Model for a Noise Matched Phased Array Feed

TL;DR

This paper introduces a novel matrix-based model for Noise Matched Phased Array Feeds that accurately predicts system performance and can guide noise optimization for radio telescopes.

Contribution

The model characterizes the PAF system with a single matrix derived from voltage covariances, enabling improved noise matching and performance prediction.

Findings

Model predictions agree well with measurements.

Noise matching can double the bandwidth of optimal performance.

The matrix approach simplifies PAF system characterization.

Abstract

We present a model for a Noise Matched Phased Array Feed (PAF) system and compare model predictions with the measurement results. The PAF system consists of an array feed, a receiver, a beamformer and a parabolic reflector. The novel aspect of our model is the characterization of the {\em PAF system} by a single matrix. This characteristic matrix is constructed from the open-circuit voltage covariance at the output of the PAF due to signal from the observing source, ground spillover noise, sky background noise and (low-noise) amplifier (LNA) noise. The best signal-to-noise ratio on the source achievable with the PAF system will be the maximum eigenvalue of the characteristic matrix. The voltage covariance due to signal and spillover noise are derived by applying the Lorentz reciprocity theorem. The receiver noise covariance and noise temperature are obtained in terms of Lange invariants such that they are suitable for noise matching the array feed. The model predictions are compared with the measured performance of a 1.4 GHz, 19-element, dual-polarized PAF on the Robert C. Byrd Green Bank Telescope. We show that the model predictions, obtained with an additional noise contribution due to the measured losses ahead of the low-noise amplifier, compare well with the measured ratio of system temperature to aperture efficiency as a function of frequency and as a function of offset from the boresight. Further, our modeling indicates that the bandwidth over which this ratio is optimum can be improved by a factor of at least two by noise matching the PAF with the LNA.

Equations

$$v_{out}=\sum _{i=1,M}{w}_i{v}_i={\mathbf{w}}^T\tilde{\mathbf{V}}$$

$$\vec{\mathcal{E}}(\vec{r})=\sum _{i=1,M}{\mathcal{J}}_{0_i}{\vec{\psi }}_i^e(\vec{r})={\mathbf{I}}_0^T{\vec{\psi }}^{\mathbf{e}},$$

$$\text{SNR}=\frac{{\mathbf{w}}_1^H{\mathbf{R}}_{signal}{\mathbf{w}}_1}{{\mathbf{w}}_1^H\mathbf{N}{\mathbf{w}}_1}=\frac{{\mathbf{w}}_2^H\mathbf{M}{\mathbf{w}}_2}{{\mathbf{w}}_2^H{\mathbf{w}}_2}$$

$$\mathbf{N}=\mathcal{N}\mathcal{N}\equiv {\mathbf{R}}_{spill}+{\mathbf{R}}_{rec}+{\mathbf{R}}_{sky}$$

$$\mathbf{M}\equiv {\mathcal{N}}^{-1}{\mathbf{R}}_{signal}{\mathcal{N}}^{-1},$$

$${\mathbf{V}}_{oc}={\int }_{A_{free}}\left({{\vec{\psi }}^{\mathbf{e}}}^T\times \mathcal{I}\overrightarrow{{\mathcal{H}}_r}-\mathcal{I}\overrightarrow{{\mathcal{E}}_r}\times {\vec{\mathcal{J}}}^{\mathbf{e}}\right)\cdot \hat{n}\text{d}A.$$

$${\mathbf{R}}_{signal}=\frac{2S_{source}}{Z_0}{\mathbf{C}}_{I\psi }$$

$${\mathbf{C}}_{I\psi }\equiv \left({\int }_{A_{pap}}{\vec{\psi }}_{\mathbf{pap}}^{\mathbf{e}}\text{d}A\right)\cdot {\left({\int }_{A_{pap}}{\vec{\psi }}_{\mathbf{pap}}^{\mathbf{e}}\text{d}A\right)}^H.$$

$$T_A$$

$${\mathbf{C}}_{C\psi }\equiv {\int }_{4\pi }{\vec{\mathbf{\Psi }}}^{\mathbf{e}}\cdot {{\vec{\mathbf{\Psi }}}^{\mathbf{e}}}^H\text{d}\Omega ,$$

$${\eta }_{ap}\equiv \frac{{\mathbf{w}}_1^H{\mathbf{C}}_{I\psi }{\mathbf{w}}_1}{A_{ap}{\mathbf{w}}_1^H{\mathbf{C}}_{C\psi }{\mathbf{w}}_1}.$$

$${\mathbf{R}}_{spill}=\frac{4k_B{T}_g}{Z_0}{\mathbf{C}}_{c\psi 1},$$

$${\mathbf{C}}_{c\psi 1}\equiv {\int }_{{\Omega }_{spill}}{\vec{\mathbf{\Psi }}}^{\mathbf{e}}\cdot {{\vec{\mathbf{\Psi }}}^{\mathbf{e}}}^H\text{d}\Omega ,$$

$$T_{spill}=T_g(1-{\eta }_{spill})$$

$${\eta }_{spill}\equiv 1-\frac{{\mathbf{w}}_1^H{\mathbf{C}}_{C\psi 1}{\mathbf{w}}_1}{{\mathbf{w}}_1^H{\mathbf{C}}_{C\psi }{\mathbf{w}}_1}.$$

