Cyber-Physical Systems: An Information-Theoretic Framework for Joint Recovery of Sparse Source Biosignals

—The aim of this study is to propose an information- theoretic framework that can be used for joint recovery of sparse source biosignals. The proposed method supports medical cyber- physical systems (CPS) that enhance the detection, tracking, and monitoring of vital signs via wearable biosensors. Speciﬁcally, we address the problem of sparse signal recovery and acquisition in wearable biosensor networks, where we develop an adaptive design methodology based on compressed sensing (CS) and independent component analysis (ICA) to reduce and eliminate artifacts and interference in sparse biosignals. Our analysis and examples offer a low-complexity algorithm design for patient monitoring systems, where sparse source biosignals can be recovered at low hardware costs and power consumption. Also, we show that, under noisy measurement conditions, the joint CS-ICA recovery algorithms can outperform standard CS methods, where a sparse biosignal is retrieved in a few measurements. By implementing the joint sparse recovery algorithms, the error in reconstructing sparse biosignals is reduced, and a digital-to-analog converter operates at low-speed and low-resolution.


Cyber-Physical Systems: An Information-Theoretic
Framework for Joint Recovery of Sparse Source Biosignals Yahia Alghorani, Member, IEEE, and Salama S. Ikki, Senior Member, IEEE.
. Abstract-The aim of this study is to propose an informationtheoretic framework that can be used for joint recovery of sparse source biosignals. The proposed method supports medical cyberphysical systems (CPS) that enhance the detection, tracking, and monitoring of vital signs via wearable biosensors. Specifically, we address the problem of sparse signal recovery and acquisition in wearable biosensor networks, where we develop an adaptive design methodology based on compressed sensing (CS) and independent component analysis (ICA) to reduce and eliminate artifacts and interference in sparse biosignals. Our analysis and examples offer a low-complexity algorithm design for patient monitoring systems, where sparse source biosignals can be recovered at low hardware costs and power consumption. Also, we show that, under noisy measurement conditions, the joint CS-ICA recovery algorithms can outperform standard CS methods, where a sparse biosignal is retrieved in a few measurements. By implementing the joint sparse recovery algorithms, the error in reconstructing sparse biosignals is reduced, and a digital-toanalog converter operates at low-speed and low-resolution.
Index Terms-cyber-physical systems, patient monitoring systems, joint recovery for sparse biosignals.

I. INTRODUCTION
R ECENT developments in patient monitoring systems have led to important applications in the field of biomedical engineering. Wearable devices integrated with cyber-physical systems (CPS) make medical monitoring of patients with chronic conditions easier and more consistent that allow clinicians and caregivers to monitor patients remotely and provide feedback to help maintain optimal health status, regardless of patient location. These systems are increasingly being used in hospitals and clinics to provide high-quality continuous care to patients in complex clinical scenarios [1]. Recent advances in wearable sensors make CPS a powerful candidate for real-time e-health monitoring that can be used across different monitoring areas, including homes, buildings, means of transport, etc. Unlike traditional embedded systems, CPS is typically designed as a network of realtime embedded computing interaction with physical elements that can provide the adaptability, autonomy, reliability and functionality of wearable biosensor networks. An essential part of the CPS is the internet of things (IoT) platforms that enable smart mobile healthcare services through which patient data is collected by wearable biosensors and clinical data aggregators This work was supported by Science Foundation Ireland (SFI) under grant 13/CDA/2199. Y. Alghorani and S. Ikki are with Electrical Engineering Department, Lakehead University, Thunder Bay, On, Canada. e-mail: yahia.alghorani@ieee.org, sikki@lakeheadu.ca Figure 1: CPS-enabled wearable biosensors platform for continuous patient monitoring, where vital signs are collected through wearable biosensors and then send to the data aggregator (e.g., smart watch/mobile device) which in turn transmits data flows to IoT-ambient sensors and medical servers for analysis (e.g., smart watches and mobile devices) to be transferred to ambient sensors and then stored in medical servers for monitoring health status. The IoT platform consists of a large number of ambient sensors that collect large amounts of data from patients in different locations and make them accessible to clinicians at any time for analysis (see, e.g., Fig.1). Big data analytics for CPS help clinicians predict illness, diagnosis, and treatment, resulting in improved quality of care and reduced cost. By exchanging medical records for patients between public and private hospitals, doctors and specialists will be able to predict where the patient is located on the spectrum of disease progression more accurately and quickly. Managing