The first model that we want to understand is a discrete-time model. A discrete-time model is a model that will allow us to relate inputs to outputs to a system. The system would be a discrete-time system. The inputs that will be provided to the system will be defined over discrete time, K. Let's call the input to the system, V. So this V is a function of K, and a different case might have different values. So let's say that this is the value of K equals zero. This is the value of K equals 1. This is K equals 2, K equals 3, K equals 4. What we expect to have is an output but to also define a discrete time instance. You may notice that's K. And the first value of the output will also depend on what is called the initial state of the discrete time system. This state is typically a vector that captures all the meaningful variables of this system in order to describe these change over time. So given an initial state and initial input, so this will be V at zero in initial state, then you can actually compute what the initial value would be. Or in some situations, you can actually assign an arbitrary, depends on the type of model you're thinking of. Let's say that the output of these is these Greek symbol, theta. Now, in order to compute the new value at time equal 1 of that output, then we will need to use the information of the input at time equals 0 of the initial state. So let's say that this value right here is the result of using these values here along with this value here. This is what provides this information here. And you can keep doing this for every K. So you can actually generate a signal of the output that will depend on the previous value of this state and the previous value of the input, and you can continue these typically over all K. So this case, remember, take value in that set. The question is, what is the most efficient or a good way to actually relate this input-output signals? And the way that we're discussing this course is a difference equation model. In a difference equation model, we are given a function. Let's call it capital G. The model of the input-output relationship is giving us the following. We will denote the state of the system as X, and the relationship between the state, the input, and the output is given by equations, where the value of the state after each iteration of time K is a function of G. So this will be G, and using the current value of X and the current value of the input. So if this is my current state and this is my current input, I will actually use the upper script plus to say or denote what the new value of the state will be. So this will correspond to the value after the first iteration or the second iteration and so on. While the output of these machine might also depend on some or this model, might also depend on the current value of the state and maybe the current value of the input. So some notation here, this X plus, so the plus denotes new value of X, and H here will also correspond to some output function. So we can either actually have them here, so give them functions H, and H, we will have a model uniquely defined by this. So this is my discrete time model. Now, one of the things that you probably want to realize is that the function can be thought of a iterative map on the value of the states and of the input. So, in the same way that we did this construction here, now we can go back and pretty much solve for what is the evolution of the state over time K. So this is what's happening inside this box. So if this is K equals 0, and this is 1, and this is 2, and this is 3, and this is 4, and we keep doing this, assume that for this model, the initial state is this value. So it's the value at zero. Then according to this law, the value of the state of time K equals to 1 will be given by G, evaluated at the value of the state at time equals 0 and of the value of the input or time equals 0. So then this value, whatever this expression gives, will be worth let's say, assigns this value right here for X. And, now, we can compute the same way G2, which will be now for X2, which will be now X at 1, V at 1. Now from this equation, we already know that X1 is equals to this, so then this will enter here, and we'll end up with G of G of X0 V0 comma V1. And you can keep doing this construction to build the different values of X for the time K.