ASYMMETRIC BEAMS IN COSMIC MICROWAVE BACKGROUND ANISOTROPY EXPERIMENTS

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1 Preprint typeset using L A TEX stye emuateapj v. 4/3/99 ASYMMETRIC BEAMS IN COSMIC MICROWAVE BACKGROUND ANISOTROPY EXPERIMENTS J. H. P. Wu 1,A.Babi,3,4,J.Borri 5,3,P.G.Ferreira 6,7, S. Hanany 8,3,A.H.Jaffe 3,9,1, A. T. Lee 1,3,9,S.Oh 3,1,B.Rabii 3,1,P.L.Richards 1,3,G.F.Smoot 1,3,4,9, R. Stompor 3,9,11, C. D. Winant 3,1 Draft version Juy 18, ABSTRACT We propose a new formaism to hande asymmetric beams in the data anaysis of cosmic microwave background anisotropy experiments. For any beam shape, the formaism finds the optima circuary symmetric equivaent and is thus easiy adaptabe to existing data anaysis methods. We demonstrate certain key points by using a simuated highy eiptic beam, and the beams and data of the MAXIMA-1 experiment, where the asymmetry is mid. In particuar, we show that in both cases the formaism does not bias the anguar power spectrum estimates. We anayze the imitations of the formaism and find that it is we suited for most practica situations. Subject headings: cosmic microwave background cosmoogy:theory arge-scae structure of the universe methods:numerica 1. INTRODUCTION A new generation of Cosmic Microwave Background (CMB) mapping experiments is beginning to produce data of unprecedented quaity (see e.g., Torbet et a. 1999; Mier et a. 1999; DeBernardis et a. ; Hanany et a. ). Much of the experimenta effort is concentrated on probing anguar scaes of about 1 arcminutes. To fuy benefit from the scientific potentia of these high resoution data sets, a new eve of sophistication is required in quantifying a possibe sources of error in the experimenta procedure and data anaysis pipeine (e.g., Ferreira & Jaffe ). Particuar care must be used to accuratey quantify the instrument response to the signa and to incude such response in the data anaysis. In a anayses of CMB data so far the experimenta beam has been assumed to have a radia symmetry. This assumption has been incorporated in most map-making and anguar power spectrum (C ) estimation agorithms (e.g., Bond, Jaffe, & Knox 1998) and is necessary because of imitations in computing capabiity. A crude symmetricbeam approximation was adequate in the past since most of the error budget was dominated by statistica and other systematic uncertainties. However, with the precision of current and future anayses, it becomes essentia to estabish a methodoogy for accuratey quantifying the degree of beam-asymmetry and propery incorporating it into the data anaysis pipeine. If the beam is incorrecty incorporated in the data anaysis pipeine, one may not ony artificiay distort the underying structure of the measured CMB signa but aso bias the estimate of the CMB anguar power spectrum. In this paper we present a new formaism for estimating the power spectrum that can hande any beam shape. We show that the formaism can be appied to a broad variety of cases which encompass most practica appications. As a consequence, the detaied shape of the antenna beam shoud no onger pose a imitation in measuring the anguar power spectrum of CMB experiments. The asymmetry of beams may arise from a variety of sources. For exampe, it may be due to the optics, or due to the finite response time of a detector which eaves imprints in the direction of the scan (e.g., Hanany, Jaffe, & Scannapieco 1998). Regardess of the origins of the asymmetry, the framework we sha present is genera, and consists of finding an equivaent symmetric beam that repaces the asymmetric beam in the anaysis of the data. Using the formaism one can assess the degree of asymmetry of a beam (see eq. [3-4]), how the asymmetry propagates through the anaysis pipeine, and how to find an azimuthay symmetrized beam that best approximates the asymmetric beam (see e.g., eq. [5-9]). The symmetrized beam is then used in the symmetric-beam approximation of the C estimation (see eq. [5-11]). The formaism quantifies the errors introduced in the C estimates because of the use of the symmetrized-beam approximation, the uncertainty in the fina C estimates resuting from the uncertainty in the beam measurement (see eq. [7-5]), and the smoothing effects due to the pixeization of the map 1 Dept. of Astronomy, University of Caifornia, Berkeey, CA, USA Dipartimento di Fisica, Università Tor Vergata, Roma, Itay 3 Center for Partice Astrophysics, University of Caifornia, Berkeey, CA, USA 4 Lawrence Berkeey Nationa Laboratory, University of Caifornia, Berkeey, CA, USA 5 Nationa Energy Research Scientific Computing Center, Lawrence Berkeey Nationa Laboratory, Berkeey, CA, USA 6 Astrophysics, University of Oxford, UK 7 CENTRA, Instituto Superior Tecnico, Lisboa, Portuga 8 Schoo of Physics and Astronomy, University of Minnesota/Twin Cities, Minneapois, MN, USA 9 Space Sciences Laboratory, University of Caifornia, Berkeey, CA, USA 1 Dept. of Physics, University of Caifornia, Berkeey, CA, USA 11 Copernicus Astronomica Center, Warszawa, Poand 1

2 (see eqs. [8-8] and [8-9]). It aso shows how to combine beams from independent experimenta photometers (see eqs. [3-5], [4-1], [4-11], and [4-1]). Some usefu conditions under which this new formaism wi be needed are aso provided (see eqs. [6-5] and [6-7]). The structure of this paper is as foows. In section, we describe the framework of CMB data anaysis for the estimation of the power spectrum, so as to iustrate the probems reated to asymmetric beams. In section 3, we define the index of asymmetry (IOA) ϖ, a usefu parameter in quantifying the eve of asymmetry of a beam. Simiary, we define the index of combined asymmetry (IOCA) W, which is usefu when combining data from photometers of different beam shapes. In sections 4 and 5, we investigate the probems associated with asymmetric beams. We introduce the average pixe-beam expansion, B pm, and the pixe-pixe beam expansion, B(eff), to provide an approximation scheme where the convoution effect of asymmetric beams is treated as circuary symmetric. The biasing effects of this approximation in the resuting estimated power spectrum C are aso considered. In