Method of iterative single-channel blind separation for qpsk signals
A method for single-channel blind separation of two QPSK (quadrature phase shift keying) signals is proposed. The method is based on the iterative maximization of a posteriori probability for mixture's components. The relations for a posteriori probabilities are derived and on its basis the ite...
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| Дата: | 2018 |
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Інститут кібернетики ім. В.М. Глушкова НАН України
2018
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| Назва видання: | Математичне та комп'ютерне моделювання. Серія: Фізико-математичні науки |
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| Цитувати: | Method of iterative single-channel blind separation for qpsk signals / V.Yu. Semenov // Математичне та комп'ютерне моделювання. Серія: Фізико-математичні науки: зб. наук. пр. — Кам’янець-Подільський: Кам'янець-Подільськ. нац. ун-т, 2018. — Вип. 17. — С. 108-116. — Бібліогр.: 6 назв. — англ. |
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irk-123456789-1622052020-01-05T01:25:35Z Method of iterative single-channel blind separation for qpsk signals Semenov, V.Yu. A method for single-channel blind separation of two QPSK (quadrature phase shift keying) signals is proposed. The method is based on the iterative maximization of a posteriori probability for mixture's components. The relations for a posteriori probabilities are derived and on its basis the iterative algorithm for the estimation of mixture's components is developed. The algorithm for the estimation of channel parameters (amplitudes, phases, time delays) is also developed. The effectiveness of method is demonstrated for various noise levels and time diversities between channels. The proposed parameters’ estimation procedure provides significant reduction of bit error rate (BER) over the case of unknown parameters. Запропоновано метод одноканального сліпого розділення двох сигналів з квадратурно-фазовою маніпуляцією (QPSK). Метод базується на ітеративному оцінюванні компонентів суміші за принципом максимізації апостеріорної ймовірності. Отримані формули для відповідних апостеріорних ймовірностей та на їх основі розроблено алгоритм оцінювання компонентів суміші. Також розроблено алгоритм оцінювання параметрів каналу (амплітуд, фаз і часових затримок). Ефективність методу перевірена при різних рівнях шуму та часового рознесення між каналами. Розроблена процедура оцінювання параметрів забезпечує суттєве скорочення бітової похибки (BER) у порівнянні з випадком невідомих параметрів. 2018 Article Method of iterative single-channel blind separation for qpsk signals / V.Yu. Semenov // Математичне та комп'ютерне моделювання. Серія: Фізико-математичні науки: зб. наук. пр. — Кам’янець-Подільський: Кам'янець-Подільськ. нац. ун-т, 2018. — Вип. 17. — С. 108-116. — Бібліогр.: 6 назв. — англ. 2308-5878 http://dspace.nbuv.gov.ua/handle/123456789/162205 654.165 en Математичне та комп'ютерне моделювання. Серія: Фізико-математичні науки Інститут кібернетики ім. В.М. Глушкова НАН України |
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A method for single-channel blind separation of two QPSK (quadrature phase shift keying) signals is proposed. The method is based on the iterative maximization of a posteriori probability for mixture's components. The relations for a posteriori probabilities are derived and on its basis the iterative algorithm for the estimation of mixture's components is developed. The algorithm for the estimation of channel parameters (amplitudes, phases, time delays) is also developed. The effectiveness of method is demonstrated for various noise levels and time diversities between channels. The proposed parameters’ estimation procedure provides significant reduction of bit error rate (BER) over the case of unknown parameters. |
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Article |
| author |
Semenov, V.Yu. |
| spellingShingle |
Semenov, V.Yu. Method of iterative single-channel blind separation for qpsk signals Математичне та комп'ютерне моделювання. Серія: Фізико-математичні науки |
| author_facet |
Semenov, V.Yu. |
| author_sort |
Semenov, V.Yu. |
| title |
Method of iterative single-channel blind separation for qpsk signals |
| title_short |
Method of iterative single-channel blind separation for qpsk signals |
| title_full |
Method of iterative single-channel blind separation for qpsk signals |
| title_fullStr |
Method of iterative single-channel blind separation for qpsk signals |
| title_full_unstemmed |
Method of iterative single-channel blind separation for qpsk signals |
| title_sort |
method of iterative single-channel blind separation for qpsk signals |
| publisher |
Інститут кібернетики ім. В.М. Глушкова НАН України |
| publishDate |
2018 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/162205 |
| citation_txt |
Method of iterative single-channel blind separation for qpsk signals / V.Yu. Semenov // Математичне та комп'ютерне моделювання. Серія: Фізико-математичні науки: зб. наук. пр. — Кам’янець-Подільський: Кам'янець-Подільськ. нац. ун-т, 2018. — Вип. 17. — С. 108-116. — Бібліогр.: 6 назв. — англ. |
| series |
Математичне та комп'ютерне моделювання. Серія: Фізико-математичні науки |
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AT semenovvyu methodofiterativesinglechannelblindseparationforqpsksignals |