$${\mathbf{R}}_{rec}$$

$$\bm{R}_{rec}=4k_{B}T_{0}\;\Big{(}R_{n}\boldsymbol{\mathcal{I}}+\\ \sqrt{R_{n}g_{n}}\;\big{(}\rho\bm{Z}+\rho^{*}\bm{Z}^{H}\big{)}+g_{n}\bm{Z}\bm{Z}^{H}\Big{)}.$$

$$\bm{R}_{rec}=4k_{B}T_{min}\;\frac{\big{(}\bm{Z}+\bm{Z}^{H}\big{)}}{2}\\ +4k_{B}T_{0}N\;\frac{\big{(}\bm{Z}-Z_{LNAopt}\boldsymbol{\mathcal{I}}\big{)}\;\big{(}\bm{Z}-Z_{LNAopt}\boldsymbol{\mathcal{I}}\big{)}^{H}}{{\rm Re}\{Z_{LNAopt}\}}.$$

$$T_{rec}=T_{min}+N{T}_0\frac{{\mathbf{w}}_1^H(\mathbf{Z}-Z_{opt}\mathcal{I})(\mathbf{Z}-Z_{opt}\mathcal{I}{)}^H{\mathbf{w}}_1}{\text{mbox}Re\{Z_{opt}\}\frac{1}{2}{\mathbf{w}}_1^H(\mathbf{Z}+{\mathbf{Z}}^H){\mathbf{w}}_1}.$$

$$\mathbf{R}\approx 2k_{B}T(\mathbf{Z}+{\mathbf{Z}}^H).$$

$$T_{rec}=T_{0}\;\frac{\parbox{158.99377pt}{$\bm{w}_{1}^{H}\Big{(}R_{n}\boldsymbol{\mathcal{I}}+\sqrt{R_{n}g_{n}}\;\big{(}\rho\bm{Z}+\rho^{*}\bm{Z}^{H}\big{)}$ \\ \hskip 90.3375pt $+g_{n}\bm{Z}\bm{Z}^{H}\Big{)}\;\bm{w}_{1}$}}{\frac{1}{2}\bm{w}_{1}^{H}(\bm{Z}+\bm{Z}^{H})\;\bm{w}_{1}}.$$

$$\displaystyle\frac{1}{2}\;\bm{w}_{1}^{H}\big{(}\bm{Z}+\bm{Z}^{H}\big{)}\bm{w}_{1}\;\frac{T_{rec}}{T_{0}}$$

$$=$$

Full text

I Introduction

The extreme faintness of celestial radio sources motivates the use of telescopes with large collecting areas. The field-of-view (FoV) of a single-dish telescope that uses a single feed to receive signals decreases in proportion to the collecting area. Traditionally, array feeds consisting of multiple horn antennas have been used to increase the FoV of such telescopes. The antennas in such arrays cannot be closely packed since the feeds are physically large as they have to illuminate the reflector efficiently. Consequently, the telescope beams corresponding to the different elements are separated by considerably more than the half-power beamwidth (HPBW). The non-overlapping beams reduce the mapping efficiency. Feeds consisting of dense, electrically small antenna arrays as a means to overcome this limitation are now of significant interest [1, 2, 3, 4, 5, 6, 7, 8]. These dense arrays sample the focal field pattern of the telescope. Multiple beams are then formed by combining the signals sampled by the array elements with complex weights that form an efficient reflector illumination. Such arrays are referred to as Phased Array Feeds (PAF). The beams formed using a PAF can fully sample the FoV. Additionally, a PAF can be used to improve spillover efficiency as well as the illumination of the dish. However, mutual coupling between array elements is a major hurdle in designing a low-noise PAF (see, for example, [9]). Mutual coupling modifies the element radiation patterns. It also couples amplifier noise between signal paths. Therefore, detailed electromagnetic, noise and network modeling are needed to design a PAF for radio astronomy applications [10].

Several research groups have analyzed and modeled the noise performance [11, 12, 13, 14, 15, 16] and electromagnetic properties of a PAF on a reflector antenna [17, 18]. A working expression for the covariance matrix of the signal due to a source at the output of the PAF was provided by Warnick & Jeffs (2008) [19] and corresponding matrix for spillover noise was derived by Warnick & Jeffs (2006) [20] (see also [21]). These matrices were then used to provide expression for aperture and spillover efficiencies[20, 19]. The receiver noise covariance matrix for a PAF was derived by Warnick & Jensen (2005) [11] (see also [12]). Hay (2010) [18] (see also [14]) analyzed the noise performance of a PAF with a lossless input matching network in terms of Lange invariants. He also provided optimum solutions for the matching network for maximum signal-to-noise ratio (SNR) and total efficiencies (see [15] and references therein for other optimization). Finally, simplification of the PAF receiver noise model has been discussed by Ivashina et al. (2008) [22] to better understand the factors affecting system sensitivity.

In this paper, we apply the Lorentz reciprocity theorem to a PAF mounted at the prime focus of a reflector telescope to derive a new expression for the signal covariance matrix. We also provide a new expression for the receiver noise covariance matrix and noise temperature in terms of the Lange invariants, which are advantageous for numerically optimizing the system performance with a lossless matching network (see also [23]). A novel aspect of our modeling is the introduction of a single matrix, referred to as the characteristic matrix of the PAF system, constructed from these covariance matrices to express the SNR at the output of the PAF. The PAF system is described in Section II. A summary of the novel aspects of our PAF model and its implementation are presented in Section III. We then compare the model predictions with measurements made with a 1.4 GHz, 19-element dual-polarized PAF on the Green Bank Telescope (GBT) [24]. The results of the comparative study are given in Section IV. The main results of the paper are summarized in the Section V. A list of mathematical symbols used in the paper is given in Appendix B.

