Category: computer Logical Organization

Boolean Algebra ”; Previous Next Boolean Algebra is used to analyze and simplify the digital (logic) circuits. It uses only the binary numbers i.e. 0 and 1. It is also called as Binary Algebra or logical Algebra. Boolean algebra was invented by George Boole in 1854. Rule in Boolean Algebra Following are the important rules used in Boolean algebra. Variable used can have only two values. Binary 1 for HIGH and Binary 0 for LOW. Complement of a variable is represented by an overbar (-). Thus, complement of variable B is represented as . Thus if B = 0 then = 1 and B = 1 then = 0. ORing of the variables is represented by a plus (+) sign between them. For example ORing of A, B, C is represented as A + B + C. Logical ANDing of the two or more variable is represented by writing a dot between them such as A.B.C. Sometime the dot may be omitted like ABC. Boolean Laws There are six types of Boolean Laws. Commutative law Any binary operation which satisfies the following expression is referred to as commutative operation. Commutative law states that changing the sequence of the variables does not have any effect on the output of a logic circuit. Associative law This law states that the order in which the logic operations are performed is irrelevant as their effect is the same. Distributive law Distributive law states the following condition. AND law These laws use the AND operation. Therefore they are called as AND laws. OR law These laws use the OR operation. Therefore they are called as OR laws. INVERSION law This law uses the NOT operation. The inversion law states that double inversion of a variable results in the original variable itself. Important Boolean Theorems Following are few important boolean Theorems. Boolean function/theorems Description Boolean Functions Boolean Functions and Expressions, K-Map and NAND Gates realization De Morgan”s Theorems De Morgan”s Theorem 1 and Theorem 2 Print Page Previous Next Advertisements ”;

Discuss Computer Logical Organization ”; Previous Next Computer Logical Organization refers to the level of abstraction above the digital logic level, but below the operating system level. At this level, the major components are functional units or subsystems that correspond to specific pieces of hardware built from the lower level building blocks. This tutorial gives a complete understanding on Computer Logical Organization starting from basic computer overview till its advanced architecture. Please enable JavaScript to view the comments powered by Disqus. Print Page Previous Next Advertisements ”;

Complement Arithmetic ”; Previous Next Complements are used in the digital computers in order to simplify the subtraction operation and for the logical manipulations. For each radix-r system (radix r represents base of number system) there are two types of complements. S.N. Complement Description 1 Radix Complement The radix complement is referred to as the r”s complement 2 Diminished Radix Complement The diminished radix complement is referred to as the (r-1)”s complement Binary system complements As the binary system has base r = 2. So the two types of complements for the binary system are 2”s complement and 1”s complement. 1”s complement The 1”s complement of a number is found by changing all 1”s to 0”s and all 0”s to 1”s. This is called as taking complement or 1”s complement. Example of 1”s Complement is as follows. 2”s complement The 2”s complement of binary number is obtained by adding 1 to the Least Significant Bit (LSB) of 1”s complement of the number. 2”s complement = 1”s complement + 1 Example of 2”s Complement is as follows. Print Page Previous Next Advertisements ”;

