CSE 2300W DIGITAL LOGIC DESIGN

Transcription

1 CSE 2300W DIGITAL LOGIC DESIGN How this class fits into ECE/CSE/EE/CS curricula: Already had at least some computer basics and one programming language. This course will emphasize some of the major inner workings of a computer. Chapter 1 Introduction Section 1.2 Analog versus Digital Analog devices and systems process time-varying signals that can take on any value across a continuous range of voltage, current, or other metric, while digital circuits and systems take on discrete values. A binary digital signal is modeled as taking on, at any time, only one of two discrete values, which we call 0 and 1 (or LOW and HIGH, FALSE and TRUE). Digital computers have been around since the 1940s and have been in widespread commercial use since the 1960s. Yet only in the past 10 to 20 years has the real digital revolution taken place, spreading computers to almost all aspects of our lives. Examples include 1. still pictures 2. video recordings 3. audio recordings 4. automotive engines 5. telephone and communications 6. motion pictures 7. others? What advantages do digital circuits have over analog? 1. Reproducibility of results. Given the same set of inputs (in both value and time sequence), a properly designed digital circuit always produces exactly the same results. The outputs of an analog circuit vary with temperature, power-supply voltage, component aging, and other factors. 2. Ease of design. Digital design, also called logic design, is logical. No special math skills are needed, and the behavior of small logic circuits can be easily visualized without any special insights about the operation of its internal components such as capacitors and transistors. 3. Flexibility and functionality. Once a problem has been reduced to digital form, it can be solved CSE 2300W Digital Logic Design Fall

2 using a set of logical steps. For example, you can design a digital circuit that scrambles your recorded voice so that it is absolutely indecipherable by anyone who does not have your "key" (password), but it can be heard virtually undistorted by anyone who does. Very difficult to do using an analog circuit. 4. Programmability. You're already familiar with digital computers and the ease with which you can design, write, and debug programs for them. Much of digital design is carried out today by writing programs, too, in what we call hardware description languages (HDLs). These are contained in the CDs that came with your textbook. These languages allow both structure and function of a digital circuit to be specified or modeled. Besides a compiler, a typical HDL also comes with simulation and synthesis programs. These software tools are used to test the hardware model's behavior before any real hardware is built, and then to synthesize the model into a circuit in a particular component technology. 5. Speed. Today's digital devices are very fast. Individual transistors in the fastest integrated circuits can switch in less than 10 picoseconds (10e-12 seconds) and a complex digital device built from these transistors can examine its inputs and produce an output in less than a nanosecond (1e-9 sec.). This means that such a device can produce a billion or more results per second (e.g. a 4 GHz up). Keep in mind that light traveling in a vacuum travels about one foot in a nsec, and one inch in 85 psec. 6. Economy. Digital circuits can provide a lot of functionality in a small space. Circuits that are used repetitively can be integrated into a single chip and mass-produced at very low cost, making possible throw-away items like calculators, digital watches, and singing birthday cards. 7. Steadily advancing technology. When you design a digital system, you almost always know that there will be a faster, cheaper, or otherwise better technology for it in a few years. Clever designers can accommodate these expected advances during the initial design of a system, to forestall system obsolescence and to add value for customers. For example, desktop computers often have "expansion sockets" to accommodate faster processors or larger memories than are available at the time of the computer's introduction. Section 1.3 Digital Devices The most basic digital devices are called gates. Gates originally got their name from their function of allowing or retarding ("gating") the flow of digital information. In general, a gate has one or more inputs and produces an output that is a function of the current input value(s). While the inputs and outputs may be analog conditions such as voltage, current, even hydraulic pressure, they are modeled as taking on just two CSE 2300W Digital Logic Design Fall

3 discrete values, 0 and 1. We will study these digital gates in more detail beginning later in the semester, but you probably already have heard of some of them name them. Section 1.5 Software Aspects of Digital Design Digital design does not require any software tools but it greatly speeds up the design process. Several different types of tools are available; 1. Schematic entry. This is the digital designer's equivalent of a word processor. It allows schematic diagrams to be drawn "on-line," instead of with paper and pencil. The more advanced schematic-entry programs check for common errors, such as shorted outputs and signals that don't go anywhere. 2. HDLs. Hardware description languages, originally developed for circuit modeling, are now being used extensively for hardware design. They can be used to design anything from individual function modules to large, multi-chip digital systems. We'll introduce three commonly used HDLs; ABEL, VHDL, and Verilog, in Ch 5 but focus on Verilog in this class. 3. HDL text editors, compilers, and synthesizers. A typical HDL software package has many components. The designer uses a text editor to write an HDL program, and an HDL compiler checks it for syntax and related errors. The designer then can give the program to a synthesizer that creates a corresponding circuit realization that is targeted to a particular hardware technology. Most often, though, before synthesis, the designer runs the HDL program on a simulator to verify the behavior of the design. 4. Simulators. The design cycle for a customized, single-chip digital integrated circuit is long and expensive. Once the first chip is built, it's very difficult, to debug it by probing internal connections, or to change the gates and interconnections. Usually, changes must be made in the original design database and a new chip must be manufactured to incorporate the required changes. Since this proces can take months to complete, chip designers are highly motivated to "get it right" on the first try. Simulators help designers predict the electrical and functional behavior of a chip without actually building it, allowing most of the bugs to be found before the chip is fabricated. 5. Test benches. HDL-based digital designs are simulated and tested in software environments called test benches. The idea is to build a set of programs around the HDL programs to automatically exercise them, checking both their functional and timing behavior. This is especially useful when small design changes are made, the test bench can be run to ensure that CSE 2300W Digital Logic Design Fall

