=Paper=
{{Paper
|id=Vol-2353/paper6
|storemode=property
|title=Method for Recognizing and Processing Complex Signals
|pdfUrl=https://ceur-ws.org/Vol-2353/paper6.pdf
|volume=Vol-2353
|authors=Sergey Rassomakhin,Vadim Serbin,Tetiana Kuznetsova
|dblpUrl=https://dblp.org/rec/conf/cmis/RassomakhinSK19
}}
==Method for Recognizing and Processing Complex Signals==
https://ceur-ws.org/Vol-2353/paper6.pdf
Method for Recognizing and Processing Complex Signals
Sergey Rassomakhin 1[0000-0003-1394-3588], Vadim Serbin 2[0000-0002-7748-1688],
Tetiana Kuznetsova 1[0000-0001-6154-7139]
1
V. N. Karazin Kharkiv National University, Svobody sq., 4, Kharkiv, 61022, Ukraine
rassomakhin@karazin.ua, kuznetsova.tatiana17@gmail.com
2
Yuzhnoye State Design Office, 3, Krivorozhskaya St., Dnipro, 49008, Ukraine
buba75@i.ua
Abstract. The methods of processing digital samples of complex structure sig-
nals with unknown parameters are considered. With the use of algebraic meth-
ods, the following tasks are sequentially solved: clock synchronization, deter-
mining the range of carrier frequencies, the multiplicity of phase modulation
and obtaining a stream of information bits. The methods for improving the qual-
ity of processing digital samples of signals based on solving special overdeter-
mined systems of linear equations are proposed. The estimation of efficiency of
the offered method is carried out by an imitation statistical modeling. The ad-
vantages of the proposed methods of signal processing for the telecommunica-
tions and radio monitoring systems are shown.
Keywords. Orthogonal Frequency Division Multiplexing, digital sampling, lin-
ear algebraic equations, correlation convolutions.
1 Introduction
The construction of effective information transmission systems is inextricably linked
with the problem of intensifying the usage of the time and frequency-energy resource
of communication channels. One of the ways to solve this problem is using the com-
plex signals with combined types of modulation in combination with the methods of
spectrum narrowing and noise-resistant coding [1-5]. In this connection, the structure
of the signals used to transmit information is becoming complicated, and, conse-
quently, the algorithms of their processing are becoming complicated as well [6-11].
The most promising type of signal code constructions in wireless networks is
OFDM (Orthogonal Frequency Division with Multiplexing) [2-6, 12]. The basic idea
of building such signals is arranging a set of mutually orthogonal frequency subchan-
nels so that, on the one hand, one subchannel does not interfere with the other, and on
the other hand, the spectra of the subchannels overlap. Due to the orthogonality of the
linear subchannels, each of them can be considered independently of the others. Er-
rors caused by the interference in one of the subchannels do not lead to errors in the
other. As a result, only a small part of the transmitted information is distorted. Error-
correcting coding being used the errors can be corrected [13-18]. The structure of
signals with multiple simultaneously operating subcarrier frequencies has well estab-
lished itself in conditions of heterogeneity of the propagation medium. In recent
years, the capabilities of systems with OFDM signals have evolved significantly.
Such signals began to be used in a wide variety of telecommunication systems operat-
ing in different radio frequency bands.
The complex structure of such signals, the a priori uncertainty of the channel prop-
erties cause significant difficulties in solving the problems of radio control and radio
monitoring. A distinctive feature of such tasks is the absence of data on the structure
and informative parameters of the measured signals. This information should be ob-
tained from the results of the study, with high accuracy and as soon as possible.
Therefore, the tasks of developing mathematical methods for analyzing complex sig-
nals based on digital measurement sequences are highly relevant.
