Fritz: All Fritz

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.T:ch.fmt .K:I. DIGITAL ARITHMETIC I. DIGITAL ARITHMETIC .I:binary Most modern computers are digital computers, based on the binary or base two number system. In a decimal or base ten number, the least significant or rightmost digit represents the number of ones, the next tens, then hundreds, and so forth. If we call the least significant digit the 0th then the Nth digit represents the number of 10Ns. In binary, the Nth digit represents the number of 2Ns. Thus 101 decimal means 1*102 + 0*10 + 1 while 101 binary is 1*22 + 0*2 + 1, or decimal five. When we refer to numbers in different bases we will identify the base by writing it as a subscript after the number. For example, 1110 is decimal eleven while 112 is the binary equivalent of decimal three. decimal: 10110 = 1 * 102 + 0 * 101 + 1 * 100 binary: 1012 = 1 * 22 + 0 * 21 + 1 * 20 = 4 + 0 + 1 = 510 .I:bit .I:byte Binary numbers use only the digits 0 and 1, represented in computers by transistors in either the on (=1) or off (=0) state. A binary digit is called a bit. A byte is an eight-digit binary number. bit: 0 byte: 11010011 (eight bits) A byte may be transmitted from one group of eight transistors to another over eight wires or lines, through each of which current will flow only if the transistor to which that line is attached is on. If we were to base a computer on the decimal number system, each transistor would have to be able to represent ten different states corresponding to the digits 0 through 9. This is done in mechanical adding machines by using gears with ten teeth but there is no easy way to do this electronically. .I:hexadecimal Binary numbers quickly get too long to write conveniently. It is much easier to write 123410 than its binary equivalent 100110100102. For this reason binary quantities are often represented in hexadecimal (base sixteen) to make them less unwieldy. This is done by grouping bits by fours starting from the right and converting each group to the corresponding hexadecimal digit as shown in Table 1-1. Since the hexadecimal number system reqires sixteen symbols for digits, the letters A through F are used to represent the decimal quantities ten through fifteen. Thus 100110100102 = 100 1101 0010 = 4D216 .I:octal Octal or base eight is also used as a shorthand for binary, but not as often as hexadecimal. A byte breaks neatly into two hexadecimal digits but makes two and a half octal digits. To convert from binary to octal, group bits by threes instead of fours and convert to digits in the group 0 through 7 (see Table 1-1). .I:base interconversion Converting back to binary from hexadecimal or octal is the same process in reverse. To convert binary to decimal, multiply each digit by the power of two it represents and add. Example: .- 11012 = 1*23 + 1*22 + 0*2 + 1 = = 8 + 4 + 1 = 1310 .+ You could convert octal or hexadecimal numbers to decimal by multiplying each digit by the appropriate power of eight or sixteen (see Table 1) and adding, but you may find it easier to convert to binary and from there to decimal. Converting decimal to binary is most easily done by repeated integer division by two. Arranging the remainders in the reverse of the order in which they were obtained gives the binary result: .- 13/2 = 6 r 1 6/2 = 3 r 0 3/2 = 1 r 1 1/2 = 0 r 1 --> 1101 = 1x8 + 1x4 + 0x2 + 1x1 = 13 .+ Binary numbers are added in the same way as their decimal counterparts. Starting from the right, each pair of digits and any carry in from the next lower place are added together to produce a single digit sum and carry out: .- step: 1 2 carry: 0 10 1st: 11 11 11 2nd: +10 +10 +10 sum: 1 101 ^ ^^ .+ .I:ones complement .I:twos complement .I:complement .I:subtraction Binary subtraction could be done like decimal subtraction, but this is not how the microprocessor does it. By first making the number to be subtracted a negative number and then adding it, the microprocessor does away with the need for separate subtraction circuitry. Fortunately it is quite easy to negate a binary number. Just change all the zeros to ones and vice-versa (forming the ones complement), then add one (forming the twos complement). Forming the ones complement of some eight digit binary number X is the same as subtracting it from 111111112 (try it). Forming the twos complement by adding one to the ones complement amounts to subtracting it from 1000000002. When we perform subtraction of a binary number by adding its twos complement, the 1000000002 reappears as a carry out of the highest place (which we discard): .