The following verilog source code and/or testbench works nicely across commercial simulators, iverilog as well as formal verification tool (yosys-smtbmc)
Please keep the complaint about `ifdef FORMAL until later. I need them to use with yosys-smtbmc which does not support bind command yet.
I am now debugging the generate coding since the multiplication (using modified baugh-wooley algorithm) does not work yet.
When o_valid is asserted, the multiply code should give o_p = i_a * i_b = 3*2 = 6 but the waveform clearly shows the code gives o_p = 0x20 = 32
test_multiply.v
// Testbench
module test_multiply;
parameter A_WIDTH=4, B_WIDTH=4;
reg i_clk;
reg i_reset;
reg i_ce;
reg signed[(A_WIDTH-1):0] i_a;
reg signed[(B_WIDTH-1):0] i_b;
wire signed[(A_WIDTH+B_WIDTH-1):0] o_p;
wire o_valid;
// Instantiate design under test
multiply #(A_WIDTH, B_WIDTH) mul(.clk(i_clk), .reset(i_reset), .in_valid(i_ce), .in_A(i_a), .in_B(i_b), .out_valid(o_valid), .out_C(o_p));
initial begin
// Dump waves
$dumpfile("test_multiply.vcd");
$dumpvars(0, test_multiply);
i_clk = 0;
i_reset = 0;
i_ce = 0;
i_a = 0;
i_b = 0;
end
localparam SMALLER_WIDTH = (A_WIDTH <= B_WIDTH) ? A_WIDTH : B_WIDTH;
localparam NUM_OF_INTERMEDIATE_LAYERS = $clog2(SMALLER_WIDTH);
genvar i, j; // array index
generate
for(i = 0; i < NUM_OF_INTERMEDIATE_LAYERS; i = i + 1) begin
for(j = 0; j < SMALLER_WIDTH; j = j + 1) begin
initial $dumpvars(0, test_multiply.mul.middle_layers[i][j]);
end
end
endgenerate
always #5 i_clk = !i_clk;
initial begin
@(posedge i_clk);
@(posedge i_clk);
$display("Reset flop.");
i_reset = 1;
@(posedge i_clk);
@(posedge i_clk);
i_reset = 0;
@(posedge i_clk);
@(posedge i_clk);
i_ce = 1;
i_a = 3;
i_b = 2;
#50 $finish;
end
endmodule
multiply.v
module multiply #(parameter A_WIDTH=16, B_WIDTH=16)
(clk, reset, in_valid, out_valid, in_A, in_B, out_C); // C=A*B
`ifdef FORMAL
parameter A_WIDTH = 4;
parameter B_WIDTH = 4;
`endif
input clk, reset;
input in_valid; // to signify that in_A, in_B are valid, multiplication process can start
input signed [(A_WIDTH-1):0] in_A;
input signed [(B_WIDTH-1):0] in_B;
output signed [(A_WIDTH+B_WIDTH-1):0] out_C;
output reg out_valid; // to signify that out_C is valid, multiplication finished
/*
This signed multiplier code architecture is a combination of row adder tree and
modified baugh-wooley algorithm, thus requires an area of O(N*M*logN) and time O(logN)
with M, N being the length(bitwidth) of the multiplicand and multiplier respectively
see [url]https://i.imgur.com/NaqjC6G.png[/url] or
Row Adder Tree Multipliers in [url]http://www.andraka.com/multipli.php[/url] or
[url]https://pdfs.semanticscholar.org/415c/d98dafb5c9cb358c94189927e1f3216b7494.pdf#page=10[/url]
regarding the mechanisms within all layers
In terms of fmax consideration: In the case of an adder tree, the adders making up the levels
closer to the input take up real estate (remember the structure of row adder tree). As the
size of the input multiplicand bitwidth grows, it becomes more and more difficult to find a
placement that does not use long routes involving multiple switch nodes for FPGA. The result
is the maximum clocking speed degrades quickly as the size of the bitwidth grows.
For signed multiplication, see also modified baugh-wooley algorithm for trick in skipping
sign extension (implemented as verilog example in [url]https://www.dsprelated.com/showarticle/555.php[/url]),
thus smaller final routed silicon area.
[url]https://stackoverflow.com/questions/54268192/understanding-modified-baugh-wooley-multiplication-algorithm/[/url]
All layers are pipelined, so throughput = one result for each clock cycle
but each multiplication result still have latency = NUM_OF_INTERMEDIATE_LAYERS
*/
// The multiplication of two numbers is equivalent to adding as many copies of one
// of them, the multiplicand, as the value of the other one, the multiplier.