and monitoring vital signs [2], such as heart and respiratory rates, blood pressure, blood flow, blood glucose, body temperature, oxygen saturation, electroencephalogram (EEG), electrocardiogram (ECG), and more requires a CPSenabled wearable biosensor platform that can capture vital signs and biometrics, and deliver data from the patient to edge computing devices (e.g., IoT-ambient sensors), as well as to the CPS cloud, for medical analysis. The main challenge is to implement remote monitoring and tracking of patients in sensors and data acquisition/detection, i.e., when vital signs (biosignals) contain noise and artifacts. For example, ECG signals are affected by patient motion and they often suffer from low signal-to-noise ratio (SNR) resulting from motion artifacts and interference effects. Artifacts can be defined as distorted signals caused by internal or external sources, e.g., muscle movement, overlapping of data transmission (where inter-biosensor interference occurs within the same wearable biosensor network) or inter-network interference (where the wearable biosensor network interferes with other nearby wireless sensor networks operating in the industrial, scientific, and medical (ISM) radio bands [3]. Due to ECG artifacts, high data acquisition is invisible to wearable biosensors, resulting in inadequate diagnosis and treatment. With a complete set of n discrete-time samples of a biosignal, the design of physical sampling devices such as digital-to-analog converters (DACs) and analog-to-digital (ADCs) becomes more complicated for wearable biosensors. Another challenge of patient monitoring systems is related to the large energy consumption of wearable biosensors and data aggregators due to continuous monitoring; therefore, the goal of this work is to design and implement a low-complexity computation algorithm that can eliminate noise and artifacts in wearable biosensor networks at a lowcost hardware implementation and power consumption.
Over the past decades, the problem of sparse signal recovery and acquisition has been a subject of great interest to signal and image processing communities (see, for example [4][5][6]). In the context of digital signal processing, the Shannon-Nyquist sampling theorem states that a specific minimum number of samples is required to capture a band-limited signal when the signal is sparse in some statistical domain (time or frequency).To find a sparse recovery solution for patient monitoring systems, compressed sensing (CS) has emerged as a robust method for the reconstruction of sparse biosignals, which enables a significant reduction in the sampling rate [7]. When sensing sparse source biosignals (e.g., many coefficients close to or equal to zero), the CS method can be applied to perform compressed measurements to recover the full source biosignal, only if the sensing matrix satisfies the restricted isometry property (RIP). The compressed measures are much lower than the typical number of samples required by the Shannon-Nyquist criterion. However, almost all DACs/ADCs work under the same principle: they follow the Shannon-Nyquist theorem which needs the sampling rate to be at least twice the highest frequency.
The mutual interference between wearable biosensors (e.g., biosensor-to-biosensor interference) due to the overlapping of physiological data transmission can reduce the received signal strength, which may result in significant degradation of signal detection. That is, the presence of noise and interference in the sparse source biosignals requires an increase in the number of CS measurements to improve signal quality, making the resolution of the sampling devices high, i.e., high-complexity acquisition/detection systems. In order to reduce the number of noisy biosignal measurements and get a high-resolution biosignal, we perform independent component analysis (ICA). ICA is a computational method that separates a multivariate signal into independent subcomponents, and it is mainly used to remove artifacts from EEG recordings. The primary driving force behind the use of ICA in patient monitoring is that solutions required to address the data sparsity problem in the presence of noise and interference. Because of biosignal artifacts, high-resolution and high bit-depth DACs/ADCs are needed to restore sparse source biosignals. In this paper, we propose an innovative approach to address the problem of sparse signal recovery and acquisition in wearable biosensor networks. The approach proposed in the block diagram is summarized in Fig.2 (where the sparse physiological signals are detected/reconstructed by digital CS). Since ICA recovery algorithms are powerful tools even in noisy environments, we aim to use these algorithms to remove noise and artifacts from sparse source biosignals, so that the CS can reduce the number of measurements (i.e., sampling rate) needed to retrieve sparse source biosignals and use low-speed and low-resolution sampling devices (i.e., low-cost implementation and power consumption of patient monitoring systems). Our analysis shows that CS-ICA can perform better than conventional CS when the sparse source biosignal contains noise and artifacts, where the CS-ICA improves the quality of the reconstructed sparse source biosignal compared with the CS.

