Signal Processing For Acoustic Neutrino Detection (A Tutorial)

1
Signal Processing For Acoustic Neutrino Detection
(A Tutorial)

• Sean Danaher
• University of Northumbria,
• Newcastle UK

2
What is Signal Processing?

• Used to model signals and systems where there is
  correlation between past and current
  inputs/outputs (in space or time)
• Two broad categories Continuous and Sampled
  processes
• A host of techniques Fourier, Laplace, State
  space, Z-Transform, SVD, wavelets
• Fast, accurate, robust and easy to implement
• Sadly also a big area. Engineers spend years
  learning the subject (Not long Enough!)
• Smart as Physicists are I cant cover a 3 year
  syllabus in 40 minutes!
• Notation (not just i and j)!

3
How Can Signal Processing be used in Acoustic
Neutrino Detection?

• At Virtually all stages of the process
• Accurate Parameterisation of distributions with
  no amenable analytical form e.g. Shower Energy
  distribution profiles
• Speed up computation e.g. Acoustic integrals by
  many orders of magnitude
• Design, Understand, simulate Hydrophones,
  Microphones and other acoustic transducers (and
  amplifiers)
• Design, Understand, simulate filters both
  analogue and digital (tailored amplitude, phase
  response, minimise computing time)
• Estimate and recreate noise and background
  spectra
• Design Optimal Filters e.g. Matched Filters
• Design Optimal Classification Algorithms
  (Separating Neutrino Pulses from Background)
• Improve Reconstruction accuracy
• Will cover 1-5 in this talk

4
Singular Value Decomposition
Decompose the Data matrix into 3 matrices. When
multiplied we get back the original data. W and V
are unitary. L contains the contribution from
each of the eigenvectors in descending order
along main diagonal.
We can get an approximation of the original data
by setting the L values to zero below a certain
threshold
Similar techniques are used in statistics CVA,
PCA and Factor Analysis Based on Eigenvector
Techniques
Good SVD algorithms exist in ROOT and MATLAB
5
SVD II

• Somewhere between vector five and seven we get
  the Best representation of the data
• Better than the observation as noise filtered out
• Highly compressed (only 1-2 of original size
  e.g. 6/500x1.5)
• Have basis vectors for data so can produce
  similar data

SVD done on Noisy data
6
Building the Radial Distribution
4 examples chosen have maximum variation in shape
Hadronic Component using Geant IV

• SVD works well with the radial distribution
• Three vectors sufficient to fit the data
• Can use a linear mixture of these three vectors
  as input to the Acoustic MC

We Reproduce both the shape and variation
7
Acoustic Integrals
Slow decay
fast thermal energy deposition

• Band limited by
• Water Properties
• Attenuation

8
Method I

• Throw a number if MC points 107
• Assume the energy at each point in the cascade
  deposited as a Gaussian Distribution.
• Calculate distance to observer from each point
• Propagate the Gaussian to the observer (an
  analytical solution is known for this)
• Sum over all the points
• Provided the width of the Gaussian is small in
  comparison to the shower radius we will get the
  pressure pulse

Expensive 5 flops per time point If using 1024
points on time axis 510241e75.1e10 flops
Need to do this thousands of times for different
distances, angles, distributions, etc.
9
The Convolution Integral

• Given a signal s(t) and a system with an impulse
  response h(t) (response to ?(t)) then y(t) to an
  arbitrary input is given by

We can use this to process all the MC points in
the cascade simultaneously
10
A Few examples
RC circuit
Mixed signal
This situation is similar to our acoustic integral


2d
11
Convolution Theorem
Convolution in the time domain is Multiplication
in the frequency domain (and visa versa)

• Will speed the process still further
• Convolution very expensive computationally (on2)
• Convert the signal and impulse response to the
  frequency domain (Fourier Transform)
• Provided the number of points n2m very efficient
  FFTs on log n
• Multiply and take Inverse Transform

12
Acoustic Integrals II

• Assume each point produces a Delta Function
• The sampled equivalent of a Delta function is 1
• The integral can be done using a histogram
  function
• (modern histogram algorithms are very efficient)
• The derivative can be done in the frequency
  domain (d/dt ? i? No overhead)
• We then convolve the response with the entire
  signal

