## Extraterrestrial Signal Design

In the movie Contact (1997) an alien race sends out a series of pulses in a numeric sequence that can not be mistaken for any naturally occurring phenomenon. The idea is that pulses made by intelligent beings should be in some sort of varying timed sequence so that it is not possible to be duplicated by any physical events. For example co-orbiting binary stars or other periodic events can generate highly regular pulses.

**Problems With the Sequence as Depicted in the Movie**

The "Contact" sequence repeats after a few groups of pulses... the total span lasted for a few seconds. Although even a small sample of the Prime Number series is highly unlikely to be a naturally occurring phenomenon, relying on a 20 second observation window might be too limiting.

What if the receiving race worked at a different time scale - at a rate where the 20 second groups would be interpreted as equal bursts. These 20 second bursts would be seen as simple repetitive noise, perhaps as a result of some spinning star emitting radiation regularly.

## Designing a Sequence for Extraterrestrials

I was thinking of how a pulse sequence could be designed to have the following criteria:

***** Simple periodic yet contain some irrational component such as Contact's sequence.

***** Equally spaced pulses with some missing at their designated time slots.

***** Must fit into a reasonable timescale. For example a resolution of the order of 1 Hz maximum rate.

***** Must run "forever". In the order of billions of years.

***** Must be somewhat immune to drift. Any internal oscillator will drift with time. Another cause of drift is Doppler shift from the generator's motion relative to the observer. This drift should not compromise an observer's interpretation. Ideally the sequence should be self-timing.

A receiver may start and end their observation in any place in the sequence for any duration. The observation start time can be anywhere in a billions of years span. Shorter observation times should give intriguing results. Longer ones even more so.

The longer the observation the more obvious it should be that the sequence does not contain a repeated section. It will never... well at least within a several billion year span.

## Proposal

I've decided to start with the __Golden Ratio__ once again. It's deemed to be "the most irrational number" and therefore a great candidate for a basis in a sequence.

**Details**

The transmitting device will have a micro processor on board to trigger the signal. It is of minimum compexity. Hopefully it is constructed rugedized for eternal duty.

We have to first establish how many bits wide the processor's registers will be. It will be doing some basic arithmetic and logic operations with these.

A number being represented in binary, consisting of n-bits being incremented once per second will be able to accommodate counts up to the following durations:

**16-bits**: 18 hours and 12 minutes

**32-bits**: 136 years

**64-bits**: 584,554,049,254 years (585 billion).

The universe is estimated to be 14 billion years old.

We will be using only integer math. Simple binary addition. The microprocessor's complexity is kept low.

**Set up the processor**

The "seed" number contains phi in binary to very high precision.

First trick is to obtain a 64-bit phi. Most common online calculators are only good for a maximum of 16 decimal digit accuracy (internally 53 bits of precision).

We use a high-precision online calculator to obtain a higher precision phi. Then convert that to binary.

__High precision online calculator__

Gives a 22-decimal digit phi: Φ = 0.6180339887498948482046

__High precision decimal/binary converter__

Converts to binary. Obtain 64+ bits and then round to 64 bits:

10011110 00110111 01111001 10111001 01111111 01001010 01111100 00010110

Store this in a 64-bit resister (permanent memory).

The microprocessor must also have a 64-bit adder / accumulator. This adder must be able to indicate an overflow condition to other components such as the radio pulse generator.

**Algorithm**

__Initialization__

- Set accumulator to zero.

__At every second do the following__

- Sum the seed number to the accumulator.

- If there was an overflow condition then make a pulse; else no pulse.

Onserving the overflow states the resulting sequence looks like this for the first 17 terms:

**0 1 0 1 1 0 1 0 1 1 0 1 1 0 1 0 1 ...**

Note that there is a higher probability of 1s. This makes the sequence self-timing, i.e. easy to synchronize with. The maximum number of adjacent pulses is 2. The maximum number of adjacent absences of a pulse is 1.

The sequence can be visualized graphically by rasterizing a line of slope phi:

Close-up view:

The rasterized line is represented by the grey blocks. A sequence of 1s and 0s can be derived from the blocks: '1' for a long block and '0' for a short block.

For more information on line rasterization see __Bresenham's line algorithm__.

## Reading the Sequence

- Starting at any position take any length subsequence.
- Count the number of 1s. Divide this by the subsequence length.

**This fraction will approach the number phi. The longer the sequence the more closer to phi it will be.**

## Conclusion

It should be clear to an intelligent race that the sequence is not a naturally occurring phenomenon. It is easily understood to be a non-random sequence. It can be observed starting at any point in the universes time scale for any duration.

If a receiver recognizes phi then they might be inclined to wait for the sequence length to equal the next Fibonacci number. This will produce the closest fraction (when divided by the previous Fibonacci number).

In a much shorter time, way below billions of years, phi will be approximated to the following accuracy over various durations. The accuracy is listed in number of base-10 digits. Other civilizations may use other bases and calculate accuracy accordingly.

Duration | Fib(n) n | decimal digits |

1 hour | 18 | 6 |

1 day | 25 | 9 |

2.5 days | 27 | 10 |

6 days | 29 | 11 |

25 days | 32 | 12 |

66 days | 34 | 13 |

280 days | 37 | 14 |

2 years | 39 | 15 |

Revised 23 Dec 2018