II. RELATED WORK A. CS
In recent years, there has been considerable focus on research on CS applications in wireless body sensor networks and telemonitoring purposes (see, for example [8][9][10][11]), where the CS hardware architecture has been divided into two main methods: the analog and digital CS. For instance, in [8], the digital CS method (where the linear CS compression is applied after the ADC) has been used to recover fetal ECG signals. In [12] and [13], the analog CS method has been applied to ECG, where the compression occurs in the analog sensor readout electronics prior to the ADC. In spite of the fact that the analog CS method reduces the cost and power consumption of sampling devices compared with the digital CS method, its demonstration still requires extensive work on the analog Sparsity is defined as a small number of the largest frequency coefficients k hat capture most of the information contained in a sparse source biosignal. sensor read-out electronics. Therefore, in the current work, we propose to address it to through the digital CS (as described in Fig.2).

B. ICA
In medical applications, it is common practice to use the igital ICA algorithm to restore EEG signals (where the ADCs are located at the reciever, the data is processed in the digital domain [14], [15]), which requires high-speed sampling rate (due to the use of full measurments, mn) to reconstruct the sparse source biosignal, resulting in high-cost and high-power devices. On the other hand, few studies have discussed the analog VLSI implementation of ICA algorithms, see [16] for intelligent hearing aids applications, where the low-energy ICA architecture has been proposed in a noisy and reverberant environment, and experiments have shown a clear separation and precise localization of two speech sources.
However, both CS and ICA methods have some limitations when used alone, and pre/post-processing techniques must be adopted. Some studies have recently discussed joint CS-ICA recovery for EEG signals in the context of a wireless body sensor network; see, e.g., [17][18][19], where the CS method is preceded by the ICA data processing method, for which the proposed method offers significant energy savings compared to other state-of-the-art methods. In other words, the CS-ICA algorithms can provide a much lower compression ratio than when using CS alone. It should be noted that none of these studies discussed the joint detection/reconstruction of EEG whether in the analog domain or the digital domain. In order to bridge the gap between the CS and ICA methods in interbiosensor interference scenarios, in this work, we provide a new analysis of CS-ICA schemes from an information theory perspective, which to the best of our knowledge, has not been studied before. The proposed framework enables efficient implementation in CS hardware as well as provides valuable guidelines for communications engineers/researchers working on the design of joint sparse recovery algorithms for wearable biosensor networks. Specifically, the paper's contributions are summarized as follows: • Developing low power transceivers through digital CS-ICA algorithms, as we address the problem of sparse signal recovery for biosensor-to-biosensor interference scenarios and improve the sampling rate of digital CS methods up to m ≥ 2 log 2 n K / log 2 (M.SN R). • Unlike conventional CS recovery methods, the CS-ICA algorithms reduce sparse signal reconstruction errors, where joint sparse recovery algorithms provide a low mean squared error compared to standard CS methods (where the noise power in the collected data is reduced to up M ) when reconstructing the sparse source biosignals.

