From Autocorrelation to Fourier
Dave Jesse on September 12, 2017

Just for fun

Here is a little interlude to get back to helicopter accidents – and do some more tricky maths in the process.


Here are two combs from the previous blog, which was about autocorrelation. You will recall that we compared one comb with another to see how good the match was.


Image 1

Fourier Series

This time we are going to do the same matching, but instead of using a blue comb we will use a sine wave like this:


Image 2


OK, it’s a rubbish match at this stage, but bear with me. If we adjust the number of cycles we can get much closer…



Image 3



…until magic happens at the right frequency…



Image 4



The sine wave matches the comb closely. How well this sine wave matches the comb can be plotted in the same manner as for the autocorrelation chart, but instead of the x-axis being displacement of one comb against the other, it is the frequency of the sinewave. The resulting graph is usually seen as a frequency response.


Image 5
          Frequency chart courtesy Sure Microphones



Of course, a microphone manufacturer is trying to capture all sounds equally, so this graph is very smooth, with minimal peaks.



Unsurprisingly, helicopters make loud noises at particular frequencies. Here is a frequency response from an accident investigation where the sound was recorded on the cockpit area microphone. The sound peaks relates to the main gearbox gear meshing, which is at a higher frequency than the relatively quiet rotor blade noise.


Image 6


Later on the same flight, the sounds become louder and at higher frequencies as the gearbox became more distressed. Analysis of these frequency responses helped the accident investigators to identify the cause, and propagation, of a defect in the gearbox.

Image 7

Health and Usage Monitoring

With helicopter transmission monitoring, we can use the fact that each gear in the transmission has a known number of teeth to predict the frequency of the vibrations. If we use the fact that the gears are round, we can use sinewaves with an exact number of cycles – just like the number of sinewaves to match a given length of comb.


If you want to do hard sums, you can go to university. If you want to avoid sums, think how well combs and waves match. If it’s two combs matching, it’s called Autocorrelation. If it’s a comb and a sine wave it’s a Fourier Transform.