What do $a_0$ ,$a_m$ and $b_m$ terms mean in the Fourier series formula?

Question

We know that a Fourier series for signal $x(t)$ is given as

$$\frac {a_0} 2 + \sum \limits _{m=1} ^\infty (a_m \cos \frac {2 \pi m t} T + b_m \sin \frac {2 \pi m t} T)$$

So my question is

what do $a_0$,$a_m$ and $b_m$ terms mean in the Fourier series formula?

How are they important in Fourier analysis and synthesis ?

Answer 1

The $a_i$'s and $b_i$'s are respectfully the real and imaginary parts of the complex number in the $i$th position of the vector $F_nx$, where $F_n \in \mathbb{C}^{n \times n}$ is the Fourier transform matrix. For example, $F_4$ is $$\begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & \omega & \omega^2 & \omega^3 \\ 1 & \omega^2 & \omega^4 & \omega^6 \\ 1 & \omega^3 & \omega^6 & \omega^9 \end{bmatrix}$$ and with $n=4$, $\omega=\exp(\frac{-2\pi i}{n})=\exp(\frac{-\pi i}{2})$.

Now if $$x=\begin{bmatrix} 1 \\ 0 \\ 1 \\ 2 \end{bmatrix}$$

Then $F_4x$ is $$\begin{bmatrix} 4 \\ 1+i \\ 0 \\ -2i \end{bmatrix}$$

So that $a_0=4, b_0=0, a_1=1, b_1=1, a_2=b_2=0, a_3=0$ and $b_3=-2$

Answer 2

One is studying periodic functions with given period $T>0$. Among these there are very special ones, the pure harmonic functions $$t\mapsto\cos{2\pi k t\over T}\quad(k\geq0),\qquad t\mapsto\sin{2\pi k t\over T}\quad(k\geq1)\ .$$ These functions have very useful algebraic and analytic properties, and we understand them very well.

It is a miracle that any reasonable $T$-periodic function $f$ can be written as a linear combination (or superposition) of these special functions: $$f(t)={a_0\over2}+\sum_{k=1}^\infty\left(a_k\cos{2\pi k t\over T}+b_k\sin{2\pi k t\over T}\right)\ .$$ The "physical" interpretation of the coefficient pair $(a_k,b_k)$ is the following: It tells you with which intensity and phase the pure harmonic $t\mapsto\cos{2\pi k t\over T}$ is present in $f$. An example: If $a_7=10$, $b_7=3$, and all other $a_k$, $b_k$ are $\ll1$, and altogether they converge to $0$ when $k\to\infty$ then the function $f$ is not far away from a pure harmonic with $7$ peaks per period.

Concerning light spectra: The frequency $\omega$ of light waves can have any real value in a certain interval $[\omega_\min,\omega_\max]$. The Fourier representation of an arbitrary (steady) light signal would then look as follows: $$f(t)=\int_{\omega_\min}^{\omega_\max} \bigl(a(\omega)\cos(\omega t)+b(\omega)\sin(\omega t)\bigr)\>d\omega\ .$$ A physical device that can separate the various frequencies, like a prism or a spectrometer, will show bright bands or dots along an $\omega$-scale at places where the corresponding values $a(\omega)$, $b(\omega)$ are large. From white light you get a broad band (which even shows the colors, but this is secondary), since all frequencies are equally present; whereas from a methane flame you only get dots at places characteristic for methane.