III. PROBLEM FORMULATION
In order to develop a smart medical CPS-based wearable biosensor platform, we outline the steps of CS used in noisy measurements, where source biosignals are sparse in the time domain (i.e., the K-sparse vector s K contains K non-zero elements and satisfies s l0 ≤ K n). If the source biosignal s ∈ R n is not sparse, we can make it sparse through CS, using an inverse discrete cosine transform (IDCT) matrix to produce a sparse vector, that is, x = Ψs, where Ψ n×n is a unitary matrix that can discard the small frequency coefficients of s (i.e., many frequency coefficients are set to zero via Ψ after adding a quantization step to the IDCT). With CS, we can employ low-speed DACs (i.e., sub-Nyquist sampling rates) to recover the sparse biosignals and reduce power consumption and cost of wearable biosensors/data aggregators.

A. CS Acquisition
In e-health monitoring applications, where interference can occur between N wearable biosensors (see, e.g., Fig.3), the receiving sparse source signal for each biosensor, y r ∈ R M ×n , r ∈ {1, 2, . . . , N }, at the M -sensor array (mixtures) of data aggregator (where M ≥ N , is expressed as where x i = Ψ i s i is n-pixel image/video/radio pulses), that is corrupted by additive white Gaussian noise (AWGN) n r ∈ R M ×n , h r ∈ R M ×1 is a constant channel vector with array elements h ij (which depends on the distance between the i -th biosensor and the j-th sensor node of the data aggregator. The received signal is then processed by a digital combiner w r ∈ R 1×M to obtainx r = w r y r , which can be rewritten aŝ where w r = h T r . After extracting the desired source biosignal, x r , and using the CS, the received signal is expressed as z r = Φ rxr , namely where z r ∈ R m is the measurements vector for the biosensor r, A r = Φ r Ψ r is the sensing matrix, Φ r ∈ R m×n is the measurement matrix (where n m K), and v r ∈ R m is the effective noise vector that contains the noise and interference component. Here the sensing matrix A r ∈ R m×n obeys , only if δ K is not too close to one and its entries are drawn from a suitable distribution, e.g. a Gaussian distribution [7]. So, if the RIP of order K is established, then it is sufficient to have m = O (K. log 2 n/K)measurements.

B. CS Recovery
In order to recover s r from z r , we consider the following optimization problem where v r l2 ≤ is the maximum noise power, s r p stands for the standard l p -norm and s r l0 counts the number of nonzero elements in s r . If we assume that δ 2K < √ 2 − 1 , the solution to the convex problem above satisfies where the constants C 0 and C 1 are typically small [20, eq.(14)]. The result in (5) indicates that the noise power in the data can significantly increase the reconstruction errors, which means that we need more measurements (m-samples) to recover the sparse source vector s r . An equivalent expression has been determined by [21], ŝ r −s r l2 < v r l2 + /1−δ 2K . Note that when δ 2K approaches one, the reconstruction process becomes infeasible because of the high error rate. It should be noted that the RIP works well when there is no noise, where the reconstruction of the sparse source biosignal is more accurate (i.e., ŝ r − s r l1 ≤ C 0 . s r − s r,K l1 , [7, eq. (10)]). Although the CS technique solves the sparse data acquisition problem, there are still some conditions and limitations that should be considered when dealing with scenarios of noise and interference (i.e., it is impossible to construct the sparse vector s r when the effective noise level, v r , is high). Therefore, the precise calculation of the digital combiner elements w r is important to reduce the high noise level of v r so that the sparse source biosignal is retrieved in a few measurements.
