USRE46692E1 - Structured low-density parity-check (LDPC) code - Google Patents
Structured low-density parity-check (LDPC) code Download PDFInfo
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- H03M13/1102—Codes on graphs and decoding on graphs, e.g. low-density parity check [LDPC] codes
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- H03M13/1188—Parity check matrix structured for simplifying encoding, e.g. by having a triangular or an approximate triangular structure wherein the parity-check matrix comprises a part with a double-diagonal wherein in the part with the double-diagonal at least one column has an odd column weight equal or greater than three
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- H03M13/6362—Error control coding in combination with rate matching by puncturing
- H03M13/6368—Error control coding in combination with rate matching by puncturing using rate compatible puncturing or complementary puncturing
- H03M13/6393—Rate compatible low-density parity check [LDPC] codes
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- H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
- H04L1/00—Arrangements for detecting or preventing errors in the information received
- H04L1/004—Arrangements for detecting or preventing errors in the information received by using forward error control
- H04L1/0056—Systems characterized by the type of code used
- H04L1/0057—Block codes
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- H04L1/004—Arrangements for detecting or preventing errors in the information received by using forward error control
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Abstract
Description
1 | 1 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 0 |
1 | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 |
1 | 0 | 1 | 0 | 1 | 1 | 1 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 |
1 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 0 |
0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 |
0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1. |
6 | 38 | 3 | 93 | −1 | −1 | −1 | 30 | 70 | −1 | 86 | −1 | 37 | 38 | 4 | 11 | −1 | 46 | 48 | 0 | −1 | −1 | −I | −1 |
62 | 94 | 19 | 84 | −1 | 92 | 78 | −1 | 15 | −1 | −1 | 92 | −1 | 45 | 24 | 32 | 30 | −1 | −1 | 0 | 0 | −1 | −1 | −1 |
71 | −1 | 55 | −1 | 12 | 66 | 45 | 79 | −1 | 78 | −1 | −1 | 10 | −1 | 22 | 55 | 70 | 82 | −1 | −1 | 0 | 0 | −1 | −1 |
38 | 61 | −1 | 66 | 9 | 73 | 47 | 64 | −1 | 39 | 61 | 43 | −1 | −1 | −1 | −1 | 95 | 32 | 0 | −1 | −1 | 0 | 0 | −1 |
−1 | −1 | −1 | −1 | 32 | 52 | 55 | 80 | 95 | 22 | 6 | 51 | 24 | 90 | 44 | 20 | −1 | −1 | −1 | −1 | −1 | −1 | 0 | 0 |
−1 | 63 | 31 | 88 | 20 | −1 | −1 | −1 | 6 | 40 | 56 | 16 | 71 | 53 | −1 | −1 | 27 | 26 | 48 | −1 | −1 | −1 | −1 | 0 |
wherein −1 represents L×L all-zero square matrix, the integer sij represents circular shifted L×L identity matrix, the amount of the shift s′ij is determined as follows:
qnormalized=(Nshortened/Npunctured)/[R/(1−R)].
where d=[d0 . . . dk−1]T is the block of data bits and p=[p0 . . . pM−1]T are the parity bits. A codeword is any binary, or in general, non-binary, N-vector c that satisfies:
Hc=Hdd+Hpp=0
p=Hp −1Hdd [equation 1]
where in 0 is the index of the column in which row 0 contains a “1”
where in 1 is the index of the column in which row 1 contains a “1”
where in M−1 is the index of the column in which row M−1 contains a “1”.
11 | 0 | 10 | 6 | 3 | 5 | 1 | 0 | −1 | −1 | −1 | −1 |
10 | 9 | 2 | 2 | 3 | 0 | −1 | 0 | 0 | −1 | −1 | −1 |
7 | 9 | 11 | 10 | 4 | 7 | −1 | −1 | 0 | 0 | −1 | −1 |
9 | 2 | 4 | 6 | 5 | 3 | 0 | −1 | −1 | 0 | 0 | −1 |
3 | 11 | 2 | 3 | 2 | 11 | −1 | −1 | −1 | −1 | 0 | 0 |
2 | 7 | 1 | 0 | 10 | 7 | 1 | −1 | −1 | −1 | −1 | 0 |
-
- −1, represents L×L all-zero square matrix, 0L, L equals 12 in this example;
- 0, represents L×L identity matrix, IL.
