CN103166901A - 32 amplitude phase shift keying (APSK) modulation method and 32 APSK demodulation soft information calculating method - Google Patents

Abstract

The invention relates to a 32 amplitude phase shift keying (APSK) modulation method and a 32 APSK demodulation soft information calculating method and belongs to the technical field of high-order modulation and demodulation in a wireless digital transmission system. The 32 APSK modulation method includes mapping 5-bit information input by a modulator to an (I+jQ) base band complex signal through a mapping relationship for post-stage transmission. As to the 32 APSK modulation method, the provided low-complexity demodulation soft information calculating method can calculate soft information corresponding to the 5-bit b4 b3 b2 b1 b0 in received signs for rear-end coding. By means of the demodulation soft information calculating method, calculation complexity of the soft information can be obviously reduced, realizability of high-speed digital hardware is improved, and the simplified calculating method does not cause obvious loss to channel capacity and error code performances.

Description

A kind of 32APSK modulation and the soft information computational methods of demodulation thereof

Technical field

The invention belongs to the high order modulation demodulation techniques field in the wireless digital transmission system, the particularly calculating of the different soft information of bit in 32APSK modulation and corresponding demodulating process.

Background technology

At present, because high-order amplitude-phase shift keying (APSK) technology spectrum efficiency is high, can save bandwidth resources, and compare with quadrature amplitude modulation (QAM) technology, linearity to backend amplifier is lower, thereby very wide application prospect is arranged in bandwidth-limited systems.In the standard of " variable higher level code and modulator approach in the HRT high-rate telemetry application " (the CCSDS 131.2-R-1) of second generation satellite digital broadcasting (DVB-S2) standard in Europe and consultative committee for space data system (CCSDS), all comprised a kind of 32APSK modulation scheme.

The soft information computational methods of the 32APSK demodulation of second generation satellite digital broadcasting (DVB-S2) standard and consultative committee for space data system (CCSDS) comprise the steps.

1) 5 bit information b of modulator input ₄b ₃b ₂b ₁b ₀As shown in table 1 with the mapping relations of (I+jQ) baseband complex signal, R wherein ₁, ρ ₁R ₁, ρ ₂R ₁Be respectively the radius of ring, middle ring and outer shroud in the 32APSK constellation point; ρ ₁, ρ ₂In being respectively, the ratio of the radius of outer shroud and interior ring, in table, sequence number illustrates 32 kinds of different mapping relations.

Table 1DVB ?the 32APSK mapping relations that propose of S2 and CCSDS

Mapping relations described according to table 1 obtain 32APSK constellation point diagram shown in Figure 1, wherein R ₁, ρ ₁R ₁, ρ ₂R ₁The radius that is respectively ring in the 32APSK constellation point, middle ring and encircles most;

2) utilize this planisphere to carry out demodulation and obtain soft information: with the symbol y=y that receives _re+ jy _imSubstitution universal calculation equation (1) calculates five soft information (LLR (b that obtain to be used for rear end decoding _n)):

$\mathrm{LLR}\left(b_n\right)=\ln \left(\frac{\underset{\mathrm{\γ}\∈{S^n}_1}{\mathrm{\Σ}}{e}^{-\frac{1}{{\mathrm{\σ}}^2}({(y_{\mathrm{re}}-{\mathrm{\γ}}_{\mathrm{re}})}^2+{(y_{\mathrm{im}}-{\mathrm{\γ}}_{\mathrm{im}})}^2)}}{\underset{\mathrm{\γ}\∈{S^n}_0}{\mathrm{\Σ}}{e}^{-\frac{1}{{\mathrm{\σ}}^2}({(y_{\mathrm{re}}-{\mathrm{\γ}}_{\mathrm{re}})}^2+{(y_{\mathrm{im}}-{\mathrm{\γ}}_{i,})}^2)}}\right)---\left(1\right)$

Y=y wherein _re+ jy _imBe the symbol that receives, S ⁿ ₀, S ⁿ ₁Be respectively b _nThe position hard decision is the constellation point set of " 0 ", " 1 ", γ=γ _re+ j γ _imBe S ⁿ ₀Or S ⁿ ₁In point, σ ²Variance for additive white Gaussian noise channel.