section 6, we derive the conditions under which one needs to empoy the new formaism for treating asymmetric beams. In section 7, we investigate the uncertainties in the C estimates resuting from the uncertainties in the measurement of beam shape. In section 8, we discuss another convoution effect due to the pixeization of the CMB map. Athough this is not a beam-reated issue, we demonstrate a simpe way to incorporate its treatment into our framework. In section 9, we numericay verify certain key points deveoped in sections 3 to 8, as we as the accuracy of the proposed approximation in treating asymmetric beams. In particuar, we use the data from the MAXIMA-1 experiment as an exampe to demonstrate the generic treatment of asymmetric beams in CMB experiments. It is shown that our formaism has no biasing effects in the resuting C estimates. Finay in section 1, we summarize the procedure in appying our formaism to experiments, discuss its avaiabiity, and draw a concusion.. THE CONVENTION AND PROBLEMS We first consider the standard procedure for the power spectrum estimation. This consists of two main steps. First, one estimates the pixeized map m p from a given time-stream d t, i.e., to transate the observation from the tempora (t) to the spatia (p) domain. Second, one estimates the power spectrum C from the map m p. In the tempora domain, what we observe is d t = γ t + n t, (-1) where γ t is the CMB signa and n t is the instrumenta noise. Traditionay we mode the CMB observation as γ t = A tp s p, (-) where we use the Einstein summation convention here and beow when appropriate (usuay over pixes and time sampes, but not over spherica harmonic indices). Here A tp is the pointing matrix giving the weight of pixe p in observation t, ands p is the CMB signa on the pixe convoved by a pixe beam B p (x): s p = B pm a m Y m (x p ), (-3) = m= where Y m are the spherica harmonics, and B pm and a m are the mutipoe expansions of B p (x) and the CMB signa respectivey. Note that we use a two-dimensiona vector x to denote ocations on the surface of the sphere, which we sha often consider in the sma-fied imit (see ater). We usuay take the pointing operator A tp to be one when observing pixe p at time t and zero otherwise. That is, we mode the signa γ t to be the same for any observation within pixe p. In effect, we take the sky to be smoothed with a top-hat of shape given by the pixe boundary. We sha see in section 8 that, as expected, this is equivaent to an extra convoution incuded in B p. With this modeing, one can thus estimate the pixeized map from the tempora data. This invoves maximizing the ikeihood of the signa given the data: L(s) Prob[d s] =(π) Nt/ { exp 1 ( n T N 1 n + Tr[n N] )}, (-4) where d d t, s s p,andn n t, a as defined in equations (-1) and (-), N t is the size of the time-stream, and N N tt = n t n T t is the time-time noise correation matrix. Here we have assumed that the noise is Gaussian and that a CMB maps are a priori equay ikey. Maximizing over s gives m p m =(A T N 1 A) 1 (A T N 1 d) = s p + n p, (-5) where A A tp as defined in equation (-) and n p is the noise in the pixe domain. One then moves on to estimate the power spectrum of the map, C = a m. This requires the maximization of the ikeihood function L = Prob[m C ]=(π) NC / { exp 1 ( m T M 1 m + Tr[n M] )}, (-6) where N C is the dimension of the parameter space of C, and M M pp = C Spp + C Npp, (-7) with C Spp = C B pm Bp m = m= Y m (x p )Ym(x p ), (-8) C Npp =(A T N 1 A) 1. (-9) Here C Spp = s p s T p is the pixe-pixe CMB signa correation matrix, and C Npp = n p n T p is the pixe-pixe noise correation matrix. We note first that in the estimation of C, athough there exists methods ike the quadratic estimator (Bond et a. 1998) which avoid a direct evauation of equation (-6), the reationship between the beam expansion B pm and the

3 3 power spectrum C remains the same and is iustrated in equation (-8). Second, if the beam is identica for a pixes and circuary symmetric, i.e., B pm = B p m = B, then equation (-8) can be greaty simpified as C Spp = = +1 4π C B P (cos θ pp ), (-1) where P is the Legendre function and θ pp = x p x p is the anguar distance between the pixes. Generay it is impractica to estimate C for a due to the constraints of finite sky coverage and computation power. Instead, one divides the accessibe -range constrained by the sky coverage and the observing beam size into severa bands {b}, and then estimates the band power C b, i.e., one approximates C in the form C C b C sh, (-11) where C sh is a chosen shape function characterizing the scae dependence in each band. For exampe, one can choose C sh 1 = ( +1), (-1) which eads to a scae-invariant form in each band, i.e., (+1)C =const b. With the approximation (-11), one can rewrite equation (-8) as where C Spp b C b K pp b[c sh,b p,b p ], (-13) K pp b[c sh,b p,b p ]= b m= C sh B pm Bp m Y m(x p )Ym (x p ). (-14) If the beam is symmetric, then one has from equation (-1) or (-13) that where C Spp b C b K b [θ pp ; C sh,b ], (-15) K b [θ pp ; C sh,b ]= b +1 4π Csh B P (cos θ pp ). (-16) In the anaysis procedure outined above, the first probem arises in equations (-) and (-3). Stricty speaking, what is convoved in reaity is not the pixe temperature in s p itsef but the CMB signa in the time-stream γ t, i.e., γ t = = m= B tm a m Y m (x t ), (-17) where B tm is the mutipoe expansion of the time-stream beam B t (x). This means that the experiment gives us a beam which moves on the sky as a function of time, t, and indeed may observe a different signa within the same pixe, p, depending on the orientation of the beam and the ocation of its center. We thus make a map which may have many different beams contributing to a singe pixe. However, in our anaysis formaism we must actuay express this map as in (-3), an observation of the sky with ony a singe pixe beam, B p. Hence, for the C estimation, we need to find a way to estimate the pixe-beam expansion B pm from the B tm, and this wi be the focus of sections 4 and 8. The second probem appears in equation (-13). If the beam is not symmetric, the summation over m and the dependence on the pixe pair make the exact computation prohibitivey expensive. To resove this probem, in section 5 we introduce the pixe-pixe beam expansion B(eff), which provides a consistent way to symmetrize asymmetric beams. This B(eff) then repaces the B in equation (-15), so as to approximate equation (-13). On genera grounds, the size of the observing beam is so sma that, when necessary, we sha use the fat sky approximation under the sma-fied imit. This means that when the size of a spherica patch is sufficienty sma, the expansion of the beam in spherica harmonics is equivaent to a Fourier transform on a fat two-dimensiona patch, i.e., B m = dωb(x)y m (x) dx B(x)e ik x = B(k), (-18) and k = k. (-19) Throughout the paper, we sha use a tide to denote the Fourier transform of a quantity. 