CN107612558A - A kind of big girth QC LDPC code building methods based on Fibonacci Lucas sequence - Google Patents
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Abstract
The present invention relates to a kind of big girth QC LDPC code building methods for being based on Fibonacci Lucas (Fibonacci Lucas) sequence, this method can cause information cross-correlation after an iteration during decoding for the presence of becate, so as to largely influence the problem of it decodes constringency performance, then Fibonacci Lucas sequences and triangle rotary structured approach is made full use of its check matrix is avoided the generation of four or six rings.Its procedure is：A Fibonacci Lucas sequence is first proposed, then in conjunction with triangle rotary method structural index matrix, exponential matrix respective value is replaced with unit matrix, cyclic permutation matrices afterwards, obtains final check matrix.By simulating, verifying using QC LDPC codes type that the building method is constructed without four or six rings and its error-correcting performance it is outstanding, computation complexity is low, and hardware is realized simple, is easy to practical application.
Description
Technical field
The invention belongs to field of signal processing, is related to channel coding, especially a kind of to be based on fibonacci-lucas sequence
Big girth QC-LDPC code constructing methods.
Background technology
With the development in epoch, people pursue the telecommunication transmission system of better reliability always, while require therein
Equipment is inexpensively quick.Low-density checksum (low-density parity-check, LDPC) code is that a kind of performance connects very much
The good code of nearly shannon limit, can provide high coding gain, can be used to largely to reduce the transmit power of wireless device simultaneously
Reduce antenna size, research LDPC code has huge practical significance and economic value.Human society no matter live by routine work
Or politics, economic, military, scientific and technological activities all be unable to do without the transmitting of information.Modern digital communication systems nearly all use
Channel error correction coding techniques, can provide certain coding gain.LDPC code can be divided into random LDPC code and structured LDPC
Code.More promising a kind of code is quasi-cyclic LDPC (quasi-cyclic LDPC, QC- in structured LDPC code
LDPC) code.Because its check matrix has special quasi- recursive nature, realized for the hardware of its codec module, only
Shift register need to be used, implements and is easier to, encoding and decoding complexity is low.In binary system awgn channel, if compared with
Good building method, can be in reliability close to the limit using the method that short code is long or middle code length QC-LDPC codes are encoded.Cause
QC-LDPC codes turn into study hotspot in coding field in recent years for this.Deposited in optic communication, deep space communication, mobile communication, information
Substantial amounts of application has been obtained in the fields such as storage, satellite digital video broadcast, radio communication, China Mobile Multimedia Broadcasting.
The periodic structure of LDPC code Tanner figures has a significant impact to error-correcting performance, and minimum ring length is referred to as in its Tanner figure
Girth, the minimum range of QC-LDPC codes increase with the increase of girth, so with the increase of girth, its error-correcting performance
Lifted therewith.In addition, when using SPA decodings, the too small error-correcting performance that can influence QC-LDPC codes of girth.As girth be 4 when,
Assuming that the message of transmission is incorrect, then the message returns to origin by iteration twice, and the external information of acquisition is seldom, so as to disappear
Breath wrong can not can only be transmitted by error correction, decline error-correcting performance.So for the building method of QC-LDPC codes, at least require
Fourth Ring is avoided that, makes girth be 8 or 10 as far as possible on the basis of this, i.e., small girth can make it that information is after an iteration during decoding
Cross-correlation, decoding constringency performance is had influence on, in order to overcome this shortcoming, the present invention utilizes Fibonacci-Lucas sequences
Mathematical thought simultaneously constructs a QC-LDPC code that four or six rings are not present with reference to triangle rotary method, and its yard of error-correcting performance is excellent
It is elegant.
The content of the invention
In view of this, it is an object of the invention to provide one kind based on Fibonacci-Lucas sequences and to combine triangle rotation
The method that robin constructs big girth QC-LDPC codes, by the ingehious design to exponential matrix, so as to reach lifting error-correcting performance,
Reduce the purpose of computation complexity.
