Performance improvement of FSO satellite downlink using aperture averaging and receiver spatial diversity
2016; Institution of Engineering and Technology; Volume: 10; Issue: 4 Linguagem: Inglês
10.1049/iet-opt.2015.0102
ISSN1751-8776
AutoresPooja Gopal, Virander Kumar Jain, Subrat Kar,
Tópico(s)Radio Wave Propagation Studies
ResumoIET OptoelectronicsVolume 10, Issue 4 p. 119-127 Research ArticlesFree Access Performance improvement of FSO satellite downlink using aperture averaging and receiver spatial diversity Pooja Gopal, Corresponding Author Pooja Gopal poojagopal.iitd@gmail.com Electrical Engineering Department, Bharti School of Telecom Technology and Management, Indian Institute of Technology Delhi, Hauz Khas, New Delhi, 110016 IndiaSearch for more papers by this authorVirander Kumar Jain, Virander Kumar Jain Electrical Engineering Department, Bharti School of Telecom Technology and Management, Indian Institute of Technology Delhi, Hauz Khas, New Delhi, 110016 IndiaSearch for more papers by this authorSubrat Kar, Subrat Kar Electrical Engineering Department, Bharti School of Telecom Technology and Management, Indian Institute of Technology Delhi, Hauz Khas, New Delhi, 110016 IndiaSearch for more papers by this author Pooja Gopal, Corresponding Author Pooja Gopal poojagopal.iitd@gmail.com Electrical Engineering Department, Bharti School of Telecom Technology and Management, Indian Institute of Technology Delhi, Hauz Khas, New Delhi, 110016 IndiaSearch for more papers by this authorVirander Kumar Jain, Virander Kumar Jain Electrical Engineering Department, Bharti School of Telecom Technology and Management, Indian Institute of Technology Delhi, Hauz Khas, New Delhi, 110016 IndiaSearch for more papers by this authorSubrat Kar, Subrat Kar Electrical Engineering Department, Bharti School of Telecom Technology and Management, Indian Institute of Technology Delhi, Hauz Khas, New Delhi, 110016 IndiaSearch for more papers by this author First published: 01 August 2016 https://doi.org/10.1049/iet-opt.2015.0102Citations: 13AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract In this study, the bit error rate performance evaluation of a free space optical (FSO) satellite downlink has been carried out. Subsequently, improvement in performance using aperture averaging and receiver spatial diversity techniques have been explored. With a single large aperture receiver or an array of small aperture receivers, the scintillation index in the downlink can be reduced to a very small value. Moreover, a single receiver gives better performance than an array of receivers (employing equal gain combining) with the total area same as that of the single receiver. The expression of channel capacity with outage for the slow fading FSO channel has been derived using the log-normal and gamma–gamma channel models. Based on maximum acceptable outage probability, the minimum required signal-to-noise ratio is determined, from which the capacity with outage is calculated. It has been observed that the maximum achievable capacity per unit bandwidth increases and the corresponding outage probability reduces with an increase in the aperture diameter for a single receiver or with the size of the array for a receiver with spatial diversity. 1 Introduction Optical satellite links provide several advantages over the conventional radio frequency (RF) satellite links like higher bandwidth, lower transmit power and high security [1]. Free space optical (FSO) inter-satellite links are well researched and working links have been established [2, 3]. However, FSO links between an Earth station and a satellite are under rigorous investigation [4]. These links fail in the presence of clouds and therefore cannot be used without an RF backup link. However, even in clear sky conditions, the problem arises because of the atmospheric turbulence. The major effects of turbulence on an FSO satellite downlink are scintillation and angle of arrival fluctuations [1]. The latter effect results in image dancing in the focal plane of an imaging system and is not of much concern in an intensity modulated/direct detection (IM/DD) communication system. In this paper, the performance evaluation of IM/DD optical communication link has been carried out considering the effects of scintillation. Subsequently, techniques to mitigate scintillation effects on the performance of an FSO downlink from a satellite in geostationary orbit (GEO) to the Earth station are explored. The scintillation effect at the receiver can be reduced by using multiple apertures, that is, spatial diversity. Aperture averaging is the simplest form of receiver spatial diversity in