Generalized Formalisms of the Radio Interferometer Measurement Equation

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by  Danny C. Price

Mon.Not.R.Astron.Soc.000,1–13(2015) Printed27January2015 (MNLATEXstylefilev2.2)
Generalized Formalisms of the Radio Interferometer
Measurement Equation
D. C. Price1(cid:63) and O. M. Smirnov2,3
1Harvard-Smithsonian Center for Astrophysics, MS42, 60 Garden Street, Cambridge MA, 01243 United States
2Department of Physics and Electronics, Rhodes University, PO Box 94, Grahamstown, 6140, South Africa
3SKA South Africa, 3rd Floor, The Park, Park Road, Pinelands, 7405, South Africa
Accepted2015January12.Received2014November04
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ABSTRACT
2
The Radio Interferometer Measurement Equation (RIME) is a matrix-based mathe-

n maticalmodelthatdescribestheresponseofaradiointerferometer.TheJonescalculus
a
it employs is not suitable for describing the analogue components of a telescope. This
J
is because it does not consider the effect of impedance mismatches between compo-
6
nents. This paper aims to highlight the limitations of Jones calculus, and suggests
2
some alternative methods that are more applicable. We reformulate the RIME with

a different basis that includes magnetic and mixed coherency statistics. We present
]
M a microwave network inspired 2N-port version of the RIME, and a tensor formalism
basedupontheelectromagnetictensorfromspecialrelativity.Weelucidatethelimita-
I
. tions of the Jones-matrix-based RIME for describing analogue components. We show
h
howmeasuredscatteringparametersofanaloguecomponentscanbeusedina2N-port
p
versionoftheRIME.Inaddition,weshowhowmotionatrelativisticspeedaffectsthe
-
o observedflux.WepresentreformulationsoftheRIMEthatcorrectlyaccountformag-
r neticfieldcoherency.Thesereformulationsextendthestandardformulation,highlight
t
s its limitations, and may have applications in space-based interferometry and precise
a absolute calibration experiments.
[

Key words: Methods: data analysis — Techniques: interferometric — Techniques:
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polarimetric
v
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4
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1 INTRODUCTION calibration was conducted in an ad-hoc manner, with each
0
polarization essentially treated separately. The framework
. Coherency in electromagnetic fields is a fundamental topic
1 was expounded in a series of follow-up papers (Sault et al.
within optics. Its importance in fields such as radio as-
0 1996; Hamaker & Bregman 1996; Hamaker 2000, 2006); re-
5 tronomy can not be overstated: interferometry and syn- cent work by Smirnov extends the formalism to the full
1 thesis imaging techniques rely heavily upon coherency the-
sky case, and reformulates the RIME using tensor algebra
: ory (Taylor et al. 1999; Mandel & Wolf 1995; Thompson
v (Smirnov 2011a,b).
et al. 2004). Of particular importance to radio astronomy
i
X is the Van-Cittert-Zernicke theorem (vC-Z, Zernicke 1938) ThisarticleintroducestworeformulationsoftheRIME,
andtheradiointerferometerMeasurementEquation(RIME, that extend its applicability and demonstrates limitations
r
a Hamaker et al. 1996). The vC-Z relates the brightness of a withtheJones-matrix-basedformalism.Bothoftheserefor-
source to its mutual coherency as measured by an interfer- mulations consider full electromagnetic coherency statistics
ometer, and the RIME provides a polarimetric framework (i.e. electric, magnetic and mixed coherencies). The first is
to calibrate out corruptions caused along the signal’s path. inspired by transmission matrix methods from microwave
While the vC-Z theorem dates back to 1938, more re- engineering. We show that this reformulation better ac-
centworksuchasthatofCarozzi&Woan(2009)extendsits countsforthechangesinimpedancebetweenanaloguecom-
applicability to polarized measurements over wide fields of ponents within a radio telescope. The second reformulation
view.TheRIMEhasamuchshorterhistory:itwasnotfor- isarelativistic-awareformulationoftheRIME,startingwith
mulateduntil1996(Hamakeretal.1996).BeforetheRIME, the electromagnetic field tensor. This formalism allows for
relativistic motion to be treated as instrumental effect and
incorporated into the RIME.
(cid:63) E-mail:[email protected] Thisarticleisorganizedasfollows.InSection2,were-
(cid:13)c 2015RAS
2 D.C. Price and O. Smirnov
viewexistingformalismsoftheRIMEandmethodsfrommi-
crowave engineering. Section 3 defines coherency matrices,
which are used in Section 4 to formulate general coherency
relationships for radio astronomy. In Section 5, we intro- p p p V
duceatensorformulationoftheRIMEbasedupontheelec- B pq
tromagnetic tensor from special relativity. Discussion and
example applications are given in Section 6; concluding re- source correlator
q q q
marks are given in Section 7.
free space antenna response analogue chain
Figure 1. Block diagram showing a simple model of an inter-
2 JONES AND MUELLER RIME ferometer that can be modelled with the RIME. Radiation from
FORMULATIONS a source propagates through free space to two telescopes, p and
q. After passing through the telescope’s analogue chain, the two
Before continuing on to derive a more general relationship signalsareinterferedinacross-correlator.
between the two-point coherency matrix and the voltage-
currentcoherency,wewouldliketogiveabriefoverviewand
derivation of the radio interferometer Measurement Equa- 2.1 Hamaker’s RIME derivation
tion of Hamaker et al. (1996). Our motivation behind this
The derivation of the RIME is remarkably simple and el-
istohighlightthatHamakeret.al.’sRIMEisaspecialcase
egant. For a single point source of radiation, the voltages
ofamoregeneral(andthuslesslimited)coherencyrelation-
induced at the terminals of a pair of antennas, p and q are
ship.
In their seminal Measurement Equation (ME) paper, vp(t)=Jpe0(t) (5)
Hamaker et al. (1996) showed that Mueller and Jones cal- v (t)=J e (t) (6)
q q 0
culusesprovideagoodframeworkformodellingradiointer-
ferometers. In optics, Jones and Mueller matrices are used Initssimplestform,theRIMEisformedbytakingtheouter
to model the transmission of light through optical elements product of these two relationships. Note that in their orig-
(Jones 1941; Mueller 1948). Mueller matrices are 4×4 ma- inal paper, the authors use the Kronecker incarnation of
trices that act upon the Stokes vector the outer product, which we will denote with (cid:63). We reserve
the symbol ⊗ for the matrix outer product of two matrices
s =(cid:0)I Q U V(cid:1)T , (1) A⊗B = ABH, where H denotes the Hermitian conjugate
transpose1. Using the Kronecker outer product, the RIME
whereasJonesmatricesareonly2×2indimensionandact
is given by
upon the ‘Jones vector’: the electric field vector in a coor-
dinate system such that z-axis is aligned with the Poynting (cid:104)v (cid:63)v (cid:105)=(J (cid:63)J )(cid:104)e (cid:63)e (cid:105)=(J (cid:63)J )e (7)
p q p q 0 0 p q 00
vector
where J and J are the Jones matrices representing all
p q
e(r,t)=(cid:0)ex(r,t) ey(r,t)(cid:1)T . (2) transformations along the two signal paths, and (Jp(cid:63)Jq)
is a 4×4 matrix. Here, e is the sky brightness of a single
00
Jones calculus dictates that along a signal’s path, any (lin-
point source of radiation. For a multi-element interferome-
ear)transformationcanberepresentedwithaJonesmatrix,
ter, every antenna has its own unique Jones matrix, and a
J:
RIME may be written for every pair of antennas.
