CN111600815A - Design algorithm of simplified factor graph equalizer under complex multipath channel - Google Patents

Design algorithm of simplified factor graph equalizer under complex multipath channel Download PDF

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CN111600815A
CN111600815A CN201910126771.8A CN201910126771A CN111600815A CN 111600815 A CN111600815 A CN 111600815A CN 201910126771 A CN201910126771 A CN 201910126771A CN 111600815 A CN111600815 A CN 111600815A
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factor graph
multipath
simplified
channel
equalizer
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张建军
董悦
曹晓冬
范玉进
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Tianjin Optical Electrical Communication Technology Co Ltd
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    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
    • H04L25/00—Baseband systems
    • H04L25/02—Details ; arrangements for supplying electrical power along data transmission lines
    • H04L25/0202—Channel estimation
    • H04L25/0204—Channel estimation of multiple channels
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
    • H04L25/00—Baseband systems
    • H04L25/02—Details ; arrangements for supplying electrical power along data transmission lines
    • H04L25/0202—Channel estimation
    • H04L25/0212—Channel estimation of impulse response
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
    • H04L25/00—Baseband systems
    • H04L25/02—Details ; arrangements for supplying electrical power along data transmission lines
    • H04L25/0202—Channel estimation
    • H04L25/024—Channel estimation channel estimation algorithms
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
    • H04L25/00—Baseband systems
    • H04L25/02—Details ; arrangements for supplying electrical power along data transmission lines
    • H04L25/03—Shaping networks in transmitter or receiver, e.g. adaptive shaping networks
    • H04L25/03006—Arrangements for removing intersymbol interference
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
    • H04L25/00—Baseband systems
    • H04L25/02—Details ; arrangements for supplying electrical power along data transmission lines
    • H04L25/03—Shaping networks in transmitter or receiver, e.g. adaptive shaping networks
    • H04L25/03006—Arrangements for removing intersymbol interference
    • H04L2025/03433—Arrangements for removing intersymbol interference characterised by equaliser structure
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
    • H04L25/00—Baseband systems
    • H04L25/02—Details ; arrangements for supplying electrical power along data transmission lines
    • H04L25/03—Shaping networks in transmitter or receiver, e.g. adaptive shaping networks
    • H04L25/03006—Arrangements for removing intersymbol interference
    • H04L2025/03592—Adaptation methods
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    • H04L2025/03611—Iterative algorithms

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Abstract

The invention provides a simplified factor graph equalizer design algorithm under a complex multipath channel, which deduces an equalization algorithm SFGE of a simplified factor graph model under the complex multipath channel and researches how to design a design criterion and a self-adaptive estimation algorithm of simplified factor graph parameters. The obtained simplified factor graph equalization algorithm obtains better performance with smaller complexity under a complex multipath channel. The reduced factor graph may reduce the multipath impact of the non-ideal satellite channel to within 1.5 dB.

Description

Design algorithm of simplified factor graph equalizer under complex multipath channel
Technical Field
The invention relates to the field of communication systems, in particular to a design algorithm of a simplified factor graph equalizer under a complex multipath channel.
Background
The multipath channel refers to a multipath despreading method of a spread spectrum communication system: the data from n channels are quantized by the A/D conversion circuit, then the quantized data are stored in the corresponding data memory, then the multi-path data of n channels are read out from different positions of each data memory according to the peak value and the time delay value of the multi-path, and then the multi-path de-spread is carried out: firstly, pre-despreading, secondly, secondary despreading and accumulation, thirdly, multiple despreading and accumulation and finally outputting multipath despreading data of n channels. In order to reduce the complexity of the factor graph equalizer, it is an urgent problem to develop a simplified factor graph equalizer design algorithm under a complex multipath channel.
Disclosure of Invention
The invention aims to solve the technical problems and provides a simplified factor graph equalizer design algorithm under a complex multipath channel.
In order to solve the technical problems, the invention adopts the technical scheme that:
a simplified factor graph equalizer design algorithm under a complex multipath channel comprises the following processes:
and assuming that the impulse response of the multipath channel is h, the factor graph parameter of the iterative equalizer is g, and the number of the nonzero elements in g is less than that of the nonzero elements in h. h and g satisfy the following relationship:
h＝g+f (1)
where the vector f is the remaining multipaths in h, except multipath g. At this time, the symbol y is receivedkCan be expressed as:
yk＝hsk+wn＝(g+f)sk+wn＝gsk+(fsk+wn) (2)
in the above formula skIs given askAssociated transmission symbol sequence, wnIs additive white Gaussian noise, wherein fsk+wnFor a factor graph equalizer, it can be considered as noise w'nI.e. by
w'n＝fsk+wn(3)
Its mean value m (w'n) And variance D (w'n) Comprises the following steps:
m(w'n)＝0 (4)
Figure BDA0001973875780000021
wherein
Figure BDA0001973875780000022
To send symbolsNumber skThe power of (c), here assumed to be 1,
Figure BDA0001973875780000023
for channel noise wnOf the power of (c).
