IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 52, NO. 3, JUNE 2003 809
A Novel Time-Domain Method of Analysis
of Pulsed Sine Wave Signals
Alireza K. Ziarani, Member, IEEE, Adalbert Konrad, Fellow, IEEE, and Anthony N. Sinclair, Member, IEEE
Abstract—Sine wave packs are used in nondestructive evalua-
tion of materials, most commonly in the form of ultrasonic waves.
An example of such methods is the use of electromagnetic acoustic
transducers (EMATs) in the evaluation of metallic structures. Re-
flected EMAT signals are often highly polluted by noise. Elimina-
tion of noise and extraction of peak amplitudeare important signal
processingtasks associated with the analysis of EMAT signals. This
paper presents a method of noise elimination and information ex-
traction for pulsed sinusoids. The functionality of the proposed
method is exemplified through noise reduction and peak detection
of EMAT signals. The proposed method offers a simple and robust
technique of signal analysis which is suitable for real-time indus-
trial applications since it requires a relatively low level of compu-
tational resources.
Index Terms—Electromagnetic acoustic transducers (EMATs),
noise elimination, peak detection, pulsed sine waves, ultrasonic
nondestructive evaluation (NDE).
I. INTRODUCTION
T
RANSMISSION of bursts of ultrasonic waves into a
medium is a well-known technique for acquisition of
useful information about the structure of the medium under
study. Ultrasonic waves are often used in nondestructive
evaluation of materials. Various techniques for generation of
ultrasonic waves for NDE of metallic structures exist among
which electromagnetic acoustic transduction has attracted
considerable attention over the years due to its favorable
noncontact testing feature. However, this method suffers from
serious shortcomings due to the poor quality of the received
signals which are often highly polluted by noise [1], [2]. A
significant amount of literature deals with the various methods
of improvement of the quality of received electromagnetic
acoustic transducer (EMAT) signals. Coil design considerations
constitute a major research trend in this regard [3]. Electromag-
netic field computation has been used to assist the design of
coil geometry [4]. Also, there has been considerable interest in
adapting signal processing methods for noise elimination [5],
[6] and flaw identification [7], [8].
In the NDE of materials using bursts of ultrasonic energy,
i.e., pulsed sine waves, it is often desirable to detect the peak
Manuscript received November 26, 2001; revised January 31, 2003.
A. K. Ziarani is with the Department of Electrical and Computer En-
gineering, Clarkson University, Potsdam, NY 13699-5720 USA (e-mail:
aziarani@clarkson.edu).
A. Konrad is with the Edward S. Rogers, Sr. Department of Electrical and
Computer Engineering, University of Toronto, Toronto, ON M5S 3G4 Canada
(e-mail: konrad@ecf.utoronto.ca).
A. N. Sinclair is with the Department of Mechanical and Industrial
Engineering, University of Toronto, Toronto, ON M5S 3G8 Canada (e-mail:
sinclair@mie.utoronto.ca).
Digital Object Identifier 10.1109/TIM.2003.814688
of the received signal, its amplitude and its time of arrival [1].
In order to detect such features of the received noisy signal,
it is necessary to improve the signal quality by elimination of
electrical noise. Time averaging of a repeated signal can re-
duce random electrical noise. Its main problem, however, is the
long measurement time needed which limits its applicability to
real-time NDE [2]. Narrow band filtering is used as the primary
method to improve signal quality. A notch filter with a sharp
notch is effective in eliminating the electrical noise, but renders
the equipment sensitive to potential frequency drifts. Moreover,
the output signal of such a filter has to be analyzed for detection
of the peak and its arrival time. Fourier transform analysis could
be used but, in this case, all the time information about the peak
position will be lost [1].
Preliminary results of the application of a recently intro-
duced time-domain method of extraction of sinusoidal signals
of time-varying nature buried under noise to EMAT signal
quality enhancement were presented in [9] by the aid of com-
puter simulations. This paper presents an improved method of
noise elimination for pulsed sine wave signals. The aim is to de-
tect the arrival time of the envelope peak which is necessary in
order to make accurate travel time measurements of ultrasonic
echoes. The structure of the proposed method is presented and
its behavior is demonstrated by the aid of computer simulations.