4 bug fixes or "improvements" in one area do not break something else. Test-bench programs may be written in the same HDL as the design itself, in C or C++, or in a combination of languages including scripting languages like Perl. 6. Timing analyzers and verifiers. The time dimension is very important in digital design. All digital circuits take time to produce a new output value in response to an input change, and much of a designer's effort is spent ensuring that such output changes occur quickly enough (or not too quickly). Specialized programs can automate the tedious task of drawing timing diagrams and specifying and verifying the timing relationships between different signals in a complex system. Section 1.6 Integrated Circuits A collection of one or more gates fabricated on a single silicon chip is called an integrated circuit (or IC). Large ICs, about the size of your palms can have up to 1 billion transistors with tens of millions of gates. This level of integration is called VLSI for Very Large Scale Integration (or ULSI for Ultra-Large Scale Integration). Section 1.7 Programmable Logic Devices There are a wide variety of ICs that can have their logic function programmed into them after they are manufactured. Most of these devices use technology that also allows the function to be reprogrammed, which means that if you find a bug in your design, you may be able to fix it without physically replacing or rewiring the device. Historically, programmable logic arrays (PLAs) were the first programmable logic devices. PLAs contained a two-level structure of AND and OR gates with user-programmable connections. Using this structure, a designer could accommodate any logic function up to a certain level of complexity using the well-known theory of logic synthesis and minimization that we'll present in Ch. 4. PLA structure was enhanced and PLA costs were reduced with the introduction of programmable array logic (PAL) devices. Today, such devices are generically called programmable logic devices (PLDs). The ever-increasing capacity of integrated circuits created an opportunity for IC manufacturers to design larger PLDs for larger digital-design applications. However, for technical reasons the basic two-level AND-OR structure of PLDs could not be scaled to larger sizes. Instead IC manufacturers devised complex PLD (CPLD) architectures to achieve the required scale. A typical CPLD is merely a collection of multiple PLDs and an interconnection structure, all on the same chip. In addition to the individual CSE 2300W Digital Logic Design Fall

5 PLDs, the on-chip interconnection structure is also programmable, providing a variety of design possibilities. About the same time that CPLDs were being invented, other IC manufacturers took a different approach to scaling the size of programmable logic. Compared to a CPLD, a field-programmable gate array (FPGA) contains a much larger number of smaller individual logic blocks and provides a large, distributed interconnection structure that dominates the entire chip. Section 1.8 Application-Specific ICs Perhaps the most interesting development in 1C technology for the average digital designer is not the ever-increasing number of transistors available per chip, but the ever-increasing opportunity to design your own chip. Chips designed for a particular, limited product or application are called semi-custom ICs or application specific ICs (ASICs). AMIS is basically an ASIC company. ASICs generally reduce the total component and manufacturing cost of a product by reducing chip count, physical size, and power consumption, and they often provide higher performance. The nonrecurring engineering (NRE) cost for an ASIC design can exceed the cost of a discrete design by $10,000 to $500,000 or more. NRE charges are paid to the IC manufacturer and others who are CSE 2300W Digital Logic Design Fall

6 responsible for designing the internal structure of the chip, creating tooling such as the metal masks for manufacturing the chips, developing tests for the manufactured chips, and actually making the first few sample chips. So, an ASIC design normally makes sense only if NRE cost is offset by the per-unit savings over the expected sale: volume of the product. There are a few approaches to ASIC design, differing in their capabilities and cost. The NRE cost to design a custom chip, a chip whose functions internal architecture, and detailed transistor-level design is tailored for a specific customer, is very high, $500,000 or more. Thus, full custom design is done only for chips that have general commercial application (e.g., microprocessors) or that will enjoy very high sales volume in a specific application (e.g. a digital watch chip, a network interface, or a bus-interface circuit for a PC). To reduce NRE charges, IC manufacturers have developed libraries of standard cells including commonly used functions such as decoders, registers, counters, memories, programmable logic arrays, and microprocessors. Custom cells are created (at added cost, of course) only if absolutely necessary. All of the cells are then laid out on the chip, optimizing the layout to reduce propagation delays and minimize the size of the chip. Minimizing the chip size reduces the per-unit cost of the chip, since it increases the number of chips that can be fabricated on a single wafer. The NRE cost for a standard-cell design is typically on the order of $250,000 or more. Well, $250,000 is still a lot of money, so IC manufacturers have gone one step further to bring ASIC design capability to the masses. A gate array is an IC whose internal structure is an array of gates whose interconnections are initially unspecified. The logic designer then specifies the gate types and interconnections. Even though the chip design is ultimately specified at this very low level, the designer typically works at a higher design level. The main difference between standard-cell and gate-array design is that the macro cells and the chip layout of a gate array are not as highly optimized as those in a standard cell design, therefore the chip may be 25% or more larger and may also cost more. Also there is no opportunity to create custom cells using the gate array approach. On the other hand, a gate-array design can be finished faster and at a lower NRE. Historical Perspective on Computing What we would a call a computer has evolved from a purely mechanical machine, to an electro- CSE 2300W Digital Logic Design Fall