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2025-07-14T14:44:04Z |
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| fulltext |
Математичне та комп’ютерне моделювання
108
UDC 654.165
V. Yu. Semenov, Ph. D.
Scientific and Production Enterprise «Delta SPE», Kiev
METHOD OF ITERATIVE SINGLE-CHANNEL BLIND
SEPARATION FOR QPSK SIGNALS
A method for single-channel blind separation of two QPSK (quad-
rature phase shift keying) signals is proposed. The method is based on
the iterative maximization of a posteriori probability for mixture's
components. The relations for a posteriori probabilities are derived and
on its basis the iterative algorithm for the estimation of mixture's com-
ponents is developed. The algorithm for the estimation of channel pa-
rameters (amplitudes, phases, time delays) is also developed. The ef-
fectiveness of method is demonstrated for various noise levels and time
diversities between channels. The proposed parameters’ estimation
procedure provides significant reduction of bit error rate (BER) over
the case of unknown parameters.
Key words: Blind Source Separation, BPSK (Binary Phase
Shift Keying), QPSK (Quadrature Phase Shift Keying).
Introduction. Blind Source Separation is rapidly evolving since
1990s and comprises wide field of problems in telecommunications. There
are a lot of existing approaches for the solution of blind source separation
problem (see, e.g. [1–6]).
A general statement of blind separation problem is shown at Fig. 1.
There are p sources which are mixed by some vector-function at additive
noise background. Having m sensors, separation algorithm has to estimate
the source signals.
Most of methods imply that the number of sensors is not less than the
number of sources. However, the more frequently found case is of one sensor
and several sources (underdetermined blind separation problem). When there
are less sensors than sources, the problem is known to be underdetermined and
turns out to be quite challenging. To remove the indeterminacy, we need to
exploit any a priori knowledge induced by the system.
Fig. 1. General statement of blind source separation problem
© V. Yu. Semenov, 2018
Серія: Фізико-математичні науки. Випуск 17
109
Fig. 2. Statement of considered underdetermined BSS problem
So, consider the problem from radio communications presented at
Fig. 2. We have two discrete sequences 1( )s n , 2 ( )s n which are both
QPSK, i.e. possess values 1 j . They are passed through two independ-
ent communication channels. Their mixture ( )x t is the observation signal.
The goal is to restore original sequences 1( )s n , 2 ( )s n .
In this paper the Bayesian approach proposed in [2] for the case of
BPSK signals is further developed. We expand this approach for the case
of QPSK signals and, besides, add channel parameters estimation proce-
dure, while in work [2] the channel parameters were assumed to be known.
Higher order modulations can be handled as well, though computational
expenses grow exponentially with the modulation order.
The organization of the paper is as follows. First, the structure of the
proposed receiver is described. Then the idea of Bayesian approach to the
estimation of original QPSK sequences as well as iterative separation algo-
rithm is explained. The following sections include estimation of channel
parameters and the experimental results.
Preliminaries. As is known, in the data communication system, the
transmitted QPSK sequences of symbols must be bandlimited using a
pulse shaping filter ( )g t before transmitting. The received mixture of two
digitally modulated signals received by one antenna in single channel can
be expressed as:
1 2( ) ( ) ( ) ( )x t x t x t w t ,
where ( ), 1,2ux t u are the signals from two sources:
( ) ( ) ( ), 1,2uj
u u u s u
n
x t a e s n g t nT u
and ( ); 1,2us n u are original QPSK sequences to be estimated; sT is a
symbol period; ua are the amplitudes; u are the phases; u are the time
shifts. ( )g t is a total channel response (assumed to be raised square-root
cosine with known roll-off), ( )w t is background noise with variance 2 .
Математичне та комп’ютерне моделювання
110
The structure of proposed receiver. In this section we derive the
separation algorithm, described in [2], but we do not limit ourselves to
BPSK modulation and show that this approach can be applied to any kind
of modulation. Structure of proposed receiver is presented at Fig. 3. The
idea is to produce two discrete sequences: 1( )y n synchronous with the
first source and 2 ( )y n synchronous with second source.
The mixture is passed through filter ( )g t . Introducing notation
( ) ( ) ( )h t g t g t for the «normal» raised cosine filter with the same roll-
off and taking into account that ( ) ( ) ( ), 1, 2,u ug t g t h t u we
have the following output of matched filter:
1 2
1 1 1 2 2 2( ) ( ) ( ) ( ) ( ).j j
s s
n n
y t a e s n h t nT a e s n h t nT
(1)
Sampling of the signal (1) at times 1( )snT and 2( )snT respec-
tively, produces two sequences:
'
' ' 1 2( ) ( ) ( ) ( ) ( ), 1,2,u uj j
u u u u u u
n
y n a e s n a e s n h t w n u
where ' 3u u denotes the channel index, opposite to u .
