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Proceedings of the 46th Scandinavian Symposium on Physical Acoustics, Geilo, 29 Jan – 1 Feb 2023
Open Access
1 ISBN 978-82-8123-023-1
Using spectrum-of-spectrum (SoS) filtering to
extract direct and multipath arrivals from a
frequency domain simulation. Comparison with
cepstrum and time-gating methods
Eivind Nag Mosland1, Per Lunde1, Jan Kocbach2
1 Department of Physics and Technology, University of Bergen, P.O. Box 7803, N-
5020, Bergen, Norway
2 NORCE Norwegian Research Centre AS, P.O. Box 22 Nygårdstangen, N-5838,
Bergen, Norway
Contact email: eivind.mosland@uib.no
Abstract
Different signal processing methods are explored to extract individual arrivals from
frequency spectra that include interfering direct and multipath arrivals. Specifically,
spectrum-of-spectrum (SoS) filtering of the frequency spectrum, cepstrum filtering,
and time-signal gating techniques are investigated for an ultrasonic transmit-receive
measurement system modeled using a frequency domain transmitter-medium-re-
ceiver finite element model. The measurement system consists of two coaxially
aligned cylindrical piezoelectric ceramic disks vibrating in air in the frequency range
of their first radial resonance. Findings show that all three processing methods are
suitable for parts of the frequency range. However, SoS filtering demonstrates the
best overall performance for the example case. Cepstrum filtering works well only
when the contribution from the direct arrival is significantly stronger than subse-
quent arrivals. Time-signal gating produces good results only when either the dis-
tance between transducers is long enough or the transducers are sufficiently
damped to achieve a steady-state signal. A recommendation is made to carefully
analyze intermediate steps, or compare different methods, to avoid plausible yet er-
roneous results.
1 Introduction
Frequency spectra containing interfering direct and multipath arrivals are encountered
in experimental measurements and numerical modelling. It is often necessary to either
remove the interference from the multipath arrivals, or to extract a single multipath ar-
rival, for further analysis. Spectrum-of-spectrum (SoS) filtering of the frequency spec-
trum [1], cepstrum analysis [2, 3], and time-signal gating [4], are examples of signal pro-
cessing methods used for these purposes.
Such processing methods are of interest in relation to a high-precision sound veloc-
ity cell for natural gas [5-8], intended for calibration of the sound velocity measurements
in subsea ultrasonic gas flow meters. This sound velocity can potentially be used to cal-
culate the gas energy and mass flow rates in sales and allocation measurement [9]. Dif-
fraction effects must be accounted for to obtain accurate sound velocity measurements,
but existing diffraction correction models, like the piston-type (e.g., [10, 11]) and the sim-
plified finite element diffraction correction models [12, 8, 4], do not provide satisfactory
descriptions of such effects [5-8]. Therefore, a more complete diffraction correction
2
model is needed to reduce and quantify the uncertainty of the sound velocity measure-
ments. The transfer functions for the direct arrival and the first multipath arrival, ex-
tracted from a frequency domain finite element model of the transmitter-medium-re-
ceiver system, can be used to calculate such an improved diffraction correction [13].
Three different signal processing methods are considered in this study. Time-signal
gating [4] is a gating in the time domain by use of Fourier synthesis that is inspired by
the common approach of time gating used in experimental measurements. The cepstrum
filtering method is based on cepstral analysis [3], dating back to the 1960s, and makes
use of the periodicity in the phase angle and natural logarithm of the magnitude of the
frequency spectrum caused by multipath interference. SoS filtering, as presented by the
present authors in [1], has much in common with cepstrum filtering, but filters the real
and imaginary parts of the frequency spectrum instead of the phase angle and natural
logarithm of the magnitude. It was developed to avoid observed limitations in time-sig-
nal gating for highly resonant transducers and short distances, and used in [13].
The objective of the present work is to compare and evaluate the performance of the
three different processing methods using an example case, relevant for the calculation of
the finite-element based diffraction correction for use in a high-precision sound velocity
cell for natural gas [5-7].