Due to their choice of outer product, Hamaker et. al.
e (r,t)=Je (r,t) (3)
out in
arrive at a coherency vector
A useful property of Jones calculus is that multiple effects
(cid:10)e e∗ (cid:11)
alongasignal’spathofpropagationcorrespondtorepeated px qx
multiplications: epq(rp,rq,τ)=(cid:104)ep(rp,t)(cid:63)eq(rq,t+τ)(cid:105)=(cid:10)(cid:10)eeppxyee∗q∗qxy(cid:11)(cid:11),
e (r,t)=J ···J J e (r,t), (4) (cid:10)e e∗ (cid:11)
out n 2 1 in py qy
(8)
which can be collapsed into a single matrix when required.
as opposed to the coherency matrix of Wolf (1954); this is
TheRIMEusesJonesmatricestomodelthevariouscor-
introducedin3.1below.Thecolumnvectorofapointsource
ruptions and effects during a signal’s journey from a source
at r is then e ; that is, p=q and τ =0. The vector e
0 00 00
rightthoughtothecorrelator.Ablockdiagramfora(simpli- is related to the Stokes vector by the transform2.
fied)two-elementinterferometerisshowninFigure1.From
lefttoright,thefigureshowsthejourneyofasignalfroma I 1 0 0 1 (cid:104)e0xe∗0x(cid:105)
sthouerscoeurricgehitstphircokuedghuptobtyhtewcoorarnetleantonra.s,Twhheicrahdwiaetihoanvefrdome- QU=10 01 01 −01(cid:10)(cid:104)ee00xyee∗0∗0xy(cid:11)(cid:105). (9)
notedwithsubscriptpandq.Theradiationfollowsaunique V 0 −j j 0 (cid:10)e0ye∗0y(cid:11)
pathtobothoftheseantennas;eachantennaalsohasasso-
Thequantity(J (cid:63)J )inEq.7canthereforebeviewedasa
ciated with it a unique chain of analogue components that p q
amplify and filter the signal to prepare it for correlation.
Each of these subscripted boxes may be represented by a 1 For a discussion on the subtleties of outer product definition,
Jones matrix; alternatively an overall Jones matrix can be seeSmirnov2011b,A.6.1
√
formed for the p and q branches (the dashed areas). 2 Here,j= −1,toavoidconfusionwithcurrent,i,usedlater.
(cid:13)c 2015RAS,MNRAS000,1–13
Generalized Formalisms of the RIME 3
Muellermatrix.Thatis,Eq.7canbeconsideredaMueller- This formalism is better capable of describing mutual cou-
calculus-basedMEforaradiointerferometer.Tosummarize, pling between antennas, beamforming, and wide field po-
Hamaker et. al. showed that: larimetry. Inthispaper,wefocusonamatrixbasedformal-
ismwhichconsidersthepropagationofthemagneticfieldin
• Jones matrices can be used to model the propagation
addition to the electric field. We then show that this for-
ofasignalfromaradiationsourcethroughtothevoltageat
mulation is equivalent to the tensor formalism presented in
the terminal of an antenna. Smirnov (2011b), but is instead in the vector space C6.
• AMuellermatrixcanbeformedfromtheJonestermsof
apairofantennas,whichthenrelatesthemeasuredvoltage
coherency of that pair to a source’s brightness. 2.4 Microwave engineering transmission matrix
methods
Showing that these calculuses were applicable and indeed
useful for modelling and calibrating radio interferometers AllformulationsoftheRIMEtodate—includingthetensor
was an important step forward in radio polarimetry. formulation—donotconsiderthepropagationofthemag-
netic field. In free space, magnetic field coherency can be
easily derived from the electric field coherency. However, at
2.2 The 2×2 RIME theboundarybetweentwomedia,themagneticfieldmustbe
considered.Here,weintroducesomeresultsfrommicrowave
Inalaterpaper,Hamaker(2000)presentsamodifiedformu- engineering which contrast with the Jones formalism.
lationoftheRIME,whereinsteadofformingthecoherency In circuit theory, the well-known impedance relation,
vectorfromtheKroneckerouterproduct((cid:63)),thecoherency V =ZI relatescurrentandvoltageoveraterminalpair(or
matrix is formed from the matrix outer product (⊗): ‘port’).However,thisrelationisspecifictoa1-portnetwork;
(cid:18)(cid:10)e e∗ (cid:11) (cid:10)e e∗ (cid:11)(cid:19) formicrowavenetworkswithmorethanoneport,thematrix
Epq =(cid:104)ep⊗eq(cid:105)= (cid:10)epxe∗qx(cid:11) (cid:10)epxe∗qy(cid:11) (10) form [V]=[Z][I] must be used:
px qx py qy
The resulting coherency matrix is then shown to be related v  Z11 Z12 ··· Z1Ni 
to the Stokes parBam=e(cid:18)teUrIs−+byjQVE00U=I+−BjQV,(cid:19)wh.ere (11) v...12=Z...21 ... ...... i...12, (16)
v i
N Z ··· ··· Z N
N1 NN
The equivalent to the RIME of Eq. 7 is
where Z is the port-to-port impedance from port a to
(cid:104)vp⊗vq(cid:105)=2(cid:10)(Jpe0)⊗(Jqe0)(cid:11)=2Jp(cid:104)e0⊗e0(cid:105)JHq port b, vanbis the voltage on port n, and in is the current. A
(12) commonexampleofa2-portnetworkisacoaxialcable,and
or more simply, a common 3-port network is the Wilkinson power divider.
The analogue components of most radio telescopes can
V =J BJH. (13)
pq p q be considered 2-port networks. The 2-port transmission, or
This approach avoids the need to use 4×4 Mueller matri- ABCDmatrix,relatesthevoltagesandcurrentsofa2-port
ces, so is both simpler and computationally advantageous. network:
Thisformisalsocleanerinappearance,asfewerindicesare
required. (cid:18)v (cid:19) (cid:18)A B(cid:19)(cid:18)v (cid:19)
1 = 2 , (17)
Smirnov(2011a)takesthe2×2versionoftheRIMEas i C D i
1 2
astartingpointandextendstheRIMEtoafullskycase.By
this is shown in Figure 2. If two 2-port networks are con-
treating the sky as a brightness distribution B(σ), where σ
nected in cascade (see Figure 2), then the output is equal
is a direction vector, each antenna has a Jones term J (σ)
p
totheproductofthetransmissionmatricesrepresentingthe
describingthesignalpathforagivendirection.Thevisibility
individual components:
matrixV isthenfoundbyintegratingovertheentiresky:
pq
ˆ (cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19)(cid:18) (cid:19)
v A B A B v
Vpq = Jp(σ)B(σ)JHq (σ)dΩ. (14) i11 = C11 D11 C22 D22 i22 , (18)
4π
as is shown in texts such as Pozar (2005). The elements in
This is a more general form of Zernicke’s propagation law.
the 2-port transmission matrix are related to port-to-port
SmirnovgoesontoderivetheVan-CittertZernicketheorem
impedances by3
from this result; we return to vC-Z later in this article.