IV. PROPOSED FRAMEWORK ICA was originally developed as a computational method that can be used in various applications including medical signal and image processing (with a non-Gaussian input) [22], where a multivariate biosignal decomposes into independent additive subcomponents without prior knowledge of source biosignals or mixing coefficients. In other words, if we have a mixture of N independent source biosignals, the ICA algorithm works to find the unmixing matrix W ∈ R N ×M , to extract the source biosignal and remove artifacts from EEG recordings (see, e.g., Fig.2, where N source biosignals are collected by a wearable biosensor network and compressed by a digital CS model to be transferred to the data aggregator (e.g., smart watch/mobile device) where the ICA scheme (equipped with M mixtures) is performed to extract the source signal of interest and then decompressed by the digital CS model. Using CS-ICA joint recovery, we can employ low-speed and lowresolution DACs (i.e., sub-Nyquist sampling rates and low bitdepths) to restore sparse source biosignals and reduce power consumption and cost of wearable biosensors/data aggregators. Linear receivers such as zero-forcing and minimum mean squared error equalizers can offer a good trade-off between performance and complexity. The main drawbacks of these receivers are 1) Any defects in the estimation of channel and covariance matrices in a large network of biosensors result in residual interference in the data aggregator output which in turn leads to performance degradation. 2) Increase the number of iterations needed for convergence. Therefore, the ICA algorithm is suggested as an alternative if the channel state information is not known where a fast convergence multi-biosensor detection scheme is achieved by extracting all independent components directly. Suppose now that the data aggregator observes the mixture signal y ∈ R M ×1 in the time domain (where the discrete-time representation of a continuous-time biosignal x (t), which lies at the heart of analogue to digital conversion [23]), determined by where n ∈ R M ×1 is the Gaussian noise vector, H ∈ R M ×N is the unknown mixing matrix of source biosignals s ∈ R N ×1 , and x (t) = Ψs (t) is the sparse vector, such that T Assuming the independence of source biosignals s i ∈ R n , the joint probability density function (PDF) of continuous random variables P (X 1 , . . . , X N ) is factorized as where x i (t) = Ψ i s i (t), and Ψ i ∈ R n×n is the unitary matrix generated by the IDCT. We define t as a time index, where 1 ≤ t ≤ T . The digital ICA algorithm estimates the sparse vector x ∈ R N ×1 using the unmixing matrix W , that is,x (t) = W y (t) , where W = H T is approached by maximum-likelihood estimation or FastICA algorithms [22]. Also, to eliminate the noise components W n (t), we use an adaptive filter (e.g., Least Mean Square (LMS) methods) to accomplish this task. In order to maximize the statistical independence of extracted components (i.e., source biosignals), we need to minimize the mutual information of estimated signalsx i and maximize non-Gaussianity, i.e., I X 1 ; . . . ;X N = H X i − H X 1 , . . . ,X N , where the differential entropy H X i of a continuous random variableX i with a density function p (x i ), is defined as [24, eq. (8.1)] where S is the support set of the variableX i . Now let X = W Y ; using the differential entropy property: H (W Y ) = H (Y ) + log |W |, we obtain In order to measure the difference in entropy between a given distribution and Gaussian distribution (containing the highest entropy) with the same mean and variance, we use negentropy which is defined as J H X g is the entropy of the Gaussian distribution with unit variance for all estimated sparse source biosignalsx i . In information theory and statistics, minimization of mutual information between multiple random variablesX i is achieved by minimizing entropy H X i or maximizing negentropy J X i , which is also equivalent to minimizing Gaussianity. Therefore, finding the optimal unmixing matrix W will help to minimize the mutual information between the variablesX i and make the extracted sparse source biosignals uncorrelated (independent) and non-Gaussian. Once the ICA extracts the sparse source biosignals with improved SNRs, the CS senses and acquires each sparse source biosignal, so the number of measurements m is minimized, and the sparse biosignal is sampled by few measurements at a rate below the Nyquist rate.