Theory for the three processing methods is given in Sec. 2, along with a presentation
of the studied example case, and the specific settings used when the processing methods
are applied to the example case. Results are presented in Sec. 3 and discussed in Sec. 4.
Conclusions are given in Sec. 5.
2 Theory
There are several different signal processing methods available for extraction of individ-
ual arrivals from a frequency spectrum containing multiple interfering arrivals. The
three methods considered in this study are presented and analyzed here for the voltage-
to-voltage transfer function of an example ultrasonic measurement system for gas. How-
ever, the methods can in principle be applied to any problem with interfering multipath
arrivals, e.g., acoustic waves in fluid and elastic/viscoelastic media with or without elec-
tromechanical coupling.
The example case is presented in Sec. 2.1. The three processing methods are briefly
presented in Sec. 2.2, along with the specific settings needed to reproduce the processing
of the example case. For method details, it is referred to [1, 2, 4].
All three methods make use of the Fourier transform. A harmonic time dependence
of is assumed and suppressed here, with forward and inverse Fourier transforms
accordingly defined as
(1)
and
(2)
3
respectively, where is the angular frequency, is the frequency, is time, and
. and can be a time series (signal waveform) and its corresponding fre-
quency spectrum (pressure, voltage, etc.), respectively, or an impulse response and the
corresponding transfer function. Both representations are used in the example consid-
ered here.
2.1 Example case
The three methods are compared for an ultrasonic measurement system like that de-
scribed in [14], with two identical cylindrical piezoelectric ceramic disks of the material
Pz27, with the dimensions 20x2 mm, used as transmitter and receiver, respectively. A
schematic of the system is shown in Figure 1. The disks are coaxially aligned, freely sus-
pended in an unbounded fluid medium and here separated by a distance = 0.10 m. The
fluid medium is air at ambient conditions, modelled without absorption. The disks op-
erate at the frequency range around their first radial resonance in the 100 kHz range. The
voltage-to-voltage transfer function of the system,
(3)
is calculated using a 3D axisymmetric frequency domain transmitter-medium-receiver
finite element model in COMSOL Multiphysics® [15]. is the open-circuit out-
put voltage spectrum at the receiver, and is the excitation voltage spectrum at the
transmitter. is modelled for the frequency range 50-200 kHz, in steps of 0.1
kHz. The unbounded fluid medium is modelled using perfectly matched layers. Model-
ling details can be found in [13]. Figure 2 shows the modelled pressure field at 99.3 kHz,
with the standing waves caused by the interfering direct and multipath arrivals.
2.2 Signal processing methods
2.2.1 Spectrum-of-spectrum (SoS) filtering
The SoS filtering technique utilize the periodicity of the oscillations in caused
by the interference from multipath arrivals. Treating the real and imaginary parts of the
transfer function like waveforms, their forward Fourier transforms can be cal-
culated to yield the spectrum of the spectrum (SoS). The features observed in the SoS are
associated with the individual arrivals, and information associated with the direct arrival
is localized in the lowest part of the SoS [1]. Lowpass filtering can therefore be used to
extract the direct arrival, and it can be applied multiple times to extract multipath arri-
vals. In practice, this SoS filtering technique is implemented in the frequency domain by
applying a digital finite impulse response (FIR) lowpass filter to the real and imaginary
parts of the spectrum, respectively. Method details can be found in [1].
The SoS filtering of the example case is performed as described in [1], using the ap-
plication-specific pre- and postprocessing described therein, i.e., performing the SoS fil-
tering on an intermediate quantity for which rapid changes in the underlying frequency
spectrum are removed. This is done to improve the performance of the SoS filtering tech-
nique around the rapid changes in the frequency spectrum, i.e., around the resonances
4
Figure 1: Schematic of the example measurement system studied here, with two coaxially aligned
cylindrical Pz27 piezoelectric ceramic disks operating in an unbounded fluid medium, air, sepa-
rated by a distance = 0.10 m. The setup is modelled using COMSOL Multiphysics®, with mod-
elling details given in [13].
