[MUSIC] In a conversion from analog to digital we have the following mechanism. So given a desired, Frequency, Of conversion, And let's denote this as 1 over the desired period, which we're going to call the Ts star. The ADC samples its input, converts it to digital, and puts that value, At the output. So you can think of having a converter, Which is an ADC, And that converter will potentially be measuring the output of the physics. Again here we have z dot equal, and then an output equal, so this output y would be what the converter is actually sampling. And that sampling and conversion and update of itself happens with a period Ts star or a frequency of 1 over Ts star. So in simple terms, this mechanism that we see here can be modeled in a simplified manner by performing the following operations. So initially we will trigger, An event, After, Every Ts* seconds. At each event, We are going to sample the input. Let's label the input to the converter as V sub s. Once we sample that input, we need to store it. And once we store it, we will need to provide some information to the output until the next store event or sample event occurs. There are many ways to do this, and a simple way it to use the idea of just keeping the output constant, or use the old value of the input sample constant until the next sample event. And that's why it's called zero order whole sampling. So when these tasks are defining what happening here, we can now consider different ways to model this system using a mathematical model. One way to do that is because we're going to be triggering an event every Ts star seconds, one potential idea, and what we're going to do, is define a timer. We're going to call it tau sub s, is the one that triggers the event So this is doing this task, okay. Every time that we take a sample, we need to store it according to this task, and here we want to have a memory state. And we're going to call it m for memory, s for sample, which is memory for samples. So since we need to count Ts star seconds every time that we want to trigger an event, I mean, we would like to do that in following that period, so that we generate this frequency. What we can do is to design the dynamics of this timer to be such that during continuous change of that timer, we have a differential equation that corresponds to evolving according to time. So tau s dot equal to 1, this means that tau s, as a function of time, is equal to its initial value plus time. And every time that there is an event which corresponds when tau s = Ts star, we will actually generate a change that is discrete, meaning we will reset the timer to 0 when tau s = Ts star. So this will describe the event triggering mechanism that would make the ADC start the conversions. If we assume that the conversions take no time and that there is no quantization, and that our converter can reproduce any real number applied to its input. Then this memory that we have here will be, at each event, will be updated to the value of the input. Denoting the input as v s, as we did up here, then the memory will be reset to the value of the input. And since we want to keep this constant, this value constant with the output, we might as well make the memory state during continuous change of a timer to have 0 derivative. So that now ms(t) = ms(0) or the first interval. Once it is reset, it will be reset to the value of the input from where it will remain constant and equal to that value after the reset, and the mechanism continues over and over. So what we have right here, by using a memory state and timer, we have arrived to a model that uses internally a memory state, ms. Its output is ms. And the dynamics of this model are given by a differential equation with two states and a difference equation with two states, okay. The events that trigger the difference equation law, the reset of tau and the update of ms is given by this condition. And then, in any other situation, whenever tau s is not equal to Ts star, we would actually allow our system to evolve according to this differential equation, and this differential equation right here. So our model of the converter is no more than a combination of differential equations with difference equations where the difference equations are executed whenever the timer has reached its value, Ts star, which is defining our frequency of sampling. [MUSIC]
