Journal of Signal and Information Processing, 2013, 4, 125-131
doi:10.4236/jsip.2013.43B022 Published Online August 2013 (http://www.scirp.org/journal/jsip)
125
Robust Low-Pow er Al gori t hm f or Ran d om S ensin g Mat rix
for Wireless ECG Systems Based on Low Sampling-Rate
Approach
Mohammadreza Balouchestani*, Kaamran Raahemifar, Sridhar krishnan
Electri cal and C omputer Engineering Depart ment , Ryerson University, Toronto, Canada.
Received April, 2013.
ABSTRACT
The main dra wbac k of curr ent E CG syst ems i s the loc atio n-specific nature of the systems due to the use of fixed/wired
applications. That is why there is a critical need to improve the current ECG systems to achieve extended patient’s mo-
bility and to cover security handling. With this in mind, Compressed Sensing (CS) procedure and the collaboration of
Sensing Matrix Selection (SMS) approach are used to provide a robust ultra-low-power approach for normal and ab-
normal ECG signals. Our simulation results based on two proposed algorithms illustrate 25% decrease in sampling-rate
and a good le vel of quali ty for the degree of incoherence between the random measurement and sparsity matrices. The
simulation resu lts also co nfirm that the Binar y Toeplitz M atrix (BT M) provides the best compressio n perfor mance with
the highes t energ y efficiency for r andom sensing matrix.
Keywords: Sensing Matrix; Power Consumption; Normal and Abnormal ECG Signal; Compre s sed Sensing; Block
Sparse Bayesian learning
1. Introduction
WBANs as a special purpose of Wireless Sensor Net-
works (WSNs) consist of tiny Biomedical Wireless Sen-
sors (BWSs) and a Gate Way (GW) to connect to the
external databases in the hospital and medical centers [1].
The WBANs are expected to be a breakthrough in
healthcare areas such as hospital and home care, Mobile
Health (MH), Electronic Health (EH), and physical reha-
bilitation. The GW could connect the BWSs, to a range
of wireless telecommunication networks. These wireless
telecommunication networks could be either mobile
phone networks, standard telephone networks, dedicated
medical center or using public Wireless Local Area
Networks (WLANs) nodes also known a Wi-Fi system
[2]. The compressed sensing is a revolutionary idea for
the acquisition and recovery of sparse signals that
enables sampling-rate significantly below the classical
Nyquist-rate (NR). The electrocardiogram (ECG) signals
are widely used in health care systems because they are
noninvasive mechanisms to establish medical diagnosis
of heart diseases. The current ECG systems suffer from
important limitations: limited patient’s mobility, limited
energy, limited on wireless applications. In order to fully
exploit the benefits of WBANs such as EH, MH, and
Ambulatory Health Monitoring Systems (AHMS) the
power consumption and sampling rate should be re-
stricted to minimum. Long-term records of ECG signals
in WBANs have become commonly used to collect in-
formation from the heart for diagnostic and therapeutic
purposes [3]. That is why the quantity of data grows sig-
nificantly and compression is required for reducing the
storage, trans missio n times, a nd p o wer consu mption. T he
ECG si gnals generally illustrate the redundancy bet ween
adjacent heartbeats due to its semi-periodic structure [4].
It is evident that this redundancy provides a high fraction
of common support between consecutive heartbeats that
is a good candidate for compression. Howe ver, the y have
low-frequency and non-stationary features and
processing noise setting strong characters, neither
time-demine nor frequency-domain based methods are
suitable for analyzing these signals. This paper presents
new algorithms with contribution of CS approach in
mind, and SMS procedure based on Dynamic Thre-
sholding Approach (DTA) to establish a robust ul-
tra-low-power for normal and abnormal ECG signals.
The CS theor y indicate that s a small number of rando m
linear measurements of sparse signals contain enough
information to collect, process, transmit, and recover the
original signal [5]. This approach emphasizes that the
signal representing sparsity in any orthogonal basis can

