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Optimum Bandwidth and Doppler Shifts/Doppler Chirps

Radobs 13

The Doppler shift of would be ETI transmissions is due to the line-of-sight
relative velocity component, while the Doppler drift or chirp is due to the
line-of-sight relative acceleration component.  The tables below show the
"natural" Doppler shift and Doppler drift components at each end of a link,
so taking a worse case scenario, the maximum shift and drifts may be twice
as large.  The calculations are based on an alien star/planet system assumed
similar to Sol/Earth.  It is not known what is the radial (line-of-sight)
acceleration of the Sun with respect to other stars.  One would assume that
it is relatively small.

The rationale for Optical SETI presupposes that the communication data rates
will be considerably larger than presently expected for Microwave SETI,
which is typically of the order of 1 Hz.  This bandwidth is partially
determined by interstellar dispersion effects and the expected microwave
Doppler drift rate.  At a frequency of a few GHz, interstellar dispersion
has been shown to place a lower limit on the usable signal bandwidth.  This
bandwidth is approximately 1 Hz.

It can be shown that the optimum detection bandwidth for a drifting
frequency in order to collect the maximum signal energy in each frequency
bin, is proportional to the square root of the drift rate.  Thus, a 1 Hz
bandwidth is just about right for a 1.5 GHz signal with a maximum relative
drift rate of 2.24 Hz/s, as might be caused by microwave transmitters and
receivers in synchronous planetary orbit (see table below).  The potentially
higher recovered signal powers afforded by optical communications, would
allow for a considerable expansion in bandwidth.  Indeed, it is probably one
of the main driving forces, besides compactness, preferring optical
communications over its radio frequency counterpart.  For an optical
transmitter or receiver in synchronous planetary orbit (see table below),
the maximum relative drift rate is about 682 kHz/s, thus implying an optimum
bandwidth of about 26 kHz.  As to what that bandwidth would be for chirp-
compensated signals which are described below, is unknown, but likely to be
determined by date-rate and SNR considerations only.

The maximum Doppler Shift is given by:

df = ---.f  Hz                                                           (1)

where v = maximum line-of-sight velocity (m),
      c = velocity of light (3 X 10^8 m/s),
      f = frequency (Hz).

The maximum Doppler Drift (Chirp) is given by:

df' = -----.f  Hz/s                                                      (2)

where w = angular velocity (rad/s),
      r = radius of planet or orbit (m).

| INTERSTELLAR DOPPLER SHIFTS                     Microwave     Visible    |
|                                                 0.20 m        656 nm     |
|                                 Velocity        1.5 GHz       457 THz    |
| Transmitter on the surface    +/-0.46 km/s    +/-2.31 kHz   +/-707 MHz   |
| of a rotating alien planet.                                              |
|                                                                          |
| Transmitter in synchronous    +/-3.07 km/s    +/-15.5 kHz   +/-4.68 GHz  |
| orbit about alien planet.                                                |
|                                                                          |
| Transmitter in orbit about    +/-29.8 km/s    +/-149 kHz    +/-45.4 GHz  |
| alien star.                                                              |
|                                                                          |
| Relative (radial) motion      +/-20.0 km/s    +/-100 kHz    +/-30.5 GHz  |
| between alien star system                                                |
| and Sol.                                                                 |

These Doppler shift and drifts will follow simple harmonic motion as the
transmitter orbits its star and rotates on or about a planet.  For a ground-
based transmitter or receiver or ones in planetary orbit, the actually
shifts and drifts will be the resultant of separate contributions from
planetary or planetary orbital rotation, solar orbital motion, and relative
motion between our respective star systems.

| INTERSTELLAR DOPPLER CHIRPS                     Microwave     Visible    |
|                                                 0.20 m        656 nm     |
|                                 Velocity        1.5 GHz       457 THz    |
| Transmitter on the surface    +/-0.46 km/s   +/-0.71 Hz/s  +/-51.4 kHz/s |
| of a rotating alien planet.                                              |
|                                                                          |
| Transmitter in synchronous    +/-3.07 km/s   +/-1.12 Hz/s  +/-341 kHz/s  |
| orbit about alien planet.                                                |
|                                                                          |
| Transmitter in orbit about    +/-29.8 km/s   +/-0.03 Hz/s  +/-9.05 kHz/s |
| alien star.                                                              |
|                                                                          |
| Relative (radial) motion      +/-20.0 km/s   +/-0 Hz/s     +/-0 Hz/s     |
| between alien star system                                                |
| and Sol.                                                                 |

The greatest Doppler frequency shifts are produced by a transmitter or
receiver in its own orbit about a star, yet the Doppler frequency drifts are
a minimum because of the very large orbital radius.