3. THE CRITERIA FOR BEAM SYMMETRY It is important to ceary define the eve of asymmetry of an antenna beam. Consider the mutipoe expansion B m of the beam. For a given, the variance of B m about its mean over m is η = B (ms) B (sm), (3-1) where B(ms) is the mean of squares over m: B (ms) = B m, (3-) and B(sm) is the square of the mean over m: B (sm) = [ B (m) ] = Bm. (3-3) Here B (m) is the mean of B m over m, and therefore can be either positive or negative. We aso note that B(ms) is the power spectrum of the asymmetric beam, and that B(sm) is equivaent to the power spectrum of a symmetric beam that is azimuthay averaged in the rea space. Based on this, one can define an index of asymmetry (IOA) as ϖ = η B (ms) = [ ] 1/ 1 B (sm) B(ms). (3-4) We see that ϖ varies from zero to one the arger the ϖ, the more asymmetric the beam. We aso note that if the beam is symmetric, then ϖ is exacty zero. Thus for a

4 4 given beam, ϖ provides us an objective measure of the eve of its asymmetry. In certain situations, we need to combine data from two or more photometers with different beam shapes. We sha use a subscript i (i =, 1,, etc.) to denote the quantities obtained from different photometers. As an anaog to equation (3-4), it proves usefu to define an index of combined asymmetry (IOCA) for a the beams as [ ] 1/ W = 1 B Σ(sm) BΣ(ms), (3-5) B pm = B tm. This is of course true when the pixe size is infinitesima, but is unikey to be fufied in reaity. Nevertheess, equation (4-1) is just the resut of the modeing and therefore not necessariy a requirement in practice. In our formaism for the power spectrum estimation, the s p is an unknown quantity to be estimated by using equation (-5), so the actua reation between B pm and B tm shoud be aso obtained through the same process. First, we substitute equation (-1) into (-5), and the CMB signa part yieds s p = C N A T N 1 γ t, (4-) where B Σ(sm) = [ i ζ i B i(m) ], (3-6) where C N C Npp as defined in equation (-9). Further substituting equations (-3) and (-17) into this resut, we obtain B pm Y m (x p )=C N A T N 1 B tm Y m (x t ). (4-3) B Σ(ms) = [ i ζ i B i(ms)], (3-7) the B i(m) and Bi(ms) are the B (m) and B(ms) of photometer i respectivey, ζ i = i t i(obs) /NET i [ ti(obs) /NET ], (3-8) i t i(obs) is the tota observation time of photometer i, and NET i is its noise equivaent temperature (NET). Here BΣ(sm) is the square of the noise-weighted mean of B i, and BΣ(ms) is the noise-weighted mean of the squares of B i assuming a the B i are fuy correated. As one can see, the W varies between zero and one the arger the W, the more asymmetric a ζ i -weighted combined beam can be (depending on the detaied orientations of the beams in the tempora sampes; we sha discuss this ater). This aso means that the IOA (ϖ ) of an average beam with a weight ζ i for each B i is aways equa to or smaer than W, athough the individua ϖ i of B i may be arger than W. If W =, then we know that a the beams are symmetric (ϖ i =),andviceversa. For the purpose of power spectrum estimation, one can empoy ϖ i (or W when combining data of different observing beams) to decide if a simpe symmetric-beam approximation is sufficient. For exampe, at s where ϖ, we expect equation (-15) to be adequate. On the other hand, at s where ϖ (or W ) deviates significanty from zero, one may need to empoy equation (-13). We sha further discuss these situations, and the use of the IOA (ϖ )andtheioca(w )ater. 4. THE AVERAGE PIXEL-BEAM EXPANSION 4.1. The pixe-beam expansion We first estimate the pixe-beam expansion, B pm, from given observing beams, B tm. A naive way to investigate this is to substitute equations (-3) and (-17) into the mode (-), eading to A tp B pm Y m (x p )=B tm Y m (x t ). (4-1) This equation hods if and ony if there exists one x p for every x t such that x t = x p. In this case, we have This equation is competey genera, and shoud be in principe satisfied when one tries to find the B pm from the given B tm. We thus see that equation (4-1) is just one of the soutions to equation (4-3), but not necessariy a requirement for the purpose of power spectrum estimation. In most cases, the noise n t in each tempora measure is neary independent from the others, so the time-time noise correation matrix N tt is diagona, with the tt eements equa to the noise variance at each time sampe, i.e., N tt = µ t δ(t t ), (4-4) where µ t is the standard deviation of time sampe t, and δ(t t ) is a Dirac Deta. This aows us to simpify equation (4-3) as B pm Y m (x p )= ξ t B tm Y m (x t ), (4-5) t p where ξ t is the noise-estimated statistica weight at t: ξ t = µ t t p µ t. (4-6) For simpicity, we sha take this white-noise assumption for further investigation. We consider the more genera case of correated noise in the Appendix, and show that this white-noise approximation is appropriate in most practica cases. The conditions for the use of this whitenoise assumption wi be aso derived in the Appendix (see eq. [1]). To further simpify equation (4-5), we assume that x t x p t p (i.e., the tempora measure γ t is thought of as a sampe of the pixe temperature s p ; see eq. [4-]), so that the pixe-beam expansion can now be obtained as B pm = ξ t B tm. (4-7) t p The assumption, x t x p, for achieving this resut wi be reaxed in section 8, where we show that ony an extra correction is required.