A=Z /Z (19)
11 21
Z Z −Z Z
B = 11 22 12 21 (20)
2.3 A generalized tensor RIME Z
21
In Smirnov (2011b), a generalized tensor formalism of the C =1/Z21 (21)
RIME is presented. The coherency of two voltages is once D=Z /Z . (22)
22 21
againdefinedviatheouterproducteie¯ ,givinga(1,1)-type
j
tensor expression:
3 Note that Z21 =0 is zero impedance, which is never satisfied
Vpqji =JpαiBαβJ¯βqj. (15) inrealcomponents,andZ21=∞representsanopencircuit
(cid:13)c 2015RAS,MNRAS000,1–13
4 D.C. Price and O. Smirnov
and (r ,t+τ), respectively. We may arrange these into a
I1 I2 single6q×6matrixB thatisequivalenttothetimeaveraged
pq
P1ort +-V1 CA DB +- V2 P2ort ospuatecretpimroedupcotinotfst(hree,lte)catrnodm(argn,ett+icτfi)e:ldcolumnvectorsat
p q
(cid:28)(cid:18) (cid:19) (cid:18) (cid:19)(cid:29) (cid:18) (cid:19)
e e E M
I1 I2 I3 Bpq = hp ⊗ hq = Npq Hpq (29)
p q pq pq
+V A1B1 +V A2B2 +V
- 1 C1D1 - 2 C2D2 - 3 This matrix fully describes the coherency properties of an
electromagnetic field at two points in spacetime. We will
refer to this matrix as the two point coherency matrix. It is
Figure 2.Top:TheABCDmatrixfora2-portnetwork.Inthis worth noting that:
diagram,voltageisdenotedwithV,andcurrentwithI.Bottom:
Connecting two components in cascade. Diagram adapted from • When rp =rq and τ =0 we retrieve what Bergman &
Pozar(2005) Carozzi (2008) refer to as the ‘EM sixtor matrix’. Bergman
&Carozzi(2008)showthissixtormatrixisrelatedtowhat
they refer to as ‘canonical electromagnetic observables’: a
Like the Jones matrix, the ABCD matrix allows a sig-
uniquesetofStokes-likeparametersthatareirreducibleun-
nal’s path to be modelled through multiplication of ma-
derLorentztransformations.Theseareusedinthetheanal-
trices representing discrete components. While the Jones
ysis of electromagnetic field data from spacecraft.
matrix acts upon a pair of orthogonal electric field com-
• Formonochromaticplanewaves,whenr =r andτ =
p q
ponents, the ABCD matrix acts upon a voltage-current
0,andwechooseacoordinatesystemwithzinthedirection
pair at a single port. As Jones matrices do not consider
ofpropagation(i.e.alongthePoyntingvector),E becomes
pq
changes in impedance (free space impedance is implicitly
whatSmirnov(2011a)referstoasthebrightness matrix,B.
assumed), it is not suitable for describing analogue compo- From here forward, we drop the subscript B = B and
00
nents. Conversely, the 2×2 ABCD matrix cannot model
shall refer to this as the brightness coherency to highlight
cross-polarization response of a telescope. In the section
its relationship with B.
that follows, we derive a more general coherency relation-
ship which weds the advantages of both approaches.
3.2 Voltage and current coherency
3 COHERENCY IN RADIO ASTRONOMY
Aradiotelescopeconvertsafreespaceelectromagneticfield
We now turn our attention to formulating a more general intoatimevaryingvoltage,whichwethenmeasureaftersig-
RIMEthatisvalidinalargerrangeofcases.Inthissection, nal conditioning (e.g. amplification and filtering). As such,
we introduce the coherency matrices of Wolf (1954), along radio interferometers measure coherency statistics between
withvoltage-currentcoherencymatrices.Thefollowingsec- time varying voltages.
tionthenformulatesrelationshipsbetweensourcebrightness Onemaymodeltheanaloguecomponentsofatelescope
and measured coherency based upon these matrices. as a 6-port network, with three inputs ports and three out-
putports.Weproposethissothatthereisaninput-output
pair of ports for each of the orthogonal components of the
3.1 Electromagnetic coherency electromagnetic field. We can then define a set of voltages
v(t) and currents i(t)
To begin, we introduce the coherency matrices of Wolf
(1954),thatfullydescribethecoherencystatisticsofanelec-
tromagnetic field. We may start by introducing e(r,t) and v(t)=(cid:0)v (t) v (t) v (t)(cid:1)T (30)
x y z
h(r,t)asthecomplexanalyticrepresentationsoftheelectric
and magnetic field vectors at a spacetime point (r,t): i(t)=(cid:0)ix(t) iy(t) iz(t)(cid:1)T . (31)
e(r,t)=(cid:0)e (r,t) e (r,t) e (r,t)(cid:1)T (23) In practice, most telescopes are single or dual polarization,
x y z
soonlythexandycomponentsaresampled.Nonetheless,it
h(r,t)=(cid:0)h (r,t) h (r,t) h (r,t)(cid:1)T (24)
x y z ispossibletosampleallthreecomponentswiththreeorthog-
The coherency matrices are then defined by the formulae onal antenna elements (Bergman et al. 2005). The voltage
responseofanantennaislinearlyrelatedtotheelectromag-
Epq(rp,rq,τ)=(cid:0)(cid:104)ek(rp,t)e∗l(rq,t+τ)(cid:105)(cid:1) (25) netic field strength (Hamaker et al. 1996), and the current
H (r ,r ,τ)=(cid:0)(cid:104)h (r ,t)h∗(r ,t+τ)(cid:105)(cid:1) (26) islinearlyrelatedtovoltagebyOhm’slaw,sowemaywrite
pq p q k p l q
M (r ,r ,τ)=(cid:0)(cid:104)e (r ,t)h∗(r ,t+τ)(cid:105)(cid:1) (27) a general linear relationship
pq p q k p l q
N (r ,r ,τ)=(cid:0)(cid:104)h (r ,t)e∗(r ,t+τ)(cid:105)(cid:1). (28) (cid:18)v(t)(cid:19) (cid:18)A B(cid:19)(cid:18)e(r,t)(cid:19)
pq p q k p l q = , (32)
i(t) C D h(r,t)
Here, k and l are indices representing the x,y,z subscripts
fromCartesiancoordinates.E andH arecalledtheelec- whereA,B, C and D areblockmatricesforminganover-
pq pq
tric and the magnetic coherency matrices, and M and all transmission matrix T(cid:48). We can now define a matrix
pq
N are called the mixed coherency matrices. The sub- of voltage-current coherency statistics that consists of the
pq
scripts p and q correspond to the spacetime points (r ,t) block matrices
p
(cid:13)c 2015RAS,MNRAS000,1–13
Generalized Formalisms of the RIME 5
measure the electromagnetic field. Following from Eq. 44,
V (τ)=(cid:0)(cid:104)v (t)v∗(t+τ)(cid:105)(cid:1) (33) we may write an equation relating voltage and current co-
pq k l herency:
W (τ)=(cid:0)(cid:104)i (t)i∗(t+τ)(cid:105)(cid:1) (34)
Kpq(τ)=(cid:0)(cid:104)vk(t)il∗(t+τ)(cid:105)(cid:1) (35) Vpq =T(cid:48)p(TpBTHq )T(cid:48)qH. (45)
pq k l
L (τ)=(cid:0)(cid:104)i (t)v∗(t+τ)(cid:105)(cid:1), (36) As the T(cid:48) and T matrices are both 6×6 , we can are both
pq k l collapse these matrices into one overall matrix. Eq. 45 then
these are analogous to (and related to) the electromagnetic becomes
coherencymatricesabove4.Inasimilarmannertothetwo-
point coherency matrix, we define V
pq V =T BTH, (46)
(cid:28)(cid:18) (cid:19) (cid:18) (cid:19)(cid:29) (cid:18) (cid:19) pq p q
v v V K
Vpq = ip ⊗ iq = Lpq Wpq , (37) whichisthegeneralformthatrelatesthevoltage-coherency
p q pq pq matrix V to the brightness coherency B.
