For Multiplication of Signed Numbers Booth’s Algorithm For Multiplication of Signed Numbers CSE 670 Girma S. Tewolde Monday, March 1, 2004

Booth’s Algorithm for multiplication of signed numbers Example on 4-bit signed numbers: 210 x -310 = 00102 x 11012 = 1111 10102

Upper level Datapath State machine for the upper level control

Sequential multiplier controller datapath Note: that ppdp is a combinational block that implements one iteration of Booth’s algorithm

One iteration of Booth’s algorithm entity ppdp is generic(width: positive); Port ( a : in std_logic_vector(width-1 downto 0); -- multiplicand b : in std_logic_vector(width-1 downto 0); -- multiplier at the start c : in std_logic_vector(width-1 downto 0); y1 : out std_logic_vector(width-1 downto 0); y2 : out std_logic_vector(width-1 downto 0); rbin : in std_logic_vector(0 downto 0); rbout : out std_logic_vector(0 downto 0) ); end ppdp; architecture ppdp_arch of ppdp is begin mp1: process(a, b, c) variable Avector: std_logic_vector(width downto 0); variable Cvector: std_logic_vector(width downto 0); variable Yvector: std_logic_vector(width downto 0); variable overflow: std_logic; variable sel: std_logic_vector(1 downto 0); variable y1v, y2v: std_logic_vector(width-1 downto 0); -- to help catch carry and overflow during add/subtract operations -- extend the width of the operation by one bit Avector := '0' & a; Cvector := '0' & c; sel := b(0) & rbin(0); --manipulate upper portion of partial product according to 'sel' case sel is when "01" => -- add multiplicand a Yvector := Cvector + Avector; overflow := Yvector(width) xor a(width-1) xor c(width-1) xor Yvector(width-1); when "10" => -- subtract multiplicand a Yvector := Cvector - Avector; -- determine overflow = Cout xor Cin from/to the MSB overflow := Yvector(width) xor a(width-1) xor c(width-1) xor Yvector(width-1); when others => Yvector := Cvector; overflow := '0'; end case; y1v := Yvector(width-1 downto 0); --shift 1 bit to the right by sign extending MSB --check the overflow flag when doing sign extension -- if overflow = '1' then, invert the sign bit for extension y1(width-1) <= y1v(width-1) xor overflow; y1(width-2 downto 0) <= y1v(width-1 downto 1); y2 <= y1v(0) & b(width-1 downto 1); -- lower partial product rbout(0) <= b(0); end process mp1; end ppdp_arch; One iteration of Booth’s algorithm