At this time, the equalization factor node function E in the equalizer factor graphkThe following steps are changed:
Figure BDA0001973875780000024
in the above formula EkIs in a known transmitted symbol sequence skAnd the factor graph parameter of the iterative equalizer is g, and the received symbol is ykThe probability of (c).
Assuming that the transmitted symbols are uniformly distributed, in the first iteration process, the jth equalization function node eqjTo the kth symbol node skDelivered foreign information
Figure BDA0001973875780000025
Can be expressed as:
Figure BDA0001973875780000026
in the above formula: { skDenotes the division s associated with the equalization function nodekSequence numbers other than the above.
And symbol skThe initial posterior probability information of (a) may be expressed as:
Figure BDA0001973875780000027
in the above formula, S is a transmission sequence including all symbols, N (S)k) Is a sum of skA collection of all equalization nodes that are relevant.
The case that each equalization factor node in the factor graph has only 3 edges is taken as an example for explanation, that is, the number of the nonzero elements in g is 3, which are respectively defined as: ga、gbAnd gc。ha、hbAnd hcCorresponding multipathDelayed by a, b and c symbol periods, respectively.
In the factor graph, the node s is connected with the symbolic variablekThe adjacent equalization factor node has Ek+a、Ek+bAnd Ek+c. These three factor nodes pass to skThe external information of (a) is:
Figure BDA0001973875780000028
Figure BDA0001973875780000031
Figure BDA0001973875780000032
then skA posteriori probability information LkComprises the following steps:
Figure BDA0001973875780000033
reliability R(s) defining confidencek)：
R(sk)＝sk×Lk(13)
The physical significance is as follows: when R(s)k)>Confidence L obtained at 0kCorrect, otherwise wrong; r(s)k) The greater the confidence LkThe more reliable is; r(s)k) The smaller, the confidence LkThe less reliable. According to about equation:
Figure BDA0001973875780000034
in equations (9) to (11), the sum of each term is 4, that is, n is 4. Then
Figure BDA0001973875780000035
The above formula is simplified to obtain:
Figure BDA0001973875780000036
at skTaking a fixed value, while the other symbols are uniformly distributed, yk+a，yk+bAnd yk+cThe expected values of (c) are: e (y)k+a)＝gask、E(yk+b)＝gbskAnd E (y)k+c)＝gcsk. The variance of confidence reliability can therefore be expressed as:
Figure BDA0001973875780000037
the value of the factor graph parameter g should maximize the lower bound of the expected value of the reliability of the posterior probability, that is:
Figure BDA0001973875780000041
order:
Figure BDA0001973875780000042
Figure BDA0001973875780000043
to pair
Figure BDA0001973875780000044
The derivation is carried out to obtain:
Figure BDA0001973875780000045
namely, it is
Figure BDA0001973875780000046
Is about
Figure BDA0001973875780000047
Is a monotonically increasing function of. In the same way
Figure BDA0001973875780000048
Also relates to
Figure BDA0001973875780000049
And
Figure BDA00019738757800000410
is a monotonically increasing function of. Namely, it is
Figure BDA00019738757800000411
And
Figure BDA00019738757800000412
when the maximum value is taken out of the range,
Figure BDA00019738757800000413
the maximum value is taken.
Due to ga、gbAnd gcIs an element in h, g, to maximize the lower bound of confidence reliabilitya、gbAnd gcThe three elements with the largest absolute value in h should be taken.
