Division in Verilog and Q factor representation

Question

I am currently working on a design of an algorithm for signal processing. I created a model in software that appears to work fine and I am now trying to translate it to verilog. Below is what I do in the software.

I get a 16 bit input, I do the following

    a. convert the hex to decimal
    b. subtract 32768 from the above
    c. divide the result with 32768
    d. convert the result to a signed number in the Q(4,20) format (4 bits for integer, 20 bits fractional)

For example,

case 1: value is 0x803f (ie, value > 0x8000)

    a. Convert 0x803f to decimal, ie, 32831
    b. 32831 - 32768 = 63
    c. 63/32768 = 0.001922
    d. convert to signed Q(4,20) 0x0007C8
    
    
case 2: value is 0x79fc (ie, value < 0x8000)

    a. Convert 0x79fc to decimal, ie, 31228
    b. 32668 - 32768 = -100
    c. -100/32768 = -0.003051
    d. convert to signed Q(4,20) 0xFFF3B7

I started with the following code to translate the software model to verilog and obtained result don't match what I expected. For Case 1 and Case 2, observed value for Y_DIV in simulation is 0x1. But I expect value that is close to Y (Ox003f for case 1 and 0xff9c for case 2) as 1/32768 is relatively small.

With regards to converting 0x003f and 0xff9c to Q(4,20), if I sign extend 0x003f, how do we achieve the target value of 0x0007C8 ?

I am not sure if the following is the right way to translate my algorithm

module test(
input clk,
input rst,
  input [15:0] a,
  input [15:0] b,
  output reg signed [15:0] y,
  output reg signed [15:0] y_div
);  
 

  always @(posedge clk) begin
    if (rst) begin
        y <= 0;
        y_div <= 0;
    end else begin
        y <= (a-b);
        // since (1/32768 = 0.000030) are we better off using a multiply instead of divide ?  
        y_div <= (a-b)/b; // returns wrong value, ie doesn't match expected value
        // Q(4,20) is yet to be done. First need to get the above working   
    end
  end
  
endmodule

Below is my test bench

module tb_test();
  reg CLK, RST;
  
  reg [15:0] A, B;
  wire [15:0] Y, Y_DIV;
  initial begin
    CLK = 0;
    forever #1 CLK = ~CLK;
  end
  
  initial begin
    RST = 1;
    #10;
    RST = 0;
  end
  
  initial begin
    A = 16'b0;
    B = 16'b0;
  #10;
    A = 16'h803f; // case 1
    B = 16'h8000;
  #10;
    A = 16'h7f9c; // case 2
    B = 16'h8000;    
  end
  
  test dut (.clk(CLK), .rst(RST), .a(A), .b(B), .y(Y), .y_div(Y_DIV));
  
endmodule

Answer 1

The Verilog '/' operator, when given integers for n/d, performs integer division which yields the integer quotient and throws the remainder away.
If the quotient is < 1, Verilog shows 0.
This is not directly useful in designs that are interested in fractional quotients.

Lets avoid the '/' operator, use >>> (arithmetic shift, sign extends from the LHS) instead.
Divide by 32768 =2**15 is a shift of 15 bits to the right.

Lets shift the a - b term 20 bits to the left to create a 20 fractional bits representation.

a - b need to be 17 bits for the 16 bit add/sub.

Synchronized the testbench to the clock edge.

The output of this module gets close to your vectors.
(Would need to quantize floating point numbers to get a bit-exact match to all the fixed point)
RTL:

module test
#(
  parameter DI_IN_WIDTH         = 16,
  parameter FRACTIONAL_OUT_BITS = 20,
  parameter OUT_BITS_4_DOT_20   = 24
)
(
  input clk,
  input rst,
  input  [DI_IN_WIDTH - 1:0] a,
  input  [DI_IN_WIDTH - 1:0] b,
  //
  output reg [OUT_BITS_4_DOT_20 - 1 :0] y
);  
  
  localparam SUM_WIDTH           = DI_IN_WIDTH  + 1;
  localparam SUM_PLUS_FRACT_BITS = SUM_WIDTH + FRACTIONAL_OUT_BITS;

  reg signed [SUM_WIDTH - 1:0]           a_minus_b;
  // 17.20 numbers
  reg signed  [SUM_PLUS_FRACT_BITS - 1:0] a_minus_b_w_fract_bits;
  reg signed [SUM_PLUS_FRACT_BITS - 1:0] a_minus_b_w_fract_bits_div;

