=Paper=
{{Paper
|id=Vol-2711/paper45
|storemode=property
|title=Spectral Analysis Based on a Discrete Binary Model for Signal Stochastic Quantization and Calculating a Weighted Correlation Function Estimate
|pdfUrl=https://ceur-ws.org/Vol-2711/paper45.pdf
|volume=Vol-2711
|authors=Vladimir Yakimov,Vitaliy Batishchev
|dblpUrl=https://dblp.org/rec/conf/icst2/YakimovB20
}}
==Spectral Analysis Based on a Discrete Binary Model for Signal Stochastic Quantization and Calculating a Weighted Correlation Function Estimate==
https://ceur-ws.org/Vol-2711/paper45.pdf
Spectral Analysis Based on a Discrete Binary Model for
Signal Stochastic Quantization and Calculating a
Weighted Correlation Function Estimate
Vladimir Yakimov[0000-0002-2245-2708], Vitaliy Batishchev[0000-0002-1448-7978]
Samara State Technical University, 244, Molodogvardeyskaya str., Samara, 443100, Russia
yvnr@hotmail.com, vib@list.ru
Abstract. The paper considers the high-performance digital algorithm for esti-
mating the power spectral density (PSD) by the correlogram method. This algo-
rithm is developed on the binary analog-stochastic quantization basis of the in-
vestigated continuous signal. The mathematical model for the discrete represen-
tation of binary analog-stochastic quantization made it possible to analytically
calculate the cosine Fourier transform of weighted window functions in the al-
gorithm development. As a result of this, the developed algorithm does not re-
quire numerical integration operations. The main computational operations of
the algorithm are the operations of summation and subtraction. The algorithm
also does not require the calculation of correlation function estimates. All this
increases the computational efficiency of the PSD estimation by the correlo-
gram method. The experimental studies results of the algorithm are given.
These results show that the proposed algorithm gives accurate PSD estimates in
the presence of additive noise. The computational efficiency of the algorithm
provides the ability to use it for estimating the PSD of complex signals.
Keywords: power spectral density, time-weighting function, binary analog sto-
chastic quantization, sign signal, digital time readout
1 Introduction
Spectral analysis methods of signals are used in various fields of science and technol-
ogy. In particular, this applies to acoustics, location, vibration diagnostics, radio fre-
quency identification, etc. Under conditions of a priori statistical uncertainty, the reg-
ularity of changing signal parameters over time is determined by probabilistic laws.
The spectral analysis of such signals involves estimating the power spectral density
(PSD) over a finite time interval. PSD characterizes the distribution of the average
signal power over frequencies within the analyzed frequency range.
One of the most common classical methods for assessing PSD is the correlogram
method. This spectral analysis method can be used to study the frequency composi-
tion of complex multicomponent signals that meet the conditions of stationarity and
ergodicity in time.
Copyright © 2020 for this paper by its authors. Use permitted under Creative Commons Li-
cense Attribution 4.0 International (CC BY 4.0). ICST-2020
At present, the correlogram method for estimating PSD is carried out primarily in
digital form. First, modern computing and software engineering allow you to create
high-tech problem-oriented systems. Secondly, digital signal processing guarantees
the accuracy of calculations. Moreover, traditionally, researchers use digital algo-
rithms that are developed on the basis of an approximate implementation of the ana-
log integration operation in discrete form by digital summation operations with a
uniform sampling interval in time [1–4]. All digital algorithms developed in this way
require a significant amount of calculated operations. Among these operations, a large
number of digital multiplication operations have to be performed. As you know, digi-
tal multiplication operations are the most time-consuming computing operations.
Using algorithms that require a large number of digital multiplication operations can
lead to significant time costs. The consequence of this is a decrease in the efficiency
of calculating PSD estimates.
A feature of classical algorithms for calculating the PSD estimate by the correlo-
gram method is the need for preliminary calculation of the correlation function esti-
mates sequence for the analyzed signal. It also reduces the computational efficiency
of obtaining PSD estimates.
The correlogram method involves the use of weighted window functions (the so-
called correlation windows) to attenuate the blurring effect of spectral components
estimates. The use of weighted window functions involves the operation of weighing
the correlation function estimates with the correlation window function samples. This
procedure leads to the need for additional digital multiplication operations. As a re-
sult, the use of correlation windows reduces the computational efficiency of digital
algorithms for estimating the PSD using the correlogram method.
