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J Med Syst (2011) 35:1349–1358
DOI 10.1007/s10916-009-9412-4
ORIGINAL PAPER
Compressed ECG Biometric: A Fast, Secured and Efficient
Method for Identification of CVD Patient
Fahim Sufi ·Ibrahim Khalil ·Abdun Mahmood
Received: 27 August 2009 / Accepted: 26 November 2009 / Published online: 6 January 2010
© Springer Science+Business Media, LLC 2009
Abstract Adoption of compression technology is often
required for wireless cardiovascular monitoring, due
to the enormous size of Electrocardiography (ECG)
signal and limited bandwidth of Internet. However,
compressed ECG must be decompressed before per-
forming human identification using present research
on ECG based biometric techniques. This additional
step of decompression creates a significant processing
delay for identification task. This becomes an obvious
burden on a system, if this needs to be done for a
trillion of compressed ECG per hour by the hospital.
Even though the hospital might be able to come up
with an expensive infrastructure to tame the exuberant
processing, for small intermediate nodes in a multihop
network identification preceded by decompression is
confronting. In this paper, we report a technique by
which a person can be identified directly from his / her
compressed ECG. This technique completely obviates
the step of decompression and therefore upholds bio-
metric identification less intimidating for the smaller
nodes in a multihop network. The biometric template
created by this new technique is lower in size compared
to the existing ECG based biometrics as well as other
forms of biometrics like face, finger, retina etc. (up to
F. Sufi (B)·I. Khalil ·A. Mahmood
School of Computer Science and Information Technology,
RMIT University, Melbourne, Australia
e-mail: research@fahimsufi.com
I. Khalil
e-mail: ibrahimk@cs.rmit.edu.au
A. Mahmood
e-mail: abdun.mahmood@rmit.edu.au
8302 times lower than face template and 9 times lower
than existing ECG based biometric template). Lower
size of the template substantially reduces the one-to-
many matching time for biometric recognition, result-
ing in a faster biometric authentication mechanism.
Keywords ECG biometric ·ECG compression ·
Patient identification ·Patient authentication
Introduction
ECG based biometric has made it possible to iden-
tify patients just from ECG samples taken from them
[1,3,4,7–9,16,20,23–26]. During a remote telecardiol-
ogy scenario, the patient is attached to miniature ECG
acquisition devices that transmits the patients ECG sig-
nals to remote locations, using mobile communication
devices [10,14,17–21]. In such a scenario, ECG signal
acquired by the ECG acquisition device is directed
to their own mobile phones or personal computers
as ECG packets (via Bluetooth, Wifi, Zigbee or Near
Field Communication protocol) and from there on it
is redirected to the patients’ Internet service providers.
Within the public Internet, the ECG packet also passes
through different nodes in a multi-hop manner and
finally reaches the doctor or hospital facility. As the
ECG packet passes through different nodes, both in
private and public network, it is susceptible to intruders
attack. Once the intruder have the undesired access to
the patients ECG signal, they will be able to under-
stand whose ECG it is by using biometric matching.
The hacker could reveal a great deal of highly sensi-
tive private health information of the patient including
patients’ cardiovascular details, state of autonomous
1350 J Med Syst (2011) 35:1349–1358
nervous system, stress level, breathing pattern etc. just
by analysing the ECG signal [18]. To preserve patients
privacy and uphold Health Information Privacy & Ac-
countability Act (HIPAA) of 1996 previous research
has obtained end to end security in wireless telecardiol-
ogy using special encoding and character shuffling [13,
18,21]. Apart from these permutation ciphers, there
are other methods for ECG obfuscation or anonymiza-
tion [22,25]. While the ECG data is being transmitted
within the multi-hop network (via multiple nodes), to
ensure security, patient’s name can be removed from
the ECG signal [20]. Detaching patients name from the
patients ECG signal within the ECG signal packet, also
serves the purpose of anonymizing the patients medical
condition [20]. However, if the patients ECG signal is
anonymized in such a way (without tagging the signal
with the patient name), then the intermediate nodes
would have no clue about where (hospital or which
cardiologist) to redirect the information. Therefore, an
important problem in transmitting and annonymizing
ECG signal is how to allow intermediate nodes to send
patient’s ECG signals to appropriate recipients with-
out including patient’s identifiable information, such as
name or tag.