Combinational Circuits ”; Previous Next Combinational circuit is a circuit in which we combine the different gates in the circuit, for example encoder, decoder, multiplexer and demultiplexer. Some of the characteristics of combinational circuits are following − The output of combinational circuit at any instant of time, depends only on the levels present at input terminals. The combinational circuit do not use any memory. The previous state of input does not have any effect on the present state of the circuit. A combinational circuit can have an n number of inputs and m number of outputs. Block diagram We”re going to elaborate few important combinational circuits as follows. Half Adder Half adder is a combinational logic circuit with two inputs and two outputs. The half adder circuit is designed to add two single bit binary number A and B. It is the basic building block for addition of two single bit numbers. This circuit has two outputs carry and sum. Block diagram Truth Table Circuit Diagram Full Adder Full adder is developed to overcome the drawback of Half Adder circuit. It can add two one-bit numbers A and B, and carry c. The full adder is a three input and two output combinational circuit. Block diagram Truth Table Circuit Diagram N-Bit Parallel Adder The Full Adder is capable of adding only two single digit binary number along with a carry input. But in practical we need to add binary numbers which are much longer than just one bit. To add two n-bit binary numbers we need to use the n-bit parallel adder. It uses a number of full adders in cascade. The carry output of the previous full adder is connected to carry input of the next full adder. 4 Bit Parallel Adder In the block diagram, A0 and B0 represent the LSB of the four bit words A and B. Hence Full Adder-0 is the lowest stage. Hence its Cin has been permanently made 0. The rest of the connections are exactly same as those of n-bit parallel adder is shown in fig. The four bit parallel adder is a very common logic circuit. Block diagram N-Bit Parallel Subtractor The subtraction can be carried out by taking the 1”s or 2”s complement of the number to be subtracted. For example we can perform the subtraction (A-B) by adding either 1”s or 2”s complement of B to A. That means we can use a binary adder to perform the binary subtraction. 4 Bit Parallel Subtractor The number to be subtracted (B) is first passed through inverters to obtain its 1”s complement. The 4-bit adder then adds A and 2”s complement of B to produce the subtraction. S3 S2 S1 S0 represents the result of binary subtraction (A-B) and carry output Cout represents the polarity of the result. If A > B then Cout = 0 and the result of binary form (A-B) then Cout = 1 and the result is in the 2”s complement form. Block diagram Half Subtractors Half subtractor is a combination circuit with two inputs and two outputs (difference and borrow). It produces the difference between the two binary bits at the input and also produces an output (Borrow) to indicate if a 1 has been borrowed. In the subtraction (A-B), A is called as Minuend bit and B is called as Subtrahend bit. Truth Table Circuit Diagram Full Subtractors The disadvantage of a half subtractor is overcome by full subtractor. The full subtractor is a combinational circuit with three inputs A,B,C and two output D and C”. A is the ”minuend”, B is ”subtrahend”, C is the ”borrow” produced by the previous stage, D is the difference output and C” is the borrow output. Truth Table Circuit Diagram Multiplexers Multiplexer is a special type of combinational circuit. There are n-data inputs, one output and m select inputs with 2m = n. It is a digital circuit which selects one of the n data inputs and routes it to the output. The selection of one of the n inputs is done by the selected inputs. Depending on the digital code applied at the selected inputs, one out of n data sources is selected and transmitted to the single output Y. E is called the strobe or enable input which is useful for the cascading. It is generally an active low terminal that means it will perform the required operation when it is low. Block diagram Multiplexers come in multiple variations 2 : 1 multiplexer 4 : 1 multiplexer 16 : 1 multiplexer 32 : 1 multiplexer Block Diagram Truth Table Demultiplexers A demultiplexer performs the reverse operation of a multiplexer i.e. it receives one input and distributes it over several outputs. It has only one input, n outputs, m select input. At a time only one output line is selected by the select lines and the input is transmitted to the selected output line. A de-multiplexer is equivalent to a single pole multiple way switch as shown in fig. Demultiplexers comes in multiple variations. 1 : 2 demultiplexer 1 : 4 demultiplexer 1 : 16 demultiplexer 1 : 32 demultiplexer Block diagram Truth Table Decoder A decoder is a combinational circuit. It has n input and to a maximum m = 2n outputs. Decoder is identical to a demultiplexer without any data input. It performs operations which are exactly opposite to those of an encoder. Block diagram Examples of Decoders are following. Code converters BCD to seven segment decoders Nixie tube decoders Relay actuator 2 to 4 Line Decoder The block diagram of 2 to 4 line decoder is shown in the fig. A and B are the two inputs where D through D are the four outputs. Truth table explains the operations of a decoder. It shows that each output is 1 for only a specific combination of inputs. Block diagram Truth Table Logic Circuit Encoder Encoder is a combinational circuit which is designed to perform the inverse operation of the decoder.