Experimental verification of the performance of the proposed
method is then exemplified by the refinement and information
extraction of EMAT signals.
II. E
XTRACTION OF SINUSOIDS IN NOISE
This section reviews the core algorithm which is employed
in the structure of the proposed method of analysis of highly
polluted signals presented in Section III. Consider a sinusoidal
signal polluted by some noise of unknown frequency composi-
tion and expressed by
(1)
where
represents the totality of the imposed noise, and
are potentially time-varyingamplitude and frequency of the sine
wave, respectively, and
is the constant phase of the sinusoid.
The total phase of the sine wave is
. If time-variations
are sufficiently slow, parameters
, , and are constant values
, , and within any short time interval.
Least squares error between the input signal
and the si-
nusoidal signal
embedded in may be min-
imized by employing a gradient descent method [10]. The re-
sult is the following set of nonlinear differential equations to
govern the dynamics of a signal processing algorithm aimed at
extracting the potentially time-varying sinusoidal signal buried
0018-9456/03$17.00 © 2003 IEEE
810 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 52, NO. 3, JUNE 2003
in without any assumption on the composition of the im-
posed noise:
(2)
(3)
(4)
where error
represents the difference between the input
signal polluted by noise and the extracted sinusoid, i.e.
(5)
In the above equations,
, , and are estimated values of am-
plitude, total phase and frequency of the extracted sinusoidal
signal
, respectively. Parameters , , and are posi-
tive numbers which determine the speed of the algorithm in the
estimation process as well as in tracking variations in the char-
acteristics of the input signal over time.
The following theorem, proved in [11], deals with the exis-
tence, uniqueness and stability of a periodic orbit for the dy-
namical system described by (2)–(5).
Theorem 1: Assume that
is given by (1) wherein all
the parameters are unknown but bounded. The dynamical
system represented by (2)–(5) has a locally unique and hyper-
bolically stable periodic orbit
in a close neighborhood of
.
This theorem guarantees i) the convergence of the solution
of the dynamical system to the periodic orbit associated with
the sinusoidal signal in
and ii) the tracking of its variations
over time. In terms of the signal processing performance of the
algorithm, it extracts a sinusoidal component of its input signal,
directly estimates its amplitude, phase and frequency, and adap-
tively tracks their variations over time.
The structure of the core algorithm is very simple and con-
sists of only a few arithmetic operations. Fig. 1 shows a block
diagram representation of the algorithm. The initial point of the
flow of the dynamics is set by the values of initial conditions
of integrators generating amplitude
, phase , and fre-
quency
. In the presence of multiple sine waves, each sinu-
soidal component is specified by its frequency. Therefore, the
initial condition of the frequency integrator (shown explicitly in
Fig. 1) is of particular importance; the algorithm extracts that si-
nusoidal signal whose frequency is closest to the pre-set initial
condition of the frequency integrator. Numerical implementa-
tion of the core algorithm is straightforward. For example, the
following is the discretized form of the governing equations of
the algorithm in which a first order approximation is assumed:
where is the sampling time and is the time step index.
The Matlab Simulink™ programming environment is used to
produce the graphs presented in this paper. In the first numer-
ical experiment, the input signal
consists only of a pure
Fig. 1. Employed core algorithm in a block diagram representation.
Fig. 2. Illustrationof the convergence of the core algorithm. The input signal is
a unit-amplitude sine wave. The top graph shows convergence of the dynamics
to the periodic orbit associated with the extracted sinusoid; the bottom graph
shows the flow of the dynamics in the time domain.
unit-amplitude sinusoid. Fig. 2 shows convergence of the algo-
rithm in response to such an input signal. The pre-set initial fre-
quency in Fig. 2 is the same as the frequency of the incoming
sinusoid. To show the frequency retrieval property of the algo-
rithm, in another numerical experiment the initial frequency of
the algorithm is deliberately set to be about 50% off the fre-
quency of the incoming sinusoid. Fig. 3 shows convergence of
the algorithm in frequency.
The core algorithm exhibits a high degree of immunity with
respect to noise. Fig. 4 shows this property. The noise present in
the input signal is of about the same energy as the polluted sinu-
soid (i.e.,
). The convergence is achieved in a few
cycles with a steady state error of about 5% in this experiment.