Within this paper, we propose an innovative tech-
nique to identify patient from their compressed ECG.
Therefore, in a mobile phone based cardiovascular
patient monitoring scenario [6,10,14,17–21], where
ECG is transmitted in compressed format, it is possible
for the intermediate nodes to identify the patient even
without decompressing the ECG signal. Every node
within the multi hop network have their own listing
of routing information. After identifying the patient
from their compressed ECG, the route listing is pulled
up internally by these nodes. Based on the routing
information the compressed ECG reaches the correct
destination. Apart from patient identification by the
intermediate routing nodes, the hospital (which is the
final destination for these compressed ECG) similarly
identifies the patient from their compressed ECG and
provides appropriate cardiovascular monitoring facility
that the patient is subscribed to.
Background
An ECG signal has three major features waves, namely
P wave, QRS complex and T wave (as seen from Fig. 1).
Cardiologists have used different features of these fea-
ture waves to assess the condition of heart. Table 1lists
some of these features.
Previous researches have used many of these fea-
tures to successfully identify person from their ECG
Time / Samples
Amplitude
P
Q
R
S
T
Fig. 1 The proposed cardiac diagnosis system
[1,8,26]. The earliest effort in ECG based biometric
is [1], where the researchers have utilized many time
domain features of ECG that were conventionally used
for cardiovascular diagnosis purposes only. Some of
the time domain features (features shown in Table 1)
demonstrate a degree of uniqueness. This uniqueness of
feature is the basis of biometric. It should be mentioned
that the features associated with QRS complex shows
the highest degree of uniqueness according to recent
findings [26]. Apart from the features shown in Table 1
(that remain unchanged over time), there are other
features that varies over time such as RR Interval,
Heart Rate Variability (HRV), Instanteneous Heart
Rate (IHR) etc. RR Interval is the time difference
between two consecutive heart beats. HRV is the recip-
rocal of the RR Interval ( 1
RR) and IHR corresponds to
heart rate calculated from a single RR Interval value
(60
RR). RR Interval, HRV, IHR features have also been
used in [3,7,26] for person identifications.
Apart from the time domain ECG features, ECG
based human identification is also possible with fre-
quency domain features. Although Statistical and data
mining techniques [11,12] have been widely used
for pattern identification, research in [4] shows that
Table 1 ECG features related to P wave, QRS complex and
T wave
P wave duration QRS complex T wave duration
duration
P wave amplitude QRS complex T wave amplitude
amplitude
P wave onset slope Q onset slope T wave onset slope
P wave offset slope Q offset slope T wave offset slope
R onset slope
R offset slope
S onset slope
S offset slope
J Med Syst (2011) 35:1349–1358 1351
Fig. 2 An ECG segment of
MIT BIH Normal Sinus
Rhythm Database (nsrdb)
entry no. 16420. X axis shows
the number of samples and Y
axis shows the corresponding
ECG amplitude in mV
(millivolt) range
0 200 400 600 800 1000 1200
–2
–1
0
1
2
3
employing wavelet based distance measurement tech-
niques, ECG based biometric can attain a higher
accuracy.
In addition to time domain and frequency domain
features based techniques, there are also other tech-
niques that revolves around curve fitting (or polynomial
based techniqes). Our earlier efforts of person identifi-
cation from ECG [9,20,23] falls into this category. In
this paper, we introduce a new technique that performs
ECG based biometric from compressed ECG. Person
identification or biometric from compressed ECG has
never been reported in literature according to the best
of our knowledge.
System design
Application of ECG compression is crucial for mobile
phone based cardiac patient monitoring, since enor-
mous amount of ECG data is required to be trans-
mitted over limited bandwidth of mobile network. The
compression algorithm reported in our earlier work
[18,21] represents an ECG signal losslessly. Therefore,
the encoding function (.) transforms the ECG signal,
Xnto a compressed ECG, Cr(Eq. 1). As the ECG
features set, Fis a subset of ECG signal Xn(Eq. 2),
therefore, feature waves are subset of Encoded ECG
Cr(Eq. 3). Innovative algorithm can be designed to
reveal these encoded ECG feature set (that represents
original ECG feature set) and then perform matching
between the enrollment and recognition data. This is
the core theory behind using encoded ECG to identify
a person.
