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Summary of a portion of a paper

Can someone summarize the materials and methods section for me please?

Application of complex discrete wavelet transfor

m

in classification of Doppler signals using
complex-valued artificial neural network

Murat Ceylan a, Rahime Ceylan a, Yüksel Özbay a,*, Sadik Kara b

a Selcuk University, Department of Electrical & Electronics Engineering,

Engineering and Architecture Faculty, 42075 Konya, Turkey
b Fatih University, Biomedical Engineering Institue, Department of

Electrical & Electronics Engineering, 34500 Istanbul, Turkey

Received 9 July 2007; received in revised form 14 April 2008; accepted 24 May 2008

Artificial Intelligence in Medicine (2008) 44, 65—7

6

http://www.intl.elsevierhealth.com/journals/aiim

KEYWORDS
Complex wavelet
transform;
Complex-valued
artificial neural
networks;
Atherosclerosis;
Carotid artery;
Doppler signals

Summary

Objective: In biomedical signal classification, due to the huge amount of data, to
compress the biomedical waveform data is vital. This paper presents two different
structures formed using feature extraction algorithms to decrease size of feature set
in training and test data.
Materials and methods: The proposed structures, named as wavelet transform-com-
plex-valued artificial neural network (WT-CVANN) and complex wavelet transform-
complex-valued artificial neural network (CWT-CVANN), use real and complex discrete
wavelet transform for feature extraction. The aim of using wavelet transform is to
compress data and to reduce training time of network without decreasing accuracy
rate. In this study, thepresented structureswereapplied to theproblemofclassificatio

n

in carotid arterial Doppler ultrasound signals. Carotid arterial Doppler ultrasound
signalswereacquired fromleft carotidarteries of38patients and40healthyvolunteers.
The patient group included 22males and 16 femaleswith an established diagnosis of the
early phase of atherosclerosis through coronary or aortofemoropopliteal (lower extre-
mity) angiographies (mean age, 59 years; range, 48—72 years). Healthy volunteerswere
young non-smokers who seem to not bear any risk of atherosclerosis, including 28males
and 12 females (mean age, 23 years; range, 19—27 years).
Results and conclusion: Sensitivity, specificity and average detection rate were
calculated for comparison, after training and test phases of all structures finished.
These parameters have demonstrated that training times of CVANN and real-valued
artificial neural network (RVANN) were reduced using feature extraction algorithms
without decreasing accuracy rate in accordance to our aim.
# 2008 Elsevier B.V. All rights reserved.

* Corresponding author. Tel.: +90 332 223 20 48; fax: +90 332 241 06 35.
E-mail addresses: yozbay@selcuk.edu.tr, yuksel.ozbay@gmail.com (Y. Özbay).

0933-3657/$ — see front matter # 2008 Elsevier B.V. All rights reserved.
doi:10.1016/j.artmed.2008.05.003

mailto:yozbay@selcuk.edu.tr

mailto:yuksel.ozbay@gmail.com

http://dx.doi.org/10.1016/j.artmed.2008.05.00

3

1. Introduction

There are no reliable blood tests for diagnosis of
atherosclerosis. When the symptoms developed,
angiography is taken into account as the gold stan-
dard to detect and quantify the stenosis. Since
angiography is invasive, noninvasive ultrasonic Dop-
pler sonography is mostly endorsed. Recent
advances in the Doppler imaging technique made
it possible to appraise the temporal and spatial flow
characteristics in the different portions of the arter-
ial system, such as aorta, carotid and peripheral
arteries [1—6].

Furthermore, recent advances in the field of
artificial neural networks (ANNs) have made them
attractive for analyzing signals. The application of
ANNs has opened a new area for solving problems
not reasonable by other signal processing techni-
ques [7,8]. Applications of ANNs in the medical field
include photoelectric plethysmography pulse wave-
form analysis [9], diagnosis of myocardial infarction
[10], electrocardiogram analysis [11] and differan-
tation of assorted pathological data [12]. However,
to date, neural network analysis of Doppler signals is
a relatively new approach [13—18]. It is expected
that complex-valued artificial neural networks
(CVANN) whose parameters (weights, threshold
values, inputs and outputs) are all complex num-
bers, will have applications in fields dealing with
complex numbers such as telecommunications [19],
speech recognition, signal and image processing [20]
with the Fourier transformation. When using the
existing method for real numbers, we must apply
the method individually to their real and imaginary
parts. On the other hand, CVANN allow us to directly
process data.

