SEARCHING FOR FAST TRANSIENTS

Transcription

1 SEARCHING FOR FAST TRANSIENTS Document number... WP TD 002 Revision... 1 Author... T.Colgate, N.Clarke Date Status... Approved for release Name Designation Affiliation Date Signature Additional Authors Submitted by: T.Colgate ICRAR Approved by: W. Turner Signal Processing Domain Specialist SPDO

2 WP TD 002 DOCUMENT HISTORY Revision Date Of Issue Engineering Change Number Comments A First draft release for internal review B C 1 01st April 2011 First release DOCUMENT SOFTWARE Package Version Filename Wordprocessor MsWord Word b WP TD 002 1_Transients Block diagrams Other ORGANISATION DETAILS Name Physical/Postal Address SKA Program Development Office Jodrell Bank Centre for Astrophysics Alan Turing Building The University of Manchester Oxford Road Manchester, UK M13 9PL Fax. +44 (0) Website Page 2 of 2

3 TABLE OF CONTENTS 1 INTRODUCTION Purpose of the document Scope of the document REFERENCES FAST TRANSIENT SEARCH USE CASE Station beamforming Signal combination Dedispersion processing Event detection Store in rolling buffer Buffer dump Off-line processing Commensal and targeted surveys SURVEY STRATEGY Maximising survey speed and minimising cost Trade-offs MODELLING FAST TRANSIENT EVENT RATES FOR SKA SKA Phase 1 (SKA1) system description Event rate per beam for different modes of beamforming The efficiency of coherent combination and subarraying Coherent combination for the ideal case Trade-off: Low frequency aperture arrays ( MHz) Trade-off: Low band dishes ( GHz) Frequency effects Trade-off: Event rate as a function of frequency for small bandwidths Processed bandwidth effects Trade-off: Event rate per beam as a function of processed bandwidth DISCUSSION Effectiveness of combination modes Frequency and bandwidth effects Application to Galactic populations CONCLUSION RECOMMENDATIONS FOR SKA PHASE ACKNOWLEDGEMENTS Page 3 of 36

4 A SIGNAL COMBINATION TECHNIQUES A.1 AA station beamforming A.2 Modes of beamforming for searching A.2.1 Incoherent combination A.2.2 Independently pointed sub-arrays, incoherently combined A.2.3 Coherent combination - array beamforming A.2.4 Correlation beam forming - fast imaging B MODELLING OF DETECTION RATES B.1 Extragalactic population B.2 Pulse broadening and correction (dedispersion) C FREQUENCY DEPENDENCE OF LOW FREQUENCY APERTURE ARRAYS D EVENT RATE AS A FUNCTION OF FREQUENCY Page 4 of 36

5 LIST OF FIGURES Figure 1 : Generic fast transients pipeline Figure 2 : Event rate per beam per second versus frequency for SKA1 AA-low and low band dishes Figure 3 : AA-low event rate per beam for 1 to 380 channels of width 1 MHz, where channel 1 is at 70 MHz Figure 4 : AA-low event rate per beam for 1 to 250 channels of width 1 MHz where channel 1 is at 200 MHz Figure 5 : Low band dish event rate per beam for 1 to 550 channels of width 1 MHz, where channel 1 is at 450 MHz Figure 6 : Incoherent combination Figure 7 : Coherent combination Figure 8 : Normalised event rate R ν, and breakdown of the frequency dependent components comprising R ν Figure 9 : Normalised event rate R ν for constant luminosity over frequency (ξ = 0), and breakdown of the frequency dependent components comprising R ν Figure 10 : Normalised S minν, and breakdown of the frequency dependent components comprising S minν LIST OF TABLES Table 1 : Radio astronomy searches of the high time resolution universe Table 2 : Sparse aperture array system details for SKA Phase Table 3 : Dish system details for SKA Phase Table 4 : Normalised event rate for some AA-low modes Table 5 : Normalised event rate for some dish modes Table 6 : Event rate per beam per second for SKA1 AA-low and low band dishes Table 7 : Maximum channel for which the improvement in event rate over the rate without that channel is greater than 0.5% Table 8 : The influence of beamforming, search costs and filling factor on preferred antenna combination modes Table 9 : Incoherent combination characteristics Table 10 : Incoherently combined subarray characteristics Table 11 : Coherent combination characteristics Table 12 : Correlation characteristics Page 5 of 36

6 LIST OF ABBREVIATIONS A e effective area A e element (station or dish) effective area AA-low low frequency aperture arrays for SKA1 ATA Allen Telescope Array ASKAP Australian SKA Pathfinder D element diameter D arr array diameter DM dispersion measure FoM figure of merit FoV field of view LOFAR Low Frequency Array K element taper K arr array taper L luminosity N number of elements N 0/sa number of elements per subarray N b number of element (station) beams N b arr number of array beams N op/s number of operations per second N pix number of pixels to fill the element field of view N pol number of polarisations n sa number of subarrays PAF phased array feed R event rate R ν event rate for bandwidth ν RFI radio frequency interference S minν minimum detectable flux density SKA SKA Phase 1 SKA SKA Phase 2 SNR signal to noise ratio SSFoM survey speed figure of merit T sys system temperature VLBA Very Long Baseline Array ν processed bandwidth ξ spectral index Ω element FoV Ω arr array beam FoV Ω proc processed FoV Page 6 of 36

