It's hard to be a computer user in this day and age, particularly one who is as technologically aware as somebody who builds their own audio gear is likely to be, and not be aware that binary arithmetic is out there.
So if you already understand this aspect of the subject, you may prefer to go away for a while, because I haven't finished writing about it, and I won't be moving on to the other stuff until I've dealt with it.
Now it may be that you've looked at some of this stuff before and gone blank, or failed to understand or retain this information for some other reason, but this wouldn't be a tutorial if I didn't attempt to explain some things.
We're constantly confronted with digital encoding. Telecommunications companies sell us bits and they sell us bytes, but what are they, and why are they the way they are? If we fail to confront these issues we will be unable to design logic circuits.
Why bytes?
A byte is eight bits wide. Or long. Why?
Guys who were looking at the situation at the time decided that was the best way to go.
8 is 2^3. 2 to the power of 3. This makes it a kind of mystical number when designing digital circuits. Digital designers love powers of 2. It's highly convenient because in this case (8 bits) it lends itself to the design of a basic comprehensive hardware computer and it's convenient for human comprehension. Eight bits provides a convenient sized decimal number to contain a wide variety of information by binary coding. Eight bits provides a convenient bridge between human and machine arithmetic via hexadecimal coding. Hexadecimal coding is the fundamental language whereby human requirements are translated into hardware results in electronics when large numbers of numbers have to be processed handraulically.
A typical ADDER takes its inputs from 2 ports, typically 8 bits wide, and outputs its result on a third port, again 8 bits wide, but with the proviso that it has in addition a carry or borrow bit. You probably recognize the idea of carrying or borrowing from the arithmetic which you're already familiar with. We'll come back to the ADDER later on.
What we're dealing with is the nitti-gritti detail of how a bunch of numbers can be made to perform a given hardware task. If you're not prepared to think long and hard about what this means, there's not much point in reading any further.
So we're going to attempt to learn binary and hexadecimal arithmetic by osmosis, by doing, and by seeing how it's a convenient way of getting things done electronically.
The good news is that much of the hard slog can be left to a calculator, the one built in to Windows will do, but a handheld calculator is a useful thing to have.
If you're not familiar with the Windows calculator, you can access it by going Start Menu -> Run and typing 'calc' <return>. You'll probably see this usage more than once here, it's common in programming and other computer applications. You type what's inside the speech marks '' and press return, or Enter, the big square key with the hooked arrow on it. There won't be many people reading this who aren't familiar with this convention.
That is how the calculator can be accessed in Win XP and earlier versions, perhaps somebody else could write in and tell us how to get it up in later versions of Windows, I noticed just today that the 'run' feature has moved from the Start Menu in WIndows 7, and I have refused to give up my existing installations which use XP, so I'm not familiar with some aspects of Win 7 & 8.
You need to put the Windows calculator in scientific mode. Go View -> Scientific. When you see these arrows (->) that means you have to use the menus at the top of the window, or in the Windows Start Menu.
Memory is organised in bytes. A byte memory is an array of gates (latches) which are capable of storing a number in the form of eight binary states. The individual states are called bits. States are another name for 1's and 0's. Such an array of latches is also called a register.
So bits and bytes actually mean the numbers which can be represented on a port, but also the hardware locations in which a number can be stored, either temporarily (volatile) or (semi)permanently (non-volatile).
We like to call this kind of dual-usage of terms 'operator overloading' in computing, and it is an aspect of abstraction, the process whereby digital representations are translated to humanly identifiable quantities for the convenience of programming. Operator overloading means the the same word is used to mean 2 different things, but you have to figure out which one is meant from the context. This makes it much easier for me to write about the subject because I don't have to keep writing 'byte in the sense of an 8-bit number' and 'byte in the sense of an 8-bit register'.
A typical address bus of an 8-bit computer system is 16 bits wide, each address comprising storage for 8 bits. This means that in such a system 65536 bytes of storage are provided. Each byte potentially contains a number between 0 and 255.
A bus is a bunch of wires connecting corresponding bits of 2 or more ports.
The address bus may access a single contiguous series of memories, or it may access an additional array of port addresses and a third array of video memory.
One other important idea we need to become familiar with is the idea of a clock.
When the number on any port or in any register needs to be moved to another part of the system its progress is synchronized by a clock.
A memory requires to be read for it's contents to be known. This means that the required address is displayed on the processor address bus and the read pin on the memory is set (to 1) (or, more usually, unset [to 0]). If the memory is a standalone chip, very often some other conditions must be met, (it's not uncommon to have 2 dedicated pins designated as Chip Enables) and when the system clock toggles the memory's contents are briefly displayed on the Data Bus.