Figure 2: Magnitude of the pressure field in the example measurement system studied here (Fig-
ure 1, modelled using COMSOL Multiphysics®. The transmitting piezoelectric disk is excited with
a monoharmonic continuous wave with the frequency = 99.3 kHz and an amplitude of 1 V.
in . The intermediate quantity is calculated by dividing by a corre-
sponding far-field voltage-to-voltage transfer function, extrapolated spherically to =
0.10 m [1].
The lowpass FIR filter has a 0.1 dB passband ripple and a passband ranging from 0
to approximately 0.43 ms, where 0.43 ms is where the filter attenuation is -0.1 dB or more.
The filter has a stopband with at least -60 dB attenuation extending from approximately
0.56 ms and upwards. The phase delay of the filter is corrected for [16].
2.2.2 Cepstrum filtering
Analysis and processing of the cepstrum dates back to the 1960s [3] and has found a
wide range of applications, for instance within speech processing, analysis of mechanical
problems, and for multipath and echo removal [2, 3]. The main feature of this method is
its ability to detect and analyze periodicities in the frequency spectrum, like SoS filtering.
There are different definitions of the cepstrum, but the initial step is always to calculate
the natural logarithm of the complex frequency spectrum. For the example case consid-
ered here, that means [2, p. 343]
(4)
where the resulting real part is and the imaginary part is , i.e.,
the phase angle of . The next step is to calculate the Fourier transform of the
5
real part to find the real cepstrum, or of the complex quantity to find the complex
cepstrum [3]. The cepstrum is defined using either the forward [2] or the inverse [3] Fou-
rier transform.
Here, the forward Fourier transform of the real part of Eq. (4) is calculated to yield
the real cepstrum, and a corresponding “imaginary cepstrum” is calculated by taking
the forward Fourier transform of the unwrapped imaginary part of Eq. (4). Moreover,
the filtering is here performed in a similar manner as for the SoS filtering, i.e., by apply-
ing lowpass FIR filters to the real and imaginary parts of Eq. (4), respectively, and by
filtering the intermediate quantity instead of filtering directly, as proposed in
[1]. The imaginary part, which is the phase angle, is unwrapped from prior to filter-
ing and rewrapped to after filtering. Identical lowpass filter settings as for the SoS
filtering, detailed in Sec. 2.2.1, are here used for cepstrum filtering.
2.2.3 Time-signal gating
Transfer functions for individual arrivals can be extracted from using Fourier
synthesis and time gating. A modified version of the time-signal gating method pro-
posed and described in [4] is used here. For each discrete frequency, , in the studied
frequency spectrum, time signal-gating is performed by the following steps:
1. Create a single-frequency sinusoidal time domain voltage excitation signal,
, of a specified length, and calculate its frequency spectrum, .
2. Multiply with to propagate the signal from the transmitter to
the receiver, yielding the spectrum of the receiver’s open-circuit output voltage
signal, . Calculate its inverse Fourier transform to yield the time signal
at the receiver, .
3. Identify the steady-state part of associated with the direct (or multi-
path) arrival, extract this, apply a window function to minimize spectral leakage,
and calculate its forward Fourier transform,
.
4. Calculate the gated for the direct (or multipath) arrival as the ratio
.
The two main modifications relative to the method described in [4] are the extraction
and use of the steady-state part of the signal, as opposed to the whole direct (or multi-
path) signal; and the application of a window function, both of which were suggested in
[4] but not implemented. In addition, it is chosen to extract the gated time signal,
, zero-pad it, and account for the phase delay, instead of using the technique
described in [4, p. 184].
The time-signal gating is here performed for each discrete frequency, , over the
frequency range 50-200 kHz, in steps of 0.1 kHz. A nominal signal length of 2
for
, corresponding to the time difference between the plane wave direct and first mul-
tipath arrivals, is used to maximize the chance for the time signal to reach steady-state
while minimizing its interference with subsequent arrivals. An integer number of peri-
ods is used in . In the Fourier transforms, all input quantities are zero-padded to a
length of prior to the calculation. The upper and lower time limits used when ex-
tracting the (assumed) steady-state part of the direct arrival in are
and
, respectively, i.e., the latter 30 % of the direct signal, excluding the ringing at the
6
end. A discrete Hann window, divided by its mean value to not affect the calculated
output voltage, is applied to this extracted time signal before
is calculated.
