### Back-end signal processing

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Dr. Seuss - The Sneetches and Other Stories
Back-end signal processing
John Tuthill | Digital Systems Engineer
25 September 2012
CSIRO ASTRONOMY AND SPACE SCIENCE
Outline
• What is “back-end signal processing”
• FX vs XF correlators
• Filterbanks
• CABB and ASKAP digital back-ends
• Calculation engines
2 | Back-End Signal Processing | John Tuthill
Back-end processing for Synthesis Imaging

E f r1 
f
R 
R
Electric field at the remote
source propagated to the
observing points
E f r 2 
r1
X
r2
down- X
conversion
Spatial Coherence function
or “visibilities”
V f r1 , r 2   E f r1   E

f
r 2 
Intensity distribution of
the source
I f l , m  
 V f u , v  e
i 2   ul  vm

.dudv
Sampling
Back-End
Digital Signal Processing Correlator
3 | Back-End Signal Processing | John Tuthill
Imaging: calibration,
2D FFT, deconvolution
Image:
Shaun Amy
Spectral Channelisation
V f r1 , r 2   E f r1   E f r 2 

• Interested in obtaining the cross-correlations (visibilities) across a
range of separate frequency channels:
•
•
•
•
Spectral line observations – narrow bandwidth
Continuum – wide, contiguous bandwidth
Excising channels with high RFI
Others? Fast transients
• Different astrophysics will have different requirements for
frequency resolution, total bandwidth and band segmentation.
The back-end signal processing has to be flexible to cater
for many conflicting science requirements.
4 | Back-End Signal Processing | John Tuthill
Correlation
x n   y n  
• Bring the desired signals up out of the noise
• Produce the visibilities for synthesis imaging

 x m  y m  n 

m  
Noise
Noise
1
0.8
0.6
Delay
1.134s
0.4
Amplitude
+
0.2
0
-0.2
+
-0.4
-0.6
-0.8
-1
0
0.5
1
1.5
2
2.5
Time (s)
5
3
3.5
4
4.5
5
6
4
4
3
2
1
Amplitude
Amplitude
2
0
0
-1
-2
-2
-3
-4
-4
-5
-6
0
0.5
1
1.5
2
2.5
Time (s)
3
3.5
4
4.5
5
0
0.5
1
1.5
Correlator
6000
Delay = 1.134
seconds
4000
2000
Cross-correlation
Note:
Temporal not spatial
coherence
0
0 seconds
delay
-2000
-4000
-6000
-5
5 | Back-End Signal Processing | John Tuthill
-4
-3
-2
-1
0
Time (s)
1
2
3
4
5
2
2.5
Time (s)
3
3.5
4
4.5
5
FX and XF Correlators
F  f t   g t   F  f t   F g t 

Convolution
theorem
FX architecture
XF architecture
v i  nT
D


D
D


v i  nT
Frequency Channelisation (eg FFT)


v j  nT

 NxD
FFT
V ij  f k  ; k  1, 2 ,  , N
• ATCA before CABB
• EVLA (FXF)
• ALMA (FXF)
6 | Back-End Signal Processing | John Tuthill
v j  nT

Frequency Channelisation (eg FFT)
V ij  f k  ; k  1, 2 ,  , N
• CABB (PFX)
• DiFX

Filterbanks: FFT vs Polyphase Filters
One sub-band
768-point FFT
12,288-tap polyphase filter
+ 768-point FFT
7 | Back-End Signal Processing | John Tuthill
Filterbanks: Polyphase decomposition
M-path Polyphase
down converter
Standard single-channel
down converter
e
x(n)
 j k n
H 0 Z 
M-to-1
Digital
low-pass filter down-sampler
H(Z)
y(n,k)
x(n)
e
H 1 Z 
 j 0 k 2 M
 j1 k 2  M
S
y(nM,k)
•Equivalency Theorem
•Exchange mixer and lowpass filter with a band-pass
filter and a mixer.
•Re-write the band-pass filter in
“M-path form”
•Noble Identity
•Move a down-sampler back
through a digital filter
H Z
8 | Back-End Signal Processing | John Tuthill
e
M
 M    M H  Z 
H M 1  Z 
e
y(nM,k)
 j  M 1  k 2  M
M-path Polyphase
channeliser
H 0 Z 
x(n)
r(nM,0)
H 1 Z 
M-point
FFT
H M 1  Z 
r(nM,1)
r(nM,M-1)
Sampling:
The Sampling Theorem: A band-limited signal having no frequency components
above fmax can be determined uniquely by values sampled at uniform intervals
1
of Ts satisfying:
TS 
2 f max
signal in
anti-alias
filter
Aliasing
-fs
-fs
fs
9 | Back-End Signal Processing | John Tuthill
2fs
Clean
fs
Aliased
2fs
Sampling: ”ideal” Analogue to Digital Converter (ADC)
signal in
anti-alias
filter
Discrete-time series of digital numbers out
at N-bits of resolution
2N-1 discrete levels
between full-scale inputs
Quantisation in
amplitude
For a full-scale sinusoidal input:
SNR
max