Robust Low-Power Algorithm for Random Sensing Matrix for Wireless
ECG Systems Based on Low Sampling-Rate Approach
126
be well reconstructed usi ng ℓ1 norm minimization, while
satisfying the Restricted Isomerty Property (RIP) condi-
tion fo r ra nd om mea s ur e me nt ma tr i x
Φ
and or tho go nal
Ψ
in any domain [6]. Our simulation results based on
two proposed algorithms illustrate 25% decrease in sam-
pling- rate and a good level of quality for the degree of
incoherence between the ra ndom measurement a nd spar-
sity matrices. The simula tion results al so c onfir m that th e
Binary Toeplitz Matrix (BTM) provides the best com-
pression performance with the highest energy efficiency
for rando m sensi ng matrix. The structure of this paper is
organized as follows: Section 2 gives an overview about
CS theory in general and specifically for WBANs. Sec-
tion 3 proposes the new algorithm based on combination
of CS theory, and SMS approach. The reminder of the
paper is categorized in the following way: the simulation
results are presented in Section 4. The conclusion is
drawn in Section 5.
2. Overview of Compressed Sensing
The conventi onal sampling app roaches have traditionally
relied on the Shannon sampling theorem. This theory
says a signal must be sampled at least twice its band-
width in order to be represented without error. The tradi-
tional approaches have two important drawbacks. First,
they generate huge samples for many applications with
large bandwidth that is not tolerated. Second, even for
low signal bandwidths such as ECG signals, they pro-
duced a large amount of redundant digital samples. That
is why it is desirable to reduce the number of acquired
ECG samples by using advantages of the sparsity. The
CS theory replaces the conventional sampling and
reconstruction operation with a general random linear
measurement process and an optimization scheme in
order to recover original signal from a small number of
rand o m mea sureme nt s.
2.1. Basic Theorem
The goal in the digital-CS theory as a new sampling
scheme is to reduce the load of sampling-rate by de-
creasing the number of samples after the Analog to Digi-
tal Convertor (ADC) required to completely describe a
signal by explo iting its compr essibility [7]. An important
aspect of CS theory is that the measurements are not
point samples but more general linear functions of the
signals. Any compressible or sparse signal
 in ℝN can
be expressed as:
1
N
ii
i
DC
=
= Ψ
∑
. (1)
The compresses signal

can also be found as:
11
[][] []
MMN N
D
× ××
= Φ
. (2)
Thus, the c ompressed s ignal is found a s :
1 11
[][] [][][] []
MMNNN NMN N
CC
×× ××××
=ΦΨ =Θ
. (3)
Fortunately,
[]Φ
and
[]Θ
have two interesting and
useful pro perties. First, they ar e incoherent with the basis
[]Ψ
. Second, they have the RIP with high probability
where is suitable condition to recover the original signal in
the receiver si de [8]. Thus , CS scen ario has two important
steps. First step i n CS offers a stable measure ment matr ix
[ ]
MN×
Φ
to ensure that the salient information in any
compressible signal is not damaged by the dimensionality
reduction from D ∈ℝN down to ℂ∈ℝM. In the second step,
the CS theory offers a reconstruction algorithm under
certain condition and enough accuracy to recover original
signal D from the compressed signal. Therefore, we can
exactly reconstruct the original signal D with high prob-
ability via 1 norm by solving the following convex op-
timization pr oblem (
1n
n
ss=
∑

):
1
min
N
s
s∈


subject to
S= ΦΨ
. (4)
There are two important conditions, which guarantee
the correctness of this recovery. Firstly, the number of
random linear measurements, the number of coefficients,
and the number of non-zero coefficients must satisfy the
following equatio n [9]:
/ (log)M KCN≤
. (5)
Secondly, for any vector
a
of the original signal
[]D
matrix
[]Φ
must satisf y the following co ndition for
some
0 ε 
:
22
1 /1aa εε − ≤Φ≤+ 
(6)
where satisfies RIP property for the random dictionary
matr ix. In order to recover K-sparsity of t he origi nal sig-
nal, now we have
MK×
system of linear equations,
with M equations and K unknowns. It is possible to find
out the K-sp arsity of the original signal, because of
MK≥
.
2.2. Compressed Sensing in WBANs
The CS theory says sparse or compressible signals such
as ECG; signals can be well recovered using to minimize
ℓ1 norm optimization, while satisfying the RIP condition
for the random measurement matrix Ф and orthogonal
basis ψ. Basically, the biomedical signals are sparse or
near sparse. To verify this condition, we exploit a con-
ventional Fast Fourier transformation (FFT) to check
signal sparsity. These signals have K non-zero coeffi-
cients and (N-K) zero coefficients with K N and can
be well recovered using M projects or measurements