- integer, r (<L), represents L×L identity matrix, IL, rotated to the right (for example) a number of times corresponding to the integer.
0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | ||
0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | ||
0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | ||
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | ||
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | ||
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | ||
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | ||
1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||
0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||
0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||
0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||
0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||
rL=(rLmax)modulo L
p0=h0,0d0+h0,1d1+h0,2d2+ . . . +h0,11d11
p1=h1,0d0+h1,1d1+h1,2d2+ . . . +h1,11d11
p2=h2,0d0+h2,1d1+h2,2d2+ . . . +h2,11d11
p3=p0+h3,0d0+h3,1d1+h3,2d2+ . . . +h3,11d11
p4=p1+h4,0d0+h4,1d1+h4,2d2+ . . . +h4,11d11
p5=p2+h5,0d0+h5,1d1+h5,2d2+ . . . +h5,11d11
p6=p3+h6,0d0+h6,1d1+h6,2d2+ . . . +h6,11d11
p7=p4+h7,0d0+h7,1d1+h7,2d2+ . . . +h7,11d11
p8=p5+h8,0d0+h8,1d1+h8,2d2+ . . . +h8,11d11
p9=p6+h9,0d0+h9,1d1+h9,2d2+ . . . +h9,11d11
p10=p7+h10,0d0+h10,1d1+h10,2d2+ . . . +h10,11d11
p11=p8+h11,0d0+h11,1d1+h11,2d2+ . . . +h11,11d11
h0,0d0h0,1d1+ . . . +h0,11d11+p0+p3=0 [Equation 2]
h1,0d0h1,1d1+ . . . +h1,11d11+p1+p4=0 [Equation 3]
h2,0d0h2,1d1+ . . . +h2,11d11+p2+p5=0 [Equation 4]
h3,0d0h3,1d1+ . . . +h3,1d11+p0+p3+p6=0 [Equation 5]
h4,0d0h4,1d1+ . . . +h4,11d11+p1+p4+p7=0 [Equation 6]
h5,0d0h5,1d1+ . . . +h5,11d11+p2+p5+p8=0 [Equation 7]
h6,0d0h6,1d1+ . . . +h6,11d11+p6+p9=0 [Equation 8]
h7,0d0h7,1d1+ . . . +h7,11d11+p7+p10=0 [Equation 9]
h8,0d0h8,1d1+ . . . +h8,11d1+p8+p11=0 [Equation 10]
h9,0d0h9,1d1+ . . . +h9,11d11+p0+p9=0 [Equation 11]
h10,0d0h10,1d1+ . . . +h10,11d11+p1+p10=0 [Equation 12]
h11,0d0h11,1d1+ . . . +h11,11d11+p2+p11=0 [Equation 13]
(h0,0+h3,0+h6,0+h9,0)d0+(h0,1+h3,1+h6,1+h9,0)d1 . . . +(h0,11+h3,11+h6,11+h9,11)d11p0+p3+p0+p3+p6+p6+p9+p0+p9=0
p0=(h0,0+h3,0+h6,0+h9,0)d0+(h0,1+h3,1+h6,1+h9,1)d1+. . . +(h0,11+h3,11+h6,11+h9,11)d11
p1=(h1,0+h4,0+h7,0+h10,0)d0+(h1,1+h4,1+h7,1+h10,1)d0+(h1,11+h4,11+h7,11+h10,11)d11
p2=(h2,0+h5,0+h8,0+h11,0)d0+(h2,1+h5,1+h87,1+h11,1)d1+. . . +(h2,11+h5,11+h8,11+h11,11)d11
p3=h0,0d0+h0,1d1+. . . +h0,11d11+p0
p4=h1,0d0+h1,1d1+. . . +h1,11d11+p1
p5=h2,0d0+h2,1d1+. . . +h2,11d11+p2
p6=h3,0d0+h3,1d1+. . . +h3,11d11+p0+p3
p7=h4,0d0+h4,1d1+. . . +h4,11d11+p1+p4
p8=h5,0d0+h5,1d1+. . . +h5,11d11+p2+p5
p9=h6,0d0+h6,1d1+. . . +h6,11d11+p6
p10=h7,0d0+h7,1d1+. . . +h7,11d11+p7
p11=h8,0d0+h8,1d1+. . . +h8,11d11+p8 [Equation 14]
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- a. Row combining: higher rate base parity check matrices are derived from an original lower rate base parity check matrix by combining rows of the base parity check matrix. Multiple strategies can be applied in order to make the resultant higher rate base matrix maintain the properties of the original matrix, i.e. the weight of each of the column in a block of rows is at most one. One way of doing row combining will be to combine (add together) rows that belong to the same block of rows. This guarantees the preservation of column weight properties, with decrease of the block row size. Another way of row combining will be to combine the rows that belong to different blocks of rows, where they don't have overlapping elements.
