You are an expert Verilog hardware designer working on a complete FPGA or ASIC project. You are given the full source code of a Verilog project. One of the modules in the project is incomplete — its implementation is missing and marked with:

// >>> Module Implementation Begin

// <<< Module Implementation End

Your task is to complete the missing implementation so that the entire project functions correctly. You must infer what the missing module or logic is supposed to do by analyzing the rest of the codebase: module names, signal connections, instantiations, comments, and usage patterns.

Follow these rules:

1. Only generate Verilog code to go between the `// >>> Module Implementation Begin` and `// <<< Module Implementation End` comments.
2. Use Verilog-2001 syntax. Ensure the code is synthesizable.
3. Assume the goal is a working, integrated hardware design — your code must make the whole system function as intended.
4. Analyze the naming, I/O ports, and how the module is instantiated or used in other parts of the project.
5. Implement correct behavior based on design context. Do not guess randomly — reason it out.
6. Avoid vendor-specific features unless explicitly required.
7. Do not output comments, explanation, or anything outside the required code region unless explicitly asked.

Your output should contain **only the Verilog code** that fits inside the marked region.

/*
 * Copyright (c) 2024 Your Name
 * SPDX-License-Identifier: Apache-2.0
 */

`default_nettype none

module tt_um_tommythorn_workshop (
    input  wire [7:0] ui_in,    // Dedicated inputs
    output wire [7:0] uo_out,   // Dedicated outputs
    input  wire [7:0] uio_in,   // IOs: Input path
    output wire [7:0] uio_out,  // IOs: Output path
    output wire [7:0] uio_oe,   // IOs: Enable path (active high: 0=input, 1=output)
    input  wire       ena,      // always 1 when the design is powered, so you can ignore it
    input  wire       clk,      // clock
    input  wire       rst_n     // reset_n - low to reset
);

   parameter	      frequency  = 50_000_000;
   parameter	      bps        =    115_200;

  // All output pins must be assigned. If not used, assign to 0.
  assign uo_out[7:2] = ui_in + uio_in;  // Example: ou_out is the sum of ui_in and uio_in
  assign uio_out = 0;
  assign uio_oe  = 0;

  // List all unused inputs to prevent warnings
   wire		      _unused = &{ena, clk, rst_n, 1'b0};


   reg [7:0]  transmit_data;
   reg	      we;
   wire	      busy;
   wire	      serial_out;
   wire	      serial_in = ui_in[0];
   wire	      attention;
   wire [7:0] received_data;
   wire	      pdm_out;
   reg [15:0] pdm_level;

   assign uo_out[0] = serial_out;
   assign uo_out[1] = pdm_out;

   always @(posedge clk) begin
      we <= attention & !busy;
      if (attention & !busy) begin
	 transmit_data <= received_data + 1;
	 pdm_level <= {pdm_level[7:0], received_data};
      end

      if (!rst_n) begin
	 we <= 0;
      end
   end

   rs232out #(bps, frequency) rs232out_inst
     (.clock(clk),
      .serial_out(serial_out),
      .transmit_data(transmit_data),
      .we(we),
      .busy(busy));

   rs232in #(bps, frequency) rs232in_inst
     (.clock(clk),
      .serial_in(serial_in),
      .attention(attention),
      .received_data(received_data));

   pdm pdm_inst(.clk(clk),
		.level(pdm_level), 
		.O(pdm_out));
endmodule

`ifdef TEST
module tb;
   reg rst_n = 0;
   reg clk = 1;
   always #5 clk = !clk;

   wire [7:0] ui_in;
   wire [7:0] uo_out;

   tt_um_tommythorn_workshop dut(.clk(clk), .rst_n(rst_n), .ui_in(ui_in), .uo_out(uo_out));

   initial begin
      $dumpfile("demo.vcd");
      $dumpvars();
      rst_n = 1;
      #1000 $finish;
      end
endmodule
`endif


module pdm(clk, level, O);
// >>> Module Implementation Begin

// <<< Module Implementation End
endmodule

 // pdm


// -----------------------------------------------------------------------
//
//   Copyright 2004,2007-2008 Tommy Thorn - All Rights Reserved
//
//   This program is free software; you can redistribute it and/or modify
//   it under the terms of the GNU General Public License as published by
//   the Free Software Foundation, Inc., 53 Temple Place Ste 330,
//   Bostom MA 02111-1307, USA; either version 2 of the License, or
//   (at your option) any later version; incorporated herein by reference.
//
// -----------------------------------------------------------------------

`timescale 1ns/10ps

module rs232in
   (// Control
    input  wire        clock,

    // Serial line
    input  wire        serial_in,
    output reg         attention = 0,
    output reg   [7:0] received_data = 0);

   parameter           bps        =    115_200;
   parameter           frequency  = 50_000_000;
   parameter           period     = (frequency + bps/2) / bps;

   reg  [16:0] ttyclk       = 0;
   wire [31:0] ttyclk_bit   = period - 2;
   wire [31:0] ttyclk_start = (3 * period) / 2 - 2;
   reg  [ 7:0] shift_in     = 0;
   reg  [ 4:0] count        = 0;

   reg         rxd          = 0;
   reg         rxd2         = 0;

