[MUSIC] So this is a model of ADC of this zero-order hold type. Let's validate this model. We have a variable Ms and we have a timer tau s. The variation we're given by this. So we count time and we memorize the state. And this is when tau s is in the range 0 to Tss star. Now when we do have an event, Then what we're going to do is to update the memory state to the input which was Vs. And we would reset the timer to 0. So now suppose that this is my model of my ADC. Again this is of the 0 or their whole type. And we're simplifying quantization delays and acquisition time effects. And we provide an input. So let's say that the input is a sinusoid signal. We would like to plot the output to the system, and make sure that the model captures the mechanism rule i2 model. So let's say that the input is a scalar, and as a function of time is given by this signal. And then continue. If we assume that the initial volume of the timer, Is equal to 0. And that this is the value of the constant Tss star, which corresponds to the frequency we would like to generate. And if we assume that the memory state initially is equal to 0, then when we execute this model for this given input, what's going to happen is that the timer is going to grow according to this differential equation. Because the timer is far away from Tss star which again is a positive number given by this quantity, this amount. And when the timer reaches Tss star, what's going to happen is a new event is going to occur. So, because Ms at 0 is equal to 0 and its derivative is equal to 0. If we were to plot Ms, it will be this signal all the way to here. So this is Ms, and whenever the event [INAUDIBLE] the sampling time tower has reached this value, then what we're going to do is to update Ms to this value. So, at this point, Ms will jump to this value, okay? That will be after the first event. The timer will reset to 0 and then we will need to wait another interval of length tau s the star so this would be 2 times tau s the star. Where, at this time, we'll have another event. Now, in between these two points, Ms was reset to this value, and now according to differential equation, it will remain constant all throughout until the next event. And at that event, we will have again, that Ms is map to the input. And this continues periodically as we expect. Each of this event is occurring at multiples of Tss star. So each of these separations is length Tss star. So this is Ms, and the input is Vs in the other color. So the model is capturing at least from this initial condition, it's capturing the behavior of sampling the input, put in the output, hold it until Tss star cycles, and then repeat. One thing that we might like to explore is when tau s initial is not 0 but is larger than 0, okay? And for this system to operate, the range of timer, as we wrote it, is for tau is to be between 0 and Tss star, okay? So we're going to keep it in that range. And let's consider the case when the memory state is equal to 0. Still, okay? So what's going to happen now is that because now my tau s has started at a different value, okay, we could actually plot the timer right up here. And I'm going to look in a minute. If the timer initially set at different value, the event would occur sooner than tau ss star. The first event after that, because the timer would be reset to 0. We will have that amount of time in between events from there on. So if we were to plot the timer for the initial conditioning equal to 0, maybe I should on white. Then if this is tau s then this is when the timer was resetting. And so on. Now with the different initial condition, what's going to happen since this value, Tss star is the same. Let's say that the timer started at half way Tss star, then the first event will occur right there. And then we'll have a reset, and we will be shifted by half a period throughout, The run. And as you expect, now the events are going to be shifted. Because now the event is going to occur here so my new Ms output will be now given by what we had earlier. But now at this very time, it's going to be reset to this value. And then remain constant all the way to that event. And then to this value and then remain constant all the way to that event. And then continue, Throughout. So it's just shifted half a period. [SOUND] So the effect of the initial condition generates an initial interval that is not necessarily a length Tss star. Is essentially that initial interval is of length Tss star- the initial value. When the initial value is 0, the length is Tss star, but if it is not, it's whatever this amount tells you. And after the first event, everything is happening at Tss star seconds. Good. Now, we can consider the other situation where now this variable is initialized at a different value than 0. 0 was convenient because it's the value of a signal and when Ms is different than 0, let's say some positive or negative number. What's going to happen is that the initial length of this amount will be equal to initial value, whatever that is. So for instance, for the case of 0 initial timer, if Ms is positive, and let's say this particular value. Initially, it will remain all the way to here. And then after the first event, it will synchronize with the plot of Ms for the case of Ms equal to 0. So you will have a transient of your output, which it seemed depending on the input value. We have control typically on how we initialize this system. The initial conditions are tau s on Ms. However, from a dynamical standpoint, we will like to make sure that our algorithm works no matter how it is initialized. We will not want to have a restriction on initialization, because that initialization might not always be possible, in particular because of perturbations. Or even of having the time to figure out what the initial value would be. [MUSIC]