Suppose the source biosignals are independent and identically distributed (i.i.d) random variables (i.e., all source biosignals X i have the same probability distribution and are mutually independent), then the source biosignal observed by the CS model can be expressed as where u i (t k ) = Φ ixi (t k ) is the received sparse biosignal vector,x i (t k ) is the estimated sparse vector that corresponds to the i-th row of W y (t k ), and v i (t k ) is the i-th residual gaussian noise vector after filtering the noise component W n (t k ). Note that t k , k ∈ {1, 2, . . . , m} are the time instants when samples (m-snapshots) are taken. In order to reconstruct the source biosignal x i , we fix Ψ i and pick up the input of the i-th sampling matrix Φ i randomly from non-Gaussian distribution with zero mean and variance 1/m (e.g., sub-Gaussian distribution [25]; which typically has a flat PDF with a strong tail decay property, so that E e −λu ≤ e −λ 2 ρ 2 /2 , for all λ ≥ 0 (11) where E [U ] = 0, Var [U ] ≤ ρ 2 , and the sensing matrix [A i = Φ i Ψ i ] satisfies the RIP condition (i.e., m ≥ c.K. log 2 n/K, where c is a positive constant). Here, the digital CS-ICA algorithm takes advantage of the RIP property by providing minimum and maximum power for the samples, ensuring that the K-sparse vectors do not fall into the null space of the sampling matrix Φ i , which in turn provides a stable recovery for the sparse source biosignals.

V. GENERALIZED RATE OF THE DIGITAL CS-ICA ALGORITHM
The joint recovery algorithm works to reduce interference and noise, where the generalized rate R i of independent sub-Gaussian random variables, i.e, U ik ∼ N 0, ρ 2 ik , is calculated in the following way: Proposition: Suppose U m i = Figure 5: Gaussian Channel i are an m-tuple of the RVs U i and Z i , which are generated from the data compression process. Now consider m independent Gaussian channels in parallel, which we can send the non-Gaussian input u m i through m -AWGN channels, and get the output z m i = u m i + v m i , where the output of each channel is z ik = u ik + v ik , and both U ik and V ik , are independent RVs. By following [24], the mutual information between two RVs U m i and Z m i , is calculated as Proof : Since V i1 , V i2 , . . . , V im are independent, Given H (Z ik ) ≤ 1 2 log 2πe ρ 2 ik + σ 2 ik , and H (V ik ) = 1 2 log 2πeσ 2 ik (where ρ 2 ik = P ik . h T i h i 2 , P ik is the i-th source signal power received at each measurement point k, so the power constraint for each source biosignal is defined as 1 m m k=1 P ik ≤ P N , h i is the i-th column of the unknown mixing matrix H, and σ 2 ik = E V 2 ik is a small noise power in the data), we obtain with equality if Z ik are independent and sub-Gaussian RVs (where Z ik ∼ N 0, ρ 2 ik + σ 2 ik , and the SNR is obtained for the k-th sample as SN R ik = P ik /σ 2 ik . Note: In (13), assuming that the ICA mixtures have equalmagnitude of channel coefficients, then we obtain h T i h i 2 = M.|h ik | 2 .
Special cases: 1) Since the ICA algorithm mitigate interference between source biosignals, we assume that a) the received SNR is too high so that we can distribute equal amounts of energy across channels using a water-filling solution (i.e., P ik ≤ P/N ), and b) the magnitude of the channel coefficient is |h ik | 2 = 1 (i.e, Tr h T i h i = M ). Using Jensen's inequality, (13) can be expressed as where SN R = P/N σ 2 .
