Sean Danaher

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Signal Processing For Acoustic Neutrino

Detection (A Tutorial)

Sean Danaher University of Northumbria,

Newcastle UK

Sean Danaher

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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 can’t cover a 3+ year

syllabus in 40 minutes!• Notation (not just i and j)!

Sean Danaher

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At Virtually all stages of the process1. Accurate Parameterisation of distributions with no

amenable analytical form e.g. Shower Energy distribution profiles

2. Speed up computation e.g. Acoustic integrals by many orders of magnitude

3. Design, Understand, simulate Hydrophones, Microphones and other acoustic transducers (and amplifiers)

4. Design, Understand, simulate filters both analogue and digital (tailored amplitude, phase response, minimise computing time)

5. Estimate and recreate noise and background spectra6. Design Optimal Filters e.g. Matched Filters7. Design Optimal Classification Algorithms (Separating

Neutrino Pulses from Background)8. Improve Reconstruction accuracy……………Will cover 1-5 in this talk……

How Can Signal Processing be used in Acoustic Neutrino

Detection?

Sean Danaher

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Singular Value Decomposition

W L VTW L VT

(((( )))) (((( ))))(((( ))))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 AnalysisBased on Eigenvector Techniques

Good SVD algorithms exist in ROOT and MATLAB

Sean Danaher

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SVD IISomewhere 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 onNoisy data

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Building the Radial Distribution

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

4 examples chosen have maximum variation in shape

Hadronic Component using Geant IV

Sean Danaher

Acoustic Integrals

fast thermal energy deposition

2

2

d

dt

( ) ( / )0

4 p

E dP t r c t dv

C r dt

Band limited by •Water Properties•Attenuation

Slow decay

Sean Danaher

Method I

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•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

Expensive5 flops per time pointIf using 1024 pointson time axis5*1024*1e7=5.1e10 flops

-0.200.22

3

4

5

6

7

8

9

10

11

12

0 2 4 6 8 10 120

0.5

1

1.5

2

2.5

3 x 104

Longitudinal Distance (m)F

req

uen

cyRadial Distribution

-5 0 5-0.5

0

0.5

Time (ps)

P (

arb

itra

ry)

x 10-4-6 -4 -2 0 2 4 6-0.1

-0.05

0

0.05

0.1

Time (s)

Pre

ssu

re (

Pa)

Need to do this thousands of times for different distances, angles, distributions, etc.

Sean Danaher

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The Convolution IntegralGiven 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

( ) ( ) ( )y t s t h t d

s(t) Bipolar Acoustic Pulse

h(t) Impulse response high pass filter (SAUND)

y(t) ElectricalPulse

We can use this to process all the MC points in the cascade simultaneously

Sean Danaher

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* =

A Few examples

RC circuit

2d

Mixed signalThis situation is similar to our acoustic integral

Sean Danaher

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Will speed the process still furtherWill speed the process still further• Convolution very expensive

computationally (o=n2)– Convert the signal and impulse response to

the frequency domain (Fourier Transform)– Provided the number of points n=2m very

efficient FFTs o=n log n– Multiply and take Inverse Transform

Convolution in the time domain is Multiplication in the frequency domain (and visa versa)

Convolution Theorem

Sean Danaher

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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

( ) ( / )

( / ) ( )

0

0 0

4

4 4

p

xyzp p

E dP t r c t dv

C r dt

E Ed dr c t dv E t

C dt r C dt

Sean Danaher

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Only Histogram depends on the number of MC points

511

512

[ ] ( ) ( ) ( ) ( )n

i nTxyz w

n

P t E w D A e

FT of Exyz

Blackman window

Derivative j

Water Attenuation

Histogram: 103 flopsScale histogram by 1/d: 103 flops (only needed in near field)FFT nLogn c 104 flopsMultiplication c 4x103 flopsIFFT nLogn c 104 flops

6 orders of magnitude less than approach 1Limitation: Will not work if volume of the integral is so great that attenuation varies significantly (Large 100+m source in near field)

Sean Danaher

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State Space AnalysisMIMO systemsMixed Mechanical/electrical models etcMatrix Based Method

X = AX +BU

Y = CX +DU

Pictured 20000 state simulation of a square acoustic membrane:(100x100 masses 2 states x and v)A Matrix 400x106 elements

All modern control algorithms use

SS methods.

Mode 1

Mode 14

Mode 100

didt

H

1

2

3

4

.

