=Paper=
{{Paper
|id=Vol-2588/paper10
|storemode=property
|title=Estimations of the Signal Information Parameters in Radio Engineering Systems
|pdfUrl=https://ceur-ws.org/Vol-2588/paper10.pdf
|volume=Vol-2588
|authors=Igor Prokopenko,Igor Omelchuk,Alina Osipchuk,Yousef Ibrahim Daradkeh
|dblpUrl=https://dblp.org/rec/conf/cmigin/ProkopenkoOOD19
}}
==Estimations of the Signal Information Parameters in Radio Engineering Systems==
https://ceur-ws.org/Vol-2588/paper10.pdf
Estimations of the Signal Information Parameters
in Radio Engineering Systems
Igor Prokopenko 1 [0000-0003-4169-3774], Igor Omelchuk 1 [0000-0000-0000-0000],
Alina Osipchuk 1 [0000-0002-9053-2072] and Yousef Ibrahim Daradkeh 2 [0000-0002-9209-0626]
1
National Aviation University, 1 Kosmonavta Komarova Ave, Kyiv, Ukraine
igorprok48@gmail.com
2
College of Engineering, Prince Sattam Bin Abdulaziz University, Department of Computer
Engineering and Networks,Wadi Addawasir, KSA
daradkehy@yahoo.ca
Abstract. Radio engineering systems are used to monitor the environment and
operate in difficult conditions with a large number of interferences of natural
and artificial origin. Contamination of the frequency space by radiation sources
places high demands on methods and means of signal processing of such sys-
tems, in particular methods of estimation of information parameters of signals,
their amplitudes, phases, frequencies, modulation depths. A new approach is
proposed to estimate the phase difference in VOR systems and the difference in
the modulation depth of the radar signals of ground beacons of the ILS instru-
mental landing system using the maximum likelihood method, and based on the
Fourier transform. One of the applications of the considered method in on-
board computers for estimating the difference in modulation depth is shown.
The stability conditions for the optimal estimation of the harmonic signal fre-
quency to action and the parameters of impulse noise are analytically deter-
mined, which was confirmed by statistical modeling.
Keywords: Estimation Of A Signal Parameters, Phase, Frequency, Impulse
Noise, Robust Method, Navigation Systems.
1 Introduction
An increase in air traffic intensity, a high saturation of aviation with radio engineering
means of communication, control, navigation and landing, far and near radar, an in-
crease in the information saturation of the radio frequency spectrum as a whole, and
an increase in industrial interference leads to a constant complication of the interfer-
ence situation in airport areas. Under these conditions, the noise immunity of the on-
board equipment of short-range navigation and landing systems is deteriorating.
The most of radio engineering systems signal processing methods are based on the
Gaussian model, for which a number of optimal solutions are obtained. However, in
most cases signal and interference distributions are different from the Gaussian mod-
Copyright © 2020 for this paper by its authors. Use permitted under Creative Commons License Attribu-
tion 4.0 International (CC BY 4.0) CMiGIN-2019: International Workshop on Conflict Management in
Global Information Networks.
el. This leads to a significant decrease in the real performance of systems compared to
theoretical.
Of particular importance is the efficiency and noise immunity of signal processing
channels in flight radio systems [8-10], in particular in short-range navigation systems
operating in radio frequency contaminated conditions [4].
The article solves the problem of synthesizing new post-detector algorithms for
evaluating the information parameters of radio signals in short-range navigation sys-
tems (navigation beacons VOR, ILS) in digital channels of radio receivers. The algo-
rithms are obtained by statistical synthesis using the maximum likelihood method and
provide high noise immunity and accuracy.
2 Digital signal processing in on-board equipment of navigation
system VOR
2.1 Features of the beacon VOR
Digital signal processing (DSP) in receivers VOR is considered as their non-linear
transformation in order to measure the phase difference of the receiver detected sig-
nals. The phase estimation algorithm is further synthesized using the statistical maxi-
mum likelihood method, which provides good noise immunity and accuracy at low
signal-to-noise power ratios (SNR).
The information signal of the VOR receiver at the output of the detector of the
high-frequency amplifier is the sum of the signals of the channel of the reference
phase (RPH) and the channel of the variable phase (VPH) [4]. The signal of the RPH
channel is a frequency-modulated oscillation of the form:
0 𝑓 0
𝑈𝑜𝑝ℎ (𝑡) = 𝑈𝑜𝑝ℎ sin(2𝜋𝑓0 𝑡 + 𝑑𝑒𝑣 cos(𝑓30 𝑡 + 𝜑𝑜𝑝ℎ ) + 𝜑) , (1)
𝑓0
0
where 𝑈𝑜𝑝ℎ - amplitude of FM oscillations; 𝑓0 - subcarrier frequency; 𝜑0 - its initial
phase; 𝑓𝑑𝑒𝑣 - subcarrier deviation frequency; 𝑓30 - frequency modulating oscillations; 𝜑
- its initial phase.
