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Memo 74 System Considerations for the SKA A. R. Thompson, J. D. Bregman 06/04/06

www.skatelescope.org/pages/page_memos.htm

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Members of the System Subgroup of the SKA Engineering Working Group include P. Alexander, G. Bower, J. D. Bregman, M. Davis, J. Dreher, M. Gatti, T. Küsel, A. R. Thompson (chair), A. van Houwelingen, and J. Welch. The purpose of a system study is to outline the properties of the instrument in terms of the performance requirements of the major components without getting into design details of the hardware, except in so far as to determine that the requirements can feasibly be met. The performance requirements follow from the scientific requirements, and are summarized in Table 1A in Appendix A. In planning a new instrument, an initial system study would usually be made to facilitate the choice of the major components, such as the type and distribution of the antennas, after which it becomes possible to outline the requirements of the signal processing elements from the low-noise amplifiers to the correlator. In the case of the SKA, this first step has been replaced by a call for proposed concepts, and since the choice between these has not been made, the initial purpose of the present study is to considers rather broadly the ways in which the requirements can be met, and the extent to which the existing concepts are able to meet them. Here we start by considering some general system characteristics which help to determine the broad choices. Sections 1-14 of this report were written in 2005, before the choice of a Reference Design for the SKA was made. The Reference Design was chosen in Jan. 2006, and is briefly described in Section 15. The earlier sections are retained essentially unchanged since they illustrate many of the relevant concerns in the choice of the Reference Design. Some thoughts on the continuing work of the System Subgroup are given in Section 16. 1. Dimensions of the Array The large collecting area and high angular resolution required for the SKA clearly indicate an instrument in the form of a large synthesis array. Baselines from a few tens of meters to a few thousand kilometers are required. The parameters for the proposed distribution of collecting area are: 20% within a central area of 1 km diameter, 50% within an area of 5 km diameter, 75% within an area of 150 km diameter, and maximum baselines extending to at least 3000 km. The minimum baseline length is 20 m, but in the case of large diameter antennas the short baseline data can, in principle, be determined from the total power responses. Thus most of the present concepts can be adapted to these dimensions. A final plan is likely to include further refinement of the scientific requirements and take account of the characteristics of the chosen site, etc. 2. Dynamic range and Number of Stations The various concepts for the SKA that have been developed are characterized almost solely by the type of antennas involved. The first consideration here is therefore to examine the antenna requirement from a systems viewpoint. A prime consideration is the number of single antennas or phased groups of antennas between which cross correlation of the received signals are formed. These units are conveniently referred to as “stations” and the number of combinations of pairs of stations defines the number of baselines. In

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the current concepts the number of stations varies between 30 and 2080. In terms of providing the required signal-to-noise ratio it is mainly necessary that the total collecting area should meet the basic requirements. Thus it may appear that the number of stations is only very broadly constrained, as the diversity of the currently proposed concepts suggests. However, considerations of dynamic range place a further constraint on the minimum number of stations. At about 1.4 GHz, the sensitivity of the SKA to discrete sources corresponds to a minimum flux density of a few tenths of a microjansky for integration times of hours, or a few tens of nanojanskys for integrations of several days. At these levels the number of sources within the specified field of view of 1 sq. deg. is estimated to be of order 106. On average, the strongest source at 1.4 GHz within any one square degree is about 100 mJy, so the required dynamic range1 is 106 to 107. A study by Lonsdale et al. (2000) throws some light on the effect of the number of stations on the dynamic range and image fidelity that is attainable. Simulations were made of observations of a square field of dimensions 50 arcseconds with a distribution of sources of brightness in the range 25 mJy per beam to 100 nJy per beam, i.e. a dynamic range approaching 106, as a representation of the sky at a wavelength of about 20 cm. Results for arrays of 100 and 200 stations were compared for observing times of 1, 4, and 12 hours, after processing the images with the CLEAN algorithm to maximize the dynamic range. In all cases the background noise level was higher than the theoretical level due to system noise. For one hour integration, the background rms in the 200 station run was approximately 1/3 of that in the 100 station run. For 12 hour integration, the background rms with 200 stations was ½ of that for 1 hour integration. The decrease in the relative improvement of the 200 station data with increasing observing time can be understood if the improvement is attributed to the increased (u,v) coverage, since in both cases the (u,v) coverage becomes more complete as the integration time is increased. Thus the overall trend was for the dynamic range to increase with the number of (u,v) samples in the data set. In the case of an array with fixed total collecting area, the area of a station would be inversely proportional to the number of stations, so if this were included in the simulation, the size of the field, and hence the number of sources within the main beam, would be halved in the 100 station case. Lonsdale et al. considered the same field size for both the 100 and 200 station cases, but conclude that the background is dominated by receiver noise and deconvolution errors, not by sidelobes of unsubtracted flux density, so reducing the size of the field in the 100 station case would not make a major difference. The superiority of the 200 station result appears to be largely attributable to the improved sidelobe levels of the dirty beam. It could be expected that these sidelobe levels would be inversely proportional to the square root of the number of (u,v) points with measured visibility values, that is they would be approximately inversely proportional to the number of antennas and the square root of the observing time. Note, however, that some experience with the Westerbork Northern Sky Survey

1 Note that the term “dynamic range”, as used here, refers to the range of flux densities that can be meaningfully observed in a deep-sky survey with an instrument in which the limit set by system noise is in the nanojansky to microjansky range. Dynamic range is also sometimes defined in terms of the response to a single discrete source, in which case it depends mainly upon the accuracy of calibration and closure relationships. However, the number and configuration of antennas becomes important as the sky complexity increases, as in the present case.

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(WENSS) shows that in combining data the improvement is approximately proportional to the number of (u,v) points, i.e. better than the square root of the number. Thus this issue is more complex than may appear from simple considerations. As a further consideration, doubling the size of the main beam in the study by Lonsdale et al. would, on average, double the total power received from background sources relative to the power from any particular source under study. Since the background sources are randomly positioned on the sky, their individual contributions to the total visibility are randomly distributed in phase. Thus the rms background fluctuations could be expected to increase by a factor of √2. However, if the background rms values from the study quoted above are increased by √2, the benefit of the larger number of smaller antennas is still substantial2. The self-cal, imaging, and cleaning processes have known artifacts which must also be taken into account in considering the dynamic range. The analysis described above shows that the standard self-cal + image + clean processing package encounters some limit that is not lowered by the square root of the number of (u,v) samples. Although more samples improve the image quality we are not able to define a hard minimum number of telescopes that would be needed if a “perfect” self-cal procedure were available. It should be realized that the current self-cal + image + clean processing package does not solve for telescope based amplitude and phase corrections that change over the primary beam, i.e. the “main” field-of-view. For instance, pointing changes of individual telescopes can change the amplitudes of source visibilities for each baseline combination differently depending on the location of the source. We can cite three examples showing how the noise floor depends on other considerations. (1) Bhatnagar et al. (2006) have recently shown that inclusion of the effect of pointing changes in the measurement equation results in thermal noise limited images that are a factor 3 more sensitive than those without this correction. (2) For the WSRT (Westerbork Synthesis Radio Telescope), solving for the ~0.1 degree rotation of the dipoles on the sky (de Bruyn 2006) increased the dynamic range from 105 to 106. (3) The current VLSS (VLA Low-Frequency Sky Survey)3 at 74 MHz has a thermal noise floor of 0.02 Jy while the 5 sigma level for the detections is set at 0.5 Jy indicating a factor 5 difference between actual and nominal sensitivity due to a number of artifacts. In these three examples the effect of more “independent” (u,v) samples would be different, which means that although adding more (u,v) samples gives a slight improvement, it does not solve the real issue of using a measurement equation that is not sufficiently complete. LOFAR (see, e.g., Noordam 2001) is expected to improve the processing procedure by more appropriate matching of the telescope design and the calibration procedures to the ionospheric requirements, and some discussion of this is given in §3.

2As an alternative consideration, if the field of view is doubled the flux density of the strongest source within the field is increased by a factor of approximately 1.6 (for the case where the number of sources per steradian on the sky for which the flux density is ≥S is proportional to S-1.4). If the limit set by the dynamic range is proportional to the flux density of the strongest background source, there is again a significant benefit in the larger number of smaller antennas. 3 See website at http://lwa.nrl.navy.mil/VLSS/