To reach above-mentioned purpose, the present invention provides following technical scheme：
A kind of big girth QC-LDPC code constructing methods based on Fibonacci-Lucas sequences, including：
1.Fibonacci sequences distribution situation be 0,1,1,2,3,5,8,13,21,34 ..., make Section 1 and Section 2
Respectively 0,1, as its primary condition, it is all first two from the remaining each single item after Section 3 and is added institute's total, for
Lucas sequences, its distribution situation be 2,1,3,4,7,11,18,29,47 ..., L (1)=2, L (2)=1, it is defined as L (n)
=L (n-1)+L (n-2) (n >=3, n ∈ N*).Fibonacci is can be regarded as each Fibonacci-Lucas sequence
A kind of popularization of ordered series of numbers and Lucas ordered series of numbers.
Fibonacci-Lucas sequence definitions：A kind of increasing sequence distribution situation be 1,3,4,7,11,18,29,
47th ... this sequence can be designated as to F [1,3] when Section 1 and Section 2 are set to 1,3 as primary condition, then, should
Fibonacci-Lucas recurrence Sequences are expressed as F (n)=F (n-1)+F (n-2) (n >=2, n ∈ N*)。
Fibonacci-Lucas sequences theorem 1：If m ＞ n, and m, n, k ∈ N*, then f (m+k)-f (m) ＞ f (n+k)-f
(n)。
2. constructing an exponential matrix E (H), E (H) dimension is J × L.F (0) has taken exponential matrix first for 1
OK, the matrix form that all rows are formed thereafter is matrix resulting after rotating.Then just like (1) formula shown in Uniformly bounded
Battle array：
It can be seen that element is positive integer in exponential matrix E (H) from (1) formula；In addition to the first row element is all F (0), its
Remaining often row is all incremented by ordered series of numbers.Make 0≤i≤J-1, any one element can be expressed as (2) formula in 0≤s≤L-1, E (H)：
E (i, s)=F (2i+s+r)+i+s (2)
3. according to the exponential matrix E (H) that constructs, the square that is obtained after the unit matrix cyclic shift for being P × P with dimension
Battle array goes each element in replacement E (H), wherein, cyclic shift number is corresponding element in exponential matrix E (H), and spreading factor P is unit
Matrix dimension, expression formula need to be met：P >=F (2J+L-3+r)+J+L-1, then, for example following (3) the formula institute of construction of check matrix H
Show：
The beneficial effects of the present invention are：Based on Fibonacci-Lucas sequences and combine triangle rotary method construction QC-
LDPC code, be avoided that the generation of Fourth Ring and six rings, there is a preferable error-correcting performance, cyclic permutation submatrix dimension can continuous value,
Corresponding parameter is set to can obtain different code lengths and code check in addition.F-L-QC-LDPC codes are constructed with this building method, are emulated
As a result show that its error-correcting performance is better than the same code length based on Fibonacci sequence combination triangle rotaries method construction with code check F-QC-
LDPC code is similarly better than the same code length using Lucas Sequence Filling exponential matrix with code check L-QC-LDPC codes and based on equal difference
The APS-QC-LDPC codes of ordered series of numbers construction.In summary, one kind provided by the present invention based on Fibonacci-Lucas sequences
Big girth QC-LDPC code constructing methods are advantageous in space etc. needed for net coding gain, storage, can preferably meet to lead to
The requirement of letter system.
Brief description of the drawings
In order that the purpose of the present invention, technical scheme and beneficial effect are clearer, the present invention provides drawings described below and carried out
Explanation：
Fig. 1 is the Technology Roadmap of the inventive method；
Fig. 2 is six kinds of structure types of six rings
Fig. 3 is the form one of six rings
Fig. 4 is the performance comparision figure of F-L-QC-LDPC codes and other yards that the code check of construction is 0.5；
Embodiment
Below in conjunction with accompanying drawing, the preferred embodiments of the present invention are described in detail.