7 mechanical one to purely electronic today. A good reference web page on computer is available at The French mathematician Blaise Pascal in 1642 designed and build a simple machine capable of adding and subtracting and was used for tax collection in France. It had two set of six dials, and each dial had ten settings. Two numbers could be added or subtracted through the action of a set of gears by turning a crank. The German mathematician Gottfried Liebnitz extended Pascal's design to include multiplication and division in 1694, by adding chains and pulleys. Many different people developed various types of mechanical calculators through the subsequent centuries. Further developments occurred in England in the early 1800's but the biggest problem remained the mass production of precision mechanical gears required by the calculators. Charles Babbage of England designed his Differential Engine in 1822 but it was never completed. He also designed his Analytical Engine in 1833 which was a true general purpose calculator. This had the first modernly recognizable calculator design containing two parts; the mill (which did the arithmetic) and the store (storage of numbers). He introduced the idea of using punched cards for data input (from the Jacquard looms). He also saw the need for a conditional branch instruction (IF-THEN-ELSE) for efficient execution. Simultaneously in the middle of the 19 th century, an English mathematician George Boole was developing a logical algebra and proved that arithmetic and other algebras could be derived from his algebra. This algebra became the foundation for modern computer calculations. The logical descendent of Babbage's design was built by Howard Aiken at Harvard with assistance from IBM. This project was started in 1937 and completed in 1944 and was called the Automatic Sequence Controlled Calculator or better known as the Mark 1. Information was stored on decimal counter wheels and could hold 72, 23-digit numbers. The machine was controlled by punched paper tape. Alan Turing helped develop the Colossus computer in 1943 which was used to crack the German Enigma code and was also based on electromechanical relays. CSE 2300W Digital Logic Design Fall

8 The next key calculator was the Mark 2 developed by Aiken s group between 1945 and All the arithmetic and transfer operations were accomplished by electromagnetic relays (13,000 of them). The machine could execute addition/subtraction in 200 msec and a multiply in 700 msec. Its storage capability was decimal digit numbers plus sign. A special purpose machine was being built during the war and was completed in 1946 at the University of Pennsylvania by L. Eckert and John Mauchly. This machine was called the Electronic Numerical Integrator and Calculator (ENIAC) and was designed to compute ballistic tables for military ordnance. The key innovation in this machine was the use of vacuum tubes in place of relays (no moving parts). 18,000 tubes were required and the device consumed 100 KW of power and required a room 100 feet in length. The machine could execute addition/subtraction in 200 usec and a multiply in 2300 usec. Its storage capability was decimal digit numbers plus sign. Simultaneously many theoreticians were working on the optimal design of future calculators / computers. Notable among these people was John von Neumann. An important idea of his was to store not only data in memory but also the sequence of instructions, where the instructions would also be represented as numbers. This concept was important in moving the calculators to true computers, and made them much more versatile. Two more importance concepts developed by him and a group of associates was the storage of data in serial form and the use of binary arithmetic. A general purpose machine which included these concepts was built in the early 1950's and was called the Electronic Discrete Variable Automatic Computer (EDVAC). In 1951 Eckert and Mauchly developed the first truly commercial computer called the Univac 1 and sold 48 of them for one million dollars each. Magnetic core memories were developed between 1951 and 1953, and by K memories were available. Transistors were discovered in the late 1940's and introduced into computers in FORTRAN was developed in 1957 to simplify the programming process. The concept of the operating system was developed in Time sharing was introduced in In 1964 IBM came out with the System 360 which introduced a family of different computers based on the same architecture. The first minicomputer was developed by the Digital Equipment Company (DEC) in 1965, the PDP-8, CSE 2300W Digital Logic Design Fall

9 and was also the first machine to run Unix. Intel introduced the 4004 microprocessor in 1971 which contained 2300 transistors. The PC revolution began in 1977 with the introduction of the Apple II and the IBM PC in There are two principle types of computers, digital and analog. We are most familiar with the digital version and that is what this course will focus on. The chief difference between the two is that an analog signal is continuous, with an infinite number of possible values (eg. temperature) while a digital signal has discrete, discontinuous values (eg. how much money you have). Analog computers are fast but of limited accuracy and not as general purpose. Digital computers have noise immunity, are accurate and have memory and control advantages. The type of digital computers we will be discussing uses binary (ON-OFF) logic since that is easy to obtain from a switch. A good definition of a digital computer is a digital system that can perform an automatic changeable succession of arithmetic and logical operations. The principle elements of a digital computer are: INPUT- > MEMORY -- > OUTPUT CONTROL < -> ARITHMETIC INPUT and OUTPUT are self-explanatory, MEMORY stores data and is passive. The ARITHMETIC unit operates on input and memory data provided to it using basic rules of logic (only a few are actually required for most mathematical operations) and the CONTROL unit selects the data from memory in the proper sequence and sends it to the appropriate unit along with the operations required to be performed on the data. It is an active unit and makes decisions. That completes the big picture for this course. Next we will introduce several basic electrical concepts which will be useful in understanding digital devices. The basic unit of electrical charge is the coulomb. One electron is approximately -1.6x10-19 C. CURRENT is the flow of charge per unit time and is analogous to the volume of water in a pipe. Current flows when there is a difference in electrical potential or voltage. VOLTAGE or electrical potential is the force that "pushes" the current and is analogous to the water CSE 2300W Digital Logic Design Fall

10 pressure in the pipe. A GROUND is always required when referring to a voltage since it is a voltage difference that causes current to flow. RESISTANCE limits the flow of current and is analogous to friction in the water pipe. V = I*R POWER - not confuse with work or energy. Is measured in watts and P=V*A ENERGY = P * sec FORCE = Energy / distance END CH. 1 Chapter 2 Number Systems and Codes Reading assignment is all of Ch 2. Section 2.1 Positional Number Systems Now look at the various important number systems in digital computers. We are most familiar with the base or radix 10 decimal system. This system, like the other 3 we will discuss (octal, hexadecimal and binary) is a positional number system. This means that a number CSE 2300W Digital Logic Design Fall

