To appear in Review of Scientific Instruments, 2005 Spectroscopy with Multichannel Correlation Radiometers

A.I. Harris Department of Astronomy, University of Maryland,

College Park, MD 20742*

Correlation radiometers make true differential measurements in power with high accuracy and small systematic errors. This receiver architecture has been used in radio astronomy for measurements of continuum radiation for over 50 years; this article examines spectroscopy over broad bandwidths using correlation techniques. After general discussions of correlation and the choice of hybrid phase experimental results from tests with a simple laboratory multi-channel correlation radiometer are shown. Analysis of the effect of the input hybrid's phase shows that a 90ffi hybrid is likely to be the best general choice for radio astronomy, depending on its amplitude match and phase flatness with frequency. The laboratory results verify that the combination of the correlation architecture and an analog lag correlator is an excellent method for spectroscopy over very wide bandwidths.

I. INTRODUCTION

Modern analog lag correlators are capable of autocor-relation spectroscopy over wide bandwidths [1]. This article examines an application of analog lag correlators,that of measuring cross-correlation functions for spectroscopy with correlation detection techniques. Althoughvariants of the correlation detection schemes are common for radio continuum radiometry, it seems that no one hasyet adapted the architecture for spectroscopy (multichannel radiometry) for high-resolution spectroscopy with asingle telescope. A cross-correlation spectrometer on a single-aperture instrument would have the same excellentstability as cross-correlation spectrometers in aperture synthesis arrays. In addition, as shown in Section IV, amultichannel correlation radiometer can share one analog cross-correlation backend spectrometer between twosky positions, providing a dual-beam system with about half the backend spectrometer cost and complexity of areceiver with dual total-power spectrometers.

Correlation radiometers have made accurate radio andmillimeter-wave continuum intensity measurements of the radio sky, the Cosmic Microwave Background, ofrapidly changing scenes, and polarimetry (e.g. [2-6]). A number of authors have described specific architecturesand examined the operation and sensitivity of correlation radiometers in absolute terms and their suppression of ef-fects from the 1/

f noise common to amplifiers [4, 7-14].In the following, Section II contains a general discussion

of correlation and Section III gives an analysis of thechoice of hybrid phase, information that is not readily available elsewhere. Section IV shows experimental re-sults from tests with a simple laboratory multichannel correlation radiometer that verify the theoretical expec-tations.

*Electronic address: harris@astro.umd.edu

II. CORRELATION DETECTION Ryle introduced correlation detection to radio astron-omy with his invention of the phase-switching interferometer [15]. Ryle's interferometer squared the sum of thevoltages from two antennas after modulating the phase of one antenna's signal. Maintaining phase sensitivity bymultiplying voltages instead of detecting the total power alone made this a correlating instead of a phased arrayof antennas. Phase switching the signal from one antenna was the key element in the method's success, as itseparated the desired cross-product of the two antenna's voltages from total power signals from the individual an-tennas. Correlation detection with rapid phase switching brought a substantial improvement in instrumental sta-bility since gain and noise fluctuations of amplifiers on the two antennas were uncorrelated in time; the onlycorrelated signal, that common to the two antennas, was from the astronomical source. Communication engineers[16, 17] had already recognized that correlation techniques were valuable for retrieving small periodic signalsin noise, and eventually came to view cross-correlation as a method of producing an optimal filter: it selects thecomponent of the input signal at one multiplier input with waveform equal to the reference signal at the othermultiplier input [18]. Viewed in this light, synchronous detection is a familiar example of a correlation receiver.Adding a four-port circuit (a "hybrid") to correlation detection combines signals from two regions of the fo-cal plane of a single aperture telescope and redistributes them before amplification. This allows the correlationtechnique to be used for single-dish observations. Figure 1 shows the signal flow through a correlation radiome-ter; the components to the right of the hybrid are equivalent to a spatial interferometer's signal path. The ter-minology for correlation detection is unfortunately muddled. Communications engineers use the term correlationreceiver to describe what radio astronomers usually think of as a spatial interferometer or a synchronous detector,while single-dish astronomical instruments take the name "correlation receiver" [10, 11] or the more apt "continu-ous comparison receiver" [4, 9]. More recently, the same

2 Figure 1: General model for a continuous comparison radiometer. The central block represents the input hybrid, with input ports numbered 1 and 4 and outputs 2 and 3. General noise and gain terms n and g and a phase shift ' complete the model.