The effect of different time gating limits is investigated briefly in Appendix A.
3 Results
Results are shown in Sec. 3.1 for the voltage-to-voltage transfer function of the example
transmitter-medium-receiver FE model described in Sec. 2.1, containing both direct and
multipath arrivals. This is processed using the three methods described in
Secs. 2.2.1, 2.2.2, and 2.2.3 aiming to extract the part of containing only the di-
rect arrival, and the results are shown in Sec. 3.2. In Sec. 3.3, parts of corre-
sponding to different multipath arrivals are shown, obtained using the SoS filtering tech-
nique.
3.1 FE modelling results:
The magnitude, phase angle, and slowly varying phase of are shown in Figure
3 for = 0.10 m over the frequency range 50-200 kHz, calculated using the FE model (see
Sec. 2.1 and [13]). The undesired oscillations caused by multipath interference are clearly
seen in both the magnitude and the slowly varying phase angle. In the magnitude, there
are two peaks around approximately 100 kHz and 115 kHz that correspond to the first
radial resonance of the cylindrical piezoelectric ceramic disks at transmission and recep-
tion, respectively.
The amplitude of the oscillations in the slowly varying phase angle is largest in the
frequency range 160-200 kHz. The irregular pattern in this frequency range is caused by
insufficient frequency resolution relative to the rapid changes in the slowly varying
phase angle and is not a physical effect. Furthermore, the seemingly large-amplitude
oscillations in the frequency ranges 80-100 kHz and 115-135 kHz are not caused by a
physical effect, but by oscillations with relatively small amplitudes close to 180° and
-180°. This results in wrapping of the phase multiple times in each period of the oscilla-
tion, which falsely appear to be oscillations with large amplitudes.
3.2 Processed results: direct arrival
The part of corresponding to the direct arrival is shown in Figure 4 and Figure
5 for = 0.10 m over the frequency ranges 50-200 kHz and 85-130 kHz, respectively. The
results are obtained using SoS filtering, cepstrum filtering, and time-signal gating, re-
spectively, of . The oscillations in the frequency spectrum are in general greatly
reduced by all three methods, indicating that the direct arrival is extracted.
For the magnitude, there is relatively close agreement between all three methods in
the lower part of the frequency spectrum, up to approximately 85 kHz, and in the fre-
quency range 130-160 kHz. Around the radial resonances, i.e., in the frequency range 85-
130 kHz, there is relatively good agreement only between the SoS filtering and cepstrum
filtering results, since the time-signal gating results exhibits some oscillations, as seen in
Figure 5a. In the frequency range 160-200 kHz, there is relatively good agreement be-
tween the SoS filtering and time-signal gating results, while cepstrum filtering yields a
larger magnitude (see Figure 4a).
7
Figure 3: Magnitude (a), phase angle (b), and slowly varying phase angle (c) of shown
for the distance = 0.10 m between the transmitter and receiver, over the frequency range 50-
200 kHz. Modelled by a frequency domain transmitter-medium-receiver FE model [13]. The slowly
varying phase is found by dividing by the plane wave phase term for the direct arrival,
.
For the slowly varying phase angle in Figure 4b and Figure 5b, there is also relatively
good agreement between the different methods, for the same methods and over the same
frequency ranges as for the magnitude results, up to approximately 160 kHz. In the fre-
quency range 160-200 kHz, there is relatively good agreement between the SoS filtering
and time-signal gating results, while cepstrum filtering yields slowly varying phase an-
gles oscillating over the full range of 180° for the frequency range 160-200 kHz. The
deviation in the cepstrum results, relative to the results obtained using the other meth-
ods, is more profound in the slowly varying phase angle than in the magnitude.