Typically, the computational efficiency of digital signal processing is enhanced in
three main ways [5–9]. Firstly, an increase in the speed of data processing programs is
ensured by an increase in the overall performance of computing systems. Secondly,
the developers of the algorithms carry out special preparation of the initial data in the
form of ordered algebraic structures. Thirdly, the computational process is accelerated
by refactoring and using various methods of optimizing program code. However,
none of these methods can solve the problem of increasing the computational effi-
ciency of digital signal processing in full. This is explained by the fact that the time
characteristics of the execution of application programs are determined largely by the
mathematical model of preparing data in digital form and by performing computa-
tional operations on this data. The use of binary quantization as the primary conver-
sion of the analyzed continuous signal into a digital code can improve the computa-
tional efficiency of digital signal processing [10, 11]. The application of binary quan-
tization has been investigated extensively in the past years [12–15]. In [16, 17], fast
digital algorithms were developed for estimating the PSD based on the correlogram
method using binary analog-stochastic quantization. The discrete-time mathematical
model for this quantization type provided analytical calculation of analog integration
operations in the development of these algorithms [18]. As a result, the need to per-
form a large number of digital multiplication operations is eliminated. The main oper-
ations of the algorithms are logical operations and arithmetic operations of summing
and subtracting discrete values of the cosine function. Algorithms also do not require
preliminary calculation of estimates of the correlation function. All this simplifies the
digital procedures for calculating the PSD estimate by the correlogram method and
reduces the overall complexity of the spectral analysis. However, the algorithms in
[16, 17] were developed without taking into account the use of weight functions (cor-
relation windows), which limits their application in practice.
Thus, the task of developing a computationally efficient algorithm for estimating
the PSD based on the correlogram method is relevant. To solve this problem, it is
fundamentally important to obtain simple computational procedures using the
weighted window functions and reduce the number of digital multiplication opera-
tions. Moreover, such an algorithm should allow obtaining the PSD estimates with the
necessary accuracy and frequency resolution.
2 Algorithm for estimating the PSD by the correlogram method
based on binary analog-stochastic quantization of the
analyzed signal
It was noted above that in [16, 17] fast algorithms for estimating the PSD based on the
correlogram method were developed using a technique that allows analytical calcula-
tion of integration operations. Let us summarize this technique for developing a digi-
tal algorithm for calculating the PSD estimates by the correlogram method using
weighted window functions.
The correlation function and weighted window functions are even functions. With
this in mind, the PSD estimate using correlogram method is calculated as follows:
T
Sˆ XX ( f ) = 2 w() Rˆ XX () cos 2fd , (1)
0
where Rˆ XX ( ) is the correlation function estimate of the analyzed signal X (t ) ;
w() is the weighted window function; T is the length of time the spectral analysis.
Let within the time intervals t [0; T ] and t [0;2T ] , the results of two inde-
pendent operations of binary analog-stochastic quantization are sign signals:
o o
z1 (t ) = sgn{ x(t ) + 1 (t )} and z2 (t ) = sgn{ x(t ) + 2 (t )} , (2)
o
where sgn{...} is the operator of the sign function; x(t ) is a centered implementa-
tion of the analyzed signal X (t ) ; 1 (t ) and 2 (t ) are auxiliary random signals.
Auxiliary signals 1 (t ) and 2 (t ) have a uniform distribution ranging from
− max to + max , where the value of max must exceed the highest possible value of
o
the implementation x(t ) with a probability close to unity [10,11,18].
As an estimate of Rˆ XX ( ) , we take an unbiased estimate [19–21]:
T
Rˆ XX () = 2maxT −1 z1 (t ) z2 (t + )dt .
0
(3)
We introduce the notation:
g (, f ) = w( ) cos 2f . (4)
Then, taking into account (3) and (4), estimate (1) is:
T T
0
Sˆ XX ( f ) = 22maxT −1 g (, f ) z1 (t ) z2 (t + )dtd .
0
(5)
In (5), we change the order of integration over the variables t and τ:
T T +t
0
Sˆ XX ( f ) = 22maxT −1 z1 (t ) z2 () g ( − t , f )ddt .
t
(6)
Further development of the digital algorithm for estimating the PSD was reduced
to the practical implementation of analog integration operations in a discrete form. An
effective calculation of these operations can be achieved using a discrete-time math-
ematical model for changing the values of the sign signals z1 (t ) and z 2 (t ) in time.
These signals are continuous functions and their values are limited in level by the
values “–1” and “+1”. Therefore, we can unambiguously represent the signals z1 (t )
and z 2 (t ) at the time intervals of their formation t [0; T ] and t [0;2T ] using the
values of z1 (t 0 ) and z 2 (t 0 ) at the initial moment of quantization time t 0 = 0 and
z z
the time samples { ti 1 } and { t j 2 } , at which they change their value, where
1 i I and 1 j J . t 0z1 = t 0z2 = t 0 = 0 , t Iz1 = T and t Jz 2 = 2T [14].
