Signal Matched Multirate Filter Bank Design for Optimum Coding Gain and its Application in Real-Time Sleep Apnea Detection

Abstract

Various authors have designed multirate filter banks for modeling a given process. The existing algorithms, however, suffer from some major drawbacks; for example, some of these filter banks are not signal adaptive; in most of the cases the, filter computations are not efficient; moreover, the existing signal-adapted filter banks is not always optimal. To address these limitations, we propose an optimal signal matched filter bank (SMFB) to maximize the coding gain. The objective is to achieve spectral majorization and complete decorrelation of the sub-band signals. The filter bank is designed here to perfectly reconstruct only the given signal, instead of every signal from the finite energy signal space, as is typically done. These objectives are achieved by whitening the outputs of the analysis filter bank, within as well as across the channels. With these orthogonal whitened signals as inputs, the synthesis filter bank is designed to reconstruct a delayed version of the given signal. A time and order recursive least squares algorithm has been developed for the SMFB, with a single realization of the signal. The recursions of the algorithm lead to a lattice structure. Since the filter coefficients are not directly available from the lattice, a fast algorithm is developed to estimate the same using the lattice parameters. Simulation results are presented to verify the efficacy of the proposed theory. We also present an automated real-time sleep apnea detection algorithm using SMFB which gives a $92.27\%$ accuracy, $88.60\%$ sensitivity, and $94.31\%$ specificity on a publicly available MIT PhysioNet Apnea-ECG dataset.

Appendices

Proof of Lemma 1: Constrained Projection

Proof

The prediction error ${\epsilon }$ is defined as

$$\begin{aligned} {\epsilon }={x}-\sum _{i=1}^{n}{{h_{i}}{s}_{2i}}. \end{aligned}$$

(30)

Since ${\epsilon }$ is confined to the orthogonal complement space of $S_1$ (from condition 1), we can write

$$\begin{aligned}&{\epsilon }={\epsilon } \mid S_1^{\perp },\nonumber \\&\quad ={x}\mid S_1^{\perp }-\sum _{i=1}^{n}{{h_{i}}({s}_{2i}\mid {S_1^{\perp }})}. \end{aligned}$$

(31)

x can be written as

$$\begin{aligned}&{x}={x} {\vert }_{{S_1}}+ {x} {\vert }_{{S_1^{\perp }}}, \end{aligned}$$

(32)

$$\begin{aligned}&{x} {\vert }_{{S_1^{\perp }}}={x}-{x} {\vert }_{{S_1}}. \end{aligned}$$

(33)

The projections in (31) can be written as

$$\begin{aligned}&{x}\mid S_1^{\perp }={x}-\sum _{k=1}^{p}{{a_{k}} {s}_{1k}}.\end{aligned}$$

(34)

$$\begin{aligned}&\quad \text{ Similarly } \quad {s}_{2i}\mid S_1^{\perp }={{{s}}_{2i}-\sum _{k=1}^{p}{b_{ik} {s}_{1k}}}. \end{aligned}$$

(35)

Substituting the above values in (31), we get

$$\begin{aligned} {{\epsilon }}=\left( {x}-\sum _{k=1}^{p}{{a}_{k}{s}_{1k}}\right) -\sum _{i=1}^{{n}}{{h}_{i}\left( {{s}}_{2i}-\sum _{k=1}^{p}{ b_{ik} {s}_{1k}}\right) }. \end{aligned}$$

(36)

$\square$

The projection error, as given by (36), represents the error in projecting x onto the $S_2$ when it is constrained to lie in the orthogonal complement space of $S_1$.