The simplified factor graph, which does not take into account the performance impact of multipath with smaller amplitude, will cause a loss in error performance, which is called the reduction loss and is expressed as the difference (dB) in signal-to-noise ratio required to achieve the same bit error rate, and is denoted as LSFG. Meanwhile, the simplified factor graph has fewer short loops than the original factor graph, so that the error code performance is improved, and the signal-to-noise ratio threshold is improved, wherein the improvement of the signal-to-noise ratio performance is called as simplified gain and is marked as GSFG. And the actual SNR threshold difference is expressed in Δ SNR, and Δ SNR ═ GSFG-LSFG。
Primitive factor graph parameter h, number of non-zero elements thereof<h>With a corresponding confidence level of
Figure BDA00019738757800000414
Reliability of R0(sk). Reduced factor graph parameter g, number of non-zero elements thereof<g><<h>With a corresponding confidence level of
Figure BDA00019738757800000415
Reliability of Rs(sk). According to ln (e)x+ey) Max (x), knowing
Figure BDA00019738757800000416
Figure BDA0001973875780000051
When the reliability of other variable nodes is large enough, the above two equations can be simplified as follows:
Figure BDA0001973875780000052
Figure BDA0001973875780000053
in the primitive factor graph, (y)k+i-hsk+i) Is a mean of 0 and a variance of
Figure BDA0001973875780000054
(ii) a gaussian distribution of; in the reduced factor graph (y)k+i-gs'k+i) Is a mean of 0 and a variance of
Figure BDA0001973875780000055
Is a gaussian distribution of
Figure BDA0001973875780000056
And
Figure BDA0001973875780000057
the mean values of (a) are respectively:
Figure BDA0001973875780000058
Figure BDA0001973875780000059
their variances are:
Figure BDA00019738757800000510
Figure BDA00019738757800000511
namely, it is
Figure BDA00019738757800000512
And
Figure BDA00019738757800000513
are all gaussian distributed as (m,2| m |). The performance loss of the reduced factor graph can therefore be analyzed by comparing the mean of the two. When the two reach the same error code performance (such as 1E-5), namely the posterior probability reaches a certain same distribution, the signal-to-noise ratio corresponding to the iterative equalization of the original factor graph is SNR0The channel noise is
Figure BDA00019738757800000514
The reduced factor graph iteratively equalizes the corresponding channel noise as
Figure BDA00019738757800000515
In this case, the following requirements are:
Figure BDA00019738757800000516
namely, it is
Figure BDA00019738757800000517
Corresponding loss of signal-to-noise ratio (L)SFG) Is composed of
Figure BDA0001973875780000061
In practical systems, when the performance loss cannot be larger than χIn dB, LSFGShould satisfy
Figure BDA0001973875780000062
I.e. g does not calculation2Should satisfy the minimum value of
Figure BDA0001973875780000063
When | | h | | non-conducting phosphor2When the normalization is 1, min { | | | g2The value of | } is
Figure BDA0001973875780000064
According to the change situation of the minimum value of the simplified factor graph parameters along with the original signal-to-noise ratio under the loss of different signal-to-noise ratios, when the original signal-to-noise ratio is higher, the multipath energy corresponding to the parameters of the simplified factor graph is closer to the multipath response of a channel; and the smaller the loss of the signal-to-noise ratio is required to be, the closer the multipath energy corresponding to the parameters of the simplification factor graph is to the multipath energy of the channel.
Therefore, the parameters of the simplified factor graph should select the multipath with larger amplitude in the multipath of the channel, and the energy of the selected multipath should be as close to the energy of the multipath of the channel as possible. According to the signal-to-noise ratio threshold of 0-15 dB in the actual satellite channel, under the condition that the loss is as small as possible, the multipath energy corresponding to the parameters of the simplified factor graph is set to be more than 95% of the multipath energy of the channel.
From the above analysis and assumptions, a parameter adaptive estimation algorithm of the simplified factor graph equalizer can be obtained:
1) estimating the response h of the multipath channel by utilizing an LMS algorithm;
2) sorting the multipath amplitudes in h from large to small;
3) selecting the multipath with the maximum multipath amplitude from the h as a main path;
4) selecting the multipaths with larger amplitude from h from large to small and adding the multipaths into the simplified factor graph parameter g;
5) detecting whether the energy of the simplified factor graph parameter g meets the condition that the energy of h is more than 95%; if yes, stopping selecting multipath from h, otherwise, continuing.
The invention has the advantages and positive effects that: the algorithm deduces an equalization algorithm SFGE of a simplified factor graph model under a complex multipath channel, and researches how to design a design criterion and a self-adaptive estimation algorithm of the parameters of the simplified factor graph; the obtained simplified factor graph equalization algorithm obtains better performance with smaller complexity under a complex multipath channel, and the simplified factor graph can reduce the multipath influence of an undesirable satellite channel to within 1.5 dB.
Drawings
FIG. 1 is a graph of impulse response amplitude for a linear distortion channel in a satellite communication system;
fig. 2 is a constellation diagram of BPSK symbols formed under a multipath channel g;
FIG. 3 is a diagram showing the variation of the minimum value of the simplified factor graph parameters with the original SNR under different SNR losses;
FIG. 4 is a graph of error performance for different reduced factor graphs;
FIG. 5 is a simplified factor graph of signal-to-noise ratio simplified loss and actual loss for iterative equalization at different error performance;
FIG. 6 is a simplified factor graph of error performance for iterative equalization with non-ideal amplitude-frequency characteristics;
fig. 7 is a simplified factor graph iterative equalization error performance graph for the case of non-ideal group delay characteristics.
