This Is A Custom Widget

This Sliding Bar can be switched on or off in theme options, and can take any widget you throw at it or even fill it with your custom HTML Code. Its perfect for grabbing the attention of your viewers. Choose between 1, 2, 3 or 4 columns, set the background color, widget divider color, activate transparency, a top border or fully disable it on desktop and mobile.

This Is A Custom Widget

This Sliding Bar can be switched on or off in theme options, and can take any widget you throw at it or even fill it with your custom HTML Code. Its perfect for grabbing the attention of your viewers. Choose between 1, 2, 3 or 4 columns, set the background color, widget divider color, activate transparency, a top border or fully disable it on desktop and mobile.

Forms of Fourier Series

///Forms of Fourier Series

Forms of Fourier Series

I was a bit sloppy in class, so I thought I’d make a quick post about the three equivalent forms for Fourier series.

First Form: Phase-Shifted Sines

The first type of form we began with could be described as a “phase-shifted sines” form:
\[
f(t)=\sum_{n=0}^N A_n\sin(2\pi nt+\phi_n).
\] Here I’m including $n=0$, which gives a constant term $A_0\sin(\phi_0)$.

Second Form: Sines and Cosines

If we use the trig identity $\sin(\alpha+\beta)=\sin(\alpha)\cos(\beta)+\sin(\beta)\cos(\alpha)$, we can rewrite the shifted sine term as
\[
\sin(2\pi nt+\phi_n) = \sin(2\pi nt)\cos(\phi_n)+\cos(2\pi nt)\sin(\phi_n).
\] Using this, we can rewrite our original Fourier series in a “sines and cosines” form, as
\[
f(t)=\sum_{n=0}^N \left(a_n\cos(2\pi nt)+b_n\sin(2\pi nt)\right),
\] where $a_n = A_n\sin(\phi_n)$ and $b_n=A_n\cos(\phi_n)$. Notice that the constant term (when $n=0$) is $a_0$. In class I mentioned that this constant term is often instead denoted $\frac{a_0}{2}$, for reasons that will slightly simplify some formulas we’ll see in the future. For now, let’s not worry about that.

Third Form: Complex Exponentials

Our third form uses the famous identity $e^{i\theta}=\cos(\theta)+i\sin(\theta)$. Notice that this equation implies
\begin{align*}
e^{2\pi int}&=\cos(2\pi nt)+i\sin(2\pi nt)\\
e^{-2\pi int}&=\cos(-2\pi nt)+i\sin(-2\pi nt)=\cos(2\pi nt)-i\sin(2\pi nt).
\end{align*}
Adding these two equations and dividing by $2$ (respectively, subtracting them and dividing by $2i$) yield
\begin{align*}
\cos(2\pi nt)&=\frac{1}{2}\left(e^{2\pi int}+e^{-2\pi int}\right),\\
\sin(2\pi nt)&=\frac{1}{2i}\left(e^{2\pi int}-e^{-2\pi int}\right)= -\frac{i}{2}\left(e^{2\pi int}-e^{-2\pi int}\right).
\end{align*}
Substituting these into the “sines and cosines” form and simplifying yields
\[
f(t)=\sum_{n=0}^N \left(\frac{1}{2}(a_n-ib_n)e^{2\pi int}+\frac{1}{2}(a_n+ib_n)e^{-2\pi int}\right).
\] We can write the above expression even more simply as
\[
f(t)=\sum_{n=-N}^N c_n e^{2\pi int},
\] where $c_0=a_0$ and for $n>0$ we have
\begin{align*}
c_n &= \frac{1}{2}(a_n-ib_n)\\
c_{-n}&= \frac{1}{2}(a_n+ib_n).
\end{align*}
In particular, notice that $c_n$ and $c_{-n}$ are complex conjugates.

What is great about this final form is that it is very nearly a (finite) geometric series, with common ratio $e^{2\pi i t}$. (It actually is a geometric series when all of the constants $c_n$ are identical.) This will make certain calculations much simpler, and is why we will often prefer this form.

By | 2017-10-07T09:03:05+00:00 October 2nd, 2017|Categories: Courses, Math 344|0 Comments

Leave A Comment