  
  // a - b
  always @ * begin
    a_minus_b = a - b;
    a_minus_b_w_fract_bits = a_minus_b << FRACTIONAL_OUT_BITS; 
  end
    
  always @(posedge clk) begin
    if (rst) begin
      a_minus_b_w_fract_bits_div <= '0;
    end else begin
      // divide by 2^15
      a_minus_b_w_fract_bits_div <= a_minus_b_w_fract_bits >>> 15;
    end
  end
  
  assign y = a_minus_b_w_fract_bits_div[OUT_BITS_4_DOT_20 - 1 : 0];
  
//   initial begin
//     $display("DI_IN_WIDTH         = %0d",DI_IN_WIDTH);
//     $display("SUM_WIDTH           = %0d",SUM_WIDTH);
//     $display("FRACTIONAL_OUT_BITS = %0d",FRACTIONAL_OUT_BITS);
//     $display("SUM_PLUS_FRACT_BITS = %0d",SUM_PLUS_FRACT_BITS);
//     $display("OUT_BITS_4_DOT_20   = %0d",OUT_BITS_4_DOT_20);
//   end
  
endmodule    

Testbench:

module tb_test();

  localparam  DI_IN_WIDTH        = 16;
  localparam FRACTIONAL_OUT_BITS = 20;
  localparam OUT_BITS_4_DOT_20   = 24;    
  
  reg CLK, RST;
  
  reg signed [DI_IN_WIDTH - 1:0] A, B;
  reg signed [OUT_BITS_4_DOT_20 - 1 :0] Y;
  
  initial begin
    CLK = 0;
    forever #1 CLK = ~CLK;
  end
  
  initial begin
    RST = 1;
    repeat(2) @(posedge CLK);
    RST = 0;
  end
  
  initial begin
    $display("Test Starting");
    A <= 16'h803f; // case 1 32831
    B <= 16'h8000; // 32,768
    repeat(3) @(posedge CLK);
    $strobe("  t= %0t, a - b = %0d, Y_hex = %h, Y_dec = %0d, Y_real_FMT_4_20 = %f",
      $time,dut.a_minus_b,Y,Y,$itor(Y)/2**20);
    //
    A <= 16'h7f9c; // case 2
    B <= 16'h8000;      
    repeat(1) @(posedge CLK);
    $strobe("  t= %0t, a - b = %0d, Y_hex = %h, Y_dec = %0d, Y_real_FMT_4_20 = %f",
      $time,dut.a_minus_b,Y,Y,$itor(Y)/2**20);
    //
    A <= 16'h8000; // case 3, max value numerator
    B <= 16'h0000;      
    repeat(1) @(posedge CLK);
    $strobe("  t= %0t, a - b = %0d, Y_hex = %h, Y_dec = %0d, Y_real_FMT_4_20 = %f",
      $time,dut.a_minus_b,Y,Y,$itor(Y)/2**20);
    //
    repeat(1) @(posedge CLK);
    $display("Test Done");
    $finish;    
  end
  
  initial begin
    $dumpfile("dump.vcd"); 
    $dumpvars;
  end
  
  test dut (
    .clk(CLK),
    .rst(RST),
    .a(A),
    .b(B),
    .y(Y)
    );

endmodule

Result:

xcelium> run
Test Starting
  t= 5, a - b = 63, Y_hex = 0007e0, Y_dec = 2016, Y_real_FMT_4_20 = 0.001923
  t= 7, a - b = -100, Y_hex = fff380, Y_dec = -3200, Y_real_FMT_4_20 = -0.003052
  t= 9, a - b = 32768, Y_hex = 100000, Y_dec = 1048576, Y_real_FMT_4_20 = 1.000000
Test Done
Simulation complete via $finish(1) at time 11 NS + 0