- Y + ~X = Y + 111111112 - X + 1 = Y - X + 1000000002 .+ In other words, to subtract X from Y, change all of the ones in X to zeroes and the zeroes to ones, add one, add the result to Y, and ignore the carry out of the highest place. Example: .- 5 0101 0101 $ëëLÉèd∩└Σ H$═ô╚ê≤ê≥ 'æµ&Db╤'$$aéd≤≡Qÿææ°díD≤≤≤HττΓ'ττττττττττττττττττττττττττττπ#ππ#'α gπππ#'τα'τΓ88x `88x8gττττττττττ╔g└└'╞└─@τ└@╟┴g╟└τ└D┴D'└@DB'└τ┴╟D'╟'└D'°8gτττττττ╟Dτ╟└τ├└┴D'└@DB'╟'└D@'┴g└┴─'└─@└τ╞Aαgτ└┴┴'┴'°8gτττττττ╞└─τ└┴─'≡g╞A≡ gτ┬F'└D'╞Aτ┴'└D─τ└└'┬D┬D@τ└┴─'°8gτττττττ┬`τ└D'└─'╞Aτ┴'╟G╟@─'─@g└┴─'└A└─ gτ└┴τ─└τ╞A┬'°8gτττττττ└'╞g┬D┬D@τα#"a±##±'└└'α#"`1##±'┴g└└'╞└─@τ└@╟┴gor unsigned numbers from 0 to 25510. Either way amounts to 25610 different numbers. Notice that the computer doesn't need to know whether the numbers we are adding are twos complement signed numbers or unsigned numbers: .- 101 either 5 or -3 +010 equals 2 (signed or unsigned) 111 either 7 or -1 .+ .I:overflow When the result of an arithmetic operation is too big to be contained in the number of bits available, overflow has occurred. For unsigned arithmetic this happens whenever there is a carry out of the highest bit. For twos complement arithmetic, this happens when the carries out of the two highest bits are not equal. Example: .- carry: 10000 (The carry out of the highest bit is lost) 1st: 1001 (9 or -7) 2nd: +1010 (10 or -6) sum: 0011 (3. Should be 19 or -13) .+ Sometimes you can get away with mixing signed and unsigned numbers, but the usual overflow tests will not work. for example: .- 1010 10 unsigned +1110 -2 signed 1000 8 unsigned .+ Binary multiplication and division work just like their decimal counterparts, but only on unsigned numbers. To perform these operations on signed (twos complement) numbers, use their absolute values and work out the sign of the result later. .- 110 101 x101 110 /11110 110 110 000 110 110 110 11110 .+ .- Problems 1. Convert each number below to the other two bases: binary dec hex 1011 110010 101101 23 47 113 2C D3 .+ 2. Convert the decimal numbers in each exercise below to binary, work the problem with the binary numbers, then convert the answer back to decimal to check your answer. Use twos complement arithmetic for negative numbers. .- a) 5+7 b) 12+16 c) 17+23 d) 53+27 e) 7-5 f) 19-3 g) 25-10 h) 48-31 i) 9*6 j) 20*13 k) 35/5 l) 144/6 .+ .K:Logic II. LOGIC .I:gates .I:logic At the heart of all digital computers lie circuits called logic gates, each made up of only a few transistors. These are combined in different ways to produce most of the electronics in a computer. Logic gates get their names from their similarity to the operators of mathematical logic. Each has one output which is turned on only for certain combinations of its inputs, each of which may be either on or off. The actions of the three basic logic gates NOT, AND, and OR reflect the common meanings of these words. .I:inverter .I:NOT .I:AND The inverter or NOT gate is the simplest logic gate. If its input is on, the output is off, and vice versa. If we let 'on' stand for the logical value 'true' and 'off' stand for 'false' then the connection with mathematical logic becomes more apparent: the output NOT A is true if the input A is false and false if it is true. Similarly, the output of the AND gate is on only if all inputs are on. Thus, for a two-input AND gate, the output A AND B is on only if both inputs A and B are on. .I:OR .I:XOR There are two kinds of OR gates derived from the inclusive and exclusive OR operators. The inclusive OR statement A OR B is true if either A or B or both are true. The exclusive OR statement A XOR B is true if either A or B but not both are true. Thus, the output of the OR gate is on if any input is but that of the XOR gate is on only if the two inputs are different. .I:NAND .I:NOR The NOR and NAND gates produce outputs which are the inverse of those produced by the OR and AND gates. They are equivalent to NOT (A OR B) and NOT (A AND B) respectively, and are