7 1 Introduction The high time resolution universe is not well sampled at radio frequencies. Opening up this parameter space represents a recognised scientific philosophy of the Square Kilometre Array (SKA); known as the Exploration of the Unknown [1]. Table 1 lists projects implemented on existing and future telescopes which will explore this parameter space. The SKA will provide at least an order of magnitude improvement in both sensitivity and field of view over these experiments, as shown in Macquart et al. [2]. Searching for fast transients involves looking for impulsive emission; this will likely be due to high energy density events. Known search targets include giant pulses, magnetars and rotating radio transients [2]. Experiment Telescope Status Reference Fly s eye fast radio transient search ATA Completed [3] Archival searches Parkes [4, 5] High Time Resolution Universe Pulsar Survey Parkes Operational [6] Pulsar ALFA (PALFA) Survey Arecibo Operational [7, 8] V-FASTR VLBA Operational [9] LOFAR Transients Key Science Project LOFAR In progress [10, 11] Commensal Real-Time ASKAP Fast-Transients (CRAFT) Survey ASKAP Planned [2] Effelsberg Northern Sky Pulsar Survey Effelsberg Planned URL a a Table 1: Radio astronomy searches of the high time resolution universe. Only experiments within SKA Phase 1 frequencies (70 MHz 3 GHz) are listed. Pulsar surveys insensitive to single pulses are excluded. Optimal, cost-effective methods to search for fast transients with the SKA are dependent on many factors. We propose a metric, number of detected events per operation, to assist with optimisation and trade-offs. This metric incorporates the choice of receptor, the cost and efficiency of the signal combination and search strategy and the dependence on observed sky direction. Our initial analysis uses event rate per beam formed and searched as a more simplistic metric. Put simply, one search strategy may have a better rate of detection than another, but the processing cost of the strategy also needs to be considered. This is especially important for Phase 1 of the SKA (SKA1), considering transient detection will only be carried out if it can be done with minimum additional cost or effort [12]. This paper finds that incoherent combination of antennas is produces the optimal event rate per beam formed and searched. Coherent combination only becomes preferable after the limit on forming dish or station beams is reached. Fly s eye is unattractive because of the need to search many more data streams and its lack of localisation capability. However, if the search cost is low, three element subarrays could usefully employ antennas unused by the primary user observation. The dependence on observing frequency and processed bandwidth needs to be considered when making event rate calculations. The recommendations for SKA Phase 1 are: Page 7 of 36

8 SKA processing needs to provide flexible search modes; the preferred receptor and combination mode depends on direction, especially for Galactic populations. Low cost fast transients searches can be implemented using existing processing chains, either internally or via dish and station beam data spigots. Initial requirements: 1. Availability of incoherent and coherent combination modes for AA-low and low band dishes. 2. Processing for low cost commensal surveys. 3. Provision of a spigot to the dish and station beam data. 4. Voltage buffering capability. Processing the full available bandwidth is not required. Bandwidths of MHz should be sufficient, but further study of the trade-offs is required. 1.1 Purpose of the document This document presents a high-level use case for searching for fast transients with SKA1 receptors: low band dishes and low frequency aperture arrays (AA-low). It undertakes a detailed analysis of the effects of antenna combination modes, observing frequency and bandwidth on the event rate per beam. This analysis is applied to the SKA1 system description to determine optimal search strategies, but is equally applicable to SKA Phase 2 (SKA2). 1.2 Scope of the document The requirement of high time resolution observations creates synergies between fast transient and pulsar searches. But most pulsar searches take advantage of pulsar periodicity to improve sensitivity, hence this document focuses on the aspects of the system exclusive to searching for fast transients and single pulse searches: maximising event detection. Here we define fast transients as those with a pulse width less the than the normal correlator integration time, hence less than 5 seconds. For longer timescales, it is conceivable that transients can be detected by comparing correlator images or visibilities. This document does not model the effects of scattering on pulse broadening. It does not discuss radio frequency interference (RFI) identification and removal. 2 References [1] P. N. Wilkinson, K. I. Kellermann, R. D. Ekers, J. M. Cordes, and T. J. W. Lazio. The exploration of the unknown. New Astronomy Reviews, 48(11-12): , Page 8 of 36

