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SNR/SNR & Incoherent/Coherent Detection ComparisonsRadobs 38The 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 detection receivers, and ideal photon-counting receivers. 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 RADOBS.38 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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