which there is a reduction in scintillation with an increase in aperture diameter [5, 6]. The Shannon/ergodic capacity for FSO links based on log-normal and gamma–gamma channel models is available in literature [7–9]. Practical achievement of this capacity would require extremely long codewords for FSO links working at Gbps data rates since the channel is slow fading. The other definition of capacity is based on outage which is more suitable for slow fading channels. The aperture averaging and spatial diversity techniques applied to terrestrial FSO links have been studied in the past [10–16]. The effect of the above mentioned techniques on FSO satellite links using the gamma–gamma channel model which is valid in all turbulence regimes has not been explored. Moreover, the derivation of expressions for capacity with outage and the study of the effect of these techniques on the achievable capacity and outage probability for the FSO satellite links are not available in the literature and hence is the subject of study in this paper. The organisation of this paper is as follows. In Section 2, the bit error rate (BER) performance of an FSO satellite downlink is studied. The reduction in scintillation effect and the improvement in BER performance by using aperture averaging and receiver spatial diversity techniques are discussed in Section 3. Link capacity evaluation is discussed in Section 4. The last section gives the conclusions of the study. 2 Performance evaluation For this study, we have considered an FSO link from GEO to Earth. The GEO satellite is in Earth's equatorial plane and the zenith angle, ζ of the link from New Delhi, India (which is at a latitude of 28.61°) is calculated to be around 30° for satellite altitude, H = 38.5 × 106 m, receiver at ground level and radius of the Earth, RE = 6371 × 103 m. Therefore, we have considered ζ = 30° in our analysis. For most of the path, the signal travels through free space where there is no atmospheric turbulence. By the time it reaches the receiver at the ground, the signal wave-front would be approximately plane. Hence the scintillation is modelled using the plane wave approximation. 2.1 Channel models For the description of atmospheric turbulence, several models namely: double Weibull [17], double generalised gamma [18], Malaga [19], gamma–gamma [20], log-normal [20], etc. are available. The first three models are accurate and also provide closed form expressions for the BER. However, the last two models are relatively easy to tackle mathematically and are as accurate as the first three models in weak scintillations [18] and the downlink scintillations mostly fall in the weak regime. Further, the results on the analysis using the gamma–gamma model for satellite links are not available in literature. Though this model is valid for all turbulence regimes, at very weak turbulence regime it does not give proper results. The log-normal model can be used in such cases. Therefore, in this paper we have considered both gamma–gamma and log-normal models. The atmospheric turbulence in the channel may vary from weak to strong. In the weak turbulence regime, the channel is modelled using the log-normal distribution. The probability density function (pdf) of channel state s (= I /〈I 〉 where I and 〈I 〉 are the instantaneous and average values of the received intensity, respectively) is given by [20] (1)where is the scintillation index. For moderate to strong turbulence regime, the gamma–gamma distribution is used. The pdf of s in the gamma–gamma model is given by [20] (2)where α and β are positive parameters related to the scintillation index as [20] (3)This model can be used in the weak turbulence regime as well. However, the approximation in numerical computation involving the Meijer-G function [21] does not give proper results in a very weak turbulence regime which corresponds to high values of α and β. In such cases, the log-normal model is preferred. The expressions for α and β for the satellite downlink are given by [1] (4a)and (4b)where is the Rytov variance and is given by [1] (5)Here, k is the wave number in m−1 corresponding to wavelength λ in m (k = 2π /λ), h0 the height in m above ground level of the downlink receiver and H the satellite altitude in m. The variation of the refractive index structure constant, with height, h for a vertical or slant path is obtained from the Hufnagel Valley Boundary model given by the following empirical formula [1] (6)where w is the rms wind speed in ms−1 and the ground level turbulence in m−2/3. 