[MUSIC] We will talk about the Properties of Fourier Transform here. The Fourier Transform is a linear operator. So if f(x,y) and F(u,v) are Fourier Transform pair. And g(x, y,) and G(u, v) transform, let's assume. And then a1f(x, y) + a2g(x, y) and then what will be the Fourier transform of this linearly combined function? And then, its Fourier transform is another linear combination of their individual Fourier transform as shown here. So a1F(u, v) + a2G(u, v). Where a1 and a2 are two constants. So this property is true. So the Fourier transform is a linear operator. You can easily prove that based on the definition of Fourier transform. And this is translation property so, let's say, f (x, y) = F (u, v) are Fourier transform pair. And then translational property states that f(x- x0, y- y0) which is a function that shaped original f(x,y) along x0 along x in the amount of x0 along x direction, and the amount on y0 along y direction. So they are just a shifted function compared to the orignal f(x,y). And they are shifting in the spatial domain, and what will happen to each frequency component? And that is multiplication of this complex of. So e to the -j2 pi(ux0 + vy0). So that is frequency dependent is multiplied to the frequency component F(u, v). So again, translation in the imaging domain makes the multiplication of frequency component which is linearly modulated. The frequency component is linearly modulated, and then that is multiplied to the original function. So linearly modulated base means, I meant ux2 + vy0. So that is the frequency component that is multiplied to the original F(u,v) And this is conjugation property, and let's assume they are a Fourier transform pair. And then conjugation property states that f * (x, y) and its Fourier transform is F* (-u, -v). So this is conjugation property. So here, this assumes that f(x, y) can be a complex signal too. So if f(x, y) is a complex signal. And if it's complex conjugated, and then we consider a full transform of its complex conjugate then it can be represented as a complex conjugate of frequent domain, and also u and frequency component polarity changed. But in many cases, just if (x, y) is real, but in case of MRI, this f(x,y) could be complex numbers. And let's assume f(x,y) is linear, if we consider this as a image, then this assumption f(x,y) is linear, is true. If that is also true for MRI, if we consider that is on the image domain, but the complex domain is generally in the complex number for the case base data in case of MRI. So, if x, y is linear then G (u, v) equals F*(-u,-v), because this is conjugate symmetry, because if f(x,y) is real and each complex conjugate will be the same as the original f(x,y). So Fourier transform should be the same, F(u,v) and F* (-u,-v) so if f(x,y) is linear, and this property is true. So because of that, magnitude of these two component will be the same. So their magnitude will be the same, so symmetric magnitude. And also their phase two component will have opposite polarity, so this is anti-symmetric phase. This conjugation property is useful, if we have complex case based data and some portion is missing, then this property can be used to recover the missing data in the case based for the MR imaging. So this property is useful for MR imaging. And this is scaling property, so if f(x,y) and F(u,v) Fourier transform pair and the scaling property states that, if we scale x and y variables with the constant a and b, and then what will happen to its frequency component? And then, in the frequencies domain, so that frequency component (u,v) up with are also scaled but it's inversely scaled as a, 1/a and 1/b as shown here. And also, the magnitude is also inversely scaled. So 1 over a and 1 over b. So it's taking the absolute values. So this is scaling property. So generally speaking, this property in the spatial domain, x and y, and the property in the frequency domain, (u,v) )with that they have an inverse relationship generally speaking. Well, if special cases a and b and -1, then f(- x, -y) then what will happen if we change the polarity of the variables x and y? And then the frequency component, (u,v) their polarity also need to be changed, okay? So reversing a signal in space also reverses its Fourier transform. So that is entirely true for the Fourier transform. Okay, so let's say x variable x, and variable y they are separable, okay. And then f (x, y) so this is imagery domain and let's assume they are separable as shown here. So, f1 (x) * f2(y). So, f(x, y) can be represented as a sample of product as shown here. And then, what will be its Fourier transform? And its Fourier transform is also a separable product. F1(u) multiplied by F2(u). So Fourier transform of a separable signal is also separable, okay? So these properties can be easily proved, by using the definition of Fourier transform. So if you have time, please spend time to prove these properties of Fourier transform.