Robust Low-Power Algorithm for Random Sensing Matrix for Wireless
ECG Systems Based on Low Sampling-Rate Approach
127
such as K≤ M<<N. As the result, the small number of
non-zero coefficients is small; the CS theory can be ap-
plied to reduce the load of sampling. Figure 1 illustrates
CS theory in WBANs .
Figure 1. CS in WBANs.
As it can be seen the biomedical signals are com-
pressed by wireless sensors. The collected compressed
biomedical data are then transmitted wirelessly to Access
Points (APs) at hospital, ambulance, and helicopter [10,
11]. The APs recover compressed biomedical data for
diagnostic and therapeutic purposes. Further mor e, the D
data vector in WBANs is a sparse vector, because the GW
needs to collect only M bits instead of N bits of data
(M≈K– spa rs e) t hr ou gh the ne t wor k. In t he W B AN s wit h
N wireless senso r, senso r
i
is acquiring a sa mple i
d of
the hum an body [9]. Th e final goal in WBANs for medical
applications is to collect Data's vector D of N wireless
senso rs in a s uit able basis Ψ= [Ψ1][Ψ2]…[ΨN] like:
1
N
ii
i
Dd
=
= Ψ
∑
. (7)
CS suggests that, under certain conditions, instead of
collecting data vector D, we can collect compressed ve c-
tor
[][][]D= Φ
 where Φ is (K×N) sensing matrix
whose entries are i.i.d random variables. In non-CS sce-
nario a node is receiving N-1 packets and sends out N
packets ((N-1) received packets plus its data) each packet
corresponding to data sample from a node. In WBANs
with CS theory the GW needs only to receive M
(M≈K-sparse) packets [9]. In order to use CS, each node
needs to know the value of Compressed Ratio (CR=N/ K)
that is constant and value of N [16]. The node i compute
K=N/CR and generate K values Φji (1≤ j ≤k) and creates a
vector Di [Φ1 i, Φ2 i… Φk i], where Di is its own data. Typ-
ically, node i would wait to receive from all its down-
stream neighbors. Each received packet carries its index
from 1 to K, so that it can be added to the data already
waiti ng in i with the same index (either locally produced
or received from a neighbor). Then node i would send
exactly K-Packets corresponding to the aggregated col-
umn vec tor s . Now the diff erence between CS and non-CS
operation becomes clear [10]: CS operation requires each
node to send exactly M packets irrespective of what it has
received, and each node needs to know CR and N and then
computes the value of (M≈K). The received vector in GW
can be written as:
11
[][] []
MMN N
D
× ××
= Φ. (8)
Consequently, the received vector in GW is a con-
densed representation of the sparse events and can be
expressed like:
1111 1
1
N
MMMN N
D
D
ΦΦ
 
 
=
 
 
ΦΦ
 

 

. (9)
Our simulation r esults sho w that b y emplo ying the C S
the WBANs can achieve a higher transmission, a lower
time delay and higher probability of success of data
transmission. Therefore, a combination of CS theory to
WBANs is an optimal solution for achieving robust
WBAN with low sampling rate and power consumption.
As it can be seen the biomedical signals are compressed
by wireless sensors. The collected com pressed biomedical
data are then transmitted wirelessly t o Access Points (APs)
at hospital, ambulance, and helicopter [10, 11]. The APs
recover compressed biomedical data for diagnostic and
therapeutic purposes.
3. Proposed Approach
To validate the performance of the considered compres-
sion schemes three performance measurements are de-
fined in this section first. Then, the proposed algorithm
for selecting the best fit for random sensing matrix is
proposed.
3.1. Performance Measure
The Compression Ratio (CR), the Structural Similarity
Index (SSI), and Percentage Root-mean-square Differ-
ence (PRD) are employed as performance measures in
our approach. The CR is found as follows [12]:
/ 100CRN M= ×
, (10)
where M and N are the number of random linear mea-
surements and number of samples in ECG signals re-
spectively. Further, our simulation results indicate that
satisfying quality of SR can be achieved when CR does
not exceed of 35%. The SSI metric is defined as [13]:
(/)100SSI D= ×
, (11)
where
D
and

are the original and recovered ECG
signals respectively. This metric measure the similarity
between the recovered and original ECG signals [14].
Higher SSI means better recovery quality. Our simulation
results will show the proposed approach has this ability to
achieve SSI with value close to 100%. The PRD is com-
puted as [14]:

Robust Low-Power Algorithm for Random Sensing Matrix for Wireless
ECG Systems Based on Low Sampling-Rate Approach
128
22
(/) 100PRD DD
=−×
 . (12)
The value of PRD shows the quality of reconstruction
approach. The relationship between the measured PRD
and diagnostic distortion is recognized on the weighted
diagnostic data for ECG signals, which classi fies the dif-
ferent va lues o f PRD based o n the signal quality obtained
by a specialist. Table 1 illustrates the resulting different
quality classes and corresponding PRD values. As de-
picted in Table 1, lower PRD means better recovery
quality.
Table 1. Different Quality Classes.
PRD
Quality of recovery
0 1%
Excellent
1 2%
Very good
2 0.85%
Good
0.85%≥
Poor
3.2. Proposed SMS Algorithm
The random measurement matrix
[]Φ
is a key compo-
nent of CS theory. Two key features are needed for a
successful implementation of CS approach: Sparsity of
the biomedical signal and incoherence between the ran-
dom sensing matrix and the sparsity basis [15]. That is
why; the random sensing matrix must exhibit a high de-
gree of incoherence with the sparsity basis
[]Ψ
. In this
part, the new SMS procedure is presen ted to select the best
fit for the random sensing matrix
[]Φ
. Herein, Bernoulli
Toeplitz, Gaussian Circulant, and Binary Toeplitz ma-
trices are examined to find out the best fit for random
sensing matrix [16]. The Toeplitz matrix is a matrix in
whic h eac h de sce ndi ng dia go nal fro m left to right is co n-
stant. The random sensing matrix in Binary form is ex-
pressed as:
01 1
10
0
1 10
n
n
n
− −+
−
−
ΦΦ Φ


ΦΦ Φ

Φ=
Φ

Φ ΦΦ



 

(13)
The Circulant matrix is a special kind of Toepliz ma-
trix where each row vector is rotated on element to the
right relative to the preceding row vector [17]. The ran-
dom sensing matrix in Circulant form is illustra te d as:
01 1
10 2
0
1 10
n
n
−
−
ΦΦ Φ


ΦΦ Φ

Φ=
Φ

Φ ΦΦ



 

(14)
In the simulation part, CS approach is applied on the
ECG data obtained from MIT-BIH database for three
sensing matrix possibilities: (1) Bernoulli Toeplitz matrix,
(2) Gaussion Circulant matrix, and (3) Binary Toeplitz
matrix. Our simulation results will further confirm that the
Binary Toeplitz matrix shows the best performance for the
random sensing matrix
Φ
. Table 2 illustrates our new
algorithm to select the best fit for random sensing ma-
trix
Φ
.
Table 2. The best fit for sensing matrix.
Algorithm: The Best Fit for Random Sensing Matrix
Φ
Enter: Raw ECG data
1: Apply Dynamic Thresholding Approach to Raw ECG data
2: Select Initial Square Matrix
3: Apply Row Se l ection Schem e (select the fir st M ro ws as the in it ia l
sensing matrix
Φ
)
4: Compare with Binary Toeplitz Matrix
5: If
Φ
is Binary Toeplitz Matrix Stop, the Algorithm is completed
6: M=M+1
7: Go to Step 4
In the step 1 of the proposed algorithm, the DTA pro-
cedure is applied to the raw ECG data. The princip al ob-
jective of the DTA is to vary the sparsity level of a raw
ECG signal to convenient level [18]. In the simulation
part, the convenient level is defined 98%. In the step 2,
the initial square matrix is used for each of the sensing
matrices in the experiments [19]. In the step 3, a Row
Selection Scheme (RSS) is applied to reduce the number
of rows from N to M [20]. Two RSSs approaches are
compared: (1) select first M rows from the initial
NN×
matrix, and (2) randomly select M rows from
the initial
NN×
matrix. The first RSS approach de-
monstrates better performance than the second RSS ap-
proach. So only the first RSS approach is utilized in the
proposed algorithm.
4. Simulation Results
The following assumptio ns w e r e made for simulation:
► Experiments are carried out over a 10-minute s E CG
signal from MIT-BIH database [21] .
► One hundred repletion’s are averaged for our simu-
lation results. To validate the simulation results ECG
signals from records 100,107,115 and 117 of MIT-BIH
are investigated.
►The mean of ECG blocks is rounded in the sliding
window to the nearest multiple of
2
L
, where
L
is the
BSBL level [22].
►To simulate SNR for ECG signals the follo wing eq-
uation is used [23].
10
20log (0.01)SNR PRD= −
. (15)
►Three sensing matrix possibilities are examined for
rand om sens ing mat rix
Φ
: (1) Bernoulli To eplitz matrix,
(2) Gaussian Circulant matrix, and (3) Binary Toeplitz