2.2 BER analysis For a PIN photodiode detector, the instantaneous and average electrical signal-to-noise ratios (SNRs) are given by and , respectively, where ℝ is the responsivity of the photodetector in AW−1 (ℝ = ηqλ /hP c where η is the quantum efficiency, q the electronic charge, hP the Planck constant and c the speed of light), A the area of the receiver aperture of diameter D (A = πD2 /4 assuming a circular aperture) and the total noise variance. The incoming information signal along with the background noise signal falls on the photodetector. Since the photodetector is a square-law device, the noise components generated at the photodetector output are: shot noise due to signal, shot noise due to background radiation, signal-background beat noise and background–background beat noise. In addition to these, there will be a thermal noise component at the receiver output. The noise power associated with the above components are , , , and , respectively, and are given by (7a) (7b) (7c) (7d)and (7e)where B is the receiver bandwidth in Hz, Nb the power spectral density (PSD) of the background radiation in WHz−1, kB the Boltzmann constant, T the receiver temperature in K and RL the load resistance in Ω. The PSD of background radiation is obtained from the following formula [22] (8)where R (λ) is the spectral radiance of the background source in Wm−3, c is the speed of light in ms−1 and m is the number of spatial modes. A single spatial mode corresponds to the light collected from the diffraction limited field of view (FOV) which is the smallest possible FOV of the lens. The receiver FOV is defined as the solid angle over which the detector is sensitive to incoming light. As shown in Fig. 1, the number of spatial modes, m is given by [22] (9)where Ωs, ΩFOV and ΩDL are the solid angles subtended by the source of background radiation, receiver FOV and the diffraction limited FOV, respectively. These are given by (10a) (10b)and (10c)where Ds is the diameter of the radiation source, θFOV the planar angle of receiver FOV and D the receiver aperture diameter. When the source of background radiation is the Earth's atmosphere extending over the entire hemisphere, clearly Ωs > ΩFOV (see Fig. 1), implying m = ΩFOV /ΩDL. The value of m and therefore Nb depends on the aperture diameter. For D = 4 cm and other parameters as given in Table 1, the values of m and Nb are 3.3275 and 1.3554 × 10−26 WHz−1, respectively. Since m is an integer, it is taken to be 3. Table 1. Parameters used in the numerical computation Parameter Value λ 1550 nm ζ 30° h0 0 m H 38.5 × 106 m w 21 ms−1 η 0.8 B 109 Hz θFOV 90 × 10−6 radians R (λ) 2.1171× 105 Wm−3 T 300 K RL 50 Ω Fig. 1Open in figure viewerPowerPoint Different fields of view for calculation of number of spatial modes The total noise variance, is given by (11)Among all the noises, for the GEO to Earth link considered we observe that , and are negligible. Moreover, except for high 〈I 〉 and large D, the shot noise is not significant and can be neglected making the receiver thermal-noise-limited. For on–off keying modulation scheme, the BER conditioned on s is given by [23] (12)since γ /μ = I /〈I 〉 = s. In order to get the unconditional BER, we have to integrate pe, c over the pdf pI(s) as (13) (i) Log-normal model: Substituting pI(s) from (1), we get (14) With and change of limits of the integration, we get (15)The above equation is similar to (16a)where (16b)For easy numerical computation, we can apply the Gauss–Hermite approximation [24] given by (17)where xi s are the roots and wi s are the corresponding weights of the Hermite polynomial [21]. Applying the above approximation to (15), we get (18) (ii) Gamma–gamma model: Substituting pI(s) from (2) in (13), we get (19) We can express and erfc(x) in terms of Meijer-G function as given in equations 03.04.26.0009.01 and 06.27.26.0006.01 in [25] as (20)The integral in the above equation is similar to the form given in equation 07.34.21.0011.01 in [25] and hence we get (21)Variations of 〈BER〉 with 〈I 〉, computed from (18) for and 5 × 10−14 m−2/3 and (21) for , 5 × 10−13 and 10−12 m−2/3, are shown in Fig. 2. All other parameters used for the numerical computation are given in Table 1. It is observed from this figure that the degradation in the link performance due to atmospheric turbulence is significant for . Fig. 2Open in figure viewerPowerPoint 〈BER 〉 against. 〈I 〉 for different for a point receiver 3 Performance improvement techniques For a point receiver (i.e. receiver with diameter less than the spatial correlation width of the atmosphere, ρc), we see from Fig. 2 that the performance degrades significantly for high values, that is, . For receiver aperture sizes greater than ρc, aperture averaging takes place. Moreover, multiple receivers separated by distances greater than ρc act like independent receivers and hence can significantly reduce the scintillation effect. In this section, we explore the effect of aperture averaging and receiver spatial diversity on the downlink performance. 