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A large number of visibility values provide the ability to solve for many unknown quantities, which include such things as reducing RFI by adaptive nulling and other methods. To make best use of the flexibility of the SKA, it is clearly important that the number of baselines should be great enough that high dynamic range can be obtained without requiring a full eight to twelve hour observation. The study by Lonsdale et al. (2000) indicates that 200 stations would be a minimum requirement and larger numbers would be advantageous. Factors that might place an upper limit on the number of stations include the complexity of the correlator, which is proportional to the square of the number of stations, and the cost of the additional fiber for signal transmission. At the present time the largest correlator is that under construction for the 64-station ALMA. Correlators capable of handling ~2000 antennas should become feasible in the next decade (D’Addario and Timoc, 2002) and should not be considered as a limiting factor for the SKA. The post-correlator processing load, including deconvolution and possibly interference mitigation, can increase with the number of antennas to the third or higher power (Perley and Clark, 2003; Cornwell, 2004) and must also be taken into account. In the most sensitive frequency range, 0.5 to 5 GHz, the specification for A/T is 20,000 m2/K so, for example, taking T ≈ 20 K the overall collecting area is 400,000 m2. For 200 stations the collecting area for each is equivalent to a single 60 m diameter dish (assuming 70% aperture efficiency). 3. Ionospheric Considerations Related to Station Size At the low frequency end of the SKA range, a lower limit on the size of a station may be appropriate. Synthesis imaging becomes more difficult when the station beamwidth is wider than the angular size of the isoplanatic patch of the ionosphere. The isoplanatic patch can be defined as the area over which the variation of excess phase due to the ionosphere is small compared with radian. If the beam is wider than the isoplanatic patch, and if it is necessary to map the full beam area, then one has to allow for variation of the phase calibration over the field of view. Experience with the WSRT indicates that at 330 MHz a typical patch size is about 2º. If we take the half-power width of the beam of a circular aperture as 1.1x (wavelength/diameter) radians, then the antenna diameter for which this is equal to 2º at 330 MHz is 26 m. Note that on some occasions the size of the isoplanatic patch is smaller than this so a larger antenna diameter would be needed to avoid variation of the phase calibration over the beam. Observations in the WSRT 117 - 175 MHz band, the WSRT 310 - 390 MHz band, the VLA 330 MHz band and the VLA 74 MHz band show that for baselines up to 15 km the ionospheric phase errors scale almost linearly with the baseline length and correspond to quasi-periodic waves with periods of order 10 minutes. Assuming a typical propagation speed of these mini Travelling Ionospheric Disturbances (TIDs) of 300 km/h at a lower-level ionospheric height of 300 km, we obtain a value of 50 km for the spatial sinewave structure. The maximum phase error then occurs on a 25 km baseline in the direction of the wave propagation, and is proportional to wavelength. For baselines up to 15 km, a linear phase slope is an acceptable approximation, which implies a source shift without distortion. Alternatively we could interpret the 15 km as the diameter of a Kolmogorov seeing cell and in such a phase screen we could use the best tip-tilt correction for an area

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with a diameter of three seeing cells resulting in a super cell still with one radian rms phase error. In the optical domain the corresponding improvement in seeing-limited resolution would be nearly a factor three. For radio wavelengths longer than 1 m we have a typical angular field size of 25 km/300 km ~4.7º in which the apparent positions of objects are shifted parallel to the baseline, in either direction, by amounts that are proportional to the wavelength. In the VLSS the phase screen is parameterized and solved for by a number of sources that are viewed by antennas with spacings less than 15 km (VLA configurations A and B) and differential refraction of ~0.01º is found. A major issue is the number of relevant parameters that can be solved for. First we need sufficient sensitivity to solve for sources and their positions. For the VLA at 74 MHz the thermal noise in a 6.6 s snapshot image with 1.5 MHz bandwidth in two polarizations is typically 0.6 Jy. In the 11.5º beam we typically have one source of 23 Jy and three of 10 Jy. The total phase screen that spans about 25 km in diameter is now sampled with 27 telescopes each with four sources which allows a good Zernike polynomial solution such that there is no dynamic range problem with the other sources in the field. Unfortunately, due to an effective aperture efficiency of about 0.15 for the 25 m dishes, the sensitivity per baseline is only 16 Jy and is not enough for self-calibration (with one source), therefore a separate calibration observation is made on a source that is sufficiently strong. For LOFAR the receiving elements are phased arrays of dipoles with which it is possible to form beams in the directions of several hundred strong calibrators and thus calibrate the ionospheric refraction every few seconds. Thus the LOFAR telescopes, compared with those of the VLA, are larger and have a much larger aperture efficiency, increasing the sensitivity by a factor of five, while the bandwidth can be increased by a factor 20. According to simulations by Jeffs et al.(2006) the “peeling” self calibration can only solve for 6 sources each with an independent complex gain factor for each telescope, while the theoretical maximum number is equal to the number of stations. This would allow for solution for only one complete seeing cell of 15 km diameter in each telescope. In the case of telescopes with spacings closer than 15 km, the beams overlap at the height of the ionospheric phase screen and we get some form of oversampling of the phase screen, which reduces the required number of parameters to be solved for independently, just as in the VLSS field calibration. Distances between stations approaching ~15 km become comparable to the size of the ionospheric structure, and the variation of ionospheric path length with position on the sky is different for different areas of the array. In this case one can no longer solve for a contiguous phase screen. For these distant stations the field of view needs to be limited to cover one seeing cell, for which the available number of solvable parameters is adequate. This requires increased station size, except for cases where the field is limited by the decorrelation of the signals as the relative time delays vary with the angle of incidence (the "delay beam" effect). For the lowest frequencies of the SKA, a possible solution to the ionospheric problem lies in beamforming using small arrays of dipoles, as in LOFAR and the Mileura Widefield Array. In cases where the receiving elements are small parabolic dishes, the requirement for close spacings to minimize sidelobes in the beams of phased groups may be at odds with the spacing requirements to avoid shadowing, so the situation is more complicated.

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Such solutions should apply, at least, for the ~50% of antennas within the inner few km of the SKA. As noted above, for fields wider than the ionospheric structure calibration of the ionospheric path may no longer be possible. However, with a multi-channel correlator it would be possible to correct for delay effects by inserting a differential phase correction per frequency channel after correlation. Note also that observations using the outermost SKA antennas are likely to be limited to individual discrete sources, for which only a single small field, or a limited number of small fields, require calibration. The size of the isoplanatic patch varies with frequency because the excess phase introduced by the ionosphere on any path is proportional to ν-1 where ν is the frequency. This results from the variation of the refractive index (proportional to ν-2) and the path length. Thus, as frequency is decreased smaller structures in the ionosphere become more important and the patch size is approximately proportional to frequency. To be more precise, the structure of the ionospheric irregularities also affects the frequency dependence, and if the structure is characteristic of Kolmogorov turbulence, the patch size is found to be proportional4 to ν-6/5. The ionospheric problem is most severe when the station FoV is wider than the isoplanatic patch. The diameter of a dish for which the beamwidth is equal to the size of the isoplanatic patch is proportional to ν-2.2. Thus the ionospheric problem rapidly disappears as the frequency increases, but remains important within the frequency range that is considered essential for the important epoch-of-reionization studies. However, caution suggests that one should not let a consideration that applies only at the very low end of the SKA frequency range weigh too heavily on the overall design: see the discussion of hybrid solutions in §7. 4. Other Parameters of Antennas Field of view The field of view (FoV) for contiguous imaging is defined in the science requirements for frequencies of 1 GHz and greater is one square degree at 1.4 GHz, with scaling such that the effective collecting area remains constant5. If we take the half-power beamwidth of a parabolic antenna to be 70 x (wavelength/diameter) degrees, the diameter corresponding to the field requirement is approximately 13 m. The effective collecting area of such an antenna is approximately 95 m2, so for antennas of this size the number required would be ~10,000 to provide the nominal 1 km2. However, with system temperature T = 20 K, the fundamental A/T specification is met with ~4000 13 m

4 This result was pointed out by J. M. Moran, and is derived as follows. The atmospheric irregularities are characterized by the structure function which gives the variance of the phase difference for two paths separated by a distance d. This is discussed for the case of Kolmogorov turbulence in the neutral atmosphere in Thompson et al. (2001: see Table 13.2). For distances up to a few tens of kilometers (i.e. small compared to the thickness of the ionosphere) it is appropriate to consider three-dimensional turbulence for which the structure function is proportional to d5/3, so the rms phase difference is proportional to d5/6. For the ionosphere the frequency variation of the refractive index also introduces a dependence on frequency: for a fixed path the phase is proportional to ν-1. Thus the rms phase difference for paths with separation d is proportional to d5/6 x ν-1. The isoplanatic patch size is represented by a constant phase difference, in which case d is proportional to ν-6/5. The dimensions of a blob in the turbulent structure are assumed to be similar in directions parallel and perpendicular to the line of sight. Note that the validity of the application of Kolmogorov turbulence to the ionosphere is not entirely clear as the blob sizes vary between day and night, and large scale travelling ionospheric disturbances also occur. 5 This scaling is appropriate for a parabolic reflector antenna with one or more discrete feeds, but would be different for antennas with phased-array feeds or cylinders.

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antennas. Considerations of cost and complexity may tend to constrain the number of stations to something less than this, that is, the station area would be larger than that of a single antenna, and consequently each station would need to form multiple receiving beams. In principle this could be achieved either by multiple feeds on antennas of diameter greater than ~13 m, or phased-arraying techniques using antennas of diameter ~13 m or less. In phased arraying, the spacing of the antennas to minimize shadowing would necessitate discontinuities in the combined aperture, resulting in sidelobes or grating responses in the array beam. Since such unwanted features can introduce responses to sources outside the field of view, it would be necessary to determine that they would not seriously degrade the dynamic range. Lonsdale et al. (2004) discuss related considerations with respect to the necessity to image the full primary beam of the antennas. Similar considerations regarding the number of elements at a station apply to aperture arrays or cylinders. For the frequency range 0.5-1.0 GHz, the specified FoV scales with frequency as λ2 and is equal to 200 square degrees at 700 MHz. (Note that since the large FoV in this frequency range is based on surveying requirements, it is assumed that it can be provided by multiple beams which do not need to be contiguous.) For a single beam, the specification could be met by a circular paraboloid of diameter ~1.9 m. At these longer wavelengths, where reflector accuracy is relaxed, larger dishes might appear to be more cost-effective, but Cornwell (2005) points out that dishes of about 1.5 m size are manufactured in large quantities for TV reception, etc. and are relatively inexpensive. He considers the possibility of stations consisting of about two thousand such antennas, about half a million antennas in all. Phased arraying of subgroups of station elements would be necessary to reduce the total number of baselines to something that could be handled by any correlator that is conceivable in the foreseeable future. Thus the full field of the individual antennas would not be usable at any instant, but the large number of antennas would result in good sidelobe suppression. The practicability of this approach will depend on the cost of the elements required in large numbers for front ends, signal and local oscillator distribution, signal phasing, etc. The aperture efficiency of small dishes at low frequencies should also be considered where a diameter of 1.9 m would be only ~4 wavelengths at 700 MHz. The possibility of using such small antennas could become important if difficulties arise with multibeaming, which is mentioned next. With larger (but still “small”) antennas, for example 12 m diameter, multibeaming using focal plane array6 (FPA) feeds, will be necessary to attain the FoV. Development of a test system of this type using phased-array feeds is in progress: see xNTD in §6. At lower frequencies the area of the isoplanatic patch of the ionosphere becomes less than the field of view, so multibeaming, rather than covering the whole FoV with one beam, should simplify the phase calibration problem. For the LAR, a feed array that produces 350 independent beams is proposed to achieve the FoV requirement of 1 sq. deg. at 1.4

6 We follow the terminology introduced in SKA Memo 69, in which “focal plane array” (FPA) feed refers to a generic array of feed elements which may be one of two types, as follows. “Multiple feed cluster” (MFC) indicates an array of feed horns, each of which produces an independent beam. “Phased-array feed” (PAF) indicates an array of feed elements in which a phased combination of a number of the elements, with appropriate amplitude weights, is used to provide each individual beam.

