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Basic digital principles
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YOU VE SEEN HOW DIGITAL SIGNALS DIFFER FROM ANALOG SIGNALS. THE MANIpulation of digital signals is known as digital logic. Digital logic consists of many sometimes innumerable pulses racing around. It can be staggering in terms of quantity, but it is simple at heart: 1 or 0, high or low, yes or no. Suppose a few electronics engineers from, say, 1950 could ride a time machine to the present day. They would be flabbergasted at the speed and compactness of digital logic circuitry. Some might be struck with disbelief. Others would say, I told you so, or I knew it would happen. And then there would inevitably be the one who would quip, Is this all the further you ve gotten Well, digital-logic circuits are getting smaller, faster, and more sophisticated every day.
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Any number, such as 35912, can be rendered in some other number base or modulus. Then it s written differently. But it s always the same quantity.
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The decimal system
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In the decimal number system, each digit 0 through 9 represents either itself, or else itself times some power of 10. The value of the digit depends on its position or place in the number. In the case of the number 35912, the digit 2 is in the units place, and has a value of simply 2. The digit 1 is in the 10s place; it takes the value of 1 101 10. The digit 9 is in the 100s place, having a value of 9 102 900. The digit 5 is in the 1,000s place, and represents 5 103 5000. The digit 3 is in the 10,000s place, and has a value of 3 104 30,000. The total value is the sum of all these: 2 10 900 5,000 30,000 35,912.
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556 Basic digital principles A decimal number can be represented in digital form, but it requires 10 different possible states. Any decimal number has a binary equivalent composed of digits that are all either 0 or 1: just two different possible states. Much simpler!
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Modulo what
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You probably learned modular arithmetic in elementary school. If so, you ought to remember working with modulo 8 (the octal system), or modulo 16 (hexadecimal), or base 4, or 12, or 20. In computer electronics, octal and hexadecimal numbers are occasionally encountered, in addition to decimal numbers. At various times, special-interest groups have tried to get the whole world to switch to some modulus besides 10. Some people think base 12 should be used (isn t a baker s dozen a more logical standard than the number of toes on a baby s two feet ). Others believe that base 20 makes the most sense (it s the total number of fingers and toes, including thumbs, on a typical baby). Still others argued for base 8 or base 16 because these values are elegant : they can be repeatedly cut in half all the way down to 1. Would you like to change the world over to, say, octal numbering because it s easy to count rhythmically by eights Or to base 12 because then a gross would be an even 100 Lots of luck. Even the metric system hasn t caught on that well.
The Binary system
Binary numbers are in modulo 2. In this system, the digits are all either 1 or 0, and the places go in powers of 2. Therefore, to the immediate left of the units place is the 2 s place (21), then the 4 s Place(22), then the 8 s place (23), and so on. In Table 30-1, powers of 2 are represented up to 215.
Table 30-1.
Exponent 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Powers of 2.
Decimal value 1 2 4 8 16 32 64 128 256 512 1024 2048 4096 8192 16384 32768
Binary number value 1 10 100 1000 10000 100000 1000000 10000000 100000000 1000000000 10000000000 100000000000 1000000000000 10000000000000 100000000000000 1000000000000000
Logic signals 557 In general, the nth place from the right has the decimal value 2n. The total decimal value, for a given binary number, is the sum of the decimal values of each of the places. Converting decimal numbers to binary form can be done using Table 30-1. Suppose, for example, that you want to convert 35912 to binary notation. First, find the largest decimal number on the table that is no greater than the decimal number you wish to convert. In this case, it is 32768 215 . From this, you know that there will be 16 digits in the binary representation of this number, one for each place 20 through 215. Mark off 16 slots or spaces on a sheet of paper (quadrille graph paper is perfect for this), and place a digit 1 in the left-most space, representing 215. Now, use Table 30-1 to determine which number can be added to 32768 to get the largest decimal number that doesn t exceed 35912. It happens to be 2048 211. Place digits 0 in the slots for 214, 213, and 212. Then place 1 in the space for 211. If you continue this process, you ll ultimately get the binary number 1000110001001000. This 16-digit binary number is equivalent to the decimal 35912 8 64 1024 2048 32768 23 26 210 + 211 215. The slots for exponents 3, 6, 10, 11, and 15 each are filled with binary digit 1; the others are filled with binary digit 0. It is possible to have fractional values in binary notation, just as it is in decimal notation. The first place to the right of the point (perhaps best called a binary point rather than a decimal point ) is the 1/2 s place (2-1). The next place is the 1/4 s place (2-2); then comes the 1/8 s place (2-3), and so on. Thus, 0.001 in binary notation represents the decimal fraction 1/8. You can think of it as repeatedly dividing the size in half as you progress towards the right.
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