what would be obtained if the outputs of the OR or AND gates were passed through an inverter. Thus, A NOR B is true only when both A and B are false and A NAND B is false only when both are true. .I:truth table It is possible to completely describe the behavior of a logic gate by listing all possible combinations of inputs and the resulting outputs in a tabular form called a truth table. Here is the truth table for the logic operators described above, shown with the symbols most commonly used to represent these operators in logical expressons. Study it. NOT AND NAND OR NOR XOR A B A A.B A.B A+B A+B A+OB F F T F T F T F F T T F T T F T T F F F T T F T T T F T F T F F .I:adder, binary Logic gates may be combined to perform more complex functions, for example, binary addition. To add a pair of bits in a binary number, we must calculate a sum (adding in any carry from the next lower bit pair) and a carry out to the next higher bit pair. Let A and B be the bits to be added, S the sum, Ci the carry in, and Co the carry out. If we let FALSE stand for zero and TRUE for one, then we can simulate the behavior of the binary functions S and Co as follows: .- S = (A XOR B) XOR Ci and Co = (A AND B) OR ((A XOR B) AND Ci) .+ To prove this, let's work out the truth tables for these expressons. First set out all the combinations of A, B, and Ci, then calculate intermediate results, and finally combine these to get S and Co: .- #1 #2 #3 A B Ci A XOR B #1 XOR Ci (=S) A AND B #1 AND Ci #2 OR #3 (=Co) 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 1 1 0 0 0 0 1 1 1 0 0 1 1 1 0 0 1 1 0 0 0 1 0 1 1 0 0 1 1 1 1 0 0 0 1 0 1 1 1 1 0 1 1 0 1 .+ A circuit to perform binary addition according to the above equations is shown in Fig. 2-1. A circuit to add two bytes can be constructed by connecting eight of these end to end. .- Problems 1. DeMorgan's laws state that A AND B = (NOT A) NOR (NOT B) and A OR B = (NOT A) NAND (NOT B) .+ Prove these by working out the truth tables for the right and left half of each equation to show they are identical. 2. A circuit whose output will be on if the two inputs are equal (both on or both off), can be derived from the compound logical expression .- ( A AND B ) OR ( A NOR B ). .+ Can you come up with a shorter solution? 3. Draw schematics for the circuits which correspond to these expressions: .- a) A AND NOT B b) A OR ( A AND B ) c) ( A NOR B ) XOR C d) A AND B AND .+ .K:III. Microcomputer system overview III. MICROCOMPUTER SYSTEM OVERVIEW .I:central processing unit .I:memory .I:I/O .I:microprocessor .I:integrated circuit .I:microcomputer What is a microprocessor? All computers, micro or otherwise, have three main components: a central processing unit which does the computing, memory to store programs and data, and I/O or input- output circuitry to communicate with the outside world. A microprocessor is a central processing unit built into a single integrated circuit. An integrated circuit is a single chip of semiconductor material, the stuff of transistors, but containing up to millions of transistors. Before the advent of integrated circuits (also called ICs or chips), the central processor typically took up many circuit boards. A computer which uses a microprocessor is a microcomputer. .I:microcomputer, single chip .I:microcontroller Integrated circuit technology has progressed to the point that it is now possible to get a central processor, memory, and I/O circuitry all on one chip. These single-chip microcomputers or microcontrollers do not have enough memory to make them useable as general purpose computers but they are very cost-effective in control applications and are widely used in cars, appliances, and industrial automation. They are also used extensively in computers as keyboard, display, disk, and other I/O controllers to relieve the main processor of time-consuming I/O tasks. .I:hardware .I:program .I:software .I:instructions .I:language, high level .I:language, machine The interaction of the microprocessor, memory, and I/O circuitry, collectively called hardware, is controlled by a program (software) which resides in memory. A program is composed of instructions, each of which directs the microprocessor to perform a specific task. The process of running a program begins with the transfer of