Energy Efficient Compression of Shock Data using Compressed Sensing

Energy Efficient Compression of Shock Datausing Compressed Sensing

Jerrin Thomas Panachakel, Finitha K.C.,

September 2, 2016

Overview

Jerrin Thomas Panachakel [email protected] 2

Introduction

⇒ Avionics components encounter shock from severalsources

⇒ Components should be tested for reliability⇒ Shock data: acceleration v/s time plot


Problem

• Compression of Shock Data• Constraints

• Computational complexity should be low• Error should be minimum

• Why CS?• Shock data is sparse in multiple domains• Has lower computational complexity• Has almost equal compression efficiency


Shock Data (1/2)

• Plot of magnitude of shock pulses v/s time• Causes1:

• Rocket motor ignition• Staging events• Deployment events

• Measured using accelerometers

1Tom Irvine. “An Introduction to the Vibration Response Spectrum”. In: Rev C,Vibrationdata (2000).


Shock Data (2/2)

source: https://www.youtube.com/watch?v=KZVgKu6v808Jerrin Thomas Panachakel [email protected] 6

https://www.youtube.com/watch?v=KZVgKu6v808

Shock Respose Spectra

• Calculated from acceleration time history• For estimating damage potential• For estimating integrity of shock data

Figure: SRS


Shock Respose Spectra- Calculation (1/2)

Figure: SRS Model

ωni =

√Ki

Mi(1)


Shock Respose Spectra- Calculation (2/2)

(a) Shock data and response toSDOF systems

(b) Shock data and Shock ResponseSpectra data

Figure: Shock data and SRS


Integrity from SRS

(a) SRS of a saturated shock data

(b) SRS of a good shock data

Figure: Shock Response Spectra of shock data signals


Compressed Sensing (CS)-Motivation

• ‘‘Why go to so much effort to acquire all the data whenmost of what we get will be thrown away ?”2

• N samples acquired but only K are required• Basic requirement, sparsity

Figure: Comparison between sparse signal and compressible signal

2Jon Dattorro. Convex optimization and Euclidean distance geometry. MebooPublishing USA, 2005.


Mathematics behind CS (1/3)

x =N∑

i=1

siψi (2)

weighing coefficients, si =< x ,Ψ >, x ∈ RN

or equivalently,x = Ψs (3)

y = Φx (4)

using (3),y = ΦΨs = Θs (5)



• Recoverable if the four columns are LI• For M measurements, all M × K sub-matrices are ideally

close to orthonormal basis• RIP: 1− ε ≤ ||Φv ||2

||v ||2 ≤ 1 + εJerrin Thomas Panachakel [email protected] 13


• Design of RIP matrix is almost impossible• RIP matrices are around us!3

• iid Gaussian• iid Bernoulli

• M ≥ cKlog( NK )

Figure: Gerhard Richer- 4096 Farben

3Boris Sergeevich Kashin. “Diameters of some finite-dimensional sets and classesof smooth functions”. In: Izvestiya Rossiiskoi Akademii Nauk. SeriyaMatematicheskaya 41.2 (1977), pp. 334–351.


Recovery (1/3)

• Solution to Φs = y lies in the translated null space of Θ

• `2 recovery:• S = argmin||s′||2, such that ΘS = y

Figure: `2 minimization


Recovery (2/3)

• `0 recovery:• S = argmin||s′||0, such that ΘS = y• Computationally complex

• `1 recovery:• S = argmin||s′||1, such that ΘS = y

Figure: `1 minimization


Recovery (3/3)

Figure: Sparse signal and its reconstruction


Sparsity Analysis

(a) Time (b) DCT

(c) Haar (d) Daubechies


Performance Metrics

• Percentage Root Mean Square Difference: [1]

PRD =

√√√√√√√√N∑

n=1(x [n]− x [n])2

N∑n=1

(x [n]− x [n])2

× 100% (6)

• Compression Ratio: [1]

CR =Norg − Ncomp

Norg× 100 (7)

• Execution time


Time Domain

(a) (b)

(c)

Figure: Original and Recovered Signal: (a) Original Signal (b)Compressed Sensing (c) Thresholding based DWT


Shock Respnse Spectra

(a) (b)

(c)

Figure: Shock Response Spectra: (a) Original (b) CS (c)Thresholding based DWT


PRD v/s CR

Figure: PRD v/s CR


Time v/s CR

Figure: Execution Time v/s CR


Conclusion

• Shock data compression performed using CS• CS inferior in terms of PRD for higher CR• CS is almost 1000 times faster than thresholding based

DWT compression• Implemented technique satisfies the requirements



Engineering

Energy Efficient Compression of Shock Data using Compressed Sensing