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GHz. At 700 MHz, 350 beams could provide a FoV of 4 sq. deg. if the positions of the effective electrical centers of the feeds could be adjusted to optimize the spacing of the beams. It is not clear whether this would be possible. In any case, it appears that meeting the 200 sq. deg. specification is not practicable in the case of such large reflector antennas. Further, it is arguable that 200 sq. deg is not necessary. Full (u,v)-coverage for any SKA concept is likely to be complete within 6 hours, so that with 200 sq. deg. instantaneous FoV, the visible sky of typically 30,000 sq. deg. would be observed within 40 days. Very deep surveys requiring averaging of many observations could be obtained by repeated observation of limited fields. Thus the 200 sq. deg. seems to imply a working lifetime of only a few years for the instrument. The specified range of SKA baseline lengths, 20 m to 3000 km, would in principle allow imaging of fields of dimensions approaching 150,000 times the angular resolution. As Lonsdale et al. (2004) point out, in practice most image sizes are likely to be no more than a few hundred times the angular resolution. For example, at 1.4 GHz, 1000 km baselines would permit angular resolution of order 100 mas, so an image dimension of ~1.5 arcmin (103x 103 pixels) could contain a large amount of information. Thus the most distant stations may not need to provide the full field of view of the stations near the array center. Polarization Dual polarization of the antennas is required for measurement of polarization of the radiation, which usually involves mapping of the Stokes’ parameters. The upper limits on instrumental polarization of -40 dB at the field center and -30 dB at the field edges (i.e. the edges of the 1 sq. deg. or the 200 sq. deg. areas) are stringent requirements that will require much care to meet. Slew Speed The specified minimum slew speed is 90º per minute. In general, mechanical slew speeds decrease as the size and weight of an antenna increases. High slew speeds place requirements on the mechanical structure and increase the cost of an antenna. In practice the slew speed is likely to be limited by cost and available electric power load. As an example, for the 25 m antennas of the VLA the slew speeds are 40º per minute in azimuth and 20º per minute in elevation. 5. Surveying Considerations Unlike most existing synthesis instruments, which have been designed mainly for imaging of fairly compact objects, the SKA design must include the capacity for imaging wider areas of the sky for surveys. In the frequency range below 1.4 GHz, four of the key science applications require imaging of a significant fraction of the sky, so the consideration of the surveying speed is important. Consider a continuum survey of an area of sky with sensitivity sufficient to detect a source of minimum flux density Smin (Wm-2Hz-1). If τ is the integration time required to reach the desired flux-density limit, the survey speed is proportional to 1/τ. The speed is also proportional to the instantaneous FoV. Thus the speed is proportional to:

FoV / τ (1)

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Consider, for example, stations that consist of a single antenna with an aperture of area A/ Ns, where A is the total array aperture and Ns is the number of stations. Then the solid angle of the station beam is equal to

λ2Ns/A, (2) and the FoV is

λ2NsNsb/A (3) where Nsb is the number of station beams. Thus the FoV is proportional to the number of feeds per unit collecting aperture. From the usual expression for the detectable flux density, Smin is proportional to

T/[A(Bτ)1/2], (4) where B is the bandwidth and T is the temperature. Hence τ is proportional to

T2/(Smin2A2B) (5)

Then from expressions (1), (3), and (5), the surveying speed is proportional to

λ2ABNsNsbSmin2/T2 (6)

Over wide frequency ranges the bandwidth increases roughly in proportion to the center frequency of the observation, so Bλ, while not a constant, varies by only a relatively small factor. A/T is determined by the specification. Thus the surveying speed is determined largely7 by λNsNsbSmin

2/T. The factor λ suggests that the speed increases with decreasing frequency, but in practice the increasing sky background temperature introduces a limit. Bregman (2005a) gives a more detailed analysis and concludes that the survey speed is broadly optimized at about 700 MHz. Note that the discussion above involves the assumption that the sensitivity is limited only by the system noise. If dynamic range is the limiting factor, as discussed in §2, then the density of the (u,v) coverage, which improves with increasing Ns and τ, may become the most important consideration. In either case performance improves with increasing Ns. Bunton (2003) has considered a figure of merit for surveying in which the information is also proportional to bandwidth, as is appropriate for observation of spectral features. This case requires an additional factor B in expressions (1) and (6). A detailed discussion of factors that are relevant to optimization of survey performance is given by Wright et al. (2006). For example, the survey speed is directly proportional to the number of beams per antenna. However, as the number of beams is increased the

7In the case of the proposed cylinder antennas (see §6), for frequencies above 1.5 GHz, the FoV is limited by the beamforming, and is inversely proportional to the frequency. Thus an additional proportionality factor λ is introduced into the survey speed.

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costs of the receiving system (including beamforming), data transportation, the correlator and post processing are also increased. Thus, for a fixed overall procurement cost, maximizing the survey speed is likely to impact the performance in targeted observation of individual objects. Achieving a satisfactory balance between the performance of the SKA in various types of observing projects should become possible as more precise information on the performance of multi-beam antennas and the associated data processing is developed. 6. Major Characteristics of the SKA Concepts, including Demonstrators. Table 1 summarizes the main features of the current concepts including demonstrators. The figures shown are those that have been given in the white papers of 2002 and 2003, and demonstrator concepts of 2004 and 2005. Only major features are included here and much more information can be found in these original papers. Details are likely to change as further developments are made. Some additional information is given below. Papers describing a number of demonstrators can be found in the special issue of Experimental Astronomy edited by Hall (2005). LAR (Large Adaptive Reflector), Canada The reflector consists of large panels, on adjustable mounts, that form a parabolic surface. The feed array is carried by a tethered aerostat positioned by computer-controlled winches. The focal length is approximately 500 m. The aperture is unblocked by the feed except for pointing angles very close to the zenith. Vivaldi elements are being considered for the feed array, with possible cooling of the transistor input stages. Several feed systems would be required to cover the 0.15-22 GHz range. Inertial stabilization of the suspended feed structure may be used for the high frequency bands. As proposed for the SKA, the reflector diameter is 200 m. For a demonstrator, it is planned to build an antenna of 300-350 m diameter with frequency coverage 100-1800 MHz, sky coverage down to 30° elevation, and a field of view of diameter 0.62°. KARST (Kilometer-square Area Radio Telescope), China The reflector is an active spherical surface of radius 300 m and aperture 500 m. The feed illuminates an area of the surface of diameter approximately 300 m, and the area under illumination is continuously adjusted to parabolic shape as the beam moves across the sky. The parabolic shape allows standard types of feeds can be used. The antennas are designed to be built in natural Karst depressions in the Guizhou Province of China which contains about 400 such depressions in a region that extends for about 400 km. Moist conditions in the region would limit the upper frequency range. It is proposed to build an antenna with a total projected diameter of up to 500 m diameter and an effective aperture of about 300 m as a demonstrator. This will have sky coverage up to 50° zenith angle and frequency coverage 15-2000 MHz with possible extension to 5 or 8 GHz. LNSD (Large Number of Small Dishes), USA The receiving elements are 4400 fully steerable parabolic dishes of diameter 12 m, assuming T = 20 K and an aperture efficiency of 80%. They may be of offset-feed design, but this is not yet decided. Those within 35 km of the array center are treated as individual stations and those in the regions beyond 35 km would be set out in clusters of 13 as multi-beam phased arrays. For