9 [2] J.-P. Macquart, M. Bailes, N. D. R. Bhat, G. C. Bower, J. D. Bunton, S. Chatterjee, T. Colegate, J. M. Cordes, L. D Addario, A. Deller, R. Dodson, R. Fender, K. Haines, P. Hall, C. Harris, A. Hotan, S. Johnston, D. L. Jones, M. Keith, J. Y. Koay, T. J. W. Lazio, W. Majid, T. Murphy, R. Navarro, C. Phillips, P. Quinn, R. A. Preston, B. Stansby, I. Stairs, B. Stappers, L. Staveley-Smith, S. Tingay, D. Thompson, W. van Straten, K. Wagstaff, M. Warren, R. Wayth, and L. Wen. The Commensal Real-time ASKAP Fast Transients (CRAFT) survey. Publications of the Astronomical Society of Australia, [3] A. Siemion, G. Bower, M. Dexter, G. Foster, W. Mallard, P. McMahon, M. Wagner, D. Werthimer, and Allen Telescope Array Team. Results from the fly s eye fast radio transient search at the Allen Telescope Array. In American Astronomical Society Meeting Abstracts #217, volume 43 of Bulletin of the American Astronomical Society, page , January [4] S. Burke-Spolaor and M. Bailes. The millisecond radio sky: transients from a blind single-pulse search. MNRAS, 402: , [5] D. R. Lorimer, M. Bailes, M. A. McLaughlin, D. J. Narkevic, and F. Crawford. A bright millisecond radio burst of extragalactic origin. Science, 318(5851): , [6] M. J. Keith, A. Jameson, W. van Straten, M. Bailes, S. Johnston, M. Kramer, A. Possenti, S. D. Bates, N. D. R. Bhat, M. Burgay, S. Burke-Spolaor, N. D Amico, L. Levin, P. L. McMahon, S. Milia, and B. W. Stappers. The High Time Resolution Universe Pulsar Survey - I. System configuration and initial discoveries. Monthly Notices of the RAS, 409: , [7] J. M. Cordes, P. C. C. Freire, D. R. Lorimer, F. Camilo, D. J. Champion, D. J. Nice, R. Ramachandran, J. W. T. Hessels, W. Vlemmings, J. van Leeuwen, S. M. Ransom, N. D. R. Bhat, Z. Arzoumanian, M. A. McLaughlin, V. M. Kaspi, L. Kasian, J. S. Deneva, B. Reid, S. Chatterjee, J. L. Han, D. C. Backer, I. H. Stairs, A. A. Deshpande, and C.-A. Faucher-Giguère. Arecibo Pulsar Survey Using ALFA. I. survey strategy and first discoveries. Astrophysical Journal, 637: , [8] J. S. Deneva, J. M. Cordes, M. A. McLaughlin, D. J. Nice, D. R. Lorimer, F. Crawford, N. D. R. Bhat, F. Camilo, D. J. Champion, P. C. C. Freire, S. Edel, V. I. Kondratiev, J. W. T. Hessels, F. A. Jenet, L. Kasian, V. M. Kaspi, M. Kramer, P. Lazarus, S. M. Ransom, I. H. Stairs, B. W. Stappers, J. van Leeuwen, A. Brazier, A. Venkataraman, J. A. Zollweg, and S. Bogdanov. Arecibo pulsar survey using ALFA: Probing radio pulsar intermittency and transients. Astrophysical Journal, 703: , [9] R. B. Wayth, W. F. Brisken, A. T. Deller, W. Majid, D. R. Thompson, S. J. Tingay, and K. Wagstaff. V-FASTR: The VLBA fast radio transients experiment [10] J. W. T. Hessels, B. W. Stappers, J. van Leeuwen, and L. Transients Key Science Project. The Radio Sky on Short Timescales with LOFAR: Pulsars and Fast Transients. In D. J. Saikia, D Green, Y. Gupta, and T. Venturi, editors, The Low-Frequency Radio Universe, [11] J. van Leeuwen and B. W. Stappers. Finding pulsars with LOFAR. Astronomy and Astrophysics, 509(A7), [12] P. E. Dewdney, J.G. bij de Vaate, K. Cloete, A. W. Gunst, D. Hall, R. McCool, N. Roddis, and Page 9 of 36

10 W. Turner. SKA Phase 1: Preliminary System Description. SKA Memo 130, URL http: // [13] N. L. Clarke, L. D Addario, R. Navarro, T.-H. Cheng, and J. Trinh. An architecture for incoherent dedispersion. In preparation, [14] J. M. Cordes and M. A. McLaughlin. Searches for fast radio transients. The Astrophysical Journal, 596:1142, [15] R. Smits, M. Kramer, B. Stappers, D. R. Lorimer, J. Cordes, and A. Faulkner. Pulsar searches and timing with the Square Kilometre Array. Astronomy and Astrophysics, 493: , [16] J.-P. Macquart. Detection rates for surveys for fast transients with next generation radio arrays. Submitted to ApJ, [17] J. M. Cordes and T. J. W. Lazio. NE2001. I. A new model for the Galactic distribution of free electrons and its fluctuations. ArXiv Astrophysics e-prints, [18] L. D Addario. ASKAP Surveys for Transients: Which Observing Mode is Best? SKA Memo 123, April URL [19] J. M. Cordes. The SKA as a Radio Synoptic Survey Telescope: Widefield Surveys for Transients, Pulsars and ETI. SKA Memo 97, URL htm. [20] R. L. Graham, B. D. Lubachevsky, K. J. Nurmela, and P. R. J. Ostergard. Dense packings of congruent circles in a circle. Discrete Mathematics, 181(1-3): , [21] D. R. Lorimer, J. A. Yates, A. G. Lyne, and D. M. Gould. Multifrequency flux density measurements of 280 pulsars. Monthly Notices of the RAS, 273: , [22] B. W. Stappers. Non-imaging processing use cases, URL org/bin/view/signalprocessing. [23] W. Turner and A. J. Faulkner. High-level SKA signal processing description, URL http: //wiki.skatelescope.org/bin/view/signalprocessing. 3 Fast transient search use case To provide context for searching for fast transients, this section outlines a generic fast transient search pipeline. The search for transient events is conducted on a data stream which is a continuous observation of the sky. Figure 1 is a schematic showing the processing pipeline. The specific implementation depends on the target or expected source population and the cost and performance factors of components in the pipeline; however the general actions are described below Page 10 of 36

11 Dish signal (1..N 0 dishes) AA-low signal Signal combination: - Incoherent combination - Array beamforming De-dispersion processing Event detection Storage Off-line processing Station beamforming (1..N st stations) Signal to be searched Positive result Buffer dump Station beam Store in rolling buffer Figure 1: Schematic for a generic fast transients pipeline, for SKA Phase 1 receptors. Rounded boxes are actions, rectangles are object nodes describing the information flow. 3.1 Station beamforming Station beamforming combines the signal of many individual elements to form one or more station beams, each with field of view (FoV) limited by station diameter. The output station beam can then be processed the same as a dish signal (see Section A.1). 3.2 Signal combination The signals to be searched can be combined coherently, incoherently or not at all (see Appendix A). Combining the signals from many dishes or stations achieves: Increased telescope sensitivity while maintaining a large FoV Source localisation capability* Ability to spatially discriminate radio frequency interference (RFI)* *These are also achievable by buffering and post-processing. Each of these characteristics are technically possible but come at a cost. This document discusses which signal combination techniques achieve optimal performance and applies this to SKA1. Further discussion of various parts of the system are in Macquart et al. [2], Wayth et al. [9]. 3.3 Dedispersion processing The signals pass through a medium of unknown dispersion measure (DM). This means that the detection needs to be trialled for many DMs, each of which has a computational cost. The DM range to be trialled depends on the location on the sky. Clarke et al. [13] discusses dedispersion for SKA1 in detail Page 11 of 36

