Sean Danaher z 1 z 1 z 1 z 1 z 1 z 1 [A] -K- [A] u1u1 u2u2 u3u3 u4u4 y2y2 y1y1 y3y3 MIMOSYSTEMMIMOSYSTEM Signal Processing For Acoustic Neutrino Detection (A Tutorial) Sean Danaher University of Northumbria, Newcastle UK

Sean Danaher z 1 z 1 z 1 z 1 z 1 z 1 [A] -K- [A] u1u1 u2u2 u3u3 u4u4 y2y2 y1y1 y3y3 MIMOSYSTEMMIMOSYSTEM 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 z 1 z 1 z 1 z 1 z 1 z 1 [A] -K- [A] u1u1 u2u2 u3u3 u4u4 y2y2 y1y1 y3y3 MIMOSYSTEMMIMOSYSTEM At Virtually all stages of the process 1.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 spectra 6.Design Optimal Filters e.g. Matched Filters 7.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 z 1 z 1 z 1 z 1 z 1 z 1 [A] -K- [A] u1u1 u2u2 u3u3 u4u4 y2y2 y1y1 y3y3 MIMOSYSTEMMIMOSYSTEM Singular Value Decomposition W L V T (())(())(()) 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

Sean Danaher u1u1 u2u2 u3u3 u4u4 y2y2 y1y1 y3y3 MIMOSYSTEMMIMOSYSTEM 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

Sean Danaher z 1 z 1 z 1 z 1 z 1 z 1 [A] -K- [A] u1u1 u2u2 u3u3 u4u4 y2y2 y1y1 y3y3 MIMOSYSTEMMIMOSYSTEM 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 Band limited by Water Properties Attenuation Slow decay

Sean Danaher Method I z 1 z 1 z 1 z 1 z 1 z 1 [A] -K- [A] u1u1 u2u2 u3u3 u4u4 y2y2 y1y1 y3y3 MIMOSYSTEMMIMOSYSTEM Throw a number if MC points ~10 7 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 5*1024*1e7=5.1e10 flops x 10 4 Longitudinal Distance (m) Frequency Radial Distribution Time (ps) P (arbitrary) x Time (s) Pressure (Pa) Need to do this thousands of times for different distances, angles, distributions, etc.

Sean Danaher u1u1 u2u2 u3u3 u4u4 y2y2 y1y1 y3y3 MIMOSYSTEMMIMOSYSTEM 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 s(t) Bipolar Acoustic Pulse h(t) Impulse response high pass filter (SAUND) y(t) Electrical Pulse We can use this to process all the MC points in the cascade simultaneously

Sean Danaher u1u1 u2u2 u3u3 u4u4 y2y2 y1y1 y3y3 MIMOSYSTEMMIMOSYSTEM * = A Few examples RC circuit 2d Mixed signal This situation is similar to our acoustic integral

Sean Danaher z 1 z 1 z 1 z 1 z 1 z 1 [A] -K- [A] u1u1 u2u2 u3u3 u4u4 y2y2 y1y1 y3y3 MIMOSYSTEMMIMOSYSTEM Will speed the process still further Convolution very expensive computationally (o=n 2 ) –Convert the signal and impulse response to the frequency domain (Fourier Transform) –Provided the number of points n=2 m 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 z 1 z 1 z 1 z 1 z 1 z 1 [A] -K- [A] u1u1 u2u2 u3u3 u4u4 y2y2 y1y1 y3y3 MIMOSYSTEMMIMOSYSTEM 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

Sean Danaher z 1 z 1 z 1 z 1 z 1 z 1 [A] -K- [A] u1u1 u2u2 u3u3 u4u4 y2y2 y1y1 y3y3 MIMOSYSTEMMIMOSYSTEM Only Histogram depends on the number of MC points FT of E xyz Blackman window Derivative j  Water Attenuation Histogram: 10 3 flops Scale histogram by 1/d: 10 3 flops (only needed in near field) FFT nLogn c 10 4 flops Multiplication c 4x10 3 flops IFFT nLogn c 10 4 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 100+m source in near field)