architecture has been dubbed the "pseudo-correlation re-ceiver," apparently based on a detail of the multiplier implementation.Continuous comparison detection with a singleaperture telescope is the complement to the two-elementspatial interferometer: an interferometer is sensitive to the correlated signals from two different regions of anaperture plane, while the continuous comparison radiometer extracts the uncorrelated part of the signalsfrom two different regions of a focal plane. The input hybrid combines voltages from the source and ref-erence positions in the focal plane,

vs and vr, with differ-ent but known phase shifts before amplification. Crosscorrelating the signals from the two amplifiers with theproper phase shift extracts the power difference between the two positions, vout / G \Gamma hv2s i - hv2r i\Delta , where the an-gle brackets denote an average over a time long compared with the reciprocal of the input bandwidth. Thesingle-aperture continuous comparison radiometer has signal paths that are as similar as possible, so subse-quent amplification and processing operate equally on the signals from both inputs. An amplifier gain fluctu-ation, for instance, has exactly the same effect on the signals from both positions in the focal plane. Corre-lated terms, including amplifier noise, average away with time as 1/pBo/ , where B is the predetection bandwidthand

o/ is the integration time. With a differential mea-surement, gain fluctuations have no large noise term to

amplify, greatly reducing the excess noise across the spec-trum. Excess noise is a particular problem for instruments with wide bandwidths because the intrinsic mea-surement noise is proportional to 1

/pB by the radiome-ter equation [19]. Fluctuations add noise in individual

channels and instrumental structure across the spectrum.A true differential measurement can greatly improve the stability of a radiometer compared with conventionaltotal power measurements. Power amplification in a typical heterodyne radiometer is about 1012, so it is notsurprising that the most common limit to the stability of radiometric measurements is electronic and optical gaininstability in time. Consider a total power radiometer with gain G, input source voltage vs, and system noisevoltage

vn. Its detector output voltage vout is proportional to Ghv2detectori, or vout / G \Gamma hv2s i + hv2ni\Delta , plusthe small uncorrelated cross term 2h

vsvni that averages

away with time as 1/pBo/ . Spatial chopping and dif-ferencing between source and reference positions on the

sky largely eliminates the relatively large noise signal onaverage, but even tiny fluctuations in system gain

G ornoise power h v2ni at the chop frequency can easily be muchlarger than the weak signal, \Delta \Gamma

Ghv2ni\Delta AE hv2s i, and candominate the integrated signal. In contrast with Dicke's

[19] scheme of sequential switching between astronomicaland reference signals with a total power radiometer, the continuous comparison technique's simultaneous treat-ment of signal and reference positions provides a true differential measurement that greatly reduces the effectsof time-variable gain fluctuations. Differencing without mechanically changing the optical system can also helpreduce instabilities induced effects from microphonics or changing standing-wave structure that affect some typesof receivers (e.g. local oscillator power modulation from a focal plane chopper or nutating secondary) and otherlow-level effects.

III. CHOICE OF HYBRID Figure 1 defines the noise and gain variables for thefollowing system analysis. Generalized noise voltages nX,Y and voltage gains gX,Y affect the voltages x and y at inputs X and Y. The hybrid's output is a phase-shifted mixture of its input voltages. Further equivalent noise voltages nA,B and voltage gains gA,B follow ateach output. Noise voltages from components following the hybrid will be in phase and in quadrature (denotedI and Q) with the signal phase, so the noise terms are n = nI/p2 + jnQ/p2, where j = p-1 and the totalnoise power is h

n2i = hn2I i + hn2Qi. Solving for all noiseand gain components in Fig. 1 is too messy to clearly

show the circuit's properties. It is clearer to solve twobasic cases, one with all gain and noise following an ideal hybrid and one with all preceding it. Nonideal effects arestraightforward to include as modifications to these ideal cases.Most astronomical continuous comparison radiometers incorporate a waveguide magic tee 180ffi hybrid [3, 13],although at least one has used a 90ffi quasi-optical hybrid [4], and good branch-line 90ffi hybrids now exist atfrequencies to hundreds of gigahertz [20, 21]. The correlator's output has significantly different behavior for the180ffi and 90ffi hybrids. A lossless hybrid's scattering matrix relates its output voltages to its input voltages, withports numbered as in Figure 1, as:2

64

vo1 vo2 vo3 vo4

375

= -j 2664

0 fi ff ej` 0 fi 0 0 -ff e-j` ff ej` 0 0 fi0 -

ff e-j` fi 0

3775 264 v

i1v

i2 vi3

vi4

375

.

