
EJASA  Part 13Page 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 SignalToNoise Ratio: Heterodyne Detection (Reception) HPrPo(MRi)^2 CNR =  (32) [e{Ri(Pr+Po+NbBo)+Ib}M^(2+x)+eIs+2Nb{HPo+NbBo}(MRi)^2+2kTF/RL]Be Homodyne Detection 2HPrPo(MRi)^2 CNR =  (33) [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 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 detector. 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 frontend, which like the case for all other noise components except that due to the local oscillator quantum shotnoise, is negligible for sufficient local oscillator power. The localoscillator (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 nonPoisson distribution of arriving photons in a powerstarved 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 powerstarved, i.e., when there are relatively few photons arriving during the signal integration time so that Bose Einstein [73] statistics apply, the nonwhite 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. [9495] 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 shotnoise limited direct detection receiver approaches the performance of the photoncounting 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 photoncounting receiver. QuantumNoise Limited SignalToNoise Ratio: The CarrierToNoise 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: eta.Pr CNR =  (34) hfBif 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 diffractionlimited 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 overestimate the CNR at very low received optical powers (Pr) and/or higher bandwidths (Be). For a huge transmitting array, the Rayleigh nearfield 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 squarelaw or synchronous, must be applied to demodulate the intelligence on the signal. For this reason, the signaltonoise 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 postdetection 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 nonrepetitive signal. Page 91 In the case of a microwave or optical receiver with square law detection and an input SNR less than unity, the SignalToNoise Ratio can be increased by (postdetection) integration of a number of detected pulses over a period of time. In such a situation, the SNR is proportional to the squareroot of (Nc), where Nc is the total pulse count during the observation integration time. [83,88] The same relationship applies to the postdetection counting of individual photons, but not to predetection. 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. PhotonCount Rate: The equivalent photoncount rate for the heterodyne receiver is given by: eta.Pr Nph =  s^1 (36) hf 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 CarrierToNoise 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 powerstarved 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 photoncount rate for a required BER: [78] ln(2.BER) m =  (37) log N 2 where m = average number of photons per bit required by an ideal NPPM (pulse position modulation) system to achieve a given BER. Page 92 The photoncount rate is simply the product of m and the bit rate. For an ideal coherent system with onoff keying (OOK) or 1PPM, 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 PhotonCount 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" 10meter 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: v df = .f Hz (39) c where v = maximum lineofsight 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 groundbased 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: w^2.r df' = .f Hz/s (40) c 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).
