1. 專題研究 BOOTH ENCODING MULTIPLIER 指導教授 吳安宇 組員 蔡詩蘅 吳明吉吉 徐國軒 張景翔

2. 1.Booth encoding What is Booth Encoding ? Modified Booth Encoding Sign Extension-compensation vector Original Multiplier Structure

3. 2.Power Dissipation Transition probability MSD Property Proposed Multiplier Structure

4. What is Booth Encoding ? Y = -2 n-1 y n-1 + 2 n-2 y n-2 +…+2 0 y 0 = -2 n-1 (y n-2 - y n-1 ) + 2 n-2 (y n-3 -y n-2 ) +……+2 0 (y 1 –y 0 )

5. Modified Booth Encoding Y = -2 n-1 y n-1 + 2 n-2 y n-2 +…+2 0 y 0 = 2 n-2 (2y n-1 - 2y n – y n-1 + y n-2 ) +…… + 2 0 (-2y 1 +y 0 ) = 2 n-2 (- 2y n + y n-1 + y n-2 ) +……

6. Radix-4 Modified Booth Encoding Table Y 2i+1 Y 2i Y 2i-1 | S i ---------------------------------------- 0 0 0 | +0 0 01 | +1 0 10 | +1 0 11 | +2 1 00 | - 2 1 01 | - 1 1 10 | - 1 1 1 1 | +0

7. Booth Multiplier Structure

8. Sign-generate Sign extension

9. Correction Factor

10. Example

11. Reduced Transitions

12. Encoding Operation

13. Architecture of Booth- encoding Multiplier

14. The Power Dissipation Power = 0.5C L V DD 2 f CLK E SW E SW :The switching activity =>The probability of a transition within a clock cycle

15. Transition probability P Δ (s) = 2p(s)(1-p(s)) p(s)=probability of s being non- zero Transition:May cause glitches in combinatorial circuits=> High switch probability->Short logic depth

16. The One Probability of A Carry Signal P(C 1 ) = p(Q m )p(Q n ) Q m,Q n are the two partial-products added in the first adder row

17. Carry output of k-th adder row P(C k ) = 0.5p(C k-1 ) + 0.5p(Q i ) C k :a carry out of the k-th adder row Q i :a partial product added into the k-th adder row (P(CARRY)=p 1 p 2 +p 2 p 3 +p 3 p 1 -2p 1 p 2 p 3 when p1=0.5)

18. MSD For a signal that has a smaller magnitude than its wordlength 000 ….xxxxx …. 111 ….xxxxx …. Repeat sign-extension bits Booth encoding (Si) as +0

19. A Signal Property P(Q i ) = p(X)p(S i ) p(X)=0.5 for random multiplicand The MSD has low p(S i ) and hence low p(Q i ) P(Q i )<0.5, P(Q i ) implies P Δ (Q i ) Fist add the MSD partial-product in a carry-save array for reduced switching

20. Proposed Architecture of Booth-encoding Multiplier