Hello and welcome to episode 2 of a podcast all about the Fourier Transform. My name is Mark Newman. Despite never being particularly good at maths at school, I managed to get a degree in Electrical and Electronic Engineering from the University of Manchester in the UK and have been working as an electronics engineer for the past 25 years. In that time, I've come to love and yes, maybe even obsess a little about the Fourier Transform and how it works.
Having always found maths a bit of a challenge, I've had to develop my own methods of understanding its unique language by employing a more visual way of looking at the subject. I'd like to share some of those methods with you via these podcasts and also through an online course I'm developing called “How the Fourier Transform works.” You can visit the course homepage at https://howthefouriertransformworks.com More about the course and how it is progressing towards the end of this episode.
In the last episode, I asked you to share with me interesting ways in which you are using the Fourier Transform. I had a number of replies and I hope to share some of them with you in future episodes. However, I received an email from one listener who expressed concern. He was afraid that in sharing the way he was using the Fourier Transform, he might be trespassing on someone else’s intellectual property.
His concern was completely valid and I would therefore like to make a request. Please make sure, before sharing an idea with me, that it is either something in the public domain, your own idea, or an idea you have permission to share. There are a couple of projects I have worked on involving the Fourier Transform that I would love to share with you on this podcast, but cannot, precisely for this reason. I've also received a number of interesting questions about different aspects of the Fourier Transform. I'll be answering one of these later on in this episode. In today's podcast, we're going to find out how the Fourier Transform can help save lives when used in the field of preventative maintenance.
What is preventive maintenance? Put simply, preventative maintenance is where certain tell-tale parameters are used to indicate the health of a machine or system. If one of those parameters begins to show some kind of drift away from a defined norm, the machine or system is taken out of service for maintenance or repair BEFORE something goes seriously wrong. It's a bit like going to the doctor for your annual check-up.
The doctor listens to your heart, checks your pulse, maybe performs a series of blood tests, all to ensure that your body is working properly. In fact, the image of the doctor with a stethoscope is quite an appropriate one considering how the Fourier Transform is used in the field of preventative maintenance. The doctor uses a stethoscope to listen to your heartbeat, your breathing, and other auditory signals that your body generates as a by-product of how they work. For example, if a patient suffers from asthma, the doctor may hear a certain wheezing sound which the air makes as it flows through the obstruction. With the aid of the stethoscope, the doctor can hear this and then prescribe the correct medication.
In the last episode, I mentioned how our ears perform a biological version of the Fourier Transform. If your ears can do it, then a microphone and a computer performing the Fourier Transform should also be able to do it too. Anything that gives out a consistent, identifiable audio signature could benefit from this sort of monitoring. If something in the audio signature changes, then maybe there's a problem.
For example, let's take a fan. Here's the sound of my desk fan when it is working properly. But say there was something wrong.
I can tell from the sound that the fan needs oiling. If I don't oil the fan, eventually it could become damaged. But if I detect the fact that it needs maintenance early on, I can oil it before any damage occurs. Now, a squeaky desk fan might be a little bit annoying.
If it broke, I would have to buy a new one. However, consider another sort of fan, that, if damaged, could not only be very costly but might even endanger lives. An airplane's jet engine for example. It was a cold winter's evening on the 8th of January 1989 when British Midland Flight 092 took off from London's Heathrow Airport en route to Belfast. Shortly after take-off, while the aircraft was still climbing to its cruising altitude of 35,000 ft, a fan blade suddenly detached itself from the port engine.
The captain and his co-pilot knew something was wrong. They could hear a pounding noise and the aircraft was vibrating alarmingly. A smell of smoke was reported by the passengers and cabin crew emanating from the ventilation system. But although everyone aboard knew that something was seriously wrong, confusing readings from the instruments and a recent modification to the routing of the ventilation system, caused the co-pilot to mistakenly identify the source of the smell in the cabin and therefore which engine was at fault. The captain radioed ground control to report the emergency and was ordered to attempt an emergency landing at East Midlands airport, the nearest airport to their current position. In order to prevent any more damage to the malfunctioning engine, he throttled back the starboard engine, the one he thought was at fault.
The smell of smoke disappeared and for a short while, they thought they had contained the problem. However, they were unaware of the reality of their situation. With the starboard engine all but taken offline, it was now only the damaged port engine that was holding the aircraft in the air. During their final approach, the pilots increased thrust from their only operating engine in preparation for landing. With the working starboard engine at minimum thrust; this was more than the damaged port engine could take and it burst into flames. Realizing their fatal mistake, the pilots attempted to restart the starboard engine, but by now, it was too late.
At just before twenty-five minutes past eight pm, the Captain made the announcement that the passengers had been dreading: “Prepare for crash landing”. Not ten seconds later, the tail and main landing gear struck the ground, and the aircraft bounced back into the air and over the stretch of the M1 motorway which passes through the village of Kegworth in Leicestershire knocking down trees and a lamp-post before crashing into an embankment just half a kilometer short of the runway. On impact, the aircraft broke into three sections instantly killing 39 of the 118 passengers aboard.