Taking into account the discrete-time representation of the sign signals z1 (t ) and
z 2 (t ) , we write the estimate (6) as a sum:
I −1 t iz+11 T + t
Sˆ XX ( f ) = 2 2maxT −1z1 (t0 ) (−1) z () g ( − t , f ) ddt .
i =0
i
2 (7)
t iz1 t
From (4) it follows that for a continuous and differentiable weighted window function
w() in the time interval [0;2T ] , the function g ( , f ) is also a continuous and
differentiable function in this time interval. Then there exists a continuous function
G ( , f ) for which the condition is satisfied [22]:
g (, f )dt = dG (, f ) . (8)
In (7), we change the order of integration. Then, taking into account (8), the integral
over the variable t can be calculated analytically. After calculating this integral, we
obtain:
I
Sˆ XX ( f ) = DXX (T , f ) + 2 2maxT −1 z1 (t0 ) (−1) i i B(tiz1 , f ) ; (9)
i =0
i = 1 , if i = 0 and i = I ; i = 2 , if 2 i I − 1 ;
D XX (T , f ) = 2(G (T , f ) Rˆ XX (T ) + G (0, f ) Rˆ XX (0)) ; (10)
tiz1 +T
B (tiz1 , f ) = z2 ()G( − ti , f )d .
z1
(11)
tiz1
Relation (11) as well as (6) can be represented as a sum of integrals:
m (i )+ r (i ) t zj+21
B (tiz1 , f ) = z 2 (tiz1 ) (−1) j − m (l )
G ( − t i , f ) d ,
z1
(12)
j = m (i ) t zj2
where t m2( i ) = t i 1 and tm2(i )+ r (i )+1 = ti 1 + T .
z z z z
The integral in (12) is defined. It can be calculated numerically:
m (i ) + r (i )
B(tiz1 , f ) = z 2 (tiz1 ) (−1) j −m(l ) j G(t ji , f ) , (13)
j = m (i )
where j = t j +2 1 − t j 2 and t ji = t j 2 − ti 1 .
z z z z
Relations (9) and (13) can be directly used to calculate the SDM estimates in dis-
crete form. Moreover, we take into account that for a given duration of the spectral
analysis time T, the maximum possible frequency resolution is f = 1 / T . Then for
f k = kf we finally get:
I
Sˆ XX ( f k ) = DXX (T , f k ) + A(max , T ) z1 (t0 ) (−1)i i B(tiz1 , f k ) , (14)
i =0
m(i ) + r (i )
B(tiz1 , f k ) = z2 (tiz1 ) (−1) j − m(l ) jG(t ji , f k ) , (15)
j = m(i )
A( max , T ) = 2 2maxT −1 = 2 2max f = Const . (16)
As follows from (14) and (15), the calculation of the PSD estimate was reduced to the
discrete processing of the function G ( , f ) . Relations (14) and (15) became the basis
for the development of a digital algorithm for calculating PSD estimates. The main
operations of this algorithm are the operations of summing and subtracting the sam-
ples of function G ( , f ) , where = t ji and f = kf .
The function G ( , f ) is primitive for the function g ( , f ) = w( ) cos 2f in the
time interval [0;2T ] , where the weighted window function w() is known.
Therefore, the function G ( , f ) is also known, since its form is determined only by
the type of the applied weighted window function w() . As an example, some of the
main weighted window functions w() used in calculating the PSD estimates and the
corresponding functions G ( , f ) are presented in Table 1 [1, 23–25].
Table 1. Weighted window functions and functions G ( , f )
Window w() G ( , f )
Rectangular 1, | | T ; sin 2f
(Box Car)
window 0, | | T . 2f
Triangular || sin 2f 1 cos 2f
1 − , | | T ;
(Bartlett) T 1 − −
window 0, | | T . T 2f T (2f ) 2
1 sin 0.5 sin 0.5
cos , | | T ; + ,
Cos(x)
window 2T
0, | | T . = (4 f − f ) , = ( 4 f + f )
0.5 1 sin sin
Hann
(Raised-
0.5 + 0.5 cos , | | T ;
2 m =0
+ ,
T
Cosine)
0, | | T . = ( 2 f − mf ) ,
window
= ( 2 f + m f )
3 Experiments and results
Experimental studies of the developed algorithm were carried out using simulation
methods. In accordance with this, a logical-mathematical model was developed for
conducting computer experiments. The purpose of the experimental studies was to
evaluate the metrological capabilities of the algorithm for calculating the PSD esti-
mate. On the basis of discrete-event modeling, special software was developed to
simulate the binary analog-stochastic quantization of a continuous signal. The dis-
crete-event model of binary analog-stochastic quantization made it possible to ab-
stract from the continuous quantization result and consider only the main events de-
termined by time instants at which it changes its values. The model of centered signal
implementation was the sum of the statistical-independent harmonic components in
additive noise e(t ) :
o M
x(t ) = Am cos(2f mt + m ) + e(t ) ,
m =1
where f m are the normalized frequencies, Am are the normalized amplitudes, m
are the initial phases.
