1ECE 274 - Digital LogicLecture 13 Lecture 13 - Chapter 5.5 to 5.8 Multiplication Other Number Representations ASCII2ECE 274 - Digital Logic Shift Left and Shift Right Operations for Unsigned Numbers Unsigned binary numbers Shift Left Operation (<<) effectively multiplies value by 2 I << 2 = I * 2 Shift Right Operation (>>) effectively divides value by 2 I >> 2 = I / 200012= 11000102= 21001002= 41010002= 81000112= 31001102= 61011002= 12103ECE 274 - Digital LogicShift Left and Shift Right Operations for Signed Numbers What about signed binary numbers? We can use shift left (<<) and shift right (>>) operations, but we must preserve the sign If we shift right, replicate the sign bit in the leftmost bit (sign extend) If we shift left, normal operation of rightmost bits with 0s 0110002= 24100011002= 12100001102= 6100000112= 3101010002= -24101101002= -12101110102= -6101111012= -3104ECE 274 - Digital Logic General Multiplication General Multiplication Start with unsigned numbers How do we perform multiplication by hand?000000next bit in multiplier is 0 (0011), 0110 * 1 = 0110pp30110right most bit in multiplier is 1 (0011), 0110 * 1 = 0110pp101100next bit in multiplier is 1 (0011), 0110 * 1 = 0110pp2each row is a partial product01100011multiplicandmultiplierxx0000000next bit in multiplier is 0 (0011), 0110 * 1 = 0110pp4++productproduct is the sum of all partial products00100105ECE 274 - Digital Logic General Multiplication Another Example Notice that product is 8-bits Notice that multiplying multiplicand by 1/0 is same as ANDing with 1/0111010111110111000000001110000(14)(11)(154)x+10011010111011110AND6ECE 274 - Digital Logic General Multiplication Generalized representation of multiplication by hand7ECE 274 - Digital LogicMultiplier – Array Style Array Style Multiplier design Array of AND gatesABP*Block symbola0a1a2a3b0b1b2b3pp1pp2pp3pp4p7..p0+ (5-bit)+ (6-bit)+ (7-bit)0000000008ECE 274 - Digital Logic Multiplying Signed Numbers Multiplication of signed numbers Scenario 1: (+A) * (+B)to represent the +14 and +11, we need an extra bitpartial products computed the samenew partial product larger by 1 bit (avoid possible overflow)011100101100011100011100(+14)(+11)(+154)x0010011010++++001010100001010100010011010000000000011100000000000000multiplicandmultiplierxproductSigned Binary (5-bits) = 01110Unsigned Binary = 1110Decimal Value = 14Signed Binary (6-bits) = 001110Signed Binary (7-bits) = 0001110Using sign extension we replicate sign bit into leftmost bits9ECE 274 - Digital Logic Multiplying Signed Numbers Multiplication of signed numbers Scenario 1: (+A) * (+B) Scenario 2: (-A) * (+B)multiplicandmultiplierxproductpartial products computed the samenew partial product larger by 1 bit (avoid possible overflow)100100101111100101100100(-14)(+11)(-154)x1101100110++++110101101110101001101100000000000001100100000000000000Signed Binary (5-bits) = 10010Unsigned Binary = n/aDecimal Value = -14Signed Binary (6-bits) = 110010Signed Binary (7-bits) = 1110010Using sign extension we replicate sign bit into leftmost bitsto represent the -14 and +11, we need an extra bit10ECE 274 - Digital Logic Multiplying Signed Numbers Multiplication of signed numbers Scenario 1: (+A) * (+B) Scenario 2: (-A) * (+B)  Scenario 3: (+A) * (-B) Scenario 4: (-A) * (-B)multiplicandmultiplierxproductConvert each term into it’s two’s complementScenario 3: (-A) * (+B)Scenario 4: (+A) * (+B)Won’t change result, back to scenario 1 & 211ECE 274 - Digital Logic Fixed Point Number Representation Fixed point number consists of a integer and fraction part Position of radix (decimal point) is assumed to be fixed234.56 = (2 * 102) + (3 * 101) + (4 * 100) + (5 * 10-1) + (6 * 10-2)= (2 * 100) + (3 * 10) + (4 * 1) + (5 * 1/10) + (6 * 1/100)234.561021011005610-110-242312ECE 274 - Digital Logic Fixed Point Number Representation of Binary Numbers Fixed point number for binary is similar100.11 = (1 * 22) + (0 * 21) + (0 * 20) + (1 * 2-1) + (1 * 2-2)= (1 * 4) + (0 * 2) + (0 * 1) + (1 * 1/2) + (1 * 1/4)= 4.7510100.11222120112-12-201013ECE 274 - Digital Logic Converting Fixed Point Decimals to Binary Decimal to binary conversion Integer portion remains same Fractional portion can be converted using multiply method 2.6521202-12-22-32-410.65 * 2 = 1.30Integer value placed in 2-1position, next consider fractional part00.30 * 2 = 0.60Integer value placed in 2-2position , next consider fractional part10.60 * 2 = 1.20Integer value placed in 2-3position , next consider fractional part00.20 * 2 = 0.40Integer value placed in 2-4position , next consider fractional part01same as before2-52-600.40 * 2 = 0.80Integer value placed in 2-5position, next consider fraction part10.80 * 2 = 1.60Integer value placed in 2-6position, only have 6-bits to represent fractional part so we are done14ECE 274 - Digital Logic Converting Fixed Point Decimals to Binary Did it work?10.101001 = (0 * 21) + (1 * 20) + (0 * 2-1) + (1 * 2-2) + (1 * 2-3) + (0 * 2-4) + (1 * 2-5) + (1 * 2-6) 2.652120102-12-2012-32-4102-502-61= (1 * 2) + (0 * 1) + (1 * 1/2) + (1 * 1/4) + (1 * 1/8) + (0 * 1/16) + (1 * 1/32) + (1 * 1/64)= 2 + 0.5 + 0.125 + 0.0156 = 2.640610not the most accurate way to store numbers15ECE 274 - Digital Logic Floating Point Number Representation Floating Point Numbers234.56 = (2 * 102) + (3 * 101) + (4 * 100) + (5 * 10-1) + (6 * 10-2)= (2 * 100) + (3 * 10) + (4 * 1) + (5 * 1/10) + (6 * 1/100)234.56 * 10431021011005610-110-2423ExponentMantissaRadixMantissa indicates significant numbers – we know how to represent thisRadix + exponent indicates where/how much to move the decimal16ECE 274 - Digital Logic Single Precision/Double Precision Float Point Number Standard Floating point binary numbers IEEE (Institute of Electrical and Electronics Engineers) has produced a standard Single precision (32 bit) floating point numbers Double precision (64 bit) floating point numbers32-bitsSign bit0 denotes “+”1 denotes “-”8-bit exponentexcess – 127 format (radix =2)23-bit mantissa234.56 * 1043ExponentMantissaRadixS E M17ECE 274 - Digital Logic Single Precision Floating Point Number - Exponent Exponent is in excess-127 format (or “biased with 127”) E = actual value + 127 Makes manipulating numbers easier Reserved values E = 0 (0000 0000) indicates exact zero E = 255 (1111 1111) indicates infinity Normal range of E