pq
which we will refer to as the voltage-current coherency ma- Equation46isacentralresultofthispaper.Itisagen-
trix. This is analogous to the ‘visibility matrix’, Vpq, of eralcasewhichrelatestheEMfieldatagivenpointinspace-
Smirnov (2011a). timetothevoltageandcurrentcoherenciesinbetweenpairs
oftelescopes.Inthesectionsthatfollow,weshowthatgen-
eralized versions of the Van-Cittert-Zernicke theorem and
4 TWO POINT COHERENCY RIME may be formulated based upon this coherency rela-
RELATIONSHIPS tionship,andthatthecommonformulationscanbederived
from these general results.
Now we have introduced the two-point coherency matrix
B and the voltage-current coherency matrix V , we can
pq pq
formulaterelationshipsbetweenthetwo.Inthissection,we 4.2 The Radio Interferometer Measurement
first formulate a general coherency relationship describing Equation
propagation from a source of electromagnetic radiation to
By comparing Eq. 13 with Eq. 44, it is apparent that the
twospacetimecoordinates.Wethenshowthatthisrelation-
Jones formulation of the RIME is retrieved by setting all
ship underlies both the RIME and the vC-Z relationship.
but the top left block matrices to zero, such that we have
V =A E AH. (47)
pq p 00 q
4.1 A general two point coherency relationship
Butunderwhatassumptionsmayweignoretheotherentries
Suppose we have two sensors, located at points rp and rq, ofEq.44?Toanswerthis,wemaynotethatmonochromatic
which fully measure all components of the electromagnetic plane waves in free space have E and H are in phase and
field.Assuminglinearity,thepropagationofanelectromag- mutually perpendicular:
netic field f0 =(cid:0)e(r0,t) h(r0,t)(cid:1)T from a point r0 to rp e(r,t)=(cid:0)e (r,t) e (r,t)(cid:1)T
and r can be encoded into a 6×6 matrices, T and T : x y
q p q
fp =Tpf0 (38) h(r,t)= c10 (cid:0)−ey(r,t) ex(r,t)(cid:1)T (48)
f =T f (39) Where c is the magnitude of the speed of light. In such a
q q 0 0
case, all coherency statistics can be derived from the 2×2
The coherency between the two signals f and f is then
p q brightness matrix B. Carozzi & Woan (2009) show that the
given by the matrix B :
pq field coherencies can be written
(cid:18)E M (cid:19) (cid:18) B BFT (cid:19)
Bpq =(cid:10)fp⊗fq(cid:11) (40) B= Nppqq Hppqq = FB FBFT (49)
=(cid:104)(Tpf0)⊗(Tqf0)(cid:105) (41) where F is the matrix
=(cid:68)Tp(f0⊗f0)THq (cid:69) (42) F = c1 (cid:18)−01 01(cid:19). (50)
0
=TpBTHq (43) Undertheseconditions,therankofBis2,sotherelationship
we can write this in terms of block matrices inEq.44isoverconstrained.Itfollowsthatthe2×2RIME
is perfectly acceptable — and indeed preferable to Eq. 44
(cid:18)E M (cid:19) (cid:18)A B (cid:19)(cid:18)E M (cid:19)(cid:18)A B (cid:19)H
pq pq = p p 00 00 q q — for describing coherency of plane waves that propagate
N H C D N H C D
pq pq p p 00 00 q q through free space.
(44)
There are numerous situations in which we cannot as-
This is the most general form that relates the coherency at
sume that we have monochromatic plane waves. This in-
twopointsr andr ,totheelectromagneticenergydensity
p q cludesnearfieldsourceswherethewavefrontisnotwellap-
at point r .
0 proximated by a plane wave; propagation through ionized
In radio astronomy, antennas are used as sensors to
gas;andsituationswherewechoosenottotreatourfieldas
a superposition of quasi-monochromatic components. Most
4 Spatial location r is no longer relevant as the voltage propa- importantly,theassumptionsthatunderliethe2×2RIME
gates through analogue components with clearly defined inputs donotholdwithintheanaloguecomponentsofatelescope,
andoutputs. where the signal does not enjoy free space impedance.