2 Mathematical model of OFDM signals
For correct choice of the methods for digital analysis of the primary parameters of
OFDM signals, a brief description of their basic properties is necessary. Arbitrary
OFDM signal S j t on j -th modulation interval T p is formed by algebraic summa-
tion of the several harmonic oscillations of the same amplitude. Each of the oscilla-
tions has m options of modulation phase shift. The value m determines the multi-
plicity of the used phase (PM) modulation and corresponds to the base of the numeri-
cal source code. Commonly, m 2k where k is the number of binary symbols (bits)
represented by the elementary signal on one modulation interval. When using relative
phase coding and a unit value of the amplitude of the oscillation subcarriers, the
mathematical model of the signal can be represented as the following sequence:
n f 1
t
sin 2 f T t T T ,
i
S j t 0 p j ,i (1)
i 0 p
where t – current time; f 0 – the lowest subcarrier frequency in the signal spectrum;
T 1 f – inverse of the minimum subcarrier spacing f ; n f – the number of fre-
quencies used ; j ,i – the value of the manipulation angle of i -th fluctuation on j -th
modulation interval. This angle can take one of m values depending on the manipula-
tion code used. The informative features in the signal described by model (1) are rela-
tive phase jumps in carrier frequencies. These jumps are measured for each of the
frequency subcarriers separately: j ,i j 1,i , i 0, , n f 1 . The time parameters
of the modulation interval used in model (1) are tied by the relation:
1
Tp T T T , (2)
f
where T - the duration of the prefix part of the signal. The prefix part (hereinafter –
the prefix) is a repeating (to the exact sign) initial part of the signal added at the end
of the modulation interval Tp . Prefix structure is used to facilitate synchronization in
the presence of channel irregularities. For OFDM synchronization violations the sig-
nal can be correctly received and processed for any segment with the duration of T
within the full modulation interval .. TP ... Commonly, the choice of the prefix dura-
tion on the modulation interval corresponds to the ratio T 0,1 0,5 T . The sign
of the prefix depends on the value of the following parameter:
P f 0 mod f . (3)
For existing OFDM standards, parameter (3) can take two values that determine
the sign of the cyclic prefix: when P 0 the prefix is positive and when P f 2 –
inverse. Fig. 1 gives a qualitative idea of the form of the signal envelope constructed
in accordance with model (1) on two adjacent modulation intervals with n f 16 and
Tp 1, 47 T . At the end of each of the intervals Tp the inverse cyclic continuation of
the signal with duration T is located, which repeats, up to a sign, the shape of the
initial segment of the signal on the modulation interval. For the example in question
P f 2 therefore the prefix part is the inverse of the initial part of the signal.
Fig. 1. Example of OFDM signal
3 The main stages of the structural analysis of OFDM signals
A comprehensive analysis of the properties of complex signals is advisable to imple-
ment on the basis of phased processing. At each stage, only a part of the signal pa-
rameters is determined. Given the fact that OFDM signals contain a prefix, it is advis-
able to use the correlation method for determining structural time parameters Tp and
T at the first stage. This technique is based on the principle of "sliding" time win-
dow. This makes it possible to determine the following parameters of an OFDM sig-
nal: the value of the orthogonality interval, the duration of the modulation interval,
and the value of the frequency spacing between the channels.
At the second stage of the analysis, the tasks of determining the number and the
values of service and information channel frequencies, as well as, the signal phase
demodulation.
The two-stage processing results in the possibility to extract an information flow
from signals of an a priori unknown structure without using traditional fast Fourier
transform algorithms (FFT – Fast Fourier Transformation).
3.1 Correlation method for determining the time parameters of OFDM
signals
We propose a correlation method for determining structural time parameters Tp
and T . The basis is the "sliding time window" principle. The most probably value of
the time interval between the most correlated segments (with the "+" or "-" sign) of
the segments from the digital sample of signal measurements is determined. The as-
sessment of Tp – the most likely period of the emergence of "bursts" of correlation in
the process of moving the viewing window on the samples of the array of measure-
ments Q q0 , q1 , is determined as well. The scheme of the calculation procedure
is presented in Fig. 2. For the correlation analysis the two vectors, each containing K
elements of array Q in two non-overlapping time observation windows of the signal
are formed,
Y0 = q j ,q j+1 ,…q j+K -1 ,
(4)
Y1= q j i K , q j i K 1 , q j i 2 K 1 ,
removed from each other by i, i 0, , M . The position of the second time window
corresponding to the vector Y 1 is determined by the successive change in the offset
index i 0, , M that ensures its "slip" along the signal sample Q at each of the
values j 0 L .
For a wide range of analyzed signals, for example, for the 0.3-3.4 KHz frequency
band with minimum quality ADC, the most universal limits of the values of these
parameters, resulting in a quick and accurate assessment, are K 10 30 ,
M 200 300 and L 1000 .
At each value of index j (moving the window slip area) the M 1 dimensional
vector is being formed
V j v0j , v1j , , vMj , (5)
the elements of which are the coefficients of mutual correlation of vectors Y 0 and
Y1 . The calculations (according to the Fig. 2) are performed after centering and nor-
malizing the vectors by the formulas:
K 1 1
1 1 K 1
Y 0 N Y 0 j
K i0 Y 0i
K i 0
Y 0i ;
(6)
1
1
K 1 1 K 1
Y 1N Y 1 j
K i 0
Y 1i
K i 0
Y 1i .