Digital Counters ”; Previous Next Counter is a sequential circuit. A digital circuit which is used for a counting pulses is known counter. Counter is the widest application of flip-flops. It is a group of flip-flops with a clock signal applied. Counters are of two types. Asynchronous or ripple counters. Synchronous counters. Asynchronous or ripple counters The logic diagram of a 2-bit ripple up counter is shown in figure. The toggle (T) flip-flop are being used. But we can use the JK flip-flop also with J and K connected permanently to logic 1. External clock is applied to the clock input of flip-flop A and QA output is applied to the clock input of the next flip-flop i.e. FF-B. Logical Diagram Operation S.N. Condition Operation 1 Initially let both the FFs be in the reset state QBQA = 00 initially 2 After 1st negative clock edge As soon as the first negative clock edge is applied, FF-A will toggle and QA will be equal to 1. QA is connected to clock input of FF-B. Since QA has changed from 0 to 1, it is treated as the positive clock edge by FF-B. There is no change in QB because FF-B is a negative edge triggered FF. QBQA = 01 after the first clock pulse. 3 After 2nd negative clock edge On the arrival of second negative clock edge, FF-A toggles again and QA = 0. The change in QA acts as a negative clock edge for FF-B. So it will also toggle, and QB will be 1. QBQA = 10 after the second clock pulse. 4 After 3rd negative clock edge On the arrival of 3rd negative clock edge, FF-A toggles again and QA become 1 from 0. Since this is a positive going change, FF-B does not respond to it and remains inactive. So QB does not change and continues to be equal to 1. QBQA = 11 after the third clock pulse. 5 After 4th negative clock edge On the arrival of 4th negative clock edge, FF-A toggles again and QA becomes 1 from 0. This negative change in QA acts as clock pulse for FF-B. Hence it toggles to change QB from 1 to 0. QBQA = 00 after the fourth clock pulse. Truth Table Synchronous counters If the “clock” pulses are applied to all the flip-flops in a counter simultaneously, then such a counter is called as synchronous counter. 2-bit Synchronous up counter The JA and KA inputs of FF-A are tied to logic 1. So FF-A will work as a toggle flip-flop. The JB and KB inputs are connected to QA. Logical Diagram Operation S.N. Condition Operation 1 Initially let both the FFs be in the reset state QBQA = 00 initially. 2 After 1st negative clock edge As soon as the first negative clock edge is applied, FF-A will toggle and QA will change from 0 to 1. But at the instant of application of negative clock edge, QA , JB = KB = 0. Hence FF-B will not change its state. So QB will remain 0. QBQA = 01 after the first clock pulse. 3 After 2nd negative clock edge On the arrival of second negative clock edge, FF-A toggles again and QA changes from 1 to 0. But at this instant QA was 1. So JB = KB= 1 and FF-B will toggle. Hence QB changes from 0 to 1. QBQA = 10 after the second clock pulse. 4 After 3rd negative clock edge On application of the third falling clock edge, FF-A will toggle from 0 to 1 but there is no change of state for FF-B. QBQA = 11 after the third clock pulse. 5 After 4th negative clock edge On application of the next clock pulse, QA will change from 1 to 0 as QB will also change from 1 to 0. QBQA = 00 after the fourth clock pulse. Classification of counters Depending on the way in which the counting progresses, the synchronous or asynchronous counters are classified as follows − Up counters Down counters Up/Down counters UP/DOWN Counter Up counter and down counter is combined together to obtain an UP/DOWN counter. A mode control (M) input is also provided to select either up or down mode. A combinational circuit is required to be designed and used between each pair of flip-flop in order to achieve the up/down operation. Type of up/down counters UP/DOWN ripple counters UP/DOWN synchronous counter UP/DOWN Ripple Counters In the UP/DOWN ripple counter all the FFs operate in the toggle mode. So either T flip-flops or JK flip-flops are to be used. The LSB flip-flop receives clock directly. But the clock to every other FF is obtained from (Q = Q bar) output of the previous FF. UP counting mode (M=0) − The Q output of the preceding FF is connected to the clock of the next stage if up counting is to be achieved. For this mode, the mode select input M is at logic 0 (M=0). DOWN counting mode (M=1) − If M = 1, then the Q bar output of the preceding FF is connected to the next FF. This will operate the counter in the counting mode. Example 3-bit binary up/down ripple counter. 3-bit − hence three FFs are required. UP/DOWN − So a mode control input is essential. For a ripple up counter, the Q output of preceding FF is connected to the clock input of the next one. For a ripple up counter, the Q output of preceding FF is connected to the clock input of the next one. For a ripple down counter, the Q bar output of preceding FF is connected to the clock input of the next one. Let the selection of Q and Q bar output of the preceding FF be controlled by the mode control input M such that, If M = 0, UP counting. So connect Q to CLK. If M = 1, DOWN counting. So connect Q