Notations

The data vector representing a discrete time signal is denoted by ${\mathbf {x}}(n)\in {\mathbb {R}}^{1\times L}$, where $L>>n$, as given below:

$$\begin{aligned} {\mathbf {x}}(n) \equiv \left[ \begin{array}{ccccccc} 0&\cdots&0&x(0)&x(1)&\cdots&x(n) \end{array}\right] . \end{aligned}$$

The $1 \times L$ vector which is obtained by first introducing a delay of i to x(n) and then downsampling by M is denoted as ${\mathbf {x}}(Mn-i)\in {\mathbb {R}}^{L}$, given as follows:

$$\begin{aligned} {\mathbf {x}}(Mn-i) \equiv \left[ \begin{array}{cccccccc} 0&\cdots&0&x(M-i)&x(2M-i)&\cdots&x(Mn-i) \end{array}\right] . \end{aligned}$$

(37)

The set of N vectors, $\{ {\mathbf {x}}(Mn-k)|i \le k \le i+N-1\}$, form a $N \times L$ matrix, denoted as $X_N^{Mn-i}$, and is given as follows:

$$\begin{aligned} {X}_{N}^{Mn-i}\equiv \left[ \begin{array}{c} {\mathbf {x}}(Mn-i)\\ {\mathbf {x}}(Mn-i-1)\\ \vdots \\ {\mathbf {x}}(Mn-i-N+1) \end{array}\right] .\end{aligned}$$

(38)

These notations are used in the development of the required geometrical framework for the proposed least squares algorithm.

Update Formulas Used in this Paper

Inner product update relation, [14], has been used extensively in the development of the proposed fast recursive least squares algorithm. We provide a brief discussion on the development of this formula, using notations from Appendix B.

The Span $\{X^{n}_{p}\}=$ Span$\{X^{n}_{p-1},{\mathbf {x}}(n-p)P^{\perp }\left[ X^{n}_{p-1}\right] \}$. In terms of projection operator, we can write the following:

$$\begin{aligned} P[X^{n}_{p}]=P[X^{n}_{p-1}] + P[{\mathbf {x}}(n-p)P^{\perp }[ X^{n}_{p-1}]]. \end{aligned}$$

(39)

Expanding the projection operator, corresponding to the second term on the R.H.S. of (39), we obtain

$$\begin{aligned}&P\left[ X^{n}_{p} \right] = P\left[ X^{n}_{p-1} \right] + P^{\perp }\left[ X^{n}_{p-1} \right] {\mathbf {x}}(n-p)^T\left[ {\mathbf {x}}(n-p)P^{\perp }[X^{n}_{p-1}]{\mathbf {x}}(n-p)^T\right] ^{-1}\nonumber \\&\quad {\mathbf {x}}(n-p) P^{\perp } \left[ X^{n}_{p-1} \right] , \end{aligned}$$

(40)

and the update formula for the projection operator corresponding to the orthogonal complement space can be written as

$$\begin{aligned}&P^{\perp }\left[ X^{n}_{p} \right] \triangleq I-{} P\left[ X^{n}_{p} \right] \nonumber \\&\quad = P^{\perp }\left[ X^{n}_{p-1} \right] - {} P^{\perp }\left[ X^{n}_{p-1} \right] {\mathbf {x}}(n-p)^T\left[ {\mathbf {x}}(n-p) {} P^{\perp }[X^{n}_{p-1}]{\mathbf {x}}(n-p)^T\right] ^{-1} \nonumber \\&\quad {\mathbf {x}}(n-p) {} P^{\perp } \left[ X^{n}_{p-1} \right] . \end{aligned}$$

(41)

The inner product update relation, as given in [14], can be obtained by pre-multiplying (41) by ${\nu }$ and post-multiplying by ${\mathbf {w}}^{T}$, as given below:

$$\begin{aligned}&{\nu }{} P^{\perp }\left[ X^{n}_{p} \right] {\mathbf {w}}^T = {\nu }{} P^{\perp }\left[ X^{n}_{p-1} \right] {\mathbf {w}}^T- {\nu } {} P^{\perp }\nonumber \\&\quad \left[ X^{n}_{p-1} \right] {\mathbf {x}}(n-p)^T\left[ {\mathbf {x}}(n-p) {} P^{\perp }[X^{n}_{p-1}]{\mathbf {x}}(n-p)^T\right] ^{-1}\nonumber \\&{\mathbf {x}}(n-p) {} P^{\perp } \left[ X^{n}_{p-1} \right] {\mathbf {w}}^T. \end{aligned}$$

(42)