12 3.4 Event detection An event detection algorithm needs to be applied to the signal from each trial DM, where optimal detection is achieved with a matched filter [14]. 3.5 Store in rolling buffer The digitised voltages from the dishes or stations are stored in a circular memory (rolling) buffer. In the case of a candidate event, the data from the buffer can be saved to another location (dumped) and processed off-line. The amount of memory required in the buffer depends on the sampling rate, sample size and the expected maximum (dispersed) pulse duration. The maximum pulse duration is a function of the maximum DM to be trialled and the range of frequencies to be captured. 3.6 Buffer dump On receipt of a trigger, the buffer will dump the data to storage for off-line processing. 3.7 Off-line processing The off-line processing of the original voltage data could include RFI filtering, analysis of the candidate detection and correlation of the dishes or stations for source localisation and imaging. 3.8 Commensal and targeted surveys Archival analysis of pulsar surveys with the Parkes radio telescope show that the high time resolution sky has been poorly observed [4, 5]. Searching for an unknown signal implies that the direction on the sky is not important, and a commensal survey may be a low cost method to increase survey time. A commensal survey is a survey that is conducted in parallel with normal telescope observation. It is passive; it uses telescope signals but does not affect normal operation. There are also specific areas of the sky, such as the Galactic plane and nearby galaxies which may be better observed with a targeted survey. 4 Survey strategy 4.1 Maximising survey speed and minimising cost Figures of merit (FoMs) encapsulate the variable parameters of a problem. A simple FoM to measure the cost effectiveness of a fast transient search method is the detected event rate (R) per beam searched Page 12 of 36

13 A more comprehensive metric is the number of detected events per operation, given by F om = R N op/s events operation 1, (1) where N op/s is the number of operations per second required to form and search the beams. N op/s is given by N op/s = N op/s bfm + N op/s combine + N op/s search, (2) where N op/s bfm, N op/s combine and N op/s search are respectively the number of beamformer (AA station or phased array feed), signal combination and search operations per second required. For commensal surveys, the cost of beamforming (N op/s bfm ) is ignored when beamforming is necessary for primary user observations and therefore not an exclusive requirement for transients observations. This figure of merit uses N op/s as a proxy for cost, in the absence of sufficiently accurate design and cost information. For both figures of merit, we want to optimise for a high value of the FoM. The detected event rate must be high enough to be of scientific benefit and open new volumes of parameter space. There are also qualitative advantages to various strategies which cannot be captured in a figure of merit. We also consider the relative merits of commensal and targeted searches. The detected event rate is effectively a survey speed. It is calculated assuming the telescope can detect an object with a flux density greater than S minν, the minimum detectable flux density. Smits et al. [15] present a frequency dependent figure of merit for survey speed (SSFoM) for dishes. It is based on surveying an area of sky, thus SSFoM is proportional to field of view and sensitivity squared. In this paper, the equivalent SSFoM is based on surveying a volume of sky and is proportional to field of view and sensitivity to the power of 3 2. It draws on event rate calculations from Macquart [16]. The event rate over a processed FoV, Ω proc, is given by R = ρ Ω proc 4π V max events s 1, (3) where V max is the maximum volume out to which an object is detectable. For homogeneous population of extragalactic objects, R ν = 1 3 ρω proc ν ( Wi,ν W ν L ν 4πS minν ) 3 2 events s 1, (4) where ρ (events s 1 m 3 ) is the event density, L ν (Jy Gpc 2 ) is the luminosity of the population, W i,ν is the intrinsic pulse width, W ν is the observed pulse width and S minν is the minimum detectable flux density. See Appendix B for more details. Because our focus is on comparing receptors and observation modes, we make a simplifying assumption of constant luminosity in the population, meaning that the probability density function is a Dirac delta function. This is acceptable for extragalactic populations; as Macquart [16] shows, event rate is independent of the cutoff in the luminosity distribution. Section 5.3 gives a more detailed analysis of the frequency dependence. We discuss Galactic objects in Section Trade-offs Section 5 considers the following trade-offs to maximise survey speed and minimise cost: Page 13 of 36

14 Comparison of event rate for coherent and incoherent beam combination. Effect of filling factor in the whole array or core on the FoM for coherent combination. Effect of observing frequency on event rate. Performance return of large bandwidth observations. Comparison of FoM for AA-low and high and low band dishes. For low frequency aperture arrays, bandwidth can be traded for beams, so which is preferable and under what conditions? The following are additional trade-offs we intend to consider: The trade-off between station size and number of stations, for low frequency aperture arrays with fixed collecting area. If the processing costs to coherently combine the array and search for transients are too expensive for the higher event rate gained, compared to incoherent combination. Incoherent dedispersion has a lower SNR (sensitivity) than coherent dedispersion, but with significantly less N op/s required. To maintain the same detection rate as coherent dedispersion, Ω proc could be increased by increasing the number of (incoherent or coherent) beams formed, and the resulting N op/s can be compared. If the gain in SNR from using a more accurate incoherent dedispersion method is worth the cost. 5 Modelling fast transient event rates for SKA1 Event rate per beam is a simple metric to model fast transient event rates in the cost constrained environment of SKA1. This section considers frequency dependence of this event rate as well as the effectiveness of searching large bandwidths. The modelling is specifically applied to the SKA1 receptors for different combination modes and spectral indices of the source population. The trade-offs in this section make the following assumptions: 1. The extragalactic object population is homogeneous and of constant luminosity. 2. Pulse broadening (τ d ) due to multipath scattering does not affect the pulse width. This may not be correct for the lower frequencies given that τ d ν 4.4 and has a dependence on sky direction [17]. 3. The pulse is optimally detected with a matched filter. 4. Events are broad-band such that the frequency bandwidth of the pulse is greater than the processed bandwidth. Thus all channels across the band contain contributing signal. 5. The beam has constant (maximum) sensitivity between the half-power beamwidth points, and zero sensitivity outside of that Page 14 of 36