(cid:13)c 2015RAS,MNRAS000,1–13
6 D.C. Price and O. Smirnov
4.3 A 2N-port transmission matrix based RIME i.e. the system has an effective Jones matrix (i.e. a voltage
transmission matrix) of
For a dual polarization telescope, a 4-port description (2-in
2-out)ofouranaloguesystemismoreappropriatethanthe Jp =Ap+BpF. (58)
general 6-port description. Using the result 49 above and
However, the Jones formalism breaks down when the sys-
Eq. 46, we can write a relationship
temiscomposedofmultiplecomponents.Forexample,with
2 components, we may naively attempt to apply Jones cal-
(cid:18)A B (cid:19)(cid:18) B BFT (cid:19)(cid:18)A B (cid:19)H culus, and describe the voltage transmission matrix of the
V = p p q q , (51)
pq C D FB FBFT C D systemasaproductofthecomponents’voltagetransmission
p p q q
matrices:
Here, all block matrices have been reduced in dimensions
from 3×3 to 2×2. This version of the RIME retains the Jp =(A2p+B2pF)(A1p+B1pF), (59)
abilitytomodelanaloguecomponents,butusestheapprox-
i.e.
imations of the vC-Z to express B in terms of the regular
2×2 brightness matrix B. The transmission matrices ma- vp =(A2pA1p+B2pFA1p+A2pB1pF +B2pFB1pF)e.
trices here are similar to the 2N-port transmission matrices (60)
as defined by Faria (2002).5 This,however,completelyneglectsthecurrenttransmission
The transmission matrices may be broken down into a properties. Applying the 2N-port transmission matrix for-
chain of cascaded components. That is, for a cascade of n malism, we can see the difference:
components, we may write the overall transmission matrix (cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19)(cid:18) (cid:19)
v A B A B e
asaproductofmatricesrepresentingtheindividualcompo- p = 2p 2p 1p 1p , (61)
i C D C D Fe
p 2p 2p 1p 1p
nents:
fromwhichwecanderiveanexpressionforthevoltagevec-
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) tor:
A B A B A B
p p = np np ··· 1p 1p (52)
C D C D C D v =(A A +B C +A B F +B D F)e, (62)
p p np np 1p 1p p 2p 1p 2p 1p 2p 1p 2p 1p
Thisissimilarto,butmoregeneralthan,Jonescalculus. which differs from Eq. 60 in the second and fourth term of
In the following section we will explore the difference. the sum, since in general
FA (cid:54)=C , FB (cid:54)=D . (63)
1p 1p 1p 1p
4.4 Limitations of Jones calculus
Tosummarize,becauseJonescalculusoperatesonvolt-
Jones calculus essentially asserts two things. Firstly, it as- agesalone,andignoresimpedancematching,wecannotuse
serts that the voltage 2-vector at the output of a dual- ittoaccuratelyrepresentthevoltageresponseofananalogue
polarizationsystemv =(v ,v )T islinearwithrespectto systembyaproductofthevoltageresponsesofitsindividual
p p1 p2
the EMF 2-vector e=(e ,e )T at the input of the system: components. The difference is summarized in Eqs. 60–63; a
x y
practicalexampleisgiveninSect.6.1.Bycontrast,the2N-
vp =Jpe, (53) port transmission matrix formalism does allow us to break
downtheoverallsystemresponseintoaproductofthecom-
where the Jones matrix J describes the voltage transmis-
p
ponentresponses,bytakingbothvoltagesandcurrentsinto
sion properties of the system. The second assertion is that
account.
forasystemcomposedofncomponents,theeffectiveJones
Since we have now shown the 2×2 form of the RIME
matrixisaproductoftheJonesmatricesofthecomponents:
to be insufficient, the obvious question arises, why have we
vp =Jnp···J1pe. (54) been getting away with using it? Historically, practical ap-
plications of the RIME have tended to follow the formula-
With 2N-port transmission matrices, we instead de-
tion of Noordam (1996), rolling the electronic response of
scribetheinputofthesystembythe4-vector[e,h]T,which
the overall system (as well as tropospheric phase, etc.) into
for a monochromatic plane wave is equal to
asingle‘G-Jones’termthatissolvedforduringcalibration.
(cid:18) (cid:19) (cid:18) (cid:19)
e e Under these circumstances, the 2×2 formalism is perfectly
=(e ,e ,−e /c ,e /c )T = , (55)
h x y y 0 x 0 Fe adequate – it is only when we attempt to model the in-
dividual components of the analogue receiver chain that its
andtheoutputofthesystemisa4-vectorof2voltagesand2
limitationsareexposed.Ontheotherhand,Carozzi&Woan
currents(v ,i )T,whichislinearwithrespecttotheinput:
p p (2009) have highlighted the limitations of Jones calculus in
(cid:18)v (cid:19) (cid:18)A B (cid:19)(cid:18) e (cid:19) the wide-field polarization regime.
p = p p . (56)
i C D Fe
p p p
Note that the output voltage is still linear with respect to
5 TENSOR FORMALISMS OF THE RIME
the input e. Indeed, if one is only interested in the voltage,
the above becomes Upuntilnow,wehavepresentedour6×6RIMEusingma-
trixnotation.Wenowbrieflydiscusshowtheworkpresented
v =(A +B F)e, (57)
p p p here is closely related to the tensor formalism presented in
Smirnov (2011b).
5 WenotethatourdefinitionhereistheinverseofthatofFaria As is discussed in Smirnov (2011b), the classical Jones
(inputandoutputareswapped). formulation of the RIME is in the vector space C2. The
(cid:13)c 2015RAS,MNRAS000,1–13
Generalized Formalisms of the RIME 7
formulation proposed by Carozzi & Woan (2009) is instead Let us now consider two antennas located at p and q.
in C3. In contrast, our Eq. 46 can be considered to work in The 2-point coherency measured in frame zero becomes
C6; that is, our EMF vector has 6 components:
[V ]αβγδ = 2c2(cid:104)[F (p¯)]αβ[FH(q¯)]γδ(cid:105)
pq 0
3 3 6 (cid:104) (cid:105)
ei = (cid:88)ejxj+(cid:88)hjxj ≡(cid:88)ejxj. (64) = Kp 2c2(cid:104)[F0(x¯0)]αβ[FH(x¯0)]γδ(cid:105) KqH, (70)
j=1 k=1 j=1
where K is the complex exponent of Eq. 69, and is the
p
The coherency of two voltages is once again defined via the directequivalentoftheK-JonestermoftheRIME(Smirnov
outerproducteie¯j,quiteremarkablygivinga(1,1)-typeten- 2011a).Thequantityinthesquarebracketsistheequivalent
sor expression identical Eq. 9 of Smirnov (2011b): of the brightness matrix B, which we’ll call the brightness
tensor:
Vpi =JpiBαJ¯β . (65)
qj α β qj
In the next section, we show an alternative RIME based
Bαβγδ =[B ]αβγδ =2c2(cid:104)[F (x¯ )]αβ[FH(x¯ )]γδ(cid:105). (71)
upon a (2,0)-type tensor commonly encountered in special 0 0 0 0 0
relativity.
