Compressive Sensing system for recording of ECoG signals in-vivo

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In order to have a clear understanding of the Compressive Sensing operation, in Fig.2.4.1.1 itcan be observed how compression takes places. As it is shown, the compression ratio can beachieved is C R = N/M, where N >> M.Figure 2.4.1.1. Sketch of Compressive Sensing operation when a) x is sparse in the identity basis; b) x issparse in the sparsifying bases Ψ.The CS premise is that under specific conditions, x can be efficiently and accuratelyreconstructed from y. In particular, this is possible if the measurement matrix Φ satisfies the K-Restricted Isometry Property (K-RIP), with constant δ K for all x Є Σ K , which is defined by theexpression [24, 25]:(6)The K-RIP ensures that all the submatrices of Φ of size M x K are close to an isometry, andtherefore distance and information preserving. Model-Based CS theory states that it is possibleto decrease M without sacrificing robustness by combining signal sparsity with structuraldependencies between the values and location of the signal coefficients. This model goesbeyond the K-RIP by establishing what has been called Restricted Amplification Property(RAmP) [22]. Summarizing, there are two key features are needed for implementing CS: a)sparsity of the sampled signal and incoherence between the sparsifying basis and b) themeasurement matrix, which ensures maximum information capture by the compression scheme.Random sensing matrices have a high degree of incoherence with sparsifying basis with highprobability [1]. Effective measurement matrix random entries are drawn from a variety ofpossible distribution, such a Bernoulli, Gaussian and Uniform Distribution. In this way, CSmethod leads to a simplification in the on-line signal acquisition phase against an increasingcomplexity in the off-line recovery of the original signal. Thereby, CS is an optimal solution inapplications in which the recording system has to be kept under restrictive limits of area andpower consumption within the chip, while the post-processing can be done out of the chip. Inthe applications in which peak amplitude and spikes location are more relevant than the exactmorphology, time-domain CS has been used after the signal has been dynamically thresholded.Similarly, frequency-domain CS can be achieved by applying a dynamic smoothing can be usedto limit the number of higher frequency components.34
2.4.2. Sparsifying basesSparsifying basis have been used are identity basis for time-domain sparse reconstruction whenthe signals which are recorded are already sparse in time-domain. The inverse Fouriertransform has been used for frequency-domain sparse reconstruction [1, 22]. The Gabor space[27] has been considered as sparsifying basis for EEG signals which are assumed to becomposed by short sinusoidal bursts. In the same way, wavelet-domain basis area good basischoice for EEG and ECoG signals which are considered as windowed, piece-wise smoothpolynomials with additive noise [34, 35, 36].Figure 2.4.2.1. Tree structure of 3-level decomposed wavelet coefficients.More concretely, in the case of wavelet basis, the wavelets use a multi-scale decomposition, i.e.the coefficients of the wavelet transform are generated in a hierarchical manner using scaledependentlow pass, h(n), and high pass, g(n), filter impulse responses. h(n) and g(n) arequadrature mirror filters, (see Fig.2.4.2.1), corresponding to the type of wavelet used [1].In the present work, the sparsifying basis which has been proposed for futures systemimprovements is the one based on Daubechies wavelets. Two important points have to beconsidered in order to create the WL basis: a) The number of samples and measurements arerecommended to be multiple of two, in order to construct a sparsifying basis whose coefficientspropagation (see Fig.2.4.2.1) perfectly fit with these dimensions; b) the number of WLcoefficients has to be selected regarding to the number of samples of the input signal, in anycase the sparsifying basis can be less sparse than the input signal; and c) in the operationshows in Fig.2.4.1.1, the basis has to be the inverse transformation, in order to respect.2.5. Reconstruction MethodsCompressed signals can subsequently be recovered by using a greedy algorithm or a linearprogram that determines the sparsest representation consistent with the acquiredmeasurements. The quality of the reconstruction depends on: a) compressibility of the signal; b)choice of the reconstruction algorithm; and c) incoherence between the measurement matrixand the sparsifying basis.35
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: 2.4. Compressive Sensing2.4.1. Comp
Page 37 and 38: It is worth mentioning that compres
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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