A memory can similarly be written by impressing the number to be stored onto the data bus, the memory address to be written on the address bus, unsetting or setting the write pin as the particular chip requires, and when the clock pin toggles, the memory is set to the number expressed on the data bus.
Because, historically, each memory location (bit) had a cost, the organisation of memory was a critical issue in the design of microprocessor systems.
It was almost instantly recognized that an 8-bit wide data bus offers the first useful plateau in performance versus width.
Hence no systems in common use do not owe a legacy to the 8-bit format for the storage of basic data, and a knowledge of how numerical data is used to represent other forms of data, such as an alphabet, is crucial to understanding how computer systems work. Binary arithmetic is fundamental to understanding this subject, because we need to know how to turn the binary numbers into the more familiar decimal numbers because those are the ones we're familiar with and we find them easier to recognize and deal with.
Hexadecimal is a half-way house between decimal numbers and binary numbers.
Binary numbers look like 00110100 01010111 10100100 11001011 00101001 00101011 10001001 10101110.
It's very difficult for a human to deal reliably with numbers when they're expressed like this, but the numbers 00000000 - 11111111 just mean the numbers 0 to 255. Somehow it's easier to comprehend the meaning of 121 than 01111001, but you can probably see that 79, the hexadecimal representation, is easier to deal with than the decimal when transcribing numbers by hand, which is sometimes unfortunately necessary, and was more so in the early days of computing.
Hexadecimal numbers look like 34 57 A4 CB 29 2B 89 AE (these correspond to the binary numbers above).
Each of these pairs of hexadecimal numbers is just another way of representing the numbers in the binary string. Each 4 bits (nybbles), upper and lower, of the byte can be represented by a number between 0 and fifteen and each of the numbers between 0 and 15 can be represented either by a number or a letter.
0000 - 0 - zero
0001 - 1 - one
0010 - 2 - two
0011 - 3 - three
0100 - 4 - four
0101 - 5 - five
0110 - 6 - six
0111 - 7 - seven
1000 - 8 - eight
1001 - 9 - nine
1010 - A - ten
1011 - B - eleven
1100 - C - twelve
1101 - D - thirteen
1110 - E - fourteen
1111 - F - fifteen
So we can take 11101010 and divide it up; 1110 1010 and look at the table and go - EA
Or we can take the number 11101010, put the calculator in bin mode (by clicking the 'bin' radiobutton), paste in the number, and click the hex radiobutton and get the hex value EA or click the dec button and see that the number is 234 in decimal (human) terms.
This is why bytes are 8 bits long, it's becaue 8 bits can easily be translated to 2 hexadecimal characters and the biggest number you can get into 8 bits is 255, which is not too big and not too small. And because 8 is a power of 2.
Let's just recap.
What we're talking about is a system for the storage, retrieval, movement and manipulation of numbers, specifically 8-bit binary numbers. These numbers range from 0 to 255 decimal, and are this size for a number of reasons, some historic, and some practical. Each number is represented by voltage levels on a port (a bunch of wires or terminals) or in a register. A port can be internal to a device or external in the case of pins. Ones (1's) are represented by 5V and zeros (0's) are represented by 0V. The bits making up a number represent decimal numbers according to their position in the byte:-
(128), (64), (32), (16), (8), (4), (2), (1)
...and the decimal number they represent can be reconstructed by adding up the decimal numbers represented by each bit, with every bit that is zero (0) counting as zero, 0.
So 1010 0101 = 128+0+32+0 .... plus 0+4+0+1 = 165 or A5 Hex.
These numbers are frequently represented in hexadecimal form, because they are more readily understood, transcribed or typed by humans when in that form.
The operations of the system are controlled by a clock. A clock is a square wave generator, or frequently the square wave itself.
Depending on the design of the individual system, operations often occur on the rising edge of the clock (the sampling instant), although systems are sometimes clocked on the falling edge, or more rarely on both rising or falling edges of the clock. Thus a number on an input port at the instant that the clock toggles may be stored in a memory location according to the voltages present on the terminals.
Clocks typically operate at frequencies of 1 million transitions per second or higher. This means that we can turn any pin in a system on or off with great precision, which is how we control events in the real world, or alternatively get information into the system about the state of affairs in the real world with great accuracy and detail.
It is for these reasons that understanding how to manipulate binary and hexadecimal numbers with a degree of fluency is critical to designing microprocessor based systems, and that is why we deal with the subject so extensively when learning to design such systems.
The systems we're going to use for learning purposes are 8-bit systems because (nearly) all subsequent systems use extensions of principles first elucidated in 8-bit systems, and the simplest Microchip PICs are 8-bit systems drawing on the design legacy of preexisting 8-bit systems.
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Edited by wakibaki - 12/12/12 at 4:19pm