15 6. The processing cost of forming and searching a beam is independent of frequency, bandwidth and signal combination modes. 5.1 SKA Phase 1 (SKA1) system description Tables 4 and 5 show the relevant system details from Dewdney et al. [12]. These are part of a system description of SKA1. However the complete system design for SKA1 is still under development and subject to a decision-making process involving trade-offs and performance and cost optimisation. Low frequency aperture arrays for SKA Phase 1 Aperture Diameter (D st ) 180 m Dual polarization (2 orthogonal) Lower Frequency 70 MHz Dual polarization (2 orthogonal) Upper Frequency 450 MHz Single element covering full range Number of antennas Per station Area per antenna 2.27 m m diameter station Dense/Sparse Transition λ 2.6 m (115 MHz) A e per element is equal to packing density Array Configuration N st = 50 stations Core (radius<0.5 km) 50% (25 stations) Fractional total number of AA stations. Inner (1< radius<2.5 km) 20% (10 stations) " Mid (2.5<radius<100 km) 30% (15 stations) In clusters of 5 stations (Total of 15 clusters) Instantaneous bandwidth per beam ( ν) 380 MHz Assumes full bandwidth is available ( MHz) Performance T rcvr 150 K Receiver Noise Table 2: Sparse aperture array system details for SKA Phase 1. Notes 1. The effective area of an aperture array station is π( A e0 = )2 = m 2 ν < 115 MHz. (5) c2 ν > 115 MHz 3ν 2 m 2 2. System temperature is the sum of the receiver noise and an approximation to the sky temperature: ( c ) 2.55 T sys = T rcvr + 60 (6) ν 3. The field of view of a station beam is given by Ω st beam = π ( ) 2 λ K st, (7) 4 D st Page 15 of 36

16 where K st =1.3 is the station beam taper. Dishes for SKA Phase 1 Aperture Diameter (D dish ) 15 m Parabolic antennas Lower Frequency 300 MHz Capability range GHz Upper Frequency 10 GHz Designed for Phase 2 Total physical aperture m Dishes Array Configuration 250 antennas Optimized for science & local geography Core (radius < 0.5 km) 50% (125 ant.) Fractional total number of antennas Inner (0.5 < radius < 2.5 km) 20% (50 ant.) " Mid (2.5 < radius < 100 km) 30% (75 ant.) In 15 clusters of 5 antennas per cluster. Core filling factor 0.03 A core = m 2 ; A 0 = m 2 Antenna RF System Only one Feed available at a time Feed/LNA low band GHz Dual polarization (2 orthogonal) Bandwidth ( ν low ) 0.55 GHz For each polarization Feed/LNA high band GHz Dual polarization (2 orthogonal) Bandwidth ( ν high ) 1.0 GHz For each polarization RF Sub-bands 2 One for each polarization Performance Antenna/Feed Efficiency 70% Average over frequency Average T sys in low band 40 K Higher at low freq. end of this band. Average array A e /T sys = 773 m 2 K 1. Average T sys in high band 30 K Average array A e /T sys = 1031 m 2 K 1. Table 3: Dish system details for SKA Phase 1. Notes 1. The field of view of the dish is given by Ω dish = π 4 ( λ K dish D dish where K dish = 66π 180 is the dish illumination factor. ) 2, (8) 5.2 Event rate per beam for different modes of beamforming This section compares the event rate per beam for the antenna combination modes discussed in Appendix A for AA-low and low band dishes, and applies these results to SKA1. We refer to a single antenna or station as an element, designated with the subscript Page 16 of 36

17 Incoherent combination of an array of N 0 elements increases the sensitivity by a factor of N 0 over a single element while retaining its FoV, Ω 0. Forming N b 0 station beams further increases the FoV by that number of beams. To even further increase the FoV, n sa groups of antennas (subarrays) are incoherently combined. Each subarray is pointed in a different direction, increasing the FoV by a factor of n sa but only increasing the sensitivity by a factor of N 0/sa over a single element, where N 0/sa is the number of elements per subarray. Fly s eye is when N 0/sa =1. Sensitivity from the coherent combination of an array of N 0 elements is higher than incoherent combination and subarraying; it increases proportional to N 0. However the FoV, Ω arr, is much smaller; it is proportional to 1 D, where D arr 2 arr is the diameter of the array of elements being combined. The FoV can be linearly increased by forming N b arr array beams. The modes can be summarised by applying these relationships to Equation 4 to give the event rate per beam for the different signal combination modes: R ν = 1 ( ) 3 3 ρ (N pol ντ) 3 Lν 2 A e0ν 4 M, (9) 4πσ2k B T sysν N b 0 Ω 0 N 3/4 0 Incoherent combination where M = N b arr Ω arr N 3/2 0 Coherent combination (10) n 1/4 sa N b 0 Ω 0 N 3/4 0 Subarraying The efficiency of coherent combination and subarraying The coherent combination mode is more effective if the dishes or stations are closely spaced, thus having a higher filling factor. This is due to the 1 D relationship with FoV. D Addario [18] considers 2 the number of coherently combined beams required to achieve an event rate FoM equivalent to one incoherently combined beam for ASKAP. We modify this analysis and apply it to R ν (Equation 4) for SKA Phase 1. Following Cordes [19], the number of pixels required to fill the antenna or station FoV with array beams is N pix = Ω 0 = Ω arr ( ) 2 K0 D arr, K arr D 0 (11) (12) where K 0 and K arr are the element and array beam tapers respectively. So for an array with elements of diameter D 0, where the number of elements is held constant, a larger array diameter means that more pixels are required. Equating the incoherent and coherent combination terms of Equation 10 gives the number of array beams required to achieve an event rate equal to R incoherent : N b arr = N b 0Ω 0 N 3/4 0 Ω arr N 3/2 0 = N b 0N pix. N 3/4 0 (13) (14) Page 17 of 36