Fig. 2. Calculation scheme
The resulting vectors Y 0 N and Y1N in normalized space have the same length,
equal to К , and the cosine of the angle between these vectors is equal to the cross-
correlation coefficient:
K 1
сos (Y 0н , Y1н ) Y 0 Y1 r (Y0 , Y1 ) .
i 0
нi нi н н (7)
Since the prefix is a repetitive (up to sign) part of the OFDM signal, ideally the
correlation coefficient between these parts is ± 1.
Fig. 3 shows the distribution of the values of the elements of the vector V j calcu-
lated according to a specific implementation OFDM signal (16 carrier frequencies, a
modulation rate – 75 bauds) at M 200 . The presence of pronounced extreme values
which are close in magnitude to unity is obvious. According to the results of calcula-
tions when j 0 L the two new vector V1 v10 , v11 , , v1L and
V 2 v 20 , v 21 , , v 2 L are formed. Their elements are calculated according to the
rules:
v1 j match min(V j ), V j K ;
0
j 0 L . (8)
v 2 j match max(V j ), V j K ;
0
Here the function match x, X calculates the indices of the elements of the vector
X equal to x , where the index 0 in the function (8) indicates a selection of the ele-
ment with a minimum sequence number, if there are several such elements in the
vector. The elements of the vector V1 represent the number of sampling intervals that
fit between the initial elements Y 0 and Y1 with minimal (negative) correlation on
j -th step of moving the observation window. Accordingly, the elements of the vector
V 2 are calculated for the maximum (positive) correlation of the vectors Y 0 and Y1 .
Simultaneous determination of the maximum and minimum is necessary to reveal the
value of function (3). It is obvious that the elements of the vectors V1 and V 2 de-
fined by expression (8), can take values only in the range K K M . To study the
statistical distribution of the values of the elements the histograms for the elements of
the vectors V1 and V 2 are formed:
H1 h10 , h11 , , h1K M ,
(9)
H 2 h20 , h21 , , h2 K M .
1
max ( V )
0.5
0.5
1 min ( V )
Fig. 3. Cross-correlation coefficients
Obtaining distributions (9) gives an opportunity to estimate the most likely value
Num – the number of sampling intervals between the initial measurements of the
segments of the digital sample Q with maximal (positive or negative) correlation:
match max H1 , H1 ;
0
max H1 max H 2
Num (10)
match max H 2 , H 2 0 .
max H1 max H 2
The value Num determines the number of sampling intervals that fit on the or-
thogonality interval of the signal T . It gives an opportunity to find two interrelated
OFDM signal parameters: the orthogonality interval and the minimum carrier fre-
quency spacing:
Num
T and f T 1 . (11)
fд
Besides, choosing the corresponding method of detecting the Num value according
to the condition specified in (10) automatically determines the value of the function
(3), and, consequently, the ratio between the frequency parameters f 0 and f . If the
1
value Num is determined by the first line of expression (10), then f 0 k f .
2
Otherwise the minimal carrier frequency is multiple to the spacing of the carrier fre-
quencies: f 0 k f where k – any positive integer.
As a result of processing histograms (9) according to (10) based on the values of
function (3), only one of the vectors V1 or V 2 is left for the further analysis, herein-
after denoted V* . This is possible because the prefix repetition sign is defined. Based
on the elements of the vector V* another vector V 3 is formed for the analysis:
V 3 match Num, V* . (12)
The elements of this vector are equal to the numbers of the elements of the vector
V in which the numbers Num are located. The feature of the vector V* provided
*
that the analyzed signal belongs to the OFDM class, is that it contains a sequence of
periodic series of numbers which are close or equal to Num . Therefore, the values of
the elements of V 3 in order of increasing their indices will be the segments (series)
of an ordinary positive integer sequence with some gaps in the sequence. Small gaps
can be observed inside the series too. A possible approximation of the sequence of the
elements of V 3 is illustrated by the following expression:
V 3 11,12,14,15,16,105,107,108,110,
, 620,
.
621, 622, 623, 624
series №1 series № 2 series № nc
The length of series of consecutive numbers (position numbers) may differ due to
measurement errors, features of the signal envelope and rounding during calculations.