(1)Here ff and fi are voltage coupling coefficients with ff2 +

3 fi2 = 1. Zeros along the diagonal indicate that the portsare perfectly matched, and zeros on the cross-diagonal indicate no coupling between isolated ports. (A losslesshybrid must have these terms equal to zero to satisfy the unitary property of a lossless network S matrix [22].) Thephase angle

` in Eq. (1) may vary arbitrarily in theory,with ` = 0 and ss/2 corresponding to realizable devices:the fully asymmetrical 180ffi and fully symmetrical 90ffi

hybrids. Both have 180ffi phase total shifts between theoutputs, but the shifts are distributed differently relative to the input signals.Computing the multiplier output

vout is easiest whenthe signals are in complex phasor notation with implicit

time dependence, v(t) j |v|ej,. Then the low frequency

correlator output is vout / hRe(vAv*B)i where vA,B arevoltages at the multiplier input and the asterisk denotes the complex conjugate.The most useful case has gain and noise following the hybrid. Solving for the multiplier's output shows thatthe interesting correlator power-difference signal

vout /(h x2i-hy2i) is largest when a term cos(iA -iB -'+`) ismaximum. Here

` is the hybrid phase defined in Eq. (1), ' is an additional system phase shift shown in Fig. 1, and iA,B are the gain phases gAg*B j Gej(iA-iB). Allowingfor a phase deviation

ffi, ideally zero, from the maximumdifference condition, this cosine term is maximum for

' = ` + iA - iB - ffi. With that substitution the expressionsbecome much simpler, and the multiplier output is

vout / hfffi \Gamma hx2i - hy2i\Delta + \Gamma fi2 - ff2\Delta hxyi + 12 ihnAI nBI i + hnAQnBQij

+ ffp2 ihxnAI i - hynBIij + fip2 ihxnBI i + hynAIijiG cos(ffi) + h 12 ihnAQnBIi - hnAInBQij + ffp2 ihxnAQi + hynBQij

+ fip2 ihxnBQi - hynAQijiG sin(ffi) (2) for a 180ffi hybrid (` = 0), and

vout / hfffi \Gamma hx2i - hy2i\Delta + 12 ihnAQnBI i - hnAInBQij

+ ffp2 ihxnAI i - hynBI ij - fip2 ihxnBQi - hynAQijiG cos(ffi) - h \Gamma fi2 + ff2\Delta hxyi + 12 ihnAI nBI i + hnAQnBQij

+ ffp2 ihxnAQi + hynBQij + fip2 ihxnBI i + hynAI ijiG sin(ffi) (3)

for a 90ffi hybrid (` = -ss/2).

Ideally, the correlator output contains only the power-difference term h

x2i - hy2i. Other uncorrelated crossterms (e.g. hxnAi) average to zero as 1/pBo/ . Not allelements within the cross-terms will be completely uncorrelated in a practical system, with the correlated portionsproducing offsets at the correlator output. Fluctuations in system gain G (Eqs. 2 and 3) scale the offsets and canproduce error terms that are large compared with the signal. An important goal for a continuous comparisonradiometer is to keep the multiplier output near zero so the influence of gain fluctuations will be small. Mini-mizing the number and amplitude of correlator offsets is important to this end.

For ground-based radio astronomy the most importantdifference between the circuits with 180ffi and 90ffi hybrids

is likely to be the response to cross-correlation in the input signals, hxyi. Atmospheric emission in the telescope'snear field and noise from the telescope's ohmic loss will provide correlated voltages between the two inputs. Sup-pressing this term requires either good hybrid amplitude balance or a flat system phase: at the correlator out-put this signal scales with (

ff2 - fi2) for the 180ffi hybridand , sin( ffi) for the 90ffi hybrid. A factor of ten sup-pression implies a hybrid power imbalance no worse than

0.45/0.55 (0.87 dB) or a maximum phase error term ofsin(5

.7ffi). Although it is difficult to build hybrids withtight amplitude matching across wide bands, wideband

hybrids can have good phase flatness [20, 21]. If the othersystem components have good phase matching then the 90ffi hybrid would be the better choice.