.

n

i H

Quantum Mechanics

Sean Danaher

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A B

DC

SS Implementation

if

if

if position velocity

if velocity acceleration

c C

L L

dVx V I C Cx

dtdi

x I V L Lxdt

x x

x x

States Inputs

States

Outputs

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

A is of size States StatesB is of size States InputsC is of size Outputs StatesD is of size Outputs Inputs

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Simple Example LC ( )c L

c L

L c L C

dV dII C V L

dt dtV V I I

1 1 2 1 1

2 2 1 2 2

C

L

x V Cx x y x

x I Lx x y x

10

1 0

0 110

C

L

A C

-1.5

-1

-0.5

0

0.5

1

1.5

Vo

ltag

e

0 5 10 15 20 25 30 35 40-1.5

-1

-0.5

0

0.5

1

1.5

Cu

rren

t

Response to Initial Conditions

Time (sec)

Am

plit

ud

e

Maths simple to implementExample shows the behaviour with an initial 1V on the Capacitor (1F) and 1A flowing through the inductor (1H)

Sean Danaher

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Speaker/MicrophoneConeFrame

Magnetassembly

Coil

S N

2

2e

d x dx xF Bi m r

dt dt s Mechanically Force produced by current

Loss to air term

dx diV B L Ri

dt dt Electrically coil moving in a magnetic field

Creates an EMF

x1=x, (position) x2=dx/dt, (velocity)x3=i, (current)

0 1 0 0 0

1 10

100

0 1 0 0 0( ) ( )

r B

sm m m mB R

LL L

X X U

Y X U

-160

-140

-120

-100

-80

-60

Mag

nitu

de (

dB)

102

103

104

105

106-180

-90

0

90

Pha

se (

deg)

Bode Diagram

Frequency (rad/sec)

0 0.5 1 1.5 2 2.5 3-0.5

0

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1

1.5

2

2.5x

10

-4 Step Response

Time (ms)A

mp

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Magnetassembly

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Simple Hydrophone Model

Massm

F=rv

F=ma

F=x/SF=KVc

VcKvIc

inV Vc

R

Vin

1 2 3, ,Cx V x v x x Heart of HydrophonePiezo electric crystalPiezo electric effect

Omni works in breathing mode

40cm

30cm

25 30 35 40 45 50-15

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5

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15

20

25

Time (eq. cm)

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)

Simulated Hydrophone signal

0

x

I I e

3rd order simulation. Do we need higher?

Gaussian cross-section

“Erlangener” water tank (Niess)

10 1

0

10 1

0 00 1 0

1 10 0 0

0 1 0 0 0

K

RC CRC

K rA B

m m Sm

C DR R

= 0.09 =5e-3

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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

0 1 0 0 0 0 0 0 . . . . . 0

-8 0 4 0 0 0 0 0 . . . . . 0

0 0 0 1 0 0 0 0 . . . . . 0

4 0 -8 0 4 0 0 0 . . . . . 0A =

0 0 0 0 0 1 0 0 . . . . . 0

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

0 0 0 0 0 0 0 0 0 0 1 0 -2 0

Enhance to 2D?Embed Hydrophone models etc.?

Sean Danaher

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Butterworth Filter

StableGood Frequency Response

1930s

0 10 20 30 40 50-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Time(s)

Ou

tpu

t

Buterworth Impulse Response

1st

order

2nd

order

4th

order

8th

order

16th

order

32nd

orderNote Impulse response •Output Delay•Oscillation

2

1

1| ( ) |

nH

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Implementation

+

-

+

-

Cascade 2nd Order sections to make Butterworth high order filters

1st Order Familiar RC

2nd Order LRC

3rd Order “”

Passive Filters

At acoustic frequencies easiest to use op-amps. Expensive to make high precision lossless inductors

1

1 1

4 3 2

2 2

s +2.6131s +3.4142s +2.6131s+1

s +0.7654s+1 s +1.8478s+1

Ra

Rb

=

2| ( ) | , 1

(3 ) 1DC a

DCDC b

A Rh s A

s A s R

=

Sean Danaher

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Recovering Phase information

0 0.2 0.4 0.6 0.8 1 1.2

x 10-3

-0.1

-0.05

0

0.05

0.1

0.15

time (s)

Am

pli

tud

e (

Arb

itra

ry)

S+Ncausalnon causal

It is trivial to design digital filters which have a constant group delay d/d and hence no phase distortionIf however we know the filter response e.g. Butterworth we can run the data through the filter backwards.

