=Paper=
{{Paper
|id=Vol-1427/paper6
|storemode=property
|title=A Narrowband Sound Signal Frequency Estimation with Impulsive Noise Filtering
|pdfUrl=https://ceur-ws.org/Vol-1427/paper6.pdf
|volume=Vol-1427
}}
==A Narrowband Sound Signal Frequency Estimation with Impulsive Noise Filtering==
https://ceur-ws.org/Vol-1427/paper6.pdf
A Narrowband Sound Signal Frequency Estimation
with Impulsive Noise Filtering
Iurii Chyrka
Mathematical Methods for Sensor Information Processing Department of Institute of Information and
Communication Technologies of Bulgarian Academy of Sciences
25 A, Acad. G. Bonchev str., 1113 Sofia, Bulgaria
Yurasyk88@gmail.com
Abstract estimation in such situation. Hence there is need to
recover an original data from the degraded observations
The new stochastic approach of impulsive before main processing stage.
noise filtering, based on the input sample Filtering of an impulsive noise generally gives
separation by modified clusterization criterion, positive effect not only on frequency estimation, but
is proposed. It is shown that preliminary also for NSI algorithms. The acoustical array is a
filtration by the proposed procedure provides multichannel system and almost simultaneous
robust narrowband sound frequency estimation occurrence of impulses in many different channels can
and eliminates failures of the estimation completely corrupt calculation results.
algorithm caused by the impulsive noise.
2. Problem Statement
1. Introduction
In view of NSI application, attention in the paper is
Digital audio systems are widely applied today in paid to a class of narrowband signals which parameters
many areas of human activity. One of the biggest class are changed slowly in time. The problem of the
of them are acoustic arrays that perform measurement instantaneous frequency estimation can be interpreted
and processing of sound field. One of the key branch in as estimation on the limited observation interval
digital sound processing is Noise Source Identification (usually less than two periods of signal) during which
(NSI) techniques. They play an important role in the the parameters are changed slowly and a narrowband
acoustic camera, which is aimed to locate and signal is considered as a harmonic one.
characterize sound sources. It fuses images, received by For description of a mixture of a digital narrowband
the sensors in a complex image. This image consists of signal si with a white Gaussian noise ηi of power σ2 and
a camera image as a background and contour lines impulsive noise ζ the following typical additive model
describing sound field as a foreground [Bil76]. of a data sample is used:
Most of NSI techniques have a “narrowband” nature. xi = si + ηi + υ i ζ = ρ i sin(ωi τ(i − 1) + ϕ 0 ) + ηi + υi ζ ,
This means that the sound map must be recalculated for
each frequency of interest. Usually calculations are i = 1, N , (1)
performed for the predefined set of bounds and where υi is a sign function υi =sgn(pi), that can get
corresponding center frequencies. The difference values –1, 1, 0 with corresponding probabilities pζ/2,
between actual frequency and defined one leads to pζ/2, 1–pζ/2; pζ is an impulsive noise appearance
inaccurate results. Therefore, knowledge of the original probability; ζ is an impulsive noise amplitude
signal frequency is important for precise estimation of (considered as constant for all impulses); ρi is a signal
sound field parameters. instantaneous amplitude; ϕ0 is an initial phase; ωi is an
The estimate can be easily obtained in the steady instantaneous angular frequency; τ is a sampling
state from long enough Fast Fourier Transform, but in interval, N is a sample size. Further, the normalized
non-stationary case it must be estimated instantly in frequency γ=ωτ is used to omit τ. Such the mixture
real time. For fast, effective and precise estimation of model is close to a Bernoulli–Gaussian model of an
any instantaneous parameter, the data sample must be impulsive noise process [Vas08].
as short as possible; therefore, appearance of impulsive The mixture sample (1) can be represented by the next
noise has the most substantial influence on the probability density function
f Ξ (x n | γ , ρ, ϕ 0 , σ, p ζ , ζ , υ n ) = (1 − p ζ ) ⋅ f η (x n | γ , ρ, ϕ 0 , σ )
Copyright © 2015 for the individual papers by the papers' authors. (2)
Copying permitted only for private and academic purposes.
+ p ζ ⋅ f ηζ (x n | γ , ρ, ϕ 0 , σ, ζ , υ n ).