Rejecting correlated input noise can also be usefulwhen a common local oscillator signal (LO) is injected

4 before the hybrid. Injecting the LO into both ports withzero phase shift and equal amplitude will suppress noise in the oscillator's wings at the signal frequency. Injectingthe LO into only one input port will pump both mixers but provides no LO noise rejection.

In any case, the 90ffi hybrid circuit always rejects corre-lated signals introduced after the hybrid better than the

180ffi hybrid circuit. These terms, with form hnAQnBQiand h

nAInBI i, are suppressed by the sin(ffi) factor. Cor-relation in these terms can arise from bias fluctuations

common to both gain paths or noise from the wings of ashared local oscillator (the noise and gain model implicit in Fig. 1 is generic and can include frequency conversionand multiple amplifiers).

For a nonideal hybrid, coupling between the outputports (2 and 3 in Fig. 2) is another important potential source of correlator offset. Noise radiated from, or signalsreflected by, devices following the hybrid can emerge from the corresponding nominally isolated port to produce acorrelated signal. A circulator following the hybrid can reduce the offset by an amount equal to the circulatorisolation at the cost of adding loss before amplification. Lack of isolation between hybrid inputs is likely to be lessof a problem since the signal reflected from the telescope or other optics is likely to be small. Further weak corre-lated terms will come from noise power emitted from the system and reflected back as an input signal (e.g. a frac-tion of

nA returns as input signal x to produce a nonzeroh xnAi). This term is suppressed if the pathlength for thereflected signal is substantially longer than the correlation length, l ' c/B, where c is the speed of light.

Phase switching is a powerful method for removing cor-relator offsets and reducing the effects from nonideal colored noise. In comparison with amplitude modulation(Dicke switching) 180ffi phase switching is very efficient because the full signal amplitude is always present at thedetector. Modulating the phase difference between the arms of a continuous comparison radiometer shifts thecorrelated signal output in frequency by an amount equal to the modulation frequency. Synchronous detection re-covers the correlated signal while rejecting noise fluctuations at frequencies other than the modulation frequency.With phase switching before amplification, ideally just following the hybrid, a judicious choice of modulationfrequency can remove much of the drift and 1/

f noise as-sociated with amplifier noise fluctuations. Since, to high

order, any residual offsets or offsets from sources outsidethe phase modulation-demodulation boundaries are independent of the telescope's pointing, Dicke amplitudemodulation by chopping on the sky (optically switching between two positions on the sky) and a final synchronousdemodulation will largely remove the remaining offsets.

Phase switching does not suppress the direct effectsof amplifier or multiplier gain or phase fluctuations, but symmetry can reduce their influence. Equations (2) and(3) show that the product of the voltage gains scales the input power difference as G(hx2i - hy2i). A small gainfluctuation common to both chains introduces an amplitude error to the difference signal at the correlatoroutput, but adds no error signal when the offsets are negligible. Multiplier gain fluctuations have the sameeffect as amplifier gain fluctuations in this case. The situation is different when the amplifier gain fluctua-tions are differential-mode instead of common-mode for the two amplifier chains. Faris [12] calculates the ef-fect of a varying gain imbalance between the two chains for g2(f, t) = [1 + a(t)]g1(f ), where g1,2(f ) are thecomplex amplifier voltage gains of the two chains and a(t) is a zero-mean random variable that describes thedifferential-mode fluctuations. Fluctuations increase the output variance by a factor of (1 + ha2(t)i) comparedwith the case of purely common-mode gain fluctuations between the two arms. Similar effects arise from differen-tial phase fluctuations between the two arms. Strategies for minimizing differential-mode fluctuations could in-clude biasing the amplifiers from a common power supply and keeping good thermal contact between correspondingparts of each chain.

The second case, with amplification preceding the hy-brid's loss, is the obvious choice for maximizing the receiver sensitivity but negates much of the continuouscomparison architecture's advantage. The maximum difference signal in this case is

vout = fffih \Gamma hx2i|gx|2 - hy2i|gy|2\Delta

+ \Gamma hn2xi|gx|2 - hn2yi|gy|2\Delta i , (4)

for both 180ffi and 90ffi hybrids. For this configuration tobe useful nearly exact matching of both the noise power and power gains would be necessary. Slight imbalances inloss and gain will produce large offsets at the correlator output.