This increases the order of the filter by a factor of 2

e.g. SAUND Data

Sean Danaher

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The Digital Filter

0 1 2 0 1 21 2 1 2[ ] [ ] [ ] .... [ ] [ ] [ ] ....a y n a y n a y n b u n bu n b u n

The digital filter is simple! Based upon sampled sequences

10 52 66 45 5.59 78 8.55 6.50 2.58 44 6

11

2[ ] [ ] [ ]y n x n x n

Simple moving average Filter

Crude low-passCalled FIR orMA

Negative coefficients =>Causal

6 68 149 237 302 324 369 459 544 589 67

Perfect Integrator

Called IIR or AR

[ ] [ 1] [ ]y n y n x n

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The Z Transform and Sampled Signals

( ) [ ]n

n

n

X z x n z

1 2

2 11

0

1 3 4 1 3 4

11

1

, , ( )

, , ,... ( )n n

n

if x X z z z

if x a a X z azaz

Geometric sequences are of prime interest because

( ) [ ]i t i ntu t e u t e is a geometric sequence

Sean Danaher

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[A]

-1

0

1

-4-3-2-10123

-1 0 1

-1

0

12 1

1

2 110

2 1 210

( )cos

[ ] [ ] cos [ ] [ ]

h zz z

y n x n y n y n

0 5 10 15 20 25 30

-15

-10

-5

0

5

10

0 5 10 15 20 25 30

-1.5-1

-0.50

0.51

1.52

0 5 10 15 20 25 30-1

0

1

0 0.2 0.4 0.6 0.8 10

5

10

15

20

Fraction of Nyquist frequency

Frequency Response

Poles must be inside the unit circle for stability

/10, 1

/ 5, 0.95

/14, 1.05

A few z transforms

Sean Danaher

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Frequency response simply determined by running around the unit circle Corresponds to the Nyquist Frequency (fs/2)

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

frequency (Nyquist=1)

|H(

)|

0.5( )

0.5[ ] [ ] 0.5 [ 1]

0.5 [ 1]

zh z

zy n x n x n

y n

-1

0

1

12

0 0.2 0.4 0.6 0.8 10

10

20

30

40

50

60

frequency (Nyquist=1)

H(

) (d

egre

es)

1

2

1 2

| ( ) |

( )

H z

H z

Simple Transfer Function

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[A]Notch Filter

5 5

5 5

( )( )( )

( 0.95 )( 0.95 )

y[n]=x[n]-1.6180x[n-1]+x[n-2]

+1.5371y[n-1]-0.9025y[n-2]

i i

i i

z e z eh z

z e z e

-1 0 1

-1

0

1

/5

0 100 200 300 400 5000

0.2

0.4

0.6

0.8

1

1.2

1.4

frequency

|H(

)|

Fs=1000Hz

0 0.05 0.1 0.15 0.2 0.25-1.5

-1

-0.5

0

0.5

1

1.5

Time

Signal buried in 100Hz

Recovered signal

Sean Danaher

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

0 0.2 0.4 0.6 0.8 10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

Frequency (relative)

Am

pli

tud

e(R

elat

ive)

fftaclpc

-150 -100 -50 0 50 100 150-0.5

0

0.5

1

Lag

Am

pli

tud

e (r

elat

ive)

Autocorrelation

Linear Prediction0 1 21 2[ ] [ ] [ ].... [ ]a y n a y n a y n e n

0 100 200 300 400 500-4

-3

-2

-1

0

1

2

3

4

Sample number

Am

pli

tud

e

First 500 of 2048 data points

AC methodUse FT of ACTo get PSD

All pole filter driven by white noise. Need to choose the order with care. But can now reproduce the spectrum

Spectral Analysis

Sean Danaher

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Spectral estimation example

64k

933

2.4k

gain vocal tractfilter

pitch

noise

voiced/unvoicedswitch

synthesizedspeech

Frequency (Hz)0 1000 2000 3000 40000

1

2

3

4

5

6

7

8spectrum of frame 48

|h(

)|

0 10 20 30 40 50-1

-0.5

0

0.5

1

1.5

2

Sample [n]h

(t)

Impulse Response

•8 bit 8kHz

•Split speech into frame 240 samples long

•Use LPC10 to estimate spectrum

•Reconstruct

Sean Danaher

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Conclusions

• SP Techniques Useful• Thanks for listening• Questions