According to 5 bit b of each constellation point ₄b ₃b ₂b ₁b ₀The value that should be judged is divided into two decision region with planisphere, and as shown in Figure 2, dark expression bit hard decision is the zone of " 0 ", and light color expression bit hard decision is the zone of " 1 ".As shown in Figure 2, decision region is asymmetric about the complex plane reference axis, can not be simply positive and negative by numerical value or polar coordinate transform judge the zone at the symbol place that receives.Therefore, said method is carrying out soft information when calculating, γ in formula _re, γ _imValue is complicated, and soft information computing formula comprises a large amount of exponent arithmetics, square operation and division arithmetic, is difficult to reduce computation complexity, is not suitable for hardware and realizes high-speed digital transmission.

Summary of the invention

The objective of the invention is that the soft Information-Based Computing Technology computing of existing 32APSK demodulation is complicated, amount of calculation is large in order to overcome, be not suitable for hardware and realize the difficult problem of high-speed digital transmission, a kind of novel 32APSK modulation and the soft information computational methods of demodulation thereof are proposed, the soft information calculation expression that obtains only comprises simple judgement, addition and subtraction, constant and variable multiplying, the soft information complexity of calculating is greatly reduced, be conducive to demodulator hardware and realize high-speed digital transmission.

A kind of 32APSK modulation and the soft information computational methods of demodulation thereof that the present invention proposes is characterized in that, comprise the following steps:

1) the 5 bit information b that modulator inputted ₄b ₃b ₂b ₁b ₀According to be mapped to (I+jQ) baseband complex signal as table 2 mapping relations, wherein, R ₁, ρ ₁R ₁, ρ ₂R ₁Be respectively the radius of ring, middle ring and outer shroud in the 32APSK constellation point, ρ ₁, ρ ₂In being respectively, the ratio of the radius of outer shroud and interior ring, in table, sequence number illustrates 32 kinds of different mapping relations;

Table 2

Under given code efficiency condition, turn to optimization aim with the channel capacity maximum, to parameter ρ ₁, ρ ₂, R ₁Carry out simulation optimization, obtain ρ ₁, ρ ₂, R ₁The simulation optimization result is as shown in table 3;

Table 3

2) at receiving terminal, the symbol that demodulator receives is y=y _re+ jy _im, σ ²Be the awgn channel noise variance, the symbol that receives is carried out the soft information of demodulation output, the simplification computing formula of soft information is as follows:

b ₄The expression formula of position (highest order) soft information is:

b ₃The expression formula of the soft information in position is:

b ₂The expression formula of the soft information in position is:

$\mathrm{LLR}\left(b_2\right)\&ap;\left\{\begin{array}{c}-\frac{2}{{\mathrm{\&sigma;}}^2}(\frac{2\sqrt{{y^2}_{\mathrm{re}}+{y^2}_{\mathrm{im}}}}{R_1+{\mathrm{\&rho;}}_1{R}_1}-1);(\sqrt{{y^2}_{\mathrm{<figure-callout\; id="3"\; label="r"\; filenames="HDA00002903821700061.png"\; state="\{\{state\}\}">r</figure-callout>}3}+{y^2}_{\mathrm{im}}}\&le;\frac{R_1+{\mathrm{\&rho;}}_1{R}_1}{2})\\ -\frac{2}{{\mathrm{\&sigma;}}^2}(\frac{\mathrm{\&pi;}}{3}-|\arctan \frac{y_{\mathrm{im}}}{y_{\mathrm{re}}}\left|\right);(\frac{R_1+{\mathrm{\&rho;}}_1{R}_1}{2}<\sqrt{{y^2}_{\mathrm{re}}+{y^2}_{\mathrm{im}}}\&le;\frac{{\mathrm{\&rho;}}_1{R}_1+{\mathrm{\&rho;}}_2{R}_1}{2})\\ -\frac{2}{{\mathrm{\&sigma;}}^2}(\frac{\mathrm{\&pi;}}{4}-|\arctan \frac{y_{\mathrm{im}}}{y_{\mathrm{re}}}\left|\right);(\frac{{\mathrm{\&rho;}}_1{R}_1+{\mathrm{\&rho;}}_2{R}_1}{2}<\sqrt{{{\mathrm{<figure-callout\; id="2"\; label="y"\; filenames="HDA00002903821700061.png,HDA00002903821700081.png"\; state="\{\{state\}\}">y</figure-callout>}}^2}_{\mathrm{re}}+{y^2}_{\mathrm{im}}})\end{array}\right.$

