[MUSIC] Let's talk about the Nyquist sampling theorem here. The concept of sampling frequency is that the sampling frequency or the sampling rate fs, is the average number of samples obtained in one second. So samples per second which typically denoted as fs that is 1 over T. So T is period, as shown here. The Fourier transform of a sampling function is one dimensional case is that, so Fourier transform of 1D sampling function is that, delta x and delta x, so delta x is spacing, so that is the period of the sampling. And this is mathematically represented as a linear summation of shifted impulses, as shown here. So graphically, they can be represented like that where the sampling period is given as a delta x. And its Fourier transform has is another summation of another shifted impulses but now this shifted impulses has a scaling with delta u minus k over delta x. So scaling in the special domain with delta x is now scaled by 1 over delta x. Okay, so that is scaled by 1 over delta x, and then also magnitude is also scaled by 1 over delta x. So this is similar to the scaling property of Fourier transform. So the relationship between the shifted impulses and Fourier transform is another shifted impulses, and the spacing is determined as a 1 over delta x and also the scale. The magnitude is also scaled by 1 over delta x. So Fourier transform of an impulse train is another impulse train with an impulse gap equal to inverse of the original impulse gap. So as I mentioned before, the Fourier transform makes the relationship in the spatial domain, and also frequency domain, to be inversely related to each other. So if we have sample in densely, if it is spacing catches smaller, then the spacing on the frequency domain gets bigger. But if we don't sample densely, then this spacing is going to be bigger and then the spacing on the frequency domain, between these impulses okay. We're actually mathematically proving that it's not that simple, okay. But please, just remember that the Fourier transform of impulse train is another impulse train. And that spacing has inverse relationship to each other and this is the extension of the previous concept into two dimensional case. So Fourier transform or 2D sampling function which can be represented mathematically as shown here. And it's a Fourier transform. Again, another impulse train, but the spacing is 1 over delta x and 1/ over delta y. And the scale is also given as 1 over delta x, 1 over delta y. Okay, so Fourier transform of an impulse train is another impulse train with an impulse gap equal to inverse of the original impulse gap. So which is same for the two dimensional case, too. And we can use this concept to talk about the important concept of sampling. Okay, let's talk about the sampling for the one dimensional case here. So this is original signal, f(x). And that is, we want to sample this original signal by multiplying the function with the sampling function, delta f(x) as shown here. And then the sample version of the signal is fs(x) that is multiplied, so that is given as original signal f(x) multiplied by sample function dela s(x) as shown here. So then what will happen? On the spectrum frequency domain, and then let's say this f(x) is band-limited signal as shown here, so maximum bandwidth is defined here and here. And then sample version of the signal and that is Fourier transform is another impulse train as shown here, so another sampling function. But its spacing is equivalent as 1 over delta x. And then in the time spatial domain, they are multiplied and then in the frequency domain, in fact they are not multiplication, they are defined as a convolution. So we should not explain much here, so the property of a convolution is that, if we convert this function with shifted impulse. And then the result is that, this spectrum is replicated at each location, okay. That is the property of convolution. And then even if I didn't explain that concept in detail, but the spectrum is replicated at each shifted impulses as shown here. So this is spectrum, resulting spectrum of this sample version of the signal. So the spectrum Fs(u) is calculated by shifting the original spectrum F(u) to the location of u equals m over delta x, okay. And that is for all integer m and adding all shifted spectra and that is resulting our spectrum. So just sampling the signal in the spatial domain makes the frequency domain spectrum to be repeated as shown here, okay. So that is the basic concept of sampling theorem. And then, how densely should they sample the signal? And that is the important concept called Nyquist Sampling Theorem. Let's say, we have band-limited signal f(x) so maximum bandwidth is defined as a w. So it's defined of the +w and -w. And then when the sampling frequency, so sampling period is delta x and if sampling frequency is 1 over delta x. And if sampling frequency is twice the maximum bandwidth and then the spacing between the spectrum will be bigger. So there will be no overlapping portion, okay? So these repeat even if the spectrum gets repeated on the spectrum frequency domain after sampling, but still we can reconstruct original spectrum by applying for low-pass filter to the sample version over the frequency domain signal. Then, applying the low-pass filter will remove all of these digitally repeated portion and then applying for inverse Fourier transform to the low-pass filter signal will give us the original band-limit signal f(x). Okay, so this is important concept. So sampling should be performed with sampling frequency at least twice the maximum bandwidth or the original signal. So that is the important concept called, Nyquist Sampling Theorem. And then what happen if our sampling frequency is not big enough. So it's twice the maximum bandwidth of signal and that it is sampled partially as shown here. And then, the distance between this replicated sample spectrum will be and then, so there will be some overlapping portion between these replicated spectrum. And then we cannot recover the original spectrum even if we apply for low-pass filter, we cannot recover the original signal. So this portion is called aliasing. So again, the Nyquist sampling theorem states that the sampling frequency must be at least twice as high as the maximum bandwidth of the original signal for the signal to be represented accurately, or signal to be reconstructed accurately. Okay, so now let's try to extend this concept in the two dimensional version. And that is the same as the one dimensional case. So we are talking about images, so it's two dimensional. Original f(x, y) is given and it has band-limited signal, so F(u,v), and then that is multiplied by sampling function. And then, we can pick up the sampled values as shown here. And its spectrum is going to be replicated as shown here. And the spacing between these replicated band is, given as 1 over delta y along y direction, and 1 over delta x along x direction. This concept is the same as the one dimensional case. And then Nyquist sampling theorem also the same. So let's say this has a band-limited signal, so maximum bandwidth W along u direction, and maximum bandwidth of Z along v direction. And then, if we sample the signal with high frequency, high enough to be within twice the bandwidth of the signal along x direction and the y direction. And then there will be overlapping portion between the replicated spectrum. And then we can apply for low-pass filter to the acquired spectrum. And then applying for inverse Fourier transform, that will give us the original band-limited signal with no loss of that information, okay. The sampling frequency, so 1 over delta x and 1 over delta y. That is not big enough compared to the twice the maximum bandwidth of the original signal, and then there will be overlapping portion as shown here and here, in this figure. And then we cannot reconstruct original spectrum without most of the information, so. So again, the Nyquist sampling theorem states that the sampling frequency must be at least twice as high as the maximum bandwidth of the original signal, so for the signal to be represented accurately. So this is the aliasing for the MR imaging. So if our sampling frequency is not big enough, then there will be an overlapping portion, as shown here. So aliasing is an effect that causes different signals to become indistinguishable when sampled. So it also refers to distortion or artifacts of that results when the signal sampling is different from the original continuous signal, okay? So that is called distortion or artifact. So this is called aliasing artifact, so this is the original structure that we want to get. These are the structure, but our sampling was not high enough along this direction. And then there will be aliasing as shown here, okay? So this is called aliasing artifact that we sometimes observe when we do the MR imaging. Okay, this is the end of the course for the third week, see you next week.
