Journal of Signal and Information Processing, 2013, 4, 439-444
Published Online November 2013 (
Open Access JSIP
Optimization of Parametric Periodograms for the Study of
Density Fluctuations in a Supersonic Jet
Catalina Elizabeth Stern Forgach, José Manuel Alvarado Reyes
Facultad de Ciencias, Universidad Nacional Autónoma de México, México City, México.
Received October 24th, 2013; revised November 20th, 2013; accepted November 25th, 2013
Copyright © 2013 Catalina Elizabeth Stern Forgach, José Manuel Alvarado Reyes. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided
the original work is properly cited.
In our research on the density fluctuations of a supersonic jet we were confronted with a quite difficult problem. In the
power spectrum obtained either with a spectrum analyzer, the peaks of the two of the modes that we wanted to iden tify
overlapped. We needed to find a signal processing method that would resolve the two main frequencies. We made a
thorough investigation of several methods and thought that parametric periodograms were the appropriate tool. The use
of parametric periodograms in signal processing requires constant training. The proper application of this tool depends
on the determination of the number of parameters that has to be used to best model a real signal. The methods generally
used to determine this number are subjective, depending on trial and error and on the experience of the user. Some of
these methods rely on the minimization of the estimated variance of the linear prediction error , as a function of the
number of parameters n. In many cases, the graph vs n doesn’t have a minimum, and the methods cannot be used.
In this paper, we show that there is a strong relationship between and the frequency resolution
Δ. That is, as we
Δ, we obtain graphs of vs n that present at least one minimum. The spectrum obtained with this opti-
mal number of parameters, always reproduces the frequency information of the original signal. In this paper, we present
basically the signal processing of the data obtained in a Rayleigh scattering experiment on a supersonic jet that has also
been designed by the authors.
Keywords: Parametric Periodograms; Density Fluctuations; Supersonic Jet
1. Introduction
Kovasznay in 1953, in a perturbation analysis of Navier
Stokes equations, classified weak fluctuations in three
independent modes: vortical, acoustic and entropic.
Acoustic and entropic modes in a jet are studied by ana-
lyzing density fluctuations inside and outside the flow
The traditional way to study aero-acoustic noise pro-
duced by a jet is through correlations of signals acquired
by a three dimensional microphone array in a far-field
[4-6], and from there extrapolate to locate the acoustic
sources inside the jet.
Besides the extreme complexity of the metho d, the in-
verse problem in acoustics does not allow to determine
uniquely the source. Also, the diffraction of the acoustic
waves by the mixing layer cannot be taken into account.
In the late seventies at the Ecole Polytechnique in
France, a non-intrusive optical technique was developed.
It can be used as a microphone for a single wave vector
[4]. Information about the density fluctuations can be
obtained by means of the light scattered by the molecules
(Rayleigh scattering) of a transparent gas in motion [4-9].
The signal that is obtained from the photo detector is
proportional to the spatial Fou rier tran sform as a functio n
of time of the density fluctuation, for a wave vector given
by the optical set-up.
Originally these signals were processed by using a
spectrum analyzer. The acoustic and entropic modes
should appear as two distinct peaks. The entropic mode
corresponds to fluctuations carried by the flow; the aco us-
tic mode should propagate at the speed of sound.
Figure 1 shows the spectrum for fluctuations travel-
Optimization of Parametric Periodograms for the Study of Density Fluctuations in a Supersonic Jet
-8.00E+06 -3.00E+06 2.00E+06 7.00E+06
Figure 1. Spectrum obtained by means of the spectrum ana-
lyzer. Starting at zero, a lobe describing the density fluc tua-
tions within the bandwidth studied may be observe d.
ling perpendicular to the flow. In this case, entropic
fluctuations should be random about zero, and acoustic
wave, for the specific set up, should be at around 2 MHz.
The spectrum analyzer did not resolve the peaks [5-7].
To ameliorate the resolution, Burg’s parametric pe-
riodograms were used as a processing tool. Figure 2
shows a comparison between the spectral density ob-
tained by the spectrum analyzer and the one obtained
with Burg’s periodograms.
The use of the periodograms allowed the identification
of both peaks [7-10].