In this paper, we propose two novel cascade
structures, called wavelet transform-complex-
valued artificial neural networks (WT-CVANN)
and complex wavelet transform-complex-valued
artificial neural networks (CWT-CVANN). Real
and complex wavelet transforms are used to
reduce the number of input samples in training
and test data. The basic idea in using wavelet
transform is to eliminate unnecessary features by
compressing Doppler signals. So, we propose using
complex-valued neural network for more efficient
classification of Doppler signals. In the implemen-
ted architectures, CVANN is integrated with fea-
ture extraction algorithms. These architectures
are composed of two subnetworks. The first sub-
network, which includes real and complex wavelet
transforms, is responsible for the compression of
signal. The second subnetwork performs the clas-
sification task using the compressed data. The WT-
CVANN and CWT-CVANN presented in this study

were trained and tested with Doppler signals
obtained from healthy and unhealthy subjects.
WT-CVANN and CWT-CVANN both achieved a cor-
rect classification rate of 100% in classification of
Doppler signals. Moreover, training time of CVAN

N

and processing complexity were reduced consid-
erably.

2. Material and methods

Doppler signals used in this study were acquired by
Toshiba PowerVision 6000 Doppler Ultrasound Unit in
the Radiology Department of Erciyes University Hos-
pital [2]. Before the data was recorded, a color and
pulsed Doppler ultrasound examination of the left
carotid artery was performed in order to exclude the
presence of a hemodynamically significant stenosis.
A linear ultrasound probe of 10 MHz was used to
transmit pulsed ultrasound signals to the proximal
left carotid artery.

2.1. Spectral analysis of carotid arterial
Doppler signals

Diagnostic significance of spectral analysis of Dop-
pler signals in arterial investigation is evolvement of
quantitative parameters of Doppler flow signals
based on spectral analysis, which can be used for
diagnostic intentions in arterial obstructive disease.
Doppler shift frequency, which is directly propor-
tional to the blood flow speed, is subjected to
spectral analysis [21].

In this study, acquired Doppler data was divided
with 50% overlap and windowed with a Hamming
window in frames of 256 data points as used in [6].
Afterwards, as seen in Fig. 1, power spectral density
of every window was calculated using Welch’s
method. Therefore, number of samples for every
subject was reduced to 129.

2.1.1. Welch method–—averaging modified
periodogram for spectral analysis
In the Welch method, L data sections of lengthM are
overlapped and the periodograms are computed
from the L windowed data sections. Also, the per-
iodograms are normalized by the factor U to com-
pensate for the loss of signal energy owing to the
windowing procedure. In fact U equates to 1=k

1=

2

2 ,

where k2 is the factor on the biasing effect of data
windows as necessary to compensate for this reduc-
tion in signal energy [22]. Thus,

U ¼

1

M

X

M�1

n¼

0

w2ðnÞ (1)

66 M. Ceylan et al.

The Welch power density spectral estimate,
PWE( f), is therefore

PWEð fÞ ¼
1

L

X

L�1

j¼0

P jð fÞ (2)

The expected value of the Welch estimate is

E½PWEð fÞ� ¼
1

L

X

L�1

j¼0

E½P jð fÞ� ¼ E½P jð fÞ� (3)

that is the same as the expected value of the
modified periodogram [22].

2.2. Real discrete wavelet transform
(DWT)

In its most common form, the DWTemploys a dyadic
grid (integer power of two scaling in a and b) and
orthonormal wavelet basis functions and exhibits
zero redundancy. Actually, the transform integral
remains continuous for the DWT but is determined
only on a dicretized grid of a scales and b locations
[23]. In practice, the input signal is treated as an
initial wavelet approximation to the underlying
continuous signal from which, using resolution algo-
rithm, the wavelet transform and inverse transform
can be computed discretely, quickly and without loss
of signal information. A natural way to sample the
parameters a and b is to use a logarithmic discretiza-
tion of a scale and link this, in turn, to the size of the
steps taken between b locations. To link b to a, we
move in discrete steps to each location b, which are
proportional to the a scale. This kind of discretization
of the wavelet has the form [23]:

cm;nðtÞ ¼
1
ffiffiffiffiffiffi

am0

p c

t� nb0a
m
0

am0

� �

(4)