11 = 2* * *10 + 4*1 + 1* * l 10-2 Each weight is a power of 10 (from the base) corresponding to the numbers position. Always add a subscript in this class when writing down numbers to distinguish between bases. ( or ). The first number (2) is the most significant digit (bit) or MSB, while the 4 is the least significant digit bit or LSB, not counting the fractional numbers. We now look at the equivalencies between the various number systems we will use this semester as shown in Table 2-1 p. 28 Binary Decimal Octal 3-Bit String Hexadecimal 4-Bit String A B C D E F 1111 Now we look at the binary number system, which is also positional; = 1x x x x2 l + 0x x * *2-3 = = In binary the base that is raised to a power is 2. CSE 2300W Digital Logic Design Fall

12 Octal and hexadecimal are also similar. NOTE, do not pronounce a 11 in any other base except 10 as eleven, always say "one one". In hex: 6A51 16 = 6x xl x16 l + 1x16 0 = 24, = 27, Octal and hex are useful because their radices (8 and 16) are powers of 2, and this makes them convenient for conversion to and from binary. Octal came into existence on early computers when lights on outside panels where grouped triplets. It is not used very much anymore. To go from binary to octal, group the numbers into triplets and convert directly: = = = Do the reverse for octal to binary = = = For conversion between binary and hexadecimal, place the binary numbers into groups of 4 and convert = = C 5 7 = C57 16 Do the reverse for hex to binary; 93E45 16 = 9 3 E 4 5 = = The best way to convert between octal and hex is to go through binary as an intermediate step. To convert from decimal to binary, we continually divide the new number system base into the original number, and keep track of the remainders. The solution is read in reverse, where the MSB is the last CSE 2300W Digital Logic Design Fall

13 remainder calculated. Convert to binary: 37 / 2 rem = 1 18 / 2 rem = 0 9 / 2 rem = 1 4 / 2 rem = 0 2 / 2 rem = 0 1 rem = 1 Solution = (remember reverse order) Do a similar procedure for conversion of decimal to octal or hex. Convert to hex: 3305 / 16 rem = 9 = / 16 rem = 14 = E rem = 12 = C 16 Solution = CE9 16 Note that hex numbers are often used to describe a coputer s memory address space. For example, a computer with 16-bit addresses might be described as having read / write memory installed at address EFFF 16, and read-only memory between addresses F000 FFFF 16. Many computer programming languages use the prefix 0x to represent a hex number, e.g. 0xBFC0000. See Table 2.2 for the conversion methods for common radices. CSE 2300W Digital Logic Design Fall

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15 Section 2.4 Addition and Subtraction of Non-decimal Numbers Next we will look at how to do the arithmetic functions of addition, subtraction, multiplication and division in non-negative binary numbers. Addition and subtraction of non-decimal numbers by hand uses the same technique that we learned in grammar school for decimal numbers; the only catch is that the addition and subtraction tables are different. Table 2-3 below is the addition and subtraction table for binary digits. To add two binary numbers X and Y, we add together the least significant bits with an initial carry (c in ) of 0, producing carry (c out ) and sum (s) bits according to the table. We continue processing bits from right to left, adding the carry out of each column into the next column's sum. One example of decimal addition and the corresponding binary addition is shown below in Figure 2-1, using a colored arrow to indicate a carry of 1. CSE 2300W Digital Logic Design Fall

16 Binary subtraction is performed similarly, using borrows (b in and b out ) instead of carries between steps, and producing a difference (d). An example is shown below. A common use of subtraction in computers is to compare two numbers. For example the operation X - Y produces a borrow out of the most significant bit position then X is less than Y; otherwise, X is greater than or equal to Y. The relationship between carries and borrows in adders and subtracters will be explored Ch. 6. Addition and subtraction tables can be developed for octal and hexadecimal digits or any other desired radix. However, few computer engineers bother to memorize these tables and the rules are similar to what we just did with binary numbers. CSE 2300W Digital Logic Design Fall

17 2.5 Representation of Negative Numbers So far, we have dealt only with positive numbers, but there are many ways to represent negative numbers. In everyday business we use the signed-magnitude system. However, most computers use one of the complement number systems that we introduce later Signed-Magnitude Representation In the signed-magnitude system, a number consists of a magnitude and a symbol indicating whether the number is positive or negative. Thus, we interpret decimal numbers +98, -57, , and -13 in the usual way, and we also assume that the sign is "+" if no sign symbol is written. There are two possible representations of zero, "+0" and "-0", but both have the same value. The signed-magnitude system is applied to binary numbers by using an extra bit position to represent the sign (the sign bit). Traditionally, the most significant bit (MSB) of a bit string is used as the sign bit (0 = plus, 1 = minus), and the lower-order bits contain the magnitude. Thus, we can write several 8-bit signedmagnitude integers and their decimal equivalents; = and = = and = The signed-magnitude system has an equal number of positive and negative integers. An n-bit signedmagnitude integer lies within the range -(2 n-1-1) to +(2 n-1-1), and there are two possible representations of zero. Now suppose we wanted to build a digital logic circuit that adds signed-magnitude numbers. The circuit must examine the signs of the addends to determine what to do with the magnitudes. If the signs are the same, it must add the magnitudes and give the result the same sign. If the signs are different, it must compare the magnitudes, subtract the smaller from the larger, and give the result the sign of the larger. This makes for a complex set of logic. Adders using complement number systems are much simpler, as we'll show next Complement Number Systems While the signed-magnitude system negates a number by changing its sign, a complement number system negates a number by taking its complement as defined by the system. Taking the complement is more CSE 2300W Digital Logic Design Fall

18 difficult than changing the sign, but two numbers in a complement number system can be added or subtracted directly without the sign and magnitude checks required above. We shall describe two complement number systems, the radix-complement and the diminished radix-complement. In any complement number system, we normally deal with a fixed number of digits (e.g. n). We further assume that the radix is r, and that numbers have the following form D = d n-l d n-2 d l d 0 The radix point is on the right and so the number is an integer. If an operation produces a result that requires more than n digits, we throw away the extra high-order digits. If a number D is complemented twice, the result is D Radix-Complement Representation In a radix-complement system, the complement of an n-digit number is obtained by subtracting it from r n. In the decimal number system, the radix complement is called the 10 s complement. Some examples using 4-digit decimal numbers are shown in Table 2-4. The first row 10 s complement is = 8151 while the 9 s complement is = CSE 2300W Digital Logic Design Fall