The signal of the variable phase (VPH) channel is harmonic oscillation with ampli-
0 0
tude 𝑈𝑣𝑝ℎ , initial phase 𝜑𝑣𝑝ℎ and frequency 𝑓30 equal to:
0 0
𝑈𝑣𝑝ℎ (𝑡) = 𝑈𝑣𝑝ℎ sin(2𝜋𝑓30 𝑡 + 𝜑𝑣𝑝ℎ ), (2)
Useful information (azimuth to the beacon), which must be detected by processing
these signals, is enclosed in the difference ∆𝜑 between the current phase of the modu-
0
lating oscillation of the RPH channel 𝜑𝑜𝑝ℎ (𝑡) = 𝑓30 𝑡 + 𝜑𝑜𝑝ℎ and the phase of the
0
VPH channel 𝜑𝑣𝑝ℎ (𝑡) = 𝑓30 𝑡 + 𝜑𝑣𝑝ℎ , then:
∆𝜑 = 𝜑𝑜𝑝ℎ (𝑡) − 𝜑𝑣𝑝ℎ (𝑡) , (3)
The nominal frequency values are as follows: 𝑓30 = 30Hz, 𝑓0 = 9960Hz, 𝑓𝑑𝑒𝑣 =
480Hz.
3
According to the recommendations of the international documents “ARINС-711”
and the requirements of the “Airworthiness Norms of airplanes NLGS-3”, the error of
azimuth measurement by VOR equipment at nominal values of the input signal with a
probability of 0.95% should not exceed 0.3 ° [4].
In the presence of interfering factors, this error may be 0.5 °. The following factors
are related to interfering factors: drift of a frequency 30 Hz, which is not more than
1.5% of the nominal value (𝑓30 = (30.00 ± 0.45) Hz); departure of the subcarrier fre-
quency from the nominal value, with 𝑓0 = (9960 ± 100) Hz; drift of frequency devia-
tion from the nominal value, with fdev = (480 ± 30) Hz; the interference, consisting of
spectral components in the range 0 - 20 kHz with a total amplitude, does not exceed
the amplitude of the useful signals.
The block diagram of the digital signal processing path implemented in the on-
board navigation equipment VOR-85-01 is shown in Fig. 1. The demodulation of FM
oscillation occurs on the basis of fixing the moments of its intersection of the zero
level. The final processing is carried out by the calculator
Fig. 1. The block diagram of the digital signal processing path implemented in the on-board
navigation equipment VOR-85-01.
It is proposed to use the maximum likelihood method for calculating the phase differ-
ence with the solution of the likelihood equation according to the Newton-Raphson
algorithm, which has high noise immunity and estimation accuracy.
2.2 Synthesis of the algorithm for estimating the phase difference by the maxi-
mum likelihood method
The statistical model of the input signal (sample of values) (x1, ..., x2) is taken as an
additive mixture of Gaussian noise with dispersion (σ2) and a useful harmonic signal
with known amplitude (U) and frequency (ω0 = 2*pi*f0). It is necessary to evaluate
the phase of the input signal from a sample of values.
The likelihood function is generally written as
n
exp xi U cos(0ti )/ 2 2
1
f ( x1,...xn , t1,...tn ) (4)
( 2 2 ) n i 1
It is required to determine the coordinate of the function maximum (4) by the pa-
rameter φ.
After differentiation, we obtain the likelihood equation
ln f ( x1 ,...xn , t1 ,...t n / ,U , 0 )
(4)
n
xi U cos(0ti )sin(0ti )U 0
1
2 i1
Solving this equation using the Newton – Raphson method, we obtain
n
xi U cos(0ti )sin(0ti k )
i 1
k 1 k (5)
n
xi U cos(0ti )sin(0ti k )
i 1 k
Transforming this expression, we obtain an iterative rule of the phase estimation algo-
rithm.
n
xi U cos(0ti )sin(0ti k )
k 1 k n i 1 (6)
xi cos(0ti ) U cos(2(0ti k ))
i 1
To determine the phase difference, it's necessary to calculate the formula (6) for each
signal (for signal of the reference phase and for signal of the variable phase) and
calculate the difference between them.