where the integers m and n control the wavelet
dilation and translation, respectively; a0 is a speci-
fied fixed dilation step parameter set at a valued
greater than 1, and b0 is the location parameter
which must be greater than zero. A common chooses
for discrete wavelet parameters a0 and b0 are 2 and
1, respectively. This power-of-two logarithmic scal-
ing of both the dilation and translation steps is
known as the dyadic grid arrangement. The dyadic
grid is perhaps the simplest and most efficient dis-
cretization for practical purposes and lends itself to
the construction of an orthonormal wavelet basis.
Substituting a0 = 2 and b0 = 1 into Eq. (4) we see that
the dyadic grid wavelet can bewritten compactly, as
[23]:

cm;nðtÞ ¼ 2�m=2
cð2�mt� nÞ (5)

Real discrete wavelet transform is formed a
filter bank included low-pass and high-pass filters
(see Fig. 2). In Fig. 2, D1 and A1 are outputs of the
first high-pass filter and low-pass filter. In this
paper, the real discrete wavelet coefficients of
Doppler signals were computed using the MATLA

B

software package. Among the various wavelet
bases, the Haar wavelet is the shortest and sim-
plest basis and it provides satisfactory localization
of signal characteristics in time domain; hence it
is ideal for short time signals analysis. Therefore,
the Daubechies-2 wavelet that is the generalized
Haar wavelet was chosen as the mother wavelet in
this study [23].

2.3. Complex discrete wavelet transform
(CWT)

Wavelet techniques are successfully applied to var-
ious problems in signal processing. Data compres-
sion [24], classification [25,26] and denoising [27]
are only some examples. It is perceived that the
wavelet transform is an important tool for analysis
and processing of signals. In spite of its efficient
computational algorithm, the wavelet transform
suffers from three main disadvantages.

Classification of Doppler signals using complex-valued artificial neural network 6

7

Figure 1 The PSDs of healthy and unhealthy with ather-
osclerosis subjects.

Figure 2 The filter bank for discrete wavelet transform.

2.3.1. Limitations of wavelet transform
Although the standard DWT is a powerful tool, it has
three major disadvantages that undermine its appli-
cation for certain signal processing tasks [28,29].

2.3.1.1. Shift sensitivity. A transform is shift sen-
sitive, if the shifting in time, for input signal causes
an unpredictable change in transform coefficients.
It has been observed that the Standard DWT is
seriously disadvantaged by the shift sensitivity that
arises from down samplers in the DWT implementa-
tion [28,30]. Shift sensitivity is an undesirable prop-
erty because it implies that DWT coefficients fail to
distinguish between input signal shifts.

2.3.1.2. Poor directionality. An m-dimensional
transform (m > 1) suffers poor directionality when
the transform coefficients reveal only a few feature
orientations in the spatial domain. Wavelet trans-
form has been poor directional selectivity for diag-
onal features. Because the wavelet filters are
separable and real.

2.3.1.3. Absence of phase information. For a com-
plex-valued signal or vector, its phase can be com-
puted by its real and imaginary projections. Phase
information is valuable in many signal processing
applications [31] such as in image compression and
power measurement [32,33].

Most DWT implementations use separable filter-
ing with real coefficient filters associated with real

wavelets resulting in real-valued approximations
and details. Such DWT implementations cannot pro-
vide the local phase information. All natural signals
are basically real-valued, hence to avoid the local
phase information, complex-valued filtering is
required [34,35].

Recent research in the development of CWTs can
be broadly classified in two groups; redundant CWTs
(RCWT) and non-redundant CWTs (NRCWT). Stan-
dard DWT decimates and gives N samples in trans-
form domain for the same N samples of a given
signal. While the redundant transform gives M sam-
ples in transform domain for N samples of given
input signal (where M > N) and hence it is expensive
by the factor M/N. The NRCWT follows the design
aim to approach towards N samples in transform
domain for a given N input samples [28,29].

The RCWT include two almost similar CWTs. They
are denoted as dual-tree DWT (DT-DWT)-based CWT
(see Fig. 3) with two almost similar versions namely
Kingsbury’s and Selesnick’s [36]. In this paper, we
used Kingsbury’s CWT [28,36] for feature extraction
of Doppler signals.