A model of this type reflects the dynamics of a real signal characterizing a wide
class of physical processes. We can set a specific superposition for the arrangement of
spectral components within a given frequency range and consider a complex signal as
a combination of narrow-band signals. We also note that this model makes it possible
to study the operability of the algorithm in the presence of noise interference and to
examine its stability in calculating the PSD estimates in the presence of random back-
ground noise.
Normalized frequencies f m were set in the range from zero to 0.5. This ensured
the constancy of the frequency range of the presentation of the results of the PSD
estimation for complex signals models that a priori occupy different frequency ranges
in width. The amplitudes Am were also interpreted as normalized. They were set in
the range from zero to unity. Initial phases k were set within the interval
− k using a generator of randomly uniformly distributed quantities. Addi-
tive noise e(t ) was white noise with zero expectation and dispersion e = 1 . In par-
2
ticular, the model contained five harmonic components. The parameters of these
components are shown in Table 2.
Table 2. Harmonic components parameters
Normalized PSD estimate, dB
Am2 / Amax
2
, Hann
m Am fm Rectangular Triangular
Cos(x) (Raised-
dB (Box Car) (Bartlett)
window Cosine)
window window
window
1 0.15 0.275 -16.48 -16.21 -16.32 -16.29 -19.52
2 0.25 0.3 -12.04 -12.07 -12.14 -12.17 -13.86
3 0.75 0.32 -2.5 -2.44 -2.47 -2.47 -3.04
4 1.0 0.33 0 0 0 0 0
5 0.35 0.375 -9.12 -9.73 -9.54 -9.55 -9.11
Using simulation, experiments were carried out in the course of which the possibil-
ity of frequency resolution and the accuracy of determining harmonic components
were checked. Figures 1-4 show the normalized PSD estimates obtained for the Rec-
tangular (Box Car) window, the Triangular (Bartlett) window, the Cos(x) window and
the Hann (Raised-Cosine). PSD estimates were calculated with a resolution of 0.0001
units of normalized frequency. The results of determining the harmonic components
are presented in Table 2.
Fig. 1. Normalized PSD estimate, Rectangular (Box Car) window
Fig. 2. Normalized PSD estimate, Triangular (Bartlett) window
Fig. 3. Normalized PSD estimate, Cos(x) window
Fig. 4. Normalized PSD estimate, Hann (Raised-Cosine) window
On all graphs, we see a steady identification of all five harmonic components. The
position of the harmonic components estimates in the spectrum corresponds to the
initial values. False spectral lines are not present. Spectral line splitting is not ob-
served. Spectral lines are clearly distinguishable. Strong harmonics do not mask weak
ones. The spectral noise estimate remains at a fairly low level in relation to the har-
monic component estimates. It practically does not exceed -40 dB. This indicates a
good resolution and high stability of the algorithm to external additive noise.
4 Conclusion
Based on binary analog-stochastic quantization, we developed a digital algorithm for
estimating the PSD by the correlogram method. During the development of the algo-
rithm, the integration operations are calculated analytically. This eliminates the meth-
odological error, which is the result of numerical integration operations. The main
computational operations of the algorithm are the operations of summation and sub-
traction. Unlike the classical digital algorithms for estimating the PSD by the correlo-
gram method, this algorithm requires calculating only two estimates of the correlation
function Rˆ XX (0) and Rˆ XX (T ) . If T kx , then Rˆ XX (T ) → 0 , where kx is the
correlation interval of the analyzed signal. Then G (T , f ) Rˆ XX (T ) → 0 , and for (10)
we will have DXX (T , f ) 2G (0, f ) Rˆ XX (0) . All this increases the computational
efficiency of the estimation of the PSD by the correlogram method.
The considered algorithm can be implemented as a functionally complete software
module. This module can find application in the composition of the metrologically
significant software of multifunctional systems for operational frequency-time analy-
sis of signals [26]. The practical use of such a module in integrated software should
increase the efficiency of solving problems requiring the processing of complex mul-
ticomponent signals.
Acknowledgments. The authors are grateful to the Russian Foundation for Basic
Research (RFBR). This work was supported by the RFBR under initiative research
project No. 19-08-00228-A.
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