Detailed Description
The following detailed description of specific embodiments of the invention refers to the accompanying drawings.
In an actual multipath channel, the number of multipaths is large, and the proportion of smaller multipaths is large. Where larger multipaths have a greater impact on performance and smaller multipaths have a lesser impact on performance. The computational complexity of a factor graph equalizer is proportional to the number of multipaths. To reduce the complexity of the factor graph equalizer, only the larger multipaths can be modeled and the effects of the smaller multipaths are equated to noise. The resulting factor graph equalizer is called a simplified factor graph equalizer (SFGE, simplifedfge). The algorithm studies how to select the parameters of the factor graph so as to maximize the reliability of the initialization posterior probability.
Figure 1 shows the impulse response amplitude distribution under equivalent multipath of a typical linear distortion channel in a satellite communication system, which shows the amplitude of the impulse response under two typical parabolic group delay and parabolic amplitude-frequency characteristic channels. In the figure, the maximum delay of the group delay is one symbol period, and the maximum distortion of the amplitude-frequency characteristic is 6 dB. Observing this figure, it can be seen that the number of extended multipaths in the satellite channel is much greater than 15, while the energy is concentrated mainly on the middle ones. Therefore, based on the above similar complex multipath channel, how to design the simplified factor graph model is studied, so that the complexity of the factor graph equalizer is greatly reduced when the equalization performance is not greatly lost.
And assuming that the impulse response of the multipath channel is h, the factor graph parameter of the iterative equalizer is g, and the number of the nonzero elements in g is less than that of the nonzero elements in h. h and g satisfy the following relationship:
h＝g+f (1)
where the vector f is the remaining multipaths in h, except multipath g. At this time, the symbol y is receivedkCan be expressed as:
yk＝hsk+wn＝(g+f)sk+wn＝gsk+(fsk+wn) (2)
in the above formula skIs given askAssociated transmission symbol sequence, wnIs additive white Gaussian noise, wherein fsk+wnFor a factor graph equalizer, it can be considered as noise w'nI.e. by
w'n＝fsk+wn(3)
Its mean value m (w'n) And variance D (w'n) Comprises the following steps:
m(w'n)＝0 (4)
Figure BDA0001973875780000081
wherein
Figure BDA0001973875780000082
For transmitting symbols skThe power of (c), assumed to be 1 in the algorithm,
Figure BDA0001973875780000083
for channel noise wnOf the power of (c).
At this time, the equalization factor node function E in the equalizer factor graphkThe following steps are changed:
Figure BDA0001973875780000084
in the above formula EkIs in a known transmitted symbol sequence skAnd the factor graph parameter of the iterative equalizer is g, and the received symbol is ykThe probability of (c).
Assuming that the transmitted symbols are uniformly distributed, in the first iteration process, the jth equalization function node eqjTo the kth symbol node skDelivered foreign information
Figure BDA0001973875780000085
Can be expressed as:
Figure BDA0001973875780000091
in the above formula: { skDenotes the division s associated with the equalization function nodekSequence numbers other than the above.
And symbol skThe initial posterior probability information of (a) may be expressed as:
Figure BDA0001973875780000092
in the above formula, S is a transmission sequence including all symbols, N (S)k) Is a sum of skA collection of all equalization nodes that are relevant.
The algorithm is described herein with respect to the case where each equalization factor node in the factor graph has only 3 edges, i.e.The number of non-zero elements in g is 3, which are respectively defined as: ga、gbAnd gc。ha、hbAnd hcThe corresponding multipaths are delayed by a, b and c symbol periods, respectively. The constellation of BPSK symbols under these three multipath channels is shown in fig. 2.
