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Beamforming and Calibration with CASPER

Learn about beamforming and calibration techniques using CASPER, an educational tool. Explore topics such as time domain beamforming, frequency domain beamforming, and gateware calibration.

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Beamforming and Calibration with CASPER

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  1. Beamforming and Calibration with CASPER Jack Hickish, University of Oxford CASPER Workshop 2009, Cape Town SA

  2. Beamforming & Gateware Calibration Introduction CASPER for Students Jack Hickish

  3. CASPER for Students CASPER for Students Beamforming & Gateware Calibration Jack Hickish

  4. CASPER for Students CASPER as an educational tool. • CASPER is not just beamformers & correlators! Communications Radio / TV Jack Hickish

  5. Beamforming & Calibration CASPER for Students Beamforming & Gateware Calibration Jack Hickish

  6. Beamforming & Calibration DIGITAL BEAM Jack Hickish

  7. Beamforming & Calibration N-element Beamforming N-element Diffraction Jack Hickish

  8. Beamforming & Calibration I(θ) sin θ Jack Hickish

  9. Time Domain Beamforming Delay by integer FPGA clock periods Delay by integer ADC clock periods Interpolation between samples > > • Delay and Sum – simple, right? • Delay precision << 1/Bandwidth Jack Hickish

  10. Time Domain Beamforming • Integer FPGA periods • FIFO / Programmable Delay • Integer ADC periods • Reordering of demuxed ADC output • Interpolation • N tap FIR filter • Implementation on CASPER hardware (Nagpal, 2006) Jack Hickish

  11. Time Domain Beamforming • Calibration • Unwanted, uncontrollable “delays” introduced into the signal chains before digitisation Jack Hickish

  12. Time Domain Beamforming • 2PAD: delays between 4 analogue channels (D. Price & S. Schediwy, 2009) Jack Hickish

  13. Time Domain Beamforming Effect of calibration errors on station beam patterns (at 1GHz) (Price, Schediwy, 2009), as simulated in OSKAR (wiki.oerc.ox.ac.uk/OSKAR) 4x4 array (≈ 2PAD) 300x300 array (≈ SKA) Jack Hickish

  14. Time Domain Beamforming For a 4x4 array, with <5° pointing error, time delays calibration error <0.03ns (11°) at 1GHz Relative delays across band for 4 2PAD channels (Price & Schediwy, 2009) Jack Hickish

  15. Frequency Domain Beamforming • Before summing analogue inputs, channelise using FFT/PFB • Multiply data streams by complex phase and sum Jack Hickish

  16. Frequency Domain Beamforming Basic frequency domain beamforming algorithm… ADC FFT 10GbE ∫ X Beam Coefficients ADC FFT iBOB: 4 x 200MHz BW X Roach: 4 x 400MHz BW Jack Hickish

  17. Frequency Domain Beamforming Time delay errors (and other phase distortions) manifest themselves as relative phases between inputs. ADC FFT Conjugate BRAM ∫ Multiply Vector Accumulator ADC FFT Calibration for frequency dependent errors Jack Hickish

  18. Frequency Domain Beamforming • Beamforming coefficients are different for each spectral channel • - Can calibrate for dispersion/non-uniform delay errors across band Software Calibration Algortihm Implemented in Software Jack Hickish

  19. Frequency Domain Beamforming • A simple calibration algorithm… • Designate one signal the “reference”. • Compare phases of other channels to the “reference” – FX correlate • Use PC software to grab correlation result. • Calculate phase of result (which is “CASPER format” complex number) (arctan operation) • Generate pure phase “CASPER format” complex coefficients (sin/cos operation) • Upload coefficients to FPGA accessible memory. Jack Hickish

  20. Frequency Domain Beamforming Jack Hickish

  21. CORDIC Operations CO-ordinate Rotation DIgital Computer Efficient calculation of (amongst others) sine/cosine/arctan Eg. Sin/Cos Rotate initial vector (1,0) using rotation matrix Express sin θ, cos θ in terms of tan θ. Choose rotation angles, ß, such that tan(ß) = ±2-n Jack Hickish

  22. Frequency Domain Beamforming Calibration in gateware Jack Hickish

  23. Frequency Domain Beamforming Gateware calibration engine: Resource use, approx 5-10% of VirtexIIP Jack Hickish

  24. Frequency Domain Beamforming • Why calibrate in gateware? • Faster, and more reliable than uploading coefficients over ethernet • Elements (eg CORDIC sin/cos) of calibration engine can likely be used for rapid calculation and application of beamforming coefficients. • Less shared BRAM – less resource waste Jack Hickish

  25. Frequency Domain Beamforming Mean Error: 0.5 degrees Jack Hickish

  26. Gateware Beamforming How easy is CASPER, really? 4 input internally calibrated F-domain beamformer - Development time << 8 weeks Jack Hickish

  27. Gateware Beamforming “ one would assume that an FFT block has a base minimum sync pulse period equal to the FFT length… How easy is CASPER, really? 4 input internally calibrated F-domain beamformer - Development time << 8 weeks ” Jack Hickish

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