By some miracle, the normally busy motorway was clear of traffic at the exact moment of the crash, but over the next few days, 8 more of the passengers succumbed to their injuries bringing the death toll of the doomed British Midland flight 092 to 47 in what became known as the Kegworth air disaster. The captain and his co-pilot, both seriously injured in the crash, were dismissed following criticisms of their actions in the Air Accidents Investigation Branch report. However, was the crash entirely their fault? They made a mistake, yes, a fatal one, but they were forced into a split-second decision based on insufficient data from their instruments. If they'd only known for certain which engine was malfunctioning, they could have acted differently and shut down the correct engine.
Could better monitoring of critical systems have prevented the crash? Perhaps the whole disastrous chain of events that started with a broken fan blade could have been avoided if only the defect had been detected sooner and the faulty fan blade replaced as part of routine maintenance before the aircraft ever took off. As technology has improved, the Fourier Transform has become a vital tool in preventing this type of disaster by acting as an early warning system. The aircraft engine makes a certain sound. Even if I were to just play the sound to you without telling you what it was, you'd likely be able to identify it because you've heard it before and you know what it sounds like.
A damaged engine however would make a slightly different sound, just like my broken desk fan did. Such a change could be detected by computer if it knew what a working engine should sound like. Now, in my example of the fan, you can hear the difference between when it works properly and when there is a fault.
But, how would a computer do it? How can we mathematically detect if two signals are not the same? Maybe the simplest way to do this would be to subtract one signal from the other. However, one problem if we do this with a signal in the time domain, is that the difference between the two signals may be very small and difficult to detect. To see what I mean, head over to the resources section of the show notes for this episode at: https://howthefouriertransformworks.com/preventative-maintenance and have a look at the images of the two time-domain signals, one for my desk fan when it was working, and the other for the broken fan.
They look very similar. How can this be if our ears can detect the squeaky fan so easily? That is because the squeak is only noticeable at certain frequencies. You can't really see this on the time-domain signal as a time-domain graph only shows us the amplitude of the signal at any given moment. If the squeak isn't particularly loud, then the amplitude doesn't really change all that much. However, if we now look at the signal in the frequency domain, like your ears do, then the squeak becomes very obvious.
The time-domain signal is sampled and broken down into blocks called frames. Each frame contains a fixed number of samples. For example, there might be 1024 samples in each frame. The sample rate is constant so a fixed frame size means that we are simply cutting up the signal into a series of blocks of equal duration. To visualize the result and show why the Fourier Transform is used, we can use a spectrogram.
A spectrogram is like a 3D graph. On the x-axis of the graph, we plot the index of each frame. On the y-axis, we plot frequency, and on the z-axis, we plot the magnitude of the contribution of each frequency to the signal.
Remember that each frame will contain the frequency spectrum for a fixed duration of the signal, so the x-axis shows how that spectrum changes over time. Sometimes, it can be a little difficult to represent the z-axis, especially if we are only drawing the graph in two dimensions. Therefore the magnitude on the z-axis is often represented by different colors like black for zero magnitude, becoming blue for a frequency that only contributes a small amount to the signal going through red for a frequency that contributes a large amount to the signal all the way to bright yellow for a very large contribution. I have posted the spectrograms for the working fan and the broken fan in the resources section of the show notes. If you compare the two spectrograms, it's very easy to see the difference between them. Now if a computer were to subtract one from the other, the difference would be easily detectable.
This is just one example of how the Fourier Transform is used in preventative maintenance in critical systems. The same method can be employed in power stations to monitor how well the electricity-generating turbines are working, as well as in countless other applications too. The advantage of this method is that, because it is non-invasive, it can be run online as well as off. This means that machinery can be tested while it is working and any fault instantly detected. Now we come to the question and answer section of the podcast where I answer a listener's questions on the Fourier Transform.
The Fourier Transform is a huge subject, and inevitably I am going to receive questions that are going to require more research before I can answer them fully. A great place for some of that research could be here on this very podcast, so I thought I would deal with such a question in this episode and share with you what I've found out so far. Maybe someone listening might be able to help me improve my answer. This question came from Ahmet Serdar from Turkey, and apologies Ahmet, if I mispronounced your name. The question was about the use of the Fourier Transform in filter banks.
It read: I have a wideband signal and I am interested in a particular band of frequencies. I need to channelize the band. The first stage of the process is a down-conversion of the band using a filter bank. I know that the DFT is used widely in filter banks but I'm unclear how? My researches led me to an article on the dsprelated.com website.
You can find a link to the article in the resources section of the show notes for this episode at: https://howthefouriertransformworks.com/preventative-maintenance Firstly, what is a filter bank? A filter bank is an array of bandpass filters that channelize or separate out the different frequencies in an input signal into a set of sub-bands. Each sub-band will contain a different range of frequencies. The 1kHz band, for example, might contain that part of the signal which exists in the frequency range from 800Hz to 1.2kHz. Filter banks are used in a range of applications such as the EQ control on your stereo system. This is typically a row of sliders that allow you to set the volume of the different frequency ranges coming through your speakers.