Full - scale RMS input
RMS quantisati
on error
 6 . 02 N  1 . 76 dB
SNR for an 8-bit converter = 50 dB
10 | Back-End Signal Processing | John Tuthill
Quantisation in
time
Sampling: the real-world (especially for high-end ADC’s )
Effective Number Of Bits (ENOB)
• Aperture delay/width
ENOB 
6.02
• Acquisition time
Dynamic performance relative to
• Aperture jitter
• Crosstalk
• Missing codes
Spurious-free dynamic range (SFDR)
• Differential/Integral nonlinearity
• Digital feed-through
• Offset and Gain error
• Intermodulation distortion
Ratio of the rms amplitude of the fundamental to the
rms value of the next-largest spurious component (excluding DC)
11 | Back-End Signal Processing | John Tuthill
Sampling…why go digital at all?
• At an instance of time, a digital signal can only represent a value from a finite
set of distinct symbols.
• By contrast, an analogue signal can represent a value from a continuous
(infinite) range.
• Surely analogue is more ‘economical’.
• So why are digital systems so common place?
12 | Back-End Signal Processing | John Tuthill
Sampling…why go digital at all?
• Digital Systems:
5V
• are, to a degree, immune
to noise.
3.3V
1.7V
Noisy digital signal
0V
5V
5V
3.3V
3.3V
1.7V
1.7V
Inverter
0V
Logic 0
0V
Noisy input
Clean output
Logic 1
• are amenable to regeneration after
noise contamination/signal dispersion,
without the introduction of errors.
• can be coded in order to facilitate error detection.
systems with repeatable and reliable functionality
Much of the effort in the design of the digital back-end
hardware/firmware is to ensure these properties hold.
13 | Back-End Signal Processing | John Tuthill
Per antenna
2GHz 4.096GS/s
bands 9-bits
f1
(6-ENOB)
Secondary filterbanks
16 overlapping windows 2048
channels/window
(resolution depends on primary
filterbank mode)
Coarse
delays
Spectral
line
Fine Delay and
Fringe rotator
“F” outputs
to
correlator
engines
Pol. A
D
Continuum
Pol. B
f2
Dual-band,
dual polarisation
down conversion
Analogue-to-Digital
e-VLBI
converters
14 | Back-End Signal Processing | John Tuthill
Primary filterbanks
up to 2048 channels
4 modes: 1, 4, 16 and
64MHz resolution
 dt
auto- and
crosspolarisation
correlations
(calibration)
CABB Correlator
•6 x (6-1)/2 = 15 baselines
•Full Stokes parameters
15 | Back-End Signal Processing | John Tuthill
CABB Configurations
CABB Configuration
Primary band
Secondary band (zoom)
CFB 1M-0.5k
1.0 MHz
0.488 kHz
CFB 4M-2k*
4.0 MHz
1.953 kHz
CFB 16M-8k*
16.0 MHz
7.812 kHz
CFB 64M-32k
64.0 MHz
31.250 kHz
* Not yet implemented
16 | Back-End Signal Processing | John Tuthill
Data
throughput
reduced by a
factor of 3
188 PAF ports
768 MS/s, 8-bits
304 x 1 MHz
channels
2Tbits/s
36 dual-polarised
beams on the sky
16,416 x 18.52kHz
channels
S
Analogue-to-Digital
converters
First stage
filterbank
Crossconnect
Narrowband
Beamformers
Second stage
filterbank
Per antenna
Off-line
beam weight
computation
Array
Covariance
Matrix
36 dual-polarised
beams from 36
antennas, 16,416
fine channels
~720 Tbits/s
17 | Back-End Signal Processing | John Tuthill
D
D
Crossconnect
Fine Delay and
Fringe rotator
 dt
Hardware
Correlator
To remote
imaging
supercomputer
To
correlator
engine
Calculation Engines: so many choices…
Hard-wired logic
Application-Specific
Integrated Circuit
Field Programmable
Gate Array
•EVLA
•ALMA
•Less flexible
•Lower power/computation
•Higher initial development
18 | Back-End Signal Processing | John Tuthill
•CABB
Stored (programmed) logic
Graphics Processing Central Processing Unit/
Unit
Digital Signal Processor
•MWA
•MeerKAT
•More flexible
•Higher power/computation
•Lower initial development
•DiFX
• H. C. Ko, “Coherence Theory in Radio-Astronomical Measurements,” IEEE Trans. Antennas &
Propagation, pp. 10-20, Vol. AP-15, No. 1, Jan. 1967.
• G. B. Taylor, G. L. Carilli and R. A. Perley, Synthesis Imaging in Radio Astronomy II, Astron.
Soc. Pac. Conf. Series, vol. 180, 2008.
• CABB
• W. E. Wilson, et. al. “The Australia Telescope Compact Array Broadband Backend (CABB):
Description & First Results,” Mon. Not. R. Astron. Soc., Feb. 2011
• D. R. DeBoer, et.al, “Australian SKA Pathfinder: A High-Dynamic Range Wide-Field of View
Survey Telescope,” Proc. IEEE, 2009.
• Filter Banks
• R. E. Crochiere and L. R. Rabiner Multirate Digital Signal Processing, Prentice Hall, 1983.
• f. j. harris, Multirate Signal Processing for Communication Systems, Prentice Hall, 2008.
• P. P. Vaidyanathan, Multirate Systems And Filter Banks, Prentice Hall, 1992.
• Beamforming
• B. D. Van Veen and K. M. Buckley, “Beamforming: A Versatile Approach to Spatial Filtering,”
IEEE ASSP Magazine, April 1988
19 | Back-End Signal Processing | John Tuthill
Thank you
CASS
Dr John Tuthill
Digital Systems Engineer
t +61 2 9372 4392
e [email protected]
w www.csiro.au/
CASS - DIGITAL SYSTEMS
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