b ₁The expression formula of the soft information in position is: $\mathrm{LLR}\left(b_1\right)\&ap;-\frac{2}{{\mathrm{\&sigma;}}^2}(1-\frac{2\sqrt{{{\mathrm{<figure-callout\; id="2"\; label="y"\; filenames="HDA00002903821700061.png,HDA00002903821700081.png"\; state="\{\{state\}\}">y</figure-callout>}}^2}_{\mathrm{re}}+{{\mathrm{<figure-callout\; id="2"\; label="y"\; filenames="HDA00002903821700061.png,HDA00002903821700081.png"\; state="\{\{state\}\}">y</figure-callout>}}^2}_{\mathrm{im}}}}{({\mathrm{\&rho;}}_1+{\mathrm{\&rho;}}_2)R_1});$

b ₀The expression formula of position (lowest order) soft information is:

Characteristics of the present invention and beneficial effect:

32APSK modulation and the soft information computational methods of demodulation thereof that the present invention proposes, its bit decision zone is symmetrical about reference axis, make soft information computing formula can obtain approximate simplification the efficiently, and proposed piecemeal for irregular decision region wherein and simplified expression formula, the soft information computational methods of demodulation with the present invention's proposition, do not comprise the computings such as index, variable division in soft information calculation expression, computation complexity significantly reduces, improved high-speed digital transmission hardware realizability, and by simulating, verifying this algorithm do not cause the significantly sacrificing of performance.

Description of drawings

Fig. 1 is the 32APSK constellation point diagram in the DVB-S2 standard.

Fig. 2 is the decision region of five bits of 32APSK constellation point diagram in the DVB-S2 standard.

Fig. 3 is the 32APSK constellation point diagram that the present invention generates.

Fig. 4 is the decision region of five bits of 32APSK constellation point diagram of generating of the present invention.

Fig. 5 is 32APSK and the soft information BER Simulation of 32APSK of the present invention comparison diagram in the DVB-S2 standard.

Fig. 6 is 32APSK and 32APSK channel capacity comparison diagram of the present invention in the DVB-S2 standard.

Fig. 7 is a typical digital communication system structured flowchart.

Fig. 8 is the 32APSK modulator approach implementing procedure figure that the present invention proposes.

Fig. 9 is the soft information computational methods of the 32APSK demodulation implementing procedure figure that the present invention proposes.

Embodiment

A kind of 32APSK modulation and the soft information computational methods of demodulation thereof that the present invention proposes is characterized in that, comprise the following steps:

1) the 5 bit b that modulator inputted ₄b ₃b ₂b ₁b ₀According to be mapped to (I+jQ) baseband complex signal, wherein R as table 2 mapping relations ₁, ρ ₁R ₁, ρ ₂R ₁Be respectively the radius of ring, middle ring and outer shroud in the 32APSK constellation point, ρ ₁, ρ ₂In being respectively, the ratio of the radius of outer shroud and interior ring, in table, sequence number illustrates 32 kinds of different mapping relations.

Table 2 modulator input-output mappings relation

Make i=1,2,3 represent respectively interior ring, middle ring and outer shroud, take energy normalized as constraints, limit

N wherein _iBe the quantity of i the upper constellation point of ring, M=32 is the constellation point sum, under the condition of given code efficiency, turns to optimization aim with the channel capacity maximum, to parameter ρ ₁, ρ ₂, R ₁Carry out simulation optimization, obtain constellation point diagram parameter ρ under different coding efficient ₁, ρ ₂, R ₁Optimal value; According to ρ ₁, ρ ₂, R ₁Optimal value and modulator input generate 32APSK constellation point diagram, the b that each constellation point is corresponding unique with the mapping relations of baseband complex signal ₄b ₃b ₂b ₁b ₀Totally 5 bit information, ρ ₁, ρ ₂, R ₁The simulation optimization result as shown in table 3.