1. illustrate that Fibonacci-Lucas sequence definitions and theorem, Fibonacci sequence distribution situations are with reference to accompanying drawing 1
0th, 1,1,2,3,5,8,13,21,34 ..., it is respectively 0,1 to make Section 1 and Section 2, as its primary condition, from Section 3
Remaining each single item afterwards is all first two and is added institute total, for Lucas sequences, its distribution situation is 2,1,3,4,7,11,
18th, 29,47 ..., L (1)=2, L (2)=1, it is defined as L (n)=L (n-1)+L (n-2) (n >=3, n ∈ N*).For every
One Fibonacci-Lucas sequence can be regarded as a kind of popularization of Fibonacci ordered series of numbers and Lucas ordered series of numbers.
Fibonacci-Lucas sequence definitions：A kind of increasing sequence distribution situation be 1,3,4,7,11,18,29,
47th ... this sequence can be designated as to F [1,3] when Section 1 and Section 2 are set to 1,3 as primary condition, then, should
Fibonacci-Lucas recurrence Sequences are expressed as F (n)=F (n-1)+F (n-2) (n >=2, n ∈ N*)。
Fibonacci-Lucas sequences theorem 1：If m ＞ n, and m, n, k ∈ N*, then f (m+k)-f (m) ＞ f (n+k)-f
(n)。
2. illustrating QC-LDPC code check matrix building methods with reference to accompanying drawing 1, the first step is fibonacci-lucas sequence
Placed according to following triangular structure, wherein line number i=1,2,3 ..., parameter r value affects the change of code length, is terrible
To long code length, then r takes higher value, on the contrary obtain short code length then r take smaller value, wherein r ∈ Z+。
I=1 F (1+1+r)+1
The F (2+1+r)+2 of i=2 F (2+2+r)+2
The F (3+1+r)+3 of+3 F (3+2+r) of i=3 F (3+3+r)+3
i F(2i+r)+i F(2i-1+r)+i...F(i+1+r)+i
45 degree of second step rotate counterclockwise, structure is as follows：
The F (3+1+r)+3 of+1 F (2+1+r) of i=1 F (1+1+r)+2
The F (4+2+r)+4 of+2 F (3+2+r) of i=2 F (2+2+r)+3
The F (5+3+r)+5 of+3 F (4+3+r) of i=3 F (3+3+r)+4
The F (6+4+r)+6 of+4 F (5+4+r) of i=4 F (4+4+r)+5
An exponential matrix E (H) is constructed, E (H) dimension is J × L.F (0) has taken exponential matrix the first row for 1,
Thereafter the matrix form that all rows are formed is matrix resulting after rotating.Then just like (1) formula shown in exponential matrix：
It can be seen that element is positive integer in exponential matrix E (H) from (1) formula；In addition to the first row element is all F (0), its
Remaining often row is all incremented by ordered series of numbers.Make 0≤i≤J-1, any one element can be expressed as (2) formula in 0≤s≤L-1, E (H)：
E (i, s)=F (2i+s+r)+i+s (2)
According to the exponential matrix E (H) constructed, the matrix that is obtained after the unit matrix cyclic shift for being P × P with dimension
Each element in replacement E (H) is gone, wherein, cyclic shift number is corresponding element in exponential matrix E (H), and spreading factor P is unit square
Battle array dimension, need to meet expression formula：P >=F (2J+L-3+r)+J+L-1, then, the construction of check matrix H is as shown in following (3) formula：
3. illustrating girth property with reference to accompanying drawing 1, first, y (i, s) is defined as the i-th row in exponential matrix, the element of s row,
Wherein 0≤i≤J-1,0≤s≤L-1 and i, s ∈ N.2K rings present in QC-LDPC codes can be expressed as y (i with expression formula0,
s0), y (i0,s1), y (i1,s1) ..., y (ik-1,sk-1), y (ik-1,s0), y (i0,s0).In order that Fibonacci- must be based on
Lucas sequences simultaneously combine triangle rotary method construction QC-LDPC codes without 2K rings, then (4) formula should be caused to set up, wherein iv≠iv+1, sv
≠sv+1：
Illustrated respectively in terms of two, six rings of no Fourth Ring and nothing.
