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COHERENT NOISE ADDITION IN A SYNCHRONOUS HARTLEY QUADRATURE MODULATION DETECTOR
10.07.2025 12:34
[3. Nauki techniczne]
Автор: Aleksandr Borisovich Kokhanov, Doctor of Technical Sciences, Assoc. Prof. Odessa Polytechnic National University, Odessa
ORCID 0000-0002-7197-6380
The distinctive feature of the synchronous detector of the signal with Hartley quadrature amplitude modulation (HQAM) [1] from the synchronous detector of the signal with quadrature amplitude modulation (QAM) [2] is the presence of adders at its output. In the synchronous detector of the QAM signal, the demodulated signals are taken from the outputs of the low-pass filter (LPF), which is a significant distinctive feature. This feature of the HQAM demodulator also leads to the summation of the noise components of the in-phase and quadrature channels, which is not present in the synchronous detector of the QAM signal.
Considering that the noise power is divided equally between the in-phase and quadrature channels and assuming that these noise process powers are equal 
, we can assume that 
, where is 
— the noise amplitude in the in-phase channel, 
— is the noise amplitude in the quadrature channel.
, we can assume that , where is — the noise amplitude in the in-phase channel, — is the noise amplitude in the quadrature channel.In [3, p. 110] it is shown that additive white Gaussian noise (AWGN) is characterized by the fact that the “values” of its amplitudes at any two arbitrarily close moments in time 
and 
are uncorrelated. In addition, in [3, p. 111] it is shown that noise components of the form 
and 
are random stationary processes since their amplitudes are random processes, and the average value of these processes does not depend on time t, and their correlation function depends on only one variable . It is also shown in [2, pp. 138-138] that the processes 
and 
are normal and statistically independent.
and are uncorrelated. In addition, in [3, p. 111] it is shown that noise components of the form and are random stationary processes since their amplitudes are random processes, and the average value of these processes does not depend on time t, and their correlation function depends on only one variable . It is also shown in [2, pp. 138-138] that the processes and are normal and statistically independent.From the above it can be concluded that 
and 
at the outputs of the low-pass filters of the in-phase and quadrature channels are random, statistically independent, normal processes.
and at the outputs of the low-pass filters of the in-phase and quadrature channels are random, statistically independent, normal processes.The amplitude of the noise component of the AWGN is equal to 
both in the in-phase channel and in the quadrature channel and is nothing more than the standard deviation 
[4, p. 412]. Considering that detection occurs in a synchronous detector, then in the output stages of the HQAM signal detector, coherent summation of the signals of the in-phase and quadrature channels occurs, which ensures the restoration of the signals 
and 
. However, there is also a coherent summation of noise, which is in an additive mixture with these signals. In non-coherent detection, as is known, the dispersions of the noise components are added, which leads to an increase in the standard deviation of the noise component.
both in the in-phase channel and in the quadrature channel and is nothing more than the standard deviation [4, p. 412]. Considering that detection occurs in a synchronous detector, then in the output stages of the HQAM signal detector, coherent summation of the signals of the in-phase and quadrature channels occurs, which ensures the restoration of the signals and . However, there is also a coherent summation of noise, which is in an additive mixture with these signals. In non-coherent detection, as is known, the dispersions of the noise components are added, which leads to an increase in the standard deviation of the noise component.In the case of coherent addition of the noise components of the in-phase and quadrature channels, the value of the standard deviation is averaged, as shown in [3, p. 412]. Figure 1 shows the time diagrams of the HQAM 16 signal without noise (a) and its AWGN noise component (b) at a signal-to-noise ratio of 5 dB after filtering using a low-pass filter. Moreover, the frequency band and the parameters of the low-pass filter fully correspond to the parameters of the low-pass filter of the in-phase and quadrature channels. In the synchronous detector of the HQAM signal, coherent addition of the signals of the in-phase and quadrature channels also occurs, and the noise components of these channels, since they are random, statistically independent, normal processes, are coherently averaged.
