Motivation for Fourier series/transform

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• MexChemE
In summary, the conversation discusses the motivation for learning Fourier series and its uses in solving the heat equation and expressing periodic functions as a sum of sines and cosines. It also explores the application of Fourier transform in various fields such as music, acoustics, signal analysis, and optics. The conversation highlights the usefulness of Fourier transform in simplifying calculations and accurately determining frequency components in measurements. Additionally, it mentions how synthesizers use Fourier analysis to mimic the sounds of different instruments. The conversation also includes references to papers and examples showcasing the practical applications of Fourier transform.
MexChemE
Hello, PF!

I am currently learning Fourier series (and then we'll move on to the Fourier transform) in one of my courses, and I'm having a hard time finding motivation for its uses. Or, in other words, I can't seem to find its usefulness yet. I know one of its uses is to solve the heat equation, which is why we're learning Fourier series in the first place, but aside from that, other applications are not so evident. Now, I'm not trying to disregard Fourier series as a useful mathematical tool, I'm actually really interested in mastering this technique and the Fourier transform, I just need help in order to see the whole picture.

For example: most textbooks, when presenting the technique, state that it is a useful tool for expressing a periodic function as a sum of sines and/or cosines, and should only be used on periodic functions. What is the point of this, if the function is already, by definition, periodic?

Then, one of my exercises asks to find the Fourier sine series of $f(x) = 1 - x$ for the interval $[0,\pi]$, which is
$$f(x) = \frac{2}{\pi} \sum_{n=1}^{\infty} \frac{1-(-1)^n+\pi(-1)^n}{n} \ \sin{nx}$$
I plotted the results in order to see what is actually going on.

As you can see, the series approximate the function the most between 0 and $\pi$ (roughly). It seems obvious that the series will approximate the function only for the given interval, is this the rule? Now, regarding this same example, what would be the point of having a function like this and obtain its Fourier series, application-wise? What can you do with it?

Regarding the Fourier transform, what I understand is that you can use it to obtain a series for a function, but for the entire real domain, that is $(-\infty,\infty)$. However, this impression might be wrong, as it also changes the independent variable, which doesn't happen when you obtain the Fourier series of the function. Speaking of which; I understand that when your independent variable is time, the Fourier transform takes you into the frequency domain. But, what happens when the independent variable is x, as in position? What is the physical meaning of the variable change?

Thanks in advance for any input, I tried to be as clear as possible, which I hope I did!

Fourier analysis is very useful in music, and in acoustics more generally. The waveforms of a violin, flute and trumpet all playing A 440Hz have the same frequency but different shapes. Those shapes are determined by having different harmonics and give the 'timbre' of the note. Fourier analysis can show what the harmonics are for the instruments and how they differ.

When synthesisers set out to mimic an instrument they use the results of Fourier analysis to generate the required waveform.

olivermsun, MexChemE and fresh_42
MexChemE said:
As you can see, the series approximate the function the most between 0 and π\pi (roughly). It seems obvious that the series will approximate the function only for the given interval, is this the rule?
Yes. By definition a Fourier series approximates a periodic function.
MexChemE said:
Now, regarding this same example, what would be the point of having a function like this and obtain its Fourier series, application-wise? What can you do with it?
There are tons of applications - what you usually do is move a phenomen from the time space to the frequency space. What is hard in one space may be easy in the other.

An example: Relating Vout to Vin in the time space is hard (it ends up being a limited derivation), but relatively easy in the frequency space (it is a simple high-pass filter). See https://en.wikipedia.org/wiki/High-pass_filter.

Just for fun: I have used Fourier transforms to find the values of some infinite sums that were hard to find in other ways. I may write an Insight about it some day.

olivermsun, MexChemE and fresh_42
The general motivation is that some calculations are easier and maybe more obvious when moved into the Fourier domain. Convolutions and correlations become multiplications etc., in much the same way that logarithms turn multiplication into addition.

olivermsun and MexChemE
In optics, Fourier transform can be used in imaging called Fourier optics. You can change the amplitude distribution of an image and make it into what you want it's like. Recently a commonly used device "Spatial light modulator" can change a beam profile of plane wave into that of arbitrary distribution of both amplitude and phase of its wave front， this transform needs the knowledge of Fourier optics.
In measurement area, Fourier transform can be used analyzing the frequency components of a time-domain signal. This is important almost in every area involving measurement as every time-domain signal has its frequency-domain components.
In short, the time and frequency domain are two different parts of one signal. The connection between them is Fourier transform.

olivermsun and MexChemE
From my point of view, there is a sum of oscillating components hiding in a lot of measurements, and the Fourier transform is the optimal tool for accurately determining the amplitude, frequency, and phase of each of those oscillating components. For example, the attached figure shows a Fourier transform of measured water levels over a year. All the frequency components that make up the tides can be discerned, along with their frequencies and amplitudes.