2) On the other hand, assuming that the mixture signal y contains Gaussian and non-Gaussian random variables, so that we can maximize the entropies H X i and minimize the negentropies J X i , hence, finding an ideal unmixing matrix W becomes impossible due to Gaussian increase, leading to a low SNR (e.g., log (1 + x) ≈ x log 2 e when x is sufficiently small); namely where SN R = P/N σ 2 3) If the number of measurements is large enough (i.e., m = n → ∞ ), by the strong law of large numbers, where Pr lim m→∞

A. Channel Coding Theorem
To extract the gain of digital CS-ICA schemes, we invoke the channel coding theorem for a discrete memoryless channel [24]. Let's say we connect through a noisy channel (U   bits are used to encode/compress the n-sample sequence S i = (S i1 , . . . , S in ), then the decoder (decompressor) does the inverse process. An 2 mR , m code for the channel U m i , p (z m i |u m i ) , Z m i consists of a compressing function U m and a decompressing function Z m (see Fig.4). The rate of the code 2 mR , m is defined by R = log 2 n K , with a probability of error P e = P r S i =Ŝ i < . So, if we assume that all entries of the sparse source vector S i are i.i.d random variables and the channel coefficients of the samples are defined as |h ik | 2 = 1, then the number of measurements for each source biosignal is calculated as where log 2 n K = 2H (K/n) = 2K log 2 n K bits when 0 ≤ K/n ≤ 1 [24]. Note that the ICA method brings an additional power gain for the CS method. The higher the SNR, the lower the number of measurements. Also, we can conclude that when the number of ICA mixtures is very large i.e., M → ∞, where SN R = P/σ 2 . By assuming that the sensing matrix A i satisfies the RIP property, i.e., , then the receiver can reconstruct the n-sample sequence, S i = S i1 , . . . , S in , in the digital domain by solving the convex problem By using the triangle inequality [14] and the theorem in (5) for v i l2 ≤ = σ 2 i , the solution to (15) obeys

VI. PERFORMANCE ANALYSIS
In order to gain further insight into the performance of digital CS-ICA algorithms, we display the error in retrieving source biosignals from noisy data as shown in Fig.6 (a) and (b), where the linear regression of modelling n data points is performed and each sparse biosignal is restored from noisy measurements (e.g., n = 1000, m = 250, and SNR = 25dB). Here, the simulation results show that the digital CS-ICA algorithms have fewer reconstruction errors than traditional digital CS methods; this is because the ICA algorithms can capture all n data points containing the Kfrequency coefficients and make the sparse source biosignal detectable at a high SNR, whereas traditional CS methods cannot guarantee whether the measured samples m contain all non-zero components K, which are randomly distributed across the sparse vector s i and are captured by increasing the number of measurements m. Fig.6 (c) compares the performance of both methods in terms of the mean squared error (MSE) with standard (digital) ICA algorithms, where the CS-circuit components are omitted and the ICA scheme must perform full measurements to create W , i.e., m = 1000. The results indicate that the performance of CS-ICA algorithms can be close to standard ICA algorithms for medium and high SNR regimes, both of which are roughly equivalent at low SNR values; which means that the digital CS-ICA algorithms can significantly reduce Gaussianity output because of the high level of additive Gaussian noise (i.e., v i l2 ≤ σ 2 i / h i , eq.(16)). According to the central limit theorem and the law of large numbers, when adding a large number of independent random variables (i.e., the number of wearable biosensors gets larger, N → ∞), the total tends to a Gaussian distribution even if the original variables (source biosignals) are not normally distributed. Since the CS methods have inherently Gaussian characteristics, they can be used instead of the ICA algorithms to retrieve sparse source biosignals. Furthermore, the results demonstrate that the joint CS-ICA recovery algorithms can perform better than traditional CS methods if tested under the same conditions (e.g., m = 300). In general, Fig.6 results can be treated as benchmarks for the CS-ICA algorithms. On the one hand, the ICA can increase the sparse biosignal strength received (by reducing artifacts and interference) and use lowresolution DACs to restore sparse source biosignals. On the other hand, the CS methods can reduce the sampling rate and make the DACs operate at low speed. This process is called an "exchange of interests" between CS and ICA methods. Fig.7 shows the joint CS-ICA recovery for two sparse source biosignals, e.g., EEG and electromyography (EMG) signals where N = 2, M = 2 , and the source biosignals are sparse in the frequency domain (e.g., the number of samples per source signal is n = 2500 and the number of spikes is K = 2) with various periods for each source biosignal (e.g., T = 1 second for the EEG signal s 1 (t) and T = 2 seconds for the EMG signal s 2 (t). The relatively few frequency coefficients capture most of the source signal energy with a small number of measurements (e.g., m = 170) taken in the time domain.