13
Only Histogram depends on the number of MC points
Derivative j?
FT of Exyz
Water Attenuation
Blackman window
Histogram 103 flops Scale histogram by 1/d 103
flops (only needed in near field) FFT nLogn c 104
flops Multiplication c 4x103 flops IFFT nLogn c
104 flops
6 orders of magnitude less than approach 1
Limitation Will not work if volume of the
integral is so great that attenuation varies
significantly (Large 100m source in near field)
14
State Space Analysis
MIMO systems Mixed Mechanical/electrical models
etc Matrix Based Method
Mode 1
Quantum Mechanics
Mode 14
All modern control algorithms use SS methods.
Mode 100
Pictured 20000 state simulation of a square
acoustic membrane (100x100 masses 2 states x and
v) A Matrix 400x106 elements
15
SS Implementation
States
Inputs

• States are degrees of freedom of the system.
  Things that store energy
• Capacitor Voltages
• Inductor currents
• Positions and Velocities of masses

A
B
States
D
Outputs
C
A is of size States ? States B is of size States
? Inputs C is of size Outputs ? States D is of
size Outputs ? Inputs
16
Simple Example LC
Maths simple to implement Example shows the
behaviour with an initial 1V on the Capacitor
(1F) and 1A flowing through the inductor (1H)
17
Speaker/Microphone
Frame
Cone
Magnet assembly
Magnet assembly
Coil
x1x, (position) x2dx/dt, (velocity) x3i,
(current)
Electrically coil moving in a magnetic
field Creates an EMF
Step Response
-4
x 10
2.5
2
1.5
Amplitude
1
0.5
0
-0.5
0
0.5
1
1.5
2
2.5
3
Time (ms)
18
Simple Hydrophone Model
Heart of Hydrophone Piezo electric crystal Piezo
electric effect
Omni works in breathing mode
40cm
30cm
? 0.09 ?5e-3
Erlangener water tank (Niess)
3rd order simulation. Do we need higher?
Gaussian cross-section
19
Acoustic Simulation

• Bipolar Pulse
• 1000 element simulation
• Two types of medium Heavy and light
• pulse reflects against walls
• mismatch between sections
• velocity on each section
• amplitude change
• dispersion

Enhance to 2D? Embed Hydrophone models etc.?
20
Butterworth Filter
1930s
Stable Good Frequency Response

• Note Impulse response
• Output Delay
• Oscillation

21
Implementation
At acoustic frequencies easiest to use op-amps.
Expensive to make high precision lossless
inductors
Cascade 2nd Order sections to make Butterworth
high order filters
Ra
Rb
22
Recovering Phase information
e.g. SAUND Data
It is trivial to design digital filters which
have a constant group delay d?/d? and hence no
phase distortion
0.15
SN
causal
non causal
0.1
0.05
Amplitude (Arbitrary)
0
If however we know the filter response e.g.
Butterworth we can run the data through the
filter backwards.
-0.05
-0.1
0
0.2
0.4
0.6
0.8
1
1.2
time (s)
-3
x 10
This increases the order of the filter by a
factor of 2
23
The Digital Filter
The digital filter is simple! Based upon sampled
sequences
Negative coefficients gtCausal
10 5 2 6 6 4 5 5.5 9 7 8 8.5 5
6.5 0 2.5 8 4 4 6
Simple moving average Filter
Crude low-pass Called FIR or MA
24
The Z Transform and Sampled Signals
25
A few z transforms
Poles must be inside the unit circle for stability
26
Simple Transfer Function
Frequency response simply determined by running
around the unit circle ? Corresponds to the
Nyquist Frequency (fs/2)
) (degrees)
w
H(
Ð
27
Notch Filter
Signal buried in 100Hz
Recovered signal
28
Spectral Analysis

• Using the Fourier Transform is seldom the best
  way to get a spectrum. Normally methods based
  around
• Autocorrelation (AC)
• Linear Prediction are used

AC method Use FT of AC To get PSD
Linear Prediction
All pole filter driven by white noise. Need to
choose the order with care. But can now reproduce
the spectrum
29
Spectral estimation example

• 8 bit 8kHz
• Split speech into frame 240 samples long
• Use LPC10 to estimate spectrum
• Reconstruct

64k
2.4k
933

Frequency (Hz)
30
Conclusions

• SP Techniques Useful
• Thanks for listening
• Questions