In the factor graph, the node s is connected with the symbolic variablekThe adjacent equalization factor node has Ek+a、Ek+bAnd Ek+c. These three factor nodes pass to skThe external information of (a) is:
Figure BDA0001973875780000093
Figure BDA0001973875780000094
Figure BDA0001973875780000095
then skA posteriori probability information LkComprises the following steps:
Figure BDA0001973875780000096
reliability R(s) defining confidencek)：
R(sk)＝sk×Lk(13)
The physical significance is as follows: when R(s)k)>Confidence L obtained at 0kCorrect, otherwise wrong; r(s)k) The greater the confidence LkThe more reliable is; r(s)k) The smaller, the confidence LkThe less reliable. According to about equation:
Figure BDA0001973875780000101
in equations (9) to (11), the sum of each term is 4, that is, n is 4. Then
Figure BDA0001973875780000102
The above formula is simplified to obtain:
Figure BDA0001973875780000103
at skTaking a fixed value, while the other symbols are uniformly distributed, yk+a，yk+bAnd yk+cThe expected values of (c) are: e (y)k+a)＝gask、E(yk+b)＝gbskAnd E (y)k+c)＝gcsk. The variance of confidence reliability can therefore be expressed as:
Figure BDA0001973875780000104
the value of the factor graph parameter g should maximize the lower bound of the expected value of the reliability of the posterior probability, that is:
Figure BDA0001973875780000105
order:
Figure BDA0001973875780000106
Figure BDA0001973875780000107
to pair
Figure BDA0001973875780000108
The derivation is carried out to obtain:
Figure BDA0001973875780000109
namely, it is
Figure BDA00019738757800001010
Is about
Figure BDA00019738757800001011
Is a monotonically increasing function of. In the same way
Figure BDA00019738757800001012
Also relates to
Figure BDA00019738757800001013
And
Figure BDA00019738757800001014
is a monotonically increasing function of. Namely, it is
Figure BDA00019738757800001015
And
Figure BDA00019738757800001016
when the maximum value is taken out of the range,
Figure BDA00019738757800001017
the maximum value is taken.
Due to ga、gbAnd gcIs an element in h, g, to maximize the lower bound of confidence reliabilitya、gbAnd gcThe three elements with the largest absolute value in h should be taken. For example, for Proakis-a channel (h ═ 0.04-0.050.07-0.21-0.50.720.360.000.210.030.07]) G should be [ 0000-0.50.720.360000 ]]。
The simplified factor graph, which does not take into account the performance impact of multipath with smaller amplitude, will cause a loss in error performance, which is called the reduction loss and is expressed as the difference (dB) in signal-to-noise ratio required to achieve the same bit error rate, and is denoted as LSFG. Meanwhile, the simplified factor graph has fewer short loops than the original factor graph, so that the error code performance is improved, and the signal-to-noise ratio threshold is improved, wherein the improvement of the signal-to-noise ratio performance is called as simplified gain and is marked as GSFG. And the actual SNR threshold difference is expressed in Δ SNR, and Δ SNR ═ GSFG-LSFG。
Primitive factor graph parameter h, number of non-zero elements thereof<h>With a corresponding confidence level of
Figure BDA0001973875780000111
Reliability of R0(sk). Reduced factor graph parameter g, number of non-zero elements thereof<g><<h>With a corresponding confidence level of
Figure BDA0001973875780000112
Reliability of Rs(sk). According to ln (e)x+ey) Max (x), knowing
Figure BDA0001973875780000113
Figure BDA0001973875780000114
When the reliability of other variable nodes is large enough, the above two equations can be simplified as follows:
Figure BDA0001973875780000115
Figure BDA0001973875780000116
in the above formula, | g | | and | | | h | | | represent the absolute values of the vectors g and h, respectively.
In the primitive factor graph, (y)k+i-hsk+i) Is a mean of 0 and a variance of
Figure BDA0001973875780000117
(ii) a gaussian distribution of; in the reduced factor graph (y)k+i-gs'k+i) Is a mean of 0 and a variance of
Figure BDA0001973875780000118
Is a gaussian distribution of
Figure BDA0001973875780000119
And
Figure BDA00019738757800001110
the mean values of (a) are respectively:
Figure BDA0001973875780000121
Figure BDA0001973875780000122
their variances are:
Figure BDA0001973875780000123
Figure BDA0001973875780000124
namely, it is
Figure BDA0001973875780000125
And
Figure BDA0001973875780000126
are all gaussian distributed as (m,2| m |). The performance loss of the reduced factor graph can therefore be analyzed by comparing the mean of the two. When the two reach the same error code performance (such as 1E-5), namely the posterior probability reaches a certain same distribution, the signal-to-noise ratio corresponding to the iterative equalization of the original factor graph is SNR0The channel noise is
Figure BDA0001973875780000127
The reduced factor graph iteratively equalizes the corresponding channel noise as
Figure BDA0001973875780000128
In this case, the following requirements are:
Figure BDA0001973875780000129
namely, it is
Figure BDA00019738757800001210
Corresponding loss of signal-to-noise ratio (L)SFG) Is composed of
Figure BDA00019738757800001211
In practical systems, when the performance loss cannot be greater than χ dB, LSFGShould satisfy
Figure BDA00019738757800001212
I.e. g does not calculation2Should satisfy the minimum value of
Figure BDA00019738757800001213
When | | h | | non-conducting phosphor2When the normalization is 1, min { | | | g | | non-phosphor2The value of is
Figure BDA00019738757800001214
Fig. 3 shows the variation of the minimum value of the simplified factor graph parameters with the original snr at different snr losses. From the above graph, it can be found that when the original signal-to-noise ratio is high, the multipath energy corresponding to the parameters of the simplified factor graph is closer to the multipath response of the channel. And the smaller the loss of the signal-to-noise ratio is required to be, the closer the multipath energy corresponding to the parameters of the simplification factor graph is to the multipath energy of the channel.