- b. Row splitting: lower rate base parity check matrices are derived from an original higher rate base parity check matrix by splitting rows of the base parity check matrix. The resultant lower rate parity check matrix shall maintain the properties of the original matrix, i.e. the weight of each of the column in a block of rows is at most one.
Hp,present _ invention(m)=T (Hp,existing,m),
-
- a. horizontal step, during which all row variables are updated at the same time based on the column variables; and
- b. vertical step, during which all column variables are updated at the same time based on row variables.
T=(K×F)/(C×I),
where K is number of info bits, F is clock frequency, C is number of cycles per iteration, and I is the number of iterations. Assuming that K, F, and I are fixed and, for example, equal: K=320 bits, F=100 MHz, and I=10, the only difference between the existing method and the present invention is derived from C, the factor which is basically a measure of the level of allowed parallelism. It can be seen, by comparing
Cexisting=16 and Cpresent _ invention=4.
Tmax,existing=200 Mbps
Tmax,present _ invention=800 Mbps
Number of | ||||
Codeword | processing | Throughput | ||
size | C | units | (Mbps) | |
Existing (FIG. 5) | 320 | 16 | 20 | 200 |
1280 | 16 | 80 | 800 | |
Embodiment of | 320 | 4 | 80 | 800 |
present invention (FIG. 17) | 1280 | 16 | 80 | 800 |
-
- a) parity part ((1−R)*24 rightmost columns) of the matrix is designed to allow simple encoding algorithms;
- b) weights of all columns in the data portion of base parity check matrix is uniform;
- c) weights of all rows in the data portion of a base parity check matrix is uniform;
- d) the parity part of the matrix allows simple encoding algorithms. For example, the encoding algorithm based on
equation 1, orequation 14.
1 | 1 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 0 |
1 | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 |
1 | 0 | 1 | 0 | 1 | 1 | 1 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 |
1 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 0 |
0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 |
0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 |
where sij is specified in the matrix definition below:
6 | 38 | 3 | 93 | −1 | −1 | −1 | 30 | 70 | −1 | 86 | −1 | 37 | 38 | 4 | 11 | −1 | 46 | 48 | 0 | −1 | −1 | −1 | −1 |
62 | 94 | 19 | 84 | −1 | 92 | 78 | −1 | 15 | −1 | −1 | 92 | −1 | 45 | 24 | 32 | 30 | −1 | −1 | 0 | 0 | −1 | −1 | −1 |
71 | −1 | 55 | −1 | 12 | 66 | 45 | 79 | −1 | 78 | −1 | −1 | 10 | −1 | 22 | 55 | 70 | 82 | −1 | −1 | 0 | 0 | −1 | −1 |
38 | 61 | −1 | 66 | 9 | 73 | 47 | 64 | −1 | 39 | 61 | 43 | −1 | −1 | −1 | −1 | 95 | 32 | 0 | −1 | −1 | 0 | 0 | −1 |
−1 | −1 | −1 | −1 | 32 | 52 | 55 | 80 | 95 | 22 | 6 | 51 | 24 | 90 | 44 | 20 | −1 | −1 | −1 | −1 | −1 | −1 | 0 | 0 |
−1 | 63 | 31 | 88 | 20 | −1 | −1 | −1 | 6 | 40 | 56 | 16 | 71 | 53 | −1 | −1 | 27 | 26 | 48 | −1 | −1 | −1 | −1 | 0 |
-
- (a) Keep the performance in terms of coding gain as high as possible. This objective translates into the following needs:
- Select the largest suitable codeword from the available set of codewords. For the LDPC codes and other block codes, the longer the codeword the more coding gain can be achieved, although at certain codeword size the point of diminishing return is reached.