// Therefore, multiplicand always have the larger width compared to multipliers
localparam SMALLER_WIDTH = (A_WIDTH <= B_WIDTH) ? A_WIDTH : B_WIDTH;
localparam LARGER_WIDTH = (A_WIDTH > B_WIDTH) ? A_WIDTH : B_WIDTH;
wire [(LARGER_WIDTH-1):0] MULTIPLICAND = (A_WIDTH > B_WIDTH) ? in_A : in_B ;
wire [(SMALLER_WIDTH-1):0] MULTIPLIPLIER = (A_WIDTH <= B_WIDTH) ? in_A : in_B ;
`ifdef FORMAL
// to keep the values of multiplicand and multiplier before the multiplication finishes
reg [(LARGER_WIDTH-1):0] MULTIPLICAND_reg;
reg [(SMALLER_WIDTH-1):0] MULTIPLIPLIER_reg;
always @(posedge clk)
begin
if(reset) begin
MULTIPLICAND_reg <= 0;
MULTIPLIPLIER_reg <= 0;
end
else if(in_valid) begin
MULTIPLICAND_reg <= MULTIPLICAND;
MULTIPLIPLIER_reg <= MULTIPLIPLIER;
end
end
`endif
localparam NUM_OF_INTERMEDIATE_LAYERS = $clog2(SMALLER_WIDTH);
/*Binary multiplications and additions for partial products rows*/
// first layer has "SMALLER_WIDTH" entries of data of width "LARGER_WIDTH"
// This resulted in a binary tree with faster vertical addition processes as we have
// lesser (NUM_OF_INTERMEDIATE_LAYERS) rows to add
// intermediate partial product rows additions
// Imagine a rhombus of height of "SMALLER_WIDTH" and width of "LARGER_WIDTH"
// being re-arranged into binary row adder tree
// such that additions can be done in O(logN) time
//reg [(NUM_OF_INTERMEDIATE_LAYERS-1):0][(SMALLER_WIDTH-1):0][(A_WIDTH+B_WIDTH-1):0] middle_layers;
reg [(A_WIDTH+B_WIDTH-1):0] middle_layers[(NUM_OF_INTERMEDIATE_LAYERS-1):0][0:(SMALLER_WIDTH-1)];
//reg [(NUM_OF_INTERMEDIATE_LAYERS-1):0] middle_layers [0:(SMALLER_WIDTH-1)] [(A_WIDTH+B_WIDTH-1):0];
//reg middle_layers [(NUM_OF_INTERMEDIATE_LAYERS-1):0][0:(SMALLER_WIDTH-1)][(A_WIDTH+B_WIDTH-1):0];
generate // duplicates the leafs of the binary tree
genvar layer; // layer 0 means the youngest leaf, layer N means the tree trunk
for(layer=0; layer<NUM_OF_INTERMEDIATE_LAYERS; layer=layer+1) begin: intermediate_layers
integer pp_index; // leaf index within each layer of the tree
integer bit_index; // index of binary string within each leaf
always @(posedge clk)
begin
if(reset)
begin
for(pp_index=0; pp_index<SMALLER_WIDTH ; pp_index=pp_index+1)
middle_layers[layer][pp_index] <= 0;
end
else begin
if(layer == 0) // all partial products rows are in first layer
begin
// generation of partial products rows
for(pp_index=0; pp_index<SMALLER_WIDTH ; pp_index=pp_index+1)
middle_layers[layer][pp_index] <=
(MULTIPLICAND & MULTIPLIPLIER[pp_index]);
// see modified baugh-wooley algorithm: [url]https://i.imgur.com/VcgbY4g.png[/url] from
// page 122 of book "Ultra-Low-Voltage Design of Energy-Efficient Digital Circuits"
for(pp_index=0; pp_index<SMALLER_WIDTH ; pp_index=pp_index+1)
middle_layers[layer][pp_index][LARGER_WIDTH-1] <=
!middle_layers[layer][pp_index][LARGER_WIDTH-1];
middle_layers[layer][SMALLER_WIDTH-1] <= !middle_layers[layer][SMALLER_WIDTH-1];
middle_layers[layer][0][LARGER_WIDTH] <= 1;
middle_layers[layer][SMALLER_WIDTH-1][LARGER_WIDTH] <= 1;
end
// adding the partial product rows according to row adder tree architecture
else begin
for(pp_index=0; pp_index<(SMALLER_WIDTH >> layer) ; pp_index=pp_index+1)
middle_layers[layer][pp_index] <=
middle_layers[layer-1][pp_index<<1] +
(middle_layers[layer-1][(pp_index<<1) + 1]) << 1;
// bit-level additions using full adders
/*for(pp_index=0; pp_index<SMALLER_WIDTH ; pp_index=pp_index+1)
for(bit_index=0; bit_index<(LARGER_WIDTH+layer); bit_index=bit_index+1)
full_adder fa(.clk(clk), .reset(reset), .ain(), .bin(), .cin(), .sum(), .cout());*/
end
end
end
end
endgenerate
assign out_C = (reset)? 0 : middle_layers[NUM_OF_INTERMEDIATE_LAYERS-1][0];
/*Checking if the final multiplication result is ready or not*/
reg [($clog2(NUM_OF_INTERMEDIATE_LAYERS)-1):0] out_valid_counter; // to track the multiply stages
reg multiply_had_started;
always @(posedge clk)
begin
if(reset)
begin
multiply_had_started <= 0;
out_valid <= 0;
out_valid_counter <= 0;
end
else if(out_valid_counter == NUM_OF_INTERMEDIATE_LAYERS-1) begin
multiply_had_started <= 0;
out_valid <= 1;
out_valid_counter <= 0;
end
else if(in_valid && !multiply_had_started) begin
multiply_had_started <= 1;
out_valid <= 0; // for consecutive multiplication
end
else begin
out_valid <= 0;
if(multiply_had_started) out_valid_counter <= out_valid_counter + 1;
end
end
`ifdef FORMAL
initial assume(reset);
initial assume(in_valid == 0);
wire sign_bit = MULTIPLICAND_reg[LARGER_WIDTH-1] ^ MULTIPLIPLIER_reg[SMALLER_WIDTH-1];
always @(posedge clk)
begin
if(reset) assert(out_C == 0);
else if(out_valid) begin
assert(out_C == (MULTIPLICAND_reg * MULTIPLIPLIER_reg));
assert(out_C[A_WIDTH+B_WIDTH-1] == sign_bit);
end
end
`endif
`ifdef FORMAL
localparam user_A = 3;
localparam user_B = 2;
always @(posedge clk)
begin
cover(in_valid && (in_A == user_A) && (in_B == user_B));
cover(out_valid);
end
`endif
endmodule