EECC341 - ShaabanEECC341 - Shaaban#1 Lec # 3 Winter 2001 12-6-2001Binary MultiplicationBinary Multiplication• Multiplication is achieved by adding a list of shiftedmultiplicands according to the digits of the multiplier.• Ex. (unsigned) 11 1 0 1 1 multiplicand (4 bits)X 13 X 1 1 0 1 multiplier (4 bits)-------- ------------------- 33 1 0 1 1 11 0 0 0 0______ 1 0 1 1 143 1 0 1 1 --------------------- 1 0 0 0 1 1 1 1 Product (8 bits)EECC341 - ShaabanEECC341 - Shaaban#2 Lec # 3 Winter 2001 12-6-2001Binary Multiplication (continued)• Instead of listing all shifted multiplicands and thenadding, we can add each shifted multiplicand to a partialproduct. The previous un-signed example becomes: 11 1101 multiplicandx 13 x 1101 multiplier 143 0000 partial product 1011 shifted multiplicand 01011 partial product 0000 shifted multiplicand 001011 partial product 1011 shifted multiplicand 0110111 partial product 1011 shifted multiplicand 10001111 productEECC341 - ShaabanEECC341 - Shaaban#3 Lec # 3 Winter 2001 12-6-2001Two’s-complement Multiplication• A sequence of of two’s-complement additions of shifted multiplicandsexcept for last step where the shifted multiplicand corresponding to MSBmust be negated.• Before adding a shifted multiplicand to the partial product, an additionalbit is added to the left of the partial product using sign extension.Ex: - 5 1011 multiplicandx - 3 x 1101 multiplier 15 00000 partial product 11011 shifted multiplicand 111011 partial product 00000 shifted multiplicand 1111011 partial product 11011 shifted multiplicand 11100111 partial product 00101 shifted and negated multiplicand 00001111 productAdded bit usingsign extensionEECC341 - ShaabanEECC341 - Shaaban#4 Lec # 3 Winter 2001 12-6-2001Binary DivisionBinary Division• Shift and subtractExample: 19 10011 quotient11 217 1011 11011001 dividend 11 1011 shifted divisor 107 0101 reduced dividend 99 0000 shifted divisor 8 1010 reduced dividend 0000 shifted divisor 10100 reduced dividend 1011 shifted divisor 10011 reduced dividend 1011 shifted divisor 1000 remainderEECC341 - ShaabanEECC341 - Shaaban#5 Lec # 3 Winter 2001 12-6-2001Binary CodesBinary Codes• Groups of binary bits are often organized in specificways or binary codes to:– Represent decimal numbers or alphabetic characters:• Binary Coded Decimal (BCD).• American Standard Code for Information Interchange (ASCII)– Detect errors:• Even parity code.• Odd parity code.– Correct errors:• CRC Codes.– Aid in transmission and storage of digitalinformation.EECC341 - ShaabanEECC341 - Shaaban#6 Lec # 3 Winter 2001 12-6-2001Binary Coded Decimal (BCD)Binary Coded Decimal (BCD)• Binary Coded Decimal (BCD) is a way to store decimal numbers inbinary. This number representation uses 4 bits to store each digitfrom 0 to 9.• For example: 126810 = 0001 0010 0110 1000 in BCD• BCD wastes storage space since 4 bits are used to store 10combinations rather than the maximum possible 16.• BCD is often used in business applications and calculators.Decimal digit 0 1 2 3 4BCD 0000 0001 0010 0011 0100Decimal digit 5 6 7 8 9BCD 0101 0110 0111 1000 1001EECC341 - ShaabanEECC341 - Shaaban#7 Lec # 3 Winter 2001 12-6-2001BCD AdditionBCD Addition• Addition of BCD digits is similar to adding 4-bitunsigned binary numbers except a correction must bemade if a result exceeds 1001 by adding 6 to the digit. 0101+ 1001 1110+ 0110 Correction add 6 0001 0100 5+ 9 14Example:1 4EECC341 - ShaabanEECC341 - Shaaban#8 Lec # 3 Winter 2001 12-6-2001Alphanumeric Binary Codes: Alphanumeric Binary Codes: ASCIIASCIIMSBsLSBs 000 001 010 011 100 101 110 1110000NUL DLE SP 0 @ P ` p0001SOH DC1! 1 A Q a q0010STX DC2“ 2 B R b r0011 ETX DC3# 3 C S c s0100 EOT DC4$ 4 D T d t0101 ENQ NAK % 5 E U e u0110ACKSYN& 6 F V f v0111 BELETB‘ 7 G W g w1000BS CAN ( 8 H X h x1001HT EM ) 9 I Y i y1010LF SUB * : J Z j z1011 VT ESC + ; K [ k {1100 FF FS , < L \ l |1101 CR GS - = M ] m }1110 O RS . > N ^ n ~1111 SI US / ? O _ oDELSeven bit codes are used to represent all upper and lower case letters, numbers, punctuation and control charactersEECC341 - ShaabanEECC341 - Shaaban#9 Lec # 3 Winter 2001 12-6-2001Error Detection Binary CodesError Detection Binary Codes• Errors can occur during data transmission. They should bedetected, so that re-transmission can be requested.– Example: For single-bit error: 0010 can be erroneouslytransmitted as 0011, or 0000, or 0110, or 1010.• For single-error detection, one additional bit is needed:§ Even parity code:§ additional bit with value to make total number of ‘1’seven.§