The FM-Mux, as it is depicted in Figure 1, was first presented by Ahmed, et al. [4]. It is an extension to the compressive multiplexer as it was proposed by Slavinsky, et al. [6] to achieve efficient compressed sensing of correlated signal ensembles. In this work, we adopted the proposed FM-Mux architecture to achieve sub-Nyquist sampling of multi-lead ECG signals. Therefore, we will give a short outline of the FM-Mux concept in principle. For a thorough mathematical analysis, the reader is referred to the original publication.

Compressed sensing of multi-lead ECG signals can be achieved by utilizing the FM-Mux as follows. First, each ECG signal *x*_{m}(*t*) is modulated with a binary waveform *d*_{m}(*t*) at rate *Ω* > *W*, where *W* is the bandwidth of the ECG signals. Second, the signals are convolved with linear time invariant (LTI) filters with impulse responses *h*_{m}(*t*) of bandwidth *Ω*. Finally, the signals are added together onto a single channel and uniformly sampled by an ADC at rate *Ω*.

The modulation waveforms *d*_{m}(*t*) randomly switch their value to ±1 at a rate of *Ω*. This embeds the signals *x*_{m}(*t*) into different subspaces of ℝ^{Ω} which facilitates their separation during reconstruction. Since the sampling process can be swapped with the signal addition, the modulation can equivalently be expressed as a multiplication of the signal samples at rate *Ω* with the modulation sequence. Thus, on the time interval *t* ∈ [0, 1) the modulation can be written as multiplication of signal *x*_{m}(*t* = *n*/*Ω*), where 0 *n* ≤ *n* ≤ *Ω* − 1, with the corresponding diagonal modulation matrix *D*_{m}, whose non-zero entries are [*d*_{m}(0), *d*_{m}(1),..., *d*_{m}(*Ω* − 1)]. The binary sequences *d*_{m}(*t*) can efficiently be generated, e.g., by linear feedback shift registers.

The LTI filters *h*_{m}(*t*) aim at dispersing the signal energy across time, making the FM-Mux effective for correlated signals that otherwise share a similar initial energy distribution. One can write the action of the LTI filter in the *m*th channel as an *Ω* × *Ω* circular matrix *H*_{m}, where

$${H}_{m}={F}^{*}{\widehat{H}}_{m}F,$$(1)with *F* and *F** as the discrete Fourier transform and its inverse respectively, and *Ĥ*_{m} as a diagonal matrix that holds the Fourier series coefficients of the LTI filter *h*_{m}(*t*) scaled by a factor of $\sqrt{\Omega}$.

Finally, after analog preprocessing, the ECG signals are added together and sampled uniformly at rate *Ω* which generates the samples of the compressed signal *y* ∈ ℝ^{Ω}.

The compressive sampling of an ECG signal ensemble by the FM-Mux can coherently be expressed by the set of equations:

$$\begin{array}{lll}y\hfill & =\hfill & {\displaystyle \sum _{m=1}^{M}{H}_{m}{D}_{m}{x}_{m}}\hfill \\ \hfill & =\hfill & [{H}_{1}D{}_{1},{H}_{2}{D}_{2},\dots ,{H}_{M}{D}_{M}]\cdot \text{vec}(X)\hfill \\ \hfill & =\hfill & [{\Phi}_{1},{\Phi}_{2},\dots ,{\Phi}_{M}]\cdot \text{vec}(X)\hfill \\ \hfill & =\hfill & {\Phi}_{\text{FM-Mux}}\cdot \text{vec}(X),\hfill \end{array}$$(2)where *Φ*_{FM−Mux} is the *Ω* × *MΩ* sensing matrix and *X* is the matrix composed of the discrete representation of *M* ECG leads sampled at rate, i.e.

$$X=[{x}_{1},{x}_{2},\dots ,{x}_{M}].$$The operator vec(*B*) returns a vector by stacking the columns of a matrix *B*.

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