5 5 4.. The average pixe-beam expansion As wi be shown, it proves usefu to remove the pixe dependence of B pm in the formaism of the C estimation. We thus consider the noise-weighted average of B pm over a pixes (c.f. eq. [-5]): B pm = H(U T C 1 N B pm), (4-8) where U U p is a contraction vector with entries a equa to unity, and H =(U T C 1 N U) 1. (4-9) We sha ca B pm the average pixe-beam expansion. We note that the subscript p in B pm does not mean the pixe dependence as in the usua convention, but indicates that this is a mean taken over a pixes. With the white-noise assumption (eq. [4-4]), the B pm can be cacuated expicity by substituting equation (4-7) into equation (4-8): where B pm = t χ t = µ t χ t B tm, (4-1) t µ t. (4-11) If the data are from a singe photometer with a constant noise eve, then equation (4-1) reduces to a simpe inear average of a time-stream beams. If the data are combined from different photometers, then the µ t can be approximated as (c.f. eq. [3-8]) µ t = NET t, (4-1) δt(obs) where NET t is the NET of the corresponding photometer at time t, and δt (obs) is the integration time of the tempora observation at t. If the integration time remains unchanged among photometers, then the µ t in equation (4-11) can be simpy taken as the NET of the corresponding photometer. We aso note that with the definition (4-1), equations (3-8) and (4-11) can be reated as ζ i = χ t, (4-13) t i meaning that ζ i is the tota noise-estimated weight of photometer i. We note that in cases where both the shape of the experimenta beam and its orientation reative to the pixe are roughy constant throughout the observation, we have a reasonabe approximation (see eq. [4-1]): B pm B tm. (4-14) In other cases, equation (4-1) wi need to be empoyed, for exampe, when the reative orientation between the asymmetric beam and the pixes changes, or when data from different photometers are combined together. We aso note that even if a the beams B i of different photometers are symmetric (i.e., ϖ i = W = ), the B pm may sti have pixe dependence due to the various reative contribution of B i within different pixes (see eq. [4-7]). In such cases, one wi need to consider equation (4-1), and a simpe formaism ike equation (-15) wi be invaid for the estimation of the CMB anguar power spectrum, since the Bp is different on each pixe. As wi be shown, the formaism we sha deveop is aso capabe of deaing with this situation Usefu Limits We now derive usefu constraints on the magnitude of the average pixe-beam expansion B pm. In the sma-fied imit, the power spectrum of B pm can be written as (see eqs. [-18], [3-], and [4-1]) B p(ms) 1 π dϕ χ t Bt (k), (4-15) π where ϕ is the phase ange of k on the ring k = k. We first consider singe-photometer experiments. In this case, if the beam pattern remains the same throughout the entire observation but with ony different orientations at different t, then we can rewrite B t as t B t = A(β t )B, (4-16) where A(β t ) is the rotation matrix, β t is the rotation ange at time t with respect to t =,andb is the shape of the time-stream beam at t =. Substituting this into equation (4-15) gives B p(ms) 1 π π dϕ π f(β)a(β) B (k)dβ, (4-17) where f(β) is the weighting function of a rotation ange β, and satisfies π dβf(β) = 1. It is then straightforward to show that the function f(β) that minimizes the right hand side of the above equation is f(β) =1/π, eading to [ B 1 π ] p(ms) dϕ min π B (k) B(sm), (4-18) where B(sm) is as defined in equation (3-3). On the other hand, the function f(β) that maximizes the right hand side of equation (4-17) is f(β) =δ(β β ) (Dirac Deta, β {, π}), and this gives B p(ms) 1 π dϕ max π B (k) B(ms), (4-19) where B(ms) is as defined in equation (3-). These resuts te us that when the pixes are scanned amost uniformy in a directions, then the resuting B p(ms) shoud be coser to B p(ms) = B(sm). When the pixes are min scanned with an amost fixed direction, then the resuting B p(ms) shoud be coser to B p(ms) = B(ms). Thus, max we have a good check of the numericay cacuated B pm from equation (4-1) (or eq. [4-17]), i.e., a constraint on the ampitude of B p(ms): B (ms) B p(ms) B (sm), (4-)

6 6 or equivaenty, 1 B p(ms) B (ms) 1 ϖ, (4-1) where ϖ is the IOA of B. For symmetric beams, a the equaity signs hod. In experiments, one can take B as the measured beam, and then use equation (3-4) to cacuate ϖ. When we combine data from two or more photometers with different beam shapes, foowing the same ine of deveopment as above gives (see eqs. [3-6], [3-7], [4-1], and [4-13]) BΣ(ms) B p(ms) B Σ(sm), (4-) or equivaenty, 1 B p(ms) BΣ(ms) 1 W, (4-3) where W is the IOCA defined in equation (3-5). We sha further discuss the use of these imits ater. 5. THE PIXEL-PIXEL BEAM EXPANSION 5.1. Formaism In the data anaysis procedure briefy demonstrated in section, the effect of asymmetric beam convoution manifests itsef in equation (-8). However, the summation over m and the dependence on the pixe pair make it computationay expensive. Therefore, we prefer to use the form of equation (-1) as an approximation. This can be achieved by repacing the B in equation (-1) with a pixe-pixe beam expansion B(eff), which we sha derive in this section. First, one can repace the B in equation (-1) with Bpp = 4π m B pmbp m Y m(x p )Ym (x p ), (5-1) ( +1)P (cos θ pp ) so that equation (-1) is equivaent to equation (-8). In the sma-fied imit, equation (5-1) becomes J [k x, ϕ ; B ] Bpp Bpp pp k = (k), (5-) J (k x) where 1 π J [k x, ϕ ; B ] pp (k) = { [ ] π dϕ R B pp (k) cos[k x cos(ϕ ϕ )] ] } I[ B pp (k) sin[k x cos(ϕ ϕ )], (5-3) x = x p x p, x = x θ pp, ϕ is the phase ange of x, J is the Besse function of the first kind of integra order, B pp (k) = B p (k) B p (k), and R and I indicate the rea and imaginary parts of B pp respectivey. We notice that J [k x, ϕ ;1] = J (k x). Therefore if the beam is circuary symmetric and remains the same on a pixes, i.e., B pp (k) B k,thenj [k x, ϕ ; B pp (k)] = J [k x, ϕ ; B k ]=J (k x) B k,sothatb pp k in equation (5-) becomes B k exacty as required. To save computation time and memory when estimating C, we need to remove the dependence of Bpp on the particuar choice of a pixe pair (x p, x p ). We achieve this by taking the average of Bpp over a possibe (x p, x p ) pairs: B(eff) = Bpp. (5-4) We ca this B(eff) the pixe-pixe beam expansion. Even with this, equation (5-4) together with equation (5-) is sti computationay expensive and may not be feasibe. Therefore we further simpify the formaism in the foowing way. First, we remove the dependence of B pp (k) in equation (5-) on pixe pairs, by repacing it with a noise-weighted average (c.f. eqs. [-5] and [4-8]) B pp =(U T C 1 N UUT C 1 N U) 1 (U T C 1 N B pp C 1 N U). (5-5) Here the subscript pp in B pp does not mean the pixe pair dependence as in the usua convention, but indicates that the mean is taken over a pixe pairs. With this repacement, equation (5-) is now ony a function of x for a given k. Thus when evauating equation (5-4), we can cassify a possibe x into severa groups of different x, each with severa subgroups of different ϕ. This gives B(eff) J [k x, ϕ ; B ] pp g( x, ϕ ) (k), (5-6) J (k x) x,ϕ where g( x, ϕ ) is the weight of the configuration ( x, ϕ ), i.e., the number of pixe pairs with x and ϕ, divided by the tota number of pixe pairs. It satisfies x,ϕ g( x, ϕ ) = 1. This agorithm can normay reduce the number of operations in equation (5-4) by severa orders of magnitude, because the eement number of {( x, ϕ )} is normay severa orders beow that of {(x p, x p )}. In addition, if the number of pixes is arge enough as in most cases, then ϕ is neary uniformy distributed between and π for every given x, depending on the reative ocations of a pixes. In this case, after the summation over ϕ at each given x in equation (5-6), the first term inside the integra [ in ] equation (5-3) (which enters eq. [5-6]) becomes R B pp (k) J (k x) and the second term vanishes. Thus the Besse function in equation (5-6) can be removed and we have B(eff) 1 π [ ] R B π pp (k) dϕ. (5-7) With carefu simpification of the rea part of equation (5-5), we aso find that [ ] R B pp (k) = B p (k), (5-8) where B p (k) B pm as defined in equation (4-8). We note that the average over a pixe pairs (the eft-hand side of eq. [5-8]) is now reduced to the average over a

7 7 pixes (the right-hand side). This further enabes us to simpify equation (5-7) as B(eff) 1 π π B p (k) dϕ B p(ms), (5-9) where the ast step uses the definition (3-), and the B p(ms) is readiy evauated in equation (4-15). When cacuating B p(ms), one can take the form of equation (4-17) to save computation time. We note that the approximation sign above wi become equaity when ϕ is uniformy distributed between and π. In section 9, we sha numericay verify this resut. With such, now we can use the form of equation (-1) to approximate equation (-8) in the presence of asymmetric beams or when combining data with different symmetric beams. In other words, we have equation (-8) being approximated as C Spp = +1 4π C B (eff) P (cos θ pp ). (5-1) Furthermore, as iustrated in equations (-11) through (-16) and the context, one normay divides the range under investigation into severa bands, due to the finite sizes of the sky coverage and the observing beam, as we as the imited computation power. Using this formaism, we can approximate equation (-13) using equation (-15) with its B repaced by the B (eff) cacuated above. This gives C Spp b C b K b [θ pp ; C sh,b (eff) ]. (5-11) 5.. Uncertainties When making the approximation (5-1), we inevitaby induce errors in the basis B(eff) P (cos θ pp ) for each pixe pair. These errors can be represented as B pp B (eff) 1 ± σ, (5-1) where σ is the normaized standard deviation of the errors. This deviation can be simutaneousy evauated whie one performs equation (5-6), i.e., σ = J [k x, ϕ ; B ] pp g( x, ϕ ) B x,ϕ (eff) J (k x) 1. (5-13) Since C appears in combination with B (eff) P (cos θ pp ) (see eq. [5-1]), we know that σ basicay quantifies the bias in C for each individua pixe pair. Nevertheess, the resuting bias in the fina C estimates by using the approximation (5-1) together with the ikeihood anaysis (see eq. [-6] and context) may be much smaer than σ, because the resuting C is a consequence of the contribution from a pixe pairs. For exampe, if a pixe pairs contribute to the ikeihood function (-6) as a inear combination of B pp P (cos θ pp ), then the resuting bias in C wi be as sma as σ /N p,wheren p is the tota number of pixes. Athough we know that reaity is not ike such a simpe case, we can sti quantify the bias of approximation (5-1) using numerica simuations. Simiary, we can consider the errors in the band power C b for each individua pixe pair, resuting from the approximation (5-11). Since C b is couped with K b (eq. [5-11]) or K pp b (eq. [-13]), the errors in C b for each individua pixe pair may be quantified by comparing K b and K pp b, as we did for B(eff) and B pp. However, as argued earier, the resut cacuated in this way quantifies ony the errors in C b for each individua pixe pair, and the rea bias of the approximation (5-11) together with the ikeihood anaysis may be much smaer. We sha quantify the rea systematic bias of this approximation in section 9, using numerica simuations. 6. SYMMETRY VS. ASYMMETRY In this section, we investigate the conditions under which one needs to empoy the formaism for treating asymmetric beams, i.e. the formaism we deveoped in the previous two sections. We first consider the case where the data to be anayzed is from ony one photometer. From equation (5-9), we know that the pixe-pixe beam expansion B(eff) can be approximated by B p(ms), and therefore shoud be aso constrained by equation (4-1), resuting in 1 B (eff) B(ms) 1 ϖ. (6-1) This impies that if we simpy use B(ms) (where B is the measured beam shape from the experiment) as the B(eff) in our formaism, then B(eff) wi be overestimated by at most ϖ /(1 ϖ ). Furthermore, we consider the errors in C resuting from this effect. In our formaism, the beam convoution appears as the mutipication of B(eff) and C (see eq. [5-1]), so the errors in C canbeexpressedas δ C = dc C = db (eff) B (eff). (6-) Taking db (eff) = B (ms) B (eff), we have from equations (6-1) and (6-) that δ C ϖ. (6-3) This means that when we use B(ms) as the B (eff) in our formaism, then the resuting C at a given wi be underestimated by at most ϖ. To share this error on both sides of a mis-estimated C,wecanchoosetheB(eff) to be B(mid) = B (ms) + = 1 ϖ B 1 ϖ (sm) /B (ms), (6-4) so that the resuting error in the C estimates is now constrained as ϖ ϖ δ C (mid) ϖ. (6-5)

