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Canonical Forms and Standard Forms

Standard Formseach term in the function can have any number of literals.Example, F1 = a +b’c + cde There are 5 variables in F1 (a,b,c,d,e)Canonical form terms should be written as m0, m1… (sum of Products) or M0, M1, M2… (Product of sums). Each term in the canonical form should have all the literals. Example: F1 = ab’c’d’e’ + abcd’e’ + abcdeMore about Canonical FormsMintermsIt is Sum of Products Canonical form is Sum of Mintermsthree variable minterms are shown belowxabcminterms0000m0=a’.b’.c’1001m1=a’.b’.c2010m2=a’.b.c’3011m3=a’.b.c4100m4=a.b’.c’5101m5=a.b’.c6110m6=a.b.c’7111m7=a.b.cMaxtermsProduct of Sum (PoS)Canonical form is Product of MaxTermsthree variable maxterms are shown belowxabcminterms0000M0=(a+b+c)1001M1=(a+b+c’)2010M2=(a+b’+c)3011M3=(a+b’+c’)4100M4=(a’+b+c)5101M5=(a’+b+c’)6110M6=(a’+b’+c)7111M7=(a’+b’+c’)usually Mi = (mj)’Express the boolean function F = A + BC in a sum of minterms. The function has three variables, so F = A + BC will be F = A(B + B’) + (A +A’) BC  [sin…

Binary Codes

bit is just called as binary digitTo represent 2n elements, n bits are neededHere are the following Binary Codes, For example, the BCD code is otherwise called as 8421 code where the 8,4,2,1 are the weights assigned to the digitsfor number 3 (0011), the weightage will be 0 * 8 + 0 * 4 + 1 * 2 + 1 *1in 84-2-1 code, the weights are arranged like this, for example, for number 2 (0110),  it is 0 * 8 + 1 * 4 + 1 * –2 + 0 * –1similarly the same case for 2421 codes.Excess – 3 is a code which is in excess of 3 in decimal numbers .Error Detection CodesBinary information is usually transmitted from one place to other through wired medium, due to the electromagnetic radiation or external noise, the information bits can be changed (ie 1 to 0 or 0 to 1), in this scenario, there is a provision to check whether the given word or byte is correct or not. Parity bits are used for that.Odd parity or even parity is adopted based on the application, but mostly even parity is adopted.Parity bit is an extra…

Radix Complement and Diminished Radix complement

Complements are used in digital computers for simplifying the subtraction and for logical manipulations. There are two types of complementRadix complements ( r complement)diminished radix complements (r-1 complement)For base 2 or binary number system the r’s complement is 2’s complement and (r-1)’s complement is 1’s complementFor base 10 or decimal system the r’s complement is 10’s complement and (r-1)’s complement is 9’s complement. 10’s ComplementLet us assume the number 3567890, for finding the 10’s complement, there is a rule3567890Most significant DigitLeast Significant Digit10’s complement can be obtained byleaving all least significant 0’s unchangedsubtracting the first non zero least significant digit from 10and subtracting all higher significant digits from 9For the above example, The 10’s complement isGiven number3567890Process9-39-59-69-79-810-9unchanged10’s Complement6432110so 10’s complement of 3567890 is 64321109’s Complement9’s complement is a diminished radix complemen…

Number Conversion

Number conversion is the fundamental operation of any digital systems.There are different bases like base2, base8, base10 and base 16Base – 10Base 2
(Binary)Base 8
(Octal)Base 16
(Hexadecimal)0000000010001011200100223001103340100044501010556011006670111077810001089100111910101012A11101113B12110014C13110115D14111016E15111117FThe above table shows the various base systems. Converting between one base to another base is of importance.  Usually all the numbering systems of day to day life is done using Base 10 or Decimal system. So it is necessary to convertOther Base to Decimal Decimal to other baseAny number represented as anan-1an-2an-3…..a1a0 . a-1a-2……..a-nWe will see one by oneDecimal to Binary (Base 10 to Base 2)(19.456)10 – (?.?)2In the above, the number 19.456 has to be converted to binary or base2 which will have only 0s and 1s.First we will take 19DivisionQuotientRemainderRemarks19/291a0=19/241a1=14/220a2=02/210a3=01/201a4=1so the conversion is 10011Sec…