IV. SPECTROSCOPY WITH AN ANALOG LAG

CROSS-CORRELATOR

Sensitive spectroscopy (multichannel radiometry) overbroad bandwidths is important for observations of wide spectral lines from distant galaxies, searches for lines atwith uncertain frequency, and measurements of pressurebroadened lines in planetary atmospheres. Spectrometerbandwidths can be tens of gigahertz with channel bandwidths of tens of megahertz. Such broad bandwidthsplace stringent requirements on system stability, so it is natural to pair wideband spectrometers with the contin-uous comparison architecture. Spectroscopy with a correlation radiometer requires measurement of the cross-correlation function over a range of time lag, or delay. Analog lag correlators use purely analog components toobtain the cross-correlation function

RAB(o/ ) as a func-tion of lag o/ :

RAB(o/ ) = limT !1 12T Z

T

-T v

A(t) * vB(t + o/ ) dt . (5)

5 Tapped transmission lines provide the time delays o/ ,transistor multipliers form the product of the two input voltages vA(t) and vB(t + o/ ), and low-frequency electron-ics integrate the multiplier output to provide the time average.

A Fourier transform of the cross-correlation functionyields the power density cross-spectrum. Transforming the correlation function to recover the spectrum isslightly more complicated than making a direct Fourier transform for analog correlators because the signal is notsampled at perfectly regular intervals. Although the mechanical spacing between the microwave signal taps alongthe transmission line is well defined by the traces on the circuit board, frequency-dependent component variationscause small erratic variations in the electrical delays between multipliers. A direct Fourier inversion to find thespectrum is not possible because the transform kernel's phase term cannot be reduced to a separable product ofthe delay time and frequency, as might be the case for simple line dispersion [1]. It is possible to correct theirregular sampling in software by measuring the spectrometer's response to a series of monochromatic signalsat known frequencies, then expanding the astronomical input signal on these measured functions [1].

An interesting effect of this calibration scheme is thatit defines the spectrometer's internal phase: by definition signals in phase with the calibration are real, andthose with a relative 90ffi phase shift are imaginary. This property can be used to eliminate one of the phases inthe usual complex cross-correlation measurement. When the calibration signals are injected at the radiometer's in-put the calibration and measured signals share the same phase shifts through the entire instrument, so the mea-sured signal is purely real; the imaginary component contains only noise. A single cross-correlator can there-fore measure the cross-correlation function. While purely real cross-correlations are unusual in most spectrometers,there is no fundamental reason that they cannot exist. A real correlation function has the convenient property ofeven symmetry in the lag domain, so the positive (or negative) lags alone contain all the necessary information torecover the spectrum. Although it is possible to build a full complex correlator and calibrate it at lower fre-quency, injecting the phase calibration signals at the input to the entire radiometer yields spectra at full spectralresolution with half the number of lags. This is not only a significant savings in spectrometer cost and complexity,but eliminates requirements on phase-matching between the two amplifier and processing chains. Digital corre-lators do not readily share this property because their topologies give them an intrinsic symmetry and phaserelated to the position of the zero-lag multiplier. A full complex correlator, or close phase matching across thereceiver band, is needed for spectroscopy with a digital system and direct transform.

A simple laboratory continuous comparison radiome-ter, sketched in Figure 2, verifies that a WASP2 (Wideband Astronomical SPectrometer) analog lag correlator

Figure 2: Block diagram of laboratory continuous comparison receiver test setup.

Figure 3: Spectra from the two inputs of a laboratory correlation radiometer with a WASP2 spectrometer configured as a cross-correlator. The dotted line is a network analyzer measurement of the filter transmission.

[1] properly produces power difference cross-spectra withthis calibration scheme. The hybrid for the experiment was an off-the-shelf stripline 90ffi 2-4 GHz device. Cableslengths between the hybrid and cross-correlator brought the zero time-lag position close to one end of the multi-plier ladder, for maximum spectral resolution, but were not otherwise trimmed for length or phase matching. Abroadband noise diode and 300 MHz filter generated an artificial spectral line that could be connected to eitherhybrid input, denoted X and Y in Fig. 2. Phase calibration signals were fed into input X alone. Figure 3 showsthat the spectrometer works as predicted. The artificial line at input X, with Y terminated, produces a positivespectral line, and the same signal at input Y, with X terminated, produces a negative spectral line. After remov-ing the noise source's intrinsic spectral shape by dividing the raw spectra by spectra of the noise source alone,the filter center frequency, shape, and loss matched network analyzer measurements (Fig. 3). This confirms thatthe cross-correlation function is purely real. The sum of the two spectra is very close to zero. Residual phase er-rors scatter power across the spectrum at about 0.5% of

6 the peak line intensity, a well-understood dynamic rangelimit rather than a noise offset [1]. This fixed-pattern structure subtracts well with beamswitching.These experimental results show that the combination of the continuous comparison architecture with a ana-log lag correlator is a very promising method for spectroscopy over very wide bandwidths.