(Xn)=Cr(1)
F⊂Xn(2)
F⊂Cr(3)
Examples will clarify the theory behind person iden-
tification with compressed ECG. Figures 2,3,4,5,6,
7,and8show ECG signals from three different in-
dividuals. All these ECG signals were collected from
MIT BIH Normal Sinus Rhythm Database (known
as nsrdb) [15] (Figs. 2–10). The sampling frequency
of the ECG samples used for our experiment (from
NSRDB) was 128 Hz with 10 bit resolution. Figures 2
to 8basically illustrate the fact that there are minute
difference in the ECG signals collected from differ-
ent individuals. These differences are mostly obvious
within the features waves (P wave, QRS Complex
and T wave). Previous research identified the feature
waves using different feature detection algorithms and
performed biometric identification based on the ECG
features. However, for fast and efficient transmission
Fig. 3 An ECG segment of
MIT BIH Normal Sinus
Rhythm Database (nsrdb)
entry no. 16773. X axis shows
the number of samples and Y
axis shows the corresponding
ECG amplitude in mV
(millivolt) range
0 200 400 600 800 1000 1200
–2
–1
0
1
2
3
4
1352 J Med Syst (2011) 35:1349–1358
Fig. 4 An ECG segment of
MIT BIH Normal Sinus
Rhythm Database (nsrdb)
entry no. 16786. X axis shows
the number of samples and Y
axis shows the corresponding
ECG amplitude in mV
(millivolt) range
0 200 400 600 800 1000 1200
–1
0
1
2
3
Fig. 5 An ECG segment of
MIT BIH Normal Sinus
Rhythm Database (nsrdb)
entry no. 16795. X axis shows
the number of samples and Y
axis shows the corresponding
ECG amplitude in mV
(millivolt) range
0 200 400 600 800 1000 1200
–1.5
–1
–0.5
0
0.5
1
Fig. 6 An ECG segment of
MIT BIH Normal Sinus
Rhythm Database (nsrdb)
entry no. 17052. X axis shows
the number of samples and Y
axis shows the corresponding
ECG amplitude in mV
(millivolt) range
0 200 400 600 800 1000 1200
–1
–0.5
0
0.5
1
1.5
Fig. 7 An ECG segment of
MIT BIH Normal Sinus
Rhythm Database (nsrdb)
entry no. 17453. X axis shows
the number of samples and Y
axis shows the corresponding
ECG amplitude in mV
(millivolt) range
0 200 400 600 800 1000 1200
–1
0
1
2
3
Fig. 8 An ECG segment of
MIT BIH Normal Sinus
Rhythm Database (nsrdb)
entry no. 16265 (used as
enrollment data). X axis
shows the number of samples
and Y axis shows the
corresponding ECG
amplitude in mV (millivolt)
range
0 200 400 600 800 1000 1200
–2
–1
0
1
2
3
J Med Syst (2011) 35:1349–1358 1353
Fig. 9 An ECG segment of
MIT BIH Normal Sinus
Rhythm Database (nsrdb)
entry no. 16265 (used as
recognition data). X axis
shows the number of samples
and Y axis shows the
corresponding ECG
amplitude in mV (millivolt)
range
0 200 400 600 800 1000 1200
–2
–1
0
1
2
3
Fig. 10 Two overlapped
ECG segments of MIT BIH
Normal Sinus Rhythm
Database (nsrdb) entry no.
16265. X axis shows the
number of samples and Y axis
shows the corresponding
ECG amplitude in mV
(millivolt) range
0 200 400 600 800 1000 1200
–2
–1
0
1
2
3
Fig. 11 Compressed ECG segment of nsrdb entry 16420 (of Fig. 2)
Fig. 12 Compressed ECG segment of nsrdb entry 16773 (of Fig. 3)
1354 J Med Syst (2011) 35:1349–1358
0 20 40 60 80 100 120 140 160
0
20
40
60
80
Fig. 13 Character frequency of compressed ECG segment in Fig.