II The PAF system

The PAF system consists of a dual-polarized dipole array followed by noise matching networks, LNAs and receiver system, a beamformer and a parabolic reflector or telescope (see Fig. 1). In Fig. 1a, we show an off-axis reflector, representing the surface of the GBT. The projected surface of the reflector in the boresight direction (i.e perpendicular to the $x_{pp}-y_{pp}$ plane in Fig. 1a) is circular with a diameter 100 m and the focal length over diameter ratio is 0.6. Further details of the GBT geometry are available in [25, 26]. The PAF is mounted at the prime focus with the plane of the array at an angle 48.46∘ from the $x_{pp}-y_{pp}$ plane (see Fig. 1a).

The dipole array consists of 19 dual-polarized elements (see Fig 2). The dipole shape was optimized for active impedance match to the LNA and for maximum sensitivity on the GBT over a FoV of $\sim$ 20 arcminutes [17, 15]. The frequency range used for optimization was $\geq$ 150 MHz centered at 1350 MHz. A balun converts the signals received by the dipoles to unbalanced signals, which are available at the output of a transmission line with characteristic impedance $z_{0}$, which is same as the 50 $\Omega$ reference impedance. The transmission line terminates at the ‘reference plane’ marked on Fig 1b (see also Fig. 2). The dipole array, balun, ground plane and the transmission line together is referred to as the PAF. Further details of the PAF is available in [24].

The PAF is followed by a two-port, lossless matching network, which in the simplest case can be a transmission line of length $L_{trans}$. The transmission line in the measurement system is the thermal transition (see Fig. 2) with $L_{trans}\sim$ 6.4 cm and characteristic impedance of 50 $\Omega$. The thermal transition is made of air-core stainless steel co-axial transmission line with the central stainless steel conductor coated with copper and gold of 5$\mu m$ thickness. As discussed in Section IV, we have used this transmission line as the matching network in the PAF model and have varied $L_{trans}$ for all dipoles between 3 and 20 cm for noise matching the PAF to the LNA. The LNAs are located at the end of the matching network. The LNAs used for the measurements were cryogenic Silicon Germanium Heterojunction Bipolar Transistor (SiGe HBT) amplifiers cooled to about 15 K [27, 24]. The $i^{th}$ LNA has an input impedance of $Z_{LNA_{i}}\sim 50\;\Omega$ and gain $\sim 38$ dB. The input impedance of the receiver system is transformed by the matching network to $Z_{in_{i}}$ at the reference plane. The noise properties of the LNA are usually specified by the noise parameters $R_{n_{i}}$, the noise resistance, $g_{n_{i}}$, the noise conductance, and $\rho_{i}$, the complex noise correlation [28] (see also Fig. 1b). An equivalent set of noise parameters used in radio astronomy are $Z_{LNAopt_{i}}$, the optimum impedance and the Lange invariants, $T_{min_{i}}$ (the minimum noise temperature) and $N_{i}$ [29, 28]. The relationship between the two sets of noise parameters is discussed in [28]. The latter set of noise parameters is advantageous when lossless networks are used for noise matching. This is because the noise parameters $T_{min_{i}}$ and $N_{i}$ remain invariant when they are transformed through a lossless network placed at the input of the LNA and $Z_{LNAopt_{i}}$ transforms to $Z_{opt_{i}}$ like a normal impedance transformation. Fig. 3 shows the noise parameters vs frequency obtained from the noise model of the LNA. The predicted temperature from the noise model of the LNA is consistent with measurements in the frequency range 1.2 to 1.7 GHz (see Fig. 3)[27]. However, the model noise temperature deviates from measurement outside this frequency range. We are in the process of improving the noise model of the LNA and the results will be presented elsewhere.

In the measurement system, the signals from the output of the LNAs are further amplified, down-converted to basebands, digitized using analog-to-digital converters (ADC) and then followed by Fast Fourier Transform (FFT) engines to create time series of complex voltage spectra (see [24] for a full description of the system). The basic signal processing done in a PAF system, referred to as beamforming, can be expressed as [30]

[TABLE]

where the elements of the voltage vector $\boldsymbol{\tilde{V}}$ are the input complex voltages $v_{i}$ from a spectral channel to the beamformer, the elements of the weight vector $\bm{w}^{T}$ are the complex beamforming weights $w_{i}$ and $M=19\times 2$ is equal to the number of elements in the PAF. Multiple beams are formed by applying different sets of beamforming weights.

For modeling the PAF, its network property is specified by the (spectral) impedance matrix $\bm{Z}$ and its electromagnetic property is specified by the embedded beam patterns [31] (see also [32, 33]). In this paper, we define the $i^{th}$ embedded beam pattern, $\vec{\mathcal{\psi}}^{e}_{i}(\vec{r})$, as the radiation pattern of the PAF when $i^{th}$ port is excited with 1 A and all other ports are open circuited. The position vector $\vec{r}$ is defined in the $x_{p}-y_{p}-z_{p}$ coordinate system (see Fig. 1a). There will be $M=19\times 2$ embedded beam patterns for the dual polarized array. The beam pattern at the far-field can be expressed as an out-going spherical wave; i.e. $\vec{\mathcal{\psi}}^{e}_{i}(\vec{r})=\vec{\Psi}^{e}_{i}(\theta,\phi)\;\frac{e^{j\vec{k}.\vec{r}}}{r}$, where $\vec{k}$ is the propagation vector and $r=|\vec{r}|$. The radiation pattern of the PAF when excited by a set of arbitrary port currents $\mathcal{J}_{0_{i}}$ is