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additional area at the lower frequencies a mesh outer surface extends the reflector diameter to 16m. The minimum antenna spacing is 16 m. No specific demonstrator is currently planned, but much experience gained in the construction and operation of the Allan Telescope Array and other current projects in which the US is involved will be directly relevant to the development of an LNSD array. Cylinder Array, Australia The cylinders are mounted with the long axis in the east-west direction. Three line feeds are mounted along the axis of each cylinder to cover three frequency bands. The FoV details are complicated (see §2.3.1 of Bunton et al. 2003). A cylinder can be subdivided in the long dimension into eight sections, each of which provides a beam of area approximately 1 sq. deg. at 1.4 GHz. For the three frequency bands the beams can be pointed independently in meridian distance (the coordinate orthogonal to elevation) but not in elevation. Separate beamformers will be used for 0.3 m sections of the linefeeds for beam forming in meridian distance. In meridian distance the angular coverage is ±60º (i.e., 120° in range) from 0.1 to 1.5 GHz. At higher frequencies the coverage is limited by the beamforming to 170°/ν from 1.5 to 7 GHz, and 51°/ν above 7 GHz, where ν is the frequency in GHz. The FoV that can be used for imaging further varies inversely with the bandwidth (due to limitation of the antenna-to-correlator bandwidth of the digital transmission system) and, for example, at 1.4 GHz the instantaneous meridian coverage per beam can vary from 120° to 0.125°. The product of the imaging FoV in square degrees and the bandwidth in GHz is equal to 76/ν 2. It thus appears that the FoV requirements at 700 MHz and 1.4 GHz can be met with reasonable bandwidths. As a point on the sky is tracked in hour angle, the direction of the meridian distance coordinate rotates on the sky to an extent that depends upon the declination of the area of sky under observation. Also the beams for the three frequency bands may differ slightly in elevation if the electrical axes of the line feeds are not coincident. Cooling of the line feed would be very difficult and expensive, so the upper limit of operation will be determined by the noise performance of ambient-temperature amplifiers at the time of construction. Some development in cylinder technology is provided by the project to update the Molonglo Observatory Synthesis Telescope (SKAMP). AA (Aperture Array), European SKA Consortium The receiving surface consists of Vivaldi elements with amplifiers and components for transmission, phasing, and combination of the signals to form multiple beams. These are mounted in sections approximately 1 m square, referred to as tiles. The tiles are arranged in circular areas on the ground to form the station antennas. At any station there will be separate antennas of this kind for each of the three frequency bands in Table 1. For each frequency band and polarization, four (RF) beams are formed by phasing the elements within each tile. These beams are independently steerable over wide angles. Within these four beams further (digital) beams can be formed by combining the digital data with various delays. Approximately 70 digital beamformers are planned. At 700 MHz, the 160-m station diameter (given in Table 1) results in a beamwidth of 0.19°, i.e. a beam area 2.7x10-2 sq deg., so approximately 7000 station (digital) beams would be required to provide a 200 sq. deg. FoV. At 1.4 GHz the station beamwidth is 0.15° and 44 beams are required to

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cover 1 sq. deg. Thus the specified FoV requirement (see Appendix A) can be met at 1.4 GHz but hardly at 700 MHz. The beam pointing is electronic except for a proposed +/-15º mechanical tilt in azimuth to increase the hour-angle coverage. Cryogenic cooling of the areas involved is impractical and the upper frequency is 1.5 GHz. An array of aperture 625 m2 is planned as a demonstrator (project EMBRACE), which will be a single-polarization instrument with multiple fields of view and full hemispherical scan range. The frequency range of the receiving system will be 400-1550 MHz.

Table 1, Main Parameters of the SKA Concepts and Demonstrators.

Number of Stations

Aperture of a

Station

Frequency Range

Sky Coverage, Field of View

LAR

60 to 100

Parabolic dish,

200 m diameter. Demonstrator,

300-350 m diameter

0.15- 22 GHz

Max zenith angle 60° (with 50% aperture decrease) Multielement feed, 350 beams, FoV 1 sq. deg. at 1.4 GHz

KARST

30

300 m diameter

0.3- 6.7 GHz

Max zenith angle ~50° Multiple feeds, FoV 0.5 sq. deg. at 1.4 GHz

LNSD

2320 (inner 35 km) + 160 (outer array) = 2480 (total)

12 m diameter (inner 35 km), subarrays of 13, 12-m dishes (outer array), (4400 12-m dishes, total) Dishes have extension to 16 m for freqs.<1.4 GHz

0.3- 24 GHz

FoV 1 sq. deg. at 1.4 GHz (The 12-m dishes have mesh extension to 16 m for freqs.<1.4 GHz.)

14

Cylindrical Reflectors

600

110 m x 15 m

0.1- 22(?) GHz

Upper limit set by line feed. FoV 54 sq. deg. at 1.4 GHz and 700 MHz bandwidth. Multibeaming

AA

100

station diameter: 160 m (0.12-0.3 GHz) 112 m (0.27-0.67 GHz) 81 m (0.6-1.5 GHz)

0.12 - 1.5 GHz

Scan ± 60° (az), ± 45° (el) 4 RF beams /pol./band. Further digitally-formed beams within the RF beams.

PPD

(Demonstrator)

9-12

12 m dishes

~0.5-8 GHz

Fully steerable antennas with single feed.

xNTD

(Demonstrator)

~20

15 m dishes (SD/PAF)

0.8-1.7 GHz

10x10-beam (phased-array feeds) provides FoV 33 sq. deg.

KAT

(Demonstrator)

20

15 m dishes (SD/PAF)

0.7-1.75 GHz

10x10-beam (phased-array

feeds) dual polarization.

APERTIF (Proposal for

Demonstrator)

14

25 m dishes (Westerbork

Array)

0.85-1.75 GHz

8x8 phased-array

feeds, dual Polarization, 25

beams.

15

PPD (Prestressed Parabolic Dish), India Light weight, 12 m dishes are designed using prestressed members for stiffness. A demonstrator array at the GMRT campus is planned xNTD (Experimental New Technology Demonstrator), Australia This system has been proposed as a demonstrator to be built at the Mileura Station site. The antennas are dishes of diameter 15 m, with phased-array feeds (SD/PAF). The 10x10 beams of each antenna would provide a FoV of 33 sq. deg8. For two polarizations per beam and a bandwidth of 250 MHz, the bit rate per antenna is 8x1011 b/s. The front ends will probably be uncooled and the goal is a system temperature <50 K. Note that this is not to be confused with the Mileura Widefield Array (mentioned in §3) which covers 80-300 MHz and is being developed at the same location. KAT (Karoo Array Telescope), South Africa It is planned to construct an SKA demonstrator at a site in Northern Cape Province. The reference design is 20 parabolic antennas, each of 15 m diameter and covering a frequency range of 0.7-1.7 GHz. Focal ratio, f/D, will be in the range 0.5-0.55. Feeds will be 10x10 phased-array type with dual polarization, providing a FoV of approximately 50 sq. deg. APERTIF (APERture Tile In Focus), Netherlands Feeds will be 8x8 phased-array type producing 25 beams, fitted to the 14 25-m antennas of the Westerbork Synthesis radio Telescope. As of Sept. ’05, this plan has the status of a proposal only. Luneburg Lens Array, Australia This approach was studied in detail during 2000-3 (Parfitt et al. 2000) but has been withdrawn as an SKA concept. However, it should be mentioned for completeness. The use of Luneberg lenses as concentrators for the antenna elements offers the attractive possibility of using two or more feeds to provide independent beams that can be pointed in widely different directions on the sky, so that different observing programs can be pursued concurrently. A complication in the fabrication is that the dielectric constant of the spherical lens is required to vary as a function of radius. More seriously, the lens material proved to be sufficiently lossy that the required system temperatures could not be achieved in the higher part of the frequency range without overly reducing the lens diameter. 7. Hybrid Solution Several considerations lead to the conclusion that no single concept provides the optimum solution over the whole frequency range of the SKA. Some of these considerations are as follows. Field of View The required field of view (see Appendix A) above 1 GHz is given as 1 sq. deg. at 1.4 GHz, scaling as λ2. In the range 0.5-1 GHz it is 200 sq. deg. at 0.7 GHz, again scaling as λ2. An antenna with an area that just fits the requirement at 1.4 GHz will have a FoV of 4º at 0.7 GHz. Thus, at about 1 GHz there is a discontinuity in the FoV requirement of a factor of 50. Design for optimum performance within each of these frequency ranges would necessarily differ, and coverage by a single instrument would 8 Figure supplied by J. Bunton.

16

involve compromise. For example, with small dishes (~12-15 m diameter), a feed mounting system to provide interchangeability between both the focal plane arrays required for FoV below 1 GHz, and the high frequency feeds with cooled receivers for the higher bands, would be mechanically difficult to design and would increase the weight and cost of the antennas. Cryogenic cooling At frequencies below about 0.5 GHz, the contribution to the system temperature due to the sky background is large compared with the noise contribution from input amplifiers operating at ambient temperature. Cryogenic cooling of amplifiers only becomes helpful as the frequency approaches 1 GHz, and at the current state of the art it gradually becomes very important as the frequency goes above about 2 GHz. Since cooled components require some kind of a Dewar enclosure, the system being cooled cannot easily be distributed over an aperture array or along a line feed. Thus the cooling requirement virtually dictates that above ~2 GHz, antennas with two-dimensional concentrators are required to bring the received power to a small focal area. Below ~1.5 GHz, a wider choice of antenna types is possible. Cost of reflector accuracy Over the frequency range 0.1 to 25 GHz, the required accuracy of a reflector varies by a factor of 250. Thus, it can be argued that it is inefficient to use antennas designed to work at 25 GHz for observations at a few hundred MHz, where much cheaper antennas would suffice. Frequencies below ~200 MHz At the lowest frequencies within the SKA range, which are important for the Epoch of Reionization (EoR)9 studies, dipole arrays similar to those used for LOFAR and the Mileura Wide-field Array are almost certainly the optimum choice. They are robust, have no moving parts, and can be used to form multiple beams that are electronically steerable. Choice of site It can be argued that since, at the lower frequencies, radio interference from devices such as mobile telephones are likely to be most serious, a low RFI level is the most important criterion in choice of site, whereas at higher frequencies the effects of atmospheric water vapor become increasingly important, favoring higher elevations. Ideally, different sites may be desirable in these cases. It may also be worth noting that self-generated interference tends to become more of a problem as the number of instruments and the frequency range covered at any one site is increased. Outermost Antennas As noted in §3 above, observations involving the longest antenna spacings are likely to be concerned with compact discrete sources for which smaller fields of view are required. Thus for the outermost antennas the FoV specifications can probably be relaxed, and for observations of single objects, large single dishes without large focal plane arrays may be adequate.