2.4. Complex-valued artificial neural
network (CVANN)

Recently, there has been an increased interest in
applications of the CVANN to process complex sig-
nals [37—39]. In this study, a complex back-propa-
gation (CBP) algorithm has been used for pattern

68 M. Ceylan et al.

Figure 3 Dual-tree complex discrete wavelet transform.

recognition. We will first give the theory of the CB

P

algorithm as applied to a multilayer CVANN. Fig.

4

shows a model neuron used in the CBP algorithm.

The input signals, weights, thresholds, and out-
put signals are all complex numbers. The activity Yn
of neuron n is defined as

Yn ¼
X

m

WnmXm þ Vn (6)

where Wnm is the complex-valued (CV) weight con-
necting neuron n and m, Xm is the CV input signal
from neuron m, and Vn is the CV threshold value of
neuron n. To obtain the CVoutput signal, the activity
value Yn is converted into its real and imaginary
parts as follows:
Yn ¼ x þ iy ¼ z (7)

where i denotes
ffiffiffiffiffiffiffi

�1

p

. Although various output func-
tions of each neuron can be considered, the output
function used in this study is defined by the following
equation:

fCðzÞ ¼ fRðxÞ þ i fRðyÞ (8)

where fR(u) is called the activation function of neural
network. One of the difficulties encountered in
applying the CBP algorithm to the complex domain
involves the appropriate choice of activation func-
tion. For a practical implementation of the complex
multilayer perceptron, it is necessary that the acti-
vation function be bounded. Several researchers
developed a set of properties that a complex activa-
tion function must satisfy in order to be useful in a
multilayer perceptron trained with the back-propa-
gation algorithm [40]. Complex activation function
that used in this study is a superposition of real and
imaginary logarithmic sigmoids, as shown by

fRðuÞ ¼
1

1þ expð�uRÞ
þ j

1

1þ expð�uIÞ
(9)

Summary of CBP algorithm:

(1) Initialisation
Set all the weights and thresholds to small

complex random values.
(2) Presentation of input and desired (target) out-

puts

Present the input vector X(1), X(2), . . ., X(N)
and corresponding desired (target) response
T(1), T(2), . . ., T(N), one pair at a time, where
N is the total number of training patterns.

(3) Calculation of actual outputs
To obtain the complex-valued output signal,

the activity value Yn is converted into its real
and imaginary parts as Eq. (7).

(4) Calculation of the stopping criteria with respect
to Eq. (10) [38].

If this condition is satisfied, algorithm is
stopped and weights and biases are frozen:

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

X

p

X

N

n¼1

jT ð pÞ
n � O

ð pÞ
n j2

v

u

u

t ¼ 10�1 (10)

where T
ð pÞ
n and O

ð pÞ
n are complex numbers and

denote the desired and output value, respec-
tively. The actual output value of the neuron n

for the pattern p, i.e. the left side of (Eq. (11))
denotes the error between the desired output
pattern and the actual output pattern. N

denotes the number of neurons in the output
layer.

(5) Adaptation of weights and thresholds
We will use Wml for the weight between the

input neuron l and the hidden neuronm, Vnm for
the weight between the hidden neuron m and
the output neuron n, um for the threshold of the
hidden neuronm, and gn for the threshold of the
output neuron n. Let Il, Hm, On denote the out-
put values of the input neuron l, the hidden
neuron m, and the output neuron n, respec-
tively. Let also Um and Sn denote the internal
potentials of the hidden neuron m and the
output neuron n, respectively. Um, Sn, Hm,
and On can be defined, respectively, as
Um ¼

P

lWmlIl þ um, Sn ¼
P

mVnmHm þ gn, Hm

= fc(Um), and On = fc(Sn). Let d
n = Tn � On

denote the error between the actual pattern
On and the target pattern Tn of output neuron n.
We will define the square error for the pattern p

as E p ¼ ð1=2Þ
PN

n¼1 jTn � Onj2, where N is the
number of output neurons.

We can show that the weights and the thresholds
should be modified according to the following equa-
tions [38]:

DVnm ¼ �e

@E p

@Re½Vnm�

� ie

@E p

@Im½Vnm�
(11)

Dgn ¼ �e

@E p

@Re½gn�
� ie

@E p

@Im½gn�
(12)

DWml ¼ �e

@E p

@Re½Wml�
� ie

@E p

@Im½Wml�
(13)

Classification of Doppler signals using complex-valued artificial neural network 69

Figure 4 A model neuron used in the complex-BP algo-
rithm.