2. X axis shows the number of encoding characters (157 charac-
ters in total [18]), Y axis shows the corresponding frequency (or
the number of occurance for that character within a compressed
ECG packet)
of ECG signals in telecardiology services, researchers
are increasingly using compressed ECG [18,21]. As
shown in Eqs. 1–3, compressed ECGs reveals ECG
features. Extracting features from compressed ECG is
generally faster than feature extraction from plain text
(uncompressed) ECG, as minimal characters are read
from compressed ECG. Therefore, faster cardiovascu-
lar diagnosis has recently been established based on
compressed ECG [18]. Within this paper, we also utilize
compressed ECG, however to fulfil a different objec-
tive of patient identification. Similar to cardiovascular
diagnosis from compressed ECG, patient identification
from compressed ECG is also very fast because of min-
imal data length (in compressed ECG) and processing.
Figures 11 and 12 show the compressed ECG of ECG
segments of Figs. 2and 3respectively.
0 20 40 60 80 100 120 140 160
0
20
40
60
80
Fig. 14 Character frequency of compressed ECG segment in Fig.
3. X axis shows the number of encoding characters (157 charac-
ters in total [18]), Y axis shows the corresponding frequency (or
the number of occurance for that character within a compressed
ECG packet)
0 20 40 60 80 100 120 140 160
0
20
40
60
80
100
120
Enrollment Data
Recognition Data
Fig. 15 Overlap of character frequency of compressed ECG
segments in Figs. 8and 9. X axis shows the number of encod-
ing characters (157 characters in total [18]), Y axis shows the
corresponding frequency (or the number of occurance for that
character within a compressed ECG packet)
In a telecardiology scenario, while the ECG seg-
ments are being routed through different nodes, each
node will quickly calculate the character frequency of
the compressed ECG. This tally of character frequency
is an inexpensive process and therefore can be easily
performed by the limited resources of the nodes. Our
experiments reveal the fact that for each individual
the distribution of encoding character set (from the
compressed ECG) is different. This fact can be ob-
served in Figs. 13,14,and15, where X axis and Y axis
correspond to individual characters and their frequency
counts respectively.
However, for a particular person, the distribution
of character frequency is identified (as seen in Fig.
15), even if their original ECG may look different at
different point in time (as seen in Fig. 10).
An algorithm for biometric feature creation
Based on the fact that character frequency distribution
from the compressed ECG taken at two different time
(enrollment and recognition data for biometric) for a
single person is similar, we can write an algorithm for
person identification with compressed ECG. However,
before performing the matching, we need to generate
a shortened feature character set that represent a per-
son. Selection of feature set is crucial for biometric
identification, as optimal selection of feature set re-
sults in faster and accurate processing of identification
task. Algorithm 1 establishes a process that generates
individual features from their compressed ECG. The
algorithm first calculates the character frequency fol-
lowed by sorting of the character frequency. After
J Med Syst (2011) 35:1349–1358 1355
the sorting, the first lnumber of frequent characters
can be collected. This ordered (sequential) character
set (referred to as in Algorithm 1) is a biomet-
ric feature that uniquely identifies a person. The two
dimensional feature set, =C
Acontains both the
selected character, Cand their respective frequencies,
A. The character set, C=C1,C2,C3,···,CNranges
from 1 to N (Length). for different entries are clearly
different according to our experiments on random MIT
BIH ECG entries (as it is seen for entry 16420, 16773,
16786, 16795, 17052, 17453 and 16265). On the other
hand, features created for the same entry (same per-
son) at different point in time are similar. As we can
observe this similarity for entry 16265, 16265Eis used
as enrollment data and 16265Ris used as recognition
data. This difference in biometric feature set across
different individuals and the similarity of feature set for
the same individual establish the basis for person iden-
tification based on compressed ECG. Figure 16 shows
the distance value, ψis minimum when Enrollment and
Recognition ECG templates of 16265 are calculated
using Eq. 4. Similarly, Fig. 17 shows correct matching
(with minimum ψvalue) of 16795.
ψ=l
n=1AE
n−AR
n2
l(4)
AE
land AR
lare the frequency counts of Enroll-
ment and Recognition (respectively) for lth template
parameter. Table 2shows the 12 different template
parameter for enrollment data of 16265. According
to our experimentation, all the template attributes
varies greatly (Table 2), except for the case when same
Fig. 16 Matching enrollment data of 16265, 17453, 17052, 16795,
16786, 16773, 16420 with recognition data of 16265. X axis repre-
sents different individual and Y axis represents ψvalue of Eq. 4.
Matching occurs with the minimum value of ψ
Fig. 17 Matching enrollment data of 16265, 17453, 17052, 16795,
16786, 16773, 16420 with recognition data of 16795. X axis repre-
sents different individual and Y axis represents ψvalue of Eq. 4.
