James A. Nessel and Roberto J. Acosta

Glenn Research Center, Cleveland, Ohio

Directivity of a Sparse Array in the Presence 

of Atmospheric-Induced Phase Fluctuations 

for Deep Space Communications

NASA/TM—2010-216241

March 2010
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James A. Nessel and Roberto J. Acosta

Glenn Research Center, Cleveland, Ohio

Directivity of a Sparse Array in the Presence 

of Atmospheric-Induced Phase Fluctuations 

for Deep Space Communications

NASA/TM—2010-216241

March 2010

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135
Available from

NASA Center for Aerospace Information

7115 Standard Drive

Hanover, MD 21076–1320

National Technical Information Service

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Alexandria, VA 22312

Available electronically at http://gltrs.grc.nasa.gov

Level of Review: This material has been technically reviewed by technical management. 
NASA/TM—2010-216241 1 

Directivity of a Sparse Array in the Presence of AtmosphericInduced Phase Fluctuations for Deep Space Communications 

James A. Nessel and Roberto J. Acosta 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

Abstract 

Widely distributed (sparse) ground-based arrays have been utilized for decades in the radio science 

community for imaging celestial objects, but have only recently become an option for deep space 

communications applications with the advent of the proposed Next Generation Deep Space Network 

(DSN) array. But whereas in astronomical imaging, observations (receive-mode only) are made on the 

order of minutes to hours and atmospheric-induced aberrations can be mostly corrected for in postprocessing, communications applications require transmit capabilities and real-time corrections over time 

scales as short as fractions of a second. This presents an unavoidable problem with the use of sparse arrays 

for deep space communications at Ka-band which has yet to be successfully resolved, particularly for 

uplink arraying. In this paper, an analysis of the performance of a sparse antenna array, in terms of its 

directivity, is performed to derive a closed form solution to the expected array loss in the presence of 

atmospheric-induced phase fluctuations. The theoretical derivation for array directivity degradation is 

validated with interferometric measurements for a two-element array taken at Goldstone, California. With 

the validity of the model established, an arbitrary 27-element array geometry is defined at Goldstone, 

California, to ascertain its performance in the presence of phase fluctuations. It is concluded that a 

combination of compact array geometry and atmospheric compensation is necessary to ensure high levels 

of availability. 

1.0 Introduction 

Widely distributed (sparse) ground-based arrays have been utilized for decades in the radio science 

community for imaging celestial objects and for various astrometric measurements. By correlating 

measurements taken by several widely separated antennas, an effective aperture area of the distance 

between the furthest separated antennas is created, drastically increasing the spatial resolution of the 

system. However, the natural “seeing” ability of a particular site will be fundamentally limited by the local 

atmospheric-induced phase fluctuations imposed on the signal. This phenomenon is a result of large 

amounts of inhomogeneous distributions of water vapor exposed to turbulent air flow conditions in 

Earth’s upper atmosphere (troposphere), which directly leads to variations in the effective electrical path 

length (phase) of a received signal. Such variations are seen as ‘phase noise’ and will inherently degrade 

the resolution of radio arrays. 

The same issues arise when developing sparse arrays for intrastellar communications. But whereas 

radio science applications impose only receive-mode requirements (i.e., imaging) and observations are 

made on the order of minutes to hours (long integration times), communications applications require both 

transmit and receive capabilities, as well as real-time corrections over time scales as short as fractions of a 

second. In the receive case, adaptive techniques have been utilized by the DSN since the 1980s to 

compensate for the atmosphere at frequencies up to X-band (Ref. 1). More recently, uplink arraying of a 

7.15 GHz signal was successfully demonstrated in an experiment with the Mars Global Surveyor (Ref. 2). 

However, since this atmospheric phase noise scales with frequency, at Ka-band (the frequencies of interest 

for future NASA DSN operations) the problem becomes much more severe and has yet to be successfully 

resolved, particularly in the uplink. In this paper, an analysis of the performance of a sparse antenna array, 

in terms of its directivity, is performed to derive a closed form solution to the expected array loss in the 
NASA/TM—2010-216241 2 

presence of atmospheric-induced phase fluctuations. The theoretical derivation for array directivity 

degradation is validated with interferometric measurements for a two-element array taken at Goldstone, 

California. Excellent agreement between theory and measurement is observed for the case of a twoelement array. With the validity of the model established, an arbitrary 27-element array geometry is 

defined at Goldstone, California, to ascertain its theoretical performance in the presence of phase 

fluctuations. 