Quick Guide ”; Previous Next Computer Logical Organization – Overview In the modern world of electronics, the term Digital is generally associated with a computer because the term Digital is derived from the way computers perform operation, by counting digits. For many years, the application of digital electronics was only in the computer system. But now-a-days, digital electronics is used in many other applications. Following are some of the examples in which Digital electronics is heavily used. Industrial process control Military system Television Communication system Medical equipment Radar Navigation Signal Signal can be defined as a physical quantity, which contains some information. It is a function of one or more than one independent variables. Signals are of two types. Analog Signal Digital Signal Analog Signal An analog signal is defined as the signal having continuous values. Analog signal can have infinite number of different values. In real world scenario, most of the things observed in nature are analog. Examples of the analog signals are following. Temperature Pressure Distance Sound Voltage Current Power Graphical representation of Analog Signal (Temperature) The circuits that process the analog signals are called as analog circuits or system. Examples of the analog system are following. Filter Amplifiers Television receiver Motor speed controller Disadvantage of Analog Systems Less accuracy Less versatility More noise effect More distortion More effect of weather Digital Signal A digital signal is defined as the signal which has only a finite number of distinct values. Digital signals are not continuous signals. In the digital electronic calculator, the input is given with the help of switches. This input is converted into electrical signal which have two discrete values or levels. One of these may be called low level and another is called high level. The signal will always be one of the two levels. This type of signal is called digital signal. Examples of the digital signal are following. Binary Signal Octal Signal Hexadecimal Signal Graphical representation of the Digital Signal (Binary) The circuits that process the digital signals are called digital systems or digital circuits. Examples of the digital systems are following. Registers Flip-flop Counters Microprocessors Advantage of Digital Systems More accuracy More versatility Less distortion Easy communicate Possible storage of information Comparison of Analog and Digital Signal S.N. Analog Signal Digital Signal 1 Analog signal has infinite values. Digital signal has a finite number of values. 2 Analog signal has a continuous nature. Digital signal has a discrete nature. 3 Analog signal is generated by transducers and signal generators. Digital signal is generated by A to D converter. 4 Example of analog signal − sine wave, triangular waves. Example of digital signal − binary signal. Digital Number System A digital system can understand positional number system only where there are a few symbols called digits and these symbols represent different values depending on the position they occupy in the number. A value of each digit in a number can be determined using The digit The position of the digit in the number The base of the number system (where base is defined as the total number of digits available in the number system). Decimal Number System The number system that we use in our day-to-day life is the decimal number system. Decimal number system has base 10 as it uses 10 digits from 0 to 9. In decimal number system, the successive positions to the left of the decimal point represents units, tens, hundreds, thousands and so on. Each position represents a specific power of the base (10). For example, the decimal number 1234 consists of the digit 4 in the units position, 3 in the tens position, 2 in the hundreds position, and 1 in the thousands position, and its value can be written as (1&times1000) + (2&times100) + (3&times10) + (4&timesl) (1&times103) + (2&times102) + (3&times101) + (4&timesl00) 1000 + 200 + 30 + 1 1234 As a computer programmer or an IT professional, you should understand the following number systems which are frequently used in computers. S.N. Number System & Description 1 Binary Number System Base 2. Digits used: 0, 1 2 Octal Number System Base 8. Digits used: 0 to 7 3 Hexa Decimal Number System Base 16. Digits used: 0 to 9, Letters used: A- F Binary Number System Characteristics Uses two digits, 0 and 1. Also called base 2 number system Each position in a binary number represents a 0 power of the base (2). Example: 20 Last position in a binary number represents an x power of the base (2). Example: 2x where x represents the last position – 1. Example Binary Number: 101012 Calculating Decimal Equivalent − Step Binary Number Decimal Number Step 1 101012 ((1 &times 24) + (0 &times 23) + (1 &times 22) + (0 &times 21) + (1 &times 20))10 Step 2 101012 (16 + 0 + 4 + 0 + 1)10 Step 3 101012 2110 Note: 101012 is normally written as 10101. Octal Number System Characteristics Uses eight digits, 0,1,2,3,4,5,6,7. Also called base 8 number system Each position in an octal number represents a 0 power of the base (8). Example: 80 Last position in an octal number represents an x power of the base (8). Example: 8x where x represents the last position – 1. Example Octal Number − 125708 Calculating Decimal Equivalent − Step Octal Number Decimal Number Step 1 125708 ((1 × 84) + (2 × 83) + (5 × 82) + (7 × 81) + (0 × 80))10 Step 2 125708 (4096 + 1024 + 320 + 56 + 0)10 Step 3 125708 549610 Note: 125708 is normally written as 12570. Hexadecimal Number System Characteristics Uses 10 digits and 6 letters, 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F. Letters represents numbers starting from 10. A = 10, B = 11, C = 12, D = 13, E = 14, F = 15. Also called base 16 number system. Each position in a hexadecimal number represents a 0 power of the base (16). Example 160. Last position in

Computer Logical Organization – Resources ”; Previous Next The following resources contain additional information on Computer Logical Organization. Please use them to get more in-depth knowledge on this topic. Useful Links on Computer Logical Organization Computer Architecture − A wikipage giving a short description about computer architecture. Computer Basics by BBC − An introduction to computers including computer parts and health and safety. Basic Computer Literacy Information − A quick go through Basic Computer Literacy Information. Useful Books on Computer Logical Organization To enlist your site on this page, please drop an email to [email protected] Print Page Previous Next Advertisements ”;

