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- DIGITAL LOGIC DESIGN - DESIGN
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**JNTU KAKINADA - ADARSH**- Computer Science Engineering
- B.Tech
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Digital Logic Design Introduction A digital computer stores data in terms of digits (numbers) and proceeds in discrete steps from one state to the next. The states of a digital computer typically involve binary digits which may take the form of the presence or absence of magnetic markers in a storage medium , on-off switches or relays. In digital computers, even letters, words and whole texts are represented digitally. Digital Logic is the basis of electronic systems, such as computers and cell phones. Digital Logic is rooted in binary code, a series of zeroes and ones each having an opposite value. This system facilitates the design of electronic circuits that convey information, including logic gates. Digital Logic gate functions include and, or and not. The value system translates input signals into specific output. Digital Logic facilitates computing, robotics and other electronic applications. Digital Logic Design is foundational to the fields of electrical engineering and computer engineering. Digital Logic designers build complex electronic components that use both electrical and computational characteristics. These characteristics may involve power, current, logical function, protocol and user input. Digital Logic Design is used to develop hardware, such as circuit boards and microchip processors. This hardware processes user input, system protocol and other data in computers, navigational systems, cell phones or other high-tech systems. A.F. Kana Digital Logic Design. Page 1

Data Representation and Number system Numeric systems The numeric system we use daily is the decimal system, but this system is not convenient for machines since the information is handled codified in the shape of on or off bits; this way of codifying takes us to the necessity of knowing the positional calculation which will allow us to express a number in any base where we need it. Radix number systems The numeric system we use daily is the decimal system, but this system is not convenient for machines since the information is handled codified in the shape of on or off bits; this way of codifying takes us to the necessity of knowing the positional calculation which will allow us to express a number in any base where we need it. A base of a number system or radix defines the range of values that a digit may have. In the binary system or base 2, there can be only two values for each digit of a number, either a "0" or a "1". In the octal system or base 8, there can be eight choices for each digit of a number: "0", "1", "2", "3", "4", "5", "6", "7". In the decimal system or base 10, there are ten different values for each digit of a number: "0", "1", "2", "3", "4", "5", "6", "7", "8", "9". In the hexadecimal system, we allow 16 values for each digit of a number: "0", "1", "2", "3", "4", "5", "6", "7", "8", "9", "A", "B", "C", "D", "E", and "F". Where “A” stands for 10, “B” for 11 and so on. Conversion among radices - Convert from Decimal to Any Base Let’s think about what you do to obtain each digit. As an example, let's start with a decimal number 1234 and convert it to decimal notation. To extract the last digit, you move the decimal point left by one digit, which means that you divide the given number by its base 10. 1234/10 = 123 + 4/10 The remainder of 4 is the last digit. To extract the next last digit, you again move the decimal point left by one digit and see what drops out. 123/10 = 12 + 3/10 The remainder of 3 is the next last digit. You repeat this process until there is nothing left. Then you stop. In summary, you do the following: A.F. Kana Digital Logic Design. Page 2