2.0 Theory 

Consider a widely distributed (d >> λ) array of N elements arbitrarily spaced on a plane, whose 

geometry is defined in Figure 1, where ܽො௥ is a unit vector in the direction of propagation, ρሬԦ௠ is the 

distance from the array origin to the mth element, ܴ is the distance the signal travels from the array origin 

to the receiver, and ݎ ௠is the distance the signal travels from the mth element to the receiver. 

Let us assume that each element has its maximum radiation intensity in the +z direction and has zero 

radiation in the lower half-space (z < 0). Further, we will utilize the well known pattern approximation of 

cos௤ሺθሻ for the E- and H- planes of the antennas to simplify the analysis. Thus, for an arbitrarily polarized 

antenna element, we can represent its far field electric field, ܧሬԦ

௠, in the upper half-space (0 ≤ θ ≤ π/2) by 

(Ref. 3) 

Ԧሬܧ

௠ሺݎ, θ, φሻ ൌ ܫ ௠ቆ

݁ି௝௞௥

4πݎ ቇ ൣી෡ܸா௠ሺθሻ൫ܽ݁௝ψ cos φ ൅ ܾ sin φ൯ ൅ φ෠ܸு௠ሺθሻ൫െܽ݁௝ψ sin φ ൅ ܾ cos φ൯൧

where, 

ܫ ௠ൌ ܥ௠݁௝φ೘ ൌ complex excitation coefficient

݇ൌ2π λ⁄ ൌ wavenumber

ܸா௠ሺθሻ ൌ cos௤ಶሺθሻ ൌ E െ plane voltage pattern approximation

ܸு௠ሺθሻ ൌ cos௤ಹሺθሻ ൌ H െ plane voltage pattern approximation

 (a, b, ψ) = polarization coefficients (see Table I) 

TABLE I.—VARIOUS FEED POLARIZATIONS 

a b ψ

Linear X 1 0 0 

Linear Y 0 1 0 

RHCP 1 2 1 2 π/2 

LHCP 1 2 1 2 –π/2 

If we translate this analysis to an array environment, the array of antennas will produce a far field 

electric field that will consist of the superposition of each individual element field plus a propagation 

delay, as determined by the geometry, relative to some specified origin (see inset of Fig. 1). This will 

result in an array far field, ܧሬԦ

௔௥௥௔௬, which can be represented by the product of the individual element far 

field and an array factor. 

Ԧሬܧ

௔௥௥௔௬ሺݎ, θ, φሻ ൌ ݂ሺθ, φሻ ሬሬሬሬሬሬሬሬሬሬሬሬሬԦ ቆ

݁ି௝௞ோ

4πܴ ቇ෍ܫ௠݁ି௔௞௝ොೝ·஡ሬԦ೘

ே

௠ୀଵ

where, 

݂ሺθ, φሻ ሬሬሬሬሬሬሬሬሬሬሬሬሬԦ ൌ ી෡ܸா௠ሺθሻ൫ܽ݁௝ψ cos φ ൅ ܾ sin φ൯ ൅ φ෠ܸு௠ሺθሻ൫െܽ݁௝ψ sin φ ൅ ܾ cos φ൯

ܽො௥ ൌ unit vector in direction of propagation

ρሬԦ௠ ൌ position vector of element relative to array origin
NASA/TM—2010-216241 3 

Figure 1.—Geometry of a planar array of radiating elements located at arbitrary positions with arbitrary polarization. 

Inset: determination of excess path delay between array elements as a result of array geometry. 

From the far field array pattern, the radiation intensity, ܷሺθ, φሻ, and radiated power, ܲ௥௔ௗ, of an array 

can be determined, whose ratio defines the array directivity. The complete derivation of the array 

directivity, which involves reducing Bessel functions of 2nd order (Ref. 3), are omitted here for 

conciseness, but if we assume the antenna elements are identical, spaced many wavelengths apart, and 

there are no pointing losses to consider, then the peak directivity, ܦ௞௔௘௣, as a function of deterministic 

phase, φ, can be approximated by 

Peak Radiation Intensity:

ܷ ቀθ௣௘௔௞, φ௣௘௔௞ቁ ൌ ௥మ

ଶη ቚܧሬԦ ቀݎ, θ௣௘௔௞, φ௣௘௔௞ቁቚଶ

ൌ ଵ

ଷଶ஠మη

∑ ∑ ݁௝൫φ೘ିφ೙ ே ൯ ௡ୀଵ

ே

௠ୀଵ

Radiated Power: ܲ௥௔ௗ ൌ ଵ

ଵ଺గη ቂ ௤ಶା௤ಹାଵ

ሺଶ௤ಶାଵሻሺଶ௤ಹାଵሻ

ቃ ܰ

Peak Directivity: ܦ ௞௔௘௣ൌ ସ஠௎ቀ஘೛೐ೌೖ,φ೛೐ೌೖቁ

௉ೝೌ೏

ൌ ܦ௡௘௠௘௟௘௧

ଵ

ே ∑ ∑ ݁௝൫φ೘ିφ೙ ே ൯ ௡ୀଵ

ே

௠ୀଵ

where, ܦ௡௘௠௘௟௘௧ ൌ ଶሺଶ௤ಶାଵሻሺଶ௤ಹାଵሻ

௤ಶା௤ಹାଵ

2.1 Array Directivity in Presence of Random Phase 

The above analysis assumes a constant deterministic phase between elements, but if we now suppose 

that random phase fluctuations are present during signal transmission, such as those induced by water 

vapor in the atmosphere, the statistical distribution of the random process can be utilized to obtain a 

closed-form solution. Let us assume (and later confirm this assumption) that the phase fluctuations 

induced by the atmosphere are normally distributed over the time scale of interest with mean zero and 

variance σ

2

. The average peak directivity of an array, in terms of the statistical ensemble average of the 

random phase fluctuations between elements m and n, can be determined, in closed form, as 
NASA/TM—2010-216241 4 

௧௘௟௘௠௘௡ܦ ൌ ۄ௞௔௘௣ܦۃ

1

ܰ ෍ ෍ۃ݁௝ሺδ೘೙ሻ

ۄ

ே

௡ୀଵ

ே

௠ୀଵ

ൌ ܦ௡௘௠௘௟௘௧

1

ۄ௡௠δ sinۃ݆ ൅ ۄ௡௠δ cosۃ෍ ෍ ܰ

ே

௡ୀଵ

ே

௠ୀଵ

ൌ ܦ௡௘௠௘௟௘௧

1

ܰ ෍ ෍න cos δ௠௡

1

௡௠ߪߨ2√

݁

ି δ೘೙

మ

ଶ஢೘೙

మ ݀δ

ஶ

ିஶ

ே

௡ୀଵ

ே

௠ୀଵ

ൌ ܦ௡௘௠௘௟௘௧

1

ܰ ෍ ෍݁ି஢೘೙

మ

ଶ

ே

௡ୀଵ

ே

௠ୀଵ

where we have defined δ୫୬ ؠ φ௠ െ φ௡. The average array loss can then be described by the difference 

(in dB) of the ideal directivity and the actual directivity achieved in the presence of the phase fluctuations. 

ܦۃ௢௟௦௦ۄ ൌ 10 logሺܰܦ௡௘௠௘௟௘௧ሻ െ 10 log൫ܦۃ௞௔௘௣ۄ൯

For a two-element (N =2) array, the above equation simplifies to 

ܦۃ௢௟௦௦ۄ ൌ 10 logሺ2ሻ െ 10 log ൭1

2 ෍ ෍݁ି஢೘೙

మ

ଶ

ଶ

௡ୀଵ

ଶ

௠ୀଵ

൱

ൌ 10 logሺ2ሻ െ 10 log

1

2 ቆ݁ି஢భభ

మ

ଶ ൅ ݁ି஢భమ

మ

ଶ ൅ ݁ି஢మభ

మ

ଶ ൅ ݁ି஢మమ

మ

ଶ ቇ

ൌ 10 logሺ2ሻ െ 10 log ቆ1 ൅ ݁ି஢భమ

మ

ଶ ቇ

3.0 Validation 

The above theoretical derivation for array loss can be considered exact, so long as the original 

assumption that differential phase fluctuations induced by the atmosphere are zero-mean, normally 

distributed random variables is true. Therefore, before further analysis can proceed, we first validate this 

assumption by investigating the probability distribution function (PDF) of differential phase fluctuations, 

as measured by a two-element site test interferometer currently deployed in Goldstone, California. 