However, the determination of the optimal number of
parameters is not an easy task. The methods that exist are
based on the number of parameters, which minimize the
estimated variance of the prediction erro r . But there
is not always a minimum. We show that it is possible to
optimize these methods, when the frequency resolution is
2. Spectral Density with Periodograms
Much has been written about periodograms. In 1988,
Kay in his book Modern Spectral Estimation, Theory &
Application [12] shows the characteristics of periodo-
grams with examples and programs. In his article Spec-
trum Analysis, a Modern Perspective [13], he describes a
detailed summary of periodograms and explains their
properties, advantages and disadvantages. Other authors
have also written about periodograms [14-17], highlight-
ing their properties in most of the literature.
John G. Proakis [14] states that “The experimental re-
sults given in the references just cited indicate that the
model-order selection criteria do not yield definitive re-
sults.” On the other hand, he states that “It is apparent
that in the absence of any prior information regarding
the physical process that resulted in the data, one should
try different model orders and different criteria and, fi-
Figure 2. Comparison of spectra. The spectrum analyzer
clearly shows one lobe, while the spectrum obtained with
the Burg parametric periodogram shows three lobes.
nally, consider the different results”. Lik ewise , Stev en M.
Kay mentions in his article [13] that “Thus the prediction
error power alone is not sufficient to indicate when to
terminate the search” and his comment that led to the
development of our work is “In the final analysis, more
subjective judgment is still required in the selection of
order for data from actual processes than that required
for controlled simulated computer processes”. Proakis
and Kay state that, even today, the use of parametric pe-
riodograms requires the experience of the user, so the
“trial and error” use of these techniques allows a very
subjective manner of calculating the number of parame-
ters required to model a given signal. This work proposes
that, by considering th e frequency resolution, s
N, and
using parameter estimation techniques, it is possible to
calculate the number of parameters required to model
objectively the spectrum of a given signal.
3. Parameter Selection Techniques
The ARMA (Auto Regressive Moving Average), AR
(Auto Regressive) and MA (Moving Average) models,
used by the group of parametric methods to develop al-
gorithms that allow us to estimate the spectral density of
a signal are varied, but they all depend on parameter es-
timation calculations. One of the most important aspects
of using the AR model is the selection of the p order. As
a general rule, very few parameters flatten the spectrum
with few lobes and too many parameters introduce low-
level spurious peaks in the spectrum [12-16].
In the continued use of periodograms, the practice and
“trial and error”, is what, until today, makes the user
skillful in the application of this type of tools. There are
methods that evaluate the “optimal” number of parame-
ters for modeling a signal, based on the determination of
the number of parameters that minimize the estimated
variance of the prediction error . Table 1 shows
some of the most employed methods [11] to determine
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Optimization of Parametric Periodograms for the Study of Density Fluctuations in a Supersonic Jet 441
the number of parameters in a parametric periodogram.
But, as we mentioned above, even by using the methods
of selection of parameters, it is not possible to determine
with certainty the appropriate number of parameters to
evaluate the spectrum with parametric periodograms.
4. Applications and Results
The following exercise will demonstrate how vulnerable
are the methods traditionally suggested to determine the
number of parameters of a parametric periodogram. For
this purpose a signal with four frequency components, 1
MHz, 1.01 MHz, 1.05 MHz and 1.1 MHz is acquired.
This signal was sampled at a frequency
and has 512 samples. This sampling frequency perfectly
meets the Nyquist theorem requirements, in addition of
having a frequency resolution . The num-
ber of parameters was evaluated by the methods given in
Table 1 and proposed in the literature [11-14]. The re-
sults are shown in Figure 3. The linear prediction error
depends on the number of samples N, and on the number
of parameters, p.
z9765 H
Both the FPE and AIC methods seem to agree in the
minimum in Figure 3; approximately 50 parameters,
which minimize . The MDL technique shows a
minimum at about 20 parameters.
Figure 4 shows the spectrum of the signal using
Burg’s parametric periodogram which is compared to the
FFT of the time signal. To obtain the Burg periodogram,
the number of parameters suggested by the techniques
described in Table 1 was used. It should be noticed, that
the expected frequency components are not observed.
This is the result of the frequency resolution [11,17] ob-
tained with the selected variables, the number of samples
and the sampling frequency; they do not allow the dif-
ferentiation of components standing close by.
To solve the problem, many processing experts could
propose to increase the sampling rate or the number of
samples. Both criteria are ambiguous and based on ex-
perience rather than obj ectivity.
Our suggestion is to consider the spectral resolution as
stated in earlier papers [11,17]. We propose to increase
the resolution to , and reevaluate the “op-
timal” number of parameters. Figure 5 shows the corre-
sponding results. From this graph it is possible to distin-
305 HzfΔ≅
Table 1. Methods for evaluation of the selection of the order p.