In: A. Bădică, M. Colhon (eds.): Proceedings of the 2015 Balkan It corresponds to a Tukey-Huber model, which is the
Conference on Informatics: Advances in ICT mostly used one for investigation of robust methods. It
40
supposes that majority of sample counts have an 3.2 Impulse noise filtering techniques
expected distribution fη and some of counts belong to
fηζ. The latter ones are outliers that produce “tails” of Conventional global filtering approach like a low-
the distribution. The aim of the paper is describing of pass filtering assumes that both corrupted and
the robust instantaneous frequency estimation uncorrupted samples must be processed. Median filters
procedure with removing of distorting outliers in (2). and other order statistics filters, that process a localized
area, also typically modify uncorrupted samples as the
transversal filtering is applied uniformly over the whole
3. Methodological Basis signal [Vas08]. In addition, median filter eliminates
changes in the input signal with a duration less than a
3.1 Brief description of NSI techniques half size of the filter window and does not properly
filter a set of consequent impulses longer than a half
The NSI techniques can be divided onto two
size. Some modifications of the classic median filter,
categories: near-field acoustic holography (NAH) and
that eliminate some its disadvantages, were developed
beamforming [Bai13]. NAH is aimed to reconstruction
[Geo11].
of a sound field in the 3D space. Beamforming gives a
Some detection methods perform processing in time
map of a sound intensity by measurement of the signal
and frequency domains simultaneously, for example
response from a variety of directions. The main concept
wavelet-based method in [Non08]. Yet another filtering
of the both NSI techniques is the next: the sound
approach uses fuzzy impulse detection, but mostly for
pressures measured by the microphones (more rarely –
images [Sch06].
particle velocities, captured by probes) are processed
Impulsive noise usually distorts a relatively small
by an imaging algorithm of either type to calculate an
amount of total counts in the sample. Since usually a
acoustic map of the sound pressure or sound intensity
relatively large part of the signal counts remain
with a snap to physical coordinates.
unaffected by the impulsive noise. Hence, it is
When NAH performs near-field imaging of noise
advantageous to replace only the noisy ones, leaving
sources, beamforming generally works in far field.
the uncorrupted counts unchanged. This ideology is
Unlike beamforming, that carries out spatial filtering
implemented by the another approach that performs
and maximizes the signal power from certain direction,
model-based two-stage filtering by using a linear
NAH provides, based on measurements over a two-
prediction system [Esq02]. For audio signals, the most
dimensional aperture, a reconstruction of the three-
often used models are autoregressive (AR) or
dimensional sound field from the source’s boundary out
autoregressive moving average (ARMA) [Oud14]. In
to the far field. Precision of reconstruction depends
this case, the system consists of two main parts:
mostly on microphone spacing distance, sound
detector and interpolator that perform individual
frequency and distance between source surface and
processing of the each element of the data set.
measurement plane. NAH operates in a low frequency
Another independent class of methods is stochastic
range, upper boundary of which is limited by a distance
ones [Bas88]. They are close to model-based ones, but
between microphones. On the other hand, beamforming
the statistical properties of data samples are used. A
works in the full frequency range but its use is
similar approach is proposed in the paper for pre-
reasonable only at high frequencies when it gives better
filtering before frequency estimation.
resolution than holography.
Beamforming provide the best performance on the
irregular arrays, which geometry (positions of 3.3 Frequency Estimation
microphones) is optimized in order to get the lowest In this paper frequency estimation is carried out with
possible level of false responses. As opposite to using of the algorithm mentioned in the previous work
beamforming, NAH is usually performed on a regular [Pro12]. It has been synthesized with using an AR
array grid that is also the requirement for the classic model for a single-tone harmonic signal mixed with a
Fourier NAH. Many modern NAH approaches perform white Gaussian noise:
calculation in the time domain and usually do not si = αsi −1 − si− 2 = 2 cos( γ ) si −1 − si− 2 , (3)
require location of microphones on some equidistant
positions. Hence, it allows reconstructing of sound where α=2cos(γ) is a parameter of auto-regression.
fields even with irregular array geometries. The algorithm is based on the solution of the
quadratic equation
41
α 2 − 2Bα − 2 = 0 , (4) minimal sum of cluster variances, which for a two-
with coefficient B calculated on the basis of the input component sample can be written as:
signal sample (x ) as: { ( ) ( )}
y ( opt ) = arg min σ l2 y ( v −1), N + σ r2 y ( v ), N .
v
(6)
(xi+1 + xi −1 )2 − 2xi2 / 2 (xi+1 xi + xi xi−1 ) .(5)
N −1 N −1
B(x ) = ∑[ i =2
] ∑
i =2
In order to increase sensitivity to presence of the
second “impulsive” subsample the modified criterion is
The equation (4) has two roots considered:
α ∗ ( + , − ) = B ± B 2 + 2 . Finally, frequency is estimated
as γ * = arccos (α ∗( + ) / 2) .