3.1 Aperture averaging The empirical formula for ρc at the downlink receiver on the ground is given by [1] (22)For ζ = 30° and λ = 1550 × 10−9 m, we get the value of ρc to be 1.13 cm. Therefore, a receiver with diameter, D = 1 cm will behave like a point receiver since D ≤ ρc. When D > ρc, the aperture averaged scintillation index is derived using the ABCD parameters and is given by [1] (23a)where (23b)is the path length. Fig. 3 shows the reduction in with increase in D for different values of . We observe that for larger aperture diameters, reduces to a very small value for all ground turbulence levels. Moreover, aperture averaging is most effective at high . Fig. 3Open in figure viewerPowerPoint against. D for downlink for different The average intensity in the receiver plane at the optical axis is 〈I 〉. It reduces as exp (−r2 /W2) where r is the distance from the optical axis and W is the beam radius in the receiver plane given by where W0 is the beam radius in the transmitter plane, Θ0 and Λ0 are the beam parameters at the transmitter given by Θ0 = 1 for a collimated beam and . For W0 = 2 cm and remaining parameters as in Table 1, we get W = 1096.78 m and therefore on aperture diameter ( cm) the intensity remains almost constant. We see that even for r = 10 m, the intensity is still 99.9915% of that at r = 0 m and hence it is safe to consider the intensity to be constant over the receiver aperture. With increase in D, the BER performance improves due to two reasons: (a) reduction in scintillation effect and (b) increase in the average received power, 〈P 〉 = 〈I 〉A. Variations of 〈BER〉 vs. 〈I 〉 for various D are shown in Fig. 4 for of (a) 10−13, (b) 5 × 10−13 and (c) 10−12 m−2/3. The values of 〈I 〉 (required to achieve a BER of 10−9) and at different for various D are given in Table 2. Lower the required 〈I 〉 to achieve a BER of 10−9, better is the performance. It is observed from Fig. 4 and Table 2 that the performance improvement due to aperture averaging is more at higher . Further, improvement is very significant as the diameter increases from 1 to 4 cm. Afterwards, with the increase in diameter, the rate of improvement gradually reduces. This is true for all values. Table 2. Required 〈I 〉 to achieve a BER of 10−9 and reduction in with increase in D at different 10−13 5 × 10−13 10−12 D, cm ↓ 〈I 〉, mW/m2 〈I 〉, mW/m2 〈I 〉, mW/m2 1 (point Rx.) 40.5713 0.1338 108.3243 0.3584 350.1712 0.5727 4 2.0844 0.0634 2.2311 0.0937 2.5141 0.1316 8 0.4994 0.0422 0.5023 0.0489 0.5141 0.0574 12 0.2152 0.0296 0.2205 0.0323 0.2192 0.0356 16 0.1181 0.0210 0.1213 0.0223 0.1238 0.0240 Fig. 4Open in figure viewerPowerPoint 〈BER 〉 against 〈I 〉 for various D at different a C2 n (0) = 10−13 m−2/3 b C2 n (0) = 5 × 10−13 m−2/3 c C2 n (0) = 10−12 m−2/3 3.2 Spatial diversity Sometimes it is not feasible to use a single large receiver aperture due to practical limitations. In such cases, improved performance can be obtained by using an array of smaller sized apertures separated by distances greater than ρc. At the downlink receiver, ρc is in the order of a few centimetres as mentioned in Section 3.1. This implies that receiver spatial diversity is a feasible option for the satellite downlink. A schematic of receiver spatial diversity in FSO satellite downlink is shown in Fig. 5. Fig. 5Open in figure viewerPowerPoint Schematic of receiver spatial diversity in FSO satellite downlink Generally, three types of linear combining methods are used to combine signals from multiple receiver apertures: selection combining (SC), maximum ratio combining (MRC) and equal gain combining (EGC) [5]. Among these, SC is the simplest but does not cause significant improvement in the average SNR. MRC is the optimal method but requires complex circuitry to get the correct weighting factors. In EGC, equal gains are applied to all the signals. In spite of its simplicity, its performance though slightly poorer, is comparable with that of MRC. Therefore in the following analysis, we have considered only the EGC method. For the composite signal from N receivers, both the signal power and the noise variance are increased by a factor of N. The average SNR considering N receivers is given by (24)Moreover, the effective scintillation index is given by [1] (25)The variation of 〈BER〉 with 〈I 〉 at for an array of receiver apertures of (a) D = 1 cm (point receiver), (b) D = 4 cm and (c) D = 8 cm is shown in Fig. 6. The arrangement of the receivers in the array is not considered in this paper. However, even in this case the received intensity can be considered to be constant over the array. Fig. 6Open in figure viewerPowerPoint 〈BER 〉 against. 