Sean Danaher u1u1 u2u2 u3u3 u4u4 y2y2 y1y1 y3y3 MIMOSYSTEMMIMOSYSTEM State Space Analysis MIMO systems Mixed Mechanical/electrical models etc Matrix Based Method Pictured state simulation of a square acoustic membrane: (100x100 masses 2 states x and v) A Matrix 400x10 6 elements All modern control algorithms use SS methods. Mode 1 Mode 14 Mode 100 Quantum Mechanics

Sean Danaher u1u1 u2u2 u3u3 u4u4 y2y2 y1y1 y3y3 MIMOSYSTEMMIMOSYSTEM AB D C SS Implementation StatesInputs 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  States B is of size States  Inputs C is of size Outputs  States D is of size Outputs  Inputs

Sean Danaher u1u1 u2u2 u3u3 u4u4 y2y2 y1y1 y3y3 MIMOSYSTEMMIMOSYSTEM 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)

Sean Danaher u1u1 u2u2 u3u3 u4u4 y2y2 y1y1 y3y3 MIMOSYSTEMMIMOSYSTEM Speaker/Microphone Cone Frame Magnet assembly Coil S N Mechanically Force produced by current Loss to air term Electrically coil moving in a magnetic field Creates an EMF x 1 =x, (position) x 2 =dx/dt, (velocity) x 3 =i, (current) x Step Response Time (ms) Amplitude Magnet assembly

Sean Danaher u1u1 u2u2 u3u3 u4u4 y2y2 y1y1 y3y3 MIMOSYSTEMMIMOSYSTEM Simple Hydrophone Model Heart of Hydrophone Piezo electric crystal Piezo electric effect Omni works in breathing mode 40cm 30cm Time (eq. cm) Amplitude (Arbitrary) Simulated Hydrophone signal 3 rd order simulation. Do we need higher? Gaussian cross-section “Erlangener” water tank (Niess) = 0.09  =5e-3

Sean Danaher u1u1 u2u2 u3u3 u4u4 y2y2 y1y1 y3y3 MIMOSYSTEMMIMOSYSTEM 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.?

Sean Danaher Butterworth Filter Stable Good Frequency Response 1930s Note Impulse response Output Delay Oscillation

Sean Danaher Implementation Cascade 2 nd Order sections to make Butterworth high order filters 1 st Order Familiar RC 2 nd Order LRC 3 rd Order “  ” Passive Filters At acoustic frequencies easiest to use op-amps. Expensive to make high precision lossless inductors RaRa RbRb = =

Sean Danaher Recovering Phase information x time (s) Amplitude (Arbitrary) S+N causal non causal It is trivial to design digital filters which have a constant group delay d  /d  and hence no phase distortion If 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 The Digital Filter The digital filter is simple! Based upon sampled sequences Simple moving average Filter Crude low-pass Called FIR or MA Negative coefficients =>Causal Perfect Integrator Called IIR or AR

Sean Danaher The Z Transform and Sampled Signals Geometric sequences are of prime interest because is a geometric sequence

Sean Danaher Fraction of Nyquist frequency Frequency Response Poles must be inside the unit circle for stability A few z transforms

Sean Danaher Frequency response simply determined by running around the unit circle  Corresponds to the Nyquist Frequency (f s /2) frequency (Nyquist=1) |H(  )| frequency (Nyquist=1)  H(  ) (degrees) Simple Transfer Function

Sean Danaher Notch Filter  / frequency |H(  )| F s =1000Hz Time Signal buried in 100Hz Recovered signal

Sean Danaher Using the Fourier Transform is seldom the best way to get a spectrum. Normally methods based around Autocorrelation (AC) Linear Prediction are used Frequency (relative) Amplitude(Relative) fft ac lpc Lag Amplitude (relative) Autocorrelation Linear Prediction Sample number Amplitude First 500 of 2048 data points AC method Use FT of AC To 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 Spectral estimation example 64k k gain vocal tract filter pitch noise voiced/unvoiced switch synthesized speech Frequency (Hz) spectrum of frame 48 |h(  )| Sample [n] h(t) Impulse Response 8 bit 8kHz Split speech into frame 240 samples long Use LPC10 to estimate spectrum Reconstruct

Sean Danaher z 1 z 1 z 1 z 1 z 1 z 1 [A] -K- [A] u1u1 u2u2 u3u3 u4u4 y2y2 y1y1 y3y3 MIMOSYSTEMMIMOSYSTEM Conclusions SP Techniques Useful Thanks for listening Questions