an instruction from memory to the microprocessor. Instructions in a high level language such as BASIC consist of a collection of English words, numbers, and symbols. In contrast, instructions in machine language, the native language of the microprocessor, consist entirely of binary numbers. Each microprocessor has its own machine language into which programs written in higher level languages must be translated before they will run. .I:address The size of the numbers used to represent machine language instructions depends on the microprocessor. All eight-bit microprocessors transfer a byte of data at a time between the microprocessor and memory. A machine language instruction will consist of one or more bytes stored in consecutive locations in memory. Each memory location has a unique address which must be supplied to memory by the microprocessor to specify the location of the byte to be transferred. Most eight-bit microprocessors use sixteen-bit addresses. .I:op code The first byte of an instruction is usually the op code, a number which specifies the operation the microprocessor is to perform. Some instructions consist only of an op code. In others, one or more bytes of additional information follow. These may be a number to be loaded into the microprocessor, an address, additional information specifying the type of operation to be performed, or some combination of these. There is no difference between the way data and op codes are represented in memory. If the microprocessor is expecting an op code, it will treat the byte it receives from memory as an op code. If it is expecting data, it will treat it as data. .I:bus Data and addresses are moved between the microprocessor and memory or I/O chips on groups of lines called buses. The data bus in an eight-bit microcomputer system consists of eight lines (wires or printed circuit traces), each of can be either on or off. In most microprocessors, a potential of 2 to 5 volts is on and 0 to .8 volts is off. The terms high and low are often used instead of on and off. Each data line and the pin or IC terminal to which it is connected are represented by the letter D followed by a number representing the magnitude of the bit it represents. D0 represents the least significant bit (the number of 20s or ones) and D7 the most significant (the number of 27s or 128s). If the binary number 10010010 is being transferred over the data bus then D7, D4, and D1 will be high and the others low. The address bus consists of sixteen lines which transfer addresses from the microprocessor to memory or I/O. .I:control signal .I:chip select signal The data bus allows two-way communication between the microprocessor and other system components. The direction of the data transfer on the data bus is indicated to memory or I/O by a control signal output by the microprocessor. This might be high for a read (from memory or I/O) operation and low for a write (to memory or I/O). Other control signals called chip select signals are necessary to direct the flow of data and addresses to the right chip. Usually these are active low, meaning that the memory or I/O chip to which the signal is connected will be selected if the line is low. .I:bus, tri-state When its chip select line is high, a memory or I/O chip is electronically disconnected from the address and data buses. Lines which can be electronically disconnected in this way are called tri-state because they can be on, off, or disconnected (high impedance). If all of the memory and I/O chips in a microcomputer system have tri-state buses, then bringing only one chip select line low at a time will result in only one memory or I/O chip being connected to the microprocessor at a time. This reduces output current requirements and prevents contention between two chips for the data bus. Problems 1. What is the difference between a microprocessor, a microcomputer, and a microcontroller? 2. What are the three main components of a computer? What does each do? 3. How does the microprocessor indicate which peripheral chip it wants to communicate with? How does it indicate whether it intends to send or receive data? 4. What do the data and address buses do? What are they connected to? 5. What are the different possible components of a machine language instruction?