Matching occurs with the minimum value of ψ
person’s Enrollment and Recognition data are used (as
shown in Fig. 16). It should be mentioned that one
person’s ECG template parameter may not be present
in another person’s selected parameter. In those cases,
the values of the missing parameters are considered to
be zero (during the calculation of Eq. 4). Experiment
were performed using the 18 entries of MIT BIH of
NSRDB [15] with no cases of misclassification. NSRDB
was chosen for our experiments, since this database
contains normal ECG signals. Most of the other MIT
BIH database contains abnormal ECG (generally, used
for validating cardiovascular abnormality detection al-
gorithms [5]). Abnormal ECG signals (from other MIT
BIH database) has almost never been used in other
existing ECG based biometric systems [1,3,4,7–9,16,
20,23–26]. To obtain a more accurate evaluation of
the algorithm presented within this paper, experiments
need to be carried out on a substantially larger sam-
ple (different ECG segments from different persons)
size. We leave this a future work, when MIT-BIH
Table 2 Standard deviations
of the ECG biometric
template values (templates
for 16265E)
Attribute Value range
p64.86 ±40.70
ˆ
U65.14 ±23.04
r31.86 ±33.08
j16.71 ±28.87
h14.43 ±24.93
t26.14 ±28.12
l18.43 ±24.89
0to50 19.14 ±19.01
f8.86 ±15.4
n12.57 ±21.96
v11.71 ±22.04
50–100 28.43 ±14.55
1356 J Med Syst (2011) 35:1349–1358
accommodates more normal ECG samples (at present,
there are only 18 entries available in NSRDB [15]).
16420 =$]@;ˆ
Uæ50to100 `
e/p0to50 ?
62 59 58 52 52 40 39 32 32 27 27 26
16773 =pˆ
U0to50 @ ræ]t50to100 ;ln
67 54 44 43 42 39 38 34 31 28 27 27
16786 =ˆ
Up$ær@]t`
u50to100 `
e´
E
81 72 67 50 42 35 34 31 30 30 28 26
16795 =prˆ
Ut jl vhnx50to100 f
124 85 80 74 66 60 57 57 52 48 46 36
17052 =ˆ
U];$æ`
e´
IpÏ?/´
E
82 63 48 46 41 37 33 32 32 32 28 26
17453 =$];@ æ 150to200 /0to50 ˆ
U`
e50to100 p
74 56 56 55 43 28 27 25 23 23 23 22
16265E=pˆ
Ur jht l0to50 fnv50 −100
114 75 73 57 57 49 45 44 34 32 29 28
16265R=pˆ
Ur j t h l0to50 n50to100 fv
110 84 54 51 44 44 42 38 36 30 26 25
An algorithm for biometric feature matching
The compressed ECG feature has three major char-
acteristics: character set, order of character set and the
frequency of the individual characters.
•Character set: Character set for each feature set
representing a particular individual shows a degree
of uniqueness. As an example, in the case of entry
17052, character ´
Iand Ïis not present in 16420,
Algorithm 1: Feature extraction from compressed ECG
//Notation Description:
//Fholds the feature set comprising of
compressed character.
Count the frequency of the encoding characters
from compressed ECG
=C1C2C3... CN
A1A2A3... AN
Sort based on the frequency Count A
in a descending order
ϒ=CpCqCr... Cs
ApAqAr... As
Where, Ap≥Aq≥Ar≥As
Create feature template by taking
first lnumber of entries from ϒ
=CpCqCr... Cl
ApAqAr... Al
16773, 16786, 16795, 17052, 17453, 16265 (both in
enrollment and recognition). Therefore, existence
of some characters reduces the domain of similar
ECGs (for biometric matching purpose). In most of
the cases, the enrollment data is found within this
limited set of similar ECGs.
•Order of characters: Once a limited subset of ECG
templates is established, representing the possible
candidates for successful biometric match, ordering
of the character set is considered. It is obvious from
the enrollment and recognition data of entry 16265
that for the first 7 characters the sequence (or or-
dering) of the characters are nearly the same, with
an exception of characters tand h. For this example,
these two characters (tand j), just swapped their
corresponding positions. Higher matching of char-
acter sequence also reduces the number of candi-
dates for biometric matching. When this candidate
subset reduces to one, this signals the completion of
the human identification task.
