We have seen these pictures before. These are the two views of the world. On the left, you see the analog world, the one we live in and it's shown by a picture of the Earth taken by the Apollo Mission. On the right side, you have a digital view of the world. This is taken by digital camera, then very strongly pixelized and the values are actually also discrete. So this is the view that you have of the world inside a computer. In short, this is a contrast between an analog continuous view versus a discrete and digital view of the world. We can show this in a block diagram form. We live in an analog world, so input and output are from and to the analog world, but inside, we process on the computer which is discrete time. The input is x of (t), the output is y of (t), but inside the box, we process sequences x of [n], y of [n]. Examples of such boxes are, for example, the processing we do for MP3 music or what we do on the digital camera. Other examples are the following: You have a digital world, for example, you synthesize an image and you process it and then show it on the computer screen. This is a typical example in computer graphics or in video games. So the input to the box is a sequence, it is processed into an output, which is a continuous time or continuous space image. Sometimes, the box is used to take a decision. So the input is analog from the analog world, the output is discrete, for example, to decide something, should we take an action or not? The processing, again, inside the box is done on sequences x[n] from the input y[n] to the output. This is typical, for example in monitoring applications, is a certain level of temperature reached, and therefore in control systems. Let's summarize. The digital worldview is one of arithmetic, of combinatorics, what is usually referred to as computer science, but also is a world of Discrete Signal Processing. The analog view is a view of calculus, integrals and so on, distribution, system theory, continuous time, analog electronics. So you can see these are two sides of a coin, because we always have to deal with one and the other when we want to process signals in the real world. To make it slightly more mathematical, the digital view is. You have a countable index, so x[n], where n is an integer. So we have sequences. And we like to have sequences that have r square summable, so x[n] is an l2 of (Z). We have a frequency that is limited between minus pi and pi. And we have a Fourier Transform called, it's a discrete time Fourier Transform which maps from l2 of (Z), to capital L2 of ([minus pi to pi]). The analog worldview is that we have a real time in second, t is measured in second, we have functions on the real line, x of (t). We like them square summable, that's what we called capital L2 of (R). So frequency is given in ratings per second and is unlimited, on likens on discrete time case. And we have a Fourier Transform that maps L2 of (R) functions into L2 of (R) functions, as we will see shortly. So how do we bridge the gap between these two worlds? So here's an example, where we have for example, computer synthesis, generates a sequence x[n]. We have a sound card on the computer. The sound card has a system clock, so every Ts second it will generate an output, based on x[n]. And this output will be driving a loud speaker. In the other direction, you are speaking into a microphone like I do right now. There is a sound card, it has a system clock, again with a sampling interval Ts. And we get samples x[n] based on what was measured by the microphone. So we can see that there is a very intimate relationship between going from continues time x(t) to x[n] as a sequence, that's called sampling. Or from x[n] the sequence to x of (t) that's called interpolation. And we have to be able to understand very clearly when we can go from one to the other, when these two are tightly related or they are not faithful images of each other.
