A digital electrical signal may have only two values: High and Low. These two values are expresses mathematically as 1 and 0, and electrically they could be, for example, +5V and 0V DC logic voltage levels. Now, I have to point out that, electrically, a digital signal may be expressed in many ways, such as: -15V; +1.5V; +5V; or +24V for a mathematical 1, and as 0V; +3V or +15V for a mathematical 0. The idea to remember is that the electrical representation of the mathematical 1 and 0 it is not important, and we could use any pair from the above presented series, or we could implement custom Low and High DC voltage levels, particular to our application. The true reference it is only the mathematical pair, 0 and 1, and please note that these two numbers are the entire set of the Base 2 Mathematical System.



Now, the beauty of this Base 2 Mathematical System is that it is very simple, and we can easily use it in electronic circuits; all we have to do is to implement the High and the Low DC logic voltage levels mentioned. Possibly this notion I use, “logic", may be confusing, and I should better explain it. The Base 2 Mathematical System uses only 1 and 0, and this set of numbers can be directly referenced to the “Mathematical Logic System", which deals also with only two notions: True, and False. All these mean that when we talk about digital electronics, logic operations, or binary numbers, we refer in fact to the same thing; in other words all of the above may be perfectly referred to the Base 2 Mathematical System.

Let’s take a practical example. Suppose that we have a field switch, and it is wired to a processor pin. When the switch contact is open, the processor will sense 0V on that pin. We use that 0V DC logic voltage level to expresses it a 0 in firmware, and, possibly, in software. When the switch is closed, a +5V DC is applied to our processor pin, and we use that status as a 1 in firmware and software.

The true beauty is that, by using the Base 2 Mathematical System, we can represent any number possible, this time belonging to other Mathematical Systems; it is just a matter of “translating" numbers from one System to the other, but the end result it is exactly the same. This brings an interesting development. We have seen that we can express, totally, the Base 2 Mathematical System numbers into electrical, digital circuits; this means that our digital circuits are in fact everything we need, in order to work with any number, and with any mathematical operation! If we want to work with a large number, say 45632, we use a set of 16 Base 2 numbers 1011 0010 0011 0101, which may also be replaced by the High and Low states of 16 digital electronic circuits—nice, aye?

This is the actual meaning of the “binary code": it refers to using sets of binary numbers, or digital circuits, in order to translate, or to convert, any Base 10 numbers. In programming, we frequently use Base 10 numbers—also known as decimal numbers—but also Base 8, and Base 16; in the end all of then are reduced to sets of Base 2 (binary) numbers.

Now, in real life, many of the signals we work with are analog in nature. This means that, if a digital signal has, say, either 0V or +5V, an analog signal has any value in this range; for example +2.71635V. We would like to work, mathematically, with analog signals, and, in order to do this, we need to convert those analog signals to Base 2 Mathematical System. For this we use a set of digital electrical circuits, for example 8 of them, and this allows us to express any voltage between 0V and +5V in 256 increments. In other words, we can measure and express our analog signal in 19.5 mV increments, which is fairly good. By using a larger set of digital circuits, say 32 of them, the increment becomes 76.295 uV (micro Volts); this is way better, but we could easily use 64 or even 128 digital circuits for this conversion.

Because hardware digital electronics and binary numbers are exactly the same thing, we can use them interchangeably. Hardware digital electronics, firmware, and software work with the same Base 2 Mathematical System, and everything we do in software or in firmware, for example, may be also done in hardware. Now comes Digital Signal Processing (DSP) to add the analog signals to the Base 2 Mathematical System.

The first thing DSP does to an analog field signal is to transform it in sets of digital numbers; this is an easy task done automatically by the Analog-to-Decimal Conversion (ADC) hardware module, which is often built-in microprocessors. Now, we need a two Parts strategy, or method, to handle this ADC conversion, and things are like this. Part 1: in order to reproduce an analog signal in digital format we need an ADC conversion frequency at least double than the frequency of the analog signal—this is the Nyquist theorem, and the conversion frequency it is named, sampling. So, if we have an ADC module working at 10 MHz we are going to measure only the analog signals having a frequency of maximum 5 MHz. Part 2 says: in order to work properly with analog signals we have to group them into frames; for example, one frame may contain 1024 ADC conversions performed at 10 MHz.

Excellent! So, one frame contains 1024 mathematical numbers, representing the analog signal transformed into digital numbers, with a 10 MHz sampling frequency—what do we do next?

It happens that analog signals are in fact built as a mixture of many of other signals, and this is due mostly to interference, noise, harmonics, etc. For example, suppose that we measure an audio signal; this is a range between 16 Hz to 16 KHz at its maximum. However, the measured signal contains frequencies between 5 Hz and 5 MHz. The idea is that we want to “clean" our audio signals, so that only the 16 Hz to 16 KHz will remain—this is called “filtering".

In hardware we can build analog filters, only that they are rather limited in performance. Here comes DSP: because we have frames of the analog signal expressed in numbers, we can use mathematical operations to filter that signal. In principle, the signals in “time domain" are “transformed" into “frequency domain" using mathematical operations; then, the frequencies are filtered mathematically, and then the signal is transformed back into “time domain". Once we have our signal properly filtered, we can use it the way it pleases us most; for example we could make it an analog signal again, using a Digital-to-Analog (DAC) module, this time. The entire Digital Signal Processing is in fact very easy to implement and master; even fun!

Another two aspects are important to note here. First; in order to understand and work with the mathematical operations related to DSP you need a good book, and I recommend: “The Scientist and Engineer’s Guide to Digital Signal Processing" by Steven W. Smith. To my knowledge, this is the best book existing on this subject, and a “must have" one. The second aspect is related to practical implementation of DSP.

The best way of learning DSP is by working with it. What you need is a DSP processor, say dsPIC30F4011 built by Microchip®, then you need to have a learning kit, or PCB, with the electronic circuitry implemented—at least partial. Next, you need to learn to control your dsPIC30F4011 machine, because it is a monster with thousands of machine states; for this you need a good tutorial book, and the HCK V2.2 (Hardware Companion Kit V2.2). Due to my shy nature and to the expected modesty, I cannot divulge the name of this excellent and unique tutorial book, or the name of the author, but I can assure you that you will find it at Corollary Theorems; my home site.

This tutorial book will show you how to perfectly control your dsPIC30F4011 machine; in fact it helps you understand the entire hardware, firmware, and software design process. Once you will read it and experiment with your HCK V2.2, you will have, again, two alternatives: one is to start working with DSP at processor level, using DSP firmware mathematical routines; the second is to use the HCK V2.2 and the dsPIC30F4011 machine only to send the frames of the analog signal to PC, and there you could use other software programs for DSP as is MatLab®. This is all you need in order to become and expert in DSP fast, and it is, possibly, the cheapest option available today, in the entire world.

Fact is, managing your way up through the Hi-Tech domain it is not easy, today; it is also very expensive, and it is rather demanding. However, there are few nice shortcuts … Find them.

All the best to you!