Continous and Discreet Signals

Continuous and Digital Signals
Most physical signals we encounter are essentially continuous. Electric voltage, radio wave magnitude, how much petrol is in your tank.

Continuous signals are not divided into units. If you were to try and express them as a number, you would need an infinite number of 0.00000's to express them accurately. Not a large number, an infinite number.

DSP is about making those continuous signals into the kind of discrete units that computers can process, and then working with them.

Discreet signals are signals that are divided by time, so say I checked how many hairs were on your head ones a second, the once a second bit makes it discreet. Its about sampling a continuous signal once every so often.

The "so often" is called a sampling period. In advanced applications it can vary but for the purposes of this course we will look at regular sampling periods, say once a second, or once an hour.

Digital, as opposed to discrete signals, have the extension that the value you take when you sample is fitted to the nearest class. I'll explain that a little, say I want you to tell me how tall you are, you might tell me 6 foot. You wouldn't tell me 6.252321ft. Even if you did that wouldn't be exactly accurate. Digitizing a signal is about "quantatizing" it to the nearest level. Quantatizing devices are called ADCs, and are discussed elsewhere, but you should know a little about their imperfections.

Noise in digitization
Digital processing, as discussed later, is used among other reasons because it can be used in ways that have much better noise immunity than analogue techniques. If you are talking to someone in a loud room, you can often hear that they are talking, but not what they are saying. If you were communicating in binary, either shouting or not shouting for one second, you would have much more luck getting your message across. I recommend you do not try this. It tends to get one ejected from restaurants. Its acceptable behaviour for digital electronics however.

Although it is often overcome by the noise immunity or facility allowed by digital techniques, it must be noted that digitization inherently introduces noise.

Some of this noise is intuitively understandable. Consider noise to be loss of useful information. If you round a number to the nearest ten, you lose some information. In digitization you choose your rounding so that it doesn't lose information that you Need, if avoidable.

Further however, real ADCs are not ideal, and introduce strange non-linearities in their conversion.

MORE HERE: OFFSET, STEP NON-UNIFORMITY, ETC

Sampling in terms of maths
In order to follow some of the later parts of DSP, you need to understand sampling from a mathematical perspective. Maths is all about modelling the world, and in order to understand say DCT, you need to know a little about Delta functions.

Dirac Delta
What's 5 times 1? 5. That's the core of the Dirac Delta functions use. Dirac delta is a mathematical tool for converting a continuous signal into an instantaneous (i.e.) discrete signals.

The dirac detla, who's symbol is $$\delta(x)$$, is a continuous signal. It has infinite magnitude for an infinity small period of time, and is otherwise zero. And it has an area of 1. It has an area of one because area is height * width, and $$ \frac{1}{0} \times \frac{1}{\inf} =  1$$

That means that if you integrate it with another function, the value you get out will be the magnitude of the function at the point where the dirac delta went high.