8 8 When ϖ 1, we have δ C (mid) <ϖ /. If the beam is symmetric, then a the equaity signs above hod and B(ms) = B (mid) = B (sm) = B (eff). Foowing the same ine of ogic, we now consider the cases where the data to be anayzed is combined from two or more photometers. In this case, it is aso straightforward to show that if we choose the B(eff) to be (see aso eqs. [3-6] and [3-7] for definitions) B Σ(mid) = B Σ(ms) + B Σ(sm) = 1 W 1 W /B Σ(ms), (6-6) then the errors in the C estimates are constrained as W W δ C (mid) W. (6-7) When W 1, we have δ C (mid) < W /. If a the beams are symmetric (ϖ i = ), then a the equaity signs above hod. As a resut, we see that if the ϖ /(orw /) is we beow the toerated maximum error of C,thenwecan use B(mid) (or B Σ(mid) )astheb (eff) in the symmetricbeam formaism, i.e., we can simpy use equation (-15) with B = B(mid) (or B Σ(mid) ), without the need of going through the procedure deveoped in sections 4 and 5. The associated errors in the fina C estimates wi be constrained by equation (6-5) (or eq. [6-7]). 7. UNCERTAINTIES FROM BEAM MEASUREMENT It is inevitabe for any experiment that there are uncertainties in the measurement of the beam shapes. It is therefore crucia to quantify the uncertainties in the fina C estimates resuting from this beam shape uncertainties. ForagivenbeamB(x), consider an uncertainty ɛ in the fu width at haf maxima (FWHM), and assume that the uncertainties at a iso-height contours of the beam is a fixed fraction of the contour sizes, i.e., dx = ɛ. (7-1) x This uncertainty in the beam shape wi then be transfered to the mutipoe space as the uncertainty in at a given height B m : d = ɛ. (7-) This resuts in the uncertainty in B at a given B = db B = B (1+ɛ) B 1. (7-3) We then consider the change in the C estimates: C = dc. (7-4) C Since the beam convoution occurs as the mutipication of B and C (see eqs. [-8] and [5-1]; here we have dropped the subscript (eff) for concise notation), we know that the resuting uncertainty in C is C = B = B (1+ɛ) B 1. (7-5) This means that if the beam size is mis-estimated by ɛ (i.e., the actua size is 1 + ɛ times the measured size), then the resuting C estimates wi be 1 + C times the rea C. Thus for a given uncertainty in the beam measurement ɛ, one can empoy equation (7-5) to estimate the resuting uncertainty in the fina C estimates. We aso note that the banding of does not affect this resut, as we sha show in section 9.3. We now investigate certain specia cases. In situations where B / is not changing much within d, i.e., B B (1+ɛ), (7-6) we can approximate equation (7-3) and thus (7-5) as C = B d B B = ɛ B B, (7-7) where equation (7-) has been empoyed. For a symmetric Gaussian beam B(x) =exp( x /ϱ ), equation (7-7) becomes C (G) C (G) = ɛϱ, (7-8) whie the condition (7-6), for ɛ 1, eads straightforwardy to (G) = 1 ɛ ϱ. (7-9) Here we have again used the sma-fied imit. When combined with the condition (7-9), we find that approximation (7-8) breaks down when C (G) is comparabe with unity. In particuar, we investigate the accuracy of approximation (7-8), by comparing it with equation (7-5). We find for ɛ < % that the approximation is accurate within 1% error if < (G:1%) =( ɛ ) (G), (7-1) where (G) is given in equation (7-9). For exampe, if ɛ = 1% and the Gaussian beam has a FWHM of 1 arcminutes (i.e., ϱ = radians), then approximation (7-8) is accurate within 1% error when < (G:1%) 94. Under the condition (7-1), one can see from equation (7-8) that, for an approximatey Gaussian beam, the resuting uncertainty in the fina C estimates increases in proportion to the uncertainty in the beam measurement ɛ, the square of the beam size ϱ, and the square of the mutipoe number. 8. DECONVOLUTION OF THE PIXEL SMOOTHING We have not deat with the smoothing effects due to the pixeization of the map, when transating the data from the tempora to the pixe domain (see eqs. [-5] and [4-3]). Because convoving a CMB map with a Dirac Deta δ(x x 1 ) wi shift the origina temperature at x to a new ocation x + x 1,weknowthatY m (x) =δ m (x 1 )Y m (x + x 1 )where δ m (x 1 ) is the mutipoe expansion of δ(x x 1 ). This aows us to rewrite equation (4-3) as B pm = C N A T N 1 B tm δ m (x p x t ). (8-1)

9 9 Substituting this into equation (4-8), we obtain B pm = HU T C 1 N diag(c N A T N 1 B δm ), (8-) where B δm B tm δ m (x p x t )isan t by N p matrix, and diag(m) is a vector whose entries are the diagona eements of the matrix M. These resuts are competey genera. Without further information about N 1 or B δm, equation (8-) can not be simpified, mainy due to the invovement of δ m (x p x t ). With the white-noise assumption (see sec. 4.1), we have equation (8-1) simpified as and equation (8-) as B Πpm = ξ t B tm δ m (x p x t ), (8-3) t p B Πpm = t χ t B tm δ m (x p t x t ), (8-4) where x p t is the centra coordinates of the pixe p that covers x t,andξ t and χ t are as defined in equations (4-6) and (4-11) respectivey. Here we use the subscript Π to distinguish these resuts from those in equations (4-7) and (4-1). In the rea space, equation (8-4) is equivaent to B Πp (x) = t χ t B t (x x p t + x t ), (8-5) meaning that B Πp is the noise-weighted average over the time-stream beams B t that are shifted by x p t x t at each time t. This impies that our formaism deveoped previousy is sti avaiabe, requiring ony a modification that takes into account the detaied ocations of the tempora hits x t with respect to the pixe centers x p t. Thus we have reaxed the assumption x t x p t that was made to achieve equations (4-7) and (4-1). In most cases, both N p and N t are arge, and the beam shape B t of each photometer does not change much within severa successive pixes. This resuts in the fact that in determining the B Πp in equation (8-5), each beam configuration A(β)B (x) (see eq. [4-16]) appears at a set of x t which have offsets x p t x t distributed within a region that is confined by the pixe shapes. If a pixes have the same shape, then this is equivaent to convoving each A(β)B (x) with a top-hat ike window whose boundary is defined by the pixe shape. As a resut, we can approximate equation (8-4) as B Πpm B pm Π