Radio Science

On quadrature components of dissimilar statistics and their relevance in antenna error analysis



[1] Statistical analysis of the feed array of a complex antenna revealed distribution functions which cannot be represented by Rayleigh or Rice distribution models. Here we present the analytical description of these distributions, which may be seen as an extension of the Rice distribution. We conclude that statistical processes involving guidance of waves with very significant random content (e.g., near nulls) result in general in quadrature components of different statistical properties.

1. Introduction

[2] In the practical world, the properties of individual antenna subassemblies are not completely deterministic, and contain by necessity an amount of error. Quantifying the departure from ideal conditions is a task that can only be performed statistically, which necessitates the introduction of statistical models for the subassemblies under an assumption of small mean square errors, which is then followed by probability analysis to make predictions of antenna performance. Such an analysis is complementary to the work done in the 1960s on nondeterministic arrays, which deal with phased arrays characterized by elements with random location; elements removed at random; and the effect of random amplitude and phase errors in the antenna aperture [Steinberg, 1976].

[3] There is a widespread notion that a Gaussian (a.k.a. Normal law or Rayleigh distribution) is a very good approximation for any random phenomena. It is less known that there are not many random processes that follow Normal probability laws precisely. Statistical simulations on a multibeam (140 beams) reflector antenna system revealed distributions that cannot be represented by a Rayleigh distribution [Davenport and Root, 1958], and do not follow the more complex and well accepted Rice distribution [Davenport and Root, 1958]. Particularly perplexing are phase distributions with two separate and very distinguished peaks. Here we elaborate on these distributions.

2. Analysis

[4] It is well known that when we deal with a complex function “g,” which is characterized by real and imaginary constituents (“b” and “c,” respectively), each of which is representable as a sum of a large number of independent random variables, none of which is large enough to dominate over the sum, the properties of the constituents “b” and “c,” are those of a Gaussian distribution. This fact follows from the famed Central Limit Theorem of Probability Theory [Parzen, 1960].

2.1. Standard Theory

[5] Let the quadrature functions “b” and “c” be Gaussian and characterized by standard deviations σb and σc, respectively; and let “g” have a standard deviation σ. It is generally argued [Steinberg, 1976] that the variances must be the same, for otherwise the standard deviation of “g” will be dependent on the choice of coordinates for the quadrature. It then follows that

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Use of this, together with the fact that “b” and “c” are uncorrelated, it can be shown that the joint probability density function (PDF) w is given by

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where as usual 〈X〉 is used to denote expectation value. Once the above quadrature components are converted into polar via b = ∣g∣ cos(θ), and c = ∣g∣ sin(θ), and upon performing the angular integration, and using “A” to denote the random amplitude function of “g,” we obtain the PDF of the amplitude “a,” which is given by [Steinberg, 1976; Davenport and Root, 1958]:

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and I0 is the modified Bessel function of zero order [Abramowitz and Stegun, 1964]. Equation (3), the Rice distribution [Davenport and Root, 1958], is commonplace in random processes, such as in the problem of multiple reception [Lange, 1967]. The PDF of phase can be obtained by similar means. Upon introduction of the angular function B, defined as

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and performing radial integration, it follows that [Davenport and Root, 1958]:

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valid for all B, and where erf(x) stands for the standard error function [Abramowitz and Stegun, 1964].

[6] The PDF corresponding to power can be obtained via use of an identity in Probability Theory [Davenport and Root, 1958, p.309] which relates the density functions of two arbitrary random processes Y and X, for Y = X2:

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Since power p is the square of the amplitude, the above, coupled with (3) results in the PDF of power being given by

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Although the individual quadrature components of “g” are Gaussian, its amplitude, phase and power are not. For the case of Gaussian components, the situation is reminiscent of a Gaussian distribution, which when viewed in the complex plane which defines the quadrature components, exhibits an elliptical cross section (see Figure 1). When observed in polar coordinates, two relative maxima will be found as the radial variable is kept fixed and the azimuth is varied from 0 to 2π. Thus this simplistic view results in a bimodal phase distribution.

Figure 1.

Elliptical Gaussian distribution with quadrature variables parallel to edges of rectangular base. This is a complex plane distribution, where in “polar” coordinates the azimuth corresponds to the phase. Two characteristic peaks in the phase distribution can be envisioned from the figure.