9 The 100 MHz low-frequency specification for the SKA is based on the requirement for EoR studies, for which the observable phenomena are expected to be found at H1 redshifts greater than 6.5, (frequencies less than 190 MHz). The lowest frequency that may be important for EoR observations is not known, and could be less than 100 MHz.

17

Considerations such as those above have led to the consideration of a hybrid solution to the SKA design problem, with different concepts for the lower and higher frequencies, and a possible break point at about 1.5 GHz. See the 2004 white papers on hybrids and several papers in Hall (2005). 8. Signal-Processing System A schematic outline of a possible arrangement for the signal path from the antenna(s) at one station to the correlator is shown in Fig. 1. For frequencies below 1 GHz multiple beams will be required to fulfill the FoV requirement. The multiple beams may be FEED RF ELECTRONICS SHORT-HAUL LINKS

Figure 1. An example of a possible system architecture for the signal path from the antennas to the correlator output. Some choices concerning the system design in the figure are noted in the text in §8. The variable delay system that compensates for the geometrical path lengths from a source to the antennas is assumed to be included in the router section of the correlator. MAC indicates the multiply and accumulate functions of the correlator. The local oscillator subsystem is not shown. From Horiuchi, Chippendale, and Hall (2004). generated by a large antenna with a focal-plane feed array, by an array of smaller antennas combined with appropriate phasing, or by some combination of these arrangements. Broadly speaking, the signals received from each feed or antenna element

18

will require amplification with a low-noise stage to preserve the SNR before they are subdivided in the beam former. The principle of the beam former is as follows. Each received signal is subdivided by up to Nsb ways so that a component of it can be incorporated into each beam, if required. For each beam the components from the receiving elements are combined with their phases adjusted so as to form a maximum in the required pointing direction. The science requirements call for 50 phased-array beams within the FoV from the antennas within the inner 5 km of the array (but also state that this number is somewhat flexible). Note, however, that the beamforming in Fig. 1 would not be needed for small dishes with single feeds operating at frequencies above ~1.4 GHz. At the higher frequencies of the SKA the beamforming would be done in IF stages, so frequency conversion would be required following the low-noise amplifiication, before the signals enter the beam former. Depending on whether the beamformer is designed to operate with signals in analog or digital form, A/D conversion may also be required after or before the beamformer. Four signals are required from each beam that is formed to provide for two orthogonal polarizations and two frequency bands, which thus requires four beamformers for each pointing direction. For the two polarizations, two outputs are required from each receiving element, but one low-noise amplifier will probably provide the signals for two bands with the same polarization. Details of whether or not frequency conversion or A/D conversion are performed ahead of the beamformer will depend on the particular details of the chosen system. For example, if the station consists of an array of small paraboloids the signals will most likely be converted to an IF band and transmitted in digital form, by optical fiber, to the beamformer. On the other hand, if the station antenna involves many small receiving elements, as in the case of an aperture array or line feed, signals from two or three adjacent elements may be combined before amplification, and later power-split and combined in various stages so that the elements of the beamformer are distributed within the network of amplifiers and combiners. Figure 1 shows the second (digital) stage of beamforming as located at the stations, although it could also be performed at the central electronics complex. Consideration of the beamformer details are unlikely to influence the choice of antenna system, rather the latter will drive the requirements of the beamformer. The points at which frequency conversion and digitization are introduced are decided by the cost effectiveness of the engineering. In Fig. 1 the signals are digitized early in the system and most of the processing functions are performed digitally. This is likely to be the preferred choice for the SKA. However, all of the pre-correlator functions, including the long- and short-haul link transmissions, could be performed on signals in analog form. The low noise input stages will almost certainly be some form of HEMT amplifiers, since these provide the best sensitivity over the full SKA frequency range. See Weinreb (2004, Figs. 3-5) for curves of noise temperatures for such amplifiers as a function of frequency for operating temperatures of 5, 80, and 300 K. Such amplifiers can be designed for bandwidths of one to several octaves, which approximately match those of broadband feeds. For amplifiers with responses up to 2.5 GHz, noise temperatures of ~25 K can be achieved for ambient temperatures of 300 K. Thus for frequencies of up to ~600 MHz, for which the galactic background adds several tens of degrees to the system temperature,

19

cryogenic cooling of the front-end stages is unnecessary. Uncooled amplifiers provide acceptable performance up to about 1 GHz, which enables aperture arrays and line feeds for cylindrical paraboloids to be practicable in this range. For 80 K ambient temperature, HFET noise temperatures of ~7 K for frequencies up to about 5 GHz are achievable. For 20 K ambient temperature, noise temperatures of 3 K up to 5 GHz, and 5 K up to 10 GHz, are achievable. Thus for the frequency range above 2 GHz, cryogenic cooling appears to be unavoidable. In the range of about 1 to 2 GHz, cooling improves the performance, but is arguably not essential in all cases. The corresponding effects on the antenna requirements are discussed above under Cryogenic Cooling in §7. 9. Signal Transmission Transmission of the signals from the station to the correlator will be by optical fiber10. To provide an estimate the bit rate from a single station, condsider antennas of diameter 13 m, which have a beam that is equal to the FoV requirement of 1 sq. deg. at 1.4 GHz. If, for example, we use T = 20 K for the system temperature, and 70% aperture efficiency for the antenna, then 5,053 antennas are required. For 200 stations, a number that is intermediate between the extremes of the current concepts, 25 antennas per station would be required. Spacing of the antennas to avoid shadowing would result in an aperture filling factor, possibly of order 1/10. Thus the phased station array could have a beam solid angle as small as 1/250 sq. deg., and Nsb = 250 phased beams could be required to maintain the FoV coverage. If the beams were formed at the stations, the total station bit rate would be 8NsbBNbit, where the factor 8 represents 2 polarizations, 2 frequency bands, and the factor of 2 for Nyquist sampling; and Nbit is the number of bits per sample. If the signals from the antennas were transmitted back to the correlator location and the beamforming performed there, the station bit rate would be 8NsaBNbit, where Nsa is the number of station antennas. For the maximum specified bandwidth B = 4 GHz, Nbit = 8, and 250 beams, the station bit rate would be 6.4 x1013 b/s for station beamforming, but 6.4 x1012 for beamforming at the correlator, which would thus be the likely choice. If each station used a single dish of diameter 13x(251/2) = 65 m it would be necessary to use focal plane arrays to provide 25 beams. As another example, consider the LAR concept in which the station beam is that of a circular reflector of diameter 200 m, and Nsb = 350, so the maximum station bit rate could be as high as 9x1013 b/s. For the 200 sq. deg. requirement at 700 MHz, B is reduced to 25% of 700 MHz =175 MHz, i.e. it is reduced by a factor of approximately 23 with respect to the cases considered above. The FoV requirement is increased by a factor of 200. However, it is not at all clear how the 200 sq. deg. FoV requirement can be accommodated in practice. As mentioned above, it is very difficult with a single large reflector at each station. At the other extreme, consider a station consisting of 1.9 m dishes, each of which would have 200 sq. deg. FoV. The number of such dishes required would be approximately 45 times that for 13 m antennas. So taking into account the smaller bandwidth, the station bit rate would be doubled to ~1.4x1013 b/s. As a rough indication of the transmission capacity of an optical fiber, we can take 20 Gb/s as the capacity of a single laser channel, and about 50 as the number of laser signals that can be transmitted on one fiber, resulting 10 The use of recording of signals from distant antennas on tape or disk, as is used in VLBI, is not envisaged for the SKA.

20

in about 1012 b/s. Thus, with current technology, ~6 or more fibers would be required to carry the data from a station to the correlator. If the outermost antennas are used mainly for imaging compact objects with small fields of view, the bits rates could be lower for the longest fiber transmissions. The compensating delays required to take account of the varying space paths to the antennas, as well as the delays in transmission from the stations to the correlator, will be inserted using random-access memory (RAM) units. Large delay capacities will be required. 10. Quantization The Science Requirements specify 8 bits, i.e. 256 levels for the quantization of the signals that are transmitted from the antennas. Such a large number of levels is not required for signal-to-noise considerations, but it allows for larger amplitude interfering signals to be recorded without distortion caused by overloading. This preserves the possibility of being able to remove the unwanted signals later in the data processing.

� ��� ��� ��� ��� � ��� ��� �������

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Figure 2. Quantization efficiency for 256-level (eight-bit) sampling of a signal with Gaussian distribution, as a function of the increment between quantization levels. The level increment is in units of the rms distribution of the input. The formulas from which the curve was derived can be found in Thompson and Emerson (2005). For 265 levels the quantization efficiency, which is defined as the SNR at the correlator output expressed as a fraction of the SNR for an equivalent analog system, has a theoretical maximum value of 0.999912 for an increment between levels of 0.0308σ, where σ is the rms level at the input of the digitizer. With a level increment of 0.3σ, for which the quantization efficiency is 0.9926, the outer levels (+/-128) represent +/-38σ, that is, interfering signals remain within the range of the quantizer so long as they do not exceed 38σ (31 dB above the noise). Similarly, a level increment of 0.5σ results in a quantization efficiency to 0.9796, and signals up to 64σ (36 dB above the noise) remain within the range of the quantizer. Figure 2 shows a graph of quantization efficiency as a function of level increment for the 256-level case.