19 If D is between 1 and r n - 1, this subtraction produces another number between 1 and r n - l. lf D is 0, the result of the subtraction is r n and note we throw away the extra high-order digit and get the result 0. Thus, there is only one representation of zero in a radix-complement system. Table 2-5 lists the digit complements for binary, octal, decimal, and hexadecimal numbers Two's-Complement Representation For binary numbers, the radix complement is called the two's complement. The MSB of a number in this system serves as the sign bit; a number is negative if and only if its MSB is 1. The decimal equivalent for a two's-complement binary number is computed the same way as for an unsigned number, except that the weight of the MSB is -2 n-1 instead of +2 n-1. The range of available numbers covers the same range-2 n-1 to CSE 2300W Digital Logic Design Fall

20 +2 n-1-1. The quickest way to obtain this representation is to complement each bit individually then add 1 to the result. Some 8-bit examples are shown below (remember 8 bit two s-complement numbers can range from -128 to +127 only = complement bits = = complement bits = = complement bits = Note a carry-out from the MSB is ignored in two s complement representations. In the two's-complement number system, zero is considered positive because its sign bit is 0. We can convert an n-bit two's-complement number X into an m-bit one, but care is needed. If m > n, we must append m - n copies of X's sign bit to the left of X. That is, we pad a positive number with 0s and a negative one with 1s; this is called sign extension. If m < n, we discard X's n m leftmost bits; however, the result is valid only if all of the discarded bits are the same as the sign bit of the result. Most computers use the two's-complement system to represent negative numbers. However, for completeness, the book discusses the diminished radix-complement and ones'-complement systems on pp Two s Complement Addition and Subtraction Addition Rules CSE 2300W Digital Logic Design Fall

21 A table of decimal numbers and their equivalents in different number systems is shown in Table 2-6 and helps reveals why the two's complement is preferred for arithmetic operations. If we start with (i.e ) and count up, we see that each successive 2 s complement number all the way to (i.e ) can be obtained merely by adding 1 to the previous one, ignoring any carries beyond the fourth bit position. Because addition is just an extension of counting, 2's-complement numbers can thus be added by ordinary binary addition, ignoring any carries beyond the MSB. The result will always be the correct sum as long as the range of the number system is not exceeded. Some examples of decimal addition and the corresponding 4-bit 2 s-complement additions confirm this CSE 2300W Digital Logic Design Fall

22 Overflow If an operation produces a result that exceeds the range of the number system, an overflow is said to occur. In our previous 4-bit counting representation a sum greater than +7 will cause an overflow to occur. Note that two numbers with different signs can never produce an overflow, but the addition of two numbers of like sign can, as shown by the following; = = -5 Fortunately there is a simple rule for detecting overflow in addition: an addition overflow occurs if the addends' signs are the same but the sum's sign is different Subtraction Rules 2 s complement numbers may be subtracted as if they were ordinary unsigned binary numbers and appropriate rules for detecting overflow may be formulated. However, most subtraction circuits for two's-complement numbers do not perform subtraction directly. Rather, they negate the subtrahend by taking its two's complement, and then add it to the minuend using the normal rules for addition. Negating the subtrahend and adding the minuend can be accomplished with only one addition operation as follows: Perform a bit-by-bit complement of the subtrahend and add the complemented subtrahend to CSE 2300W Digital Logic Design Fall

23 the minuend with an initial carry (cin) of 1 instead of 0. Examples are given below: 1 for carry in for carry in Overflow in subtraction can be detected by examining the signs of the minuend and the complemented subtrahend, using the same rule as in addition. Or, using the technique in the preceding examples, the carries into and out of the sign position can be observed and overflow detected irrespective of the signs of inputs and output, again using the same rule as in addition. The table below summarizes the rules we have just presented plus the ones that we skipped in the text. CSE 2300W Digital Logic Design Fall

24 2.8 Binary Multiplication In grammar school we learned to multiply by adding a list of shifted multiplicands computed according to the digits of the multiplier. The same method can be used to obtain the product of two unsigned binary numbers. Forming the shifted multiplicands is trivial in binary multiplication since the only possible values of the multiplier digits are 0 and 1. An example is shown below: multiplicand x 13 x 1101 multiplier shifted multiplicands product CSE 2300W Digital Logic Design Fall

25 Instead of listing all the shifted multiplicands and then adding, in a digital system it is more convenient to add each shifted multiplicand as it is created to a partial product. Applying this technique to the previous example, four additions and partial products are used to multiply 4-bit numbers: 1011 multiplicand x 1101 multiplier 0000 partial product 1011 shifted multiplicand next partial product 0000x shifted multiplicand partial product 1011xx shifted multiplicand partial product 1011xxx shifted multiplicand product In general, when we multiply an n-bit number by an m-bit number, the resulting product requires at most n + m bits to express. The shift-and-add algorithm requires m partial products and additions to obtain the result, but the first addition is trivial, since the first partial product is zero. Although the first partial product has only n significant bits, after each addition step the partial product gains one more significant bit, since each addition may produce a carry. At the same time, each step yields one more partial product bit, starting with the rightmost and working toward the left, that does not change. The shift-and-ad: algorithm can be performed by a digital circuit that includes a shift register, an adder, and control logic. Multiplication of signed numbers can be accomplished using unsigned multiplication and the usual grammar-school rules: Perform an unsigned multiplication of the magnitudes and make the product positive if the operands had the same sign, negative if they had different signs. In the two's-complement system, obtaining the magnitude of a negative number and negating the unsigned product are nontrivial operations. Conceptually, unsigned multiplication is accomplished by a sequence of unsigned additions of the shifted multiplicands; at each step, the shift of the multiplicand CSE 2300W Digital Logic Design Fall