The estimation accuracy is a function of the degree of the pollu-
tion in the input signal on the one hand and the desired conver-
gence speed on the other hand. For example,the estimation error
in the experiment of Fig. 4 can be reduced to 0.5% by reducing
the values of parameters
, by a factor of about 10. This
results in a convergence time of about 10 X longer. Therefore,
the estimation accuracy is fully controllable by the adjustment
ZIARANI et al.: NOVEL TIME-DOMAIN METHOD OF ANALYSIS 811
Fig. 3. Illustration of the frequency tracking property of the core algorithm.
The top graph shows convergence of the algorithm to the periodic orbit
associated with the extracted sinusoid; the bottom graph shows the same
phenomenon in the time domain.
Fig. 4. Performance of the core algorithm in the extraction of a sinusoid and
its amplitude, present in a highly noisy input signal.
of parameters , , and in a trade-off with convergence
speed. For each particular application, one can choose a suitable
set of parameters. In general, as the frequency of the operation
increases, proportionally higher values of parameters
and
have to be used to retain the same convergence speed in terms
of the required number of cycles for convergence. Therefore, it
is reasonable to divide the values of these two parameters by
the nominal frequency of the input signal when expressing their
values. A typical set of parameters, used in the simulations of
this chapter, is
, and , where the
values of
and are normalized with respect to the nominal
frequency of the incoming signal. It is noteworthy that the al-
gorithm is very robust with respect to variations in the values
of parameters; variations of up to 50% of magnitude in the pa-
rameters have been observed to have negligible effect on the
performance.
Fig. 5. Performance of the core algorithm in the extraction of a pulsed sinusoid
and its amplitude, present in a noisy input signal.
Fig. 6. Illustration of the convergence delay of the core algorithm in detecting
the peak of the input signal. The line style (solid line for the input, dotted line
for the output, and dash-dotted line for the amplitude) is used consistently in all
the figures of this paper.
The presented core algorithm performs very well when the si-
nusoidal component of the input signal is amplitude modulated.
The algorithm basically looks for a sine wave; in its absence,
i.e., when its amplitude is zero, it returns zero for the estimated
amplitude and generates a zero-amplitude signal. Fig. 5 illus-
trates this point. Notice that the output signal follows the sinu-
soidal component of the input signal with a delay which is due
to the convergence time of the algorithm. This effect is more
clearly illustrated in Fig. 6 where the algorithm is excited by a
short-time (half cycleof a sine wave)signal. Observe that the de-
tected place of the peak is delayed. This time delay is a complex
function of parameters
, , and the values of initial con-
ditions; fortunately, it is a relatively flat function of frequency;
for each parameter setting this delay is a constant number. The
value of this delay, most conveniently measurable by simulation
as done in Fig. 6, can be used to correct the arrival time of the
812 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 52, NO. 3, JUNE 2003
Fig. 7. Gain and phase characteristics of the inlaid notch filter. The center
frequency of the notch filter in these graphs is assumed to be
f
=
100 KHz
for ease of visualization.
peak; it is sufficient to just subtract this constant number from
the arrival time detected by the algorithm.
III. P
ROPOSED METHOD
The proposed method of time-domain signal analysis consists
of a) the elimination of noise from the input signal by passing
it through the core algorithm, b) the estimation of amplitude of
the extracted sinusoid, and c) the comparison of instantaneous
amplitude with a defined threshold to determine the peak. Note
that b) is accomplished without further effort since the noise
elimination algorithm automatically provides a direct estimate
of the amplitude. The time of arrival of the peak is conveniently
detectable by time-gating of the peak detection scheme; the esti-
mated arrival time is then to be reduced by a constant time delay.
In order to further enhance the noise immunity of the algo-
rithm, the use of a simple second order band pass filter at the
input of the core algorithm is proposed. This filter is imple-
mented in digital form. If the nominal frequency of the input
signal is
, the transfer function of the band pass
filter is given by
This filter improves the signal to noise ratio (SNR) of the input
signal of the core algorithm. However, it also introduces unde-
sired attenuation and phase delay (see Fig. 7), especially if any
drift from the nominal frequency occurs, which may happen as
a result of equipment aging and other reasons. Due to the ca-
pability of the algorithm to estimate all the parameters of the
extracted sinusoid, both these effects can be compensated for as
demonstrated in Fig. 8.