Along with (42), update formulas as proposed in [41] and [42], have also been used in the development of the proposed algorithm. For more information, refer [41] and [42]. For a quick reference, these relations are given as follows:

Corollary D

from rm [41] Let ${\nu }$, ${\mathbf {w}}$ and ${\mathbf {v}}$ be row vectors $\in {\mathbb {R}}^{M}$ and $\left[ {} {X_{1:n}}\mid {\mathbf {v}}\right]$ is an augmented set defined as follows:

$$\begin{aligned} \left[ {} {X_{1:n}}\mid {\mathbf {v}}\right] =\left[ \begin{array}{ccccc} {{\mathbf {x}}_{1}}&{{\mathbf {x}}_{2}}&\cdots&{{\mathbf {x}}_{n}}&{\mathbf {v}} \end{array}\right] ^T. \end{aligned}$$

(43)

An update relation, given as Corollary 1.1 in [41], is stated as follows:

$$\begin{aligned}&{\nu }{} P^{\perp }\left[ {\mathbf {v}}\right] {} P^{\perp }\left[ X_{1:n}P^{\perp }\left[ {\mathbf {v}}\right] \right] {\mathbf {w}}^T ={\nu } {} P^{\perp }\left[ X_{1:n}\right] {\mathbf {w}}^T- {\nu } {} P\left[ {\mathbf {v}} {} P^{\perp }\left[ X_{1:n}\right] \right] {\mathbf {w}}^T. \end{aligned}$$

(44)

Pseudo-Inverse Update Relation from [42]:

Let ${\mathbf {z}}$ and ${\mathbf {v}}$ be row vectors belonging to ${\mathbb {R}}^{M}$ and ${} X_{1:n}$ is the matrix as discussed above. Consider, the term ${} K_{n}[v]$ defined as follows:

$$\begin{aligned}&{K_{n}[{\mathbf {v}}]={} X_{1:n}^{T}[{\mathbf {v}}][{} X_{1:n}[{\mathbf {v}}]{} X_{1:n}^{T}[{\mathbf {v}}]]^{-1}}, \end{aligned}$$

(45)

$$\begin{aligned}&\quad \text{ where, }\nonumber \\&X_{1:n}[{\mathbf {v}}]\triangleq \left[ {}{(P^{\perp }[{\mathbf {v}}]{{\mathbf {x}}}_1^T)}\ldots {}{(P^{\perp }[{\mathbf {v}}]{{\mathbf {x}}}_n^T)} \right] ^{T},\end{aligned}$$

(46)

$$\begin{aligned}&K_{n}[0]=X_{1:n}^{T}[X_{1:n}X_{1:n}^T]^{-1}. \end{aligned}$$

(47)

${} K_{n}[{\mathbf {v}}]$ is the pseudo-inverse of ${} {X_{1:n}[{\mathbf {v}}]}$. An update relation for ${}{K_{n}[{\mathbf {v}}]}$, as given in [42] is:

$$\begin{aligned}&{{\mathbf {z}} {} K_n[{\mathbf {v}}]}=[{{\mathbf {z}} {} K_{n-1}[{\mathbf {0}}]}\mid 0] +{{\mathbf {z}} {} P^{\perp }[X_{1:n-1}] {x}_n^T} {[{x}_n {} P^{\perp }[X_{1:n-1}] {x}_n^T]^{-1}} .{[(-{x}_{n} {} K_{n-1}[{\mathbf {0}}])}\mid 1] \nonumber \\&\quad +{{\mathbf {z}} {} P^{\perp }[X_{1:n}] {\mathbf {v}}^T[ {\mathbf {v}} {} P^{\perp }[X_{1:n}] {\mathbf {v}}^T]^{-1}}. {(-{\mathbf {v}} {} K_n[0])}. \end{aligned}$$

(48)

It can be observed from the above update equation that the first two terms on the right-hand side update the matrix ${K_{n-1}[0]}$ to ${K_{n}[0]}$ and the last term updates ${K_{n}[0]}$ to ${K_{n}[v]}$. For more information and derivation of (48), refer [42].