## 3 Technologies: Internet of Medical Things
### Algorithms for IoMT Edge Devices  
#### 3.2.1 Sub-Nyquist Sampling  
Compressive sensing, or sub-Nyquist sampling, is an effective signal processing technique that has been used to reduce the power consumption for the processing of ECG signal (signal acquisition and compression) [28]. It has been advocated to reduce the sampling rate for redundant signals by taking advantage of the signal sparsity. Mathematically, given a signal \(x\) of dimension \(N\), that is \(x\in R^{N}\), the compressive sensing seeks to transform the signal \(x\) into a signal \(y\) with \(L\) non-zero elements, referred to as L-sparse (cf. (1)).  
\[y=Ax, \tag{1}\]  
where \(A\in R^{M\times N}\) is the compressive sensing matrix that should satisfy the restricted isometry property shown in (2).  
\[(1-\delta_{L})||x||_{2}^{2}\leq||y||_{2}^{2}\leq(1+\delta_{L})||x||_{2}^{2}, \tag{2}\]  
where \(\delta_{L}\in(0,1)\) is an isometry constant.  
The reconstruction of the signal is achieved by solving the convex optimization problem illustrated in (3)  
\[\begin{split}\underset{X}{\text{minimize}}&||x||_{2} \\ \text{subject to}&||Ax-y||_{2}\leq\delta,\end{split} \tag{3}\]  
where \(\delta\) is an arbitrary small and positive number.  
Figure 4: The PULP cluster with four cores in [26].  
There are three families of algorithm that solve (3): Thresholding, Greedy, and \(\ell_{1}\)-minimization algorithms. Table 5 categorizes these algorithms. Details of those algorithms are reported in [29].  
Compressive sensing, CS, has been recently implemented in a plethora of medical wearables. In [30], the authors implemented the CS algorithm for real-time compression of ECG signals using a wearable device. The scheme is based on three processing stages. In the first stage is linear transformation, the second stage is specification, and the final one is encoding. Three types of compressed sensing matrix were evaluated in terms of latency and percentage root-mean square difference (PRD). The first matrix was generated using quantized normal number generation, the second one was generated using a pseudorandom matrix obtained by processing one stored random vector, and the last one (sparse binary) used a sub-Gaussian random matrix that satisfies the \(RIP_{p}\) property. To further reduce the redundancy between packets, Huffman coding with a codeword of size 512 was used. The reconstruction algorithm was implemented on the desktop PC. The reported results showed that the compressive sensing algorithm dissipated 37.1% less energy compared with the discrete wavelet transform.  
In [31] an IoT platform is proposed for the real-time monitoring of an ECG signal. The platform is composed of a wearable device that collects the ECG signal, compresses it using the CS algorithm, and sends this wirelessly to an IoT edge device for reconstruction, classification, and analysis. The authors considered two reconstruction algorithms: subspace pursuit and orthogonal matching pursuit. The edge devices were designed using a commercial processor that has eight cores distributed as follows: a cluster of four high-performance cores and a cluster of four low-power cores.