m, (8-6) where B pm is as defined in equation (4-1), and Π m is the mutipoe expansion of Π(x) = t δ(x p t x t ). (8-7) The same aso appies to the simpe case where a timestream beams B t are the same. We thus see that the Π m in equation (8-6) serves as an extra convoution (apart from the time-stream beam convoution) of the CMB signa due to the pixeization of the map. With such, we can now easiy incorporate this extra smoothing effects into our formaism by repacing our B(eff) with (see aso eq. [5-9]) B Π(eff) B Πp(ms) (8-8) B p(ms)π (ms) (8-9) B(eff) Π (ms), (8-1) where B Πp(ms) is the power spectrum of the B Πp defined in equation (8-5), and Π (ms) is the power spectrum of the Π(x) defined in equation (8-7). We note that in the imiting case where x t = x p t,wehaveπ pm equa to unity for a and m (since the mutipoe transform of δ(x) is unity), so the smoothing effects disappear, and we have exacty BΠ(eff) = B (eff) (see eq. [8-1]). If the pixes do not have exacty the same shape, as in the case on any arge patch of the sphere (e.g., pixeized by HEALPix, Gorski et a. 1999, or by Igoo, Crittenden & Turok 1998), then we can use equation (8-8) together with equation (8-5) to obtain BΠ(eff). If the pixe beam or the pixe shape remains roughy the same for a pixes, then we can use equation (8-9) together with equations (5-9) and (8-7) to cacuate BΠ(eff). If a the pixes have the same shape which is a reguar square of size ς in radians, then we have Π (ms) (ς) = 8 π dφ sin [(ς cos φ) /] sin [(ς sin φ) /] π 4 ς 4 (cos φ)(sin. φ) (8-11) An accurate approximation to this resut is [ ] Π (ms) (ς) exp (ς).4 [1.7 1 (ς) ] (8-1) The accuracy of this fit is within.3% error for <1.4π/ς. For exampe, if the pixe size is 5 5 square arcminutes (i.e., ς = 5 arcminutes radians), then the above fit is at 99.7% accuracy for < NUMERICAL VERIFICATIONS 9.1. The pixe-beam expansion In this and the foowing three subsections, we wi empoy an eiptic Gaussian beam with a short-axis FWHM of 5 arc-minutes and a ong-axis FWHM of arc-minutes, to demonstrate certain key points deveoped previousy. We first investigate the pixe-beam expansion of a given pixe resuting from different scanning strategies, i.e., to investigate the dependence of the pixe-beam power spectrum (4-17) on the function f(β), and to verify the resuts given in equation (4-). We note that athough those resuts are given for the average pixe-beam expansion, we expect the pixe-beam expansion to carry the same property since equation (4-7) has exacty the same form as equation (4-1). Figure 1 shows two different configurations of beam scanning on a given pixe. In case A, the pixe was hit twice by the same beam pattern, but with different orientations of a separation ange α. That is f(β) [δ(β)+δ(β α)]/. We sha investigate the cases α = 15, 45, and 9 degrees. In case B, the pixe was hit eveny in four different directions. That is f(β) [δ(β)+δ(β 45 )+δ(β 9 )+δ(β 135 )]/4.

10 1 α ϖ and the B of a pixe when the pixe is scanned by a same beam pattern with more different directions, the eve of the effective beam asymmetry (ϖ ) decreases, and so does the power spectrum of the pixe-beam expansion (B ). 1 Case A Case B FIG. 1. Beam configurations on a given pixe, resuting from different scanning strategies. B B (ms) Case A: α=15 (top) Case A: α=45 (midde) Case A: α=9 (bottom) Case B B (sm) Figure shows the IOA of the pixe beams in cases A andb,asdefinedinequation(3-4). Asonecansee,the pixe beam has the argest asymmetry (argest ϖ )when the pixe is hit by a beam with ony one direction (the dashed ine). When the pixe is hit by beams of two different directions (case A in Figure 1), the asymmetry decreases (ϖ decreases) if the separation ange of the two directions α is coser to 9 degrees (see the dotted ines in Figure ). When the pixe is scanned with four different directions (case B in Figure 1), the resuting effective beam is neary symmetric (ϖ ) up to 1, and has the owest eve of asymmetry (the smaest ϖ ) Origina beam Case A: α=15 (top) Case A: α=45 (midde) Case A: α=9 (bottom) Case B FIG. 3. The power spectra B of the pixe-beam expansions with different scanning strategies. We aso note that according to equation (6-5), the IOA of the origina time-stream beam B (the dashed ine in Fig. ) aso tes us beyond what we need to worry about the asymmetry of the beam. For exampe, when < 6, we see that ϖ <.3, giving ϖ / <.5. This means that if we simpy use the B(mid) =/[B (ms) + B (sm) ] (eq. [6-4]) as the B(eff) in the formaism (5-1), then the maxima error in the fina C estimates is guaranteed to be within about ±5% for < 6. ϖ FIG.. Indices of asymmetry of the pixe-beam expansions, as functions of the mutipoe number. Figure 3 shows the power spectra of the pixe-beam expansions with different scanning strategies. As we can see, the power spectrum of the pixe-beam expansion has a maximum given by equation (3-) (see aso eq. [4-19]), when the pixe was scanned with ony one direction. On the other hand, the power spectrum of the pixe-beam expansion has a minimum given by equation (3-3) (see aso eq. [4-18]), when the pixe was scanned eveny in a directions (note that the dot-dashed ine in Figure 3 amost coincides with the soid ine). This verifies our resuts given in equation (4-). By comparing Figure 3 with Figure, we aso earn that there is a strong correation between 9.. The pixe-pixe beam expansion We now use the eiptic Gaussian beam of 5 by arcminutes to verify some important resuts in section 5 mainy equation (5-9). Consider a square map of size 1 1, with a square pixe size of 5 arcminutes. Referring to equation (5-6) with such a map, Figure 4 shows how ϕ is distributed at each x. In the figure, each dot abes the ( x, ϕ ) that is samped by the map. As one can see, ϕ is neary uniformy distributed for any given x, except when x is cose to the boundaries constrained by the pixe and fied sizes. Because of this neary uniform distribution, we achieved equation (5-9) from equation (5-6). More precisey, we carried out equation (5-6) to obtain B(eff), and cacuated the right hand side of equation (5-9) to obtain B(ms). Here we have used the eiptic Gaussian beam directy as the B p (k) B pm in the due cacuations. We found that B(ms) agrees with B (eff) with more than 99% accuracy for =.

