Multipliers - 4 Wallace tree multiplier based implementation

Bloggers recent study adder multiplication where knowledge when learning multiplier booth encoding plus Wallace tree compression, compression found when the partial product uses carry-save adder (Carry the Save Adder) , the addition of such bloggers is not very understanding, but dragged on for a long time, I always thought carry-save adder is a general serial adder, finally some understanding here today to share with you.

　　First, we give an example, for four unsigned multiplication 1110 and 1011, we get the following equation:

 　　　　　　　　 1  0  1  1
              x   1  1  1  0
        ------------------------------
                  0  0  0  0             PP1
               1  0  1  1                PP2
            1  0  1  1                   PP3
         1  0  1  1                      PP4
        ------------------------------   
　　乘积项中间结果有4个：PP1、PP2、PP3、PP4。由于进位保存加法器只能进行三个数的相加，因而可采用两个进位保存加法器实现。第一个进位保存加法器实现PP1、PP2、PP3的加法，第二个进位保存加法器将PP3和上一级的S和C相加。最终通过一个并行加法器输出积。计算过程如下：

第一个进位保存加法器
PP1:                    0  0  0  0
PP2:                 1  0  1  1  0
PP3:              1  0  1  1  0  0
---------------------------------------
sum1              1  1  1  0  1  0
c1                0  0  0  1  0  0   
第二个进位保存加法器 
carry_in:         1  0  1  1  0  0  0
sum1              0  1  1  1  0  1  0                   
c1                0  0  0  1  0  0  0
---------------------------------------
sum2:             1  1  0  1  0  1  0
c2：              0  0  1  1  0  0  0
并行加法器：
sum2:             1  1  0  1  0  1  0
c2：              0  1  1  0  0  0  0
---------------------------------------
result            1  0  0  1  1  0  1  0  

　　我们注意到，在参与下一级运算的过程中，由于进位要用于高位的求和，故c1，c2均应左移一位求。算出来的结果为：10011010（154）。
    所谓“进位保留”，便是进位不进上去，只在本位之间计算。
   基于Wallace树的4位乘法器实现Verilog代码如下所示：

module wallace_multiplier(
input [3:0] a,
input [3:0] b,
output [7:0] result
);

wire [7:0] pp1;
wire [7:0] pp2;
wire [7:0] pp3;
wire [7:0] pp4;

wire [7:0] sum1;
wire [7:0] carry1;
wire [7:0] sum2;
wire [7:0] carry2;

assign pp1 = b[0] ? a : 'd0;
assign pp2 = b[1] ? a : 'd0;
assign pp3 = b[2] ? a : 'd0;
assign pp4 = b[3] ? a : 'd0;

assign result = sum2 + (carry2 << 1);

carry_save_adder u1_carry_save_adder(
pp1 ,
pp2 << 1'b1,
pp3 << 2'd2,
sum1 ,
carry1
);

carry_save_adder u2_carry_save_adder(
pp4 << 2'd3 ,
sum1 ,
carry1 << 1'b1,
sum2 ,
carry2
);
endmodule

module carry_save_adder
#(
parameter DATA_WIDTH = 7
)
(
input [DATA_WIDTH:0] a,
input [DATA_WIDTH:0] b,
input [DATA_WIDTH:0] cin,
output [DATA_WIDTH:0] sum,
output [DATA_WIDTH:0] cout
);

assign sum = a ^ b ^ cin;
assign cout = a & b | (a | b) & cin;

endmodule

testbench：

`timescale 1ns / 1ns

module wallace_multiplier_tb();

reg [3:0] a;
reg [3:0] b;
wire[7:0] result;

wallace_multiplier u_wallace_multiplier(
.a (a ),
.b (b ),
.result (result )
);

initial begin
a = 4'b0010;
b = 4'b0110;
#10;
a = 4'b0001;
b = 4'b1000;
#10;
a = 4'b1010;
b = 4'b0111;
#10;
a = 4'b1110;
b = 4'b0010;
#10;
a = 4'b1111;
b = 4'b1111;
#10;
a = 4'b0110;
b = 4'b1001;
end

endmodule

　　仿真结果如下图：

　　以上就是Wallace树型4位乘法器的实现了，我认为关键在于进位保留加法器（Carry Save Adder）的理解，希望能给大家带来帮助。　　

Multipliers - 4 Wallace tree multiplier based implementation

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