[TABLE]

where $\bm{I}_{0}$ is the vector of port currents and $\boldsymbol{\vec{\mathcal{\psi}}^{e}}$ is the vector of embedded beam patterns. The $\theta,\phi$ dependence of the far-field beam pattern can be written in a similar fashion: $\vec{E}(\theta,\phi)=\bm{I}_{0}^{T}\boldsymbol{\vec{\Psi}^{e}},$ where $\boldsymbol{\vec{\Psi}^{e}}$ is a vector with elements $\vec{\Psi}^{e}_{i}$. The unit of $\vec{\mathcal{E}}$ is V/m, that of $\vec{\mathcal{\psi}}^{e}_{i}$ is V/A/m, that of $\vec{E}$ is V and that of $\vec{\Psi}^{e}_{i}$ is V/A.

The voltages, currents and field amplitudes are considered harmonic quantities and their values are specified as peak values ($e^{j\omega t}$ term is omitted for simplicity) for the derivation of the open circuit voltage (Eq. 6) by applying Lorentz reciprocity theorem. When applying this result to the PAF system, we need to consider radiation field from the astronomical and thermal sources. These fields and the induced voltages at the output of the PAF as well as the noise due to the LNA are stochastic quantities. We treat these signals in the quasi monochromatic approximation and their amplitudes are taken as RMS (root mean square) values.

In this treatement, for example, the flux density of a stochastic radiation field $\vec{E}_{inc}$ is $S=\frac{1}{Z_{0}}\langle\vec{E}_{inc}\cdot\vec{E}_{inc}^{*}\rangle$ and will have units W/m2/Hz.

The open circuit voltage covariance $\langle\bm{V}_{oc}\bm{V}_{oc}^{H}\rangle$, where $\bm{V}_{oc}$ is the induced voltage due to stochastic field or noise, is average power dissipated by the noise in unit conductance and unit bandwidth or W/Hz/$\Omega$.

III PAF model

One of the novel aspects of our PAF model is the introduction of a new matrix, referred to as the characteristic matrix, to concisely represent the PAF system. Further, we apply the Lorentz reciprocity theorem [34] to derive a new expression for the signal covariance matrix. The receiver covariance matrix is obtained in terms of the Lange invariants so that it is suitable for noise matching the array feed to the LNA. We summarize below these new aspects of the PAF theory and also briefly describe its implementation.

III-A PAF Theory Revisited

The question posed to develop our model for a lossless PAF is: what is the maximum signal-to-noise ratio that can be obtained with a PAF system when observing a compact source (point source) at some angle $\theta_{s}$, $\phi_{s}$ from the boresight direction ? The answer to this question is given by the following theorem:

Theorem

Given (a) the (spectral) impedance matrix, $\bm{Z}$, and the embedded beam patterns, $\boldsymbol{\vec{\mathcal{\psi}}^{e}}$, of the PAF, (b) the LNA noise parameters, $T_{min_{i}}$, $N_{i}$ and the transformed impedance, $Z_{opt_{i}}$ and $Z_{in_{i}}$, (c) the telescope geometry and ground temperature, (d) the flux density and direction $\theta_{s}$, $\phi_{s}$ of the observing source and (e) the off-source sky temperature, one can construct a characteristics matrix $\bm{M}$ for the PAF system. Then the best signal-to-noise ratio on the source is the maximum eigenvalue, $e_{max}$, of $\bm{M}$.

III-A1 Characteristic Matrix and Proof of the Theorem

The SNR when observing a source with the PAF system is defined as the ratio of increase in power spectral density at the output of a beam due to the source relative to the off-source spectral density (see Eq. 6 [19, 30]). This SNR at the beamformer output can be expressed in terms of the covariance of the open circuit voltages at the reference plane shown in Fig. 1 as

[TABLE]

where $\bm{R}_{signal}$ is the open circuit covariance matrix due to signal from the source,

[TABLE]

is the sum of the open circuit noise covariance matrices due to spillover,$\bm{R}_{spill}$, receiver $\bm{R}_{rec}$, and sky background radiation, $\bm{R}_{sky}$,

[TABLE]

is defined as the characteristics matrix. The relationship between the weight vectors are $\bm{w}_{1}=(\bm{Z}_{in}(\bm{Z}+\bm{Z}_{in})^{-1})^{H}\bm{G}^{H}\bm{w}$ and $\bm{w}_{2}=\boldsymbol{\mathcal{N}}\bm{w}_{1}$, $\bm{Z}_{in}$ is a diagonal matrix of elements $Z_{in_{i}}$ and $\bm{G}$ is a diagonal matrix with elements as the overall system gain. It follows from Eq. 3 that the maximum SNR is given by the maximum eigenvalue, $e_{max}$, of $\bm{M}$. The weight vector that needs to be applied in the beamformer can be obtained from the eigenvector corresponding to $e_{max}$.