Proved in the absence of Fourth Ring：Understand that to obtain no Fourth Ring causes (5) formula to set up according to (4) formula：
y(i0,s0)-y(i1,s0)+y(i1,s1)-y(i0,s1) ≠ 0modp (5) makes i0＜ i1, s0＜ s1, and can by (2) formula
：
y(i0,s0)-y(i1,s0)+y(i1,s1)-y(i0,s1)
=F (2i0+s0+r)+i0+s0-(F(2i1+s0+r)+i1+s0)
+F(2i1+s1+r)+i1+s1-(F(2i0+s1+r)+i0+s1)
It can be derived from again by theorem 1：
F(2i1+s1+r)+i1+s1-(F(2i1+s0+r)+i1+s0) ＞
F(2i0+s1+r)+i0+s1-(F(2i0+s0+r)+i0+s0)
0 ＜ y (i0,s0)-y(i1,s0)+y(i1,s1)-y(i0,s1) ＜ p
Presence of the check matrix without Fourth Ring for understanding construction is derived by above formula.
Proved without six rings：In Tanner figures, form existing for six rings has six kinds as shown in Figure 2.
Six kind of six ring structure in Fig. 1 can be divided into two classes, the first kind is preceding four kinds of forms shown in Fig. 1, if having pushed away to obtain it
A kind of middle form, three kinds can then be derived and be obtained by its deformation in addition.Second class is latter two form shown in Fig. 1, can also be mutual
Deformation derives.Therefore one form of which need to only be derived per a kind of.
Understand to obtain needing (6) formula to set up without six rings according to (4) formula：
y(i,s)-y(i,t)+y(j,t)-y(j,g)+y(k,g)-y(k,s)≠0modp (6)
For Fig. 1 first kind, situation as shown in Figure 3 is likely to occur as i=0, will be proven below the six ring first kind in Fig. 3
In the first form, other situations similarly can be derived from.
By (2) formula and (6) Shi Ke get：
Y (i, s)-y (i, t)+y (j, t)-y (j, g)+y (k, g)-y (k, s)=
1-1+F(2j+t+r)+j+t-(F(2j+g+r)+j+g)+
F (2k+g+r)+k+g- (F (2k+s+r)+k+s)=
F(2j+t+r)+j+t-(F(2j+g+r)+j+g)+
F(2k+g+r)+k+g-(F(2k+s+r)+k+s)
S=g-m, t=g-n and m ＞ n are made, then above formula can turn to：
Y (i, s)-y (i, t)+y (j, t)-y (j, g)+y (k, g)-y (k, s)=
F(2j+g-n+r)+j+g-n-
(F(2j+g+r)+j+g)+F(2k+g+r)+k+g-
(F (2k+g-m+r)+k+g-m)=
F(2j+g-n+r)-n-F(2j+g+r)+
F(2k+g+r)-(F(2k+g-m+r)-m) (7)
It can be obtained by theorem 1 again：
F (2k+g+r)-(F (2k+g-m+r)-m) ＞
F(2j+g+r)-(F(2j+g-n+r)-n)
Then：0 ＜ (7) ＜ P are former formula ≠ 0modp.
When i ≠ 0：
Y (i, s)-y (i, t)+y (j, t)-(y (j, g)+y (k, g)-y (k, s)) ＞
Y (i, s)-y (i, t)+y (j, t)-(y (j, g)+y (j, g)-y (j, s))=
Y (i, s)-y (i, t)+y (j, t)-y (j, s) ＞ 0
Y (i, s)-y (i, t)+y (j, t)-(y (j, g)+y (k, g)-y (k, s)) ＜
Y (k, s)-y (i, t)+y (j, t)-(y (j, g)+y (k, g)-y (k, s))=
Y (j, t)-y (i, t)+y (k, g)-y (j, g) ＜
Y (j, g)+y (k, g)-y (j, g)=y (k, g) ＜ p
Folder forces definition of theorem：F (x) and G (x) are continuous and same limit A, i.e. x → X be present0When, limF (x)=
LimG (x)=A, if then function f (x) is in X0Certain field in, perseverance has F (x)≤f (x)≤G (x), then as x convergences X0, there is limF
(x)≤limf (x)≤limG (x) is A≤limf (x)≤A, therefore limf (X0)=A.
