Compressive Sensing system for recording of ECoG signals in-vivo

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Figure 2.5.1. l 1-minimization approaches to l 0-minimization approach.Candès, Tao, Romberg and Donoho have formalized the CS view of the world [26] by statingthat under certain assumptions there is a correspondence between the solution which isobtained from the l 0 -minimizer and the l 1 -minimizer (see Fig.2.5.1). This finding is relevant dueto the fact that a l 1 -minimization is a linear programming problem which can be solved byefficient computer algorithms, unlike the l 0 -minimization which is a NP-hard problem (Nondeterministicpolynomial time). Keeping this point in mind, it can be stated that randommeasurement matrices serve a double purpose: a) providing the easiest set of circumstancesunder which l 1 -minimization is provably equivalent to l 0 -minimization; and b) ensuring that theset of measurements vectors are as dissimilar to the sparsifying basis as possible. When it isgiven a random measurement of a sparse signal, , it generates a subspace of possible signals(green) that could have produced such a measurement. Within that subspace, the vector withsmallest l 1 -norm, is usually equal to [26].The strictest measure of sparsity is the l 0 -norm of the signal defined as the number of non-zerocoefficients of the signal. Unfortunately, the l 0 -norm is combinatorially complex to optimize andso CS enforces sparsity by minimizing the l 1 -norm of the reconstructed signal, which has beenprobed as an equivalent solution. Thereby, the minimization problem is summarized in theexpression:(7)A very important issue is that any real world sensor is subject to at least a small amount ofnoise, so in the cases in which this error margin can be approximated, it is recommendable tomodify the recovery algorithm in order to make the method stable and widely applicable,because small perturbations in the observed data should induce small perturbations in thereconstructed signal. So the expression above can be adapted by including an error ε andmaking consistent the reconstruction with the noise level:(8)36
It is worth mentioning that compressible signals are more realistic to consider than sparsesignals, and under this condition even the l 0 -minimizer does not match the signal exactly, sothere is no hope for the l 1 -minimizer to be correct.Mathematicians have developed new faster algorithms to solve the l 1 -minimization problem. Inthe [30] the main recovery algorithms are available to be downloaded.3. ECoG and AP Compressive Sensing System Design3.1. Power Consumption AnalysisIn Appendix B, two models with an analog and digital approach to be applied over a similarneural acquisition system are included. This comparison has been presented in [3], and it canbe considered the most significant example of power analysis of both, analog and digitalapproach, to the CS problem for neural signals compression in the limited existing literature.Due to this fact, this has been taken as the main reference to study the power consumption of amultichannel acquisition system and figure out which are the blocks susceptible to me improvein terms of power saving. As the bound of this project is to achieve a novelty in the analog pathimplementation for a CS system, the power analysis is focussed in the calculation considered inthe analog implementation shown in Appendix B. These equations has been inferred by takinginto account that as we are integrating over N samples, the instantaneous voltage on theintegrator can be expected to grow by an average factor of √N and cannot be allowed to exceedthe available maximum differential ADC input range. This condition can be summarized in Eq.9:(9)By applying this condition, the resulting power gain becomes NG 2 A instead of G 2 A . However,contrary to what is stated in [3], as sparse signals are integrated, the growth factor of √N is anoverestimation of how the integrated signal can increase, and consequently the total gain canbe approximated depending on the output noise at the output of the integrator and the total ADCquantization noise which can be allowed in the path. That is expressed in Eq. 10:(10)By substituting this condition in the total power consumption estimation for a complete analogpath:(11)37
Page 1: August, 31 th , 2012Compressive Sen
Page 5: The present project has been submit
Page 9: AbstractThe project “Compressive
Page 13: RiassuntoIl progetto “Compressive
Page 17 and 18: IndexAcknowledgments 7Abstract 9Gan
Page 19 and 20: E.2. Signal Digital Conversion 84Gl
Page 21 and 22: Figure 6.2.1.2. Input signal Spectr
Page 23: Index of TablesTable 2.1.1. Summary
Page 26 and 27: Figure 1.1. Energy a power costs fo
Page 29 and 30: 2. State of the art2.1 Neural Signa
Page 31 and 32: patterns in response to visual targ
Page 33 and 34: 2.4. Compressive Sensing2.4.1. Comp
Page 35: 2.4.2. Sparsifying basesSparsifying
Page 39 and 40: y specially considering the integra
Page 41 and 42: Figure 4.1.1.1. LSFR parallel (top)
Page 43 and 44: 4.1.3. Serial Implementation with t
Page 45: 4.1.4. Randomness CheckingIn order
Page 48 and 49: een designed independently as it is
Page 50 and 51: time, which has been approximated a
Page 52 and 53: Figure 5.2.2. Original and reconstr
Page 54 and 55: 6. Analog Path Design6.1. Design Di
Page 56 and 57: The spectra of the input signal hav
Page 58 and 59: Figure 6.2.1.4. LASSO reconstructio
Page 60 and 61: Figure 6.2.2.3. Compressed signals
Page 62 and 63: In this way, the pole frequency is
Page 64 and 65: A can be floating when the mixing s
Page 66 and 67: Figure 6.2.4.4. BPDN method reconst
Page 68 and 69: Figure 6.2.5.3. BPDN method reconst
Page 71 and 72: 7. Conclusions7.1. RemarksAs it is
Page 73: AppendixAppendix AConvexOptimizatio
Page 76 and 77: The total power cost for the digita
Page 78 and 79: Figure B.2.2.1. Block diagram for a
Page 80 and 81: 80Figure B.2.3.2. Binary-weighted S
Page 82 and 83: The main model is shown in Fig.C.1.
Page 84 and 85: Appendix EE.1. AmplificationThe sma
Page 87 and 88:
GlossaryADC: Analog-to-Digital Conv
Page 89 and 90:
References[1] D. Gangopadhyay et al
Page 91:
[35] M.F. Duarte et. al., Recovery
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Compressive Sensing system for recording of ECoG signals in-vivo

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