At either end of the frequency range, 50-200 kHz, there are gradually decaying rip-
ples in the results obtained using all three processing methods, but to a varying degree.
The ripples at either end decay as the frequency increases from 50 kHz and decreases
from 200 kHz, respectively.
8
Figure 4: Magnitude (a) and slowly varying phase angle (b) of before processing and
the part of corresponding to the direct arrival obtained using SoS filtering, cepstrum
filtering and time-signal gating, respectively. Shown for = 0.10 m over the frequency range 50-
200 kHz.
Figure 5: The same results as shown in Figure 4 but over the frequency range 85-130 kHz, around
the cylindrical piezoelectric disks’ first radial resonance. Magnitude (a) and slowly varying phase
angle (b) of before processing and the part of corresponding to the direct ar-
rival obtained using SoS filtering, cepstrum filtering and time-signal gating, respectively.
9
3.3 Processed results: direct and multipath arrivals
The magnitude of the parts of containing only the direct arrival and the first,
second, and third multipath arrivals, respectively, obtained using SoS filtering, are
shown in Figure 6 for = 0.10 m over the frequency range 50-200 kHz. The iterative pro-
cedure for multipath extraction described in [1] is used.
Up to approximately 160 kHz, the direct arrival appears to correspond to a moving
average of the magnitude of before filtering. This is not so over the frequency
range 160-200 kHz, where the magnitudes of the multipath arrivals increase significantly
relative to the direct arrival. This frequency range is where cepstrum filtering yields de-
viating results compared to the other two methods (see Sec. 3.2). For the frequency range
177-200 kHz, the first multipath arrival has a larger magnitude than the direct arrival.
4 Discussion
The results indicate that all the three studied processing methods, i.e., SoS filtering,
cepstrum filtering, and time-signal gating, can be used to extract the frequency spectrum
of the direct arrival from a frequency spectrum containing interfering multiple arrivals.
However, the methods’ performance varies for the example case of a frequency domain
transmitter-medium-receiver FE model with cylindrical piezoelectric ceramic disk trans-
ducers operating in air over the distance = 0.10 m. The SoS filtering method seems to
yield the best overall results, but both cepstrum filtering and time-signal gating performs
equally well for large (but different) parts of the frequency spectrum. The accuracies of
the methods cannot be quantified since reference solutions of the example case contain-
ing separate arrivals are not available. However, the relatively close agreement of results
obtained in different parts of the frequency spectrum using different processing methods
gives a strong indication that fair estimates are achieved.
The two filtering methods (SoS, cepstrum) make use of the periodicity, or oscilla-
tions, in the frequency spectrum containing multiple arrivals, such as those caused by
standing waves between the transducers in the frequency domain simulations here. One
of the main limitations of such methods is that the undesired oscillations and the under-
lying frequency spectrum should be separable in the Fourier transformed spectrum. This
may not be the case, especially when the underlying frequency spectrum have sharp
resonances or other rapid changes. These will extend over a wide time band in the Fou-
rier transformed spectrum and may therefore be adversely affected by the filtering. It is
especially challenging when the time between the consecutive arrivals decreases, since
this moves the features in the Fourier transformed spectrum associated with the separate
arrivals closer together. This is discussed for SoS filtering in [1], and also applies to
cepstrum filtering. Filtering of an intermediate quantity without rapid changes in the
frequency spectrum can improve the results [1], and this has been successfully applied
here for both the SoS filtering and the cepstrum filtering. Other filtering-related chal-
lenges are the effects of the passband ripple and the gradual roll-off of the filter [1].
The gradually decaying ripples at either end of the frequency spectra in Figure 4
and Figure 6 are effects resembling the Gibbs effect, both for the filtering methods and
the time-signal gating results. In Figure 6, they increase for each subsequent arrival,
probably due to the iterative procedure used which inherits these effects from the pre-
ceding arrivals. The ripples are an inherent feature of the filtering process and may be
10
Figure 6: Magnitude of and the parts of containing the direct, first multipath,
second multipath, and third multipath arrivals, respectively, obtained using SoS filtering, shown
for = 0.10 m over the frequency range 50-200 kHz.
difficult to reduce. However, they were omitted from the results presented in [13] simply
by modelling and processing a slightly larger frequency band than what was presented
in the figures.