III-A2 Open circuit voltage covariance

Application of the Lorentz reciprocity theorem provides an expression for the open circuit voltage $\bm{V}_{oc}$ at the output of the PAF as [32, 33]

[TABLE]

Here $\boldsymbol{\vec{\mathcal{J}}^{e}}$ is the vector of magnetic field patterns corresponding to the embedded beam pattern $\boldsymbol{\vec{\mathcal{\psi}}^{e}}$, $\vec{\mathcal{E}_{r}}$ and $\vec{\mathcal{H}_{r}}$ are the incident electric and magnetic fields respectively, $\boldsymbol{\mathcal{I}}$ is the identity matrix and $\hat{n}$ is the unit normal inward to the elementary area $\textrm{d}A$. The surface of integration $A_{free}$ is part of a closed surface outside the PAF.

III-A3 $\bm{R}_{signal}$

We consider a compact astronomical source in the direction $\theta_{s}$, $\phi_{s}$ from the boresight. For simplicity we assume that the source is unpolarized, which implies the following relationship between the flux density, $S_{source}$, and electric field of the incident plane wave, $\vec{E}_{inc}$, due to the source: $\frac{S_{source}}{2}=\frac{1}{Z_{0}}\langle E_{inc,x}E_{inc,x}^{*}\rangle=\frac{1}{Z_{0}}\langle E_{inc,y}E_{inc,y}^{*}\rangle$, where $E_{inc,x}$ and $E_{inc,y}$ are the two linearly polarized components of $\vec{E}_{inc}$ and $Z_{0}$ is the free space impedance. An appropriate closed surface, with the projected aperture plane (i.e. the telescope aperture plane projected onto a plane perpendicular to the direction of the source) as part of this surface can be considered for the evaluation of the integral in Eq. 6. The embedded beam patterns are propagated to the projected aperture plane for the evaluation of the integral (see Section III-B). The major contribution to the integral over $A_{free}$ comes from the projected aperture plane [32]. The open circuit voltage covariance due the source can be obtained as

[TABLE]

where

[TABLE]

Here $\boldsymbol{\vec{\mathcal{\psi}}^{e}_{pap}}$ is the vector of the propagated embedded beam pattern on the projected aperture plane $A_{pap}$. Note that $\bm{C}_{I\psi}$ is a function of source position $\theta_{s},\phi_{s}$. The unit of $\bm{R}_{signal}$ is power spectral density per unit conductance.

We also provide an expression for the antenna temperature, $T_{A}$, which is a generalization of the equation for $T_{A}$ for a reflector antenna with a single feed. The power spectral density due to the source, which is proportional to $\bm{w}_{1}^{H}\bm{R}_{signal}\bm{w}_{1}$, can be expressed as a physical temperature by comparing it with the power spectral density when the PAF system is at thermal equilibrium with the reference temperature $T_{0}$ [35] (see also [21]):

[TABLE]

where

[TABLE]

and the aperture efficiency, $\eta_{ap}$, is defined in terms of the overlap integrals of the embedded beam patterns and the physical area of the telescope aperture, $A_{ap}$, as (see also [19])

[TABLE]

III-A4 $\bm{R}_{spill}$

An expression for the covariance matrix of the voltage due to ground spillover radiation seen by the PAF is obtained in [19, their Eq. 9]. Applying Eq. 6 and following the arguments in [19] $\bm{R}_{spill}$ can be readily expressed in terms of ground temperature, $T_{g}$ (= 300 K), and the overlap integral of the embedded beam patterns as [32, 33]

[TABLE]

where

[TABLE]

$\Omega_{spill}$ is the solid angle over which the PAF embedded beam patterns are receiving radiation from ground. Eq. 12 differs by a factor of 2 compared to Eq. 9 in [19] because $\bm{R}_{spill}$ is the power spectral density per unit conductance (see Section II) and [19] provides the peak voltage covariance.

The spillover temperature, $T_{spill}$, as in the case of antenna temperature discussed above, is a generalization of the equation for $T_{spill}$ for a reflector antenna with a single feed [32, 33]:

[TABLE]

where

[TABLE]

In [19], Twiss’s result [36] has been re-written in terms of the peak voltage covariance and so Eq. 15 and Eq. 16 in [19] give the same result.

III-A5 $\bm{R}_{rec}$

The open circuit voltage covariance due to the LNA noise in the absence of a matching network is given by [15]

[TABLE]

Here $\langle\bm{E}_{n}\bm{E}_{n}^{H}\rangle$, $\langle\bm{I}_{n}\bm{I}_{n}^{H}\rangle$ and $\langle\bm{I}_{n}\bm{E}_{n}^{H}\rangle$ are diagonal matrices of noise fluctuations $\langle e_{n,i}^{2}\rangle$, $\langle i_{n,i}^{2}\rangle$ and their correlations $\rho_{i}$ (see Fig. 1b). For identical LNAs connected to the PAF, i.e. $R_{n_{i}}=R_{n}$, $g_{n_{i}}=g_{n}$, $\rho_{i}=\rho$, $T_{min_{i}}=T_{min}$, $N_{i}=N$ and $Z_{LNAopt_{i}}=Z_{LNAopt}$ for all $i$, we get

[TABLE]

We re-write $\bm{R}_{rec}$ in terms of the Lange invariants, and the optimum impedance (see Appendix A),

[TABLE]

The unit of $\bm{R}_{rec}$ is power spectral density per unit conductance. When a matching network is introduced, $\bm{R}_{rec}$ at the reference plane can be obtained by replacing $Z_{LNAopt}$ with the transformed impedance $Z_{opt}$ in Eq. 18.

The receiver temperature, $T_{rec}$, at the reference plane can be obtained from the power spectral density, which is proportional to $\bm{w}_{1}^{H}\bm{R}_{rec}\bm{w}_{1}$, as (see also [21])

[TABLE]

Eq. 19 is a generalization of the expression for $T_{rec}$ versus source impedance for a single antenna connected to a receiver (see Eq. 11 in [28]).