2.2. Evidence of Strange Distributions

[7] In the course of the statistical investigations of the multibeam antenna, we found cases of random signals which do not follow known distribution laws. This is especially true when we deal with low-level signals (signal comparable to noise) such as those at the output of the so-called Matrix power Amplifier (MPA) when the terminal under consideration is not directly excited (cross-coupling). (The MPA we are referring to here is a multiple beam system of the type previously referred to as “hybrid transponder” [Egami and Kawai, 1987], which is similar to a system using a Butler Matrix [Sandrin, 1974].) Especially perplexing was the existence of phase distributions with two separate and very distinguished peaks, which clearly violate intuitive expectations that call for a simple smear about the mean value of the signal. Here we mention two simple examples where numerical experiments were found in excellent agreement with a simple analytical explanation. We shall briefly mention these cases, as space limitations do not allow for a detailed description. (1) The first case was an interferometer type experiment consisting of an ideal 3dB, 90° hybrid, under quadrature signal excitation. For inputs with either Gaussian or homogeneous random phase distributions, the output phase distribution was found to possess two clear peaks. (2) The second case was when the transfer function of a transmission line of uncertain length (described by a random distribution) was found to have quadrature components of very dissimilar statistics.

2.3. Nonstandard Approach

[8] It is not only intuitive that quadrature components of a complex function must have the same variance, but is also based on our experience, as both components are one and the same representation of a time harmonic signal. In real life, however, we do not have two components, and they are essentially a mathematical artifice (such as complex numbers are). We conclude that we in general do not have to force the quadrature components to be statistically identical. This is a departure from traditional analysis. An analysis with full generality follows.

[9] When “b” and “c” are different Gaussian processes, quadrature components of the complex function “g,” the joint PDF acquires the form

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where the function Ψ is given by

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and A0 stands for the magnitude of the expectation value of “g,” as given by (4). In addition, the angle dependent deviation is defined as

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2.4. Phase Distribution Function

[10] We will first concentrate in the calculation of phase. Via integration over the amplitude variable, the above results in a PDF of phase that can be evaluated in closed form to yield

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Two cases are of special interest here: small and large amplitude signals. When the amplitude is much larger than the deviation, β(θ)/σ(θ) is large and (12) reduces to a Rayleigh form:

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provided 2σΨ2 ≪ 1 is satisfied, and where the deviation is given by

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For low level signals (amplitude much smaller than the deviation) the arguments of the exponentials in (12) as well as the error function are essentially zero, and (12) becomes

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Thus, in view of (11), for small amplitudes we expect to see two distinguished peaks π radians apart. Note that in this level of approximation the distribution is independent of θ0. As will be seen shortly, (16) is an accurate description of a bimodal distribution.

[11] The numerical evaluation of (12) and (13) is presented in Figures 2a–2d which correspond to (σb/A0 = 0.2, σc/A0 = 0.1), (σb/A0 = 0.5, σc/A0 = 1), (σb/A0 = 5, σc/A0 = 10), and (σb/A0 = 5, σc/A0 = 45), respectively. The figures show that for small deviations from equal statistics, the shape is essentially Gaussian, with peak amplitude (and therefore width) strongly dependent on θ0. As the deviation values become comparable to A0, the magnitude of the expectation value of “g,” the distribution acquires skewness (i.e., becomes asymmetric), and/or becomes characterized by negative kurtosis (or platykurtic, i.e., flatness relative to normal distribution), depending upon the value of θ0. It can also be seen that the function starts to exhibit a two-peak behavior. For even larger values of deviation, as measured in units of A0, but provided the ratio σcb is not large, the function exhibits two well developed peaks (see Figure 2c for σcb = 2), the values of which are not very sensitive to the values of θ0. Finally, for larger values of deviation, such that their ratio is also large, two very sharp peaks can be noted (see Figure 2d for σcb = 9), with the whole function being essentially independent of θ0.

Figure 2.

PDF of Phase fΘ for quadrature components of different statistics: (a) σb/A0 = 0.2, σc/A0 = 0.1, (b) σb/A0 = 0.5, σc/A0 = 1, (c) σb/A0 = 0.5, σc/A0 = 10, (d) σb/A0 = 5, σc/A0 = 45. In all cases, θ0 ∈ [0, π/4, π/2, 3π/4, π].

[12] Thus, statistically dissimilar quadrature components yield phase distributions that for large signals can be represented as Gaussians. Small amplitudes on the other hand give rise to bimodal distributions. We conclude that statistical processes involving guidance of waves with very significant random content result in quadrature components of different statistical properties.