SoS filtering and cepstrum filtering yields almost identical results for the example
case over the frequency range 50-160 kHz, as seen in Figure 4 and Figure 5Error! Refer-
ence source not found.. Upwards from 160 kHz, the magnitude in the cepstrum results
starts to deviate from the SoS filtering and time-signal gating results, and the phase angle
results starts to oscillate. The reason for this is that the cepstrum method inherently as-
sumes a direct arrival (much) stronger than the subsequent arrivals [2]. The limitations
caused by this assumption are discussed in Appendix B. From the increasing deviation
between the cepstrum results and the SoS filtering and time-signal gating results in Fig-
ure 4, it can be inferred that the multipath arrivals become gradually too strong to fulfill
this requirement, starting from approximately 160 kHz. For the frequency range 177-200
kHz, the magnitude of the first multipath arrival is larger than that of the direct arrival,
as seen in Figure 6, possibly due to near-field effects.
Time-signal gating may be more intuitive than the filtering methods, and it allows
for analysis of the intermediate time domain signals. Relatively good results are obtained
for most of the studied frequency band for the example case, except for the frequency
band 90-125 kHz, see Figure 4 and Figure 5. At these frequencies, the part of
associated with the direct arrival does not reach steady-state before the arrival of the first
multipath signal, for the case considered here with resonant transducers at a relatively
short distance ( = 0.10 m). The modelled needs to include a relatively large
frequency range for the Fourier synthesis to yield an accurate description of the time
domain signal at the receiver. The frequency resolution of is inversely propor-
tional to the length of the time traces, which must be long enough to include the multi-
path arrivals of interest. A long time trace also minimizes the effects of folding in the
time domain. Time gating limits are here chosen so that only the assumed steady-state
11
part of the signal is used to calculate the gated transfer function. Although this is as-
sumed to be the best approach when steady-state is reached, it is not necessarily the best
choice when this is not the case, as discussed in Appendix A.
Only SoS filtering is used to find the multipath results in Figure 6, since it is only
SoS filtering that yields relatively good results for the entire studied frequency range, for
the example case with = 0.10 m. However, it is assumed that cepstrum filtering would
yield similar multipath results as SoS filtering, for frequencies up to approximately 160
kHz, and that time-signal gating would yield similar multipath results as SoS filtering
outside the frequency band around the resonances.
It should be noted that inferior performance of SoS filtering, cepstrum filtering, or
time-signal gating is not always apparent from the processed results alone. Critical re-
view of the results is therefore important, for instance by analyzing intermediate results,
such as the Fourier transform of the frequency spectrum for SoS filtering and cepstrum
filtering, or the propagated time signals for time-signal gating. It may also be beneficial
to compare results obtained using different methods.
The SoS filtering method is shown to have the best overall performance for the ex-
ample case studied here, i.e., best performance for the special case where the direct arri-
val is not significantly stronger than the subsequent arrivals. It is expected that the
cepstrum filtering method will perform similar to the SoS filtering method for cases
where the multipath arrivals are successively decreasing in amplitude, given that filter-
ing of an intermediate quantity, as proposed in [1], is applied for both filtering methods
when required (see Sec. 2.2).
An electroacoustic example was studied here, but the processing methods may be
applied to problems with any type of acoustic waves (in fluid or elastic media), with or
without electromechanical coupling, as long as there are interfering multipath arrivals.
5 Conclusions
The SoS filtering, cepstrum filtering, and time-signal gating methods for extraction of
individual arrivals from a frequency spectrum that contains interfering multiple arrivals
have been evaluated for an example case of a frequency domain transmitter-medium-
receiver FE model of two coaxially aligned cylindrical piezoelectric ceramic disks oper-
ating in air in the frequency band of the first radial resonance, over a distance = 0.10 m.