Each element of the brightness tensor gives the coherency
between two components of the EMF observed in the cho-
senreferenceframe(‘framezero’).Nominally,thebrightness
5.1 A relativistic RIME tensorhas44 =256components,butonly36areuniqueand
non-zero (given the 6 components of the EMF). Carozzi &
Another potential reformulation of the RIME involves the
Bergman(2006)showthatthebrightnesstensormaybede-
electromagnetic tensor of special relativity. The classical
composed into a set of antisymmetric second rank tensors
RIME formulation implicitly assumes that both antennas
(‘sesquilinear-quadratic tensor concomitants’) that are irre-
measuretheEMFinthesameinertialreferenceframe.Ifthis
ducible under Lorentz transformations. The physical inter-
is not the case (consider, e.g., space VLBI), then we must
pretation of the 36 unique quantities within the brightness
in principle account for the fact that the observed EMF is
matrix is discussed in Bergman & Carozzi (2008), with re-
altered when moving from one reference frame to another,
gards to the aforementioned irreducible tensorial set.
andinparticular,thattheeandhcomponentsintermix.In
Whilethebrightnesstensorhasredundancynotpresent
special relativity, this can be elegantly formulated in terms
in the tensorial set of Carozzi & Bergman (2006), we shall
of the electromagnetic field tensor, which represents the 6
continuetouseitasabasistoourrelativisticRIMEforclar-
independent components of the EMF by a (2-0)-type ten-
ityofanalogytothebrightnesscoherencymatrixofEq.46,
sor:
and as it leads to a relativistic RIME for which we can
define transformation tensors analogous to Jones matrices.
 
0 −ex/c −ey/c −ez/c Theredundancycanbedescribedbyanumberofsymmetry
Fαβ = ex/c 0 −hz hy , (66) propertiesofthebrightnesstensor:itis(a)Hermitian with
 ey/c hz 0 −hx  respect to swapping the first and second pair of indices:
e /c −h h 0
z y x
Bαβγδ =B¯γδαβ, (72)
TheadvantageofthisformulationisthattheEMFten-
sorfollowsstandardcoordinatetransformlawsofspecialrel- (b) antisymmetric within each index pair (since the EMF
ativity.Thatis,foradifferentinertialreferenceframegiven tensor itself is antisymmetric, i.e. Fαβ =−Fβα):
by the Lorentz transformation tensor Λα , the EMF tensor
transforms as: α(cid:48) Bαβγδ =−Bβαγδ =−Bαβδγ (73)
F(cid:48)αβ =Λα Λβ Fα(cid:48)β(cid:48). (67) Toseethedirectanalogytothebrightnessmatrix,consider
α(cid:48) β(cid:48)
againthecaseofthemonochromaticplanewavepropagating
The measured 2-point coherency between [Fp] and [Fq] can alongz (Eq.48).TheEMFtensorthentakesaparticularly
be formally defined as the average of the outer product: simple form:
[Vpq]αβγδ =2c2(cid:104)[Fp]αβ[FqH]γδ(cid:105), (68)  0 −ex −ey 0 
where·H representstheconjugatetensor,i.e.[FH]γδ =F¯δγ. Fαβ = 1 ex 0 0 ex , (74)
Afactorof2isintroducedforthesamereasonsasinSmirnov c  ey 0 0 ey 
0 −e −e 0
(2011a),andthereasonforc2 willbeapparentbelow.Note x y
that the indices in the brackets should be treated as labels, andthebrightnesstensorhasonly82 =64non-zerocompo-
while those outside the brackets are proper tensor indices.
nents, with the additional ‘0-3’ symmetry property:
Letusnowpickareferenceframeforthesignal(‘frame
zero’),anddesignatetheEMFtensorinthatframeby[F ], B0βγδ =B3βγδ Bα0γδ =Bα3γδ
0
or [F0(x¯)] to emphasize that this is a function of the four- Bαβ0δ =Bαβ3δ Bαβγ0 =Bαβγ3 (75)
position x¯ = (ct,x). The [F (x¯)] field follows Maxwell’s
0
equations;inthecaseofamonochromaticplanewaveprop- Four unique components can be defined in terms of the
agating along direction z¯ = (1,z), this has a particularly Stokes parameters:
simple solution of
B0110 =I+Q B0220 =I−Q
[F (x¯)]=[F (x¯ )]e−2πiλ−1(x¯−x¯0)·z¯. (69) B0120 =U +iV B0210 =U −iV (76)
0 0 0
(cid:13)c 2015RAS,MNRAS000,1–13
8 D.C. Price and O. Smirnov
01 02 10 13 20 23 31 32 Finally, let us note the equivalent of the ‘chain rule’
for Jones tensors. If the signal chain is represented by a se-
01 −I I I −I
quence of Jones tensors [J ],...,[J ] (including K terms,
02 −I I I −I p,n p,1
10 I −I −I I Λterms,andallinstrumentalandpropagationeffects),then
13 I −I −I I the overall response is given by
20 I −I −I I
23 I −I −I I [Jp]ααβ(cid:48)β(cid:48) =[Jp,n]ααβnβn[Jp,n−1]ααnn−βn1βn−1···[Jp,1]αα2(cid:48)ββ(cid:48)2. (85)
31 −I I I −I
Equations80,81and85aboveconstitutearelativisticrefor-
32 −I I I −I
mulation of the RIME (RRIME). The RRIME allows us to
Table 1. The non-zero components of the brightness tensor incorporate relativistic effects into our measurement equa-
Bαβγδ for an unpolarized plane wave. Rows correspond to αβ, tion. That is, we can treat relativistic motion as an ‘instru-
columnstoγδ.
mental’effect.Someinterestingobservationalconsequences,
andtherelationoftheRRIMEtoworkfromotherfieldsare
and conversely, treated in the discussion below.
B0110+B0220 B0110−B0220
I = Q= ,
2 2
6 DISCUSSION
B0120+B0210 B0120−B0210
U = V = . (77)
2 2i The formulations presented here highlight the relationship
Theothernon-zerocomponentsofthebrightnesstensorcan between the seemingly disparate fields of microwave net-
bederivedusingtheHermitian,antisymmetryand0-3sym- working and special relativity to measurements in radio as-
metry properties. Finally, for an unpolarized plane wave, tronomy.Thisisaremarkableillustrationofhowthesefields
only 32 components of the brightness tensor are non-zero areintrinsicallyrelatedbyunderlyingfundamentalphysics.
and equal to ±I. As these will be useful in further calcula- Indeed,ithaslongbeenknownthatJonesandMueller
tions, they are summarized in Table 1. transformation matrices are related to Lorentz transforma-
So far this has been nothing more than a recasting of tions by the Lorentz group. Wiener was aware as early as
the RIME using EMF tensors. Consider, however, the case 1928 that the the 2×2 coherency matrix could be written
whereantennaspandq measurethesignalindifferentiner- in terms of the Pauli spin matrices (Wiener 1928, 1930).
tialreferenceframes.TheEMFtensorobservedbyantenna More recently, Baylis et al. (1993) presents a more general
p becomes geometric algebra formalism that unifies Jones and Mueller
matrices with Stokes parameters and the Poincar´e sphere.
