IEEE Circuits and Systems Magazine - Q1 2020 - 50

In particular, the priors we consider concern supp (p ) =
" j ; p j ! 0 ,, i.e., the positions of the non-zero elements in
p, or the magnitudes of the non-zero elements in p.
1) The "dec-ZZ" Method in [8]
This approach assumes that the signal is not only sparse
but also block-sparse. Blocks are subvectors p [ j] of p
containing adjacent entries so that one may partition
<
<
p in b blocks p < = 6p [0] f p [b - 1] @. The signal is blocksparse if supp (p ) is always contained in the union of a
number of blocks % b.
Block-sparsity is a stronger prior than simple sparsity [42] and can be paired with Bayesian learning [8] to
yield effective reconstruction algorithms. The core idea
is to assume that each block follows a parameterized
multivariate Gaussian distribution p [ j] + N (0, c j R p [ j]) and
is independent of the other blocks so that p j + N (0, R p)
with R p = diag (c 0 R p [0], f, c b - 1 R p [b - 1]). With this model,
c j = 0 implies that the block does not cover supp (p )
and thus controls the block sparsity.
Decoding is then divided in two steps. In the first step
the parameters c j and R p [ j] are learnt for j = 0, f, b - 1.
In the second step they are used to decode p by means
of straightforward Maximum-A-Posteriori estimation
pt =

/ B < c v 2 I m + B / B < m-1 y
p

p

Depending on the different strategies for learning the

for example, D is a wavelet-like orthonormal basis that
decomposes the signal into a sequence of approximation/detail pairs whose typical decay is known by design
of can be identified.
This decay information can be plugged into BPDN
by altering the sparsity promoting norm from p 1 to
W -1 p 1 where W is the diagonal matrix aligning the
coefficients modeling the decay.
The method appears to be most effective when BPDN
is solved considering its lasso relaxation, i.e., in
mind ^1 - a h W -1 p
p!R

1

+ a y - Bp

2
2

where the parameter a administers the weight of the
two components in the relaxation.
3) The "dec-P" Method in [9], [44]
In OMP, the new columns of B to be inserted in the set
that is deemed to be necessary to reproduce the measurements are selected for their alignment with the
residual measurement. A prior of the kind used in the
previous method can alter this selection by altering
the opportunity of choosing a column depending on
the decay coefficient of the corresponding entry in p.
This can be done by changing Line 5: in Table I with
j = argmax k B <$ , k 6^1 - a h W -1 + aI d@ Ty , where the parameter a administers the trade-off between following
the correct decay and aligning with the residual.

c j and R p [ j], this approach gives rise to different meth-

ods that differ in computational complexity more than
in final performance.
2) The "dec-JZ" Method in [43]
In this case one assumes that, when p j ! 0 then its average magnitude varies with j. This is most natural when,

Table I.
Pseudo-code of OMP.
1: g ! [ ]

2 initialize the vector that will contain the
non-zero components of pt

2: J ! [ ] 2 initialize the vector that will contain supp ` pt j
3: repeat
Ty ! y - B ·,J g 2 B ·,J is the submatrix of B with
4:
columns indexed by J
j = argmax kuB ·<, k Tyu 2 column B ·,k of B that best
5:
matches the measurements residual
J ! [J j ] 2 include it in J
6:
2 re-estimate g by pseudo-inversion
7:
g ! B ·+,J y
8: until convergence
g k if j = J k
2 put non-zero components
9: pt j ! '
back into pt
0 otherwise
50

IEEE CIRCUITS AND SYSTEMS MAGAZINE

VI. Back to ECGs
The ECG signals we mentioned in the introduction offer the opportunity of testing the effect of adaptation at
both the encoder and decoder side. In fact, they can be
given an approximately sparse representation along, for
example, a Daubechies-6 wavelet basis D that has a dyadic scaling and thus induces a blockwise halving decay
in the typical magnitude of the coefficients.
We consider windows containing n = 512 samples taken at 360 sample/s from a classical procedure generating
clean and realistic ECG tracks [45] and superimpose a noise
vector o + N (0, 10 -6 I m) to the measurement vector y.
Figure 7 compares the performance of the three above
decoding strategies against non-adapted BPDN. In all
cases A is made of independent Gaussian random variables. Though with different profiles, all methods give
definite advantages over non-adapted BPDN.
With reference to the hearth monitoring device briefly sketched in the introduction, we may also evaluate
what could be the overall impact of adaptation both at
the encoder and decoder side.
Since we are dealing with a portable device in which
the resources for the computation of y are limited, we
constrain A = ! m -1/2.
FIRST QUARTER 2020
IEEE Circuits and Systems Magazine - Q1 2020

Table of Contents for the Digital Edition of IEEE Circuits and Systems Magazine - Q1 2020
