((n+1)x)|_{0}^\pi }_{ (-1)^{n+1}-1 })$
$a_{n}=-\frac{1}{\pi} \frac{1}{n+1}(-1)^{n+1}+\frac{1}{\pi} \frac{1}{n+1}+\frac{1}{\pi} \frac{1}{-(1-n)}(-1)^{n+1}+\frac{1}{\pi} \frac{1}{1-n}$
2023-11-26 18:14:37 -07:00
$a_{n}=-\frac{1}{\pi} \frac{1}{n+1} (-1)^{n+1}+\frac{1}{\pi} \frac{1}{n+1}+\frac{1}{\pi} \frac{1}{n-1}(-1)^{n-1}-\frac{1}{\pi} \frac{1}{n-1}$
2023-12-04 23:21:10 -07:00
Assuming that $n\ne0,1$. (note: $n=-1$ is a non-issue since negative coefficients are never considered when taking a Fourier transform.)
So what is $a_{0}, a_{1}$?
$a_{0}=\frac{2}{\pi}\int _{0}^\pi \sin(x) \, dx=\frac{4}{\pi}$
2023-11-26 18:14:37 -07:00
$a_{1}=\frac{2}{\pi}\int _{0}^\pi \sin(x)\cos(x) \, dx=\frac{1}{\pi}\int _ {0}^\pi \sin(2x)\, dx=0$
2023-12-04 23:21:10 -07:00
< i > "zero is a very very special number it took humanity many numbers of years to invent zero"< / i > referring to when dividing by 0.
Additionally we know that the terms cancel when:
2023-11-26 18:14:37 -07:00
$a_{2k-1}=0$ for $k=1,2,\dots$
2023-12-04 23:21:10 -07:00
$a_{2k}=\frac{2}{\pi} \frac{1}{2k+1}-\frac{2}{\pi} \frac{1}{2k-1}$ for $k=1,2,\dots$
2023-11-26 18:14:37 -07:00
then:
2023-12-04 23:21:10 -07:00
$$\bar{f}(x)=\frac{2}{\pi}+\frac{2}{\pi}\sum_{k=1}^\infty\left( \frac{1}{2k+1}-\frac{1}{2k-1} \right)\cos(2k\pi x)$$
Even with 10 terms, we get a pretty good approximation:
2023-12-06 23:40:58 -07:00
Here's a little script that generates an audible waveform of this Fourier series!
2023-12-07 14:16:26 -07:00
< html >
2023-12-06 22:48:05 -07:00
< head >
< script src = "https://code.jquery.com/jquery-2.1.1.min.js" > < / script >
< meta charset = "utf-8" >
< / head >
< body >
2023-12-07 14:16:26 -07:00
< label for = "real_coff" > $a_n$ formula:< / label >
< input type = "text" id = "realc" name = "real_coff" size = "50" value = "real[2&starn] = (2/Math.PI)&star( 1/(2&starn+1)-1/(2&starn-1) )" >
< / br >
< label for = "imag_coff" > $b_n$ formula:< / label >
< input type = "text" id = "imagc" name = "imag_coff" size = "50" value = "" >
< button id = "playButton" > Play the sound< / button >
< / br >
2023-12-06 22:48:05 -07:00
< canvas id = 'scope' > < / canvas >
< canvas id = 'spectrum' > < / canvas >
< div class = "slidecontainer" >
< p > Number of harmonics: < span id = "textHarmonics" > < / span > < / p >
2023-12-06 23:08:23 -07:00
< input type = "range" min = "0" max = "100" value = "50" style = "width: 500px;" id = "sliderHarmonics" >
2023-12-06 22:48:05 -07:00
< / div >
< div class = "slidecontainer" >
2023-12-07 14:16:26 -07:00
< p > Frequency (for $n=1$): < span id = "textFreq" > < / span > < / p >
2023-12-06 23:08:23 -07:00
< input type = "range" min = "0" max = "440" value = "220" style = "width: 500px;" id = "sliderFreq" >
2023-12-06 22:48:05 -07:00
< / div >
< script src = "playaudio.js" > < / script >
< / body >
< / html >
2023-12-07 14:16:26 -07:00
You can generate any arbitrary Fourier series with this. Try putting
imag[n] = (2*Math.PI/n)*(-1)**(n+1)+4/((n**3)*Math.PI)*((-1)**(n)-1)
for the $b_{n}$ formula and delete the $a_{n}$ formula. Now it will show the fourier sin series of $x^2$ we derived earlier!
you can also make some nasty sounds, try this one:
imag[Math.floor(n**1.1)] = (-1)**(n)*1/(2*n)+Math.random()/10
2023-12-06 22:48:05 -07:00
2023-11-26 18:14:37 -07:00
We have prepared ourselves now, now we start solving PDE's. He's encouraging us to attend the lectures in these last two weeks. He's making it sound like PDE's are hard.