FPE: Final
Prediction Error
AIC: A Kaike
Information Criter ion
MDL: Minimum
Description Length
MDLln ln
Figure 3. Evaluation of parameter selection techniques,
Table 1, for a signal, with four frequency components,
and 512=N 5MHz.f=
Figure 4. Burg spectra using the number of parameters
suggested by parameter evaluation methods, using Table 1
we calculate the number of parameters, and compared
against the spectrum obtained by FFT.
Figure 5. Parameter prediction methods, using Table 1 we
calculate the number of parameters, for a signal with four
frequency components: 1 MHz, 1.01 MHz, 1.05 MHz and
1.1 MHz.
guish the minimum , which for the FPE and AIC
techniques is around 700 parameters, while with the
MDL technique is minimi zed wi t h 13 0 param et e rs.
Using both suggestions, we obtain the spectra shown
in Figure 6.
It is important to note two facts in this exercise.
N 1) The frequency resolution criterion, and not the tra-
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Optimization of Parametric Periodograms for the Study of Density Fluctuations in a Supersonic Jet
ditional “trial and error” method, is considered when
evaluating the number of parameters. The acquired signal
fully meets the Nyquist criterion, but the freq uency reso-
lution had to be considered also to obtain a reliable in-
terval of the number of parameters using the equations in
Table 1.
2) The frequency resolution allowed certainty in the
techniques used to evaluate the number of parameters
and gives a wide interval of parameters that give a good
result. This is reflected in Figure 5, which shows a
minimum value for all methods and in Figure 6 where
the four peaks are identified.
5. Frequency Resolution: In Turbulent
Signals from a Supersonic Jet
The previous exercise, in which we used deterministic
signals, showed that if frequency resolution is taken into
account, the results obtained from the parameter evalua-
tion techniques give more precise results. In the follow-
ing exercise, we will demonstrate that this result is also
valid for random signals. The signals presented below
come from the laser light scattered by a high speed jet.
The signal that comes out of the photo detector is pro-
portional to spatial Fourier transform as a function of
time, of the density fluctuation for a wave vector deter-
mined by the optical set-up. The signal should contain
then the entropic and the acoustic modes described in
Section 2. Figure 7 shows the spectral density of such a
signal obtained with a spectrum analyzer. The acoustic
peak should appear around 2 MHz. The frequency of the
entropy peak depends on the local speed and on the di-
rection of the wave vector. When the latter is in the di-
rection of the flow, the frequency corresponds to the
Doppler frequency and is higher than the acoustic. If the
wave vector is perpendicular to the flow, the average
frequency is around zero. Previous papers [5-11], have
reported an unexpected third peak, which can be identi-
fied as Mach-3C in Figure 2, and whose origin is still
Figure 6. Spectrum of a signal with four components, ob-
tained with the Burg parametric periodogram, using the
parameters suggested by the prediction methods.
-95 ..... . ..
Figure 7. Spectral density displayed in a spectrum analyzer.
subject of investigation.
In Figure 7, the peaks mentioned are no t easily identi-
fied by the untrained eye. The signal was acquired with
an oscilloscope Agilent Infinitum Model 54830B. The
sampling frequency was and the number
of samples was 65,553 samples, giving a frequency
resolution , and then fed to the spectrum
analyzer. The acquisition parameters were automatically
determined by the instrument, without possibility of any
control from the user.
40 MHz
610 Hz
The techniques presented in Table 1 were applied to
the signal; the evaluation of the number of parameters
needed for a Burg parametric periodogram is shown in
Figure 8. The Burg periodogram was chosen because it
has good resolution for low-amplitude and low energy
components [10,11].
In Figure 8 one can observe that for the AIC and FPE
models, there is no tendency to minimize the estimated
variance of the linear prediction error, while the MDL
model predicts that the variance sh ould show a minimum
for around 160 parameters.
Figure 9 shows the various spectral graphs of the
above mentioned time signal obtained with the Burg pe-
riodogram with different number of parameters consid-
ering those predicted by the techniques. The interval of
parameter variation is between 90 and 300. The number
suggested by the MDL is within the range. Figure 9
shows p eaks that app ear and disappear when the number
of parameters changes, without any particular tendency
in relation to the behavior of the peaks.
