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Chapter 2 - Part 1 - PPT - Mano & Kime - 2nd EdSlide 1 Lecture 17: Signed ArithmeticLecture 17: Signed Arithmetic
Slide 2 OverviewOverview Signed Integers Sign-magnitude Complement representations Binary adder-subtractors Signed binary addition and subtraction Overflow Binary multiplication Other arithmetic functions Design by contraction
Slide 3 Signed IntegersSigned Integers Positive numbers and zero: Use unsigned n-digit, radix r numbers Negative numbers: Need a sign bit (+ or -) By convention, the MSB is the sign bit: s an 2 a2a1a0 where: s = 0 for Positive numbers s = 1 for Negative numbers and ai = 0 or 1 represent the magnitude in some form.
Slide 4 Signed Integer RepresentationsSigned Integer Representations Signed-Magnitude here the n 1 digits are interpreted as a positive magnitude. Signed-Complement here the digits are interpreted as the rest of the complement of the number. Signed 1's Complement Uses 1's Complement Arithmetic Signed 2's Complement Uses 2's Complement Arithmetic
Slide 5 Signed Integer Representation ExampleSigned Integer Representation Example r =2, n=3 Number Sign - Mag. 1's Comp. 2's Comp. +3 011 011 011 +2 010 010 010 +1 001 001 001 +0 000 000 000 0 100 111 1 101 110 111 2 110 101 110 3 111 100 101 4 100
Slide 6 Signed-Magnitude ArithmeticSigned-Magnitude Arithmetic Algorithm is covered in the book Awkward Complex rules for correction Not covered in detail You are not expected to know this You are expected to know that it is possible Instead, use complement-based representations One s complement Two s complement
Slide 7 Signed-Complement ArithmeticSigned-Complement Arithmetic Addition: Use conventional unsigned adder Sign bit is produced correctly, except Overflow can occur: If the sign bits were the same for both numbers and the sign of the result is different, an overflow has occurred. Subtraction: Form the complement of the number you are subtracting and follow the rules for addition.
Slide 8 Signed 2 s Complement ExamplesExample 1: 1101 + 0011 0000 Example 2: 1101 1101 - 0011 1101 1010 Signed 2 s Complement Examples
Example 1: Result is 0000. The carry out of the MSB is discarded. Example 2: Complement 0011 to 1101 and add. Result is 1010. The carry out of the MSB is discarded.
Slide 9 Signed 1 s Complement ExamplesExample 1: 1101 + 0011 0000 1 0001 Example 2: 1101 1101 - 0011 1100 1001 1 1010 Signed 1 s Complement Examples 1 1
Example 1: Adding, the result is 0000 and the carry from the MSB is 1. This carry is added to the LSB to give final result 0001 Example 2: Take the 1s complement of 0011 to obtain 1100 and add. The result is 1001 and the carry from the MSB is 1. Adding this carry to the LSB, the final result is 1010.
Slide 10 2 s Complement Adder/Subtractor2 s Complement Adder/Subtractor Subtraction can be done by addition of the 2's Complement. 1. Complement each bit (1's Complement.) 2. Add 1 to the result. The circuit shown computes A + B and A B: For S = 1, subtract Invert B with XOR Add 1 by setting C0 For S = 0, add, B is passed through unchanged Adobe Systems
Slide 11 Overflow DetectionOverflow Detection Overflow occurs if n + 1 bits are required to contain the result from an n-bit addition or subtraction Unsigned addition 5 + 6 > 7: 101 + 110 => 1011 Carry out from MSB indicates unsigned overflow Signed addition 2 + 3 = 5 > 4: 010 + 011 = 101 =? 3 < 0 Sum of two positive numbers should not be negative -1 + -4: 111 + 100 = 011 > 0 Sum of two negative numbers should not be positive
Slide 12 Binary MultiplicationBinary Multiplication The binary digit multiplication table is trivial: This is simply the Boolean AND function. Form larger products the same way we form larger products in base 10. (a × b) b = 0 b = 1 a = 0 0 0 a = 1 0 1
Slide 13 Review - Decimal Example: (237 149)10Review - Decimal Example: (237 × 149)10 Partial products are: 237 × 9, 237 × 4, and 237 × 1 Note that the partial product summation for n digit, base 10 numbers requires adding up to n digits (with carries). Note also n × m digit multiply generates up to an m + n digit result. 2 3 7 × 1 4 9 2 1 3 3 9 4 8 - + 3 5 3 1 3 2 3 7 - -
Slide 14 Binary Multiplication AlgorithmBinary Multiplication Algorithm We execute radix 2 multiplication by: Computing partial products, and Justifying and summing the partial products. (same as decimal) To compute partial products: Multiply the row of multiplicand digits by each multiplier digit, one at a time. With binary numbers, partial products are very simple! They are either: all zero (if the multiplier digit is zero), or the same as the multiplicand (if the multiplier digit is one).