[Fp]αβ =[Λp]αα(cid:48)[Λp]ββ(cid:48)[F0]α(cid:48)β(cid:48), (78) Hanetal.(1997b)showthattheJonesformalismisarepre-
sentationofthesix-parameterLorentzgroup;furthertothis
where[Λ ]αistheLorentztensorcorrespondingtothetrans-
p µ Han et al. (1997a) show that the Stokes parameters form a
formbetweenthesignalframeandtheantennaframe.Since
Minkowskian four-vector, similar to the energy-momentum
thesameLorentztensoralwaysappearstwiceintheseequa-
four-vector in special relativity.
tions (due to F being a (2,0)-type tensor), let us designate
In contrast, the RRIME presented here is novel as it
[Λp]ααβ(cid:48)β(cid:48) =[Λp]αα(cid:48)[Λp]ββ(cid:48) (79) fully describes interferometric measurement between two
points, instead of simply describing polarization states and
for compactness. The measured coherency now becomes
defining a transformation algebra. Similarly, the 2N-port
[V ]αβγδ =K [Λ ]αβ [B ]α(cid:48)β(cid:48)γ(cid:48)δ(cid:48)[Λ ]γδ KH. (80) RIME specifically shows how microwave networking meth-
pq p p α(cid:48)β(cid:48) 0 q γ(cid:48)δ(cid:48) q
ods can be incorporated into a ME.
TheequivalentofJonesmatriceswouldbe(2,2)-type‘Jones
What do we gain from using these more general mea-
tensors’ Jαβ , so a more general formulation of the above
α(cid:48)β(cid:48) surement equations in lieu of the simpler 2×2 RIME? For
would be
most applications, it will suffice to simply be aware of the
[V ]αβγδ =[J ]αβ [B ]α(cid:48)β(cid:48)γ(cid:48)δ(cid:48)[JH]γδ , (81) standard RIME’s limitations, and to work in a piecemeal
pq p α(cid:48)β(cid:48) 0 q γ(cid:48)δ(cid:48)
fashion. For example, one can quite happily use Jones ma-
where tensor conjugation ·H is defined as: tricestodescribefreespacepropagation,andmicrowavenet-
[JH]αβ =J¯βα . (82) workmethodstodescribediscreteanaloguecomponents.An
α(cid:48)β(cid:48) β(cid:48)α(cid:48) overall‘systemJones’matrixmaybederivedtodescribein-
NotethatboththeK andΛtermsofEq.80canbeconsid- strumental effects, but this matrix should never be decom-
ered as special examples of Jones tensors. The K term can posed into a Jones chain.
beexplicitlywrittenasa(2,2)-typetensorviatwoKronecker Similarly, special relativity describes relativistic effects
deltas: throughLorentztransformationsactingupontheEMFten-
[K ]αβ =K δαδβ , (83) sor. We can treat relativistic motion as an instrumental ef-
p α(cid:48)β(cid:48) p α(cid:48) β(cid:48) fectbyusingtheRRIME,orwecanapplyspecialrelativistic
whiletheΛtermisa(2,2)-typetensorbydefinition(Eq.79), corrections separately, as required. The effect of relativistic
noting that boosts on the Stokes parameters are considered in Cocke &
Holm (1972).
[ΛH]αβ =[Λ ]αβ , (84)
p α(cid:48)β(cid:48) p α(cid:48)β(cid:48) In the subsections that follow, we present some poten-
since the components of any Lorentz transformation tensor tial use cases that highlight the usefulness of the 2N-port
Λα are real. and relativistic RIME formulations.
α(cid:48)
(cid:13)c 2015RAS,MNRAS000,1–13
Generalized Formalisms of the RIME 9
6.1 Modelling real analogue components
The 2N-port RIME may have practicality in absolute flux B
calibration of radio telescopes. Generally, interferometer K A
ADC
data is calibrated by assuming that the sky brightness is sky propagation antenna response
known, or at least approximately known. If we wish to do G
absolute calibration of a telescope without making assump- analogue chain digitizer
D L
tions about the sky, we must instead make sure that we
accuratelymodeltheanaloguecomponentsofourtelescope.
Aparticularlyubiquitousmethodofdevicecharacterization noise diode RF load
withinmicrowaveengineeringistheuseofscatteringparam-
Figure 3.Blockdiagramofathree-stateswitchingexperiment.
eters; we briefly introduce these below before incorporating
In addition to the antenna path, an RF load can in series with
them into the 2N-port RIME. a noise diode can be selected. The noise diode can be turned on
and off, giving three possible states: antenna, load, and diode +
load.Thesethreestatesareusedforinstrumentalcalibration.
6.1.1 Scattering parameters
Voltage,currentandimpedancearesomewhatabstractcon- simple, and the off-diagonal entries of the block matrices of
cepts at microwave frequencies, so engineers often use scat- T will no longer be zero.
tering parameters to quantify a device’s characteristics.
Scatteringparametersrelatetheincidentandreflectedvolt-
6.1.2 Scattering matrix example
age waves on the ports of a microwave network. The scat-
tering matrix, S, is given by We now present a simple illustration of the differences be-
tweenassigningacomponentascalarvalue(asisdoneinthe
v− S11 S12 ··· S1nv+ Jones formalism), and by modelling it as a 2-port network.
vv...12−−=S...21 ... ...... vv...12++, (86) Consider a component(cid:18)w0i.t1h∠a0◦sca0tt.9er∠in0g◦(cid:19)matrix
n Sn1 ··· ··· Snn n S = 0.9∠0◦ 0.1∠0◦ (92)
where vn+ is the amplitude of the voltage wave incident on foragivenquasi-monochromaticfrequency.Here,valuesare
portn,andvn−istheamplitudeofthevoltagewavereflected presentedinanglenotationtoemphasizethattheyarecom-
from port n. plexvalued.Indecibels,theS andS parametershavea
11 22
For a dual polarization system (we will label the po- magnitude of -10 dB, and the S and S parameters have
12 21
larization x and y) with negligible crosstalk, we can model amagnitudeofabout−0.5dB.Nowsupposewehavethree
theanaloguechainforeachpolarizationasadiscrete2-port identicalcopiesofthiscomponent,andweconnectthemto-
network. Assuming that the analogue chains have the same gether in cascade. If one only considered the forward gain
number of components (but not that the components are (S ), one would arrive at an overall S of
21 21tot
identical), the transmission matrix for each pair of compo-
S =S S S =0.729 (93)
nents is 21tot 21 21 21
A˜ 0 B˜ 0  Incontrast,usingthestandardmicrowaveengineeringmeth-
x x
T =C˜0x A˜0y D˜0x B˜0y, (87) o91dsa,bwoevec)a,nanfodrmtheantrcaonnsvmeritsstihoinsmbaactkriixnt(ousainngoveeqruaalltimonastr8i8x-,
0 C˜ 0 D˜ Stot. By doing this, one finds
y y
(cid:18)0.25∠0◦ 0.75∠0◦(cid:19)
where the elements are from the ABCD matrices of the x Scas = 0.75∠0◦ 0.25∠0◦ . (94)
and y polarizations, and are given by the relations
The difference becomes more marked as S and S in-
11 22
1+S S +S −S (1+S )
A˜= 12 21 22 11 22 (88) crease;asS11 andS22 approachzero,thetwomethodscon-
2S12 verge.