2.2 The research of noise immunity of the Newton-Raphson method
To evaluate the noise immunity of the Newton – Raphson method, a software model
was developed where the input signal is an additive mixture of the harmonic signal
S(t) and white noise η
x(t ) S (t ) U cos(0t ) (7)
where U - amplitude; 𝜔0 = 2 ∙ 𝜋 ∙ 𝑓0 - frequency; φ- its initial phase; .
The mathematical model of white Gaussian noise was modeled as
2 D ln( RND(1)) cos(2 RND(1)) M , (8)
5
where M – mathematical expectation, D – dispersion, RND (1) - values of the random
number generator with uniform distribution on the interval (0 ... 1).
Results of modelling for various levels of the Signal-to-Noise Ratio (SNR) are
shown in Fig.2.
Also, the calculations were performed for the input signal with a different initial
phase shift. For example, with φ0 = 60.0 ° and a noise variance of D = 0.3, a result of
60.90783° was obtained at the first iteration. Note that none of the analog phase cap-
ture methods can provide such accuracy with such a high noise level.
Phase Estimation Δφ, deg.
SNR, dB
Fig. 2. Results of modelling.
3 Synthesis Of A Frequency Estimation Algorithm In A
Frequency Detector
3.1 The Need To Take Into Account The Impact Of Pulsed Interference On
Frequency Estimation
The traditional solution for a digital frequency detector includes an algorithm for
estimating the signal frequency by counting zero crossings for a certain period of
time. For its simplicity, this method has insufficient noise immunity and is inoperative
at low signal-to-noise ratios.
Improving the accuracy and noise immunity of a digital frequency detector can be
obtained by statistical synthesis of the optimal algorithm for estimating the frequency
of a harmonic signal
In the actual operating conditions of the radio technic systems, one of the important
problems is to ensure the stability of algorithms for estimating the information param-
eters of useful signals to the effect of chaotic impulse interference (HII) [1], since
their appearance can lead to complete distortion of the calculation results.
This applies in particular to the quasi-optimal frequency estimation (QOF) algo-
rithm of the harmonic signal (hereinafter referred to as the "signal"), which is pro-
posed in [6]. This paper uses an equidistant model of the signal state in recurrent form
si si 1 si 2 , i 1, n (n is the sample size) in which the parameter
2 cos(2f ), ti ti 1 associated with the instantaneous signal frequency f,
and the observational model based on additive Gaussian noise (hereinafter referred to
as "noise") i is presented as
xi si i , i 1, n (9)
Synthesized by the maximum likelihood method, the algorithm of frequency esti-
mation is based on the solution of a quadratic equation 2 B 2 0 , in which
only the coefficient B depends on the input counts. This coefficient is a statistic of the
form
N N
B x1 , ..., xN xi xi -2 2 xi2-1 / xi xi -1 xi -2 xi 1
2
(10)
i 3 i 3
By one of the roots of the equation (10)
B B2 (11)
* 2
2 4
The estimated frequency of the signal is determined: * arccos( * / 2) . The ab-
solute radial frequency estimate of a signal * 2 f is uniquely related to its nor-
malized value as
*
* (12)
where is the sampling interval, so in the following we will consider only the normal-
ized frequency, for which in this section we use the abbreviated term "frequency".
The frequency detector consists of an analog-to-discrete converter (ADC), a sliding
window SW of size n, in which a bundle of signal is formed, an instantaneous fre-
quency value * calculator and an instantaneous and carrier frequency 0 difference
calculator - * .
7
Fig. 3. Block diagram of frequency detector based on frequency estimator in sliding window
In Fig. results of FM signal processing by frequency detector based on frequency
estimator (10-12) and frequency estimator based on counting of zero-level crossings
in a sliding window of size n = 200 are presented. Carrier frequency 1MHz, frequency
deviation 486 Hz. The upper graph shows the modulating signal, the average signal of
the frequency estimator (*) and the lower signal of the estimation-cha based on the
counter 0-sections. Noiseless situation was simulated. The estimation error for the
algorithm (10-12) was 1.8%, for the 0-crossing algorithm - 10%.
In Fig. results of operation of frequency detectors in the situation with noise
SNR = 11dB are given. The estimation error in this case for algorithm (10-12) was
7%, for algorithm 0-sections - 22%.
Fig. 4. Simulation of frequency detectors based on the frequency estimator and
0-section counter in a sliding window (n = 200) without noise
Fig. 5. Simulation of frequency detectors based on frequency estimator and 0-section
counter in sliding window (n = 200) with noise. SNR = 11dB
Fig. 6. Implementation of a mixture of FM signal, Gaussian noise and chaotic impulse
interference. SNR = 11dB, RMS HIP = 5V, probability of HIP action p = 0.001.