However, when the vector length is sufficient, averaging results converges to the true
estimation in accordance with the law of large numbers. To exclude "fragmentation"
of the series, small gaps between adjacent numbers of the series must be ignored. It
has been empirically found that in most cases the number Ni should be considered to
belong to the current series of numbers if Ni Ni 1 where 3 5 . In general,
the structure of the vector V 3 can be depicted as shown in Fig. 4.
v 3 0 , , v 3 k , v 3 n , , v 3 k , , v 3 n , , v3k , v 3 n n , , v 3 k n
nc 1 nc 1 c c
1 2 2
series № 1 series № 2 series № n 1
c series № n
c
Fig. 4. Example of the V 3 series structure
In Fig. 4: v3n j , v3k j – initial and final elements of the j -th series of the consecu-
tive numbers in the vector V 3 ; nc – the total number of the identified series. Using
the presented structure and values of the elements of the vector V 3 , it is possible to
determine the number of sampling intervals that fit between adjacent pairs of mutu-
ally correlated segments of signal measurements, i.e. a period of "bursts" of correla-
tion:
Num1
v3
k n 1 v3k2
c
v3
n n 1 v3n2
c
. (13)
2 nc 2
Using the obtained value Num1 , we can determine the average value of the modu-
lation interval T p and, therefore, the average modulation rate W :
Num1
Tp , W T p 1 . (14)
fд
For the final determination of the time-frequency structure of the signal, we must
find the number of carrier frequencies n f and the vector of their nominal values
F f 0 , , f n f 1 . This can be done on the basis of previously obtained values
T p , T , f when the position of the element of the array Q corresponding to the be-
ginning of the first full modulation interval is determined correctly. The beginning of
a reliably identified clock interval could be most correctly associated with the begin-
ning of the second series of maximal responses of correlators in the vector V 3 , since,
due to the randomness of the beginning of the observation, the first series may be
incomplete. It should be taken into account that the beginning of a series of maximal
responses of correlation of the segments from K samples must appear before the next
modulation interval actually begins. Therefore, to fall within the interval with the
duration T p (taking into account that demodulation can be performed on any seg-
ment T within T p ) it is necessary to add the number K 2 to the starting sample, at
least. Then the beginning of the modulation interval can be assumed to coincide with
the next element number in the sequence
f K
nT0 round v3n2 mod д . (15)
W 2
Here round x – the rounding function to the nearest integer. The lowest possible
frequency f 0 in the group of carrier frequencies is determined by the value of func-
tion (3) and the fulfillment of the corresponding condition in (10):
1
f , max H1 max H 2 ;
f0 2 (16)
f ,
max H1 max H 2 .
The maximal number of subcarrier frequencies (or half the number of quadrature
components) that can fit in the channel band Fef is
Fef f 0
n f max round 1 . (17)
f
3.2 Determining the amount and nominal values of subcarrier frequencies of
OFDM signals
The correlation method, considered above, allows making a reliable assessment of the
main structural parameters of OFDM – T and TP . The value of f T 1 uniquely
defines the spacing of adjacent subcarrier frequencies. The minimal value of the sub-
carrier frequency and the maximal possible number of subcarriers placed within the
signal bandwidth n f max are determined from (16) and (17).
The number of samples N taken into account when analyzing a signal on one
modulation interval, as well as, the harmonic quadrature 2 n f max define the di-
mensions of the matrix of the linear algebraic equations system (SLAE) which can be
compiled and solved to estimate the frequency range. Depending on the ratio of the
vertical and horizontal dimensions of the matrix of coefficients, the system of equa-
tions can be overdetermined N 2 n f max , determined N 2 n f max or under-
determined N 2 n f max . The simplest one is the N 2 n f max case because
then the SLAE is a joint one almost every time. The number of equations that
matches the number of used elements of the digital sample Q equals to the number
of unknowns 2 n f max determining the amplitudes of quadrature components in the
spectrum of carrier frequencies OFDM. For the correct solution of SLAU
2 n f max uniformly spaced sample counts Q starting from the point of beginning
of the observation of the first complete clock interval of signal nT0 should be selected
on i -m modulation interval. For this the following rule is used:
nTi nT0 round i where TP tд . (18)
Square matrix of coefficients for unknown SLAE with size 2 n f max 2 n f max
composed for quadrature components of subcarrier frequencies is formed according to
the rule:
A1 ai , j , i, j 0, , 2 n f max 1 ;
ai , j Sin j ti , 0 j n f max 1; (19)
ai , j Cos j ti , n f j 2 n f max 1;
where
2 f0 j f , j 0, , n f max , ti nT i tд .
i
j j n
f max
The column-matrix of free members is formed as a vector of signal measurements
on the duration of one orthogonality interval:
B1 b0 , , b 2n
f max 1
; bi qi , i tд TP .
i
(20)
Normal solution of normally defined SLAE
A1 X1 B1 X1 A11 B1 (21)
gives an estimation of the amplitude vector of quadrature components
X1 x10 , , x12n
f max 1
which corresponds to the permissible values of carrier fre-
quencies.