18 This relationship is frequency independent. Achieving the highest possible event rate is desirable, but this must be tempered by the cost of searching multiple beams. The relative event rate per beam searched depends on the array filling factor and is given by and R coherent beam 1 = Ω arrn 3/2 0 Ω 0 N 3/4 0 (15) = N 3/4 0 R incoherent beam 1 (16) N pix R subarray beam 1 = n 3/4 sa R incoherent beam 1, (17) where n sa subarray beams are searched Coherent combination for the ideal case Assuming a best case scenario of an array entirely filled with stations of equal diameter and K 0 = K arr, N pix = N 0. (18) Substituting into Equation 14, the theoretical number of array beams required to achieve an event rate equal to R incoherent is N b arr = N b 0 N 1/4 0 (19) and the relative event rate per beam (Equation 16) is R coherent beam 1 = N 1/4 0 R incoherent beam 1. (20) This estimation is optimistic towards coherent combination, given that it is not physically possible to entirely fill a circular array with circular stations. Regardless of this, for N 0 > 1, the incoherent combination will always produce a better event rate per beam searched than coherent combination, and this difference increases with N Trade-off: Low frequency aperture arrays ( MHz) Applying Equations 14, 16 and 17 to SKA1 shows that of the coherently combined arrays, the core gives the best event rate, due to its high density of collecting area. 3 array beams are required to achieve the event rate of the incoherently combined beam. The full results are shown in Table 4. Although the fly s eye mode produces a higher event rate by a factor of =2.6, the normalised rate per beam searched is less than the coherently combined core. The number of equivalent beams will be higher if the dense packing is not be achievable, as noted in Dewdney et al. [12]. Indeed, using optimisations from Graham et al. [20], the optimal packing of 25 congruent circles of diameter 180 m in a circle results in a minimum core diameter of 1036 m. This is larger than the 1000 m diameter in Dewdney et al. [12] and excludes any spacing that may be required for infrastructure Page 18 of 36

19 Mode Input parameters Calculated values D arr (km) N st N pix Num. beams required a Normalised R per beam b Coherently combined: total array Coherently combined; inner + core Coherently combined: core Incoherently combined: total array Fly s eye: total array - 50 subarrays a For an event rate equivalent to the incoherent combination mode. b Normalised to the incoherent combination event rate per beam. Table 4: Normalised event rate for some AA-low modes, using parameters from Dewdney et al. [12]. The two polarisations are summed and an averaging time of τ = 1 ms is used Trade-off: Low band dishes ( GHz) Table 5 applies the same calculations to dishes. Again, of the coherently combined arrays, the core gives the best event rate. 93 of these beams are required to achieve an event rate equivalent to the incoherently combined array. The event rate of the fly s eye mode is higher by a factor of =4and the normalised rate per beam searched is similar to the coherently combined core. 5.3 Frequency effects This section investigates the change in event rate due to the frequency dependence of luminosity, minimum detectable flux density and field of view. It is important to understand this frequency dependence because of the wide fractional bandwidths of the SKA. For the extragalactic population, R ν = 1 3 ρω proc ν ( Lν 4πS minν Looking at each of these dependencies in turn: ) 3 2. (21) The processed field of view depends on whether the beams are formed incoherently or coherently, but either way is proportional to λ 2 : Ω procv = π 4 ( ) 2 Kλ, (22) D where K is the feed illumination factor or station or array beam taper, and D is the diameter of the dish or station for incoherent combination or array for coherent combination Page 19 of 36

20 Mode Input parameters Calculated values Diameter (km) N ant N pix Num. beams required a Normalised R per beam b Coherently combined: total array Coherently combined; inner + core Coherently combined: core Incoherently combined: total array Fly s eye: total array subarrays a For an event rate equivalent to the incoherent combination mode. b Normalised to the incoherent combination event rate per beam. Table 5: Normalised event rate for some dish modes, using parameters from Dewdney et al. [12]. The two polarisations are summed and an averaging time of τ = 1 ms is used. Because we are looking for an unknown population, we do not know how the luminosity varies with frequency. However, spectral indices for pulsars have been measured. We use ξ = 1.6 [21], a value typical of the pulsar population, such that L v = L 0 ( ν ν 0 ) ξ, (23) where L 0 is the luminosity at reference frequency ν 0. For aperture arrays, S minν is a function of T sysν and A eν. Substituting from the notes to Table 2 and assuming the station beams are coherently combined, σ2k B T sysν S minν = (24) N st A e0ν Npol ντ σ2k B (T rcvr + 60 ( ) ) c 2.55 ν = N st A, (25) N pol ντ π 4 where A = D2 st ν < ν transition (26) N ant/st c2 3ν ν > ν 2 transition. D st = 180 m is the station diameter, N st is the number of stations, N ant/st = is the number of antennas per station, c is the speed of light in a vacuum and ν transition = 115 MHz Trade-off: Event rate as a function of frequency for small bandwidths Figure 2 shows how different modes and spectral indices compare. It plots R ν per beam formed and searched at 1 MHz intervals for centre frequency ν and processed bandwidth ν = 1 MHz (Equation 21). It spans the frequency range of AA-low and low band dishes (70 MHz ν 1000 MHz) using Page 20 of 36