〈I 〉 for different N and a D = 1 cm (point receiver) b D = 4 cm c D = 8 cm at When we consider an array of finite sized apertures, the effects of both aperture averaging and spatial diversity occur. Presuming all receivers in the array to be of same size, first the aperture averaged scintillation for a single receiver is calculated and then it is reduced by a factor of N to account for the spatial diversity effect. It is seen from this figure that a very large array of point apertures is required to match the performance of single large diameter aperture. Further, an array of finite sized apertures instead of point apertures considerably improves the performance. Table 3 gives the values of 〈I 〉 (required to achieve a BER of 10−9) and for different N considering an array of 1 cm (point), 4 and 8 cm diameter apertures. An interesting and practically useful observation can be made from Tables 2 and 3. From these tables, we observe that a single aperture of 12 cm diameter achieves a similar performance as an array of 5 apertures of 8 cm diameter (corresponding entries in both tables are highlighted in grey). The single aperture has an area of 113.10 cm2 as compared with a total area of 251.33 cm2 for the array. It may be noted that the total receiver noise (thermal noise) in the array is N = 5 times greater than that of the single aperture. Table 3. Required 〈I 〉 (to achieve a BER of 10−9) and reduction in at with increase in N for different D 1 (point Rx.) 4 8 N ↓ 〈I 〉, mW/m2 〈I 〉, mW/m2 〈I 〉, mW/m2 1 350.1721 0.5727 2.5142 0.1316 0.5144 0.0574 5 17.0315 0.1145 0.9124 0.0263 0.2196 0.0115 10 10.3914 0.0573 0.6222 0.0132 0.1541 0.0058 15 8.2314 0.0382 0.5041 0.0088 0.1252 0.0038 20 7.0459 0.0286 0.4361 0.0066 0.1092 0.0029 4 Link capacity evaluation We have considered that channel pdf is known at both the transmitter and the receiver ends. If there is no channel state information (CSI) available at either the transmitter or the receiver end, then we have to find an input distribution that maximises capacity. This procedure can be very complex for the FSO channel distributions. When the CSI is available at the receiver end, there are two definitions of link capacity: Shannon capacity and capacity with outage. The Shannon or ergodic capacity is given as [26] (26)where p (γ) is the pdf of the channel with the instantaneous SNR, γ as the random variable. Since the transmitter is not aware of the CSI, it maintains a constant rate. In order to achieve the above capacity, we need to send a sufficiently long code word which gets affected by all the fading states. The coherence time of the atmosphere is in the order of milliseconds [20]. For a signal with data rate in the order of Gbps, the transmitter and the receiver would become extremely complex. Hence achieving this capacity is not feasible for the FSO satellite downlink with a slow fading channel. The second definition is the capacity with outage, Co which is suitable for slow fading channels. The fixed channel capacity will be . This implies that all data with γ ≥ γmin is decoded with negligible probability of error. However, data at the receiver with γ < γmin results in an outage whose probability is given by pout = p (γ < γmin). The average capacity with outage is therefore given by (27)In this equation, pout is a design parameter based on acceptable outage. For a given pout, we can determine γmin. The average capacity with outage, Co is calculated using pout and γmin from (27). In the previous section, we have seen that by using aperture averaging and receiver spatial diversity techniques, can be reduced to a very small value for all turbulence conditions. This corresponds to very high values of α and β for which the Meijer-G approximation of the gamma–gamma distribution does not give proper results. Hence both the log-normal (which is accurate for low ) and the gamma–gamma models are used here for the derivation of capacity with outage. The pdf of both log-normal and gamma–gamma models given by (1) and (2) are in terms of the channel state, s. To get pdf in terms of the instantaneous SNR, γ, we use the transformation (28a) (28b)since relation between γ and s is γ = μs. 4.1 Log-normal model Using (1) and (28), we can write p (γ) as (29)The outage probability is given by (30a) (30b)If we take , it gives . In that case, the outage probability in terms of x will be (31a) (31b)where . Using Gaussian