## Fourier Transform and its applications

I gave a speech at my university about Fourier transform and this was my last official talk at my university. I have learned quite a lot while preparing a talk. First of all the math analysis I have learned so far has found its applications in a speech. It was a pleasure to come back to math analysis and differential equation course notes.  The talk was only a bit revision of theory and concise guide on numerical methods of computing Fast Fourier Transform. Look at the photo below to feel what’s inside my power point slides from the link below.

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Fourier transform is very versatile. It allows to analyze function’s symmetry, and how the function behaves. Its various applications are remarkable. If you have to digitize the continuous function (most real-world nature signals) you will get in touch with sampling, quantization and Fourier Transform. Remember, that in case you are analyzing some natural phenomena and want to deal with Fourier Transform you had better find reciprocal relationship in it. Once you get it you can convert it to digital computer world and marvelous things. This post shows some interesting applications of FFT.

## Telecommunication

It’s a bit obvious for to write that there would be digital telecom without Nyquist-Shannon theorem and Fourier Transform. Both are extremely important. Fourier transform has greatly improved the way we are sending/collecting data. Note that mp3 file format uses transform. Transforms are used in file compression: for instance cosinus transform is also used in jpeg file compression. There are numerous applications in telecoms, and therefore I decided not to dwell on them and go further.

## Automotive

Car dashboard with its distance pointers that in Poland are often manually reversed so that a car can be sold for higher price

In post-lecture discussion we have been talking about software solution that measures how long distance has in fact en car engine. A car is going to be have an engine test stand. A computer with multiple microphone is going to measure voice of the engine the car produces. The software analyses it and gives output that shows us how devastated engine is and what is its real distance. This is pretty cool, and comes in handy. The problem of chating the car dashboard pointers in Poland is quite big, so you have better watch carefully if you buy a second-hand car. This kind of software would set limits on this regrettable procedure.’

A system of Vibro-acoustic signal analysis (acceleration of the unsprung mass of passenger’s car suspension actuated to vibration by harmonic, kinematic vibration) that analized the suspension and adjusted it automatically to the current road condition. You can see more here.

Adaptive cruise control. The cruise control is a systems that controls your speed car. There are many improvements to this system. One is that it uses FFT while controlling speed. You can read a sample article about it here

Voice recognition system. Driving a car requires a lot of attention. Sometimes it’s difficult to put the hands off the steering wheel and change the radio station. If we have to push one button, it is far less problematic than choosing a street number in a GPS receiver. The automotive companies are working on a voice recognition systems. You can check the following articles that describes the advancement of this kind of systems and what to expect from them.

## Hearing devices

Beltone hearing device

Let’s have a look at hearing devices right now. It’s marvellous how good they are. You have from 9 up to 16 directed microphones, it minimizes and filters out the noise of a street. It is self adjusting – this means you get different voice intensification /amplification in different environments. This means that no mater where you are – in laboratory or in pub, you can always talk and hear the person you are talking with. If you have a better one device – you can talk with your phone despite having the disability. This allows people for normality in their lives. Everything is packed in a small device,  which you can hardly see. That’s marvellous.

MRI device

## Medicine – imaging and MRI/CT

I have prepared a report to the talk about graph structures used to represent geometry from the medical device. MRI is using fourier transform to obtain an image from the device measurments. Note that there are two possible ways of storing image: the first one – colour based, that we are used to and the second one – using spectrum. You can switch between them using Fourier transform. This is important in image processing. There are numerous explanations on how the MRI works. Check for instance:

## Voice recognition systems

Voice recognition

There are attempts to perform automatic voice recognition system in customer support. The goal is to eliminate the extremely boring procedure in veryfing the right user asking questions like: what is mother’s maiden name, what is your postal code etc. The system may verify the user by analyzing its voice using Fourier Transform. It is right research direction in my opinion and will greatly facilitate our live.

## Optics

Optics

There is a very nice conversation about Fourier Optics applications available here
You can also check some information about Fourier transform spectroscopy at wikipedia

## Classical Mechanics

Damping and vibrations

The Fourier transform of the square wave generates a frequency spectrum that presents the magnitude of the harmonics that make up the square wave (the phase is also generated, but is typically of less concern and therefore is often not plotted). The Fourier transform can also be used to analyze non-periodic functions such as transients (e.g. impulses) and random functions. With the advent of the modern computer the Fourier transform is almost always computed using the Fast Fourier Transform (FFT) computer algorithm in combination with a window function.

## Geology

Nuclear explosion

The major exploration tool used by the oil and gas industry for locating new hydrocarbon reserves is the seismic reflection method. This technique enables sub-surface sedimentary rock layers to be mapped, from measurements of the amplitude and reflection times of events obtained by the reflection of seismic (sound) waves from the layers. The data recorded at the receivers under goes extensive DSP to produce a seismic section which is a representation of a cross section of the Earth. This is then interpreted in terms of what the research/data was looking for. The Fast Fourier transform is used to convert the signal into the frequency domain, where convolution between two signals is obtained simply multiplying them together. Once again the Fourier Transform is used as a tool in Digital Signal Processing to perform other operations, rather than producing an end result itself.

## Military

Submarine requires extremely reliable communication

The main point of Fourier Transform in military applications is: perform reliable and safe communication. Using fourier transform you can decode your message in a special way. Please remember, that cosinus tranform is used in jpeg file (->data compression). But the point is to sent a message under the noise level. That’s what Fourier transform can be used for.