According to the analysis, the parameters of the simplified factor graph should select the multipath with larger amplitude from the multipath of the channel, and the energy of the selected multipath should be as close to the energy of the multipath of the channel as possible. According to the signal-to-noise ratio threshold of 0-15 dB in an actual satellite channel, the multipath energy corresponding to the parameters of the simplified factor graph is set to be more than 95% of the multipath energy of the channel under the condition that the loss is as small as possible.
From the above analysis and assumptions, a parameter adaptive estimation algorithm of the simplified factor graph equalizer can be obtained:
1. estimating the response h of the multipath channel by utilizing an LMS algorithm;
2. sorting the multipath amplitudes in h from large to small;
3. selecting the multipath with the maximum multipath amplitude from the h as a main path;
4. selecting the multipaths with larger amplitude from h from large to small and adding the multipaths into the simplified factor graph parameter g;
5. detecting whether the energy of the simplified factor graph parameter g meets the condition that the energy of h is more than 95%; if yes, stopping selecting multipath from h, otherwise, continuing.
Fig. 4 shows the error performance curves of the iterative equalization of the original factor graph and the simplified factor graph. Take Proakis-a channel as an example, h ═ 0.04-0.050.07-0.21-0.50.720.360.000.210.030.07])，g＝[0 0 00 -0.5 0.72 0.36 0 0 0 0]. Then | h | non-calculation21.001, | g | | | 0.8980. When the iterative equalization of the primitive factor graph reaches 1E-5, the SNR of the threshold value03.5dB, noise power
Figure BDA0001973875780000131
Performance loss L of the reduced factor graphSFG1.29 dB. If g ═ 0000-0.50.7200000]Number of non-zero elements thereof<g>2, the performance loss L of the simplified factor graphSFG＝4.92dB。
It can be found that when the 1E-5 error performance is achieved, the delta SNR of the two-path simplified factor graph is-4.5 dB, and the delta SNR of the three-path simplified factor graph is-1 dB. Finally, the G of the two-path simplified factor graph can be knownSFG0.42dB, G of the three-path reduced factor graphSFG0.29 dB. Because the short loop of the two-path factor graph is less than that of the three-path factor graph, the simplified gain of the two-path factor graph is greater than that of the three-path factor graph; meanwhile, the error introduced by the two-path factor graph is larger. Fig. 5 shows the reduction loss and the actual loss of the reduction factor graph equalization under different error performance.
Fig. 6 shows the error performance of SFGE in the case of non-ideal amplitude-frequency characteristics as shown in fig. 1, and its performance compared with LMS linear equalization. Parameters in the SFGE algorithm are estimated by the adaptive algorithm presented above, and the number of non-zero elements in the estimated SFGE parameter g is 3(< g > -3). Although SFGE only adopts a 3-path model, the equalized performance of the SFGE is very close to the error code performance without ISI, and only differs by 0.25 dB. And SFGE performance is about 1dB better than that of the LMS linear equalizer, where the order of the filter in the LMS linear equalizer is 11.
Figure 7 shows the error performance of SFGE under the group delay channel condition as shown in figure 2, and its performance compared to LMS linear equalization. Parameters in the SFGE algorithm are estimated by the adaptive algorithm presented above, and the number of non-zero elements in the estimated SFGE parameter g is 5(< g > -5). SFGE adopts a 5-path model, and the equalized performance of the SFGE is close to the error code performance without ISI and has a difference of about 1.5 dB. And SFGE performance is about 0.25dB better than that of the LMS linear equalizer, where the order of the filter in the LMS linear equalizer is 11. Fig. 6 and 7 demonstrate the effectiveness of the simplified factor graph equalizer in complex multipath channels.
The embodiments of the present invention have been described in detail, but the description is only for the preferred embodiments of the present invention and should not be construed as limiting the scope of the present invention. All equivalent changes and modifications made within the scope of the present invention should be covered by the present patent.