Acknowledgments The author thanks J. Kooi and T.G. Phillips for sug-gestions and several useful discussions about LO noise

rejection and practical hybrid designs. The author alsothanks the Caltech Submillimeter Group for their hospitality while this paper was written. This work was sup-ported in part by grant AST-9819747 from the National Science Foundation.

[1] A. Harris and J. Zmuidzinas, Rev. Sci. Inst. 72, 1531

(2001). [2] C. Haslam, W. Wilson, D. Graham, and G. Hunt, Astron.

Astrophys. Suppl. 13, 359 (1974). [3] N. Jarosik, C. Bennet, M. Halpern, G. Hinshaw,

A. Kogut, M. Limon, S. Meyer, L. Page, M. Pospiezalski, D. Spergel, et al., Astrophys. J. Suppl. Ser. 145, 413 (2003). [4] C. Predmore, N. Erickson, G. Huguenin, and P. Goldsmith, IEEE Trans. Microwave Theory Tech. MTT-33, 44 (1985). [5] K. Rholfs and T. Wilson, Tools of Radio Astronomy

(Springer Verlag, 2000), chap. 4.5, 3rd ed. [6] O. Koistinen, J. Lahtinen, and M. T. Hallikainen, IEEE

Trans. Instrumen. Meas. IM-51, 227 (2002). [7] R. Fano, Signal to noise ratio in correlation detectors,

M.I.T. Res. Lab. Elec., Rep. RLE-186 (1951). [8] S. Goldstein, Jr., Proc. IRE 43, 1663 (1955), with further

correspondence and errata in Proc. IRE 48, 365, 1956. [9] 'E.-J. Blum, Ann. Astrophys. 22, 140 (1959). [10] M. Tiuri, IEEE Trans. Antenn. Propag. AP-12, 930

(1964). [11] R. Colvin, Ph.D. thesis, Stanford University (1961), also

Stanford Radio Astronomy Institute Publ. No. 18A. [12] J. Faris, J. Res. Nat. Bur. Stand.-C 71C, 153 (1967). [13] M. Seiffert, A. Mennella, C. Burigana, N. Mandolesi,

M. Bersanelli, P. Meinhold, and P. Lubin, Astr. Astrophys. 391, 1185 (2002). [14] A. Mennella, M. Bersanelli, M. Seiffert, D. Kettle,

N. Roddis, A. Wilkinson, and P. Meinhold, Astr. Astrophys. 410, 1089 (2003). [15] M. Ryle, Proc. R. Soc. Lon. A 211, 351 (1952). [16] Y. Lee, T. Chetham, and J. Wiesner, The application of

correlation functions in the detections of small signals in noise, M.I.T. Res. Lab. Elec., Rep. RLE-141 (1949). [17] Y. Lee and J. Wiesner, Electronics 23, 86 (1950). [18] P. Peebles, Probability, Random Variables, and Random

Signal Processes (McGraw-Hill, 1993), p. 310, 3rd ed. [19] R. Dicke, Rev. Sci. Inst. 17, 268 (1946). [20] S. Srikanth and A. Kerr, Waveguide quadrature hybrids

for ALMA receivers, ALMA Memo 343 (2001), URL http://www.alma.nrao.edu/memos/index.html. [21] J. Kooi, A. Kovacs, B. Bumble, G. Ghattopadhyay,

M. Edgar, S. Kaye, H. LeDuc, J. Zmuidzinas, and T. Phillips, in Millimeter and submillimeter detectors for astronomy II, edited by J. Zmuidzinas, W. S. Holland, and S. Withington (SPIE-The International Society for Optical Engineering, 2004), vol. 4587 of Proc. SPIE, pp. 332-348. [22] D. M. Pozar, Microwave Engineering (J.W. Wiley &