Multiplication

Multiplier Notation Partial Products Logical-AND

Shift and Add Paradigm

Shift and Add Examples

Programmed Multiplication

Programmed Multiplication (cont.)

Hardware Shift and Add (right)

Hardware Shift and Add

Hardware Shift and Add (left)

Signed Number Multiplication (positive case)

Signed Number Multiplication (negative case)

Booth’s Recoding (or encoding) Developed for Speeding Up Multiplication in Early Computers When a Partial Product of 0 Occurs, Can Skip Addition and Just Shift Doesn’t Help Multipliers Where Datapaths Go Through Adder Such as Previous Examples Does Help Designs for Asynchronous Implementation or Microprogramming Since Shifting is Faster Than Addition Variable Delay – Depends on Number of One’s in Booth Observed that a String of 1’s May be Replaced as:

Booth’s Recoding Example xn xn-1 ... xi xi-1 ... x0 (0) yi=xi-1 - xi yn ... yi ... y0 EXAMPLE 0011110011(0) 0100010101

Booth’s Recoding Maps Words With Digit Set [0,1] to Those With [-1,1]

Sequential Multiplication X 1101 (-310) Y 0111 (recoded) (-1) Add –A 0101 Shift 00101 (+1) Add +A 1011 11011 Shift 111011 001111 Shift 0001111 (+1510)

Booth Multiplier Example

Booth’s Recoding Drawbacks Number of add/sub Operations are Variable Some Inefficiencies EXAMPLE 001010101(0) 011111111 Can Use Modified Booth’s Recoding to Prevent Will Look at This in Later Class

Sign Extension Consider 6-bit 2’s Complement Number s=0 Positive Value; s=1 Negative Value Show Sign Extension Works: Definition of 2’s Complement

Sign Extension Example Y 010101 (recoding) 11111101010 (neg. A) 0000000000 (0 A) 111101010 (neg. A) 00000000 (0 A) 0010110 (neg. A) 000000 (0 A) 00011110010 (24210)

Sign Extension Example Same Trick as Before, Complement Original Sign Bit Add 1 to Column 5 1 001010 (neg. A) 100000 (0 A) 001010 (neg. A) 100000 (0 A) 110110 (neg. A) 100000 (0 A) 00011110010 (24210)

Methods for Fast Multiplication Reduce Number of Partial Products to be Added Group Multiplier Bits Together Higher Radix Multiplier Add the Partial Products Faster

Radix-r Shift and Add

Radix-4 Multiplication Shifter is Multi-bit No Longer a Simple AND of xi with a Need 4:1 MUX with 0, a, 2a, 3a as Inputs

Partial Product Selection 0, a and 2a are easy 3a=a+2a  Requies an Adder! Need a Way to Compute 3a Efficiently

Example With 3a Availability

Computing 3a One Way is to Precompute 3a and Store in Register Initially Another Way is When 3a Occurs Add -a Send Carry of 1 to Next into Next Radix-4 Digit of Multiplier Causes Incoming Multiple to be [0,4] Versus [0,3] – 4 Because incoming carry to 112 Causes Digit 1002 Multiples 0, 1, 2 Handled Easily Multiple 3 Converted to –1 With Outgoing Carry of 1 Multiple 4 Converted to 0 With Outgoing Carry of 1 Requires Extra Cycle of Computation Since MSD May Have Carry

Example With 3a Availability

Using Radices >4 Could Also Use Radices of 8, 16, ... Bit Groupings of Size 3, 4, ... Multiple Generation Hardware Becomes More Complex Must Precompute 3a, 5a, 7a, .... Or Use 3a With a Carry Scheme Carry Scheme Converts Multipliers 5a, 6a, 7a to –3a, -2a, -a, etc. Carry Digit in This Form Becomes a 1

Booth Recoding Modern Arithmetic Circuits DO NOT Apply Booth Recoding Directly Useful in Understanding Higher-radix Versions of Booth Recoding No Consecutive 1’s or –1’s Occur Using Previously Seen Booth Recoding Booth Recoding in Radix-4 Results in the Following: Only Multiples of a or 2a are Required These are Easily Obtained Using Shifting and Complementation

Modified Booth Recoding Booth Recoding Results From xi and xi-1 Radix-4 Multiplier Digits Implies Booth Recoding Based on xi+1, xi and xi-1 Similar to Classical Booth Recoding, Modified Booth Recoding Encodes Multipliers into [-2,2]

Modified Booth Recoding

Example Modified Booth Recoding

Example Multiplication with MBR

Hardware MBR Example