21

Eight-bit samplers with a clock rate of 2 GHz, which allow Nyquist sampling of a bandwidth of 1 GHz, are currently available and are being used in the EVLA program. It is reasonable to expect that higher clock rates will be available when the SKA is under construction. In some cases where wideband IF systems are coming into use in radio astronomy, the digitized signals are passed through FIR filters before cross correlation to sharpen band edges or cut out interference, etc. This requires requantization after filtering to reduce the number of levels to the original value. Further quantization noise is introduced, resulting in a further small loss in SNR. The cross correlation of the two sets of quantization noise is likely to be small, in which case, the variances combine additively. If it were possible to remove all interference by filtering or similar processes prior to cross correlation of the signals, the number of levels could be reduced after the removal to simplify the correlator. However, some important mitigation techniques, e.g. adaptive nulling (Leshem et al., 2000), are applied in the post-correlation processing. 11. Local Oscillator System Tunable local oscillator signals are required at the antennas to reduce the received signals to IF bands. Synchronous signals to drive the sampler clocks are also part of the LO system. In existing large synthesis arrays, local oscillators at the antennas are synchronized in phase with a master oscillator at the array electronics center by sending out reference frequencies to the antennas. The effect of variation in the transmission path lengths is calibrated by returning the oscillator signals to the central location and measuring any changes in the round-trip phase path. The reference signals that are transmitted to the antennas can be at fixed frequencies, in which case frequency synthesizers are required at the antennas to provide the necessary tuning capability. Alternatively, the synthesizers at the master oscillator location can be used to generate subharmonics of the required local oscillator frequencies and these are transmitted to the antennas to simplify generation of the required frequencies there. Fixed reference frequencies have advantages for round-trip phase correction, but either case should work satisfactorily for distances up to ~50 km from the center of the SKA. For distances greater than about 50 km, the attenuation in optical fiber (0.2-0.4 dB per km) necessitates the use of amplifiers11. For two-way propagation on a single fiber, two amplifiers are necessary, and optical circulators have to be used to separate the signals in the two directions and pass them to the appropriate amplifier (see, e.g., Coleman et al. 2003). In this case the signal paths are not identical in the two directions since they include different amplifiers. Further development is needed in this area to determine the limits of performance. Experience with antennas at distances of hundreds and thousands of km, as used in VLBI, has mostly involved the use of hydrogen masers at the remote antennas. At a cost of roughly $200,000 each, these may be too expensive for use in the quantities that would be required for the SKA. However, it is possible that less expensive alternatives with sufficient frequency stability may emerge. Distribution of reference signals by geostationary satellite is technically feasible and would only require narrow bandwidth channels. Correction for the varying time delays caused by atmospheric effects would be necessary, but feasible by using some scheme for measurement of round-trip phase (Clark 2002). Tests with such a system in which an rms time error less 11 Erbium-doped fiber amplifiers (EDFAs).

22

than 3 ps was measured for integration times within the range 1-2000 s are described by Bardin et al. (2004) and Bardin (2005). In comparison with a hydrogen maser, the Allan standard deviation for Bardin’s satellite system is about a factor of 6 times greater for 1 s integration, but steadily improves with integration time and is equal to that of a maser at 500 s. Fringe rotation and phase switching are somewhat minor details of the LO system, but deserve brief consideration. In connected-element arrays, fringe rotation is commonly applied in the local oscillator at the antennas, and in VLBI it is usually applied digitally at the correlator. With the SKA, the cross correlation for all of the baselines is to be performed in real time, so fringe rotation in the LO system is the appropriate procedure12. Phase switching is helpful in suppressing the effects of certain types of spurious signals, and mitigating certain voltage offsets in the quantizer thresholds. For phase switching, a number of orthogonal switching waveforms equal to the number of stations is required. Walsh functions are convenient and widely used. The slowest switching sequency is set by the shortest integration time, and the highest sequency is therefore set by the number of stations. With a very large number of stations the highest switching rated can become inconveniently high. Phase switching is generally considered unnecessary in VLBI since the high fringe rates help to remove unwanted effects. Thus it may be possible to leave unswitched the outermost antennas that are only involved in baselines with high fringe rates. 12. Data Rates The design of the correlator is not considered here, but it is interesting to consider the data rates that the correlator should be able to handle. In an XF correlator the rate at which cross multiplications are performed is proportional to Ns

2Nsb, that is, the square of the number of stations times the number of beams per station. For a given total collecting area, A, the FoV of a single station beam is proportional to Ns, and the total station FoV is proportional to NsNsb. Thus NsNsb is determined by the FoV requirement, and to minimize the number of cross multiplications we could minimize Ns. However, following the discussion in §2, the effect on the dynamic range must be considered in choosing the number of stations. The 200 stations discussed in §9 result in a station bit rate of 6.4x1012 b/s. Since the collecting area of a station is inversely proportional to Ns, the solid angle of the station beam is proportional to Ns. The number of station beams required is inversely proportional to the solid angle of each such beam. Thus the station bit rate is inversely proportional to Ns, and the total bit rate at the correlator from all stations is therefore independent of Ns. (It is assumed that the bit rate numbers are driven by the diffraction-limited FoV requirement.) As an example of the capacity of a correlator that is large by current standards we can take the one under construction for ALMA. This array has one antenna at each of 64 stations. The maximum bandwidth is the same as for the SKA, and the number of bits per sample is 3, resulting in a bit rate from each station of 9.2x1010 b/s, that is, about 1/70 of the rate estimated for an SKA

12 In VLBI the FoV may be limited by the “delay beam” effect, and it is thus sometimes useful to be able to rerun the data for a slightly different field center. Fringe rotation at the correlator is advantageous in this case. Since, however, the pre-correlation data will not be recorded for the SKA, fringe rotation at the correlator offers no advantage, but would increase the size and cost of the correlator.

23

station. The biggest factor in the difference between these two cases results from the fact that ALMA does not include a FoV specification like that for the SKA and it generates only a single beam at each station. These remarks are consistent with the proportionality between the FoV and the number of receiver chains noted in §5. D’Addario and Timoc (2002) have considered in some detail the design of a correlator for 2560 stations (similar to that required for the LNSD concept in Table 1), with 3.2 GHz total bandwidth, and conclude that it is feasible. 13. Points for which Further Study or Experience is Required DynamicRange The sole reason for the very large aperture of the SKA is to provide the high sensitivity characterized by item 5 in Table 1A of the Appendix. This is derived from the science requirements on the assumption that the sensitivity will be limited entirely by the system noise. Before any design concept can be chosen, it is imperative to understand the extent to which its dynamic range may also be a limiting factor in the sensitivity. As discussed in §2, further studies of the effect of the number of stations are needed. This requires estimating the expected flux density distribution of sources down to the limits set by the science projects, and modeling the responses for arrays with different numbers of stations but a constant total collecting area, that is, a detailed computer simulation of a “software telescope” is required. This is not intended to imply doubt of the results of Lonsdale et al. (2000), but since some arbitrary choices are unavoidable in selecting the model parameters, an independent verification of such a basic parameter as the number of stations is highly desirable. Estimates of the source population at the sensitivity limit of the SKA can be found in Hopkins et al. (1999). Experience with LOFAR and the Mileura Array will also help to throw some light on this requirement. For stations that use phased arrays, as discussed in §4, the possible effect of the sidelobes of the beam on the dynamic range could be investigated in such a study. It is fairly generally assumed that it is necessary to image and deconvolve the entire main beam area of the antenna elements (i.e. the main beam of the individual antennas in the case of phased-array stations), see, e.g. Perley and Clark (2003). However, Lonsdale et al. (2004) conclude that the necessity to image the full primary beam to high fidelity is not well justified. They consider several effects including the introduction of a weighting in the (u,v) data, i.e. effectively convolving the point spread function with a function that reduces the sidelobes. This is referred to as controlling the correlator FoV and Lonsdale et al. illustrate it by a simplified example, from which they conclude that high station sidelobes resulting from phased arraying are not likely to compromise performance. They note, however, that the calibration requirements for high dynamic range require further study and can be expected to introduce a serious computational burden. One can conclude that post-processing area needs detailed simulation to better understand how to maximize the dynamic range, including related constraints on the number and configuration of antennas, the use of phased arraying of antennas or phased-array feeds for multibeaming, etc. Field of View below 1 GHz Only one of the current concepts (Cylinders) appears capable of precisely meeting the requirement of 200 sq. deg. at 0.7 GHz. It appears to be