{ ( ) ( )}
y ( opt ) = arg min σ l4 y ( v −1), N + σ r4 y ( v ), N .
v
(7)
Generally, this algorithm is robust in many cases It allows correct treatment of mixtures that contain
with only the impulsive noise and without the Gaussian small amount of impulses.
one [Pro09], but it becomes highly sensitive when Finally, the proposed impulse filtering procedure can
Gaussian and impulsive noises occur simultaneously be represented in the form of four consequent steps of
and are multiplied during calculations. Therefore, calculations. On the first step of processing in order to
removing of impulses is a necessary condition before facilitate detection of the sample with bipolar pulse it is
estimation procedure starts. necessary to take the absolute value z n = xn ,
n = 0, N − 1 . Taking into account that time order of
4. Impulsive noise detection discrete values (1) is insignificant, it can be
()
transformed via y = ℜ z into an ordered statistics
The NSI Robust parameters estimation usually y = ( y(1), N , y ( 2 ), N ,... y( N ), N ) , (8)
consists of the next processing stages [Hub09]:
1) Data “errors” or corrupted counts detection; which must be separated onto two parts with
2) Processing of detected counts by removing them corresponding standard deviation values: σl, σr.
from the sample or their restoration with the help of The next step includes the calculation of values of
neighbor ones; the above mentioned criterion for each discrete rank
3) Pure robust estimation using the restored sample. (v),N. (the function (7) is used here) The separation
Here steps 1) – 2) belong to impulse noise filtering. threshold is simply found as argmin among all these
The new stochastic approach based on the cluster values. When the threshold is obtained, all counts that
analysis theory [Eve11] is proposed for performing exceed its value are marked as defective or i. e.
impulses detection. The key idea is to apply containing a noise impulse.
clusterization methods for analysis of the input sample On the last stage of processing the detected impulses
probability distribution and separate all counts in the must be replaced by interpolation or extrapolation
sample onto (two) independent clusters: the subsample using “good” neighbor counts. In view of assumption
with normal counts and relatively small subsample with on small duration of the impulses, only the simplest
impulsive noise counts. The important assumption here linear interpolation is used in this paper.
is that the signal amplitude is relatively small in
comparison to impulse amplitude, that allows to 5. Simulation results
distinguish these clusters. The similar approach was
The effectiveness of impulsive noise filtering was
proposed earlier by the author in application to
analyzed in connection to its influence on the frequency
electroencephalogram signals analysis [Pro13].
estimation process by the aforementioned algorithm.
Generally, clusterization is carried out on the basis
The flowchart of the frequency estimation process
of some optimization criterion or an objective function.
including the impulse filtering one can see in Fig. 1.
Usually it is an extent of objects density inside the
Statistical simulations by the Monte-Carlo approach
cluster or an extent of distance between different
were done under the next conditions: a signal sample
clusters. Actually, most of cluster separation
size N=50, the sample contains one period of the signal,
procedures can be considered as exact or approximate
hence the normalized frequency γ = 2π / 50 ≈ 0.126 ,
algorithm of some objective function optimization and
SNR=20 dB, signal to impulsive noise ratio is –20 dB,
finding a threshold.
number of numerical simulations for each plot is
The most widespread methods for optimal threshold
10000.
v calculation [Kor89] are based on the criterion of the
42
Figure 2: Comparison of algorithm failure
probabilities for different filtration methods
Figure 1: Flowchart of the frequency
estimation process
Fig. 2 shows plots of dependencies of estimation
algorithm failure probability on impulsive noise
appearance probability. Three cases are regarded:
without filtering, with filtration by a classic median
filter and by using the proposed separation. From the
presented figure one can see that the appearance of the
impulsive noise with big power significantly worsen
performance of the frequency estimation algorithm.
Small decreasing of failure rate at higher noise
probability can be explained by the fact that some
impulses start to appear one after another and such a
sequence has less influence on the algorithm.
The median filter of size 5 makes a situation better
and substantially decreases probability of failure, at low
appearance probability in particular, but its rate is still
Figure 3: Comparison of precision indicators of
high enough. Longer filter window allows getting
additional suppression of many impulses but there must frequency estimations for different filtration methods
be tradeoff between noise filtering and the signal
Similarly to situation with failures, the estimation
deformation due influence of the filter. The proposed
algorithm does not give reliable estimates, when the
separation procedure for impulse noise filtration makes
probability of noise appearance is high. The median
failures caused by impulses action almost impossible.
filter reduces the deviation of a mean in about two
In addition, the comparison of the precision of
times. The separation procedure provides a mean and a
frequency estimation was carried out in three
standard deviation close to constant values.
aforementioned cases without any filtering, with
The carried out statistical studies have shown that
filtration by the classic median filter and by using the
the proposed method works well with relatively big
separation. The plots of mean and standard deviation
impulses, when signal to impulsive noise ratio is –10
are shown in Fig. 3.
43
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