11 11 φ (degrees) x (arcminutes) FIG. 4. The distribution of ϕ as a function of x. Each dot abes the ( x, ϕ ) that is samped by a square map of 1 1, with a pixe size of 5 arcminutes Uncertainties from beam measurement In this section, we wi numericay verify the resuts in section 7. First, we use a symmetric Gaussian beam with a FWHM of 1 arcminutes, to investigate the approximation C (G) given by (7-8), as a comparison to the exact resut C (G) given by (7-5). Here we take the uncertainty in the beam measurement to be ɛ = 1% (eq. [7-1]). As one can see in Figure 6, the approximation breaks down towards the imit (G) given by equation (7-9). For (G), the approximation C (G) reproduces the correct resut C (G). By comparing C (G) and C (G), we cacuate the 1% accuracy imit (G:1%) (the dot-dashed ine), at which C (G) / C (G) 1 = 1%. In addition, by varying the vaue of ɛ between ±%, we obtain the resut presented in equation (7-1). That is, for a symmetric Gaussian beam with an uncertainty of ɛ in size, the approximation (7-8) for the resuting uncertainty in C is accurate within 1% error for < (G:1%) =( ɛ ) (G). We have aso cacuated the average deviation σ of Bpp from B(eff) for each individua pixe pair, using equation (5-13). The resut is shown in Figure 5. First, we see many spikes in σ. This is due to the zeros of the Besse function J, which appears at the bottom of equation (5-13). These spikes shoud be negected, as in reaity no such singuarities appear in our anaysis pipeine. We note that these spikes have the same origin as those presented in Hanany et a. (1998), where a simiar situation was considered. Second, as addressed previousy, athough the σ obtained from equation (5-13) can be as arge as comparabe to unity, the rea errors in the fina C estimates by using the formaism (5-11) with the approximation (5-9) wi be much smaer than this vaue. This is because the σ here tes ony the mean discrepancy of B(eff) for each individua pixe pair, and may average out when a pixe pairs come into account in the ikeihood anaysis. In section 9.5, we wi numericay justify this and thus the accuracy of empoying equation (5-11) with (5-9). σ FIG. 5. The mean discrepancy σ of B pp from B (eff) for each individua pixe pair C (G) * C (G) * (G) * (G:1%) FIG. 6. The uncertainty in C (soid ine; given by eq. [7-5]), resuting from an uncertainty of ɛ = 1% in the beam shape measurement of a Gaussian beam with a FWHM of 1 arcminutes. Aso potted is the approximation (7-8) (dashed ine). The dotted vertica ine indicates the imit of the approximation (G) given by equation (7-9), whie the dot-dashed vertica ine shows the 1% accuracy imit (G:1%), which is we fitted by equation (7-1). Now we investigate the case where the beam is asymmetric. We use an eiptic Gaussian beam, whose ongand short-axis FWHM s are and 5 arcminutes respectivey. This beam is first convoved onto a simuated CMB map of size 1 1, with a pixe size of 1 arcminutes. The underying cosmoogy is an infationary mode with (Ω b, Ω cdm, Ω Λ,n,h)=(.7,.61,.3, 1,.6), normaized to the COBE DMR. A random Gaussian noise of 1µK is then added into each pixe. We ca this simuation (1). We repeat the same procedure again except that this time the beam size is increased by 1%, i.e., ɛ = 1%, to obtain a simuation (), where the CMB and noise reaizations are exacty the same as those used in simuation (1). We then anayze both simuations using the procedure outined in section, with the approximation (5-11). The

12 1 resuting uncertainty in C can thus be cacuated using equation (7-4) as C (a) = C () 1, (9-1) C (1) where the subscripts (1) and () indicate resuts from the two simuations. The resuts are shown as crosses in Figure 7. Aso potted is the resut using equation (7-5) (the soid ine), which we abe with a subscript (b). It is obtained directy by varying the beam shape with ɛ = 1%. As one can see, the crosses are highy consistent with the soid ine. This means, first, that the asymmetry of the beam does not affect our resut given by equation (7-5). Second, the banding of does not affect the resut, so we can use equation (7-5) as an estimate for the uncertainty in the band power C b resuting from that in the beam measurement. This is aso an important support to the fact that the banding of does not affect the genera reation C B (see eqs. [5-1] and [5-11]). We have aso verified that the sizes of the error bars in the C estimates between simuations (1) and () do not change by more than 4% for <1. Thus we know that when C is sma, the uncertainty in the beam shape measurement does not affect the sizes of error bars significanty, but does affect the ampitudes of the C estimates. On the other hand, when C is arge (comparabe to one), the signa to noise ratio may be affected and so may the error bar sizes. eiptic Gaussian beam of 5 by arcminutes in FWHM (same as the one used in previous sections). For each tempora sampe, we then add Gaussian random white noise n t with 5% in RMS ampitude. In this run, we require the tempora sampes to be exacty at the centers of each pixes, i.e., N t = N p and x t = x p t, such that m = d (see eq. [-5]). We ca this simuation (). In a second run, the procedure is the same except that the CMB tempora sampes now have offsets with respect to the centers of each pixes, i.e., N t = N p but x t x p t with x t x p t randomy distributed within a square of size 1 arcminutes. We ca this simuation (1). In third, fourth, and fifth runs, the procedures are the same as simuation (1), except that the numbers of tempora sampes in each pixes are now 3, 1, and (i.e., N t =3N p,1n p,and N p ) respectivey, instead of one. We denote these as simuations (3), (1), and () respectivey. A these runs are then anayzed in the same way, using the procedure outined in section, with the approximation (5-11) and B(eff) = B p(ms) (eq. [5-9]). Therefore the ratio C (j) C (), j =1, 3, 1,, (9-) wi quantify the smoothing effect due to the pixeization of the map. We pot this ratio in Figure 8, as a comparison to the Π (ms) given in equation (8-1)..5.1 C(a) C(b) Π C (1) / C () C (3) / C () C (1) / C () C () / C () FIG. 7. The uncertainty in C resuting from that in the beam measurement. The horizonta axis is the mutipoe number. The crosses, C (a), are resuts based on simuations using equation (7-4) (oreq.[9-1]), whie the soid ine, C (b), is obtained directy from the beam shape using equation (7-5). An eiptic Gaussian beam, with ong- and short-axis FWHM s of and 5 arcminutes respectivey, is used. The uncertainty in the beam shape is ɛ = 1% Deconvoution of the pixe smoothing We now test the formaism of deconvoving the smoothing effect due to the pixeization of a map. This is to verify equation (8-9), with equation (8-1) as an approximation in cases where the pixes are reguar squares. We consider a square CMB map of size 1 1, with reguar-square pixes of size 1 arcminutes. We first simuate a timestream of the CMB signa γ t, that is convoved with an FIG. 8. The smoothing effect due to the pixeization of a CMB map. The ratios C (j) /C () are compared to the approximated smoothing window Π (ms),withj = 1, 3, 1, representing the number of tempora sampes per pixe in different runs. The horizonta axis is the mutipoe number. The square at the bottom-eft corner shows the Π(x) forj = 1. As one can see and expect, the smoothing effect approaches the top-hat-window approximation when the number of tempora sampes per pixe increases. When it is arger than 1, as in most rea situations, the top-hatwindow approximation appears to be a good one. Aso potted at the bottom-eft corner is the Π(x) givenby equation (8-7) for j = 1. The neary uniform distribution of x p t x t shows the appropriateness of the tophat-window approximation. Thus we have verified that