The peak that we are interested in should appear at
around 2 MHz, so following Nyquist’s Theorem, the
signal was acquired properly. It could be suggested that
filters and other processing tools (decimation, windows,
etc.) might improve the acquired signal in order to obtain
more precise frequency information. However, once the
signal has been acquired, we cannot observe more than
what the frequency resolution allows. Therefore, the de-
sired frequency resolution has to be taken into account
before the acquisition of the signal [10,11].
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Optimization of Parametric Periodograms for the Study of Density Fluctuations in a Supersonic Jet 443
Figure 8. Using Table 1 we calculate the number of pa-
rameters, for the evaluation of the signal whose spectrum is
shown in Figure 7.
Figure 9. Burg periodograms evaluated with different pa-
rameters; using the minimum values of the graphs pre-
sented in Figure 8. Modeling the spectrum of Figure 7.
In exactly the same experimental conditions, a new
signal was acquired considering, this time, a frequency
resolution of . The evaluation of the
number of parameters in this case is shown in Figure 10.
It is interesting to note that the three parameters predic-
tion methods show a minimum in the estimated variance
of the linear prediction error. Two of these predict 150
122070 HzfΔ=
Figure 11 shows the corresponding spectrum calcu-
lated with a Burg periodogram using 150 parameters.
The three expected peaks may clearly be ob ser ved [11].
This result clearly shows that there must be a close re-
lationship between the frequency resolution of the ac-
quired signal and the number of parameters. Therefore,
the parameter prediction methods should explicitly con-
sider the frequency resolution.
As a final proof of the importance of the frequency
resolution, Figure 12 shows the spectrum of the signal
evaluated with different number of parameters. In this
Figure 12, the three lobes appear in a wide interval of
parameters. The fact that three modes of density fluctua-
tions inside the flow can be resolved is a new and impor-
tant result. Two modes were expected and the origin of
the third one is under study.
Figure 10. Evaluation of the number of parameters for a
turbulent signal with . Using Table 1, we
calculate the number of parameters.
fΔ=122070 Hz
Figure 11. Spectrum evaluated considering the criteria re-
ferred to in this paper. Using the minimum values of the
graphs presented in Figure 10.
Figure 12. Spectra of a same signal, , with
a different number of parameters.
fΔ=122070 Hz
Figures 11 and 12 prove the importance of taking into
account the frequency resolution before acquiring a sig-
nal; Nyquist’s theorem alone is not enough. As it may be
seen in Figure 12, when a good acquisition is made, the
shape of the spectrum does not vary over a wide parame-
ter interval, thus ensuring the shape of the spectru m. This
is valid in turbulent as well as in deterministic signals.
6. Summary and Conclusions
We presented two different kinds of signals for which we
could determine the optimal number of parameters thr ough
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Optimization of Parametric Periodograms for the Study of Density Fluctuations in a Supersonic Jet
Open Access JSIP
the minimum in the graph vs p. To do so, we had
to choose the frequency resolution
Δ. This choice is
simple when the signals are well known as in the first
example. The four frequencies of the original signal were
clearly identified. For the second example, where the
signal is random, the resolution depends on previous
knowledge of the experiment. In our case, entropic and
acoustic peaks expected from the literature, were re-
solved. Furthermore, a third low frequency non expected
peak, was studied with respect to the position in the jet
where the signal came from.
This paper gives a more objective way to determine
the number of parameters and a better spectral density. It
is surprising that even though it is well known that high
temporal resolution implies low frequency resolution, wh en
sampling a signal, the Nyquist theorem is applied blindly
without taking into accoun t the final use of the data. This
is an important result in signal processing that can be
seen by comparing Figure 7 with Figure 11.
A Rayleigh scattering technique combined with the
heterodyne detection of light scattered by the molecules
of a transparent gas was used to detect density fluctua-
tions. The periodograms helped resolve various frequent-
cies and gave more insight on the internal structure of the
The periodograms are parametric signal processing
tools that allow the modeling of a signal. To properly
implement them, it is necessary to determine the optimal
number of parameters, which ensures that the signal has
been modeled correctly. The theory predicts that many
parameters could add spurious peaks, and that few pa-
rameters may not reproduce the signal properly. The
results presented in this paper show that the resolu-
Δcould be an important factor in finding the opti-
mal number of parameters required to model a signal by
using param e t ri c periodograms.
In the future, we expect to determine clearly how to
obtain the optimal number of parameters directly from
the frequency resolution .
7. Acknowledgements
To the PAPIIT IN117712 project “Propagación de ondas
a través de interfaces”.
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