B˜ =−Z 1+S11−S12S21+S22+S11S22 (89) Whendesigningacomponent,S11andS22aregenerally
0 2S12 optimised to be as small as possible over the operational
C˜ = 1 −1+S11+S12S21+S22−S11S22 (90) bandwidth6.Nevertheless,theiraffectontheoverallsystem
Z 2S is often non-negligible.
0 12
1+S +S S −S −S S
D˜ = 11 12 21 22 11 22 (91)
2S
12 6.1.3 Absolute calibration experiments
Here, Z is the characteristic impedance of the analogue
0
Nowwehaveshownhowscatteringparameterscanbeused
chain, which in most telescopes is set to 50 or 75 ohms. We
withinthe2N-portRIME,weturnourattentiontothechal-
haveaddedtildestotheABCD parameters,asweareusing
the inverse definition to that in Pozar (2005).
If the system has significant crosstalk between polar- 6 AnotableexceptionisRFfilters,forwhichS11isoftencloseto
izations, the analogue chain is more accurately modelled as unityout-of-band.InsuchcasesS11 isstronglydependentupon
a 4-port network. In this case the relationships are not so frequency.
(cid:13)c 2015RAS,MNRAS000,1–13
10 D.C. Price and O. Smirnov
lenges of absolute calibration of radio telescopes. Measure- This effect is a prime example why one must not use Jones
mentoftheabsolutefluxofradiosourcesisfiendishlyhard; matrices to describe analogue components separately; an
as such, almost all calibrated flux data presented in radio example showing a standing wave present on three-state
astronomy literature are pinned to the flux scale presented switchcalibratedspectrumfromaprototypeLEDAantenna
in Baars et al. (1977), either directly or indirectly (Keller- is shown in Figure 4.
mann2009).Recentexperiments,suchastheExperimentto We may instead write the equations above in terms of
Detect the Global EoR Step (EDGES, Bowman & Rogers 2N-port transmission matrices, and form MEs for the three
2010), and the Large-Aperture Experiment to detect the states:
Dark Ages (LEDA, Greenhill & Bernardi 2012), are seek-
V = G(T B TH+B )GH (100)
ing to make precision measurement of the sky temperature A A sky A rx
(i.e. total power of the sky brightness) as a function of fre- V = G(B +B )GH (101)
L L rx
quency. The motivation for this is to detect predicted faint V = G(T B TH+B +B )GH. (102)
D L D L L rx
(mK) spectral features imprinted on the sky temperature
duetocouplingbetweenthemicrowavebackgroundandneu- Here,wehavereplacedthescalarpowersPA,L,D withcorre-
tralHydrogenintheearlyuniverse.Forsuchinstruments— sponding voltage-coherency matrices VA,L,D, temperatures
and other instruments with wide field-of-views — this flux TA,L,sky,rx arereplacedwithbrightnessmatricesBA,L,sky,rx,
‘bootstrapping’ method is not sufficient. andGkB isinsteadrepresentedbyasystemgainmatrixG.
For experiments such as EDGES a thorough under- We have added a transmission matrix for the antenna TA,
standingoftheanaloguesystemsisvitaltocontrolthesys- andatransmissionmatrixTL fortheloadinserieswiththe
tematicsthatconfoundcalibration.Thecalibrationstrategy noisediode.NotethattherelationofEq.99isnowencoded
used for EDGES is detailed in Rogers & Bowman (2012), into the ME by the matrix TA, and that we have dropped
and consists of a novel three-state switching and rigorous antennanumbersubscriptsasp=q forautocorrelationmea-
scattering parameter measurements with a Vector Network surements.
Analyzer (VNA) to remove the bandpass and characterize It is immediately apparent that the cancellations that
the antenna. occur in the ratio of Eq. 98 will not in general occur for
Our 2N-port RIME allow for scattering parameters to the equivalent ratio R=(VA−VL)(VD−VL)−1. We can
be directly incorporated into a measurement equation that however retrieve the result of Eq. 98 by treating the two
relatesskybrightnesstomeasuredvoltagesatthedigitizer. polarizations separately, setting G=GI and TL =I, with
Thiscouldbeusedeitherto(a)inferthescatteringparame- TA =TA(1−|Γ|2)I.
tersofadevicegivenaknownskybrightness,or(b)inferthe OurequationsEq.100-102allowforbothpolarizations
sky brightness from precisely measured scattering parame- to be treated together, which will be important if cross-
ters. The EDGES approach is the latter, but via an ad-hoc polarization terms are non-negligible. Also, we may expand
method without formal use of a ME. Eq. 100 to include ionospheric effects. This ability to com-
bine all effects into a single ME may simplify data analysis
and improve calibration accuracy for such experiments.
6.1.4 A three-state switching measurement equation
The three-state switching as described in Rogers & Bow-
man (2012) involves switching between the antenna and a
6.2 RRIME and Lorentz boosts
referenceloadandnoisediode,andmeasuringtheresultant
powerspectra(P ,P ,andP ,respectively).Thisisshown
A L D To illustrate how the RRIME can be used to describe rel-
as a block diagram in Figure 3. The power as measured in
ativistic effects, let us first consider the ‘simple’ case of a
each state is then given by:
relativisticallymovingsource.Supposewehaveanunpolar-
P = Gk ∆ν(T +T ) (95) izedpointsource(Table1),withantennaspandqlocatedin
A B A rx
thexyplaneperpendiculartothedirectionofpropagationz
P = Gk ∆ν(T +T ) (96)
L B L rx (sotheK componentbecomesunity),bothmovingparallel
PD = GkB∆ν(TD+TL+Trx) (97) to the x axis with velocity v. The Lorentz transformation
where k is the Boltzmann constant, T and T are the tensor from the signal frame to the antenna frame is then
B D L
diode and reference load noise temperatures, T is the re-
rx
ceiver’s noise temperature, G is the system gain, and ∆ν is  
γ −βγ 0 0
bandwidth.OnecanrecovertheantennatemperatureT by
T =T PA−PL +T , A(98) [Λp]αα(cid:48) = −0βγ γ0 10 00 , (103)
A DPD−PL L 0 0 0 1
where the diode and load temperatures, T and T , are
D L
where
known.Antennatemperatureisthenrelatedtotheskytem-
perature by v 1
β = , γ = . (104)
T =T (1−|Γ|2), (99) c (cid:112)1−β2
sky A
where the reflection coefficient Γ ≡ S . Failure to account TheLorentzfactorγ isunity1atv=0,andgoestoinfinity
11
forreflections(i.e.impedancemismatch)resultsinanunsat- with v → c. This case can be analyzed without invoking
isfactorycalibration,duetostandingwaveswithinthecoax- coherencies.ConsidertheEMF,whichinthesignalframeis
ialcablethatmanifestasasinusoidalrippleonT =T (ν). a plane wave (Eq. 74). In the antenna frame it becomes
A A
(cid:13)c 2015RAS,MNRAS000,1–13