21 the SKA system description (Tables 2 and 3). The four lines are permutations of an assumed spectral index of ξ =0or ξ = 1.6, and incoherent combination of the array or coherent combination of the core. A breakdown of the frequency dependencies of AA-low is shown in Appendix C. For the ξ =0case, the event rate for AA-low has its maximum at the sparse-dense transition frequency of 115 MHz incoherent total, ξ = 1.6 incoherent total, ξ =0 coherent core, ξ = 1.6 coherent core, ξ =0 Rν (events beam 1 s 1 ) Frequency (MHz) Figure 2: Event rate per beam per second versus frequency for SKA1 AA-low and low band dishes. A rate volume density of ρ = 1 event s 1 Gpc 3 and L 0 = 1 mjy Gpc 2 at ν 0 = 1000 MHz is used, with ν = 1 MHz. Solid lines show incoherent combination, dashed lines coherent combination. Black lines show a spectral index of -1.6 and red lines 0. The loss of signal to noise at low frequencies due to pulse broadening is not shown. The event rates shown in Figure 2 are scalable with the number station (N b 0 ) or array (N b arr ) beams. Dishes with single pixel feeds (the solid line from MHz) are only scalable with N b arr. From this plot, incoherent combination of dishes would be most efficient for searching for sources with a low spectral index, while AA-low would be more efficient for high spectral indices, assuming the scattering effects at low frequency are not severe. Multiple beams (either incoherently or coherently formed) would also increase the event rate. 5.4 Processed bandwidth effects Processed bandwidth is the bandwidth of the astronomical signal at the detection system. In radio astronomy an increase in processed bandwidth is usually assumed to produce a ν increase in SNR. From Equation 4, this would produce a ν 3 4 increase in event rate. However this section shows that this does not hold for large processed bandwidths, due to frequency dependent effects Page 21 of 36

22 The event rate can be calculated numerically for a processed bandwidth of ν = N ch ν ch : R v = ( Nch i R 4/3 i ) 3 4 ( = 1 Nch 3 ρ ( Ω 2/3 i L i proc i S min,i ) 2 ) 3 4, (27) where N ch is the number of frequency channels of width ν ch. The derivation of this is shown in Appendix D. Table 6 makes these calculations for different modes and spectral indices, for AA-low and low band dishes. Strikingly, the contribution from the dishes in the combined AA-low and low band dish event rate is only noticeable for incoherent combination where ξ =0, and even then the improvement is not significant. Mode Spectral AA-low R ν index ξ (beam 1 s 1 ) Low band dish AA-low and low (beam 1 s 1 ) (beam 1 s 1 ) Incoherently combined total array Coherently combined core Table 6: Event rate R ν (beam 1 s 1 ), for a rate volume density of ρ = 1 event s 1 Gpc 3 and L 0 = 1 mjy Gpc 2 at ν 0 = 1000 MHz with incoherent and coherent combination and spectral indices of 0 and ν = 380 MHz for AA-low and 550 MHz for low band dishes. R ν band dish R ν Trade-off: Event rate per beam as a function of processed bandwidth Equation 27 sums individual, frequency dependent channel contributions shown in Figure 2 into an event rate over a large processed bandwidth. As ν increases, the contribution of each channel to R ν decreases. The result of this decreasing contribution is that the increase in event rate is less than ν 3 4. The only exception is in the dense AA regime (<115 MHz), where the opposite occurs at low spectral indices. Given ν = N ch ν ch, plotting R ν as a function of the number of contributing channels shows the decreasing contribution of higher frequency channels to the event rate. Figures 3-5 show three cases: 1. AA-low where the whole 380 MHz bandwidth is available (Figure 3); 2. AA-low where only the MHz bandwidth is available, because the low frequencies are unusable due to pulse broadening (Figure 4); 3. Low band dishes where the whole 550 MHz bandwidth is available (5). The ideal case of a ν 3/4 increase over the ν = 1 MHz event rate is also shown in each figure. These plots show no optimal bandwidth. This is because the event rate will always be higher with the contribution of additional channels, even if the contribution is small. Additionally, to a first-order approximation, N op/s scales linearly with ν. So even for the ideal case, the event rate per operation is proportional to ν 1 4, which implies that an infinitesimal bandwidth would be optimal Page 22 of 36

23 Rν (events beam 1 s 1 ) incoherent total, ξ = 1.6 incoherent total, ξ =0 coherent core, ξ = 1.6 coherent core, ξ = Number of channels ν 3/4 Figure 3: AA-low event rate per beam for 1 to 380 channels of width 1 MHz, where channel 1 is at 70 MHz. Thick lines show the calculated R ν versus processed bandwidth ν for a rate volume density of ρ = 1 event s 1 Gpc 3 and L 0 = 1 mjy Gpc 2 at ν 0 = 1000 MHz with incoherent and coherent combination with spectral indices of 0 and Thin lines show the expected ν 3/4 increase in event rate over R ν=1 MHz. Due to the sparse-dense transition, R initially increases faster than ν 3/4 when ξ =0. To quantify the problem a threshold can be arbitrarily set beyond which additional channels contribute very little to the event rate. Thus the channels of increasing frequency are included while the following is true: R ν+ νch R ν > threshold. (28) For example, say the threshold is set to 0.5%. Then for the ideal case, N ch = 150 channels contribute to R ν. Adding a 151 st channel will contribute less than 0.5 % to R ν. The number of contributing channels for the other cases is shown in Table 7. To interpret Table 7, compare the 3 rd and 5 th rows. For Case 2 (ξ = 1.6), 54 channels or ν = 54 MHz bandwidth is above the threshold. For Case 3 (ξ = 1.6), the same threshold is achieved with ν = 87 MHz bandwidth. This implies that the AA-low processed bandwidth become less useful more quickly. This is expected, given the steeper spectral dependence of AA-low over dishes, shown in Figure Page 23 of 36