III-A6 $\bm{R}_{sky}$ and Losses in the PAF system

The sky background radiation has components due to cosmic microwave background (CMB), galactic and extragalactic radiation. We also include atmospheric radiation along with the sky contribution. When comparing the model results with measurement, it is also required to account for the losses in the PAF system ahead of the LNA. These losses are not included in the PAF theory and characterizing them is an ongoing research. In our model, the open circuit voltage covariance matrices due to these components are taken as [36]

[TABLE]

For $\bm{R}=\bm{R}_{sky}$, the covariance matrix due to sky radiation, $T=T_{cmb}+T_{atmo}+T_{bg,\nu_{0}}\left(\frac{\nu}{\nu_{0}}\right)^{-2.7}$, where $T_{cmb}=2.7$ K is the cosmic microwave background temperature, $T_{atmo}=2$ K [37] is the atmospheric radiation temperature (assumed to be frequency independent near 1.4 GHz) and $T_{bg,\nu_{0}}=0.7$ K is the galactic background radiation temperature at $\nu_{0}=1.42$ GHz [38] and $\nu$ is the frequency at which $\bm{R}_{sky}$ is computed. For $\bm{R}=\bm{R}_{loss}$, the covariance matrix due to losses, $T=T_{loss}$, which is the physical temperature corresponding to the noise due to losses in the PAF system ahead of the LNA (see Section III-C). When comparing the model results with the measurements we have added $\bm{R}_{loss}$ to $\bm{N}$ in Eq. 4.

III-B The embedded beam pattern and the Impedance matrix

For the implementation of the theory presented above, the electromagnetic and network properties of the PAF were obtained using CST microwave studio (CST MWS222 https://www.cst.com/products/cstmws). A 3D model of a dual polarized element and balun were created in CST MWS by importing the Autodesk Inventor333 https://en.wikipedia.org/wiki/Autodesk_Inventor CAD model of the dipole assembly and the array was created by placing copies of them. A rectangular ground plane of dimension 114.3 cm $\times$ 95.3 cm was added at a distance of 5.8 cm below the dipole elements. The full-wave 3D electromagnetic solver (transient solver in CST MWS) was used to get radiation patterns and scattering matrices of the PAF with hexahedral mesh size of $\lambda/20$. The solver would excite ports one at a time while all the other ports were terminated with the port impedance of 50 $\Omega$ to obtain the radiation pattern and scattering parameter. The final outputs of the CST simulation were these radiation patterns and scattering matrices for the different frequencies. The coordinate system that was used to obtain the radiation patterns corresponded to the $x_{p}-y_{p}-z_{p}$ system defined in Section II (see Fig. 1a). A MATLAB program was developed to compute the embedded beam patterns, as defined in Section II, from the CST outputs. The impedance matrix from the scattering parameter was also obtained using this program.

The embedded beam patterns need to be propagated to the projected aperture plane to compute $\bm{R}_{signal}$ (see Eq. 7). We used geometric optics approximation to obtain the fields in the aperture plane [39]. The reflected field due to each incident embedded beam pattern on the telescope surface is first computed after accounting for the spherical spreading loss and then propagated to the aperture plane. Note that the geometric phases due to the location of feed elements (or in other words the excitation current distribution) relative to the coordinate origin were included in the computed embedded beam patterns (see [32] for further details).

III-C The losses ahead of the LNA

The receiver temperature of the PAF was measured in the outdoor test facility at the Green Bank Observatory (GBO) [24]. The hot/cold load method was used to measure the receiver temperature [21, 40, 41]. A warm absorber, placed close to the dipole array, formed the hot load. The array was pointed vertically to observe a cold sky region, which formed the cold load. The effect of ground scattered radiation was mitigated to some extent by placing a metallic cone around the array during measurement. These measurements provide the receiver noise covariance matrices for different frequencies which are then converted to temperature covariance matrices using the hot and cold load measurements [24]. The off-diagonal terms in the receiver temperature matrix have contributions from the mutual coupling in the array and the residual ground scattered radiation. These correlations can be canceled out to a large extent by taking the minimum eigenvalue of the temperature covariance matrix. This minimum eigenvalue we refer to as the minimum receiver temperature, $T_{rec,min}$. The measured $T_{rec,min}$ as a function of frequency is shown in Fig. 8. The minimum difference of $\sim$ 2.5 K between $T_{rec,min}$ and the measured amplifier noise temperature provides an upper limit to the losses ahead of the LNA [24]. The contributions to the uncertainty in the measured upper limit include (a) thermal uncertainty of the measurement ($\sim$ 0.3 K), (b) uncertainty in the LNA noise measurement ($\sim$ 0.3 K) and (c) uncertainty in the measured receiver temperature due to error in the hot and cold load temperatures ($\sim$ 1 K). Thus the net 1$\sigma$ uncertainty of this upper limit is $\sim$ 1.1 K.