Dum ¼ �e

@E p

@Re½um�
� ie

@E p

@Im½um�
(14)

Eqs. (11)—(14) can be expressed as

DVnm ¼ HmDgn (15)

Dgn ¼ e
Re½dn�ð1� Re½On�ÞRe½On�
þi Im½dn�ð1� Im½On�ÞIm½On�

� �

(16)

DWml ¼ Il Dum (17)

Dum ¼ e

ð1� Re½Hm�ÞRe½Hm�

�
X

n

Re½dn�ð1� Re½On�Þ
Re½On�Re½Vnm�
þIm½dn�ð1� Im½On�Þ
Im½On�Im½Vnm�

0

B

B

@

1

C

C

A

2

6

6

6

6

4

3

7

7

7

7

5

� ie

ð1� Im½Hm�ÞIm½Hm�

�
X

n

Re½dn�ð1� Re½On�Þ
Re½On�Im½Vnm�
�Im½dn�ð1� Im½On�Þ
Im½On�Re½Vnm�

0

B

B

@

1

C

C

A

2

6

6

6

6

4

3

7

7

7

7

5

(18)

where z̄ denotes the complex conjugate of a com-
plex number z.

3. The results of numerical
experiments

A significant number of data used for applications
are naturally available in representations that are
difficult to learn. Transforming the data into a
more appropriate representation can facilitate
the learning process. For instance, using a smaller
number of parameters, which are often called
features, to represent the signal under study is
particularly important for recognition and diag-
nostic purposes. Given any set of features for data
representation, it is therefore important to esti-
mate the difficulty of learning the underlying
concepts using that training data. The learning
system should then seek to transform the repre-
sentations into a space that is easier for learning
purposes [41]. The studies in the literature
show that the degree of difficulty in training a
neural network is inherent in the given set of
training examples. By developing a technique
for measuring this learning difficulty, they devise
a feature construction methodology that trans-
forms the training data and attempts to improve
both the classification accuracy and computa-
tional times of artificial neural network (ANN)
algorithms. The fundamental notion is to organize
data by intelligent preprocessing, so that learning
is facilitated [41,42]. For this purpose, in this
paper, a classification and feature extraction-
based approach is adopted for classifying Doppler
signals.

3.1. The proposed structures and
training/test data

Training and test data set used in this study are
carotid arterial Doppler ultrasound signals acquired
from left carotid arteries of 38 patients and 40
healthy volunteers. The subjects had no clinical
and echocardiographic evidence of valvular disease
or heart failure. The patient group included 22
males and 16 females with an established diagnosis
of the early phase of atherosclerosis through cor-
onary or aortofemoropopliteal (lower extremity)
angiographies (mean age, 59 years; range, 48—72
years). Healthy volunteers were young non-smokers
who seem to not bear any risk of atherosclerosis,
including 28 males and 12 females (mean age, 23
years; range, 19—27 years). The hardware used in
recording of Doppler signals and recording system’s
features were given in studies of Kara and Latifoğlu
[5] and Ceylan et al. [6] as detailed.

Doppler signals were recorded in 2 s at 44,100 Hz,
there were 88,200 samples in one segment of train-
ing/test data. Samples (88,200) are too many for
taking a better performance of classification.
Accordingly, firstly, power spectral densities (PSD)
of Doppler signals were calculated using Welch
method, therefore 88,200 samples in one segment
of training and test data were reduced to 129 using
Hamming window with 256 data points. In this study,
data set was separated into two subsets. Each set
includes 20 healthy subjects and 19 unhealthy sub-
jects. First subset was used for training and remain-
ing second subset was used for testing. Then, the
same procedure was performed changing used sub-
sets. So, twofold cross-validation was done for
obtaining a better network generalization.
Obtained training and test errors were averaged.

In this study, three structures were formed using
two different feature extraction methods, real dis-
crete wavelet transform and complex discrete
wavelet transform. These structures were WT-
CVANN, WT-RVANN and CWT-CVANN (Fig. 5). Here,
classification tasks were performed by complex-
valued artificial neural network and real-valued
artificial neural network for comparison.