integral and error function as given by (32a) (32b)in (31), we get (33) 4.2 Gamma–gamma model From (2) and (28) (34)With the outage probability can be written as (35)where (36)Let γ1 = γ /γmin. This implies γ = γmin γ1 and dγ = γmin dγ1. Changing the variable in Int from γ to γ1 we get (37)where H (x) is the Heaviside function. The Meijer-G function equivalents of xv H (x − 1) and are respectively [21]. Substituting this in (35) and using the Meijer-G integration formula given by equation 07.34.21.0011.01 in [25], we get (38)In Figs. 7 a and b, variations of Co /B vs. γmin and pout vs. γmin, respectively, are given for different D, for the case of aperture averaging. The maximum achievable Co /B is higher for larger D as seen from Fig. 7 a. From Fig. 7 b, it is observed that the rate of increase in pout with an increase in γmin is higher for larger D. Similar behaviour is observed in Figs. 8 a and b for the receiver spatial diversity where maximum achievable Co /B is higher for larger N. For spatial diversity, the diameter of each receiver is taken to be D = 4 cm. In both cases, the average received SNR, μ is taken to be 10 dB and . Fig. 7Open in figure viewerPowerPoint Co /B and pout against. γmin for different D for aperture averaging at a Co /B against γmin b pout against γmin Fig. 8Open in figure viewerPowerPoint Co /B and pout against γmin for different N for receiver spatial diversity with D = 4 cm at a Co /B against γmin b pout against γmin In Table 4, the values of γmin and pout corresponding to maximum achievable Co /B are given. When pout is taken to be the design parameter, in Table 5 the values of γmin and Co /B corresponding to two values of pout = 1% and pout = 10% are given for both the techniques. From Table 4, we observe that for all cases of aperture averaging and spatial diversity, the value of pout corresponding to maximum achievable Co /B is >1%. Hence, if the maximum acceptable pout = 1%, then the values of γmin corresponding to pout = 1% are chosen instead of those corresponding to maximum achievable Co /B. However if the acceptable pout = 10%, then γmin corresponding to maximum achievable Co /B can be chosen for all cases except the case of D = 1 cm which has pout = 22.7% as seen from Table 4. In this case γmin corresponding to pout = 10% is chosen. Table 4. Values of γmin, Co /B and pout corresponding to the maximum achievable capacity at for aperture averaging and receiver spatial diversity technique with aperture diameter, D = 4 cm Technique Aperture averaging Spatial diversity Parameter Aperture diameter, D, cm No. of receivers, N 1 4 8 12 16 1 5 10 15 20 Co /B 1.85 2.51 2.77 2.89 2.98 2.51 2.96 3.08 3.14 3.18 γmin, dB 6.25 7.55 8.15 8.45 8.68 7.52 8.60 8.95 9.11 9.21 pout 22.7% 8.3% 4.7% 3.4% 2.9% 8.2% 2.4% 2.1% 1.8% 1.4% Table 5. Values of γmin and Co /B corresponding to pout = 1% and pout = 10% at for aperture averaging and receiver spatial diversity technique with aperture diameter, D = 4 cm Technique Aperture averaging Spatial diversity Parameter Aperture diameter, D, cm No. of receivers, N 1 4 8 12 16 1 5 10 15 20 γmin (dB) for pout = 1% 1.29 6.12 7.47 8.04 8.42 6.22 8.36 8.86 9.07 9.19 corresponding Co /B 1.21 2.34 2.69 2.84 2.96 2.34 2.93 3.07 3.13 3.17 γmin (dB) for pout = 10% 4.57 7.71 8.56 8.89 9.09 7.72 9.04 9.33 9.46 9.55 corresponding Co /B 1.75 2.51 2.70 2.81 2.86 2.50 2.86 2.93 2.99 2.99 5 Conclusions For a point receiver in optical satellite downlink, the link performance degrades significantly for . The improvement in performance using aperture averaging and receiver spatial diversity techniques have been explored in this work. The scintillation index can be reduced to a very small value for all for a single large aperture diameter due to aperture averaging or with an array of small fixed diameter receivers due to spatial diversity. In receiver spatial diversity, a very large array of point receivers are required to match the performance of a single large diameter receiver. However considering an array of finite sized receivers instead of point receivers considerably improves the performance. It has been observed that a single receiver gives better performance than an array of receivers whose total area is same as that of the single receiver. This is because the total receiver noise in the array is N times greater than that in the single receiver, but the signal power in both the cases remains almost same. The capacity with outage per Hz, that is, Co /B increases with the increase in the aperture diameter for a single receiver or with size of the array in the receiver with spatial diversity. 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