2.5. Amplitude Distribution Function

[13] Simulations showed that the distributions arising from cross-coupling (i.e., weak amplitudes), even though able to be approximated by the exotic amplitude functions of (3), were indeed of a different nature (more of a right triangle than Gaussian). Calculations showed that the family of radial functions arising out of quadrature components of different statistical properties provides a much better fit to the data of the numerical simulations. The PDF of amplitude is given by

equation image

and due to its complexity has not been identified in terms of special functions. A form alternative to (17) can be obtained via harmonic expansion of all exponential terms in the integrand other than the one involving the cos(θ) term. On use of an integral representation for In(z), the modified Bessel function of order n [Abramowitz and Stegun, 1964, equation (9.6.19)], together with contour manipulation, yields

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This constitutes a representation alternative to (17). For small deviations from the condition of quadrature components of equal statistics, τ → 0, and the only contribution to Ξ is coming from modified Bessel functions of zero order, and (18) reduces to

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this upon comparison with (3) indicates that under these conditions, the distribution is essentially one of equal statistics, with an equivalent variance σeq given by

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Away from the neighborhood of equal statistics, asymptotic evaluations of (17) can be done by standard means; however, the complexity of (17) prevents simple expressions. For instance, even the case of small deviations leads to complex stationary phase points described by transcendental equations. Thus, for practical reasons, we will not elaborate on (17) any further.

[14] The data contained in Figures 3a and 3b correspond to the PDF of amplitude (17), for small standard deviations (σb/A0 = 0.2, σc/A0 = 0.1), and large deviations (σb/A0 = 4, σc/A0 = 20), respectively. For small deviations and as expected from (21), the shapes are essentially of the modified Bessel type (3) but with peak amplitude depending on θ0. For large deviations on the other hand, we obtained shapes that are essentially independent of θ0, to the extent that on a graph curves corresponding to different θ0 are hard to distinguish. These shapes are new, and strongly dependent of the ratio of the deviations of the quadrature components σcb. Figure 3b depicts the case σcb = 5, which corresponds to a shape resembling a right triangle. Such a shape cannot be duplicated by the distributions corresponding to equal quadrature statistics (3).

Figure 3.

Normalized PDF of amplitude fA(a) as a function of a/A0, for quadrature components of different statistics: (a) σb/A0 = 0.2, σc/A0 = 0.1, and θ0 ∈ [0, π/4, π/2]; (b) σb/A0 = 4, σc/A0 = 20, and θ0 = 0.

3. Verification

[15] Here we present distributions calculated via inclusion of random amplitudes and phase errors in subassembly parameters in an 8 × 8 MPA, including the effect of random errors in Input and Out put Networks (with errors at the level of a hybrid, which were made to satisfy energy conservation laws), Matrix Power Amplifiers, Switch Matrices, the Beam Forming Network (BFN), filter/antenna element insertion errors, as well as tolerances in all the associated cables (available from vendors). The statistical model of the feed system is available elsewhere [Monzon, 1996].

[16] Particularly interesting is the cross coupling simulations (the statistical parameters reflect fictitious over life conditions, and are not typical on the multibeam feed array). A data point here refers to a set of randomly generated network parameters resulting in a cross coupling transfer function. Due to the complexity of the system, a typical point involves the generation of thousands of random parameters. The data so obtained is used to generate histograms. Several sample sizes were used with excellent agreement, indicating converging results. Here we show the case of 16,312 points. The histograms corresponding to simulations of cross coupling in the MPA are shown in Figures 4a and 4b, which correspond to amplitude and phase, respectively.

Figure 4.

Histograms from actual numerical simulations of cross coupling in the MPA. (a) Amplitude distributions. (b) Bimodal phase distribution.

[17] The phase distribution is a bimodal distribution, of the type presented in Figure 2d, and can be described accurately by the small signal approximation (16) (peak separation ∼180°). The amplitude curve resembles a right triangle with a particularly flat “hypotenuse” that cannot be duplicated by the distributions corresponding to equal quadrature statistics. This example validates the analytical development, and indicates that more complete statistical models must be employed in dealing with small signal scenarios. Aside from the qualitative verification of the simulated cross coupling data, the need and validity of the assumption of quadrature components of unequal statistics has been demonstrated.

4. Conclusions

[18] Extensive statistical analysis of the feed array of a complex antenna system revealed distribution functions which cannot be represented by either the Rayleigh or Rice models. The existence of these new functions was traced to the violation of the assumption of quadrature components of identical statistics. Here we presented the analytical description these new functions, and validate the concept with statistical data obtained from cross-coupling MPA simulations. We found that statistical processes involving guidance of waves with very significant random content result in general in quadrature components of different statistical properties. This concept may eventually result in a need for reexamination of the theory of random antenna, random array, sidelobes, etc.


[19] The author would like to thank C. Profera, G. Hesselbacher, and D. Nguyen of the Antenna Division, Lockheed Martin, East Windsor, New Jersey, for their interest and support in this study. Thanks are extended to the anonymous reviewers who helped improve the presentation.