24 Rν (events beam 1 s 1 ) incoherent total, ξ = 1.6 incoherent total, ξ =0 coherent core, ξ = 1.6 coherent core, ξ =0 ν 3/ Number of channels Figure 4: AA-low event rate per beam for 1 to 250 channels of width 1 MHz, where channel 1 is at 200 MHz. Other parameters as per Figure Rν (events beam 1 s 1 ) incoherent total, ξ = 1.6 incoherent total, ξ =0 coherent core, ξ = 1.6 coherent core, ξ = Number of channels ν 3/4 Figure 5: Low band dish event rate per beam for 1 to 550 channels of width 1 MHz, where channel 1 is at 450 MHz. Other parameters as per Figure Page 24 of 36

25 Case Spectral index ξ Maximum contributing channel number 1: AA-low, ch 1 = 70 MHz : AA-low, ch 1 = 200 MHz : Low band dish, ch 1 = 450 MHz Table 7: Maximum channel for which the improvement in event rate over the rate without that channel is greater than 0.5%. 6 Discussion 6.1 Effectiveness of combination modes Section shows that incoherent combination will always produce a better event rate per beam searched than coherent combination, and the difference increases with an increased number of elements or a reduced filling factor. The per beam analysis is important because it captures the processing cost for each beam. For this reason, although fly s eye gives the highest total event rate, it comes at the expense of searching many more beams. There are cases when coherent combination will produce a higher total event rate than incoherent combination and arises when the limit on forming more station beams is reached. The increase in total event rate, R total, for coherent combination is proportional to the number of array beams. For incoherent combination, R total is proportional to the number of station beams. With AAs, more station or array beams can be formed, while only array beams can be formed with single pixel feed dishes. However, if phased array feeds (PAFs) are available on dishes, then multiple dish beams can be formed, linearly increasing R total for incoherent combination. So we must consider do we want to form and search more array or station beams? Although exact costs are unavailable, Table 8 generalises how the preferred combination mode is influenced by station beamforming (for AAs), array beamforming, search costs and the array filling factor. For coherent combination to be preferred, an array with a high filling factor and low array beamforming and search costs is required. Even then, for AAs, the number of array beams formed and searched must be greater than the maximum possible number of station beams formed. Fly s eye and subarraying is only preferable when the beam search cost is low or n sa is small. A fly s eye mode excludes the buffering and source localisation advantages of an array and commensality with most observations (see Section 3). However, a three element subarray counters this problem and could employ antennas unused by the primary user observation. For example, splitting 24 AA-low stations outside the core into 8 of these three element subarrays results in a total event rate approximately equal to the incoherent combination of 50 stations. The SKA1 AA-low results show that there is a weak preference for more stations beams: 1 station beam Page 25 of 36

26 Preferred mode Station beamforming Array beamforming Beam search Filling factor cost cost cost Coherently combined: core Low Low High Incoherently combined: total Low High Low array Fly s eye: total array Low Low Low Table 8: The influence of beamforming, search costs and filling factor on preferred antenna combination modes. The cost of incoherent combination is always low. gives an event rate equivalent to 3 array beams, assuming the stations are very closely packed. However the number of station beams formed is restricted by the performance limitations of forming station beams within the antenna FoV. If station beams are formed at no cost for normal array observing, or the cost of array beamforming is high, using station beams would be preferable. Other effects such as RFI mitigation and station and array beam quality also need to be taken into account. For SKA1 low band dishes, 93 array beams are required to achieve an event rate equivalent to the incoherent combination mode. Coherent combination with dishes would only be optimal for hundreds of array beams, and even then it is likely that the processing power would be more effectively spent on AAlow. However, if scattering effects rule out the lower frequencies of AA-low, an incoherent commensal survey with low band dishes may be a cost-effective method to cover parameter space. 6.2 Frequency and bandwidth effects Figure 2 shows the frequency dependent event rate per beam. The slope on the dish event rate is due to the spectral index and changing FoV; for AA-low a frequency dependent T sys is also a factor. In general, R is better for AA-low, although for sources of low spectral indices, this advantage is lessened. For areas of the sky where the lower frequencies are unusable due to pulse broadening, and only a few station or core beams are available, the incoherent combination of the low band dishes gives a higher R total. If phased array feeds (PAFs) are available in SKA Phase 1, this would be advantageous for the dish array. A low frequency turnover in the source spectra would also render AA-low less attractive. Lower frequencies also have an increased memory cost for dedispersion. The effects of frequency and bandwidth are interdependent as shown in Section 5.4. Processing more channels and hence bandwidth increases R, however beyond a threshold bandwidth the processing is more effectively used to form and search more beams (increasing R total through FoV), trial more DMs or increase the detection SNR (see [13]). The threshold depends on the costs of forming and searching the beams, hence an optimal frequency range for searching for fast transients may be smaller than the full band of the receptor Page 26 of 36