Quotient Remainder ----------------------------1234/10 = 123 4 --------+ 123/10 = 12 3 ------+ | 12/10 = 1 2 ----+ | | 1/10 = 0 1 --+ | | |(Stop when the quotient is 0) | | | | 1 2 3 4 (Base 10) Now, let's try a nontrivial example. Let's express a decimal number 1341 in binary notation. Note that the desired base is 2, so we repeatedly divide the given decimal number by 2. Quotient Remainder ----------------------------1341/2 = 670 1 ----------------------+ 670/2 = 335 0 --------------------+ | 335/2 = 167 1 ------------------+ | | 167/2 = 83 1 ----------------+ | | | 83/2 = 41 1 --------------+ | | | | 41/2 = 20 1 ------------+ | | | | | 20/2 = 10 0 ----------+ | | | | | | 10/2 = 5 0 --------+ | | | | | | | 5/2 = 2 1 ------+ | | | | | | | | 2/2 = 1 0 ----+ | | | | | | | | | 1/2 = 0 1 --+ | | | | | | | | | |(Stop when the | | | | | | | | | | | quotient is 0) 1 0 1 0 0 1 1 1 1 0 1 (BIN; Base 2) Let's express the same decimal number 1341 in octal notation. Quotient Remainder ----------------------------1341/8 = 167 5 --------+ 167/8 = 20 7 ------+ | 20/8 = 2 4 ----+ | | 2/8 = 0 2 --+ | | | (Stop when the quotient is 0) | | | | 2 4 7 5 (OCT; Base 8) Let's express the same decimal number 1341 in hexadecimal notation. Quotient Remainder ----------------------------1341/16 = 83 13 ------+ 83/16 = 5 3 ----+ | 5/16 = 0 5 --+ | | (Stop when the quotient is 0) | | | 5 3 D (HEX; Base 16) In conclusion, the easiest way to convert fixed point numbers to any base is to convert each part separately. We begin by separating the number into its integer and fractional part. The integer part is converted using the remainder method, by using a successive division of the number by the base until a zero is obtained. At each division, the reminder is kept and then the new number in the base r is obtained by reading the remainder from the lat remainder upwards. The conversion of the fractional part can be obtained by successively multiplying the fraction with the base. If we iterate this process on the remaining fraction, then we will obtain successive significant digit. This methods form the basis of the multiplication methods of converting fractions between bases Example. Convert the decimal number 3315 to hexadecimal notation. What about the hexadecimal equivalent of the decimal number 3315.3? A.F. Kana Digital Logic Design. Page 3

Solution: Quotient Remainder ----------------------------3315/16 = 207 3 ------+ 207/16 = 12 15 ----+ | 12/16 = 0 12 --+ | | | | | C F 3 Product Integer Part -------------------------------0.3*16 = 4.8 4 0.8*16 = 12.8 12 0.8*16 = 12.8 12 0.8*16 = 12.8 12 : : Thus, 3315.3 (DEC) --> CF3.4CCC... (Stop when the quotient is 0) (HEX; Base 16) (HEX; Base 16) 0.4 C C C ... | | | | ----+ | | | | | ------+ | | | | --------+ | | | ----------+ | | ---------------------+ (HEX) - Convert From Any Base to Decimal Let's think more carefully what a decimal number means. For example, 1234 means that there are four boxes (digits); and there are 4 one's in the right-most box (least significant digit), 3 ten's in the next box, 2 hundred's in the next box, and finally 1 thousand's in the left-most box (most significant digit). The total is 1234: Original Number: 1 2 | | How Many Tokens: 1 2 Digit/Token Value: 1000 100 Value: 1000 + 200 or simply, 3 | 3 10 + 30 4 | 4 1 + 4 = 1234 1*1000 + 2*100 + 3*10 + 4*1 = 1234 Thus, each digit has a value: 10^0=1 for the least significant digit, increasing to 10^1=10, 10^2=100, 10^3=1000, and so forth. Likewise, the least significant digit in a hexadecimal number has a value of 16^0=1 for the least significant digit, increasing to 16^1=16 for the next digit, 16^2=256 for the next, 16^3=4096 for the next, and so forth. Thus, 1234 means that there are four boxes (digits); and there are 4 one's in the right-most box (least significant digit), 3 sixteen's in the next box, 2 256's in the next, and 1 4096's in the left-most box (most significant digit). The total is: 1*4096 + 2*256 + 3*16 + 4*1 = 4660 In summary, the conversion from any base to base 10 can be obtained from the formulae x1 0 = n −1 ∑d b i =− m i i Where b is the base, di the digit at position i, m the number of digit after the decimal point, n the number of digits of the integer part and X10 is the obtained number in decimal. This form the basic of the polynomial method of converting numbers from any base to decimal Example. Convert 234.14 expressed in an octal notation to decimal. 2*82 + 3*81 + 4*80+1*8-1 + 4*8-2 A.F. Kana = 2*64 +3*8 +4*1 +1/8 +4/64 =156.1875 Digital Logic Design. Page 4

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