26 corresponds to the weight of the multiplier bit. The bits in a two s complement number have the same weights as in an unsigned number, except for the MSB, which has a negative weight. Thus, we can perform two's-complement multiplication by a sequence of two's-complement of shifted multiplicands, except for the last step, in which the shifted multiplicand corresponding to the MSB of the multiplier must be negated before it is added to the partial product. Our previous example is repeated below, this time interpreting the multiplier and multiplicand as two's-complement numbers (i.e -5 * -3) 1011 multiplicand x 1101 multiplier partial product shifted multiplicand next partial product 00000x shifted multiplicand partial product 11011xx shifted multiplicand partial product 00101xxx shifted and negated multiplicand product Handling the MSBs is a little tricky because we gain one significant bit at each step and we are working with signed numbers. Therefore, before adding each shifted multiplicand and k-bit partial product, we change them to k + 1 significant bits by sign extension, as shown in red above. Each resulting sum has k + 1 bits, and any carry out of the MSB of the k + 1-bit sum is ignored. 2.9 Binary Division The simplest binary division algorithm is based on the shift-and-subtract method that we learned in grammar school. Table 2-8 gives examples of this method for unsigned decimal and binary numbers. CSE 2300W Digital Logic Design Fall

27 In both cases, we mentally compare the reduced unsigned division with multiples of the divisor to determine which multiple of the shifted divisor to subtract. In the decimal case, we first pick 11 as the greatest multiple of 11 less than 21, and then pick 99 as the greatest multiple less than 107. In the binary case, the choice is somewhat simpler, since the only two choices are zero and the divisor itself. Division methods for binary numbers are somewhat complementary to binary multiplication methods. A typical division algorithm takes an (n + m)-bit dividend and an n-bit divisor, and produces an m-bit quotient and an n-bit remainder. A division overflows if the divisor is zero or the quotient would take more than m bits to express. In most computer division circuits, n = m. Division of signed numbers can be accomplished using unsigned division and the usual grammar school rules: Perform an unsigned division of the magnitudes and make the quotient positive if the operands had the same sign, negative if they had different signs. The remainder should be given the same sign as the dividend. As in multiplication, there are special techniques for performing division directly on two's- CSE 2300W Digital Logic Design Fall

28 complement numbers; these techniques are often implemented in computer division circuits (see References) Binary Codes for Decimal Numbers Even though binary numbers are the most appropriate for the internal computations of a digital system, most people still prefer to deal with decimal numbers. As a result, the external interfaces of a digital system may read or display decimal numbers, and some digital devices actually process decimal numbers directly. The human need to represent decimal numbers doesn't change the basic nature of digital electronic circuits, they still process signals that take on one of only two states that we call 0 and 1. Therefore, a decimal number is represented in a digital system by a string of bits, where different combinations of bit values in the string represent different decimal numbers. For example, if we use a 4-bit string to represent a decimal number, we might assign bit combination 0000 to decimal digit 0, 0001 to 1, 0010 to 2, and so on. A set of n-bit strings in which different bit strings represent different numbers or other things is called a code. A particular combination of n bit-values is called a code word. As we'll see in the examples of decimal codes in this section, there may or may not be an arithmetic relationship between the bit values in a code word and the thing that it represents. Furthermore, a code that uses n-bit strings need not contain 2 n valid code words. At least four bits are needed to represent the ten decimal digits. There are billions and billions of different ways to choose ten 4-bit code words, but some of the more common decimal codes are listed in Table 2-9. CSE 2300W Digital Logic Design Fall

29 Perhaps the most "natural" decimal code is binary-coded decimal (BCD). which encodes the digits 0 through 9 by their 4-bit unsigned binary representations, 0000 through The code words 1010 through 1111 are not used. Decimal codes can have more than four bits; for example, the biquinary code in Table 2-9 uses seven. The first two bits in a code word indicate whether the number is in the range 0-4 or 5-9, and the last five bits indicate which of the five numbers in the selected range is represented. One potential advantage of using more than the minimum number of bits in a code is an error-detecting property. In the biquinary code, if any one bit in code word is accidentally changed to the opposite value, CSE 2300W Digital Logic Design Fall

30 the resulting code word does not represent a decimal digit and can therefore be flagged as an error. Out of 128 possible 7-bit code words, only 10 are valid and recognized as decimal digits; the rests can be flagged as errors if they appear. A 1-out-of-10 code, such as the one shown in the last column of Table 2-9, is the sparest encoding for decimal digits, using 10 out of 1024 possible 10-bit code words Gray Code In electro-mechanical applications of digital systems, such as machine tools, auto braking systems, and copiers, it is sometimes necessary for an input sensor to produce a digital value that indicates a mechanical position. For example, Figure 2-5 is a conceptual sketch of an encoding disk and a set of contacts that produce one of eight 3-bit binary-coded values depending on the rotational position of the disk. The dark areas of the disk are connected to a signal source corresponding to logic 1, and the light areas are unconnected, which the contacts interpret as logic 0. CSE 2300W Digital Logic Design Fall