Theorem is forced to obtain according to folder：Former formula ≠ 0modp.
For first form in the classes of Fig. 1 second, work as i=0：
Y (i, s)-y (i, t)+y (k, t)-(y (k, g)+y (j, g)-y (j, s))=
1-1+F(2k+t+r)+k+t-(F(2k+g+r)+k+g)+
F (2j+g+r)+j+g- (F (2j+s+r)+j+s)=
F(2k+t+r)+k+t-(F(2k+g+r)+k+g)+
F(2j+g+r)+j+g-(F(2j+s+r)+j+s)
S=g-m, t=g-n and m ＞ n are made, then above formula can turn to：
Y (i, s)-y (i, t)+y (k, t)-(y (k, g)+y (j, g)-y (j, s))=
F(2k+g-n+r)-n-F(2k+g+r)+
F(2j+g+r)-(F(2j+g-m+r)-m) (8)
It can be derived from again by theorem 1：
F (2j+g+r)-(F (2j+g-m+r)-m) ＜
F(2k+g+r)-(F(2k+g-n+r)-n)
Then：- P ＜ (7) ＜ 0 is former formula ≠ 0modp.
When i ≠ 0：
Y (i, s)-y (i, t)+y (k, t)-(y (k, g)+y (j, g)-y (j, s)) ＜
Y (j, s)-y (j, t)+y (k, t)-(y (k, g)+y (j, g)-y (j, s))=
Y (j, g)-y (j, t)+y (k, t)-y (k, g) ＜ 0
Y (i, s)-y (i, t)+y (k, t)-(y (k, g)+y (j, g)-y (j, s)) ＞
Y (i, g)-y (j, t)+y (k, t)-y (k, g) ＞ y (k, t)-y (k, g) ＞-p
Theorem is forced to obtain according to folder：Former formula ≠ 0modp.
Analysis, which can obtain six kind of six loop type, more than to occur, so, as P >=F (2J+L-3+r)+J+L-1
When, no Fourth Ring and six rings, the property that girth is at least 8 are proved.
4. error-correcting performance is analyzed, a QC-LDPC is constructed based on fibonacci-lucas sequence combination triangle rotary method
Code, row weight is J=3, capable weight L=6, parameter r=2, and its exponential matrix is as follows：
P value is：P >=F (2J+L-3+r)+J+L-1=430, P=450 and P=430 is selected herein, then code length point
Wei 2700 and 2580.It is 6 that row weight, which can finally be respectively obtained, and row weight is 3, and code check is 0.5 F-LQC-LDPC (2700,1352)
Code and F-L-QC-LDPC (2580,1292) code.
Based on fibonacci-lucas sequence and the QC-LDPC codes of triangle rotary method construction are combined, are used for Matlab
Software is emulated, and is arranged on White Gaussian channel, while carries out binary phase shift keying modulation, and uses BP decoding algorithms,
Decoding iteration number is chosen for 50 times, is then again based on F-L-QC-LDPC codes analogous diagram with code check with same code length respectively
QC-LDPC codes that Fibonacci sequence combination triangle rotary methods construct, the ring of nothing four or six based on Lucas ordered series of numbers construction
QC-LDPC codes and based on arithmetic progression construction QC-LDPC codes be analyzed.Simulation result is as shown in Figure 4.
The error-correcting performance curve of pattern is constructed as shown in figure 4, being 10 in BER-6When, based on Fibonacci-Lucas sequences
And F-L-QC-LDPC (2700,1352) code-phases of triangle rotary method construction are combined compared with based on Fibonacci ordered series of numbers and with reference to three
The same code length of angle rotary process construction increases at least 8 with code check F-QC-LDPC (2700,1352) code, girth, net coding gain
About 1.0dB is improved, compared to the same code length constructed using Lucas Sequence Filling exponential matrix with code check L-QC-LDPC
(2700,1353) code, four or six rings are equally avoided, but Fibonacci-Lucas sequences are ingenious simultaneously combines triangle rotary method
So that other yard of bit, which is calculated information, provides bigger help, error-correcting performance is improved, net coding gain improves about
1.6dB, it is 10 equally in BER-6When, the F-L-QC- based on Fibonacci-Lucas sequences and combination triangle rotary method construction
LDPC (2580,1292) compares with the APS-QC-LDPC (2580,1292) based on arithmetic progression construction, and net coding gain improves
About 1.0dB.