Robust Low-Power Algorithm for Random Sensing Matrix for Wireless
ECG Systems Based on Low Sampling-Rate Approach
129
matrix [24].
►The SPARCO toolbox is used for testing sparse re-
const ructio n algorithm.
►The SPGL1 (Spectral Projected Gradient for 1 mi-
nimization) toolbox is used to determine Large-scale
one-norm regularized least squares in the following equ-
ation:
min 1
N
c
c∈


subject to
D= Φ
. (16)
►To validate the simulation results, the BPBQ (Basis
Pursuit DeQuantizer) toolbox is used for recovery of
sparse signals from quantized random measurements to
solve [25]:
argmin 1
N
c
c∈


subject to
p
D−Φ 
for
2p≥
. (17)
►The simulation results were obtained for an input
signal of N=512 samples and a 12-bits resolution for the
input signal and t he me asure ment signal

.
►To simulate the SMS approach, the DTA framework
is used to vary the sparsity level [26].
Figure 2 illustrates the sampling-rate for random bi-
nary matrix with CS theory for different values of
non-zero entries
K
for specific records of ECG sig na ls.
Figure 2. Sampling-rate.
Based on the results of Figure 2 and suitability of the
rand om bina ry matri x, the sampling rate can be reduced
by 75% of NR without sacrificing the performance. Fig-
ure 3 shows simulation results on power consumption
for random binary matrix with CS theory in terms of
Compressed Ratio (CR) for specific records of ECG sig-
nals.
As depicted in the Fig. 3, the power consumption can
be reduced by 65% by employing CS theory. Table 3
compares the simulation results on sampling rate and
power consumption for random binary matrix with CS
theory.
Figure 3. The pow er consumpti on.
Table 3. Comparing SR and PC.
N in ECG CR SR PC
1024 10.24 25%*(NR) 30%*(PC in non-CS)
2048 20.48 28%*(NR) 35%*(PC in non-CS)
3074 30.74 32%*(NR) 40%*(PC in non-CS)
Table 2 indicates that satisfying quality on sampling
rate and power consumption can be achieved when CR
does not exceed of 30.
5. Future Work
We have simulated the benefit of CS to wireless ECG
systems for some recodes of ECG signals. Our future
work involves developing the CS theory to other records
of ECG signal, including abnormal records for wireless
ECG syst ems.
6. Conclusions
The ECG signal is widel y used in WB ANs becau se it is a
noninvasive way to provide medical diagnosis of heart
disea ses. This paper has presented new algorithm with a
contribution of CS approach, and SMS procedure based
on DTA aapproach to establish a robust ultra -low-power
for normal and abnormal ECG signals. The works by
Alvarado [2] and Baheti [14] focus only CS theory for
normal ECG signal with only random Gaussian matrix.
While the present study offers a new algorithm to select
the best fit for random sensing matrix of the CS approach
for normal and abnormal ECG signals. Our simulation
results validate the suitability of a new algorithm for a
real-time energy-efficient ECG compression on re-
source-constrained in WBANs. The simulation results
also confirm that t he Binary Toep litz matrix pro vides the
best compression performance with the highest energy
efficiency for random sensing matrix. Advanced ECG

Robust Low-Power Algorithm for Random Sensing Matrix for Wireless
ECG Systems Based on Low Sampling-Rate Approach
130
systems based on CS will be able to deliver healthcare
not only to patients in hospita l; but also in t heir homes.
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