31 The encoder in Figure 2-5 has a problem when the disk is positioned at certain boundaries between the regions. For example, consider the boundary between the 001 and 010 regions of the disk; two of the encoded bits change here. What value will the encoder produce if the disk is positioned right on the boundary? Since we're on the border, both 001 and 010 are acceptable. However, because the mechanical assembly is not perfect, the two right-hand contacts may both touch a "1" region, giving an incorrect reading of 011. Likewise a reading of 000 is possible. In general, this sort of problem can occur at any boundary where more than one bit changes. The worst problems occur when all three bits are changing, as at the and boundaries. The encoding-disk problem can be solved by devising a digital code in which only one bit changes between each pair of successive code words. Such a code is called a Gray code and a 3-bit Gray code is listed in Table CSE 2300W Digital Logic Design Fall

32 The encoding disk now is redesigned using this code as shown below. CSE 2300W Digital Logic Design Fall

33 Now only one bit changes at each boundary, so boundary readings give us a value on one side or the other of that particular boundary. Note we can create any-sized Gray code using the following two rules; 1. The first 2 n code words of an (n + 1 )-bit Gray code equal the code words of an n-bit Gray code, written in order with a leading 0 appended. 2. The last 2 n code words of an (n + l)-bit Gray code equal the code words of an n-bit Gray code, but written in reverse order with a leading 1 appended. If we draw a line between rows 3 and 4 of Table 2-10, we can see that rules 2 and 3 are true for the 3-bit Gray code. CSE 2300W Digital Logic Design Fall

34 2.12 Character Codes Note that a string of bits does not have to represent a number. One important character code is ASCII (American Standard Code for Information Exchange). Here each character is represented by a 7-bit string for a total of 128 possibilities. CSE 2300W Digital Logic Design Fall

35 2.15 Codes for Detecting and Correcting Errors An error in a digital system is the corruption of data from its correct value to some other value. An error is caused by a physical failure. Failures can be either temporary or permanent. For example, a cosmic ray or CSE 2300W Digital Logic Design Fall

36 alpha particle can cause a temporary failure of a memory circuit, changing the value of a bit stored in it. Letting a circuit get too hot or zapping it with static electricity can cause a permanent failure, so that it never works correctly again. The effects of failures on data are predicted by error models. The simpler error model, which we consider here, is called the independent error model. In this model, a single physical failure is assumed to affect only a single bit of data; the corrupted data is said to contain a single error. Multiple failures may cause multiple errors, two or more bits in error, but multiple errors are assumed to be less likely than single errors Error-Detecting Codes Recall from our definitions in Section 2.10 that a code that uses n-bit string-need not contain 2 n valid code words; and this is important for the codes we now consider. An error-detecting code has the property that corrupting or garbling a code word will likely produce a bit string that is not a valid code word. A system that uses an error-detecting code generates, transmits, and store-only code words. Thus, errors in a bit string can be detected by a simple rule, if the bit string is a code word, it is assumed to be correct; if it is a non-code word then it contains an error. An n-bit code and its error-detecting properties under the independent error model are easily explained in terms of an n-cube. A code is simply a subset of the vertices of the n-cube. In order for the code to detect all single errors, no code-word vertex can be immediately adjacent to another code-word vertex. For example Figure 2-10(a) below shows a 3-bit code with five code words. Note code word 111 is immediately adjacent to code words 110, 011, and 101. Since a single bit failure could change 111 to 110, 011, or 101, this code does not detect all single bit errors. CSE 2300W Digital Logic Design Fall

37 If we make 111 a non-code word, we obtain a code that does have the single error-detecting property, as shown in (b) above, where no single bit error can change one valid code word into another. Stated more formally, a code detects all single errors if the minimum distance between all possible pairs of code words is 2. In general we need n + 1 bits to construct a single-error-detecting code with 2 n code words. The first n bits of a code word, called information bits, may be any of the 2 n n-bit strings. To obtain a minimumdistance-2 code, we add one more bit, called a parity bit, that is set to 0 if there are an even number of 1s among the information bits, and to 1 otherwise. This is illustrated in the first two columns of Table 2-13 for a code with three information bits. A valid (n + l)-bit in even number of 1s, and this code is called an even-parity code. Note we can do the same thing with odd parity. These are both examples of 1-bit parity codes. CSE 2300W Digital Logic Design Fall

38 1-bit parity codes do not detect 2-bit errors, since changing two bits does not affect the parity. However, the codes can detect errors in any odd number of bits. For example, if three bits in a code word are changed, then the resulting word has the wrong parity and is a non-code word. This doesn't help us much, though. Under the independent error model, 3-bit errors are much less likely than 2-bit errors, which are not detectable. Thus, practically speaking, the 1-bit parity codes' error-detection capability stops after 1- bit errors. Other codes, with minimum distance greater than 2, can be used to detect multiple errors Error-Correcting and Multiple-Error-Detecting Codes By using more than one parity bit, or check bits, according to some well-chosen rules, we can create a code whose minimum distance is greater than 2. Before showing how this can be done, let's look at how such a code can be used to correct single errors or detect multiple errors. Suppose that a code has a minimum distance of 3. Figure 2-11 shows a fragment of the n-cube for such a CSE 2300W Digital Logic Design Fall

39 code. As shown, there are at least two non-code words between each pair of code words. Now suppose we transmit code words and assume that failures affect at most one bit of each received code word. Then a received non-code word with a 1-bit error will be closer to the originally transmitted code word than to any other code word. Therefore, when we receive a non-code we can correct the error by changing the received non-code word to the nearest code word, as indicated by the arrows in the figure. Deciding which code word was originally transmitted to produce a received code word is called decoding and the hardware that does this is an error-correcting decoder. A code that is used to correct errors is called an error-correcting code. In general, if a code has minimum distance 2c + 1, it can be used to correct errors that affect up to c bits (c = 1 in the preceding example). If a code's minimum distance is 2c + d + 1, it can be used to correct errors in up to c bits and to detect errors CSE 2300W Digital Logic Design Fall