The two-element interferometer developed by NASA Glenn Research Center utilizes a digital I and Q 

receiver which monitors an unmodulated beacon signal at 20.199 GHz from a geostationary satellite, Anik 

F2, with a baseline separation distance of 256 m and an elevation angle of 48.5°. A localized 10 MHz 

GPS-disciplined rubidium oscillator provides the reference timing for all operations and data collection. A 

more detailed description of the system hardware and setup can be found in (Ref. 4). The signal is 

sampled at 3.64 MHz with an integration time of 144 ms and recorded every second. Since our 

measurements limit the resolution to which we can observe phase fluctuations to time scales greater than 

1 second, we must make several assumptions as to the characteristics of these fluctuations at time scales 

comparable to a symbol period (<< 1 sec), the time scale of interest for communications applications. 

3.1 Statistics of Phase Fluctuations 

A representative plot of the calibrated differential phase, as measured by our two-element 

interferometer, is shown in Figure 2 for September 1, 2007.1

 

1

Note: This day was arbitrarily chosen from a set of data blocks that possessed no erroneous data points. The 

statistics derived from this data is representative of all data collected, thus far. 
NASA/TM—2010-216241 5 

We analyze the statistics for a block of data from 02:00 to 03:00 GMT (where the atmosphere appears 

to be mildly turbulent) and 12:00 to 13:00 GMT (where the atmosphere appears to be calmer) at different 

time scales. Figure 3 shows the PDF at time scales of 1 hr, 30 min, and 10 min for each of these times. 

From the plot, we observe that the statistics do appear to follow a zero-mean normal distribution as we 

extend to shorter and shorter time periods, but notice that the rms of the phase fluctuations as we move 

towards shorter time scales does possess some variance. Since the rms phase is the metric by which array 

degradation is measured, characterizing these changes at time scales comparable to a symbol period is 

paramount to determining array performance for communications applications. 

From this analysis, we validate that the distribution of phase fluctuations induced by the atmosphere is 

indeed Gaussian for time scales larger than approximately 10 min. Due to the low sampling frequency, we 

assume that the trend observed in the statistics for this process continue to extend to shorter time scales; 

that is, at time scales comparable to a symbol period, the PDF of phase fluctuations remains a zero-mean, 

normally distributed process. To truly verify this assumption, the sampling rate of recorded phase data 

would need to be increased to obtain enough information at sub-second intervals. 

3.2 Two-Element Array Loss: Measurement Versus Predicted 

To compare the theoretically predicted array loss with measured array loss for a two-element array, 

we must begin with the assumption that the phase difference induced by the atmosphere is an ergodic 

random process. In this way, we can take the time-averaged directivity and directly compare it to the 

ensemble average directivity loss determined above. From the equation for peak directivity for a twoelement array, 

ܦ௞௔௘௣ሺݐሻ ൌ ܦ௡௘௠௘௟௘௧

1

2 ෍ ෍݁௝൫φ೘ሺ௧ሻିφ೙ሺ௧ሻ൯

ଶ

௡ୀଵ

ଶ

௠ୀଵ

The time-averaged directivity for a given time interval, ΔT, can be described by 

ܦ ௞௔௘௣തതതതതതതത ൌ ܦ௡௘௠௘௟௘௧

1

2∆ܶ න ቂ2 ൅ 2݁௝൫φమሺ௧ሻିφభሺ௧ሻ൯ቃ ݀ݐ

௧ೖା∆்

௧ೖ

ൌ ܦ௡௘௠௘௟௘௧ ቈ1 ൅

1

∆ܶ න ݁௝ஔమభሺ௧ሻ݀ݐ

௧ೖା∆்

௧ೖ

቉

where δଶଵሺݐሻ ؠ φଶሺݐሻ െ φଵሺݐሻ. 

Figure 2.—Differential phase time series measured by the two-element site test interferometer at Goldstone, 

California, on September 1, 2007. 

0 2 4 6 8 10 12 14 16 18 20 22 24 -150

-100

-50

0

50

100

150

GMT Time (hrs)

Differential Phase (deg)

Differential Phase Time Series for September 1, 2007
NASA/TM—2010-216241 6 

 

(a) (b) 

Figure 3.—PDF’s of phase fluctuations for 1 hr (top left, 1), 30 min (top right, 2), and 10 min (bottom center, 6) at (a) 

02:00 to 03:00 GMT and (b) 12:00 to 13:00 GMT on September 1, 2007. 