ρ under table 3 different coding efficient ₁, ρ ₂, R ₁Optimum results

Obtain 32APSK constellation point diagram shown in Figure 3, wherein R according to described mapping relations ₁, ρ ₁R ₁, ρ ₂R ₁Be respectively

The interior ring of 32APSK constellation point, middle ring and the radius that encircles most;

According to 5 bit ρ of each constellation point ₁, ρ ₂, R ₁Each bit value that should be judged, respectively planisphere is divided into five kinds of different decision region: namely five kinds of different hard decisions are the zone of " bit value is 0 " and the hard decision zone for " bit value is 1 ", as shown in Figure 4, wherein dark expression bit hard decision is the zone of " 0 ", and light color expression bit hard decision is the zone of " 1 ".As shown in Figure 4, decision region is symmetrical about the complex plane reference axis, can be simply positive and negative by numerical value or polar coordinate transform judge the zone at the symbol place that receives.Due to S ⁿ ₀, S ⁿ ₁In constellation point symmetrical about reference axis, therefore γ in formula (1) _re, γ _imEquate or opposite number each other, by numerical simulation, this formula be similar to abbreviation, obtain simple soft information calculation expression:

2) at receiving terminal, it is y=y that demodulator receives symbol _re+ jy _im, σ ²Be the awgn channel noise variance, the symbol that receives is carried out the soft information of demodulation output, the simplification computing formula of soft information is as follows:

b ₄The expression formula of position (highest order) soft information is:

_bThe expression formula of 3 soft information is:

b ₂The expression formula of the soft information in position is:

$\mathrm{LLR}\left(b_2\right)\&ap;\left\{\begin{array}{c}-\frac{2}{{\mathrm{\&sigma;}}^2}(\frac{2\sqrt{{y^2}_{\mathrm{re}}+{y^2}_{\mathrm{im}}}}{R_1+{\mathrm{\&rho;}}_1{R}_1}-1);(\sqrt{{y^2}_{\mathrm{<figure-callout\; id="3"\; label="r"\; filenames="HDA00002903821700061.png"\; state="\{\{state\}\}">r</figure-callout>}3}+{y^2}_{\mathrm{im}}}\&le;\frac{R_1+{\mathrm{\&rho;}}_1{R}_1}{2})\\ -\frac{2}{{\mathrm{\&sigma;}}^2}(\frac{\mathrm{\&pi;}}{3}-|\arctan \frac{y_{\mathrm{im}}}{y_{\mathrm{re}}}\left|\right);(\frac{R_1+{\mathrm{\&rho;}}_1{R}_1}{2}<\sqrt{{y^2}_{\mathrm{re}}+{y^2}_{\mathrm{im}}}\&le;\frac{{\mathrm{\&rho;}}_1{R}_1+{\mathrm{\&rho;}}_2{R}_1}{2})\\ -\frac{2}{{\mathrm{\&sigma;}}^2}(\frac{\mathrm{\&pi;}}{4}-|\arctan \frac{y_{\mathrm{im}}}{y_{\mathrm{re}}}\left|\right);(\frac{{\mathrm{\&rho;}}_1{R}_1+{\mathrm{\&rho;}}_2{R}_1}{2}<\sqrt{{{\mathrm{<figure-callout\; id="2"\; label="y"\; filenames="HDA00002903821700061.png,HDA00002903821700081.png"\; state="\{\{state\}\}">y</figure-callout>}}^2}_{\mathrm{re}}+{y^2}_{\mathrm{im}}})\end{array}\right.$

b ₁The expression formula of the soft information in position is: $\mathrm{LLR}\left(b_1\right)\&ap;-\frac{2}{{\mathrm{\&sigma;}}^2}(1-\frac{2\sqrt{{{\mathrm{<figure-callout\; id="2"\; label="y"\; filenames="HDA00002903821700061.png,HDA00002903821700081.png"\; state="\{\{state\}\}">y</figure-callout>}}^2}_{\mathrm{re}}+{{\mathrm{<figure-callout\; id="2"\; label="y"\; filenames="HDA00002903821700061.png,HDA00002903821700081.png"\; state="\{\{state\}\}">y</figure-callout>}}^2}_{\mathrm{im}}}}{({\mathrm{\&rho;}}_1+{\mathrm{\&rho;}}_2)R_1});$

b ₀The expression formula of position (lowest order) soft information is:

For the 32APSK modulation-demo-demodulation method that proposes in DVB-S2 and this 32APSK modulation and the soft information computational methods of demodulation thereof, introduce simultaneously error correcting code coding and decoding simulated environment, carry out emulation testing, obtain respectively the error-correcting performance of soft information in two schemes, result as shown in Figure 5.As seen from Figure 5, two ber curves overlap substantially, and 32APSK modulation and the soft information computational methods of demodulation thereof that the present invention proposes are described, do not cause the significantly sacrificing of error performance.

