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Multi-operand Addition

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].

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Multi-operand Addition

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  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] 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. Parallel Implementation – bit level

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

  10. Dot Notation 3:2 Compressor 2:2 Compressor

  11. Example Tree

  12. Example Tree (cont)

  13. Tabular Form Representation

  14. Adder Tree Bus Sizes

  15. Serial Carry Save Adder

  16. Serial Carry Save Adder

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