   /*
    * The theory: look for a negedge, then wait 1.5 bit period to skip
    * start bit and center in first bit.  Keep shifting bits until a full
    * byte is collected.
    *
    *        Start                        Stop
    * data   ~\__ B0 B1 B2 B3 B4 B5 B6 B7 ~~
    * count        8  7  6  5  4  3  2  1
    */
   always @(posedge clock) begin
      attention <= 0;

      // Get rid of meta stability.
      {rxd2,rxd} <= {rxd,serial_in};

      if (~ttyclk[16]) begin
         ttyclk <= ttyclk - 1'd1;
      end else if (count) begin
         if (count == 1) begin
            received_data <= {rxd2, shift_in[7:1]};
            attention     <= 1;
         end

         count       <= count - 1'd1;
         shift_in    <= {rxd2, shift_in[7:1]}; // Shift in from the left
         ttyclk      <= ttyclk_bit[16:0];
      end else if (~rxd2) begin
         // Just saw the negedge of the start bit
         ttyclk      <= ttyclk_start[16:0];
         count       <= 8;
      end
   end
endmodule


// -----------------------------------------------------------------------
//
//   Copyright 2004,2006-2008 Tommy Thorn - All Rights Reserved
//
//   This program is free software; you can redistribute it and/or modify
//   it under the terms of the GNU General Public License as published by
//   the Free Software Foundation, Inc., 53 Temple Place Ste 330,
//   Bostom MA 02111-1307, USA; either version 2 of the License, or
//   (at your option) any later version; incorporated herein by reference.
//
// -----------------------------------------------------------------------

`timescale 1ns/10ps

module rs232out
   (// Control
    input wire         clock,

    // Serial line
    output wire        serial_out,

    input  wire  [7:0] transmit_data,
    input  wire        we,
    output wire        busy);

   parameter           bps         = 0;
   parameter           frequency   = 0;
`ifndef __ICARUS__
   parameter           period      = (frequency + bps/2) / bps;
`else
   // One of the very few simulation artifacts we have to deal with at the source level.
   parameter           period      = 0;
`endif
   parameter           TTYCLK_SIGN = 16; // 2^TTYCLK_SIGN > period * 2
   parameter           COUNT_SIGN  = 4;

   reg  [TTYCLK_SIGN:0] ttyclk      = 0; // [-4096; 4095]
   wire [31:0]          ttyclk_bit  = period - 2;
   reg  [8:0]           shift_out   = 0;
   reg  [COUNT_SIGN:0]  count       = 0; // [-16; 15]

   assign               serial_out  = shift_out[0];
   assign               busy        = ~count[COUNT_SIGN] | ~ttyclk[TTYCLK_SIGN];

   always @(posedge clock)
      if (~ttyclk[TTYCLK_SIGN]) begin
         ttyclk     <= ttyclk - 1'd1;
      end else if (~count[COUNT_SIGN]) begin
         ttyclk     <= ttyclk_bit[TTYCLK_SIGN:0];
         count      <= count - 1'd1;
         shift_out  <= {1'b1, shift_out[8:1]};
      end else if (we) begin
         ttyclk     <= ttyclk_bit[TTYCLK_SIGN:0];
         count      <= 9; // 1 start bit + 8 data + 1 stop - 1 due to SIGN trick
         shift_out  <= {transmit_data, 1'b0};
      end
endmodule


Reference solution:
rtl/generated.v: 
   parameter N = 16;  /* The resolution in bits */


   input     wire         clk;
   input     wire [N-1:0] level;
   output    wire          O;

   /*
    * A completely ridiculous side project.  Pulse Density Modulation
    * for controlling led intensity.  The theory is pretty simple:
    * given a desired target level 0 <= T <= 1, control the output O
    * in {1,0}, such that O on on average is T.  Do this by
    * integrating the error T - O over time and switching O such that
    * the sum of (T - O) is finite.
    *
    * S = 0, O = 0
    * forever
    *   S = S + (T - O)
    *   if (S >= 0)
    *      O = 1
    *   else
    *      O = 0
    *
    * Check: T=0, O is never turned on; T=1, O is always on; T=0.5, O toggles
    *
    * In fixed point arithmetic this becomes even simpler (assume N-bit arith)
    * S = Sf * 2^N = Sf << N.  As |S| <= 1, N+2 bits is sufficient
    *
    * S = 0, O = 0
    * forever
    *   D = T + (~O + 1) << N === T + (O << N) + (O << (N+1))
    *   S = S + D
    *   O = 1 & ~(S >> (N+1))
    */

   reg [N+1:0] sigma = 0;
   assign O = ~sigma[N+1];
   always @(posedge clk) sigma <= sigma + {O,O,level};