\require{AMSmath} \require{eqn-number}

# Digital Signal Processing (DSP)

(last update:   17 Feb 2013)

Digital Signal Processing (DSP) is ubiquitous. If you have a cell phone, you have a device which uses some pretty sophisticated DSP techniques. But DSP has found its way into appliances large, and small alike. If you own a modern automobile, that too is crammed full of DSP.

DSP covers a broad range of signal processing. The question might be where to start?

Perhaps the best place to start is with an example of a signal, and how DSP might be applied to it.

However, before I do that it will be important to keep something fundamental in mind.

#### Numeric vs. Analytic Approaches To DSP

Much of the grunt work of DSP is done numerically. This may sound obvious, since the "D" in DSP stands for Digital. However, the concepts used to perform the digital processing of DSP are rooted in analysis.

Analysis goes way back. Long before the Internet. Long before computers, Long before the telephone.

Among others, a guy named Fourier was trying to figure things out. Doesn't really matter what he was working on, you can look that up for yourself. Fourier came up with a way of breaking things up, mathematically, such that something complicated became simple. We'll dig into one of his ideas/techniques shortly. His technique, as well as techniques developed by others, made it possible to describe things in a way, such that it was possible to completely (more or less) account for all aspects of the problem being worked on. Prior to Fourier, doing so was either most difficult or impossible.

It may help to think of analysis as something smooth, with no roughness at all. Like a pond or lake when there is no wind, and not a ripple on it. No gaps... no lumps... bumps or anything. Just a smooth surface...

Now pick up a stone and toss it into the pond. Don't look at the splash, just look at the ripples after the splash. The smooth undulations spreading out in all directions. Analysis can describe those ripples mathematically. You can picture them in your imagination easily but describing them mathematically is tough.

Why describe them mathematically? So that you can take related phenomenon, and predict what may happen when you use them, without going to great lengths to produce them. You can also extend the analysis to other things, far more complicated than ripples on a pond.

The only problem with analytic solutions are that they too can become intractable. They may be good for getting insight into complex phenomena but the analytic solution may be next to useless when it comes to building something. It may not be much use as an engineering solution.

Analytic approaches point us in the correct direction. Numeric approaches get us where we want to go, in that direction... approximately.

Let's go back to the ripples on the pond.

With the exception of the ripples, the pond is featureless... right? Absolutely smooth. No roughness, no graininess. Just the ripples. For the purposes of this discussion, analysis doesn't allow for texture... sort of.

But wait... what is water made of? Hydrogen and oxygen, H_2O. That means there is some amount of graininess after all. There isn't an endless regression of finer, and finer steps. The "featureless" surface of the pond, other than the ripples, isn't so featureless after all.

If it isn't featureless, that implies we can count individual things (H_2O molecules). There may be quite a few of them but we can assign a number to each individual one.

Now we have a collection of NUMBERS... the flat smooth, featureless pond, with ripples, is now a collection of numbers. It is numeric in its nature after all.

If we have a bunch of numbers, we can add, subtract, multiply, and divide them. We can also group them. We can associate some group, with another group. A group can have few members or many members. You can have a group of one H_2O molecule. You can have more than one group of one H_2O molecule. You can have a single group composed of all the H_2O molecules in the pond.

Thanks to Analysis, we can do much more than that. Why? Because Analysis works on groups as well as on entire ponds.

All we need to do is choose our groups properly. If we do that, we can use numbers to analyze the features of the pond. The pond, with ripples, which was easy to visualize but hard to quantify, now has a means by which it can be numerically described.

That means we can predict complex behavior. It may take a whole lot of calculating but computers are good at that. We just need to tell the computer what to do, and then let it do its thing.

DSP is a set of techniques we can use to tell computers how to do complex analysis, with a known degree of accuracy.