In electronics, for instanc,

you have a lot of equations that describe the inner workings of electronic circuits.

Here you have an RC network, and

the voltage at the output is described by this law of charge or

discharge of the capacitor that would be something like this.

So what makes digital signal processing interesting?

In these two examples you can see that the mathematical model that we used to

describe the signals, is that of a function of a real variable time.

This is a standard model for what we call analog signals.

But if we go back to the motion of a project dial we see that this equation

has only one degree of freedom.

The only thing you can choose is the initial velocity, but

then the shape of the signal will always be the same.

And similarly in an RC network, you can choose the value for the resistor and for

the capacitor, but the signal will always have the same exponential decaying shape.

Now if I show you a wave form like this, you probably have seen this before and

you can guess that it is some sort of representation of a sound or speech.

And if we were to apply this mathematical model to this kind of information.

The question is, what is the function that describes the sound?

Well there's no really an easy answer for that.

What I can try to do is record this information, and

people have invented extremely sophisticated devices to do so.

So for sound, for instance, I could come up with a record player, I could come up

with a tape recorder, and then if I want to measure a temperature signal.

Then I would have to come up with a mechanical system that

drags a pencil on a piece of paper to record the evolution of temperature.

And to capture photographs, I will have to invent a camera.

But you see every device is specific to a certain signal.

It will record any information but

it will not let me manipulate this information easily and in a generic way.

In other words, the recording device will give me something like a picture,

something like this.

But it will not answer the question,

what is the function that describes the phenomenon?

This is the problem with analogue signals.

The big paradigm shift and the power inherent to digital signal processing,

is that we're moving away from an analog model for signals like this.

So we're not asking, what is the function that represents the signal anymore,

we're just moving to a recording of the values of this function.

And represent the phenomenon just as a series of numbers.

And in particular, for digital signals, these numbers are integers.

The so called digital paradigm is composed of two fundamental ingredients,

discrete time and discrete amplitude.

And let's start with discrete time because this is really the paradigm shift in

the way we perceive the world.

When you were in elementary school,

you were probably asked to run a little experiment.

You recorded the temperature every morning, and

then plotted it on a piece of graph paper.

And you obtained a plot that probably looked like this, so

you look at a little point on the mercury scale.

And plot it in the right position on the graph paper.

And when you do this for a number of days, the resulting representation,

it seems pretty reasonable.

It makes sense to measuring the temperature once, or at most, twice a day,

is a pretty descriptive representation of the weather.

But can we really do that, can we really splice up

time into a series of discrete instance, and not lose information?

The question is really tied to the very nature of time.

And this is something that philosophers have been grappling with

since the beginning of intellectual speculation.

One of the most famous analysis of the problem of time was conducted by

Augustine of Hippo in the 5th century.

And his analysis led to, fundamentally, a negation of the existence of time.

Because the past has already happened, the future is unknown, and the present instant

cannot be pinpointed the moment I mention it, it's already become part of the past.

And so Augustine said time does not exist.

Centuries later, one of the last philosophers to try and

systematize the world in philosophical terms, Immanuel Kant.

Solved the problem of time, but says that time and

space are fundamental categories of the spirit.

Fundamentally we are hardwired to project a notion of time and

space on everything that we observe.

But perhaps the best known philosophical reflection on the concept of time,

was performed by Zeno of Elea in 5th century BC.