For further comparison of standard deviations, the AWGN noise (b) was extracted from the HQAM 16 signal and passed through a low-pass filter with the same bandwidth and characteristics as the in-phase and quadrature channel low-pass filters. This made it possible to ensure identical frequency bands occupied by the noise components shown in Figure 1, where the values of standard deviations for each process are also given.
a) HQAM 16 signal without noise.
b) Noise component at a signal-to-noise ratio of 5 dB after the low-pass filter.
c) Noise component at the output of the common-mode channel after the low-pass filter.
d) Noise component at the output of the quadrature channel after the low-pass filter.
f) The sum of the noise components of the in-phase and quadrature channels.
Fig. 3 — HQAM 16 signal and noise components of the synchronous detector with a signal-to-noise ratio of 5 dB (the parameters of all low-pass filters are identical).
With coherent summation of the noise components of the AWGS [3, p. 412], one can write the sum for the total noise component at the output of the corresponding channel of the synchronous detector after passing the low-pass filter during coherent detection (coherent averaging) as
(1)because 
, where 
— number of realizations of the noise process [3, p. 415] (Z=2, since there are two channels — in-phase and quadrature). In this case, the noise dispersion will also decrease 
, which is nothing more than the noise power at the detector output.
, where — number of realizations of the noise process [3, p. 415] (Z=2, since there are two channels — in-phase and quadrature). In this case, the noise dispersion will also decrease , which is nothing more than the noise power at the detector output.Relationship (1) is well confirmed by the simulation results shown in Figure 1. The ratio of the standard deviation of the noise component at the input of the synchronous detector (Fig. 1 b) is 
(V), and the standard deviation of the noise component at the output of the synchronous detector is 
(V). Ratio
(V), and the standard deviation of the noise component at the output of the synchronous detector is (V). Ratio 
(2)almost corresponds to the meaning 
with an error of 1.1%, which, in turn, agrees with (1). Deviations in the simulation modeling arise due to the fact that a short pseudo-random sequence with the number of symbolic intervals N=32 is used. It is also known that pseudo-random sequences (i.e., the length of which is limited) are used when generating random signals in programs. To obtain a more accurate result, 10 simulations of the processes were carried out. The results of the obtained relationships are given in Table 1.
with an error of 1.1%, which, in turn, agrees with (1). Deviations in the simulation modeling arise due to the fact that a short pseudo-random sequence with the number of symbolic intervals N=32 is used. It is also known that pseudo-random sequences (i.e., the length of which is limited) are used when generating random signals in programs. To obtain a more accurate result, 10 simulations of the processes were carried out. The results of the obtained relationships are given in Table 1.Table 1 — Values of standard deviations at the input and output of the synchronous detector.
From Table 1 it can be seen that the absolute error has an important value 
, which corresponds to a relative error of 0.31%.
, which corresponds to a relative error of 0.31%.Thus, the obtained theoretical results and the results of simulation modeling of coherent summation of noise (coherent averaging) in a synchronous detector of signals with quadrature amplitude modulation (HQAM) are well confirmed by the obtained results of this modeling.
Literature:
1. A. Kokhanov. Formation of a signal with Hartley amplitude-phase modulation as the sum of two signals with Hartley amplitude modulation / Praci Odesa Polytechnic University, 2025. Issue. 1(71). ISSN 2076-2429 (print). DOI: 10.15276/opu.1.71.2025.01
2. Prokis J. Digital communication: trans. from English / J. Prokis; trans. from English. edited by D.D. Klovsky. - M .: Radio i svyaz, 2000. - 800 p.: ill. - ISBN 5-256-01434-X.
3. Levin B.R. Theoretical foundations of statistical radio engineering. - 3rd ed. Revised and additional — M. Radio and Communications, 1989.-656 p.: ill.
4. Lyons R. Digital Signal Processing: Second Edition / Richard Lyons. – M.: OOO Binom-press, 2011. – 656 p.
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