Fig. 3. The structure of the receiver
Let us assume that the impulse response ( )h t is essentially non-zero
only for (2 1)l symbols (a common example is 2l ). Using this as-
sumption, the model of observations transforms to:
1 2
2 1
1 1 1 2 2, 2 1
2 2 2 1 1, 1 2
( ) ( ) ( ) ( ),
( ) ( ) ( ) ( ),
j j T
j j T
y n a e s n a e h s n w n
y n a e s n a e h s n w n
(2)
where
, '{ ( ), ... ,
( ) { ( )}, ... .
u s u u
u u
h h kT k l l
s n s n k k l l
Серія: Фізико-математичні науки. Випуск 17
111
So, as can be seen from observation model (2), the observed signals
1( )y n and 2 ( )y n include first signal plus weighted tail of the second and
second signal plus weighted tail of the first. We assume that the observa-
tion noises 1 2,w w possess the same variance 2 . So, having observations
1( )y n and 2 ( )y n , our goal is to find estimates of 1( )s n and 2 ( )s n .
Bayesian estimation of original QPSK sequences. The main idea
of proposed approach is to maximize maximum a posteriori probability of
transmitted symbols for each time instant 1,...,n N :
1..4
max ( ( ) / ( )), 1,2; { 1 }.u m u mm
P s n S y n u S j
Similarly to technique, implemented in [2], one can show that a posteriori
probability for u -th signal is connected with that of the opposite signal:
' '
( ( ) / ( ))
( ( ) / ( ), ( ) ) ( ( ) ).
l
u m u
u m u u u
s S
P s n S y n
P s n S y n s n s P s n s
(3)
Assuming that the observation noise is Gaussian,
2 2
' , ,( ( ) / ( ), ( ) ) exp( 0.5 ( ))u m u u u m sP s n S y n s n s d n
(we dropped the denominator of Gaussian density for simplicity), where a
priori discrepancy is given by
, , ' ' ',( ) ( ) ( exp( ) exp( ) )T
u m s u m u u u u ud n y n S a j a j h s
and
' ' 1
...
( ( ) ) ( ( ) )u u k l
k l l
P s n s P s n k s
.
Thus, formula (3) turns to
2
, , ' 1
...
( ( ) / ( ))
exp( 0.5 ) ( ( ) ).
l
u m u
u m s u k l
s S k l l
P s n S y n
d P s n k s
(4)
As can be seen, relation (4) describes interdependence of a posteriori
probabilities for the opposite signals. This gives a hint to construct itera-
tive algorithm:
( ) 2
, , ',, 1
...
( ) exp( 0.5 ) ( )
l
m
u m u iu i
k l l
p n d p n k
s
s S
,
where i is a number of iteration. So, the iterative algorithm for the
estimation of sequences 1( )s n and 2 ( )s n is as shown at Fig. 4. The
iterations stop when average probabilities on adjacent iterations do not
differ too much.
Математичне та комп’ютерне моделювання
112
Fig. 4. The algorithm for the restoration of original QPSK sequences
Channel parameters estimation. There is a common practice to in-
sert predefined symbols (unique words) into the transmitted sequences. In
our modeling, we use 32-symbol (64-bit) sequences denoted by U . The
position of the unique word can be identified by its cross-correlation with
received signal (see Fig. 5). Once we have detected the position of the
unique word, we analyze its peak value
max
( )u
iR ( 1, 2u ). Then the ampli-
tude, phase and time delay can be estimated approximately as follows:
max
2( )ˆ / ,u
u ia R U ,
max
( )ˆ arg u
u iR ,
max max
max max max
( ) ( )
1 1
( ) ( ) ( )
1 1
ˆ
2
u u
i i
u u u u
i i i
R R
R R R
Серія: Фізико-математичні науки. Випуск 17
113
The last formula for the delay comes from the parabolic interpolation
of correlation function (see Fig. 6).
Fig. 5. Detection of unique words in the mixture of two signals
Fig. 6. Determination of channel parameters from cross-correlation peak value
Experimental results. In this section the performance of proposed algo-
rithm at different signal-to-noise ratios is analyzed. We take in these experi-
ments the following values of parameters: 1 2 1a a , 1 2 0.35 . The
value of time diversity 1 2| | was allowed to take different values and
we examined algorithm's performance for different .
Fig. 7 shows the performance of proposed algorithm when the channel
parameters are assumed known and Fig. 8 shows the performance of proposed
algorithm when the channel parameters are estimated as was discussed above.