- Adjust properly the amount of shortening and puncturing, as this directly affects the performance, and the efficiency of the transmission.
- (b) Use as few of the modulated symbols as possible. This in turn means that it is desirable to utilize transmit power economically. This is especially important for battery operated hand-held wireless devices by keeping the air time at minimum.
- (c) Keep the overall complexity at a reasonable level. This usually translates into a requirement to operate with a relatively small set of codewords in different sizes. In addition, it is desirable to have a code designed in such a way that various codeword lengths can be implemented efficiently. Finally, the actual procedure defining concatenation rules should be simple.
- (a) Keep the performance in terms of coding gain as high as possible. This objective translates into the following needs:
qrate _ preserved=R/(1−R)
-
- (a) specifying general rules for shortening and puncturing patterns;
- (b) providing mechanism for q >qrate
—preserved ; - (c) establishing a limit on the amount of puncturing; and
- (d) providing an algorithmic method for finding the optimal solution within the range of given system parameters.
-
- 1. Based on the data packet size determine the minimum number of required modulated symbols;
- 2. Select the codeword length from the set of available codeword lengths;
- 3. In an iterative loop determine required amount of shortening and puncturing and corresponding estimated performance and add additional modulated symbol(s), if necessary;
- 4. Distribute amount of shortening and puncturing across all constituent codewords efficiently; and
- 5. Append padding bits in the last modulated symbol if necessary.
Referring toFIG. 21 , these steps are more fully shown in the flow chart in which the process starts 211 and various parameters areinput 212 including: - Data packet size in bits, L;
- Set of codewords of size Ni (i=1, 2, . . . , number_of_codewords) for the derived code rate R;
- Number of bits carried by a modulated symbol S; and
- Performance criteria.
-
- NNmax=2304, NNmin=576, NNinc=576: maximum, minimum and increment of codeword lengths, effectively resulting 4 codeword lengths: 576, 1152, 1728, 2304;
- pmax: maximum puncturing percentage, which is defined as:
- number of punctured bits/total number of parity bits (%).
-
- R: target code rate;
- NCBPS: number of data bits in OFDM symbol;
- AggregationFlag: Boolean signaling whether PSDU is an aggregate of MPDUs (AggregationFlag=1),
- HT_LENGTH: number of payload octets (AggregationFlag=0), or number of OFDM symbols (AggregationFlag=1)
-
- NN: code length to use;
- NCodeWords: number of codewords to use;
- KKS,KKS _ Last: number of information bits to send in first codeword(s), and in last codeword;
- Np, Np _ Last: number of bits to puncture in first codeword (s), and in last codeword;
- NOFDM: number of OFDM symbols used;
- NPaddingBits: number of bits the last OFDM symbol is padded;
if(AggregationFlag == 0) { |
NInfobits=8×HT_LENGTH; |
//in non-aggregation case HT_LENGTH is the number of payload octets |
NOFDM=ceil(NInfobits/ (NCBPS×R)); |
// minimum number of OFDM symbols |
} |
else { |
NOFDM= HT_LENGTH; |
// in aggregation case HT_LENGTH is the number of OFDM symbols |
NInfoBits=NOFDM×NCBPS ×R; |
// number of info bits includes padding;MAC will use its own delineation |
//method to recover an aggregate payload |
} |
NCodeWords = ceil(NCBPS× NOFDM/NNmax); |
//number of codewords is based on maximum codeword length |
NN = ceil(NCBPS× NOFDM/(NCodeWords×NNinc))× NNinc; |
// codeword length will be the larger of the closest one |
// to NCBPS× NOFDM/NCodeWords |
KK=NN×R; |
// number of information bits in codeword chosen |
