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EJASA - Part 13

                                                                     Page 88

    The total noise produced is proportional to the electrical post-
    detection bandwidth Be.  To an approximation at high avalanche gain,
    the surface dark current component Is, which is not subject to gain, is
    sometimes ignored, and Ib is called Id.

    Coherent Signal-To-Noise Ratio:

    Heterodyne Detection (Reception)

 CNR = ---------------------------------------------------------------- (32)

    Homodyne Detection

 CNR = ---------------------------------------------------------------- (33)

    The electrical signal power is proportional to Pr and the optical
    mixing efficiency H, and the noise components proportional:

    1.  To the quantum noise produced by the signal photons.

    2.  To the shot noise produced by the local oscillator.

    3.  To the fluctuation noise produced by the background radiation Pb
        (NbBo).  This noise is also proportional to the optical bandwidth
        and its ratio to the quantum shot noise is effectively inversely
        proportional to the local oscillator power Po.

    4.  To the shot noise produced by the bulk dark current in the photo-

    5.  To the shot noise produced by the surface leakage dark current.

    6.  To the background radiation beating with the local oscillator,
        which is very small, the noise being proportional to the noise
        spectral density and independent of optical bandwidth.

    7.  To the background noise spectral density squared, which is again
        very small, the noise being proportional to the optical bandwidth.

    8.  To the thermal kT noise of the optical front-end, which like the
        case for all other noise components except that due to the local-
        oscillator quantum shot-noise, is negligible for sufficient local-
        oscillator power.

    The local-oscillator (L.O.) is assumed to have negligible excess
    intensity noise or it is balanced out, so that the Relative Intensity
    Noise (RIN) is at the theoretical quantum noise level.

                                                                     Page 89

    Note, the excess noise due to a non-Poisson distribution of arriving
    photons in a power-starved situation, is not included in this expres-
    sion.  Poisson statistics imply that sufficient photons arrive during
    the observation time to take the probability of the arrival of a photon
    as being given by a binomial distribution. [83]  In situations where
    the optical receiver is power-starved, i.e., when there are relatively
    few photons arriving during the signal integration time so that Bose-
    Einstein [73] statistics apply, the non-white noise associated with
    statistics of the photon arrival times will lower the effective CNR.

    The total noise produced is again proportional to the electrical post-
    optical detection bandwidth Be.  Usually Po >> Pr and Pb, and thus
    other multiplicative noise components relating to Pr and Pb are not
    included in these expressions, since they are negligible.  For this
    application the nearest star is several light years away, Po is much
    larger the background Pb, and the latter component is also negligible
    for all optical bandwidths, unlike the case for incoherent detection.
    This is also generally true for large diffraction limited telescopes
    operating in daylight.  For SETI to be practical, the EIRP needs to be
    extremely high, but since the star is distant, the background Nb is
    very small.  However, for communications within the solar system, these
    background noise components (from the Sun or reflected light from Earth
    or another planet) can be significant. [94-95]

    For the Amateur Optical SETI analysis, a more conservative approach for
    assessing the performance of various receiving systems has been
    employed.  Account has been made for the efficiencies of atmospheric
    transmission, telescope aperture, monochromator (incoherent systems
    only) and in the case of coherent receivers, an allowance for the
    optical (heterodyne or homodyne) mixing efficiency.

    Expression (31) relates to incoherent detection, while (32) and (33)
    relate to coherent detection.  The ideal shot-noise limited direct
    detection receiver approaches the performance of the photon-counting
    receiver at higher received powers.  For substantially cooled photon-
    counting receivers, the dark currents Is and Ib may be taken as zero,
    and thermal noise is insignificant.  In the quantum noise limit, the
    CNR of the homodyne system is 3 dB more than the heterodyne, which is
    itself 3 dB more than the direct detection or photon-counting receiver.

    Quantum-Noise Limited Signal-To-Noise Ratio:

    The Carrier-To-Noise Ratio in a perfect quantum noise limited (656 nm)
    optical heterodyne system where the L.O. has negligible intensity and
    phase noise, and where the shot noise from the L.O. swamps all other
    sources of noise, is given by:

    CNR = ------                                                        (34)

    where  Pr  = received optical power (1.6 nW),
           Bif = Intermediate Frequency bandwidth (30 MHz).

                                                                     Page 90

    One of the major advantages of using the normalized CNR approach is
    that we can express the CNR for the perfect diffraction-limited
    ten meter diameter symmetrical heterodyne system, for any transmitter
    power, range and electrical bandwidth, in the form:

    |                                                      |
    |  CNR = 54 + 10.log(Pt) - 20.log(R) - 10.log(Be)  dB  |            (35)
    |                                                      |
    where  Pt = transmitter power (kW),
           R  = range (L.Y.),
           Be = I.F. bandwidth (Hz).

    For Pt = 1 GW, R = 10 L.Y., and Be = Bif = 30 MHz:

                                  CNR = 19 dB

    Again, it should be remembered that this relationship (Equ. 35) only
    holds out to distances where interstellar attenuation is insignificant,
    and will over-estimate the CNR at very low received optical powers (Pr)
    and/or higher bandwidths (Be).  For a huge transmitting array, the
    Rayleigh near-field range may be so large (Equ. 7), that the 20.log(R)
    term disappears from the above expression, and the 54 dB constant has
    a higher value.