Математичне та комп’ютерне моделювання
114
In both cases 15 iterations of the algorithm were used. It can be seen that the
case of known parameters has a slight advantage in terms of BER over the
case when parameters are estimated by proposed procedure.
Fig. 7. Performance of proposed algorithm when the channel
parameters are assumed known
Fig. 8. Performance of proposed algorithm when
the channel parameters are estimated by proposed method
As can be seen from the figures 7 and 8, the higher time diversity
leads to better separation performance of the algorithm. For example, with
0 we have no diversity and the components of the mixture cannot be
separated. On the opposite, the best performance is achieved when time
diversity takes its maximum value 0.5 sT . This shows that the algorithm
may properly exploit the diversity induced by the delay between channels.
To understand better the effect of parameters estimation procedure,
we consider the case 1/ 3 sT . Fig. 9 shows the comparison of BER for
several cases:
Серія: Фізико-математичні науки. Випуск 17
115
1. Amplitudes 1 2,a a are assumed to be known, but the phases 1 2, and
the delays 1 2, take random values from their area of definition
(«partially-known» case).
Fig. 9. Performance of proposed parameters’
estimation method for the case 1 / 3 sT
2. All channel parameters are assumed unknown and they are estimated
by the proposed procedure.
3. All channel parameters are assumed known beforehand («ideally-
known» case).
From the Fig. 9 it can be seen that the proposed estimation procedure
crucially improves the BER of the algorithm providing improvement over the
case 1 («partially-known» parameters) from 1.3 times for 0/ 0bE N dB to
112 times for 15SNR dB. At the same time, the ratio between proposed
method estimation method and case of ideally known parameters is not large:
the obtained BERs are always of the same order, the maximum ratio between
them is from 1.02 times for 0SNR dB to 2.7 times for 15SNR dB. The
similar conclusions are confirmed for other values of .
Conclusions. In this paper the new method for the single-channel
separation of two QPSK signals based on iterative maximization of a pos-
teriori probability for transmitted symbols is presented. The best perfor-
mance of the method is achieved when time diversity between channels
takes its maximum value, namely half of a symbol period. The essential
advantage over the previously proposed approach is due to proposed pro-
cedure of channel parameters’ estimation. For the case 1/ 3 sT it was
shown that the BER is improved from 1.3 to 112 times (for different
0/bE N ) in comparison with the case of partially known parameters. At
the same time, the BER values for the proposed estimation procedure are
of the same order as for the case of ideally known parameters.
Математичне та комп’ютерне моделювання
116
References:
1. Wu C. Single-Channel Blind Source Separation of Co-Frequency Overlapped
GMSK Signals Under Constant-Modulus Constraints / C. Wu, Z. Liu,
X. Wang, W. Jiang, X. Ru // IEEE Communications Letters. — March
2016. — Vol. 20. — № 3. — P. 486–489.
2. Gouldieff V. MISO Estimation of Asynchronously Mixed BPSK Sources /
V. Gouldieff, J. Palicot // Proc. IEEE Conf. EUSIPCO. — 2015. — P. 369–373.
3. Arulampalam M. S. Particle-Filtering-Based Approach to Undetermined Blind
Separation / M. S. Arulampalam, S. Maskell, N. Gordon, T. Clapp // Advances in
information Sciences and Service Sciences. — 2012. —Vol. 4. — P. 305–313.
4. Pan B. Blind Separation of Two QPSK Signals Based on Lattice Reduction /
B. Pan and S. Tu // 2017 International Conference on Information Science and
Control Engineering (ICISCE). — Changsha, 2017. —P. 1437–1440.
5. Warner E. Single-channel blind signal separation of filtered MPSK signals /
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tion. — 2003. — Vol. 150. — № 6. — P. 396–402.
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МЕТОД ІТЕРАТИВНОГО ОДНОКАНАЛЬНОГО
СЛІПОГО РОЗДІЛЕННЯ QPSK-СИГНАЛІВ
Запропоновано метод одноканального сліпого розділення двох си-
гналів з квадратурно-фазовою маніпуляцією (QPSK). Метод базується
на ітеративному оцінюванні компонентів суміші за принципом мак-
симізації апостеріорної ймовірності. Отримані формули для відповід-
них апостеріорних ймовірностей та на їх основі розроблено алгоритм
оцінювання компонентів суміші. Також розроблено алгоритм оціню-
вання параметрів каналу (амплітуд, фаз і часових затримок). Ефекти-
вність методу перевірена при різних рівнях шуму та часового розне-
сення між каналами. Розроблена процедура оцінювання параметрів
забезпечує суттєве скорочення бітової похибки (BER) у порівнянні з
випадком невідомих параметрів.
Ключові слова: сліпе розділення, BPSK (двійкова фазова маніпу-
ляція), QPSK (квадратурна фазова маніпуляція).
Отримано: 24.05.2018
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