Beamforming and Calibration with CASPER
Jack Hickish, University of Oxford
CASPER Workshop 2009, Cape Town SA
Introduction
Beamforming &
Gateware Calibration
CASPER for Students
Jack Hickish
jack.hickish@astro.ox.ac.uk
CASPER for Students
CASPER for Students
Beamforming &
Gateware Calibration
Jack Hickish
jack.hickish@astro.ox.ac.uk
CASPER for Students
CASPER as an educational tool.
• CASPER is not just beamformers & correlators!
Communications
Radio / TV
Jack Hickish
jack.hickish@astro.ox.ac.uk
Beamforming & Calibration
CASPER for Students
Beamforming &
Gateware Calibration
Jack Hickish
jack.hickish@astro.ox.ac.uk
Beamforming & Calibration
DIGITAL BEAM
Jack Hickish
jack.hickish@astro.ox.ac.uk
Beamforming & Calibration
N-element
Beamforming
N-element
Diffraction
Jack Hickish
jack.hickish@astro.ox.ac.uk
Beamforming & Calibration
I(θ)
sin θ
Jack Hickish
jack.hickish@astro.ox.ac.uk
Time Domain Beamforming
• Delay and Sum – simple, right?
• Delay precision << 1/Bandwidth
Delay by integer
FPGA clock periods
Jack Hickish
>
Delay by integer
ADC clock periods
>
Interpolation
between
samples
jack.hickish@astro.ox.ac.uk
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
jack.hickish@astro.ox.ac.uk
Time Domain Beamforming
Calibration
• Unwanted, uncontrollable “delays” introduced into the
signal chains before digitisation
Jack Hickish
jack.hickish@astro.ox.ac.uk
Time Domain Beamforming
2PAD: delays between
4 analogue channels (D.
Price & S. Schediwy,
2009)
Jack Hickish
jack.hickish@astro.ox.ac.uk
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)
Jack Hickish
300x300 array (≈ SKA)
jack.hickish@astro.ox.ac.uk
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
jack.hickish@astro.ox.ac.uk
Frequency Domain Beamforming
• Before summing analogue inputs, channelise using FFT/PFB
• Multiply data streams by complex phase and sum
Jack Hickish
jack.hickish@astro.ox.ac.uk
Frequency Domain Beamforming
Basic frequency domain beamforming algorithm…
ADC
FFT
X
∫
10GbE
Beam Coefficients
iBOB: 4 x 200MHz BW
ADC
FFT
X
Roach: 4 x 400MHz BW
Jack Hickish
jack.hickish@astro.ox.ac.uk
Frequency Domain Beamforming
Time delay errors (and other phase distortions) manifest themselves as
relative phases between inputs.
ADC
FFT
Conjugate
Multiply
∫
BRAM
Vector
Accumulator
ADC
FFT
Calibration for frequency dependent errors
Jack Hickish
jack.hickish@astro.ox.ac.uk
Frequency Domain Beamforming
• Beamforming coefficients are different for each spectral channel
- Can calibrate for dispersion/non-uniform delay errors across band
Software Calibration
Implemented
in Software
Algortihm
Jack Hickish
jack.hickish@astro.ox.ac.uk
Frequency Domain Beamforming
A simple calibration algorithm…
1. Designate one signal the “reference”.
2. Compare phases of other channels to the “reference” – FX correlate
3. Use PC software to grab correlation result.
4. Calculate phase of result (which is “CASPER format” complex
number) (arctan operation)
5. Generate pure phase “CASPER format” complex coefficients (sin/cos
operation)
6. Upload coefficients to FPGA accessible memory.
Jack Hickish
jack.hickish@astro.ox.ac.uk
Frequency Domain Beamforming
Jack Hickish
jack.hickish@astro.ox.ac.uk
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
jack.hickish@astro.ox.ac.uk
Frequency Domain Beamforming
Calibration in gateware
Jack Hickish
jack.hickish@astro.ox.ac.uk
Frequency Domain Beamforming
Gateware calibration engine: Resource use, approx 5-10% of
VirtexIIP
Jack Hickish
jack.hickish@astro.ox.ac.uk
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
jack.hickish@astro.ox.ac.uk
Frequency Domain Beamforming
Mean Error:
0.5 degrees
Jack Hickish
jack.hickish@astro.ox.ac.uk
Gateware Beamforming
How easy is CASPER, really?
4 input internally calibrated F-domain beamformer - Development time << 8
weeks
Jack Hickish
jack.hickish@astro.ox.ac.uk
Gateware Beamforming
How easy is CASPER, really?
4 input internally calibrated F-domain beamformer - Development time << 8
weeks
“
Jack Hickish
one would assume that an FFT block has a base minimum
sync pulse period equal to the FFT length…
”
jack.hickish@astro.ox.ac.uk