24

virtually impossible for the large dish concepts (LAR and KARST) to do so. The FoV specification is perhaps the most difficult to meet in the antenna design. The xNTD would provide a FoV about ¼ of the present specification. Most programs could presumably be done with this smaller FoV, but would take four times as long to complete. Is this an acceptable compromise? Even with the reduced specification, multi-beaming is required, so it is necessary to develop prototype phased-array feeds (PAFs) to demonstrate the required beamforming capability while maintaining the specified bandwidth and low system temperature. Phased Arrays as Station Elements In the case of stations that consist of a phased array of Ns small dishes or other elements, the spacing of the elements to avoid shadowing at large zenith angles13 will result in grating lobes or extended sidelobes, depending on whether the antennas are placed in lines with uniform spacing or in some other arrangement. Also, to obtain the same FoV as a single element, the required number of beams of the phased array is roughly equal to Ns divided by the aperture filling factor. Clearly one would like the filling factor to be as close to unity as possible. If the number of stations is large enough that the (u,v) plane is well sampled over a limited range of hour angle (say ± 3 h), then limiting the hour angle coverage would allow closer spacing of the antennas. However, such a limitation could be detrimental in the case of observations of transient phenomena, or for observations including the most distant antennas for which the difference in longitude also limits the possible hour angle coverage. Some thought should be given to an optimum value of the hour angle range for phased arrays. The currently proposed LNSD concept avoids the phased-arraying problem by cross correlating all individual antennas within the inner 35 km radius. LO References for stations beyond ~100 km For stations within ~50 km of the array center, LO references can be transmitted on fiber with round-trip phase correction. For longer distances it is likely to be necessary to include amplifiers in the links, which will introduce slight differences in the outgoing and incoming path lengths. Thus some deterioration in the phase accuracy with increasing distance may be expected. The maximum practicable distance over which LO distribution by fiber can be used does not appear to be known. Other techniques for distant stations include the use of a maser standard at each station, or possibly distribution of reference signals by satellite as mentioned above in §11. Development for the New Mexico Array (phase 2 of the extended VLA) could provide some help in finding the optimum techniques. Mitigation of RFI In recent years numerous techniques for removal or attenuation of radio-frequency interference have been suggested, and in some cases demonstrated, but most are not yet in regular use. Use of such techniques will certainly influence computing requirements, and possibly even choice of sites. More experience is needed with these techniques before they can be included as a feature in SKA design. At the present time it is unclear to what extent the residual interference responses will reduce the sensitivity relative to that of the corresponding interference-free situation. 13 The use of mesh extensions to increase the low-frequency area, as mentioned in §6 under LNSD, exacerbates the shadowing problem at the higher frequencies.

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LOFAR is designed to operate in a hostile RFI environment and in this respect is a pathfinder for the SKA. The adopted strategy is to use analog-to-digital converters with 12 bits so that signals of large amplitude can be accommodated without overloading. Observing bands of width 100 MHz are digitized and then processed by a digital filter bank that creates sub-bands with extremely low out-of-band responses. In the initial implementation of LOFAR, a limited set of bands with 16 bit data format is transported to the central processing location for spectral filtering and cross-correlation. There the RFI-free channels are selected for further processing and imaging. At this stage the RFI can be detected in each snapshot and subtracted just like other strong sources. It is anticipated that after sufficient experience with this procedure the selection of RFI-free sub-bands can be performed at the antenna stations. Since these sub-bands need only 4 bits to preserve adequate sensitivity, the number of sub-bands or station beams that can be transmitted to the central processing location can be increased. Procedures to perform spatial filtering in the beam forming process at each station have been investigated but require further development. 14. Some Conclusions The responses to the call for proposals for the SKA indicate that no single concept meets all the science requirements in a practical and cost effective manner, and a hybrid solution is therefore indicated. As mentioned in §7, for frequencies below about 300 MHz, arrays of dipoles of the general type developed for LOFAR provide the optimum approach for the SKA. For frequencies above a few GHz, cryogenic cooling of receivers is required to obtain the necessary sensitivity. This requires antennas in which the received radiation is brought to a compact focal area, as in dish-type reflectors. In the intermediate range from a few hundred MHz to a few GHz, concepts based on multi- beam dishes, cylinders, and aperture arrays have been proposed, with multiple beams for large FoV, and uncooled front ends. The choice between these possibilities requires development of prototypes for an accurate comparison of performance versus cost. It can be argued that this intermediate range is the one that deserves the most immediate attention with regard to the SKA, since observations for four of the six key science requirements fall within it. Further, it is a part of the spectrum in which commercial usage is rapidly expanding. Demonstration of the multibeaming capability is important. For further discussion of effective frequency ranges and relative costs of different concepts see Bregman (2005b). A hybrid approach with three different antenna technologies, as indicated above, could be an effective solution covering the entire range 0.1-25 GHz. Studies involving simulation are particularly needed in the area of dynamic range and how it is related to the number of stations and the sizes of the antennas (as discussed in § 13 under Dynamic Range. Development of effective methods of interference mitigation is also essential for the success of the SKA. Achievement of dynamic range and mitigation of interference are likely to be major factors in determination of the post processing load.

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15 Reference Design In January 2006, a reference design for the SKA (SKA Memo 69) was chosen. The intention of this step is to concentrate the general development effort toward a system most likely to be optimum with regard to engineering practicality and scientific capability. This more clearly defined vision of the SKA is required to provide a basis for detailed cost accounting, for science policy makers, and for both public and scientific outreach. In addition to the Reference Design for the full SKA, a Phase 1 instrument was also specified. This will provide a starting point for development of the detailed design, and when operating will verify the practicality of the full instrument. The basic parameters for the Reference Design and Phase 1 are given in Table 2. The only major change in the specifications discussed during 2004-5 is the reduction of the maximum field of view for frequencies above 300 MHz from 200 sq. deg. to 50 sq. deg. The full reference design is comprised of three parts. (1) The major part is composed of small dishes of diameter of order 10 m with an operating frequency range of 0.3 to 25 GHz. Feeds for 0.3-1 GHz will use a phased-array technology to provide multiple beams covering 50 sq. deg. in the 0.3-1.0 GHz range and 1-10 sq. deg. in the 1-3 GHz range. From 3 to 25 GHz the emphasis is on wide-bandwidth, single-beam feeds which would provide fields of (3/f)2 sq. deg. where f is the frequency in GHz, e.g. approx. 0.36 sq. deg. at 5 GHz. The longest baselines would exceed 3000 km. (2) A part of the central core of the array will consist of Aperture Array (AA) tiles covering the frequency range 0.3-1.0 GHz. The electronic pointing and beamforming will allow, for example, frequent monitoring of known burst-type sources as well as providing additional aperture below 1 GHz. The fraction of the total aperture provided by the AA tiles will be determined by cost and performance considerations as development proceeds. (3) An array of broadband dipole antennas of the type used in LOFAR or the Mileura Widefield Array, covering the frequency range ≤0.1-0.3 GHz. The major scientific study for this frequency range is the Epoch of Ionization (EoI) H-line survey. In the Phase 1 instrument the goal is to provide approximately 10% of the aperture of the full SKA. A large part of this aperture would be within the inner 5 km, but with some elements providing baselines up to 50 km. A possible configuration that is being considered is a central array of dishes and AA elements with further clusters of dishes spaced at intervals along five spiral arms. The frequency range for Phase 1 is 0.3-10 GHz, and does not include the dipole arrays in part (3) of the full SKA. This is omitted because LOFAR will provide approximately 10% of the full SKA aperture in this frequency range, so including it in Phase 1 would not provide a new science capability, (unless the H-line feature is primarily found in the FM broadcast bands in which the heavy usage of the spectrum might preclude useful observations). The choice of the antenna types to cover the frequency range of the Reference Design follows largely from the considerations discussed in the earlier part of this report. At the highest frequencies dishes provide the only feasible approach, and the field of view

27

Table 2. Ref. Design, Full SKA Ref. Design, Phase 1 Frequency Range 0.1-25 GHz 0.3-10 GHz Aeff/Tsys 0.1-0.3 GHz 5000 m2K-1 0.3-1 20000 2000 m2K-1 1-3 20000 2000 3-10 20000 2000 10-25 10000 Field of view 0.1-0.3 GHz 200 sq. deg. 0.3-1 50 50 sq. deg. 1-3 1-10 1-10 3-10 0.33 (3/f)2 1 (1.4/f)2 10-25 0.33 (3/f)2 Maximum Baseline >3000 km 50 km requirements eliminate the possibility of using large dishes. Cylinders can provide fields of sufficient angular size, but these are elongated on the sky and (unless the antennas are located at the Earth’s equator) rotate in angle as the hour angle varies. As a result, the (u,v) coverage varies over the field, and deconvolution to obtain high dynamic range becomes, at best, very difficult. Dishes with diameters of order 10 m cover the frequency range from 0.3 GHz upward satisfactorily, and the main technical problem is the development of phased-array feeds to provide multiple beams covering up to 50 sq. deg. at 1 GHz. The AA tiles are limited in practice to frequencies not much higher than 1 GHz, but introduce new technology with particular agility in pointing. The use of dipole arrays for 100-300 MHz is relatively straightforward since antennas of a similar type are in use in LOFAR. 16 Systems Issues of the Reference Design Details of the antennas continue to be a central issue in ongoing development of the Reference Design. At this early stage in the development of multi-beam, phased-array feed systems for small dishes, the extent to which the desired performance can be achieved in the 0.3-3.0 GHz range is one of the major uncertainties of the Reference Design. However, the choice of dishes as a major component allows, if necessary, fall-back to simpler, well understood feed clusters with more limited instantaneous sky coverage. The achievable bandwidth of any phased-array feed system is perhaps not much more than one octave, so a number of feeds would be required to cover the 0.3-3.0 GHz decade. These would then share space on a feed mounting system with one or more broadband, single-beam feeds for 3.0-25 GHz. The number of feeds that it is practicable

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to accommodate will be limited by their weight and the tolerable blockage of the small-dish aperture. When the limitations on the achievable field of view with phased-array feeds and the limits on the size and weight of the feed system are better understood, it may be necessary to consider the optimum balance between frequency coverage and instantaneous sky coverage. The desirability of continuous frequency coverage from 0.3 to 25 GHz results largely from the requirement to observe spectral lines over a wide range of redshifts, in particular H1, OH, and the maser lines of H2O. Following the progress of the various demonstrator projects will thus be an ongoing concern of the System Subroup. It will become possible to specify details of the number and configuration of the antenna stations as the optimum reflector size (in terms of cost and performance) and the achievable field of view are determined. The choice of site is also a consideration in this respect. The fraction of the total aperture in the form of AA tiles can be more clearly determined when the cost and performance of EMBRACE and other AA demonstrators are known. Much of the design of the receiving and signal transmission hardware, other than the phased-array feeds, can be based on practice that is well understood from existing instruments and demonstrators. Detailed choices for the SKA will depend on the cost and estimated maintainability of the hardware. A few questions remain open such as the synchronization of timing at the most distant stations, e.g., through use of masers or signals transmitted by optical fiber or satellites. The area of data reduction is one in which requirements and capabilities will continue to evolve. Computing requirements for high dynamic range in widefield imaging are discussed, for example by Lonsdale et al. (2004) and Cornwell (2004). The numerous strategies for RFI mitigation (see, e.g. Ellingson 2004), will impose requirements of varying degree on the computer processing. For example, adaptive nulling, which is a promising procedure for instruments with multiple antennas, can involve processing of large matrices. Studies based on simulations as well as practical experience gained from observations with demonstrators are required to define more precisely the range of data processing necessary for the SKA. A final point to be borne in mind is that the ability to make deep field observations with high dynamic range depends on the ability to control the systematics of the instrumentation, that is, the instrumental factors that result in systematic (as distinct from random) errors in the data. The stability of the instrumental characteristics and the ability to control or calibrate them accurately are of fundamental importance and should take precedence over most other considerations. Acknowledgements. Helpful reviews and comments have been provided by J. Bunton, L. R. D’Addario, K. I. Kellermann, R. C. Walker and S. Weinreb.

