So, so far we've been considering continuous functions. But in structural biology, we're always going to be working with digitized functions. So, just to recall in our minds what we're talking about, we have analog versus digital images. So, imagine that we have something as a function of x and we want to, to take a picture, for instance, of an object whose electron density goes from low to high in this kind of a pattern. That's an analog signal. But, in fact, we're going to image it on a detector full of pixels. And, so as a result we will measure a digital signal. And, so we can imagine that our detector has a series of pixels. Here I've numbered them from zero to ten. Say it's a CCD camera with ten pixels across. And so what we record is we sample the analogue signal in each pixel and give it a discrete value. So for instance, in this pixel the average value of the signal across that pixel is something like 0.3. And then we go to the next pixel. And we find the average value of the function in that pixel, and we assign it that value and et cetera and et cetera. We sample the function at each of these different pixels and we build up a quantized image, a digital image. And so our digital image, right, will be just a series of ten discrete numbers. So let's think about that digital image. It's a one dimensional image. A one dimensional array with ten values. And we can, it with ten, ten numbers. We can calculate its Fourier transform. If we send that array to a Fourier transform routine in the computer, it will return very nearly the same number of values, ten numbers plus two extra, and I'll talk more about that. And what it sends back to us from the Fourier transform routine, is a series of amplitudes and phases. So I, I list those as A and P. And there are in this this return, six sets. Zero, amplitude and phase of the first. Amplitude and phase of the second, the third, the fourth, and the fifth. So these numbers give us the amplitudes and phases of five sine waves, and these are the five sine waves that are needed to reproduce this digital function. The first term, A naught, is the amplitude of that sine wave with an infinite wavelength. Which is essentially telling us what is the average value of the function all the way across the array? And that is often called the DC component, because in electrical signal processing there are direct currents and there's alternating currents, and a direct current has a, a constant value. And so from that field, we, we borrow the terminology that this first term is the DC component and it has a particular amplitude. The next number here, and by the way, the phase of that DC component doesn't matter much, because the wave is flat, and so it doesn't matter if it's shifted left or right. It's the same contribution no matter where it is, so I didn't highlight that, but the computer will send back a number in that, that position of the array as the first phase. Then the next number is the amplitude of the fundamental frequency, namely the sine wave that has one wavelength across the array, or across the box that we're reproducing. The next number is the phase of that fundamental frequency component. Telling us how to shift that sine wave left or right to best match this original array. The next number, amplitude sub two here is the amplitude of the, the first harmonic, which is the name that we use to represent the wave that has exactly two wavelengths across the box, or across the array. So this is a sine wave that oscillates up and down twice in this period. And the next term is the phase of that second harmonic. The next number that comes back is the amplitude of the second harmonic, or the sine wave with three wavelengths across the array, across the box. The next term is the phase of that second harmonic and, of course, the pattern just continues until the end. Now this last term in the array that the computer sends back is an important one. It's called the Nyquist frequency, named after a person named Nyquist, and so the last numbers that the computer sends back are the amplitude and phase of the Nyquist component. Now what makes it special is that, as you can see by its index, it is the wave that oscillates up and down five times across this box, this array, with ten values inside of it. In other words, a sine wave that oscillates five times within ten pixels is a sine wave that is going up in one pixel and down in the next. And up in the one pixel and down in the next. And up and down. And up and down. And up and down. Turns out, that that is the highest frequency component that is present in the arrimage, in the image. There are no higher com, higher frequency components, because they couldn't be represented in a digital image with only 10 pixels. You can't have within a digital image with 10 pixels, you can't have for instance, a wave that's oscillating many, many times all within a pixel. How could that ever be represented without sampling it more finely? So the Nyquist frequency is the, the sine wave that oscillates up and down every other pixel in your image. In this case, because there's only ten pixels in the original image, the Nyquist frequency is a sine wave with a 5 wavelengths across that image. But of course if our image had 1000 pixels across it, the Nyquist frequency would be the sine wave that had 500 oscillations across the image. It's every other pixel. Now an important thing to notice is that as the Fourier transform of our original image, which had ten numbers in it, the computer sends back as the Fourier transform a set of ten numbers plus two. These five amplitudes and phases are the amplitudes and phases of the fundamental, the first harmonic, the second harmonic, the third harmonic, and the last harmonic present, which is always called the Nyquist frequency. So the amplitude and phases of five sine waves, plus we get an amplitude of a DC component, which is just the average. Density within the image and its phase which really is irrelevant. And so we basically, the Fourier transform of an array with ten values is very nearly ten values, except that instead of actual density values per pixel. If we get the ten number back into Fourier transform, are the amplitudes and phases of the five sine waves that are present in this image. And the two forms of represent, representing the function are interchangeable. In other words, if I have these numbers, I can do an inverse Fourier transform and perfectly recover my original array. And of course, if I have the original array, I can calculate its Fourier transform. So the Fourier transform is a different representation of the very same function. It's like saying the same thing in two different languages. The Fourier transform comes in the language of sine waves, amplitudes and phases, and the original function is usually densities, as a function of position. Now, we can obviously plot the value of the original function. For instance, if this is x, and this is the value of the function y, or. And the function looks something like that, it turns out that we can make an analogous plot of at least the amplitude components in the Fourier transform. And the way we'll do it is we'll