On the basis of this solution, it is possible to determine the power distribution vec-
tor of the signal between the harmonic oscillations of the carrier frequencies:
, y x x
2
Y y0 , , y n
f max 1
i
1 2
i
1
i n f max
, i 0, , n f max 1 . (22)
The case of insufficiently defined SLAE N 2 n f max is interesting for analyz-
ing small samples of the signal. To solve such a SLAE, the pseudo inverse matrix
method Moore-Penrose can be used. It is known that there is the normal solution of an
underdetermined SLAE and is the only one. It is found by: X1 A 1 B1 where A 1
– Moore-Penrose pseudo inverse matrix of size 2 n f max 2 n f max which is deter-
mined by the ratio: A1 A 1 A1 A1 . In practice A 1 can be found by the formula:
A1 C D C (C C )1 (D D ) 1 D .
The representation of the matrix A1 in the form of a product of two matrices with
the size of N r and r N is used:
d1,1 d1,r c1,1 c1, N
A1 D C .
d
N ,1 aN ,r cr ,1 cr , N
With various skeletal decompositions of the matrix A the same solution for A
which can be written in the form: X1 A 1 B1 is derived. It is a pseudo solution
giving a zero residual: X1 A 1 B1 0 .
The case of N 2 n f max is the most advantageous for the maximal recording of
signal information on the modulation interval. Due to using additional signal meas-
urements from the sample Q the system which contains more equations with the
same number of unknowns is formed. To form the matrix A 2 and the vector B 2 the
maximal number of signal measurements determined by Num TP tд on the dura-
tion TP is used:
A 2 ai , j , i 0, , ( Num 1), j 0... 2 n f max 1 ;
ai , j Sin j ti , 0 j n f max 1; (23)
ai , j Cos j ti , nmax j 2 n f max 1;
B 2 b0 , , b Num 1 , bk qk , k 0, , Num 1 . (24)
SLAE has the form:
A 2 X2 B2 , (25)
and as a rule, has many solutions. To select the only one we need to use some criteria.
In practice, the maximum likelihood criterion is used more often. In the case of a
normal distribution of the vector B 2 it is equivalent of the least square’s criterion:
X*2 ( A 2T A 2 ) 1 A 2T B 2 . (26)
An approximate solution of system (26) gives a more accurate result than a strict
solution of (25). The noise immunity of the solution is achieved by averaging the
disturbing effect of interference when the number of signal measurements exceeds the
required minimum. The obtained vector of amplitudes of the quadrature components
X*2 , as well as X*1 gives a possibility to calculate the power distribution signal in
carrier frequencies using expression (22), wherein xi2 is used instead of x1i .
For any type of SLAE determining the actual list of carrier frequencies in the
OFDM spectrum is performed by comparing the elements of power distribution histo-
grams with a threshold value. The obtained nominal values of frequencies determine
the last structural time-frequency parameter of the analyzed signal – the vector of
working subcarrier frequencies F .
Thus, the previously obtained signal parameters T p , T , f , W and the obtained in
this subsection vector F identifies completely its structural properties and makes
signal demodulation possible.
4 Conclusions
The considered statistical method for analyzing the structure and demodulation of
OFDM signals under conditions of a priori uncertainty of solving radio monitoring
tasks has been practically tested. It has demonstrated the high accuracy of parameter
identification. The relatively low computational complexity of correlation and alge-
braic analysis makes it possible to identify the structure and the parameters of signals
practically in seconds.
The noise immunity of the analysis is achieved by solving a SLAE with rectangu-
lar overdetermined matrixes of coefficients. To eliminate phase errors generated by
asynchronous, with respect to the clock intervals of modulation, sampling the method
for calculating phase corrections which takes into account the time-frequency parame-
ters of the signal structure is proposed. The application of the phase correction
method provides ideal conditions for identifying the modulation type of subcarrier
oscillations. Mathematical formalization of solving the problem of determining the
modulation multiplicity, based on generating the multimodal reference functions and
sequential calculating the degree of mutual correlation, allows us completely auto-
mate the process of identifying the secondary parameters which are necessary for
demodulating the signals of subcarrier frequencies.
The further researches can be focused on the generalization of the method for any
structures of mono and poly frequency signals including those with a linear frequency
modulation and also proposed author’s method can be used in some different other
areas [19-23].
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