Claims (3)

1. A design algorithm of a simplified factor graph equalizer under a complex multipath channel is characterized by comprising the following steps:
assuming that the impulse response of a multipath channel is h, the factor graph parameter of the iterative equalizer is g, and the number of nonzero elements in g is less than that of the nonzero elements in h; h and g satisfy the following relationship:
h＝g+f(1)
wherein the vector f is the remaining multipaths in h except multipath g;
at this time, the symbol y is receivedkCan be expressed as:
yk＝hsk+wn＝(g+f)sk+wn＝gsk+(fsk+wn)(2)
in the above formula skIs given askAssociated transmission symbol sequence, wnIs additive white Gaussian noise, wherein fsk+wnFor a factor graph equalizer, it can be considered as noise w'nI.e. by
w'n＝fsk+wn(3)
Its mean value m (w'n) And variance D (w'n) Comprises the following steps:
m(w'n)＝0(4)
Figure FDA0001973875770000011
wherein
Figure FDA0001973875770000012
For transmitting symbols skThe power of (c), here assumed to be 1,
Figure FDA0001973875770000013
for channel noise wnThe power of (d);
at this time, the equalization factor node function E in the equalizer factor graphkThe following steps are changed:
Figure FDA0001973875770000014
in the above formula EkIs in a known transmitted symbol sequence skAnd the factor graph parameter of the iterative equalizer is g, and the received symbol is ykThe probability of (d);
assuming that the transmitted symbols are uniformly distributed, in the first iteration process, the jth equalization function node eqjTo the kth symbol node skDelivered foreign information
Figure FDA0001973875770000015
Can be expressed as:
Figure FDA0001973875770000016
in the above formula: { skDenotes the division s associated with the equalization function nodekA sequence number sequence of outer;
and symbol skThe initial posterior probability information of (a) may be expressed as:
Figure FDA0001973875770000021
in the above formula, S is a transmission sequence including all symbols, N (S)k) Is a sum of skA set of all equalization nodes that are relevant;
the case that each equalization factor node in the factor graph has only 3 edges is explained, that is, the number of the nonzero elements in g is 3, which are respectively defined as: ga、gbAnd gc，ha、hbAnd hcCorresponding multipath delays a, b and c symbol periods respectively;
in the factor graph, the node s is connected with the symbolic variablekThe adjacent equalization factor node has Ek+a、Ek+bAnd Ek+c(ii) a These three factor nodes pass to skThe external information of (a) is:
Figure FDA0001973875770000022
Figure FDA0001973875770000023
Figure FDA0001973875770000024
then skA posteriori probability information LkComprises the following steps:
Figure FDA0001973875770000025
definition confidenceReliability of degree R(s)k)：
R(sk)＝sk×Lk(13)
The physical significance is as follows: when R(s)k)>Confidence L obtained at 0kCorrect, otherwise wrong; r(s)k) The greater the confidence LkThe more reliable is; r(s)k) The smaller, the confidence LkThe more unreliable; according to about equation:
Figure FDA0001973875770000026
in equations (9) to (11), the sum of each term is 4, that is, n is 4; then
Figure FDA0001973875770000031
The above formula is simplified to obtain:
Figure FDA0001973875770000032
at skTaking a fixed value, while the other symbols are uniformly distributed, yk+a，yk+bAnd yk+cThe expected values of (c) are: e (y)k+a)＝gask、E(yk+b)＝gbskAnd E (y)k+c)＝gcsk(ii) a The variance of confidence reliability can therefore be expressed as:
Figure FDA0001973875770000033
the value of the factor graph parameter g should maximize the lower bound of the expected value of the reliability of the posterior probability, that is:
Figure FDA0001973875770000034
order:
Figure FDA0001973875770000035
Figure FDA0001973875770000036
to pair
Figure FDA0001973875770000037
The derivation is carried out to obtain:
Figure FDA0001973875770000038
namely, it is
Figure FDA0001973875770000039
Is about
Figure FDA00019738757700000310
A monotonically increasing function of; in the same way
Figure FDA00019738757700000311
Also relates to
Figure FDA00019738757700000312
And
Figure FDA00019738757700000313
a monotonically increasing function of; namely, it is
Figure FDA00019738757700000314
And
Figure FDA00019738757700000315
when the maximum value is taken out of the range,
Figure FDA00019738757700000316
obtaining a maximum value;
due to ga、gbAnd gcIs an element in h, g, to maximize the lower bound of confidence reliabilitya、gbAnd gcTaking three elements with the maximum absolute value in h;