IV. E
XPERIMENTAL VERIFICATION
Data from an experimental setup for conducting EMAT
tests are used to demonstrate the performance of the proposed
method in signal refinement and analysis. The modulator
Fig. 8. Block diagram of the enhanced algorithm for noise elimination and
peak detection of pulsed sinusoids.
Fig. 9. Performance of the proposed method in noise elimination and peak
detection of a highly noisy EMAT signal. The top graph shows the input signal.
The refined EMAT signal and its amplitude are shown in the bottom graph.
generating high current/voltage signals to feed the EMAT trans-
mitter produces a smoothly curved pulsed sinusoidal signal at
about 1.8 MHz. The received EMAT signal is amplified and
sent to a digital oscilloscope to measure the voltage across the
receiver coil. Digitization is done at a high sampling frequency
(102.4 MHz) to preserve signal integrity.
Fig. 9 shows the performance of the proposed method for
a set of 1024 points of recorded data. The received sine wave
pack is reflected back from the bottom of a metallic cube under
examination. In order to produce a worst-case scenario, no at-
tempt was made to obtain good signal quality while recording
theEMAT signals. The SNR of the input signal is estimated to be
. It is observed that the proposed method is fully capable
of removing the electrical noise. However, the ripples detected
during the periods of absence of the pulsed sine wave are some-
what undesirable. They are not electrical noise as will be later
justified by reference to the frequency spectra; such ripples may
be due to the pulse generator, with a possible contribution of ul-
trasonic echoes from small flaws in the test specimen. Fig. 10
compares the frequency spectrum of the input signal to that of its
refined variant. It is clear that the algorithm does not affect the
frequency content of the desired signal and acts on the electrical
noise only. Therefore, it is justified that what is passed through
the signal refiner is in fact of sinusoidal shape at the EMAT op-
erating frequency, whatever its interpretation may be.
In an attempt to provide a “correct” version of the received
EMAT signal, the signal received by the EMAT receiver has
been averaged 2048 times by a digital oscilloscope. The resul-
tant averaged signal has 1024 data points. This averaged signal
ZIARANI et al.: NOVEL TIME-DOMAIN METHOD OF ANALYSIS 813
Fig. 10. Frequency spectrum of the noisy input and clean output signals.
Fig. 11. Performance of the proposed method in further noise elimination and
peak detection of an EMAT signal whichhas been averaged 2048 times. The top
graph shows the input signal and the output signal and its detected amplitude.
The frequency spectra of the input and output signals are shown in the bottom
graph.
was observed to be still a bit noisy. It was then used as the input
signal to the proposed algorithm. Fig. 11 shows further refine-
ment achievedby the proposed algorithm. It goes without saying
that the time of arrival as detected by the algorithm in both cases
of Figs. 9 and 11 has to be shortened by the amount of the con-
vergence time-delay, which was numerically determined to be
about one cycle for this setting of parameters.
V. C
ONCLUSION
The effectiveness of a novel time-domain method of signal
analysis is presented by demonstrating elimination of noise, es-
timation of arrival time, and detection of peak amplitude value
for EMAT signals. However, the methodology is general and
may be employed in signal conditioning and information ex-
traction of any pulsed sinusoidal signal which is highly polluted
by noise and whose frequency may also vary with time. The
main features of the proposed method are its high noise im-
munity and robustness while demanding limited computational
resources due to the simplicity of its structure. These features
render the proposed algorithm favorable for industrial applica-
tions where real-time operation is desirable.
R
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Alireza K. Ziarani (S’99–M’02) received the B.Sc. degree in electrical and
communication systems engineering from Tehran Polytechnic University,
Tehran, Iran, in 1994, and the M.A.Sc. and Ph.D. degrees in electrical
engineering from the University of Toronto, Toronto, ON, Canada in 1999 and
2002, respectively.
He joined the Department of Electrical and Computer Engineering, Clarkson
University, Potsdam, NY, in 2002, where he is presently an Assistant Professor.
He is the co-founder of Signamatic Technologies, Toronto. His research inter-
ests include nonlinear adaptive signal processing, biomedical engineering, em-
bedded systems design, and the theory of differential equations.