•Character frequency: If the previous steps of char-
acter set and order of characters still leaves few
candidates, then the frequency of each characters
are matched. For the ECG pairs, having closer
match according to the character frequency is given
preference for selection as biometric recognition /
identification.
Discussion
This paper reports an algorithm for selecting biometric
feature template from compressed ECG and a method-
ology for performing matching. The ECG biometric
technique presented here is particularly useful for mo-
bile phone based cardiovascular monitoring in the fol-
lowing respects (as seen from Fig. 18).
•Intermediate nodes: The template (feature) match-
ing algorithm is simple enough to be implemented
within the limited computational resources of an
intermediate node that is responsible for relaying
the compressed ECG within a multihop network.
By successfully identifying the person from com-
pressed ECG, the intermediate node knows the
destination of that compressed ECG packet, pro-
vided that the intermediate node possess the knowl-
edge of subscriber listing for different cardiovascu-
lar monitoring services (or hospitals).
•Hospital or cardiovascular monitoring service
provider: The monitoring facility can also iden-
tify a person from the compressed ECG sent to
them. The primary goal of identification is to check
J Med Syst (2011) 35:1349–1358 1357
Patient A Patient B
Compressed ECG transmission
to multihop network
Multihop network nodes
utilizes ECG biometric and
knows which patient the
data is coming from and
where to route the message
ABC Hospital XYZ Hospital
Hospital uses ECG based biometric from compressed data and knows
what type of facilities the patient should receive from the hospital
Fig. 18 ECG based biometric (from compressed ECG) being
used by the multihop network node as well as the hospital
whether that person is actually subscribed to the fa-
cility. If the person is found to be a valid subscriber,
then the list of subscribed services for that person is
retrieved. This biometric authentication and autho-
rization is made possible by ECG biometric from
compressed ECG.
The compressed ECG based biometric technique
presented in this paper, is the first of its kind for ECG
based biometric authentication. This technique has sev-
eral advantages over existing ECG based biometric
authentication. This technique is fast and efficient. Fast
execution speed during matching depends on the size
of the template. According to Table 3the presented
Table 3 Comparison of template sizes
Biometric data type Size in bytes
Iris [27] 512
Face [27] 153600–307200
Voice [27] 2048–10240
ECG [26] 600
ECG (WDM) [4] 1371
ECG (PRD / CC) [4] 2210
ECG (PDM) [20,23] 340
ECG (Proposed Method) 37
technique has minimal template size. Compared to
the face recognition biometric template size of 307200
bytes [27], the presented ECG biometric template is
approximately 8302 times smaller in size. On the other
hand, compared to the Polynomial Distance Measure-
ment (PDM) technique of ECG based biometric, the
proposed biometric template is at least 9 times smaller
in template size.
As an individual grows older, his or her ECG signals
may change (and template pattern) slightly. Therefore,
the biometric template may be subject to a validity date.
After the validity date is over, a new biometric template
might need to be enrolled from the individual. How-
ever, the exact time frame of the validity (or longevity)
of an acquired ECG biometric template requires rigor-
ous research.
Conclusion
This paper reports a novel mechanism for identifying
patient just from their compressed ECG. By adopting
this compressed ECG based biometric technique, pa-
tient can be easily identified by intermediate nodes in a
multihop internetwork, as well as by the medical service
provider. Utilizing this innovative patient identification
mechanism for wireless cardiovascular patient moni-
toring, ensures better security in preserving patient’s
private health information (as mandated by HIPAA
regulations [13]). Compared to earlier attempts of
telecardiology in [2], where both patient name and
private health conditions were allowed to be trans-
mitted simultaneously, the presented architecture (Fig.
18) only transmits the compressed ECG. Compressed
ECG inherently has the benefit of faster and effective
transmission (and efficient storage for health service
providers). If by any chance the compressed ECG falls
into the wrong hand (by spoof attack), the attacker
only gets compressed and encrypted ECG messages
([18,21]) without the patient’s name. Therefore, the
private health information is not compromised like
existing telecardiology architectures [2,6,10].
Other than protecting patient privacy and upholding
HIPAA regulations, the presented ECG based biomet-
ric promises substantially faster matching and reduced
template storage (approximately 9 to 8302 times).
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