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` `
```The following formulas predict the post-detection Signal-To-Noise Ratios
(SNRs) or the pre- (electrical) detection Carrier-To-Noise Ratios (CNRs) in
the three basic types of receiver.  These expressions have been used in
Symphony spreadsheets 9106-001.WR1, 9106-002.WR1, 9106-003.WR1,
9106-008.WR1, 9106-013.WR1, and 9106-015.WR1.```
```The following expressions are rather more detailed than is usually shown in
standard texts on optical detection systems.  Though I have found some
books and papers somewhat lacking in detail and contradictory on the exact
forms of these expressions, I believe that the following expressions are
accurate.  Readers are welcomed to point out any errors.
```
`SYMBOL DEFINITIONS:`
```For the microwave system:
Pr = received power,
T  = effective system temperature (K),
Be = electrical intermediate frequency bandwidth (Hz).
```
```For the optical systems:
H  = optical mixing efficiency,
Pr = received optical power (W),
Po = local oscillator power (W),
M  = avalanche gain,
Ri = unity gain responsivity (W/A),
e  = electronic charge (1.6 X 10^-19 C),
Nb = background radiation spectral density (W/Hz),
Ib = bulk dark current at unity gain (A),
Is = surface dark current (A),
x  = excess noise factor,
k  = Boltzmann's constant (1.38 X 10^-23 J/K),
T  = front-end amplifier temperature (K),
F  = front-end amplifier noise figure,
RL = front-end load (Ohms),
Bo = optical pre-detection bandwidth (Hz),
Be = electrical bandwidth (Hz).
```
```RADIO FREQUENCY:
Microwave Heterodyne Reception```
```       Pr
CNR = ----                                                              (1)
kTBe
```
```The dimensions of all signal and noise components in the format of the
optical expressions are in units of amperes^2.  The numerator is
representative of the electrical signal power in the photodetector load,
while the denominator represents the electrical noise power in the
photodetector load.  For the incoherent systems, the pixel has a
diffraction limited field-of-view (FOV) corresponding to the Airy disk,
i.e., (2.44)Wl/d radians, where Wl = wavelength, and d is the aperture
diameter.  For coherent systems, a smaller FOV is employed; that
corresponding to the FWHM response, i.e., (1.02)Wl/d radians.  The latter
pixel size is smaller because of the requirement to reduce the amount of
local-oscillator power that does not beat with the signal but only induces
excess quantum shot-noise.```
```For coherent receivers, dual-balanced photodetection is assumed so that all
the received signal power is utilized, and so that the noise floor is not
raised by excess intensity noise on the local-oscillator laser.  It is
further assumed that the linewidths of the received signal and local-
oscillator laser are sufficiently small compared to the modulation and/or
detection bandwidths, as to not raise the noise floor.
```
```INCOHERENT OPTICAL:
Direct Detection and Photon-Counting```
```                                Pr^2(MRi)^2
SNR = --------------------------------------------------------------   (2)
[2e{Ri(Pr+NbBo)+Ib}M^(2+x)+2eIs+2Nb{Pr+NbBo}(MRi)^2+4kTF/RL]Be
```
```The electrical signal power is proportional to Pr^2, and the noise
components proportional:```
`1.   To the quantum noise produced by the signal photons,`
```2.   To the fluctuation noise produced by the background radiation Pb
(NbBo).  Notice that this noise is proportional to the optical
bandwidth, and the ratio of this noise to the quantum noise component
is inversely proportional to the received optical power,```
```3.   To the shot noise produced by the bulk dark current in the
photodetector,```
`4.   To the shot noise produced by the surface leakage dark current,`
```5.   To the background radiation beating with the signal, which is
independent of optical bandwidth.  The noise spectral density is the
important factor here,```
```6.   To the noise beating with noise, which is proportional to both the
noise spectral density squared and the optical bandwidth.  The latter
two noise components are insignificant, and may be safely omitted for
this application where the background is very small,```
```7.   To the thermal kT noise in the photodetector load and front-end
amplifier, and may be neglected for shot noise limited direct

```
```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.  For the purposes of the spreadsheet analyses there is
not a separate Is term.  For higher modelling accuracy in the spreadsheets,
the unity gain photodetector dark current can be assumed to have a value Id
= Ib+Is/M^(2+x).
```
```COHERENT OPTICAL:
Heterodyne Detection (Reception)```
```                                 HPrPo(MRi)^2
CNR = ----------------------------------------------------------------  (3)
[e{Ri(Pr+Po+NbBo)+Ib}M^(2+x)+eIs+2Nb{HPo+NbBo}(MRi)^2+2kTF/RL]Be
```
`Homodyne Detection (Synchronous)`
```                                2HPrPo(MRi)^2
SNR = ----------------------------------------------------------------  (4)
[e{Ri(Pr+Po+NbBo)+Ib}M^(2+x)+eIs+2Nb{HPo+NbBo}(MRi)^2+2kTF/RL]Be
```
```The electrical signal power is proportional to Pr, 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
photodetector,```
`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 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.  In this SETI application, the star is distant, and Nb is very
small.  However, for communications within the solar system, these
background noise components (from the Sun or reflected light from the Earth
or a planet) can be significant, and should be accounted for.```
```In the analyses, I have used a more conservative approach for assessing the
performance of various receiving systems, by accounting for atmospheric
transmission efficiency, telescope aperture efficiency, spectrometer
efficiency (incoherent systems only) and in the case of coherent receivers,
an allowance for the optical (heterodyne or homodyne) mixing efficiency.
This accounts for any discrepancies in the performance of 10 m heterodyne
receivers given in previous files.
```
`The received signal intensity just outside the Earth's atmosphere is:`
```       EIRP
Ir = --------                                                           (5)
4.PI.R^2```
```where  EIRP = effective isotropic radiated power (W),
R    = range (10 L.Y. = 9.461 X 10^16 m).
```
```At a range of 10 light years, an EIRP = 1.13 X 10^23 W produces an
intensity Ir = 10^-12 W/m^2.
```
`The optical power reaching the photodetector is given by:`
`Pr = Ir.At.Ae.A.SE                                                      (6)`
```where  Ir = intensity just outside atmosphere (W/m^2),
At = atmospheric transmission (0.4),
Ae = antenna efficiency (0.7),
A  = antenna aperture area (0.52 m^2),
SE = spectrometer efficiency (0.5).
```
```For the 0.81 m diameter (32") Perkins telescope, and the above parameter
values, the received signal Pr = 7 X 10^-14 W.
```
```Expression (2) relates to incoherent detection, while (3) and (4) 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 & 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.```
```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.
```
```June 30, 1991
BBOARD No. 585
```
```* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
* Dr. Stuart A. Kingsley                       Copyright (c), 1991        *
* AMIEE, SMIEEE                                                           *
* Consultant                            "Where No Photon Has Gone Before" *
*                                                   __________            *
* FIBERDYNE OPTOELECTRONICS                        /          \           *
* 545 Northview Drive                          ---   hf >> kT   ---       *
* Columbus, Ohio 43209                             \__________/           *
* United States                            ..    ..    ..    ..    ..     *
* Tel. (614) 258-7402                     .  .  .  .  .  .  .  .  .  .  . *
* skingsle@magnus.ircc.ohio-state.edu         ..    ..    ..    ..    ..  *
* CompuServe: 72376,3545                                                  *
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *```

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