Fourier analysis

This video just has a plain transcript, not time-aligned to the videoWe're now going to use a series expansion approach to get from a digital signal in the time domain to its frequency domain representation.
All the signals are digital.
We understand how series expansion works, in a rather abstract way.
We're now going to make that concrete, as Fourier analysis.
Let's just recap series expansion.
We saw how it's possible to express a complex wave, for example this one, as a sum of simple basis functions.
We wrote the complex wave as equal to a weighted sum of sine waves.
Let's write that out a little more correctly.
We have some coefficient - or a weight - times a basis function.
These basis functions have a unit amplitude, so they're scaled by their coefficient and then added together.
We add some number of basis functions in a series, to exactly reconstruct the signal we're analysing.
So this is the analysis: the summation of basis functions weighted by coefficients.
Notice how those basis functions - those sine waves - are a series.
Each one has one more cycle in the analysis frame than the previous one.
These are coefficients and we now need to think about how to find those coefficients, given only the signal being analysed and some pre-specified set of basis functions.
Here is a series of basis functions: just the first four to start with.
I want you to write down their frequencies and work out the relationship between them.
Pause the video.
The duration of the analysis window is 0.01 s.
The lowest frequency basis function makes one cycle in that time, meaning it has a frequency of 100 Hz.
I'm going to start writing units correctly: we put a space between the number and the units.
The second one makes two cycles in the same amount of time, so it must have a frequency of 200 Hz.
The next one makes three cycles, that's 300 Hz.
And 400 Hz.
I hope you got those values.
It's just an equally-spaced series of sine waves, starting with the lowest frequency and then all the multiples of that, evenly spaced.
If I tell you now that the sampling rate is 16 kHz, there's lots more basis functions to go.
What's the highest frequency basis function that you can have?
Pause the video.
Well, we know from digital signals that the highest possible frequency we can represent is at the Nyquist frequency, which is simply half the sampling frequency.
The Nyquist frequency here would be 8 kHz.
We'd better zoom in so we can actually see that.
There we go: this waveform here has a frequency of 8 kHz.
We can't go any higher than that.
Fourier analysis simply means finding the coefficients of the basis functions.
We need somewhere to record the results of our analysis, so I've made some axes on the right.
This horizontal axis is going to be the frequency of the basis function.
Because we're going to go up to a basis function at 8000 Hz (that's 8 kHz), I'll give that units of kHz.
On the vertical axis, I'm going to write the value of the coefficient.
I'm going to call that magnitude.
Here's the lowest frequency basis function.
It's the one at 100 Hz.
So I'm going to plot on the right at 100 hertz (that's 0.1 kHz, of course) how much of this basis function we need to use to reconstruct our signal.
How do we actually work that amount out?
We're going to look at the similarity between the basis function and the signal being analysed.
That's a quantity known as correlation.
That's achieved simply by multiplying the two signals sample by sample.
So we multiply this sample by this sample and add it to this sample by this sample, and this sample by this sample, and so on and add all of that up.
That will give us a large value when the two signals are very similar.
In this example, if I do that for this lowest frequency basis function, I'm going to get a value 0.1.
Let's put some scale on this.
Then I'll do that for the next basis function
That's going to be at 0.2 kHz and I do the correlation and I find out that I need 0.15 of this one.
Then I do the next one, 0.3 kHz, and I find that I need 0.25 of this one.
Then the next one, 0.4 kHz, and I find that I need 0.2 of that one; and so on.
I've plotted a function on the right.
Let's just join the dots to make it easier to see.
This is called the spectrum.
It's the amount of energy in our original signal at each of the frequencies of the basis functions.
We now need to talk about a technical but essential property of Fourier analysis, where the basis functions are sine waves (in other words pure tones).
They contain energy at one and only one frequency.
That means that any pair of sine waves in our series are orthogonal.
Let's see what that means.
Take a pair of basis functions: any pair.
I'll take the first two, and work out the correlation between these two signals.
So multiply them sample by sample: this one by this one, this one by this one, and so on, and work out that sum.
For this pair, it will always be zero.
We can see that simply by symmetry.
These two signals are orthogonal.
There is no energy at this frequency contained in this signal, and vice versa.
The same thing will happen for any pair.
The correlation between them is zero.
There is no energy at this frequency in this waveform, and vice versa.
This property of orthogonality between the basis functions means that when we decompose a signal into a weighted sum of these basis functions, there is a unique solution to that.
In other words, there is only one set of coefficients that works.
That uniqueness is very important.
It means that there's same information in the set of coefficients as there is in the original signal.
It's also easy to invert this transform.
We could just take the weighted sum of basis functions and get back the original signal perfectly.
So Fourier analysis, then, is perfectly invertible, and gives us a unique solution.
We could go from the time domain to the frequency domain, and back to the time domain as many times as we like, and we lose no information.
We've covered, then, the essential properties of Fourier analysis.
It uses sine waves as the basis function.
There is a series of those from the lowest frequency one (and that frequency will be determined by the duration of the analysis window) up to the highest frequency one (and that will be determined by the Nyquist frequency).
We said Fourier 'analysis', but this conversion from time domain to frequency domain is often called a 'transform'.
So from now on we'll more likely say the 'Fourier transform'.
The Fourier transform is what's going to now get us into the frequency domain.
That's one of the most powerful and widely used transformations in speech processing.
We do a lot more processing in the frequency domain than we ever do in the time domain.