In the first structure, WT-CVANN, feature vectors
of training and test patterns whose lengths are 129
samples, were calculated using real discrete wavelet
transform. Three different feature vectors with dif-
ferent lengths (66 samples, 34 samples and 18 sam-
ples) were formed to present as inputs to CVANN. FFT
values of obtained feature vectors were calculated
for arising of real and imaginary components. Finally,
the new training sets included FFT results (66
samples � 39 subjects, 34 samples � 39 subjects
and 18 samples � 39 subjects) were classified using

70 M. Ceylan et al.

the CVANN. The networks trained by these training
setswerenamedasWT-CVANN1,WT-CVANN2andWT-
CVANN3, respectively. The complex-valued back pro-
pagation algorithm was used for training of the net-
works. In training phase, the weights and biases of
CVANN was initialised with small random complex
numbers. An error goal (stopping criteria threshold of
10�1) was specified (see Eq. (11)). The training ofWT-
CVANN was stopped when the error goal was
achieved. After that, the performance of WT-CVANN
was tested by presenting test subjects. The optimum
numbers of hidden nodes were determined as 12 via
experimentation for all networks with the highest
classification accuracy of 99%. Learning rate was
chosen as 0.7 for WT-CVANN1 and WT-CVANN2 in
training via experimentation, it was chosen as 0.9
forWT-CVANN3. The optimumnetwork was chosen as
WT-CVANN2, so the optimum number of input sam-
ples obtained with real discrete wavelet transform
was found as 34 (Fig. 6).

In the second structure, WT-RVANN, feature vec-
tors of training and test patterns whose lengths are
129 samples, were calculated using real discrete
wavelet transform. Then, FFT of feature vectors

were calculated. The new training sets included
FFT results (66 samples � 39 subjects, 34
samples � 39 subjects and 18 samples � 39 sub-
jects) used by real discrete wavelet transform were
classified using the RVANN by accepting real and
imaginary components as different two inputs
(Fig. 7a). On the other hand, complex-valued neural
networks allow us to directly process data (Fig. 7b).
The networks trained by these training sets were
named as WT-RVANN1, WT-RVANN2 and WT-RVANN3,
respectively. The real-valued back propagation
algorithm was used for training of the networks.
In training phase, the weights and biases of RVANN
was initialised with small random real numbers. An
error goal was specified as 10�1. The training of WT-
RVANN was stopped when the error goal was
achieved. After that, the performance of WT-RVANN
was tested by presenting test subjects. The opti-
mum numbers of hidden nodes were determined as 6
for WT-RVANN1 and WT-RVANN2, while it was deter-
mined as 4 for WT-RVANN3 with the highest classi-
fication accuracy of 99% via experimentation.
Learning rates were chosen as 0.9, 3.0 and 5.0 for
WT-RVANN1, WT-RVANN2 and WT-RVANN-3 in train-

Classification of Doppler signals using complex-valued artificial neural network 71

Figure 5 The block representation of (a) WT-CVANN, (b) WT-RVANN and (c) CWT-CVANN.

ing via experimentation, respectively. The optimum
network was chosen as WT-RVANN2, so the optimum
number of input samples obtained with real discrete
wavelet transform was found as 34 (Fig. 6).

In the third structure, CWT-CVANN, feature vec-
tors of training and test patterns whose lengths are
128 samples (first 128 samples of 129 samples), were
calculated using complex discrete wavelet trans-
form. In forming of feature vectors, dual-tree com-
plexdiscretewavelet transform (CWT) [36]wasused.
The sourcecodesofdual-treecomplexwavelet trans-
form is taken from web. Three different feature
vectors with three different lengths (32 samples,
16 samples and 8 samples) were formed to present
as inputs toCVANN. Finally, theobtainednewtraining
sets formed by CWT (32 samples � 39 subjects, 16
samples � 39 subjects and 8 samples � 39 subjects)
were classified using the CVANN. These networks
trained by obtained new training sets were named
as CWT-CVANN1, CWT-CVANN2 and CWT-CVANN3,
respectively. Training and testing processes were
performed as like first structure. The optimum num-
bers of hidden nodes were determined as 4, 30 and 6
via experimentation for CWT-CVANN1, CWT-CVANN2
andCWT-CVANN3with thehighest classificationaccu-
racy of 99%, respectively. Learning rateswere chosen
as 4, 3 and 2 for CWT-CVANN1, CWT-CVANN2 and WT-
CVANN3, respectively. The optimum network was
chosen as CWT-CVANN3, so the optimum number of
input samples obtained with real discrete wavelet
transform was found as 8 (Fig. 6).

