I am trying to explain harmonic analysis to students without a calculus background by using the averages of trigonometric multiplication of periodic waves.
I have attached my writeup so far with the Latex source code.
Is this a feasible argument or is there a better way to explain the matter with references maybe?
Any suggestions or corrections will be most welcome.
While a detailed treatment of Fourier series is beyond the scope of this text, we will state some basic principals and present some basic calculations that can be used to extend our understanding of harmonic analysis.
If a periodic waveform has a fundamental frequency $f_1$, then the sinusoidal components present in the waveform will be integer multiples of $f_1$. These integer multiple frequencies are known as the harmonic components (or overtones) with $nf_1$ being called the $n$th harmonic component. Thus the periodic waveform
\begin{IEEEeqnarray}{rCl}
a(t) & = & A_0 + \sum_{n=1}^\infty A_n \cos(n\omega_1 t +\phi_n)
\end{IEEEeqnarray}
where $\omega_1= 2\pi f_1$ and $A_n$ is the magnitude of the harmonic component of frequency $nf_1$, and $\phi_1$ is its phase. $A_0$ is simply the constant or $\dc$ component of a(t). For any given waveform, the $n$ values, $A_n$ and $\phi_n$ can be calculated and and their values serve to describe the spectrum of the waveform.
It is important to note that for any periodic waveform $a(t)$ the fundamental can be extracted by multiplying the waveform $a(t)$ with the corresponding fundamental frequency wave and calculating the averages,
\begin{IEEEeqnarray}{rCl}
C_1 & = & \sqrt{2}\left[\overline{a(t)\cdot\cos(\omega_1 t)}\right]\\
\text{and}\nonumber\\
D_1 & = & \sqrt{2}\left[\overline{a(t)\cdot\sin(\omega_1 t)}\right]\\
\text{giving}\nonumber\\
a_1(t) & = & C_1\cos(\omega_1 t) + D_1\sin(\omega_1 t).
\end{IEEEeqnarray}
By recalculating $C_1$ and $D_1$ we can confirm that
\begin{IEEEeqnarray}{rCl}
C_1 & = & \sqrt{2}^2\left[\overline{C_1\cos^2(\omega_1 t)}\right] = 2 C_1\left[\overline{\frac{1}{2} + \frac{1}{2}\cos(2\omega_1 t)}\right]= C_1\nonumber\\
\text{and}\nonumber\\
D_1 & = & \sqrt{2}^2\left[\overline{D_1\sin^2(\omega_1 t)}\right] = 2D_1\left[\overline{\frac{1}{2} - \frac{1}{2}\cos(2\omega_1 t)}\right]= D_1
\end{IEEEeqnarray}
are both indeed reasonable values if the assumption that \begin{IEEEeqnarray}{rCl}
& ~ &\cos(m\omega_1 t+\phi_m)\cdot\cos(n\omega_1 t+\phi_n) = 0~ \forall ~m \ne n
\end{IEEEeqnarray}
which is left as an exercise for the reader.
I have now added the example below, and I seem to be close - to a factor of $\frac{1}{n-1}$. I know I must still fix $\overline{\sin(\omega t)} = \frac{2}{\pi}$.
I have managed to find a very simple way to explain the Fourier series using only averages and almost no calculus. (No calculus is used if the average of a half-cycle sine wave is defined as $\frac{2}{\pi}$ without derivation.)
The full Fourier series simplified is posted, rather than snippets, to show the context.
Any comments or references to similar papers would still be extremely helpful.
I am also willing to past the full Latex source code in this answer if requested.