Adopt awgn channel calculation of capacity formula, the channel capacity of the 32APSK constellation point diagram that in contrast DVB-S2 standard, 32APSK and this method generate is carried out emulation, and its channel capacity is with bit signal to noise ratio (E _b/ N ₀) change as shown in Figure 6.Both curves overlap as seen from Figure 6, illustrate that channel capacity does not have obvious decline.

The present invention is further described below in conjunction with accompanying drawing and embodiment, and the present invention is applied in common 32APSK modulation /demodulation digital communication system, and the below describes as an example of a simple digital communication system example.The structured flowchart of this communication system as shown in Figure 7, information source produces bit stream input error correcting encoder, every 5 one group of the bit stream of encoder output is sent into this 32APSK modulator, the mapping method of this 32APSK modulator by describing in table 2, produce the symbol of (I+jQ) complex space, be modulated to radio frequency band by radio-frequency modulator.

Radiofrequency signal can be introduced noise through after awgn channel, and the symbol that radio-frequency (RF) demodulator demodulates is to comprise noisy symbol.The 32APSK demodulator can judge the cut zone under this symbol, and computing formula is simplified in substitution accordingly, calculates b ₀b ₁b ₂b ₃b ₄Corresponding soft information, send into decoder and carry out decoding respectively, and decode results is delivered to the stay of two nights, completes communication process.

For 32APSK baseband modulation module in Fig. 7, detailed modulation implementing procedure as shown in Figure 8.For the soft information computing module of demodulation in Fig. 7, detailed implementing procedure as shown in Figure 9.

The bit information that the below produces with an actual information source illustrates that as embodiment method of the present invention is as follows:

1) suppose that encoder encodes efficient is 4/5,5 bits of modulator input are 10101;

2) according to the optimum results ρ of parameter ₁=2.72, ρ ₂=4.87, R ₁=0.260, and the mapping relations of table 2, the output symbol that obtains modulator is y=-0.184+j0.184.

3) if modulator continues the new bit of input, repeat so 2) in process calculate.

An actual symbol of inputting take a demodulator is as example explanation demodulating process:

The noisy baseband signalling that contains of supposing the demodulator input is y=-0.45+j0.7794, y _re=-0.45, y _im=0.7794.

Utilize expression formula that the present invention provides to calculate the soft information of each bit in 5 bits by above-mentioned reception information:

$\mathrm{LLR}\left(b_4\right)\&ap;-\frac{2}{{\mathrm{\&sigma;}}^2}{y}_{\mathrm{re}}=\frac{0.9}{{\mathrm{\&sigma;}}^2}$

$\mathrm{LLR}\left(b_3\right)\&ap;-\frac{2}{{\mathrm{\&sigma;}}^2}{y}_{\mathrm{im}}=-\frac{1.5588}{{\mathrm{\&sigma;}}^2}$

Because $\sqrt{{{\mathrm{<figure-callout\; id="2"\; label="y"\; filenames="HDA00002903821700061.png,HDA00002903821700081.png"\; state="\{\{state\}\}">y</figure-callout>}}^2}_{\mathrm{re}}+{{\mathrm{<figure-callout\; id="2"\; label="y"\; filenames="HDA00002903821700061.png,HDA00002903821700081.png"\; state="\{\{state\}\}">y</figure-callout>}}^2}_{\mathrm{im}}}=0.900$ Satisfy $(\frac{R_1+{\mathrm{\&rho;}}_1{R}_1}{2}<\sqrt{{{\mathrm{<figure-callout\; id="2"\; label="y"\; filenames="HDA00002903821700061.png,HDA00002903821700081.png"\; state="\{\{state\}\}">y</figure-callout>}}^2}_{\mathrm{re}}+{{\mathrm{<figure-callout\; id="2"\; label="y"\; filenames="HDA00002903821700061.png,HDA00002903821700081.png"\; state="\{\{state\}\}">y</figure-callout>}}^2}_{\mathrm{im}}}\&le;\frac{{\mathrm{\&rho;}}_1{R}_1+{\mathrm{\&rho;}}_2{R}_1}{2}),$ Therefore

$\mathrm{LLR}\left(b_2\right)\&ap;-\frac{2}{{\mathrm{\&sigma;}}^2}(\frac{\mathrm{\&pi;}}{3}-|\arctan \frac{y_{\mathrm{im}}}{y_{\mathrm{re}}}\left|\right)=\frac{\mathrm{\&pi;}}{{3\mathrm{\&sigma;}}^2}$