3.2. Test results

After the training phase, all of the networks were
tested with the remaining patterns by using twofold

cross-validation. As noted, the trained network with
optimum parameters was used in the test to achieve
best results. The test results for all of networks are
shown at Table 1. It was shown that the best test
results forWT-CVANNs andWT-RVANNswere obtained
by 34 input samples. Although WT-RVANN2 achieved
to less test error than WT-CVANN2, the both of two
networks obtained 100% sensitivity, specificity and
average detection rate. However, the optimum
number of input sample was found as ‘8’ by the
CWT-CVANN structure in this study. So, the optimum
CWT-CVANN structure was expressed as CWT-
CVANN3. In this case, considering test results, the
CWT-CVANN structure produced more good results
than WT-CVANN. But, if the number of iteration was
considered, the best results were obtained by WT-
CVANN2 and WT-RVANN2 structures.

In this study, as seen in Table 1, RVANN and CVANN
was trained and tested to classify Doppler signals.
According to obtained results, the training time and
test error of RVANN were higher than those of
CVANN, while the number of hidden nodes in RVANN
was less than that of CVANN. Training and test errors
given in Table 1 were calculated according to pub-
lished paper of Özbay et al. [43].

72 M. Ceylan et al.

Figure 7 Presenting of complex-valued inputs to RVANN
and CVANN. (a) For RVANN and (b) for CVANN.

Figure 6 Optimum number of real and complex discrete
wavelet coefficients for WT-CVANN, WT-RVANN and CWT-
CVANN architectures.

For evaluation of network’s performance, sensi-
tivity, specificity and average detection rate for
optimum networks and RVANN/CVANN were deter-
mined. As seen in Table 2, all of networks in this
study were achieved 100% sensitivity, specificity and
average detection rate.

Although numerical errors were obtained
(Table 1), proposed methods were classified all
subjects, successfully, as seen in Table 2, because
numerical errors only indicate convergence of
actual outputs to target outputs. These error values
were mentioned about distance of targets and
actual outputs but classification success was eval-
uated using proposed algorithms in Section 3.3.

3.3. Calculation of training and test errors

Method we used to calculate the numbers of correct
and incorrect classified complex-valued data and
real-valued data are given below in detail. Further-
more, the performances of the ANN algorithms were
calculated using measurements of sensitivity, spe-
cificity and average detection rate [45].

3.3.1. Calculation of number of correct and
incorrect classified complex-valued data
We developed an algorithm to evaluate the classi-
fication results of WT-CVANN outputs for training
and test data in complex plane. Desired values are

coded ‘‘i’’ and ‘‘1 + i’’ for healthy and unhealthy
data, respectively. The number of correct classified
data in WT-CVANN was calculated according to the
following algorithm [44,45]:

In this algorithm, if output of the node is
0 � output of WT-CVANN � 0.5 then this output is

Classification of Doppler signals using complex-valued artificial neural network 73

Table 1 Training and test results for all structures

Method Optimum
architecture

Optimum
learning rate

Averaged
iteration
numbers

Training time
(averaged
second)

Training error
(% averaged)

Test error
(% averaged)

RVANN 258:10:2 0.1 120 13.65 0.25 1.11
CVANN 129:80:1 1.0 4.5 8.51 0.02 0.04
WT-CVANN1 66:12:1 0.7 16.5 1.04 0.09 0.48
WT-CVANN2 34:12:1 0.7 22 1.45 0.1 0.36
WT-CVANN3 18:12:1 0.9 7 1.11 0.15 0.47
WT-RVANN1 132:6:2 0.9 36 1.56 0.25 0.73
WT-RVANN2 68:6:2 3.0 6.5 1.07 0.06 0.06
WT-RVANN3 36:4:2 5.0 10 1.21 0.22 0.59
CWT-CVANN1 32:4:1 4.0 15.5 1.642 0.23 1.285
CWT-CVANN2 16:30:1 3.0 13.5 1.888 0.243 0.816
CWT-CVANN3 8:6:1 2.0 71 7.351 0.256 0.257

Table 2 The comparative representations of test results to belong to the optimum structures and RVANN/CVANN

Measurement of
classifier performance

RVANN CVANN WT-CVANN2 WT-RVANN2 CWT-CVANN3

Sensitivity, % (SEN) 100 100 100 100 100
Specificity, % (SPE) 100 100 100 100 100
Average detection

rate, % (ADR)
100 100 100 100 100

determined as ‘‘0’’; if output of the node is
0.5 � output of WT-CVANN � 1 then this output is
determined as ‘‘1’’ for real and imaginary parts of
WT-CVANN outputs. The graphical representation of
classification regions can be seen in Fig. 8.

