1. Multi-operand Addition
Consider the Following Addition: SUM = a[0]; for (i=1; i<N; i++) { SUM = SUM + a[i]; }
a[7]
a[6]
a[5]
a[4]
a[3]
a[2]
a[1]
a[0]
a[7]+a[6]
a[5]+a[4]
a[3]+a[2]
a[1]+a[0]
a[7]+a[6]+a[5]+a[4]
a[3]+a[2]+a[1]+a[0]
a[7]+a[6]+a[5]+a[4]+a[3]+a[2]+a[1]+a[0]

2. Multi-operand Addition
a[7]+a[6]
a[5]+a[4]
a[3]+a[2]
a[1]+a[0]
a[7]+a[6]+a[5]+a[4]
a[3]+a[2]+a[1]+a[0]
a[7]+a[6]+a[5]+a[4]+a[3]+a[2]+a[1]+a[0]
O(lg2N) – Lower Bound – Theoretical Lower Limit
This is “Binary Reduction” Operation
Theoretical Time to Add Two Values
O(n) – Carry Ripple Operation
O(lg2n) – CLG/CLA tree/Prefix/Carry Skip/Carry Select
O(1) – Avizienis/Takagi Signed Digit Arithmetic

3. Multiplication Multiplication Requires Multi-operand Addition
Dot Product Requires Multi-operand Addition
Defer Carry Assimilation
Represent Intermediate Sums Redundantly

4. Implementation Serially

5. Implementation with Pipelining

6. Parallel Implementation

7. Parallel Implementation – bit level

8. Carry Save Adders
FA Used in This Configuration is Also Known as a 3:2 Compressor

9. Dot Notation
3:2 Compressor
2:2 Compressor

10. Example Tree

11. Example Tree (cont)

12. Tabular Form Representation

13. Adder Tree Bus Sizes

14. Serial Carry Save Adder

15. Wallace Tree Previous Example is 7 Input Wallace Tree
n-input Wallace Tree Reduces k-bit Inputs to Two (k + log2n - 1)-bit Outputs
CSA Reduces Number of Operands by Factor of 1.5
Smallest Height h(n) For an n-input Tree Can be Given by a Recurrence Relation

16. Wallace Tree h(n) = 1 + h(2n/3)
Ignoring Ceiling Operator Write as: h(n) = 1 + h(2n/3)
Can Get Lower Bound on Tree Height: h(n)  log1.5(n/2)
Equality for n = 2, 3 only

17. Wallace Tree Height
Can Also Consider n(h) – Number of Inputs for a Tree of Height h
Recurrence is: n(h) = 3n(h-1)/2
Ignoring Floor Operator Can get Bounds
Lower Bound: n(h) > 2(3/2)h-1
Upper Bound: n(h) < 2(3/2)h
Exact Values for 0  h  20 in Table

18. Tree Levels

19. Wallace Versus Dadda Trees
Reduce the Number of Operands at Earliest Opportunity
m Dots Per Column – Apply m/3 Full Adders to Column
Tends to Minimize Overall Delay by Making CPA CPA as Short as Possible
Delay of Fast CPA is Generally Not Smoothly Increasing Function of Word Width
EXAMPLE: CLA Has Essentially Same Delay for Widths of Bits
Dadda Tree Reduces Number of Operands to Next Lower Number Using the Fewest FAs and HAs as Possible
Justification is No Need to Reduce Number of Operands to Next Lower n(h) in Tree Since A Faster Tree Would Not Result

20. Wallace Tree

21. Dadda Tree

22. Parallel Counters
Single-bit Full Adder Referred to as (3:2) Counter (or Compressor)
Meaning is it “Counts” the Ones in 3 Input Bits
Can be Generalized to (n : log2(n+1) Counter
Has n Inputs
Produces a log2(n+1)-bit Binary Output Representing the Number of 1’s Among the n Inputs
Next Example Shows a (10:4) Counter

23. (10:4) Parallel Counter

24. Generalized Parallel Counters
Parallel Counter Reduces Number of Dots in a Column (same Radix Position)
Output Dots are Placed into Different Positions (one each)
Can Generalize This Notion
Generalized Parallel Counter Receives “Dot Patterns” as Input (not Necessarily in Same Bit Position)
Converts Them to Other Dot Patterns (not Necessarily one in Each Column)
If Output Dot Pattern Has Fewer Dots Than Input, the Counter is a Compressor and Can be Used for a Tree

25. Generalized Parallel Counters
Characterized by Number of Dots in Each Input Column and Output Column
Book Limits to Class of Counters that Output a Single Dot in Each Column
Limitation Allows Output to be Characterized by Single Integer Representing Number of Columns Spanned by Output
Input Side is Characterized by Integer Sequence Corresponding to Number of Inputs in Various Columns

26. (5,5 : 4) Parallel Counter Dot Notation for (5,5 : 4) Counter
(5,5 : 4) Counters to Compress 5 Numbers to 2 Numbers
Can Have Other Forms, eg. ( 4,6 : 4) Counter
Receives 6 bits of weight 1 and 4 bits of weight 2
Delivers the Weighted Sum in the Form of a 4-bit Binary Number
This Type Requires Sum of Output Weights to Equal or Exceed Sum of Input Weights

27. Generalized Parallel Counters
Powerful Concept – 4-bit Binary Full Adder Can be Viewed as (2,2,2,3 : 5)-counter
Goal is to Reduce n Numbers to 2 Numbers in Carry-Save Adder
Sometimes Notation of (n : 2)-counter is Used Although it Strictly Doesn’t Make Sense for n > 3
(n : 2)-counter is Shorthand Notation for a Slice of a Circuit
When Slice is Replicated, n Values are Reduced to 2 Values
Slice i Receives n Input Bits in Position i Plus Transfer (or Carry) Bits From One or More Positions to Right (i - 1, i - 2, etc.)
Slice i Produces Output Bits in Positions i and i + 1 Plus Transfer Digits Into Higher Positions (i + 1, i + 2, etc.)
yj Denotes Number of Transfer bits From Slice i to i + j

28. (n : 2) Parallel Counters
Must Satisfy This Inequality for Scheme to Work
3 Represents Maximum of 2 Output Bits
eg. (7 : 2)-counter can be Built Allowing y1 = 1 - Transfer bit From Position i to i and y2=2 - Transfer bit into Position i + 2

29. Adding Multiple Signed Numbers
Must Sign Extend 2’s Complement Numbers to Final Result Width
Appears Sign Extension Could Dramatically Increase Complexity of CSA Tree for Large n
Trick is to Take Advantage of Fact that all Sign Extension bits are Identical
Use a Single Full Adder to do Job of Several Full Adders
Allows CSA Internal Widths to be Marginally Increased

30. Hardware Sharing Method
Single Full Adder Used Here With Result Fanned Out

31. Negative Weight Interpretation
Recall That 2’s Complement Values May be Interpreted as:
Replace Negative Sign Bit by it’s Complement and Put a -1 in Sign Column
Multiple –1’s Can be Combined Each Pair Placed in –1 in Next Higher Column
A Solitary –1 in a Column is Replaced by a +1 in That Column and a –1 in the Next Higher Column

32. Negative Weight Interpretation
Complement Three Sign Bits and Place –1’s in Sign Column
Replace Three –1’s by a +1 in Sign Position and Two –1’s in Next Higher Position
These Two –1’s are Removed and Single –1 is Inserted in Position k + 1
Latter –1 is in Turn Replaced by a + 1 in Position k + 1 and a – 1 in Position k + 2
Finally a –1 Moves Out of the Resultant Sum Width and the Procedure Stops