# What is the significance of the Fourier coefficients?

Let us take an example, a white ray (which is composed of bunch of frequency components) is passed through a prism, the ray gets split (decomposed) into its elementary vibgyor colours (i.e.different colour frequencies) and we get rainbow like pattern.

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

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

Here, $a_0$,$a_m$ and $b_m$ terms are termed as Fourier coefficients in the Fourier series formula.

But,which information do they provide about the signal $x(t)$? Also, What is the significance of the Fourier coefficients?

• Would Mathematics or Signal Processing be a better home for this question? – Qmechanic May 8 '15 at 18:57
• @Qmechanic the question asks for some meaning of the terms. That's vague enough that a physicist can give it a physicsy answer. – DanielSank May 8 '15 at 19:16
• The $a_0$ gives you an offset from the 'f=0' axis, which might be necessary to model certain functions. – Time4Tea May 9 '15 at 2:31

Consider a single value of $m$. The Fourier series for just that $m$ gives $$a_m \cos(2\pi m t / T) + b_m \sin(2\pi m t / T) \, .$$ This can be rewritten as $$M_m \cos (2\pi m t / T + \phi_m)$$ where $$M_m = \sqrt{a_m^2 + b_m^2} \qquad \text{and} \qquad \phi_m = \tan^{-1}(-b_m/a_m) \, .$$ So, you can see that $a_m$ and $b_m$ are just the cartesian coordinate equivalent of the signal's amplitude and phase.

When you have the full series with all the terms it's just a sum of many sinusoids each with their own amplitude and phase. In other words, we could equivalently write $$\frac{a_0}{2} + \sum_{m=1}^\infty M_m \cos(2\pi m t / T + \phi_m)$$ using the relations above.

Anyway, the point is that the physical meaning of the coefficients in the Fourier series is that they tell you the amplitude and phase at each frequency.

What is color? Color has two possibilities. It is what our eye retina perceives as red, blue, yellow ... and its study belongs to biology. For example mixing blue paint and yellow paint gives the green color identified by our retina.

In the rainbow each frequency is displayed according to the strength coming from the white light source, and our retina identifies it as a color . If one takes a white beam of light and generates the spectrum , if fitted with fourier functions, the intensity of each line would be the constants infront of the sine and cosines.

Information comes from the dominant terms: they tell the dominant frequencies, the larger the coefficient of a sine/cosine the stronger is that frequency/color in the mix of frequencies of white light.

• I will edit, as you seem not to know what colors are. – anna v May 9 '15 at 9:18
• Maybe this will help you, facstaff.bucknell.edu/mastascu/eLessonsHTML/Freq/… , it is into signal processing. The frequency spectrum of white light by itself is not time dependent . – anna v May 9 '15 at 11:31
• So what can we infer from the analysis to the rainbow? I guess the frequencies in the exponents are the wave frequencies, the coefficients are the intensities, the sum is the interference of them. Am I correct? – Ooker Nov 5 '17 at 19:04
• @Ooker Yes, for a spectropgraph. The perception by humans of the colors has to involve the biology of the retina. – anna v Nov 6 '17 at 3:54