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Appendix A SKA Parameters Determined by the Science Requirements Required SKA parameters and brief descriptions of the related science requirements are given by Jones (2004), from which the following Table is taken.

Table 1A – Expanded SKA Science Requirements (version 9) 1. Frequency range 100 MHz - 25 GHz Goal: 60 MHz - 35 GHz 2. Simultaneous independent observing bands14 2 pairs (2 polarizations at each of two independent frequencies, with same FoV centers) 3. Max. freq. separation of observing bands Factor of 3 between observing band center frequencies (same FoV centers) 4. Instantaneous bandwidth of each observing Full width = 25% of observing band center frequency, up to a band up to a maximum of 4 GHz BW for all frequencies above 16 GHz 5. Sensitivity at 45 degrees elevation (A/T) Goal: 2500 at 60 MHz, 5000 at 200 MHz, 20000 between 0.5 and 5 GHz, 15000 at 15 GHz, and 10000 at 25 GHz Goal: 5000 at 35 GHz 6. Configuration Minimum baselines 20 meters, 20% of total collecting area within 1 km diameter, 50% of total collecting area within 5 km diameter, 75% of total collecting area within 150 km diameter, maximum baselines at least 3000 km from array core (angular resolution < 0.02 / fGHz arcsec)

7. Image quality Dynamic range > 106 and image fidelity > 10

4 between 0.5 and

25 GHz, over a range of 90 degrees in declination and 100 in angular resolution 8. Contiguous imaging field of view (FoV) 1 square degree within half power points at 1.4 GHz, scaling as

λ2, 200 sq. deg. within half power points at 0.7 GHz, scaling as

λ2 between 0.5-1.0 GHz

9. Number of separated fields of view 1 with full sensitivity Goal: 4 with full sensitivity 10 simultaneous sub-arrays 10. Correlator and post- correlation processing Input bandwidth 25% of center frequency for frequencies below 16 GHz and 4 GHz for frequencies above 16 GHz (per observing band) Imaging of 1 square degree at 1.4 GHz with 0.1 arcsec angular resolution Imaging of 200 sq. degrees at 0.7 GHz with 0.2 arcsec angular resolution

imaging of 104 separate regions within the FoV, each covering

at least 105beam areas at full (maximum baseline) angular

resolution

Spectral resolution of 104 channels per observing band per

baseline

14An “observing band” is a contiguous set of frequencies that pass through all processing steps simultaneously.

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Minimum sampling interval 0.1 ms for wide-field pulsar searches 11. Beamformer capability 50 simultaneous summed (phased array) beams within FoV, inner 5 km diameter of array. No time averaging, 8 bits/sample.

12. Survey speed FoV x (A/T)2 x BW = 3 x 10

17 deg

2 m

4 K

-2 Hz

-1

at 1.5 GHz

FoV x (A/T)2 x BW = 1.5 x 10

19 deg

2 m

4 K

-2 Hz

-1

at 0.7 GHz 13. Antenna pointing and slewing Blind pointing < 0.1 HPBW, move between adjacent sky Positions separated by 0.5 HPBW in 3 sec, move between sky positions sep. by 90 deg. in < 60 s 14. Instrumental polarization Polarization error / total intensity –40 dB at FoV center, -30 dB out to FoV edge (after routine calibration)

15. Spectral dynamic range 104(flatness of bandpass response after calibration)

16. Total power calibration Total power (zero-spacing) flux density measured with 5% error within 1 hr.

References Bardin, J. C., Local Oscillator Distribution using a Commercial Geostationary Satellite, MSc Thesis, Dept. of E.E., U. of Cal., Los Angeles, 2005. Bardin, J., Weinreb, S., and Bagri, D., Local Oscillator Distribution Using a Geostationary Satellite, Exp. Astron. 17, 193-199, 2004, reprinted in Hall (2005). Bhatnagar, S., Cornwell, T. J., and Golap, K., Correction of Errors due to Antenna Power Patterns during Imaging, EVLA Memo 100, 2006. Bregman, J. D., System Optimization of Multi-Beam Aperture Synthesis Arrays for Survey Performance, Exp. Astron., 17, 365-380, 2005a, reprinted in Hall (2005). Bregman, J. D., Cost Effective Frequency Ranges for Multi-beam Dishes, Cylinders, Aperture Arrays and Hybrids, Exp. Astron., 17, 407-416, 2005b, reprinted in Hall (2005). Bunton, J., Figure of Merit for SKA Survey Speed, SKA Memo 40, 2003. Bunton, J. et al., Panorama of the Universe, a Cylindrical Reflector SKA, white paper, May 2003. Clark, B. G., A Satellite LO System, EVLA Memo 36, NRAO, March 2002.

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Coleman, J., et al., A Time Standard for Radio Astronomy, the New Mexico Array, EVLA Memo 57, 2003. Cornwell, T. J., SKA and EVLA Computing Costs for Wide-Field Imaging, Exp. Astron., 17, 329-343, 2004, reprinted in Hall (2005). (Also SKA Memo 49, 2004.) Cornwell, T. J., LNSD Reconsidered – the Big Gulp option, SKA Memo 61, July 2005. D’Addario, L. R. and Timoc, C., Digital Signal Processing for the SKA: A Strawman Design, SKA Memo 25, August 2002. De Bruyn (2006), private communication. Ellingson, S. W., RFI Mitigation and the SKA, Exp. Astron., 17, 261-267, 2004, reprinted in Hall (2005) Hall, P. J., Editor, The Square Kilometer Array: An Engineering Perspective, Springer, 2005, (reprint of 36 papers from Experimental Astronomy, Nos. 1-3, 2004). Hall, P. J. et al., Reference Design for the SKA, SKA Memorandum 69, Jan 2006. Hopkins, A., Windhorst, R., Cram, L., and Ekers, R., What Will the SKA See at 1.4 GHz?, Perspective on Radio Astronomy—Science with Large Antenna Arrays, M. P. van Haarlem Ed., Netherlands Foundation for Radio Astronomy, 1999. Horiuchi, S., Chippendale, A., and Hall, P., SKA System Definition and Costing: A First Approach, SKA Memo 57, 2004. Jeffs, B. D., van der Tol, S., and van der Veen, A-J, Direction Dependent Self Calibration of Large Distributed Sensor Arrays, IEEE Int. Conf. on Acoustics, Speech and Signal Processing (ICASSP), May 15-19, 2006, Toulouse, France. Jones, D. L., SKA Science Requirements: Version 2, SKA Memo 45, 2004. Leshiem, A., van der Veen, A.-J., and Boonstra, A.-J., Multichannel Interference Mitigation Techniques in Radio Astronomy, Astrophys. J. Supp., 131, 355-373, 2000. Lonsdale, C. J., Doelman, S. S., Capallo, R. J., Hewitt, J. N., and Whitney, A. R, Exploring the Performance of Large-N Radio Astronomical Arrays, Radio Telescopes, H. R. Butcher, Ed., Proc. SPIE, 4015, 126-134, 2000. Lonsdale, C. J., Doelman, S. S., and Oberoi, D., Efficient Imaging Strategies for Next- Generation Radio Arrays, Exp. Astron.17, 345-362, 2004, reprinted in Hall, (2005). See also, SKA Memo 54. Noordam, J. E., Report ASTRON-LOFAR -11227, 2001.

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Parfitt, A., James, J. G., Kot, J., Hall, P., A case for the Luneburg Lens as the Antenna Element for the Square Kilometer Array Radio Telescope, Radio Science Bulletin, No. 293, 32-37, June 2000. Perley, R. and Clark, B., Scaling Relations for Interferometric Post-Processing, EVLA Memo 63, 2003. Thompson, A. R. and Emerson, D. T., Quantization Noise, EVLA Memo 88, 2005 (see Appendix). Thompson, A. R., Moran, J. M., and Swenson, J. W., Interferometry and Synthesis in Radio Astronomy, 2nd Ed., Wiley, NY, 2001. Weinreb, S., Noise Temperature Estimates for a Next Generation Very Large Microwave Array, SKA Memo 27, 2004. Wright, M. and Members of SKA Task Force on Survey Optimization, SKA Survey Optimization, SKA Memo in preparation, 2006.