$\mathrm{LLR}\left(b_1\right)\&ap;-\frac{2}{{\mathrm{\&sigma;}}^2}(\frac{{\mathrm{\&rho;}}_1{R}_1+{\mathrm{\&rho;}}_2{R}_1}{2}-\sqrt{{{\mathrm{<figure-callout\; id="2"\; label="y"\; filenames="HDA00002903821700061.png,HDA00002903821700081.png"\; state="\{\{state\}\}">y</figure-callout>}}^2}_{\mathrm{re}}+{{\mathrm{<figure-callout\; id="2"\; label="y"\; filenames="HDA00002903821700061.png,HDA00002903821700081.png"\; state="\{\{state\}\}">y</figure-callout>}}^2}_{\mathrm{im}}})=-\frac{0.1734}{{\mathrm{\&sigma;}}^2}$

Because $\sqrt{{{\mathrm{<figure-callout\; id="2"\; label="y"\; filenames="HDA00002903821700061.png,HDA00002903821700081.png"\; state="\{\{state\}\}">y</figure-callout>}}^2}_{\mathrm{re}}+{{\mathrm{<figure-callout\; id="2"\; label="y"\; filenames="HDA00002903821700061.png,HDA00002903821700081.png"\; state="\{\{state\}\}">y</figure-callout>}}^2}_{\mathrm{im}}}=0.900$ Satisfy $(\frac{R_1+{\mathrm{\&rho;}}_1{R}_1}{2}<\sqrt{{{\mathrm{<figure-callout\; id="2"\; label="y"\; filenames="HDA00002903821700061.png,HDA00002903821700081.png"\; state="\{\{state\}\}">y</figure-callout>}}^2}_{\mathrm{re}}+{{\mathrm{<figure-callout\; id="2"\; label="y"\; filenames="HDA00002903821700061.png,HDA00002903821700081.png"\; state="\{\{state\}\}">y</figure-callout>}}^2}_{\mathrm{im}}}\&le;\frac{{\mathrm{\&rho;}}_1{R}_1+{\mathrm{\&rho;}}_2{R}_1}{2}),$ Therefore

$\mathrm{LLR}\left(b_0\right)\&ap;-\frac{2}{{\mathrm{\&sigma;}}^2}\left(\right|\arctan \frac{y_{\mathrm{im}}}{y_{\mathrm{re}}}|-\frac{\mathrm{\&pi;}}{6})=-\frac{1}{{\mathrm{\&sigma;}}^2}.$

Can calculate thus corresponding 5 bit b ₀b ₁b ₂b ₃b ₄The soft information of each, the decoder that soft information is used for the rear end carries out the decoding judgement, completes the function of modulation /demodulation.And the computation complexity of soft information reduces greatly.If demodulator receives new symbol, repeat above-mentioned steps.

Claims (1)

1. a 32APSK modulates and the soft information computational methods of demodulation, it is characterized in that, comprises the following steps:

1) the 5 bit information b that modulator inputted ₄b ₃b ₂b ₁b ₀According to be mapped to (I+jQ) baseband complex signal as table 2 mapping relations, wherein, R ₁, ρ ₁R ₁, ρ ₂R ₁Be respectively the radius of ring, middle ring and outer shroud in the 32APSK constellation point, ρ ₁, ρ ₂In being respectively, the ratio of the radius of outer shroud and interior ring, in table, sequence number illustrates 32 kinds of different mapping relations;

Table 2

Under given code efficiency condition, turn to optimization aim with the channel capacity maximum, to parameter ρ ₁, ρ ₂, R ₁Carry out simulation optimization, obtain ρ ₁, ρ ₂, R ₁The simulation optimization result is as shown in table 3;

Table 3

2) at receiving terminal, the symbol that demodulator receives is y=y _re+ jy _im, σ ²Be the awgn channel noise variance, the symbol that receives is carried out the soft information of demodulation output, the simplification computing formula of soft information is as follows:

b ₄The expression formula of position (highest order) soft information is:

b ₃The expression formula of the soft information in position is:

b ₂The expression formula of the soft information in position is:

b ₁The expression formula of the soft information in position is:

b ₀The expression formula of position (lowest order) soft information is:

。

Images