Computer Logical Organization – Overview ”; Previous Next In the modern world of electronics, the term Digital is generally associated with a computer because the term Digital is derived from the way computers perform operation, by counting digits. For many years, the application of digital electronics was only in the computer system. But now-a-days, digital electronics is used in many other applications. Following are some of the examples in which Digital electronics is heavily used. Industrial process control Military system Television Communication system Medical equipment Radar Navigation Signal Signal can be defined as a physical quantity, which contains some information. It is a function of one or more than one independent variables. Signals are of two types. Analog Signal Digital Signal Analog Signal An analog signal is defined as the signal having continuous values. Analog signal can have infinite number of different values. In real world scenario, most of the things observed in nature are analog. Examples of the analog signals are following. Temperature Pressure Distance Sound Voltage Current Power Graphical representation of Analog Signal (Temperature) The circuits that process the analog signals are called as analog circuits or system. Examples of the analog system are following. Filter Amplifiers Television receiver Motor speed controller Disadvantage of Analog Systems Less accuracy Less versatility More noise effect More distortion More effect of weather Digital Signal A digital signal is defined as the signal which has only a finite number of distinct values. Digital signals are not continuous signals. In the digital electronic calculator, the input is given with the help of switches. This input is converted into electrical signal which have two discrete values or levels. One of these may be called low level and another is called high level. The signal will always be one of the two levels. This type of signal is called digital signal. Examples of the digital signal are following. Binary Signal Octal Signal Hexadecimal Signal Graphical representation of the Digital Signal (Binary) The circuits that process the digital signals are called digital systems or digital circuits. Examples of the digital systems are following. Registers Flip-flop Counters Microprocessors Advantage of Digital Systems More accuracy More versatility Less distortion Easy communicate Possible storage of information Comparison of Analog and Digital Signal S.N. Analog Signal Digital Signal 1 Analog signal has infinite values. Digital signal has a finite number of values. 2 Analog signal has a continuous nature. Digital signal has a discrete nature. 3 Analog signal is generated by transducers and signal generators. Digital signal is generated by A to D converter. 4 Example of analog signal − sine wave, triangular waves. Example of digital signal − binary signal. Print Page Previous Next Advertisements ”;

Octal Arithmetic ”; Previous Next Octal Number System Following are the characteristics of an octal number system. Uses eight digits, 0,1,2,3,4,5,6,7. Also called base 8 number system. Each position in an octal number represents a 0 power of the base (8). Example: 80 Last position in an octal number represents an x power of the base (8). Example: 8x where x represents the last position – 1. Example Octal Number − 125708 Calculating Decimal Equivalent − Step Octal Number Decimal Number Step 1 125708 ((1 × 84) + (2 × 83) + (5 × 82) + (7 × 81) + (0 × 80))10 Step 2 125708 (4096 + 1024 + 320 + 56 + 0)10 Step 3 125708 549610 Note − 125708 is normally written as 12570. Octal Addition Following octal addition table will help you to handle octal addition. To use this table, simply follow the directions used in this example: Add 68 and 58. Locate 6 in the A column then locate the 5 in the B column. The point in ”sum” area where these two columns intersect is the ”sum” of two numbers. 68 + 58 = 138. Example − Addition Octal Subtraction The subtraction of octal numbers follows the same rules as the subtraction of numbers in any other number system. The only variation is in borrowed number. In the decimal system, you borrow a group of 1010. In the binary system, you borrow a group of 210. In the octal system you borrow a group of 810. Example − Subtraction Print Page Previous Next Advertisements ”;

Binary Arithmetic ”; Previous Next Binary arithmetic is essential part of all the digital computers and many other digital system. Binary Addition It is a key for binary subtraction, multiplication, division. There are four rules of binary addition. In fourth case, a binary addition is creating a sum of (1 + 1 = 10) i.e. 0 is written in the given column and a carry of 1 over to the next column. Example − Addition Binary Subtraction Subtraction and Borrow, these two words will be used very frequently for the binary subtraction. There are four rules of binary subtraction. Example − Subtraction Binary Multiplication Binary multiplication is similar to decimal multiplication. It is simpler than decimal multiplication because only 0s and 1s are involved. There are four rules of binary multiplication. Example − Multiplication Binary Division Binary division is similar to decimal division. It is called as the long division procedure. Example − Division Print Page Previous Next Advertisements ”;