3.3.2. Calculation of number of correct and
incorrect classified real-valued data
We used an algorithm to evaluate the classification
results of WT-RVANN outputs for training and test
data in real plane. Desired values are coded ‘‘0’’ and
‘‘1’’ for healthy and unhealthy data, respectively.
The number of correct classified data in WT-RVANN
was calculated according to the following algorithm
[45]:

4. Conclusions and discussion

ANN is a practicle and valuable tool in the medical
field area for the development of decision support
systems. The actual implementation of ANN analysis
of Doppler signals involves several stages of varying
complexity. Acquisition of data during a routine
Doppler ultrasound examination by means of
tape-recording or employing directly the digital,
takes rather short time and does not excessively
prolong duration the examination. A substantial
amount of training data whichmust be preprocessed
off-line into a suitable format for the presentation
to ANN are required. Following the existing trans-
form methods (Fast Fourier transform, complex
wavelet transform, etc.) used for real numbers,
the conventional classification method must be
applied to the new outcoming complex numbers’
real and imaginary parts separately. However,
CVANNs allow us automatically the advantage of
capturing good rotational behaviour of complex
numbers.

In this paper, the CWT-CVANN andWT-CVANN have
been developed and presented to classify Doppler
signals. In these systems, real/complex discrete
wavelet transform for feature extraction were used
to make an existing CVANN system more effective. A
comparative assessment of the performance of
RVANN, CVANN, WT-CVANN, WT-RVANN and CWT-
CVANN show that more reliable results are obtained
with the WT-CVANN for classification of Doppler
signals. CVANNs are still able to generalize with
good accuracy. However, they take longer time to
train. The aim in developing WT-CVANN and CWT-
CVANN was to achieve better results with relatively
few signal features. All of the structures succeeded
to classify Doppler signals with 100% sensitivity,
specificity and accurarcy rate.

In this study, complex discrete wavelet transform
was used firstly with CVANN in application of bio-
medical signal classification and 100% correct clas-
sification rate and 99% accuracy rate were achieved.
We hope that the performance of proposed net-
works will be better, if the number of healthy and
unhealthy subjects used in training and test data are
increased. In future studies, complex-valued wave-
let neural network [45] can be used in spite of
CVANN for classification of the feature vectors
formed with real and complex discrete wavelet
transforms.

In this study, the results show that a new expert
system developed for the interpretation of the car-
otid artery Doppler signals using presented struc-
tures. Proposed new structures have advantages
over conventional methods such as fast diagnosis,
operating convenience and cost effectiveness. This

74 M. Ceylan et al.

Figure 8 The graphical representation of classification
regions for complex-valued data.

system has better clinical application over others,
especially for earlier survey of the population.

For the future studies, other classification meth-
ods (support vector machine, combined NN,
genetic-trained ANN, etc.) can be used to classify
Doppler signals, and obtained results can be com-
pared with the proposed method in this study.

Acknowledgment

This work is supported by the Coordinatorship of
Selcuk University’s Scientific Research Projects.

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76 M. Ceylan et al.

• Application of complex discrete wavelet transform in classification of Doppler signals using �complex-valued artificial neural network

Introduction

Material and methods

Spectral analysis of carotid arterial Doppler signals

Welch method-averaging modified periodogram for spectral analysis

Real discrete wavelet transform (DWT)

Complex discrete wavelet transform (CWT)

Limitations of wavelet transform

Shift sensitivity

Poor directionality

Absence of phase information

Complex-valued artificial neural network (CVANN)

The results of numerical experiments

The proposed structures and training/test data

Test results

Calculation of training and test errors

Calculation of number of correct and incorrect classified complex-valued data

Calculation of number of correct and incorrect classified real-valued data

Conclusions and discussion

Acknowledgment

References