Figure 4.—Measured versus theoretical array loss for varying rms phase. 

The plot of Figure 4 shows the comparison between the predicted ensemble average array 

degradation, ܦۃ௢௟௦௦ۄ, for a given rms phase and the measured time-averaged directivity loss, ܦ௢௟௦௦ തതതതതത, for a 

two-element array. As 10 min is the smallest interval in which we possess enough data points to establish 

a normal distribution, 10 min averages were used. From the plot, we observe extremely good agreement 

between the two curves, indicating the correctness of the theoretical derivation for array loss in the 

presence of atmospheric-induced phase fluctuations, as well as confirming the ergodicity of the 

atmospheric-induced random process. 
NASA/TM—2010-216241 7 

4.0 Typical Array Loss at Goldstone, California 

To determine the typical array loss at Goldstone, California, we generate the cumulative distribution 

function (CDF) of the phase rms based on 1-yr data collected (2007 to 2008). To normalize our analysis, 

the data has been transformed to zenith and an operating frequency of 32 GHz, where the transformation 

to zenith is performed by multiplying by 1/air mass (sinሺ݈݁݁ݒܽݐ݅݋݊ ݈ܽ݊݃݁ሻ) and the rms phase scales 

linearly with frequency. The baseline separation distance is 256 m. The resulting zenith rms phase CDF is 

shown below in Figure 5. 

From the CDF, we observe that 90 percent of the time, the rms phase is better than 35.2° (at zenith), 

which corresponds to an array loss of only 0.39 dB (2.2 dB at 20° elevation).2

 Though this value appears 

low, recall that this result is for a simple two-element array. We can extrapolate this value to N elements, 

given a particular array geometry, by scaling the phase rms to different baselines (Ref. 5). Note that the 

results of this analysis will be extremely geometry dependent (Ref. 6). As a simple example, let us 

consider an array geometry similar to the Very Large Array (VLA) in Socorro, New Mexico, first, with an 

antenna spacing of 250 m between individual elements (circles in Fig. 6(a)), and one with a spacing of 

50 m (x’s in Fig. 6(a)). Since maximum directivity is only a function of number of elements (in widelyspaced arrays), these two geometries can be readily compared. In our analysis, we assume the theoretical 

Kolmogorov turbulence root phase structure function exponent of 5/6 (d < 1 km) and 1/3 (d > 1 km) to 

scale the phase rms to different baselines (Ref. 7). We further assume that the average rms phase between 

antenna elements is similar for identical baseline separations, regardless of orientation or reference. 

Calculating the array loss curve based on the theoretical derivation (Fig. 6(b)), we observe that for the 

250-m baseline array geometry in Goldstone, California, we will need a margin of approximately 2.8 dB 

at zenith (12.3 dB at 20° elevation) to maintain 90 percent availability. This margin can be reduced, by 

reducing the baseline separation to the 50-m geometry, which only requires 1.1 dB at zenith (5.2 dB at 20° 

elevation). This is due entirely to the fact that small-scale fluctuations will contain much less energy than 

larger scale fluctuations, which would directly impact large baseline arrays. Thus, for communications 

applications, it will be desirable to maintain the most compact geometry possible to minimize array loss 

due to atmospheric phase fluctuations, as the furthest extent of the array will dominate this factor. 

Figure 5.—CDF of rms phase for Goldstone, California, during first 

year of data collection. 

 

2

Where the air mass (1/ sin ߠ (is used to convert rms phase at zenith to another elevation angle 

0 20 40 60 80 100 120

10-2

10-1

100

101

102

RMS Phase (deg)

% Time Abscissa Exceeded

CDF of RMS Phase (at Zenith, 32 GHz) for Goldstone, CA: Year 1
NASA/TM—2010-216241 8 

 (a) (b) 

Figure 6.—(a) Model array geometries for Goldstone, California, array loss calculation example with 250-m baseline 

geometry (o) and a 50-m baseline geometry (x), and (b) resulting theoretical array loss versus rms phase curve. 

 Figure 7.—(Top) Time series phase fluctuations, (middle) rms phase time series, and (bottom) 

calculated array loss for model arrays for July 25, 2007 measured data at Goldstone, California. 