the simplified factor graph, which does not consider the performance influence of multipath with smaller amplitude, will cause a loss of error performance, which is called the simplified loss and is expressed by the difference (dB) of the required signal-to-noise ratio when the same bit error rate is achieved, and is marked as LSFG(ii) a Meanwhile, the simplified factor graph has fewer short loops than the original factor graph, so that the error code performance is improved, and the signal-to-noise ratio threshold is improved, wherein the improvement of the signal-to-noise ratio performance is called as simplified gain and is marked as GSFG(ii) a And the actual SNR threshold difference is expressed in Δ SNR, and Δ SNR ═ GSFG-LSFG；
Primitive factor graph parameter h, number of non-zero elements thereof<h>With a corresponding confidence level of
Figure FDA0001973875770000041
Reliability of R0(sk) (ii) a Reduced factor graph parameter g, number of non-zero elements thereof<g><<h>With a corresponding confidence level of
Figure FDA0001973875770000042
Reliability of Rs(sk) (ii) a According to ln (e)x+ey) Max (x), knowing
Figure FDA0001973875770000043
Figure FDA0001973875770000044
When the reliability of other variable nodes is large enough, the above two equations can be simplified as follows:
Figure FDA0001973875770000045
Figure FDA0001973875770000046
in the primitive factor graph, (y)k+i-hsk+i) Is a mean of 0 and a variance of
Figure FDA0001973875770000047
(ii) a gaussian distribution of; in the reduced factor graph (y)k+i-gs'k+i) Is a mean of 0 and a variance of
Figure FDA0001973875770000048
Is a gaussian distribution of
Figure FDA0001973875770000049
And
Figure FDA00019738757700000410
the mean values of (a) are respectively:
Figure FDA00019738757700000411
Figure FDA00019738757700000412
their variances are:
Figure FDA0001973875770000051
Figure FDA0001973875770000052
namely, it is
Figure FDA0001973875770000053
And
Figure FDA0001973875770000054
are all gaussian distributed as (m,2| m |); the performance loss of the reduced factor graph can be analyzed by comparing the mean of the two; when the two reach the same error code performance, namely the posterior probability reaches a certain same distribution, the SNR corresponding to the iterative equalization of the original factor graph is SNR0The channel noise is
Figure FDA0001973875770000055
The reduced factor graph iteratively equalizes the corresponding channel noise as
Figure FDA0001973875770000056
In this case, the following requirements are:
Figure FDA0001973875770000057
namely, it is
Figure FDA0001973875770000058
Corresponding loss of signal-to-noise ratio (L)SFG) Is composed of
Figure FDA0001973875770000059
In practical systems, when the performance loss cannot be greater than χ dB, LSFGShould satisfy
Figure FDA00019738757700000510
I.e. g does not calculation2Should satisfy the minimum value of
Figure FDA00019738757700000511
When | | h | | non-conducting phosphor2When the normalization is 1, min { | | | g | | non-phosphor2The value of is
Figure FDA00019738757700000512
2. The reduced factor graph equalizer design algorithm under a complex multipath channel as claimed in claim 1, wherein: according to the change situation of the minimum value of the simplified factor graph parameters along with the original signal-to-noise ratio under the loss of different signal-to-noise ratios, when the original signal-to-noise ratio is higher, the multipath energy corresponding to the parameters of the simplified factor graph is closer to the multipath response of a channel; the smaller the loss of the signal-to-noise ratio is required to be, the closer the multi-path energy corresponding to the parameters of the simplified factor graph is to the multi-path energy of the channel; the parameters of the reduced factor graph should select the multipath with larger amplitude from the multipath of the channel, and the energy of the selected multipath should be as close to the energy of the multipath of the channel as possible.
3. The design algorithm of the simplified factor graph equalizer under the complex multipath channel according to claim 2, characterized in that, according to the signal-to-noise ratio threshold of 0-15 dB in the actual satellite channel, under the condition of the minimum loss, the multipath energy corresponding to the parameters of the simplified factor graph is set to be larger than 95% of the multipath energy of the channel, and the parameter adaptive estimation algorithm of the simplified factor graph equalizer is characterized in that the process is as follows:
1) estimating the response h of the multipath channel by utilizing an LMS algorithm;
2) sorting the multipath amplitudes in h from large to small;
3) selecting the multipath with the maximum multipath amplitude from the h as a main path;
4) selecting the multipaths with larger amplitude from h from large to small and adding the multipaths into the simplified factor graph parameter g;
5) detecting whether the energy of the simplified factor graph parameter g meets the condition that the energy of h is more than 95%; if yes, stopping selecting multipath from h, otherwise, continuing.
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