set up a plot where this axis is amplitude, and this axis is called spatial frequency. And if we say that the frequency of our first component 1 here, this first amplitude A1. If, if it had a value of say here, we could say, we could plot its, its value here. And then we could, on this axis of spatial frequency, we could plot. Let's say there's just a little bit of that second, of the first harmonic. Let's say there's a lot of the second harmonic. There's a medium amount of the third harmonic. And let's say there's a lot of the Nyquist frequency. We could be plotting these values. So that's A1. That would be A2. This would be a A3. In other words, each of these amplitudes present in the Fourier transform, we could plot them along an axis and show graphically how much of each of these components was present in the original function. Well, if we do that, we call this kind of a graph real space. And we call this graph, we call this reciprocal space. Okay, let's move on. And look at some specific examples of one dimensional functions and their transform. Transforms are also, are often also called spectra. And so let's start with this one. Here is a function, F of x. Here's the x-axis. And the function is simply an amplitude times cosine of x. And so it's just a cosine function. And we can ask, what is its Fourier transform? Now if we were to plot the Fourier transform, now this vertical axis is essentially amplitude. And now, usually, what we plot is not actually the amplitude, but the amplitude squared. Or the intensity, or the power of each of those components, more or less. So we're plotting intensity along this axis, and this axis is now spatial frequency. And so among all the spacial frequencies that could possible be present in this function, it turns out there's only really one term in this function. Because it's a simple cosine term cosine function itself. There's just one spatial frequency that has that is present there. And it's present at the specific spatial frequency k naught. Because k naught here is the frequency of the cosine term. And the amplitude turns out, or the intensity, turns out to be pi times this amplitude. That's the intensity that's required in order to get this function. Now, if we make a plot of the components present in this original function, it turns out that we really can't tell whether this is a cosine wave traveling this direction. Or a cosine wave traveling the opposite direction. In other words, if we regard a cosine wave moving in this direction as having a positive spatial frequency, we can't tell that from a cosine wave. That would be moving in the opposite direction, which we could regard as having a negative spatial frequency. So, if we were to plot what waves are present in this original function, there are really two equivalently good answers. One, is that there's one wave present, with a positive spatial frequency plus k naught. And there's another one present, at a negative spatial frequency, minus k naught. And so, you'll always see a mirror image across the origin. It just means that it's really one wave, but we can't tell which, which wave it is, either the positive direction, or the negative direction. So let's look at another example, slightly more complicated. Here we're going to look at the function f of x is equal to an amplitude times the cosine of 3 k naught times x. So here again, an amplitude of A, but now it's oscillating three times as quickly as this function. And the question is, what will its Fourier transform look like? And the answer is this. Here again, we'll plot intensity along this axis, and spatial frequency along this axis. Now there's only one wave present here. It's not the sum of a lot of waves, it just one wave. And it's a single cosine wave, whose frequency is 3 k naught. And so as a result, its Fourier transform just has one, one component here at the spatial frequency, 3 k naught. Now there's also another one at minus 3 k naught that it could also have been. And so here's, here's the mirror image on the other side of the origin. Now how about this one? Here f of x is equal to A. So it's a constant function, just straight across. What is its Fourier transform? Well, if you'll remember, a flat line can be representeded, represented as a wave with an infinite wavelength. And so its Fourier transform can be represented this way, as a single delta function at the origin, meaning a wave with 0 spatial frequency. Zero spatial frequency is the same as saying infinite wavelength, which means that it doesn't oscillate at all. It's just flat. And so, this is the Fourier transform of a flat function. A slightly more complicated example. Here we have a function, which is an amplitude, times 1 plus a cosine wave. So basically, it's just a cosine wave, but it's been lifted off the x axis, so that its av, average value is now A. What will be, what will its Fourier transform be? The answer is that there are basically two different components present in this function. First of all, because the average value of the function is non zero, meaning that the function is up above the X axis. One component present, is a sine wave with infinite wavelength, which is just a flat line that lifts, that expresses the fact that the average value is not, is non zero. And then there's one cosine wave superimposed upon that. And it has a spatial frequency of K9, and so that component shows up in the plot here, and then it's negative equivalent over there. And so, when you have a more complicated function, with multiple components, the Fourier transform separates those components, and displays them individually in reciprocal space. Now this can be done with, with any periodic function. So for instance here is a step function. And we, we previously looked at how you could calculate a Fourier transform of a step function, and separate into an infinite number of, of components or sine waves. Those can be represented in a plot in reciprocal space. So here we have intensity on the vertical axis, and we have spacial frequency K along this axis, and it turns out that if we were to plot each different component present here. We would need a lot of this one. And less of this one. And a little bit less of this one, and less and less. And then, we would need to have, an inverted amount of a few of these components here. And in the limit of a continuous function, or an infinite number of samplings. And so, we're sampling this function infinitely finely, what we would find is that the amplitudes would follow a, a curve like this. The amplitudes that were needed in a course, it looks symmetric across the vertical axis because, we can't tell whether any of these components are going forward or backwards. And so we express this by saying the Fourier transform of the original function, F of X is equal to, we typically represent the Fourier transform as a capital F, whereas the original function in real space had a small F before a transform capital left as a function of spatial frequency is actually this curve. Okay, and this can be done with other curves as well, here for instance a pyramid function, or triangle function,. It's Fourier transform looks like this.