III-D A computational model

A MATLAB444https://www.mathworks.com/ program was developed to apply the theory presented in Section III-A to the PAF system described in Section II. The program starts with the computation of $\bm{R}_{rec}$, $\bm{R}_{sky}$, $\bm{R}_{spill}$ and $\bm{R}_{signal}$. These matrices are computed for a set of frequencies where impedance matrices and embedded beam patterns are available. $\bm{R}_{signal}$ is computed for a set of directions to the source within the desired FoV. The characteristic matrix $\bm{M}$ is then computed after adding $\bm{R}_{loss}$ to the noise matrix. The maximum SNR and the corresponding weight vectors are obtained from $\bm{M}$ for the set of frequencies and positions. The measured performance of the PAF system is usually expressed in $T_{sys}/\eta$, where $T_{sys}$ is the system temperature and $\eta$ is approximately the aperture efficiency $\eta_{ap}$ of the telescope (assuming that the radiation efficiency of the PAF is close to unity). The model maximum SNR is converted to $T_{sys}/\eta$ using the flux density of the source and the physical area of the telescope aperture [24]. The flux density models (typical uncertainty in flux density $<$ 5%) of the observed sources are taken from Perley and Butler (2017) [42].

The model also provides the full polarization field patterns in the aperture plane and the far-field pattern of the telescope. These are provided for a specified set of weights. Further, the model computes the antenna temperature, aperture efficiency, spillover temperature and spillover efficiency for the specified weights.

IV Comparison with measurement

The performance of a cryogenic 19-element dual-polarized PAF was measured on the GBT in March 2017 [24]. The observations that were made can be categorized into two groups. (1) Measurement of the performance of the boresight beam as a function of frequency. For this measurement, on-source and off-source voltage covariance on a set of calibrators were obtained. The covariances were measured for a set of frequencies in the range 1200 to 1500 MHz, each averaged over a bandwidth of $\sim$ 300 KHz. (2) Measurement of the system performance over the FoV. Voltage covariances on a grid of positions centered on a strong calibrator source Virgo A were obtained for these measurements at 1336 MHz.

The SNR on a source was obtained from the on-source and off-source measurements for both classes of observations. The beamformer weights were obtained by maximizing the SNR. The maximum SNR is expressed as the ratio of system temperature over efficiency, $T_{sys}/\eta$. The current measurements, however, do not provide separate values for $T_{sys}$ and aperture efficiency.

The Y-polarization data set were affected by two faulty signal paths and telescope pointing offset. For comparison here, only a subset of Y-polarization data that are not severely affected by these problems are used (see [24] for further details).

IV-A Boresight $\frac{T_{sys}}{\eta}$

The PAF model $T_{sys}/\eta$ vs frequency for the boresight direction are obtained for different lengths, $L_{trans}$, for the thermal transition, which is used as the noise matching network in the model. These results are obtained with an additional contribution to noise matrix corresponding to $T_{loss}$ = 0, 1, and 2 K to account for the losses ahead of the LNA (see Section III-C). The model results for $L_{trans}$ = 9.1 cm and for the three values of $T_{loss}$ are shown in Fig. 4. The measured X polarizations $T_{sys}/\eta$ vs frequency for the boresight taken from Roshi et al. (2018) [24, their Fig. 14] are included in Fig. 4 for comparison. As seen from Fig. 4, a reasonable fit to the data points is obtained with $T_{loss}$ = 1 K and $L_{trans}$ = 9.1 cm. The physical length of the thermal transition in the PAF is about 6.4 cm. We attribute the discrepancy between the two physical lengths to unaccounted electrical length in the signal path.

The model predictions for different $L_{trans}$ values for the boresight $T_{sys}/\eta$ vs frequency for the X and Y polarizations are shown in Figs. 5a and 5b, respectively. The model results are plotted for 3 representative values for $L_{trans}=$ 6.4, 9.1 & 17.1 cm and $T_{loss}$ = 1 K. The measured values are also shown in these figures for comparison. As seen from Fig. 5a and 5b, the bandwidth performance of the existing system can be increased to $\geq$ 300 MHz by noise matching the PAF to the LNA.

The model aperture efficiency, spillover efficiency and receiver temperature as a function of frequencies are shown in Fig. 5c, 5d and 6a. The aperture efficiency obtained is $\sim$ 60% for the three models near 1350 MHz, while the spillover efficiency is $\sim$ 98.7%. These values are consistent with the preliminary model results presented in Roshi et al. (2018) [24]. The beamformed receiver temperature is $\sim$ 5.5 K for the new amplifier model and for $L_{trans}=9.1$ cm at 1350 MHz. This receiver temperature is about 2 K smaller than the earlier result (see [24]).

The reported aperture efficiency of room temperature PAFs on parabolic reflectors are $>$ 70% [43, 44]. The weights for beamforming were obtained by maximizing the SNR for both room temperature PAF observations and our observations. In such maximization, the achieved aperture and spillover efficiencies are function of receiver temperature (see also [30]). Generally the aperture efficiency (non-linearly) increases with increase in receiver temperature. Thus the lower aperture efficiency of $\sim$ 60% obtained for our system is a result of lower receiver temperature for the cryogenic PAF. The aperture efficiency of the cryogenic PAF is comparable to a corrugated horn with an edge taper of $\sim$ 22 dB placed at the prime focus of the GBT (S. Srikanth, NRAO, private communication). The spillover efficiency of the PAF, on the other hand, is a factor of $\sim$ 1.8 times better than that achieved with the same corrugated horn.