Maybe you want a more bassy sound to your music. In such a case, you would turn up those sliders controlling the lower frequencies of the signal. Alternatively, maybe, every time the drummer hits the cymbals, they are hurting your ears.
You could turn down the 1kHz range slider and take the edge off the cymbal crash. How is the DFT used in a Filter Bank? Depending on what the filter bank is used for, the DFT; or the Discrete Fourier Transform, can be used in a number of different ways. In fact, the FFT, or Fast Fourier Transform is more often used in these filters. The FFT is a more efficient version of the DFT which computers can perform very fast. I deal with the DFT and FFT in my course on How the Fourier Transform works. A filter bank uses bandpass filters to split up the signal.
In the digital world of computers, it is much easier to perform this operation on a signal in the frequency domain than it is in the time domain. The time domain is where we look at how the amplitude of a signal changes over time, like you would see on an oscilloscope. Time is on the x-axis and amplitude is on the y-axis. The frequency domain is where we look at each frequency within a signal and how much it contributes to the overall signal, like on a spectrum analyzer. Now the frequency is on the x-axis and magnitude is on the y-axis. The Fourier Transform converts a signal from the time-domain into the frequency-domain.
Once the whole spectrum of frequencies is laid out in front of us in this way, it becomes much easier to modify individual frequencies or even whole groups of frequencies like in the bandpass filter of a filter bank. You simply multiply the magnitude of each frequency by the desired response of your filter. You could attenuate (turn down) certain frequencies, or amplify them (turn them up).
So to create a bandpass filter centered around 1kHz, for example, I could multiply the whole spectrum by a bell curve with the peak of the bell curve centered on the 1kHz frequency. However, this is not the only way that the Fourier Transform can be used. The question mentions down-converting the band. Down-conversion is the process whereby we shift the frequency range of a signal to a lower set of frequencies.
Think of AM radios. AM stands for Amplitude modulation. At the radio station, the DJ’s audio signal, also known as the baseband signal, is modulated onto a high-frequency carrier wave so that it can be transmitted through the air.
In an AM transmitter, the audio signal changes the amplitude of the carrier wave producing a carrier signal. If we look at what is happening to the signal in the frequency domain, amplitude modulation shifts the spectrum of the baseband signal up along the x-axis to a band of frequencies centered around the higher frequency of the carrier wave. This is known as up-conversion.
Your radio receiver reverses this process. It shifts the spectrum of the carrier signal back down along the x-axis to the lower baseband frequencies thereby recovering the DJ’s audio signal. This is known as down-conversion. Once we have our signal in the frequency domain, it is much easier to shift the signal down along the x-axis to perform the down-conversion and recover the baseband signal. You can then use the inverse Fourier Transform to convert the frequency-domain data back into a time-domain signal. Do you think you could improve my answer? Maybe you have your own Fourier Transform-related question? Please contact me at: podcast@howthefouriertransformworks.com.
And so to the final section of this episode and an update on where the “How the Fourier Transform works” online course stands at the moment. Visitors to the course website may have noticed that the advanced part of the course covering the Fourier Transform, the DTFT, the DFT, and the FFT is now available to patrons only. This is because I have begun adding the new video lectures to this section. More videos will be added as I complete them. The videos, the course, the podcast, and all the other material I produce on the Fourier Transform take a lot of time and effort to produce and it is only through the support of patrons that I can devote the time and research needed.
The patrons-only section grants early access to the course as it is being built as a thank you to those of you who are helping me on a monthly basis to provide this resource. So far I've completed three videos covering the changes which were made to the Fourier Series to turn it into the Fourier Transform. I'm now working on the fourth video all about the Discrete-Time Fourier Transform. I'm trying to think of a great location in which to film this lecture. So far, I've demonstrated the idea of infinite limits by delivering the lecture from an asteroid in space. I've demonstrated the difference between a discrete and continuous functions by first walking across a spotlit stage and then across a floodlit one.
I'm trying to choose a setting for each lecture that will illustrate visually what that lecture is about. My current idea for the Discrete-Time Fourier Transform is to deliver the lecture from inside a clock. Big Ben would be a cool place to film, although I don't think I would ever get permission to film there, so I'm looking for some plate shots or a 3D model of the interior and I'll overlay myself into place using green screen.
I’m also planning a complete overhaul of the patron section of the course to include worksheets and answers covering the whole course so you can practice what you are learning in the lectures. For non-patrons, the basic course containing videos covering Fourier’s history, the core concepts of the Fourier Transform, and up as far as convolution and how the Fourier Series works, is still freely available on the website. If you would like to become a patron of the course, please visit my Patreon page at https://howthefouriertransformworks.com/patreon
To receive updates about the course, notification when the next podcast episode is available, and other freebies which I release from time to time, please subscribe to the mailing list at: https://howthefouriertransformworks.com/mailing-list. So that's it for this episode. In the next, we'll be taking a look at how the Fourier Transform is used in the world of sound engineering. Until then, keep your questions, ideas, and suggestions coming by contacting me at podcast@howthefouriertransformworks.com
I look forward to hearing from you.
2021-08-20