All three methods can be used for selected parts of the frequency range, but based on the
presented results it is concluded that SoS filtering of the frequency spectrum has the best
overall performance for the example case. The example case is challenging for the
cepstrum method due to frequency bands where the first multipath arrival is stronger
than the direct arrival, and for the time-signal gating method due to resonant transduc-
ers and a relatively short distance. The short distance and resonant transducers are also
challenging for the SoS and cepstrum filtering methods, but this is countered by the fil-
tering of an intermediate quantity, as proposed in [1] (see Sec. 2.2). When determining
which method to use for a specific application, it is recommended to study intermediate
results, such as the spectrum of the spectrum (SoS), and compare results obtained using
different methods.
12
Acknowledgements
This work is part of the SUBFLOW project, Subsea ultrasonic gas flow metering, funded by
the Research Council of Norway under grant 303169. The authors thank Magne Ves-
trheim, University of Bergen, Norway, for useful discussions and feedback during this
work.
Appendix A: Effect of time window used in time-signal gating
As stated in Sec. 2.2.3, the time gating window can affect the time-signal gating results.
This is shown here with time gating defined so that either the signal from
to
,
denoted time window A, or from
to
, denoted time window B, is extracted and
used to calculate the part of associated with the direct arrival. The time gating
windows corresponds to the whole and latter 30 % of the excitation signal, excluding the
ringing at the end. Due to the large dynamic range of the compared to the os-
cillations in the time-signal gating results in Figure 5, Figure 7 shows results for the rel-
ative quantity
. Here
is the time-signal gating result
with different time gating limits, and
is the SoS filtered result. It is seen from
Figure 7 that for the example case here, with = 0.10 m, the deviation is largest in both
magnitude and phase angle when time window A is used. However, the deviation covers
a smaller frequency range than when tine window B is used, i.e., close to the two reso-
nance peaks in . The results indicate that the two different time gating win-
dows yield similar results well outside the resonances. If the distance was increased or
more broadband transducers were used, it is expected that using the steady-state part of
the signal would yield the best results close to the resonances.
Figure 7: Deviations of magnitude (a) and phase angle (b) in the part of associated with
the direct arrival, when comparing time-signal gating results with two different window time-limits
to the SoS filtering result for = 0.10 m over the frequency range 85-130 kHz. Note that the
processing is performed using the 50-200 kHz frequency range, but only the frequency range
around the disks’ first radial resonance is shown.
13
Appendix B: Cepstrum filtering assumption
The cepstrum filtering method presumes that the multipath arrivals are weaker than the
direct arrival [2, p. 341]. Figure 6 shows that this is not always the case, and in such cases
the cepstrum filtering method fails. A simplified plane wave representation of the exam-
ple case is used here to explain this limitation and consequent failure of the cepstrum
filtering method. By considering only the direct arrival and the first multipath arrival,
and reflection at each interface as from a rigid plane wall of infinite extent, the total pres-
sure can be given as
(1)
where
is the plane wave pressure of the direct arrival, consisting of
an amplitude and a phase term
. Similarly,
is the plane
wave pressure of the first multipath arrival. A harmonic time dependence of is as-
sumed and suppressed. Calculating the natural logarithm of divided by the
plane wave phase term for the direct arrival yields
(2)
where
is the magnitude and
is the phase angle
of the total pressure divided by the plane wave phase term for the direct arrival.
Focusing on the real part of Eq. (2) as an example, it can be written as
(3)
Inside the natural logarithm and the square root in Eq. (3), there are an
term, an
term, and an oscillating term. This indicates that the condition needs to be ful-
filled for cepstrum filtering to yield a good estimate of , since otherwise the
term
will not be removed by the filtering.
Additional conditions apply, following a similar line of reasoning as above, when
estimating in the presence of a second multipath arrival,
. If the
iterative approach discussed for the SoS filtering in [1] is used, the first step is to estimate
under the conditions and . To determine , the estimated
is subtracted from the total pressure before performing cepstrum filtering, under
the condition . In general, it is required that and that is
much stronger than the sum of all subsequent multipath arrivals to successfully estimate
for the th arrival.
14
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