Slide 15 Example: (101 x 011) Base 2Example: (101 x 011) Base 2 Partial products are: 101 × 1, 101 × 1, and 101 × 0 Note that the partial product summation for n digit, base 2 numbers requires adding up to n digits (with carries) in a column. Note also n × m digit multiply generates up to an m + n digit result (same as decimal). 1 0 1 × 0 1 1 1 0 1 1 0 1 0 0 0 0 0 1 1 1 1
Slide 16 Multiplier Boolean EquationsMultiplier Boolean Equations An n × m block multiplier forms partial products. Example: 2 × 2 The logic equations for each partial-product binary digit are shown below: We need to "add" the columns to get the product bits P0, P1, P2, and P3. Note that some columns may generate carries. b1 b0 ´ a1 a0 (a0 . b1) (a0 . b0) + (a1 . b1) (a1 . b0) P3 P2 P1 P0
Slide 17 Multiplier Arrays Using AddersMultiplier Arrays Using Adders An implementation of the 2 × 2 multiplier array is shown: C 0 C 3 HA HA C 2 C 1 A 0 A 1 B 1 B 0 B 1 B 0
Slide 18 Other Arithmetic FunctionsOther Arithmetic Functions Convenient to design the functional blocks by contraction - removal of redundancy from circuit to which input fixing has been applied Functions Incrementing Decrementing Zero Fill and Extension
Slide 19 Design by ContractionDesign by Contraction Contraction is a technique for simplifying the logic in a functional block to implement a different function The new function must be realizable from the original function by applying rudimentary functions to its inputs Contracted version specialized for a fixed input Logic can be much simpler
Slide 20 Design by Contraction ExampleDesign by Contraction Example Contraction of a ripple carry adder to incrementer for n = 3 Set B = 001 The middle cell can be repeated to make an incrementer with n > 3. Adobe Systems
Slide 21 Incrementing & DecrementingIncrementing & Decrementing Incrementing Adding a fixed value to an arithmetic variable Fixed value is often 1, called counting (up) Examples: A + 1, B + 4 Functional block is called incrementer Decrementing Subtracting a fixed value from an arithmetic variable Fixed value is often 1, called counting (down) Examples: A - 1, B - 4 Functional block is called decrementer
Slide 22 Zero FillZero Fill Zero fill - filling an m-bit operand with 0s to become an n-bit operand with n > m Filling usually is applied to the MSB end of the operand, but can also be done on the LSB end Example: 11110101 filled to 16 bits MSB end: 0000000011110101 LSB end: 1111010100000000
Slide 23 Sign ExtensionSign Extension Sign Extension maintaining sign bit for a complement representation Copies the MSB of the operand into the new positions Positive operand example - 01110101 extended to 16 bits: 0000000001110101 Negative operand example - 11110101 extended to 16 bits: 1111111111110101 Results in equivalent number in a wider representation
Slide 24 SummarySummary Signed Integers Sign-magnitude Complement representations Binary adder-subtractors Signed binary addition and subtraction Overflow Binary multiplication Other arithmetic functions Design by contraction
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