Fig. 7. Modeling of frequency detectors based on frequency estimator and 0-section
counter in sliding window (n = 200) with noise. SNR = 11dB and impulse interfer-
ence with parameters: RMS HIP = 5V, probability of HIP action p = 0.001.
The influence of chaotic impulse interference on the efficiency of signal frequency
estimation is investigated. Under these conditions, the observation model looks like:
xi si i i , i 1, n (13)
9
Impulse interference i is distributed by law
p x2 ,
f p ( x) (1 p) ( x) exp( )
2 2 2
where 2 is the conditional variance of the impulse noise, p is the probability of
occurrence of the impulse noise.
The analysis of the results of simulation of frequency detectors shows the significant
advantages of the proposed FD over the FD based on the 0-section counter.
4 Algorithms For Estimating The Modulation Coefficient Of
Signals In Navigation-Landing Systems
4.1 Signals Of Instrumental Landing Systems
The position of the aircraft relative to a given descent path is determined by the angu-
lar deviations Δφ, Δθ in the horizontal and vertical planes. Angular deviations are
measured relative to the course and glide path planes, the intersection of which gives
a predetermined path, using the dependence of the spatial modulation depth coeffi-
cient. The spatial dependence of the modulation depth on the angles φ, θ is set by
course and glide path beacons with the corresponding radiation pattern forms.
The task of the onboard radio is to isolate the signal, filter it and determine the
modulation coefficients.
A signal is generated in space using radio beacons of Instrumental Landing
Systems (ILS):
S(t) = U0. (1+M1cos(Ω1t+ φ1) + M2cos(Ω2t + φ2)).cos(ω0t + φ0), (14)
where: U0 is the signal amplitude, which depends on the radiation pattern at the re-
ceiving point;
M1, M2 - spatial modulation depth coefficients;
Ω1, Ω2 are the frequencies of the modulating signals 150 Hz and 90 Hz;
φ1, φ2 are the phases of the corresponding signals;
ω0, φ0 - frequency and phase of the carrier wave 330 MHz;
The informative parameter that needs to be determined is the difference in the
depth of spatial modulation (DDM)
DDM = М1 - М2.
4.2 Algorithm for estimation of the informative parameters of ILS
The received signal at the input of the radio receiver is an additive mixture of
the useful signal and Gaussian interference
X(t) = S(t) + , (15)
The task is to isolate the information parameter from this mixture.
To solve it, we use a statistical approach based on a method of the maximum like-
lihood. We have an input sample of values xi = X(ti). To simplify the calculations, we
assume that the amplitude U0, the frequencies Ω1, Ω2, ω0 and the corresponding phas-
es φ1, φ2, φ0 of the signal are known.
We compose the likelihood function with according to the vector of parameters
M (M1, M 2 ) .
n 2
1
xi S (ti | M )
(15)
f ( x1 , x2 ,..., xn / M ) exp i 1 .
( 2 2 ) n 2 2
We optimize these equations by finding the maximum of the likelihood func-
tion by differentiating the logarithm of the function (15). We obtain the following
system of likelihood equations:
M ln f1 ( x1,...xn , / M1 )
1
1 n
xi U 0 1 + M1cos(1t i ) + M 2cos( 2 t i ) cos(0 t i ) cos(1t i ) cos(0 t i ) 0, (16)
2 2 i 1
ln f ( x ,...x , / M )
n
M 2 2 1 2
1 x U0 1 + M1cos(1t i ) + M 2cos(2 t i ) cos(0 t i ) cos(2 t i ) cos(0 t i ) 0.
n
2 2 i 1 i
or
n
x U0 1 + M1cos(1t i ) + M 2cos(2 t i ) cos(0 t i ) cos(1t i ) cos(0 t i ) 0,
i 1 i (17)
n
xi U 0 1 + M1cos(1t i ) + M 2cos( 2 t i ) cos(0 t i ) cos( 2 t i ) cos(0 t i ) 0.
i 1
The use of traditional methods for solving such a system of equations (17) is very
laborious, since they can be used under conditions of parametric definiteness of the
problem and with a small sample; otherwise, calculations are difficult to perform, and
their speed decreases. Therefore, we further consider the features of using numerical
methods for solving nonlinear equations.
4.3 Simple Iteration Method
This method consists of two stages: determining the initial approximation and
the iterative process itself. Considering that the modulation coefficient can take a
value from 0 to 1, we take M = 0 as the initial approximation. Next, we substitute this
value into the equation and calculate the correction. For the next approximation, we
take the obtained value of the modulation coefficient.