40 up to d additional bits Hamming Codes In 1950 R. W. Hamming described a general method for constructing codes with a minimum distance of 3, now called Hamming codes. In the minimum-distance-3 code we can see that all 1-bit and 2-bit changes to a code word will always yield a non-code word. For any value of i, his method yields a (2 i - l)- bit code with i check bits and (2 i i) information bits. The bit positions in a Hamming code word can be numbered from 1 through (2 i - l). In this case, any position whose number is a power of 2 is a check bit, and the remaining positions are information bits. Each check bit is grouped with a subset of the information bits, as specified by a parity-check matrix. As shown in Figure 2-13(a), each bit is grouped with the information positions whose numbers have a 1 in the same bit when expressed in binary. The key to all of his systems was to have the parity bits overlap, such that they managed to check each other as well as the data. CSE 2300W Digital Logic Design Fall

41 For example, check bit 2 (010) is grouped with information bits 3 (011), 6 (110), and 7 (111). For a given combination of information-bit values, each check bit is chosen to produce even parity, that is, so the total number of 1s in its group is even. Traditionally the bit positions of a parity-check matrix and the resulting code words are rearranged so that all of the check bits are on the right as in Figure 2-13(b). CSE 2300W Digital Logic Design Fall

42 Why this arrangement? The three parity bits (1,2,4) are related to the data bits (3,5,6,7) as shown below. In this diagram, each overlapping circle corresponds to one parity bit and defines the four bits contributing to that parity computation. For example, data bit 3 contributes to parity bits 1 and 2. Each circle (parity bit) encompasses a total of four bits, and each circle must have EVEN parity. Given four data bits, the three parity bits can easily be chosen to ensure this condition. It can be observed that changing any one bit numbered 1 7 uniquely affects the three parity bits. Changing bit 7 affects all three parity bits, while an error in bit 6 affects only parity bits 2 and 4, and an error in a parity bit affects only that bit. The location of any single bit error is determined directly upon checking the three parity circles. For example, the message would be sent as , since: CSE 2300W Digital Logic Design Fall

43 When these seven bits are entered into the parity circles, it can be confirmed that the choice of these three parity bits ensures that the parity within each circle is EVEN, as shown here. The first two columns in the table below list the resulting code words for a Hamming (7,4) code. CSE 2300W Digital Logic Design Fall

44 Next we can design an error-correcting decoder for a received minimum-distance-3 Hamming code word. First, we check all of the parity groups; if all have even parity, then the received word is assumed to be correct. If one or more groups have odd parity, then a single error is assumed to have occurred. The pattern of groups that have odd parity (called the syndrome) must match one of the columns in the parity- CSE 2300W Digital Logic Design Fall

45 check matrix; the corresponding bit position is assumed to contain the wrong value and is complemented. If an error occurs in any of the seven bits, that error will affect different combinations of the three parity bits depending on the bit position. For example, suppose the above message is sent and a single bit error occurs such that the codeword is received: transmitted message received message > BIT: BIT: The above error (in bit 5) can be corrected by examining which of the three parity bits was affected by the bad bit: The pattern from the parity bits is 101 or decimal 5. The correction is then to complement bit 5 and the resulting code word becomes Another example, suppose we receive the word (note the book on p. 64 uses this example but switches bits 3 and 4) which is a non-code word. Groups B bits 7/6/3 (010) and C bits 7/6/5 (010) have odd parity as shown below BIT CODEWORD 0 x 0 x 1 x 1 Group A EVEN PARITY? OK! 0 LSB 0 1 x x 1 1 x Group B EVEN PARITY? NO! x x x Group C EVEN PARITY? NO! 1 MSB The pattern from the parity bits is 110 or decimal 6. The correction is then to complement bit 6 and the resulting code word becomes CSE 2300W Digital Logic Design Fall

46 CRC Codes There are many other error detecting and correcting codes besides the Hamming codes. One group, which includes Hamming, is the cyclic-redundancy-check (CRC) codes. These are typically used in disk drives and data networks. In a disk drive, each block of data (typically 512 bytes) is protected by a CRC code, so that errors within a block can be detected and sometimes corrected Codes for Serial Data Transmission Parallel and Serial Data Most computer and other digital systems transmit and store data in a parallel format. In parallel data transmission, a separate signal line is provided for each bit of data, and in parallel data storage, all of the bits of a data word can be written or read simultaneously. But parallel formats are not cost effective for some applications. For example, parallel transmission of data bytes over the telephone network would require 8 phone lines, and parallel storage of data bytes on a magnetic disk would require a disk drive with eight separate read/write heads. Serial formats allow data to be transmitted or stored one bit at a time. Even in board-level design and interfacing, serial formats can reduce cost and simplify certain problems. Figure 2-16 illustrates some of the basic ideas in serial data transmission. CSE 2300W Digital Logic Design Fall

47 A repetitive clock signal, named CLOCK below, defines the rate at which bits are transmitted, one bit per clock cycle. Thus, the bit rate in bits per second (bps) equals the clock frequency in cycles per second (hertz, or Hz). The reciprocal of the bit rate is the bit time and numerically equals the clock period in seconds. This amount of time is reserved on the serial data line named (SERDATA) for each bit that is transmitted. The time occupied by each bit is sometimes called a bit cell. The format of the actual signal that appears on the line during each bit cell depends on the line code. Regardless of the line code, a serial data transmission or storage system needs some way of identifying the significance of each bit in the serial stream. In order to determine where the 8-bit byte starts, a synchronization signal (SYNCH) is used in the previous figure. We therefore need a minimum of 3 signals to recover a serial data stream; clock, synch and the data. END Ch. 2 CSE 2300W Digital Logic Design Fall