ECE 4110–5110 Digital System Design

1. ECE 4110–5110 Digital System Design Lecture #22 • Agenda • MSI: Multipliers • Announcements • Quiz 2, Nov 15th.

2. Multipliers • Multipliers- binary multiplication of an individual bit can be performed using combinational logic:A * BP0 0 00 1 0 we can say that: P = A·B1 0 01 1 1 - for multi-bit multiplication, we can mimic the algorithm that we use when doing multiplication by hand ex) 1 2 this number is the "Multiplicand"x 3 4 this number is the "Multiplier" 4 8 1) multiplicand for digit (0)+ 3 6 2) multiplicand for digit (1) 4 0 8 3) Sum of all multiplicands - this is called the "Shift and Add" algorithm

3. Multipliers • "Shift and Add" Multipliers- example of Binary Multiplication using our "by hand" method 11 1 0 1 1 - multiplicandx 13 x 1 1 0 1 - multiplier 33 1 0 1 1 11 0 0 0 0 - these are the individual multiplicands 1 0 1 1 + + 1 0 1 1 1 4 3 1 0 0 0 1 1 1 1 - the final product is the sum of all multiplicands - this is simple and straight forward. BUT, the addition of the individual multiplicand products requires as many as n-inputs. - we would really like to re-use our Full Adder circuits, which only have 3 inputs.

4. Multipliers • "Shift and Add" Multipliers - we can perform the additions of each multiplicand after it is created- this is called a "Partial Product"- to keep the algorithm consistent, we use "0000" as the first Partial Product 1 0 1 1 - Original multiplicandx 1 1 0 1 - Original multiplier 0 0 0 0 - Partial Product for 1st multiply 1 0 1 1 - Shifted Multiplicand for 1st multiply 1 0 1 1 - Partial Product for 2nd multiply 0 0 0 0  - Shifted Multiplicand for 2nd multiply 0 1 0 1 1 - Partial Product for 3rd multiply 1 0 1 1  - Shifted Multiplicand for 3rd multiply 1 1 0 1 1 1 - Partial Product for 4th multiply 1 0 1 1  - Shifted Multiplicand for 4th multiply 1 0 0 0 1 1 1 1 - the final product is the sum of all multiplicands

5. Multipliers • "Shift and Add" Multipliers - Graphical view of product terms and summation

6. Multipliers • "Shift and Add" Multipliers - Graphical View of interconnect for an 8x8 multiplier. Note the Full Adders

7. Multipliers • "Sequential" Multipliers - the main speed limitation of the Combinational "Shift and Add" multiplier is the delay through the adder chain. - in the worst case, the number of delay paths through the adders would be [n + 2(n-2)] ex) 4-bit = 8 Full Adders 8-bit = 20 Full Adders- we can decrease this delay by using a register to accumulate the incremental additions as they take place.- this would reduce the number of operation states to [n-1] • "Carry Save" Multipliers- another trick to speed up the multiplication is to break the carry chain- we can run the 0th carry from the first row of adders into adder for the 2nd row- a final stage of adders is needed to recombine the carrys. But this reduces the delay to [n+(n-2)]

8. Multipliers • "Carry Save" Multipliers

9. Signed Multipliers • Multipliers- we leaned the "Shift and Add" algorithm for constructing a combinational multiplier- but this only worked for unsigned numbers- we can create a signed multiplier using a similar algorithm • Convert to Positive- one of the simplest ways is to first convert any negative numbers to positive, then use the unsigned multiplier- the sign bit is added after the multiplication following: pos x pos = pos Remember 0=pos and 1=neg is 2's comp so this is an XOR pos x neg = neg neg x pos = neg neg x neg = pos

10. Signed Multipliers • 2's Comp Multiplier- remember that in a "Shift and Add', we created a shifted multiplicand- the shifted multiplicand corresponded to the weight of the multiplier bit- we can use this same technique for 2's comp remembering that - the MSB of a 2's comp # is -2(n-1)- we also must remember that 2's comp addition must - be on same-sized vectors - the carry is ignored- we can make partial products the same size as shifted multiplicands by doing a "2's comp sign extend" ex) 1011 = 11011 = 1110111- since the MSB has a negative weight, we NEGATE the shifted multiplicand for that bit prior to the last addition.

11. Signed Multipliers • 2's Comp Shift and Add Multipliers - we can perform the additions of each multiplicand after it is created- this is called a "Partial Product"- to keep the algorithm consistent, we use "0000" as the first Partial Product 1 0 1 1 - Original multiplicandx 1 1 0 1 - Original multiplier 0 0 0 0 0 - Partial Product for 1st multiply w/ Sign Extension1 1 0 1 1 - Shifted Multiplicand for 1st multiply w/ Sign Extension1 1 1 0 1 1 - Partial Product for 2nd multiply w/ Sign Extension0 0 0 0 0  - Shifted Multiplicand for 2nd multiply w/ Sign Extension1 1 1 1 0 1 1 - Partial Product for 3rd multiply w/ Sign Extension1 1 0 1 1  - Shifted Multiplicand for 3rd multiply w/ Sign Extension1 1 1 0 0 1 1 1 - Partial Product for 4th multiply w/ Sign Extension 0 0 1 0 1  - NEGATED Shifted Multiplicand for 4th multiply w/ Sign Extension1 0 0 0 0 1 1 1 1 - the final product is the sum of all multiplicands ignore Carry_Out

12. Division • Division - "Repeated Subtraction"- a simple algorithm to divide is to count the number of times you can subtract the divisor from the dividend- this is slow, but simple- the number of times it can be subtracted without going negative is the "Quotient"- if the subtracted value results in a zero/negative number, whatever was left prior to the subtraction is the "Remainder"

13. Division • Division - "Shift and Subtract" - Division is similar to multiplication, but instead of "Shift and Add", we "Shift and Subtract"