digital filtering - cern
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Digital FilteringDigital Filteringin the ATLAS Calorimeter Triggerin the ATLAS Calorimeter Trigger
Jan JongmannsJan Jongmanns
HighRR BiWeekly MeetingHighRR BiWeekly Meeting14.06.201714.06.2017
14.06.2017 Digital Filter 4
-Analogue Filter-Once upon a Time (2 weeks ago...)
● HighRR Hands-On:We looked at a bandpass filter
● Today: Digital Signal Filters
14.06.2017 Digital Filter 5
-From Analogue to Digital-The Difference
AnalogueAnalogue DigitalDigitalContinuous signal:
Continuous in time:
Discrete signal:
Discrete in time:
Am
pli
tud
e
Time
14.06.2017 Digital Filter 6
-From Analogue to Digital-Digital Definitions 1/2
● Digital processing systems transform series of input values to series of output values
● For purpose of this talk, consider three processing blocks:
++xx
DelayDelay
14.06.2017 Digital Filter 7
-From Analogue to Digital-Digital Definitions 2/2
● Note: similar to analogue s-domain, digital circuit can be described in z-domain
● Transfer function is defined as usual
● For this talk: mostly focus on time domain
14.06.2017 Digital Filter 8
-Digital Filter-Filter Definition
● A digital filter maps a sequence of input values, to a sequence of output values,
● Define impulse response as filter output for
● Two categories of filters:
- Finite Impulse Response (FIR):
- Infinite Impulse Response (IIR):
FilterFilter
14.06.2017 Digital Filter 9
-Digital Filter-FIR Filter Implementation
● Finite Impulse Response Filter:
● Defining the Filter Coefficients
DelayDelay
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xxxx
DelayDelay
++
xx
DelayDelay
++
xx
14.06.2017 Digital Filter 10
-Digital Filter-IIR Filter Implementation
● Infinite Impulse Response Filter:
● Realised via feedback loop:
FIRFIR
DelayDelay
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xx
DelayDelay
xx
DelayDelay
xx
++++
14.06.2017 Digital Filter 11
-Digital Filter-Properties
FIR FilterFIR Filter
guaranteed stable
can have linear phase
no analogue equivalent
less quantisation effects
easier to implement
less effective at same order
IIR FilterIIR Filter
can be unstable
only approx. linear phase
derived from analogue
quantisation effects from feedback
more effective at same order
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++++
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● In general, FIR filter preferred due to stability and linearity
● IIR filter used if FIR would be too large
14.06.2017 Digital Filter 12
-Digital Filter-Designing IIR Filter
● Coefficients of IIR filters typically derived from analogue circuit
● General procedure:
- Design analogue circuit with desired bandwidth properties
- Write down transfer function
- Convert to digital transfer function
- Analyse to extract and
14.06.2017 Digital Filter 14
-ATLAS Level-1 Calorimeter Trigger-One-Slide Overview
● ~7200 cables from calorimeterto L1Calo trigger
- Analogue signals give energy deposited in small region of calorimeter
● L1Calo analyses signals
- PreProcessor produces digital information from analogue input
- Object-finding processors analyse depositions to find high- particles
PreProcessorPreProcessor Signal digitisation, identification calibration
ProcessorsProcessors
EMClusters
Jets & Sums
14.06.2017 Digital Filter 15
-ATLAS Level-1 Calorimeter Trigger-Input Signals
● Bi-polar signals from calorimeters
- Shaped from triangular ionisation pulse to optimise signal-to-noise
● Typical length 450~600 ns
- Short positive pulse followed by long undershoot
- Sampled at 40 MHz, i.e. T = 25 ns
● Both maximum amplitude and integral of positive part correspond to energy deposited in calorimeter region
14.06.2017 Digital Filter 16
-ATLAS Level-1 Calorimeter Trigger-Input Noise
● Distinguish between thermal noise and pile-up noise
- For Run-2, pile-up dominates most calorimeter regions
- Pile-Up strongest in forward region and EM calorimeter
14.06.2017 Digital Filter 17
-ATLAS Level-1 Calorimeter Trigger-Peak-Finding
● L1Calo Preprocessor identifies signal by finding peak sample
- Peaks identified by comparison of samples:
- Noise threshold removes peaks with small amplitudes
- Also cuts low energy signals -> filter to improve signal-to-noise
14.06.2017 Digital Filter 18
-ATLAS Level-1 Calorimeter Trigger-FIR Noise Filter
● FIR Filter in L1Calo Preprocessor uses 5 samples
- Corresponds to signal peak
- Design coefficients to improve signal-to-noise
14.06.2017 Digital Filter 19
-Optimal Filter-Derivation 1/3
● Given:
- Signal amplitude and shape
- Noise contribution
● Find:
- Coefficients that maximise Signal-to-Noise
14.06.2017 Digital Filter 20
-Optimal Filter-Derivation 2/3
● Rewrite:
● Introducing Autocorrelation Matrix
14.06.2017 Digital Filter 21
-Optimal Filter-Derivation 3/3
● Set derivative to zero to find maximum:
● Solve:
● Rewrite in vectorial form:
● Solution:
14.06.2017 Digital Filter 22
-Optimal Filter-Discussion
● The filter coefficients optimise Signal-to-Noise
- Expected signal shape
- Noise correlation matrix
● The structure of the noise dictates the filter coefficients
- For thermal (white) noise, noise samples are uncorrelated
-> is diagonal -> is diagonal -> = Matched FilterMatched Filter
- Pile-up follows signal shape -> pile-up noise strongly correlated
-> non-diagonal -> differs from = Autocorrelation FilterAutocorrelation Filter
14.06.2017 Digital Filter 25
-Summary-
● Digital Filters fall into two categories:
- Finite Impulse Response filters depend only on input signals
- Infinite Impulse Response filters have feedback loops
● IIR filters have higher selectivity at the same order, and are usually designed starting from analogue filter
● FIR filters allow for different type of filtering
- e.g. Signal-to-Noise optimisation in ATLAS L1Calo trigger
- improved signal identification efficiency in high-pile-up environment
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