VII. EXPERIMENTAL RESULTS
This example obviously shows the gain acquired by the digital CS-ICA algorithms, where the sparse source biosignals can be restored in only a few measurements. Fig.8 compares the performance of CS-ICA algorithms and standard CS methods to two source ECG signals (operating in the 28 GHz band [26]). The mm-wave pulse train is sparse in the time domain (e.g., the pulse repetition period ( t) is 10 ns for the first source ECG signal and 3 ns for the second source ECG signal, N = 2), where each sparse source biosignal has the same frame size, for example, T = 30 ns for the sparse ECG signals s 1 (t) and s 2 (t). The reason is that we assume that the mmwave signals are sparse in the time domain rather than the frequency domain to simplify the construction of the IDCT matrix Ψ n×n (where n is very large) required to produce the sparse source vectors x 1 (t) and x 2 (t). Here, the sampling rate is calculated as f s = 4×28 giga samples/sec, and the total number of samples (data points) taken to perform a Fourier transform, is n = 3361. Also, in this example, assume that the wearable ECG sensors have line-of-sight (LOS) paths with the data aggregator sensors M = 2 (e.g., SN R = 20dB), where each sparse ECG signal is retrieved from a few measurements that are captured in the frequency domain (e.g., m = 300). In this setup, the mixing matrix elements of H ∈ R 2×2 are uniformly distributed and the unmixing matrix W = H T is calculated through the FastICA algorithms [22]. Our experimental results show that the digital CS-ICA algorithms can provide an accurate and rapid recovery of ECG signals compared to CS methods. Another practical example of realtime patient monitoring systems can be found in wearable magnetoencephalography (MEG) scanners which are used for mapping brain activity. Fig.9 illustrates a phantom recovery experiment that exactly mimics MEG images that can be contaminated with outside noise and artifacts. In this example, we restore a large image size of 256x256 pixels of noisy measurements (e.g., SN R = 15dB), where the sparse source biosignal of interest s 1 is extracted from an unwanted source biosignal s 2 (e.g., N = 2, M = 2). The sparse MEG image is recovered using the min-total variation (TV) method [27], with 25 non-zero frequency components distributed uniformly in the Fourier domain, where a few measurements are created to restore the desired MEG image (e.g., m = 6213). Compared with standard digital ICA methods, the joint CS-ICA recovery methods can reduce noise and number of measurements (e.g., from 65,536 to 6213 snapshots), allowing receivers to use lowspeed/low-resolution DACs to retrieve the sparse source MEG signal. Furthermore, the experiential results show that the digital CS-ICA algorithm can achieve high-resolution of MEG images compared to standard digital CS method. However, it should be noted that standard digital ICA algorithms have poor resolution of highly correlated brain sources.
VIII. CONCLUSION In this paper, we present an information-theoretic framework that can address the sparse signal recovery and acquisition problem in CPS-enabled wearable biosensor platforms.
The min-total variation (TV) solution is used for the large-scale 2Dstructure instead of min −l 1 , which ensures stable recovery for sparse MEG images with high accuracy. Using digital CS-ICA implementation in wearable biosensor devices, we can reduce the number of measurements, increase the noise robustness, and improve the accuracy and efficiency of standard digital CS methods. Our results showed that the CS-ICA algorithms can perform better than standard CS methods when a sparse source biosignal contains noise and artifacts, where we can recover K-sparse biosignals/images from just m ≥ 2 log 2 n K / log 2 (M.SN R) noisy measurements, thereby improving the quality of the reconstructed sparse source biosignals. Compared to standard ICA algorithms, the CS-ICA algorithms can reduce the sampling requirements for digital-to-analog converters as well as the computational complexity of recovery at mm-waves, so that we can reduce the power consumption of wearable biosensors and data aggregators, and retrieve sparse source biosignals in a few measurements.
The proposed sensing framework will have a significant impact on the healthcare sector by improving the efficiency, reliability and accuracy of patients' continuous monitoring systems, resulting in better patient diagnosis and treatment options. The sensing method will generally provide significant environmental and economic benefits to healthcare delivery systems, allowing clinicians to monitor patient conditions directly and reduce the burden of healthcare costs.