During the early years of SETI, it was believed that ETIs would help us to
detect and receive their signals, by compensating for their "natural" local
contributions to Doppler shift and drift.  In more recent years, that
rationale has fallen out of favor as electronic techniques have been
developed which allows us to efficiently analyze a microwave spectrum, and
observe a signal drifting in frequency.  Indeed, the present MultiChannel
Spectrum Analyzer (MCSA) systems make it easy to spot a drifting signal even
in the presence of considerable noise.

Because visible light frequencies are over five orders of magnitude greater
than the low microwave frequencies, the Doppler shifts and drifts are
increased by the same factor.  Fortunately, wider transmission bandwidth
systems are better able to cope with large drifts in frequency.  However,
for Optical SETI we must go back to the earlier SETI rationale, and assume a
minimum requirement that the aliens de-chirp their transmission frequency. 
It may not be necessary to remove some or all of the Doppler shift offset in
frequency, if we assume the availability of local-oscillator lasers in the
receiver that can be precisely tuned to any frequency.  Since we presently
don't know the "magic" optical frequencies, it may not matter about precise
frequencies being maintained, particularly if the Galactic communications
spectrum is not crowded.  What is the probability that one ETI signal could
interfere with another with so much space and so much spectrum available
(the optical cosmic haystack), notwithstanding the important question as to
if there are any ETIs out there, let alone many, signalling in our
direction?  What would the Galactic FCC say about the lack of stability in
transmission frequencies!

What is far more important, is to remove the Doppler drift or chirp because
this smears out the energy over a wide bandwidth, thus making it very
difficult to detect.  However, once a signal is acquired, the use of
automatic frequency control (AFC) on the local-oscillator laser can keep the
signal in lock.  By not worrying about Doppler shifts, there is then only
the requirement to remove Doppler drifts.  Since the aliens will know very
precisely their transmitter acceleration at any time along the line-of-
sight, they can apply a de-chirping signal to their laser which will almost
completely cancel the induced chirp.  Similarly, we can do the same at our
end of the link, and de-chirp the local-oscillator to remove our local
component of acceleration along the line-of-sight.

Notwithstanding the targeting problem, neither of us needs to know anything
about each other's line-of-sight accelerations or even velocities with
respect to some galactic frame of reference.  It is very unlikely that even
ATCs could know our receiver accelerations and velocities because they
wouldn't know where on Earth or in space our receiver was located.  Many
orders of magnitude of de-chirping should be achievable, which would
hopefully reduce the residual chirp to something of the order of the
linewidth of the lasers and the spectral spreading caused by any
interstellar dispersion effects.  Thus, the monochromaticity of the received
ETI signal will be maintained.  Computers controlling the transmitter and
receiver can be programmed to vary the amount of de-chirping as a function
of direction.  In an optical "search" program, we would superimpose our
local de-chirp on any tuning strategy that we might adopt to search through
the optical frequency spectrum.

Even without any de-chirping, it is clear that if aliens wanted to keep the
"natural" transmitter frequency drift to a minimum, they would put their
transmitter in its own orbit about their star.  This may be a requirement
anyway if their laser is solar or nuclear pumped.  There are also safety
considerations with powerful lasers in the MW to GW range, so they probably
wouldn't want to locate it close to large populations or in areas with heavy
space traffic.

December 26, 1990
BBOARD No. 287

* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
* Dr. Stuart A. Kingsley                       Copyright (c), 1990        *
* AMIEE, SMIEEE                                                           *
* Consultant                            "Where No Photon Has Gone Before" *
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