Methods and systems for transmitting and receiving data using non-periodic functions

A method for communicating comprising:

• receiving, by a receiver, transmittable waveforms constructed using non-periodic functions transmitted over a communication channel;
• demodulating, by a demodulator, the transmittable waveforms to yield output communication symbols;
• and
• communicating the output communication signals to at least one of a user and an electronic device,
• wherein the at least one transmittable waveforms includes a head function and a tail function, the head function and tail function being different functions and where the head function conforms to a general spiral formula of the form:
• f
• m
• ⁡
• (
• t
• )
• =
• [
• κ
• 0
• ⁢
• e
• i
• ⁢
• ω
• 0
• ]
• ⁢
• e
• [
• κ
• 1
• ⁢
• e
• i
• ⁢
• ⁢
• ω
• 1
• ]
• ⁢
• (
• t
• +
• t
• 0
• )
• ⁢
• i
• [
• κ
• 2
• ⁢
• e
• i
• ⁢
• ⁢
• ω
• 2
• ]
• ⁢
• (
• 2
• 2
• -
• m
• )
• ,
• wherein
• k0 is a first level amplitude coefficient,
• ω0 is a first level frequency constant,
• i is an imaginary constant square root of minus one,
• k1 is a second level amplitude coefficient,
• ω1 is a second level frequency constant,
• t is a time parameter,
• t0 is an initial time,
• k2 is a third level amplitude coefficient,
• ω2 is a third level frequency constant, and
• m is a waveform shape term.

A system for communicating, comprising

• at least one of a computer-readable media, a computer, a satellite communication device and a mobile device, wherein the at least one of a computer-readable media, a computer, a satellite communication device and a mobile device to perform at least one of:
• supply input communication symbols;
• obtain, by an encoder, the input communication symbols;
• convert, by the encoder, the input communication symbols into the transmittable waveforms constructed using non-periodic functions, wherein at least one of the waveforms having a head function and a tail function, the head function and tail function being different functions, where the head function conforms to a general spiral formula of the form:
• f
• m
• ⁡
• (
• t
• )
• =
• [
• κ
• 0
• ⁢
• e
• i
• ⁢
• ω
• 0
• ]
• ⁢
• e
• [
• κ
• 1
• ⁢
• e
• i
• ⁢
• ⁢
• ω
• 1
• ]
• ⁢
• (
• t
• +
• t
• 0
• )
• ⁢
• i
• [
• κ
• 2
• ⁢
• e
• i
• ⁢
• ⁢
• ω
• 2
• ]
• ⁢
• (
• 2
• 2
• -
• m
• )
• ,
• wherein
• k0 is a first level amplitude coefficient,
• ω0 is a first level frequency constant,
• i is an imaginary constant square root of minus one,
• k1 is a second level amplitude coefficient,
• ω1 is a second level frequency constant,
• t is a time parameter,
• t0 is an initial time,
• k2 is a third level amplitude coefficient,
• ω2 is a third level frequency constant, and
• m is a waveform shape term;
• transmit, by a transmitter, the transmittable waveforms over a communication channel;
• receive, by a receiver, the transmittable waveforms from the communication channel;
• demodulate, by a demodulator, the transmittable waveforms to yield output communication symbols;
• and
• communicate output communication symbols to at least one of a user, a secondary computer-readable media, a secondary computer, a secondary satellite communication device and a secondary mobile device.