Zhongkai Chen

 Appears in: VLSI Design, Automation and Test, VLSI-DAT International Symposium on Date:25-27 April 2007  Zi-Yi Zhao, Chien-Hung Lin, Yu-Zhi Xie, Yen- Ju Chen, Yi-Jie Lin, and Shu-Chung Yi.  National Changhua University of Education, Changhua, Taiwan 500, ROC

 Introduction  BA (Binary to Abacus) Module  PA (Parallel Addition) Module  TB (Thermometric to Binary) Module  Example  Extension to 4n-Bit Adder  Results

 The first digital arithmetic circuits employing the method of the Chinese abacus were proposed on  The most drawback is the delay time due to serial addition of each bead by the Shift-Up module

 (a) Chinese abacus coding with base 10 of the decimal number 6  (b) the proposed Chinese abacus adder coding with base 16 of the decimal number

Block diagram of the 4-bit abacus adder

 This module converts a 4-bit binary number (I 3 I 2 I 1 I 0 ) 2 into an abacus representation (H 2 H 1 H 0 | L 2 L 1 L 0 ) abacus.  H2H1H0 and L2L1L0 are determined by: H 2 =I 3 I 2, H 1 =I 3, H 0 =I 3 +I 2 L 2 =I 1 I 0, L 1 =I 1, L 0 =I 1 +I 0 Where 0≤H 2 ≤H 1 ≤H 0 ≤1, and 0 ≤L 2 ≤L 1 ≤L 0 ≤1. H 2 H 1 H 0 :Higher Beads L 2 L 1 L 0 : Lower Beads

 This block can parallel add two abacus numbers simultaneously.  The sum of (X 2 X 1 X 0 ) and (Y 2 Y 1 Y 0 ) will then be represented as the thermometric transformation (K 5 K 4 K 3 K 1 K 0 ), where 0 ≤K i ≤K j ≤1 for i>j.

 The behavior of PA module is modeled in the following equations: 1.This module acts similarly as a multiplexer. The addend X 2 ~X 0 2.are as signal selectors to modify the configuration of augend Y 2 ~Y 0. 3.The thermometric results will be then represented in K 5 ~K 0. 4.There are only four configurations of addend (X 2 X 1 X 0 ), i.e., (000), (001), (011), or (111).

 This module transforms thermometric representation to binary numbers. The outputs O 1, O o and C 0ut are determined by the following equations:  O 0 =K 0 ∙C in +K 1 ∙K 0 ∙C in +K 2 ∙K 1 ∙C in +K 3 ∙K 2 ∙ C in +K 4 ∙K 3 ∙C in +K 5 ∙K 4 ∙C in +K 5 ∙C in  O 1 =K 5 +K 4 ∙C in +K 2 ∙K 0 ∙C in +K 4 ∙K 1 ∙C in  C out =K 3 + K 2 ∙C in

 9=>(1001)13=> (1101)  (1001)+(1101)  BA Module:  BA1: (1001) 2 =(011|001) abacus  BA2: (1101) 2 =(111|001) abacus  PA Module:  PA1: X2=0 X1=1 X0=1 Y2=1 Y1=1 Y0=1  PA2: X2=0 X1=0 X0=1 Y2=0 Y1=0 Y0=1

 PA1: X2=0 X1=1 X0=1, Y2=1 Y1=1 Y0=1, f1=X2X1=1, f2=X1X0=0 K0=1, K1=1, K2=Y0=1, K3=Y1=1, K4=Y2=1 K5=0

 PA2: X2=0 X1=0 X0=1, Y2=0 Y1=0 Y0=1, f1=X2X1=0 f2=X1X0=1 K0=1 K1=Y0=1 K2=Y1=0 K3=Y2=0 K4=0 K5=0

 TB Module:  TB2:  Cin=0, K0=1 K1=1 K2=0 K3=0 K4=0 K5=0  O0=…= =0  O1=…= =1  Cout=…=0  O 0 =K 0 ∙C in +K 1 ∙K 0 ∙C in +K 2 ∙K 1 ∙C in +K 3 ∙K 2 ∙C in +K 4 ∙K 3 ∙C in +K 5 ∙K 4 ∙C in +K 5 ∙C in  O 1 =K 5 +K 4 ∙C in +K 2 ∙K 0 ∙C in +K 4 ∙K 1 ∙C in  C out =K 3 + K 2 ∙C in  O 0 =K 0 ∙C in +K 1 ∙K 0 ∙C in +K 2 ∙K 1 ∙C in +K 3 ∙K 2 ∙C in +K 4 ∙K 3 ∙C in +K 5 ∙K 4 ∙C in +K 5 ∙C in  O 1 =K 5 +K 4 ∙C in +K 2 ∙K 0 ∙C in +K 4 ∙K 1 ∙C in  C out =K 3 + K 2 ∙C in

 TB1  Cin=0, K0=1, K1=1, K2=1, K3=1, K4=1 K5=0  O0=…= =1  O1=…= =0  Cout=…=1+0=1  Final Answer: (10110) 2 =22  O 0 =K 0 ∙C in +K 1 ∙K 0 ∙C in +K 2 ∙K 1 ∙C in +K 3 ∙K 2 ∙C in +K 4 ∙K 3 ∙C in +K 5 ∙K 4 ∙C in +K 5 ∙C in  O 1 =K 5 +K 4 ∙C in +K 2 ∙K 0 ∙C in +K 4 ∙K 1 ∙C in  C out =K 3 + K 2 ∙C in  O 0 =K 0 ∙C in +K 1 ∙K 0 ∙C in +K 2 ∙K 1 ∙C in +K 3 ∙K 2 ∙C in +K 4 ∙K 3 ∙C in +K 5 ∙K 4 ∙C in +K 5 ∙C in  O 1 =K 5 +K 4 ∙C in +K 2 ∙K 0 ∙C in +K 4 ∙K 1 ∙C in  C out =K 3 + K 2 ∙C in

 Generally, conventional fast adders of high bit number are all connected by 4-bit unit adders. The Chinese abacus adder can also extend to 4n-bit adder by CLA-like or conditional-sum-adder-like methods. Replace it with Abacus Adder Extend both A and B to 4 bits The C out of CLA: C i+1 =G i +P i C i The C out of Abacus Adder: C out =K 3 +K 2 C in Compared the above two equations, the 4-bit basic unit abacus adder can be extended to a 4n-bit CLA-like abacus adder

 The delay of the 8-bit abacus adders are 22%, and 14% less than those of CLA adders for 0.35um, and 0.18um technologies, respectively.  The power consumption of the abacus adders are 30%, and 60% less than those of CLA adders for 0.35um, and 0.18um technologies, respectively.  The transistor count of abacus adder is similar to that of CLA

 The delays of the 32-bit abacus adder are 17%, and 12% less than those of CLA adder for 0.35um, and 0.18um technologies, respectively.

Thank you. Questions?