To investigate the transient behavior of the model array, we can analyze the time-domain array 

performance for a particularly turbulent day at Goldstone, California. Figure 7 shows the phase 

fluctuations observed by the two-element interferometer on July 25, 2007, as well as the resulting rms 

0 20 40 60 80 100 120 140 160 180 0

5

10

15

RMS Phase (deg)

Array Loss (dB)

250-m Baselines

 50-m Baselines

0 5 10 15 20

-200

-100

0

100

200

GMT Time (hrs)

Phase Fluctuations (deg)

Phase Fluctuations Time Series: 07/25/07

0 5 10 15 20

0

20

40

60

80

100

120

GMT Time (hrs)

RMS Phase (deg)

RMS Phase Time Series: 07/25/07

0 5 10 15 20

0

5

10

15

GMT Time (hrs)

Array Loss (dB)

Array Loss Time Series: 07/25/07

250-m Baselines

 50-m Baselines
NASA/TM—2010-216241 9 

phase and calculated array loss (for both the wide and compact array geometries described above). During 

extremely turbulent times (beginning of the day), array degradation can exceed 10 dB with a mean array 

loss of 4.4 dB for the entire day (250-m baseline geometry).3

 An approximate 2 dB improvement, on 

average, can be realized for the more compact array design (50-m baseline geometry). 

5.0 Conclusions 

Herein we report on the theoretical performance of a sparse array whose signal degradation is primarily 

due to atmospheric-induced phase fluctuations. The ensemble average directivity of an N element array in 

the presence of phase noise was derived theoretically and validated with measured data. Further, it is shown 

that the measured phase differential between two elements is indeed normally distributed (to the resolution 

limits defined by the experimental setup) and ergodic, the fundamental assumption which allows the 

prediction of a theoretical array’s performance. It is observed that the performance of an array in the 

presence of atmospheric-induced phase fluctuations is limited by the furthest extent of the array elements, 

and, for communications applications, this geometry should remain as compact as possible. Finally, the time 

series performance of an arbitrary array is shown for a particularly turbulent atmospheric day. For the 

geometries described, there is still significant array losses observed and to prevent these losses, some form of 

compensation is necessary, particularly during transmit. 

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42–175, Nov. 15, 2008, p.1–7. 

7. C. Coulman, “Fundamental and Applied Aspects of Astronomical Seeing,” Ann. Rev. Astron. 

Astrophys., Vol. 23, pp. 19–57, 1985. 

 

3

Recall that the phase data recorded at Goldstone, California, is for an elevation angle of 48.5° 
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01-03-2010 

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Technical Memorandum 

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4. TITLE AND SUBTITLE

Directivity of a Sparse Array in the Presence of Atmospheric-Induced Phase Fluctuations for 

Deep Space Communications 

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6. AUTHOR(S)

Nessel, James, A.; Acosta, Roberto, J. 

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WBS 439432.04.04.07 

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National Aeronautics and Space Administration 

John H. Glenn Research Center at Lewis Field 

Cleveland, Ohio 44135-3191 

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13. SUPPLEMENTARY NOTES

14. ABSTRACT

Widely distributed (sparse) ground-based arrays have been utilized for decades in the radio science community for imaging celestial objects, 

but have only recently become an option for deep space communications applications with the advent of the proposed Next Generation Deep 

Space Network (DSN) array. But whereas in astronomical imaging, observations (receive-mode only) are made on the order of minutes to 

hours and atmospheric-induced aberrations can be mostly corrected for in post-processing, communications applications require transmit 

capabilities and real-time corrections over time scales as short as fractions of a second. This presents an unavoidable problem with the use of 

sparse arrays for deep space communications at Ka-band which has yet to be successfully resolved, particularly for uplink arraying. In this 

paper, an analysis of the performance of a sparse antenna array, in terms of its directivity, is performed to derive a closed form solution to 

the expected array loss in the presence of atmospheric-induced phase fluctuations. The theoretical derivation for array directivity 

degradation is validated with interferometric measurements for a two-element array taken at Goldstone, California. With the validity of the 

model established, an arbitrary 27-element array geometry is defined at Goldstone, California, to ascertain its performance in the presence of 

phase fluctuations. It is concluded that a combination of compact array geometry and atmospheric compensation is necessary to ensure high 

levels of availability. 

15. SUBJECT TERMS

Antenna arrays; Deep Space Network; Directivity; Phase noise; Atmosphere 

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