[MUSIC] Let's talks about the basic signal functions here. So complex signals can be simplified as a linear combination of certain basic functions. So these basic functions are useful to understand the concept of Fourier transform and sampling that we want to discuss in this week. Okay, let's talk about the basic functions. So the most important basic function that you want to discuss is unit impulse function, so which can be defined with a rectangular function as shown here. So this rectangular function, so P delta tao t is our rectangular function, with height is 1 over delta tau and the width is delta tau, from here to here. So if we take delta tau to be 0, so make this delta tau infinitely small. So this area is just 1, because the width is delta tau, and the height is 1 over delta tau. So the area is going to be 1, and if we take limitation of delta tau to be infinitely small, 0. And then what will happen to this rectangular function? And then its width is going to be infinitely small, but its height is going to go to the infinity. But the area of this signal is not 0, so it's going to be 1. So area is 1. So this function is called unit impulse function, or delta function, or it can be called many different names. But it's typically denoted as this delta t. And also typically presented by this arrow, and we represent the value. So this value is the area of this arrow. So arrow does not have area, typically. So line does not have area. But this looks like line, but it has an area. So that's a little bit confusing to understand initially. So this is conceptual function, so that does not literally exist. So infinite at t = 0, 0 when t is not 0, the sum of total area is 1. So it's often called Dirac delta function, denoted as delta t. So any post that has area 1 and is too short and too high to be displayed, can be used to define this delta function, or unit impulse function. The division of delta t doesn't have to be based on a rectangular shape, or it can be used as a Gaussian or sinc. Also mathematical function, that does not use this in the real world and it's useful to understand the concept of sampling. Okay, next function is unit step function. So this unit step function, which is typically represented as a ut, and it has an age of value 0, if t less than 0. And it has a value 1 if t is greater than 0. And it has a value of 0.5, 1 over 2, when t equals 0. And this is defintion of a unit step function. And if we take derivative to this unit step function, then we can get delta function, okay? So if we take derivative and then 2ut then there will be 0 up to 0 here, and also 0s here too. Well we will have non-zero variables at t equals 0, if we take a derivative to this function. But that value is going to infinity, but it goes to infinity. But its value is going to be 1, if we take integration, okay? So delta t can be considered as a derivative of a unit step function. In other words, so this unit step function, u(t), can be derived by taking integration over delta t from minus infinity to time point t. The next function is rectangular function, it's typically denoted as rect function, rect, this one here, so the definition is as shown in this figure. If absolute t is greater than 1 over 2, then it's 0. And if it's less than 1 over 2, then it's 1. When [INAUDIBLE] equals 1 over 2, then its value is 1 over 2, okay? So this rect function can be represented by combination of two shifted unit step function, so unit step function of t = u (t + 1 over 2) okay? That is shifted toward left hand side, so this unit step function can be presented by this function and u(t- 1 over 2) is another unit step function shifted in step function, which is shifted like that. Okay, and subtraction between these two unit step functions can be considered as rect function t, as show here. Or it can be considered as a multiplication of two shifted unit step functions. In this case, opposite polarity, okay? So it has opposite polarity as shown here, like that, okay? So unit step functions u(1 over 2- t) can be considered as this function. So that is multiplied by this unit step function as shown here. And these two multiplications of these two unit step functions will give this rectangular function, which is obvious. And rect t over tau, so now time t is scaled by tau. And then that is also represented by combination of two shifted unit step function, as shown here. And also multiplication of two unit step functions, as shown here, which is obvious too. Okay, the next function is a sinc function. So this division of a sinc function can be represented in two different forms. So one is a sinc. So sinc function is typically represented s-i-n-c, which we read as sinc. So sinc(x), so that is defintion, is sin pi x divide by pi x. So this is typically used in the engineering field. Sometimes in some mathematical field, in some textbooks sinc x is defined as a sin x divided by x. But let's assume the sinc function is sin times pi x in this course. So this sinc function can be also used to define delta function. So delta t can be also defined as just taking limitation of delta tau to go to 0, and 1 over delta tau, sinc t over delta tau, okay? This can be also used as a definition of delta function. So Fourier transform of this rectangular function is sinc function. And you can prove that by using the definition of Fourier transform. And please try to do that, it's not that difficult. And the delta function can be also defined using the sinc function I mentioned. Okay, this is the table for the basic Fourier transform pairs. So for the one dimensional case, if we have signal, delta function, delta t. Then each Fourier transform is going to be just 1. And which is obvious because definition of Fourier transform is taking integration from minus infinity to infinity, original signal, f(t), which is replaced by delta t here. Delta t, e to -j 2 pi ftdt. And then delta t has variables only at t = 0. Otherwise if t is not 0, then all the signals become 0. So that integration is going to be just scaled to be if we put t = 0 and the a to the -a 2pi, 0 becomes just 1. So that integration gives the value just 1. And if it goes integration of delta t is going to be 1. If signal is just 1, it's constant and each Fourier transform is delta u. Okay, you can maybe prove it. You can use definition of inverse Fourier transform. Then you can prove this relationship too. So just delta t and just 1, they are Fourier transform pair. You may want to remember that. And delta t- t0, your shifted delta function. It's Fourier transform is complex sinusoidal signal e to minus the 2 pi u t0. So this can be considered shifting in the spatial domain, just to mention, in the property of Fourier transform. So shifting in the spatial domain is multiplying this phase into my own state, e to- piut2. So this is related to the shifting translational property for the Fourier Transform that we just measured, they are related to this property of Fourier transfor, for the shift in the delta function too. And Fourier transform or complex sinusoidal, which has a very complex sine, so that is also can be fourier transform is delta(u-u_0) shifting in the time domain is multiplying this complex sinusodial signal. Shifting in the frequency domain is multiplying complex sinosudoian in the time domain. Okay, and Fourier transform of sinusodial signal, sine or cosine, is going to be a linear combination of two shifted delta functions. And the reason, it can be proved easily because the sine function, cosine function can be represented as a linear combination of two sinusoidal signals based on the order of equation. So it can be easily also proved there. And rectangle function, rect (t) and its Fourier transform is a sinc function, and Fourier transform of sinc function is also rectangular function, as shown here. So Fourier transform can be considered as a pair, as shown here. So rect function and sinc function, they are Fourier transform pair. And delta function, shifted delta function, and also complex sinudsodial signal, they are also Fourier transform pair. And delta t and just constant value 1, they are also Fourier transform pair. Okay, here is mathematical formula that is used to prove the Fourier transform concept. Please go through step by step as shown here, just putting t = 2 and then delta t, take integration delta t, that is of the 1. And also this property can be also proved if we go through this mathematical procedure, which is not that difficult. And sinusodial signal can be represented as a summation of two complex sinusodial signal, as shown here. And that can be converted as shown here by using this property. And rectangular functions, sinc function, that can also be proved very easily. This is rectangular function, and taking integration and then we can easily calculate. So the result is sin pi u over pi u, and that is a sinc function. I believe you can easily follow these mathematical relations. So this is the extension of Fourier transform pairs in the two dimensional case. Which is obvious, because as I mentioned before, so two dimensional Fourier transform can be considered as a one dimensional figure apply for one dimensional Fourier transform. So these two are Fourier transform pair, and it's very similar, just extension to the two dimensional case.