    We see that one advantage of coherent detection for this application is
    that the effective bandwidth determining the relative level of detected
    background noise is the electrical bandwidth Be, not the optical
    bandwidth Bo.  Since Be can be much less than Bo, coherent receivers
    have a considerable sensitivity advantage over incoherent receivers in
    the presence of weak signals and/or significant background radiation,
    besides being able to allow for the demodulation of phase or frequency-
    modulated signals.  In the case of the heterodyne receiver, Be
    corresponds to the I.F. bandwidth, and the signal has still to be
    demodulated.  A further stage of "detection", either square-law or
    synchronous, must be applied to demodulate the intelligence on the
    signal.  For this reason, the signal-to-noise ratio for the radio
    frequency heterodyne and optical heterodyne systems is denoted as CNR
    and not SNR.

    Signal Integration:

    In practically all SETI systems, what is being looked for is an ETI
    beacon.  In such systems, the sensitivity of the receiver is enhanced
    by post-detection signal integration, perhaps over many seconds.  This
    increases the detected signal level, and reduces the noise level; both
    at the expense of increasing the search time.  This can only be done
    for detecting the presence of a signal beacon, not for the demodulation
    of a continuously and rapidly changing non-repetitive signal.

                                                                     Page 91

    In the case of a microwave or optical receiver with square law
    detection and an input SNR less than unity, the Signal-To-Noise Ratio
    can be increased by (post-detection) integration of a number of
    detected pulses over a period of time.  In such a situation, the SNR is
    proportional to the square-root of (Nc), where Nc is the total pulse
    count during the observation integration time. [83,88]  The same
    relationship applies to the post-detection counting of individual
    photons, but not to pre-detection.  That is why the quantum limited
    CNRs (SNRs) for both incoherent and coherent optical detection systems
    are proportional to the photon count rate.  See Equ. 36 below.

    Photon-Count Rate:

    The equivalent photon-count rate for the heterodyne receiver is given

    Nph = ------  s^-1                                                  (36)

    Alternatively, this can be expressed as CNR.(Bif).  For the 1 GW
    transmitter that results in a CNR = 19 dB re 30 MHz:

                            Nph = 2.64 X 10^9 s^-1

    This count rate is more than adequate for the photon arrival (and
    detection) statistics to be taken as Gaussian (Poisson), and hence the
    CNR expressions should give an accurate figure for the Carrier-To-Noise
    Ratio.  This is reasonably true even for the 1 kW transmitter, where
    on average, only 5,280 photons arrive per second, of which on average,
    2,640 photons are detected every second.  However, the method of
    expressing CNRs in this analysis, even in the power-starved case,
    allows for a simple linear extrapolation for CNR at any received
    optical power (Equ. 35).

    Bit Error Rate (BER):

    This analysis has concentrated on optical signal detectability in terms
    of SNR not Bit Error Rate (BER), as would be applicable for a digital
    system.  For the sake of completeness, the following expression may be
    used to predict the photon-count rate for a required BER: [78]

    m = ----------                                                      (37)
          log N

    where m = average number of photons per bit required by an ideal N-PPM
              (pulse position modulation) system to achieve a given BER.

                                                                     Page 92

    The photon-count rate is simply the product of m and the bit rate.  For
    an ideal coherent system with on-off keying (OOK) or 1-PPM,
    BER = 10^-9, and very small extinction (light off/light on) ratio,
    m = 10 photons/bit.  However, a more realistic value is nearer to
    20 photons/bit.  Thus, for a 1 GHz (approx. 1 GBit/s) channel:

                  Minimum Photon-Count Rate = 2 X 10^10 s^-1

    The modelled 1 GW system is a little deficient in being able to achieve
    this goal, since this required count rate is an order of magnitude
    greater than the calculated value of Nph.  With digital compression
    techniques, the 1 GW transmitter is capable of supporting a late
    Twentieth Century digital HDTV signal, compressed into a 10 MHz
    bandwidth. [87]

    Range Equation:

    Instead of expressing the CNR as a function of transmitter power,
    range and bandwidth, we can express the quality of the optical
    communications link in terms of its maximum range.  As before, if we
    ignore interstellar absorption, the range (in light years) required to
    reduce the quantum limited CNR to 0 dB for the "perfect" 10-meter
    diameter 656 nm symmetrical Professional Optical SETI system defined by
    Equ. 35, can be express in the form:

    Rmax = 10^[{54 + 10.log(Pt) - 10.log(Be)}/20]                       (38)

    where  Pt = transmitter power (kW),
           Be = I.F. bandwidth (Hz).

    For Pt = 1 GW (EIRP = 2.29 X 10^24 W) and Be = 1 MHz:

                                Rmax = 500 L.Y.

    Doppler Shift:

    The maximum Doppler Shift is given by:

    df = -.f  Hz                                                        (39)

    where  v = maximum line-of-sight velocity (29.8 km/s),
           c = velocity of light (3 X 10^8 m/s),
           f = frequency (4.57 X 10^14 Hz).

                                                                     Page 93

    For a ground-based receiving telescope, the maximum local Doppler Shift
    at 656 nm due to the orbit of Earth around the Sun:

                               df = +/- 45.5 GHz

    Doppler Drift:

    The maximum Doppler Drift (Chirp) is given by:

    df' = -----.f  Hz/s                                                 (40)

    where  w = angular velocity (7.27 X 10^-5 rad/s),
           r = radius of planet or orbit (6,378 km).

    For a receiving telescope on the equator, the maximum local Doppler
    Drift at 656 nm due to Earth's rotation is:

                              df' = +/- 51 kHz/s

    Fortunately, for Amateur Optical SETI observations, the Doppler Drift
    during reasonable observations times is insignificant with respect to
    the bandpass of the incoherent optical filter (approximately 100 GHz).


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