IV-B $\frac{T_{sys}}{\eta}$* over the FoV*

The radial distributions (i.e. offset from the boresight) of the measured $T_{sys}/\eta$ for X and Y polarizations along with model results are shown in Fig. 7a and 7b respectively. The measurements are made at 1336 MHz. The model results for $L_{trans}=9.1$ cm and $T_{loss}$ = 1 K are shown for frequencies 1.2, 1.3 and 1.5 GHz. As seen from Fig. 7a and 7b, the model results for 1.3 GHz compare well with measurements. The model aperture efficiency, spillover efficiency and receiver temperature as a function of offset from boresight are shown in Fig. 7c, 7d and 6b respectively. The performance of the system degrades beyond $\sim$ 5′ because the Airy pattern shifts to the edge of the array and hence there are not enough elements to form a high sensitivity beam (see [24] for further details). Thus the field of view of the PAF is limited by the size of the dipole array.

IV-C Receiver temperature

We compare the model receiver temperature with measurements made at the outdoor test facility at the GBO. As mentioned in Section III-C, these measurements are affected by ground scattered radiation. The value of the receiver temperature depends on the weights used to obtain them (see Eq. 19 and also [24]). The receiver temperature that contributes to the system temperature when the PAF is on the telescope can be obtained from the knowledge of the beamformer weights and complex system gain. However, determining the on-telescope receiver temperature is currently not possible since the measurement system has different system gain compared to that on the telescope. Therefore, here we compare model results with the minimum receiver temperature, $T_{rec,min}$ (see Section III-C). The reference plane and the length of the thermal transition used for these comparisons are, respectively, at the input of the thermal transition (see Fig. 1b) and $L_{trans}=9.1$ cm.

We plot $T_{rec,min}$ obtained from the measurements (see [24]) and that obtained using Eq. 19 vs frequency in Fig. 8. The model values and measurements differ by $\sim$ 1.5 K near 1350 MHz. The difference between model values and measurements increases to $\sim$ 5 K near 1200 and 1500 MHz (see Fig. 8). We believe part of this discrepancy may be due to any uncanceled ground scattering in the measured $T_{rec,min}$. Understanding the discrepancy and investigating methods to improve the receiver temperature measurements are ongoing research activity.

IV-D Uncertainty in the model results

The model uses a set of input parameters and results obtained from electromagnetic simulation of the dipole array. Currently, the uncertainties in both these quantities are not well determined. Further, the accuracy of the measured results against which model predictions are compared are currently limited by systematic errors.

We expect that the accuracy of the model results is dominated by the uncertainty in the input parameters to the computational model. The embedded beam patterns and impedance matrix are obtained from the output of the CST MWS simulation with solver mesh size $\lambda/20$. We found that our CST simulation results have comparable values with results obtained from HFSS555http://www.ansys.com/products/electronics/ansys-hfss (Warnick, private communication) as well as those obtained from the CST with a coarser mesh ($\lambda/10$) size. However, a direct comparison of the field patterns and S-parameters with measurements is currently not available. The secondary fields due to reflections from the telescope surface are computed using a geometric optics approximation, but we have not quantified the inaccuracies in this computation. The uncertainties in the other input parameters are discussed in Section II, III-C & III-D. To obtain an uncertainty for the model results, we estimate the results by changing the input parameters by an amount equal to their uncertainties. The estimated variation in the model values for $T_{sys}/\eta$ for $L_{trans}=9.1$ cm and $T_{loss}$ = 1 K is $\sim$ 15% near 1350 MHz. This uncertainty in the model result is valid over a frequency range of $\sim$ 150 MHz centered at 1350 MHz.

V Conclusion

In this paper, we have presented a model for noise matched phased array feed system. The PAF model predictions were then compared with measurement results. The measurements were made with a cryogenic 19 element dual polarized PAF mounted on the GBT. This PAF was designed to operate over a frequency range $\geq$ 150 MHz centered at 1350 MHz. Our main results were:

We have derived a new expression for the covariance matrices for signal by applying the Lorentz reciprocity theorem. An expression for the receiver noise covariances matrix and noise temperature were derived in terms of the Lange invariants, which were advantageous for noise matching the PAF for optimum performance. 2. 2.

We have shown that the PAF model predictions compare well with the measured $T_{sys}/\eta$ made on the GBT, both as a function of frequency and as a function of offset from boresight. 3. 3.

Our modeling have indicated that the bandwidth performance of the PAF could be improved by a factor of two (i.e. $\geq$ 300 MHz) compared to the current performance by noise matching the PAF to the LNA. 4. 4.

The PAF model results differ by $\sim$ 5 K (maximum) from the measured minimum receiver temperature. Investigation to understand this discrepancy and to improve receiver temperature measurements are underway.

Appendix A

The power spectral density due to receiver noise can be converted to a physical temperature using the definition of noise figure [35] (see also [21]). In the absence of a matching network, the receiver temperature of the PAF can be obtained using Eq. 17 as [32]

[TABLE]

Equation 21 can be rewritten as

[TABLE]

[TABLE]

Appendix B

[TABLE]

Acknowledgment

The authors would like to acknowledge very useful discussions and suggestions from Matt Morgan during the initial phase of the development of the PAF model. The possibility of an analytical solution for PAF problem was pointed out to D. A. Roshi by Stuart Hay, CSIRO. The authors acknowledge the efforts of Bob Simon, Steve White and other Green Bank Observatory and NRAO Technology Center staff in successfully building and testing the PAF receiver. We thank Marian Pospieszalski, Anthony Kerr for useful discussions on the noise properties of the receiver, Srikanth for discussions on the computation of the GBT aperture field, Wavley Groves and Matt Morgan for providing the amplifier noise model, Robert Dickman and S. K. Pan for the support and useful discussions during the course of this work. We thank the referees for providing very useful comments and suggestions, which has significantly improved the paper.