Finally illustrate, preferred embodiment above is merely illustrative of the technical solution of the present invention and unrestricted, although logical
Cross above preferred embodiment the present invention is described in detail, it is to be understood by those skilled in the art that can be
Various changes are made to it in form and in details, without departing from claims of the present invention limited range.
Claims (3)
1. one kind is based on the quasi-circulating low-density parity check of Fibonacci-Lucas (Fibonacci-Lucas) sequence
(quasi-cyclic low-density parity-check, QC-LDPC) code constructing method, it is characterised in that：For becate
Presence can cause information cross-correlation after an iteration during decoding, decode constringency performance so as to can largely influence its
The problem of, a Fibonacci-Lucas sequence is designed first, then in conjunction with triangle rotary method structural index matrix, Zhi Houyong
Unit matrix, cyclic permutation matrices replace exponential matrix respective value, obtain final check matrix.
2. the QC-LDPC code constructing methods based on Fibonacci-Lucas sequences according to requiring right 1, it is characterised in that：
F (n)=F (n-1)+F (n-2) (n >=2, n ∈ N are defined using Fibonacci-Lucas recurrence Sequences*), and Fibonacci-
Lucas sequence theorems, f (m+k)-f (m) ＞ f (n+k)-f (n), wherein m ＞ n, m, n, k ∈ N*, constructing check matrix can have
The generation for avoiding four or six rings of effect.
3. according to a kind of QC-LDPC code constructing methods based on Fibonacci-Lucas sequences described in the requirement of right 1 or 2, it is special
Sign is：Based on Fibonacci-Lucas sequences, then in conjunction with triangle rotary method structural index matrix, through unit matrix, circulation
The check matrix that permutation matrix obtains after replacing eliminates becate, and its error-correcting performance is outstanding, and decoding convergence is very fast, needs storage element
Few, computation complexity is low, and hardware is realized simple.
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| CN108832936A (en) * | 2018-05-30 | 2018-11-16 | 东南大学 | A kind of building method and system of LDPC code |
| CN108768593A (en) * | 2018-06-12 | 2018-11-06 | 中国地质大学（武汉） | A kind of QC-LDPC code coding and decoding method and systems based on DMT modulation |
| CN108768593B (en) * | 2018-06-12 | 2020-07-07 | 中国地质大学（武汉） | QC-LDPC code encoding and decoding method and system based on DMT modulation |
| CN109088850A (en) * | 2018-06-22 | 2018-12-25 | 陕西师范大学 | Batch cloud auditing method based on Lucas sequence positioning wrong file |
| CN109088850B (en) * | 2018-06-22 | 2021-06-15 | 陕西师范大学 | Lot cloud auditing method for positioning error files based on Lucas sequence |
| CN109756232A (en) * | 2019-01-09 | 2019-05-14 | 重庆邮电大学 | A method of big girth rule QC-LDPC code is constructed based on Fibonacci-Lucas ordered series of numbers |
| CN109802689A (en) * | 2019-03-13 | 2019-05-24 | 重庆邮电大学 | A kind of QC-LDPC code constructing method that the girth based on Hoey sequence is 8 |
| CN111030705A (en) * | 2019-12-19 | 2020-04-17 | 重庆邮电大学 | QC-LDPC code construction scheme based on AP and ETS elimination |
| CN111211789A (en) * | 2020-01-08 | 2020-05-29 | 重庆邮电大学 | Quick-coding QC-LDPC code construction method with girth of 8 |
| CN111211789B (en) * | 2020-01-08 | 2023-08-04 | 重庆邮电大学 | Quick coding QC-LDPC code construction method with girth of 8 |
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