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FPGA Elementary Cellular Automata

A 1D cellular automata implemented on a Basys 3 board for CPE 133 final

This is the report I submitted for the final project, transcribed into Markdown by hand, transformed into HTML by the static site generator, and rendered into pixels by your browser for your eyes to see.

A Basys 3 running the program


What is an elementary cellular automata?

An elementary cellular automata is a very simple model for computation. See this Wikipedia article for more information, or read my explanation here.

There is an array of cells that can have one of 2 states: either dead or alive. Every iteration, a rule is applied to every cell that looks somewhat like this:

Current Pattern000001010011100101110111
New state for center cell10010001

where the current pattern is [left][center][right]. Assuming that the bordering cells are surrounded by 0, this is what will happen in one step of this rule:

State 0110011
State 1100010

Purpose of this project

This project simulates a 1-dimensional cellular automata displayed on the LEDs. It provides an interface for the user to configure the initial state using the switches. Additionally, provides the user an interface to change the simulation rule and adjust the simulation rate through the 7-segment display.

Top-Level Black Box

A schematic black box named top. Inputs: CLK, dec, dig_next, dig_prev, inc, reset, sw_IBUF[15:0], and toggle_BUFG. Outputs: an_OBUF[3:0], dp_OBUF, led_OBUF[15:0], seg_OBUF[6:0], and toggle_reg.

Top-Level Structural Diagram

The structural diagram.


  • DBNC is the debouncing circuit found in
  • SelRule is the submodule found in
  • Divider is the variable clock divider circuit found in
  • DisplayDrv is the submodule found in

Simulation Results

Xilinx Vivado simulation results.

What this simulation does is:

  • 0s to 10s: Set the rule to 16 by using inc, dec, and the digit cursor.
    • Rule is changed, but active_rule is not changed until Reset is pressed.
  • 12s to 60s: Increase the speed.
    • The simulation clock accelerates during this time.
    • The simulation clock divider N decreases during this time.
  • 65s to 120s: Decrease the speed.
    • The simulation clock decelerates during this time.
    • The simulation clock divider N increased during this time.

Appendix A: Code

The code can be found on GitHub at

Appendix B: Efficient, adder-only approximately constant-increment frequency selection


I wanted to make it so that if you increment, you increase the frequency at a constant rate. Say we started at a clock divider output period of 1000ms1000ms, or 1Hz1Hz, and we wanted to increase the frequency by 0.1hz every time the increment button is pressed. We would have to change the period to 11.1Hz=909ms\frac{1}{1.1Hz} = 909ms, then 11.2Hz=833ms\frac{1}{1.2Hz} = 833ms, then 11.3Hz=769ms\frac{1}{1.3Hz} = 769ms, etc.

If I’m being completely honest, I didn’t need to do any of this, but I thought it would be cool at the time.


Division is inefficient, so I decided to use an approximation to convert frequency to period. In the graph, the black curve is the function T(f)=1fT(f)=\frac{1}{f} on f[14Hz,2Hz]f \in [14 Hz,2 Hz], the bounds I want to restrict the simulator’s frequency to. The green curve is T1(t)T_1(t), which is the same function, but rescaled to [0,1][0,1] and centered on the origin.

Approximation of T(f)T(f)

Let the third-order Taylor series approximation of T1(t)T_1(t) be the cubic function g(t)=at3+bt2+ct+dg(t)=at^3+bt^2+ct+d. The blue function on the graph is the Taylor series approximation of it. I decided that the simulator would have 16 increments, so the actual function used is g(t)g(t) rescaled to [0,16][0,16] centered at 8:


Optimization of g1(n)g_1(n)

Since multiplication is slow and exponentiation is even slower, I decided to optimize the polynomial calculation using a third-order recursive sequence:

s0(n)=s0(n1)+s1(n1) (period)s_0(n)=s_0(n-1)+s_1(n-1) \text{ (period)}

s1(n)=s1(n1)+s2(n1) (first-order difference)s_1(n)=s_1(n-1)+s_2(n-1) \text{ (first-order difference)}

s2(n)=s2(n1)+s3 (second-order difference)s_2(n)=s_2(n-1)+s_3 \text{ (second-order difference)}

s3=Constant (third-order difference)s_3=Constant \text{ (third-order difference)}

A third-order recursive sequence like this creates a sequence of numbers that lie on a cubic curve.

This allows us to calculate the result in a single clock cycle, and only using a few adders when going forward, or doing a small amount of subtraction when going backward.

I set s0(0)=g1(12)s_0(0)=g_1(\frac{1}{2}) and found the formulas for the initial values for these functions using a computer algebra system (specifically, a TI-92+):





The code that implements these formulas is in

Date20 Feb 2020 to 5 Mar 2020

GitHub Github


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