11
M 1 ( j 1) M 1 ( j )
n
1 xi U 0 1+M1cos(1t i ) + M 2cos( 2 t i ) cos(0 t i ) cos(1t i )cos(0 t i )
i i 1 (18)
M 2 ( j 1) M 2 ( j )
1 n
xi U 0 1+M1cos(1t i ) + M 2 cos( 2 t i ) cos(0 t i ) cos( 2 t i )cos(0 t i )
i i 1
The criterion for the end of the iterative process (18) is to select a condition until the
approximate value M i( k ) differs from the previous one M i( k -1) by an acceptable value
of accuracy E.
i Mi
M (k) (k-i)
max E
1in M (k)
i
The simple iteration method has fast convergence and computation speed.
4.4 Newton-Raphson Method
We use yet another numerical Newton-Raphson method to solve the system of
equations (17). We can write the iterative equation in the form
M 1 ( j 1) M 1 ( j )
(19)
xi U 0 1+M1 (j)cos(1t i ) + M 2 (j)cos(2 t i ) cos(0 t i ) cos(1t i )cos(0 t i )
n
i 1
n
xi U 0 1+M1 (j)cos(1t i ) + M 2 (j)cos(2 t i ) cos(0 t i ) cos(1t i )cos(0 t i )
M 1 i 1 M1 M ( j )
Transforming it, we obtain an algorithm for estimating the modulation depth
coefficient:
M 1 ( j 1) M 1 ( j )
xi U 0 1+M1 (j)cos(1t i ) + M 2 (j)cos( 2 t i ) cos(0 t i ) cos(1t i ) cos(0 t i ) (20)
n
i 1
( U 0 ) cos(1t i )cos( 0 t i ) cos(1t i )cos( 0 t i )
n
i 1
The value of M2 is similar. Accordingly, to determine the difference in the modula-
tion depth coefficient, it will be necessary to calculate the formula (19) for each coef-
ficient and calculate the difference between them DDM = M1 - M2.
The calculations were carried out for M1 = 0.2 and M2 = 0.2.
4.5 Fourier Transform
The use of the Newton-Raphson method for solving systems of equations in radio
engineering problems allows one to obtain a sufficiently high accuracy in estimating
information parameters. However, it is time-consuming and difficult to build a com-
puting process.
Considering that when assessing the modulation depth coefficient in the ILS-85 in-
strumental landing system, an allowable error in the estimation of signal parameters
of at least 2% was established, we can use simpler and more easily implemented cal-
culators methods that provide a given accuracy. One such approach to determining the
parameters of radio engineering signals is to use the Fourier transform.
In the ILS-85 instrument landing system, the task of determining the modulation
depth coefficient is simplified by the fact that the carrier frequency and modulation
frequencies are known in advance and do not need additional determination. First, we
take the phases of the corresponding signals φ1, φ2, φ0 known. Using the Fourier trans-
form, we find the signal spectrum and harmonic amplitudes corresponding to the car-
rier frequency Ω0 and frequencies Ω0 ± Ω1, Ω0 ± Ω2. The modulation depth coefficient
is found from the ratio of the amplitudes of these harmonics.
The research results are presented in the form of curves of the root-mean-square
deviation (RMS) of the DDM estimate from the sample size, the signal-to-noise ratio
U²/2σ² and from the value of the estimated DDM parameter itself.
When changing the signal sample, it can be seen that even with a small number of
signal samples (400-500 samples), the DDM estimate is 0.001, i.e. It is within ac-
ceptable limits with a sufficient margin of accuracy. As can be seen from Fig. 8, when
the SNR decreases, the DDM estimate changes slightly and at the level of 20 dB it
equals 0.0005, which is an order of magnitude higher than the permissible value.
Thus, the RMS of the DDM assessment is no more than 0.0005 and, accordingly,
does not exceed the permissible value. This gives us a sufficient margin of accuracy
and reason to use the considered methods for assessing DDM in the design of radar
landing systems.
It should be noted that the RMS of DDM estimates obtained using the maximum
likelihood method is less than when using the Fourier transform of the signal, but they
require a lot of time.
Fig. 8. RMS for DDM assessment when changing SNR
13
RMS
N
Fig. 9. RMS of the DDM assessment when changing the signal sample
RMS
DDM
Fig. 10. RMS of the DDM assessment when changing the value of the DDM
Relative error
Fig. 12. The relative error of the estimate of the DDM from the value of the DDM
5 Conclusions
Using presented methods of radio signals estimation (phase, frequency, depth modu-
lation coefficient) allows to increase the technical characteristics of radio engineering
systems with respect to signal processing. So, these processing algorithms allows to
evaluate information parameters in more difficult interference conditions.
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