See: https://arxiv.org/pdf/1507.01832v1.pdf

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olivermsun and MexChemE
In other cases, the frequency of an oscillation may be known, but a more precise measurement of its amplitude might be of interest. Everyone knows about the annual ripple in the Mauna Loa CO2 measurements. The period is exactly one year. But the attached figure shows that the amplitude of that oscillation is changing over the decades, suggesting as CO2 levels rise, the amount of carbon cycled into the biomass each year is also increasing.

See Fig 4 of: https://arxiv.org/pdf/1507.01832v1.pdf

Lots of practical examples of Fourier transform usefulness in the linked paper.

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olivermsun and MexChemE
andrewkirk said:
When synthesisers set out to mimic an instrument they use the results of Fourier analysis to generate the required waveform.
Hey, this sounds pretty cool, I would have never guessed synthesizers emulated other instruments with the help of Fourier analysis, but now that I think about it, it actually seems very intuitive.
Svein said:
Just for fun: I have used Fourier transforms to find the values of some infinite sums that were hard to find in other ways. I may write an Insight about it some day.
Dr. Courtney said:
View attachment 107641 In other cases, the frequency of an oscillation may be known, but a more precise measurement of its amplitude might be of interest. Everyone knows about the annual ripple in the Mauna Loa CO2 measurements. The period is exactly one year. But the attached figure shows that the amplitude of that oscillation is changing over the decades, suggesting as CO2 levels rise, the amount of carbon cycled into the biomass each year is also increasing.

See Fig 4 of: https://arxiv.org/pdf/1507.01832v1.pdf

Lots of practical examples of Fourier transform usefulness in the linked paper.
I found your posts very interesting, and I really like the idea of using the transform of some data set in order to obtain insights which would be otherwise hard to find. And thanks for the links, I'm about to check them out.

Thank you for all your great responses! My view on Fourier analysis is a bit clearer now, though I still have so much to learn.

Fourier analysis is not only an invaluable tool for solving equations, but it also gives us a very powerful insight into the physical world. I will try to explain the sheer significance of Fourier analysis to engineering by drawing on real world examples. I remember being a bit perplexed by Fourier series when first learning about it, but it really grew on me. It really is a fantastic tool and it opens up a new way to think about phenomena.

Consider the refraction of light in a prism. As you have learned, any waveform can be decomposed into individual frequency components. In this case, light is refracted by the medium. In a sense, the physical material performs a Fourier transform on the incident light and spitting out a spectrum of different frequencies. Fourier analysis gives us the analytical tools necessary to describe this mathematically, and with the help of physics we can really begin to understand the interaction between waves and crystal structures.

How can cancer cells be hit through the skin without burning the skin? In just the same way as light passes through a window - the lasers used for burning cancer cells emit a monochromatic beam of photons with a frequence which is absorbed by the cancer cells, but which is absorbed to a much smaller degree by skin and healthy tissue. A side effect of this observation is that this frequency response tells us about the molecular structure whatever we shine a light at. Related to this is the practice of spectroscopy; analyzing the scattering of light from a substance in a controlled environment gives us the ability to deduce the molecular structures of whatever we shine a light on, as long as out measuring equipment is good enough. The discovery of the structure of DNA was made possible by Fourier analysis.

An application of the Fourier series which I find very interesting is compression of data. Consider a digital picture, W pixels wide and H pixels high. A digital picture is often composed of three components - red, green and blue. By letting the brightness of each pixel represent the value of a function at each point, we can simply invoke the power of Fourier analysis and decompose this picture into frequency components. The number of frequency components stored directly affects the quality of the reconstructed picture, and this technique lays the foundation of the widely used JPEG standard. This is applicable to any set of data

When a satellite sends a signal to a receiving antenna on earth, the signal goes through a terrible transformation on its way through the atmosphere. First of all, the electromagnetic waves has to pass through the atmosphere. Atmospheric gases absorb some of the power and parts of the power will be scattered. Secondly, the receiving equipment comes with its own imperfections giving rise to noise. The received signal may look unrecognizable when looking at the time domain representation, and no useful information can be obtained. However, if we know the original frequencies of the signal (which we do, as the transmitter were made to emit certain signals) we can apply a Fourier series analysis and look at the amplitudes of each frequency component.

Also, because the atmospheric attenuation of signals is different for different frequencies, only a limited range of frequencies can be used. Although a gross oversimplification, almost all of these examples have common elements, and not just the Fourier transform. From an engineering perspective, they are all examples of systems that take some input and gives some output. And in all these cases, the output is frequency dependent.

Put a pendulum in motion, measure the length of the string and observe the frequency of oscillation. This tells you about the mass of the pendulum.
Put a sinusoidal sound signal on a speaker and record it from the other side of a wall. Vary the frequencies and measure the attentuation of the sounds for each frequency. Now, play a song and listen to how the sound is muffled. Then, take a sample of the song, perform a Fourier transform of the song, multiply it by the frequency spectrum you have found, then transform back to the time domain. You will most likely hear the same muffled sound, but coming directly from your speaker.

There are countless examples, and I could have gone on and on about them, but I hope I was able to put some perspective on the value of Fourier analysis.

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MexChemE, beamie564, olivermsun and 2 others