MM=NN−KK; |
// number of parity bits in codeword chosen |
NParityBits_requested=NCodeWords× MM; |
// total number of parity bits allocated in NOFDM symbols |
NParityBits =min(NOFDM× NCBPS− NInfoBits,NParityBits_requested); |
//in non-aggregation case allow adding extra OFDM symbol(s) to limit |
//puncturing |
if(AggregationFlag==0) { |
while(100×(NParityBits_requested−NParityBits)/ | |
NParityBits_requested>Pmax) { |
NOFDM= NOFDM+1; |
// extra OFDM symbol(s) are used to carry parity | |
NParityBits =min(NParityBits + NCBPS,NParityBits_requested); | |
} |
} |
// Finding number of information bits to be sent per codeword(s), | |
//KKS, KKS_Last, and number of bits the codeword(s) will be punctured |
NP, |
//and NP_Last. Making sure that last codeword may only be shortened | |
// more then others, and punctured less then others. |
KKS=ceil(NInfoBits/ NCodeWords); |
KKS_Last =NInfoBits − KKS ×( NCodeWords −1); |
MMP =min(MM, floor(NParityBits/CodeWords); |
MMP_Last = min(MM, NParityBits − MMP ×(NCodeWords −1)); |
NP =MM − MMP; |
NP_Last =MM− MMP_Last; |
// Finally, calculating number of padding bits in last OFDM symbol |
NPaddingBits = NOFDM × NCBPS − NInfoBits − NParityBits; |
-
- Bits selected for puncturing should be chosen such that each one is connected to as few check nodes as possible. This can be equivalently stated as follows: bits selected for puncturing should not be the ones corresponding to the heavy-weight, or strong columns, i.e. columns containing large number of non-zero elements, 1's in this particular case.
- Bits selected for puncturing should be chosen such that they all participate in as many parity check equations as possible.
-
- 241: Shortened=0 Punctured=216 Infobits 3 strong columns
- 242: Shortened=0 Punctured=216 Infobits 3 weak columns
- 243: Shortened=0 Punctured=216 Infobits random
- 244: Shortened=0 Punctured=216
Parity columns - 245: Shortened=0 Punctured=216 Parity random and
- 246: Shortened=0 Punctured=216
Parity columns
qnormalized=(Nshortened/Npunctured)/[R/(1−R)],
Ppuncture=100×(Npuncture/M),
-
- Shortening, or combined shortening and puncturing is applied in order to provide a large range of codeword sizes from a single parity check matrix.
- The effective code rate of the code defined by the parity check matrix modified by shortening and puncturing is equal to or less than the original code rate.
- Shortening is performed so that the column weight distribution of the modified matrix is optimal for the new matrix size and code rate. Another solution is to keep the column weight distribution of the new matrix only approximately optimum.
- Shortening is performed so that the approximately uniform row weight is preserved.
- Puncturing is performed so that each of the bits selected for puncturing is connected to as few check nodes as possible.
- Puncturing is performed so that the bits selected for puncturing all participate in as many parity check equations as possible.
- Puncturing is performed so that the approximately uniform row weight is preserved.
- A suboptimal but computationally efficient method is to first rearrange the columns of the data portion of the parity check matrix, Hd, by applying the preceding rules assuming that shortening is applied to a group of consecutive bits of the data portion of the parity check matrix and puncturing is applied to another group of consecutive bits of the data portion of the parity check matrix as illustrated by the
example matrix 250 shown inFIG. 25 . - Performance of the new code, which is obtained by applying both the shortening and puncturing, can be kept at the level of the original code by setting the normalized shortening to puncturing ratio, qnormalized=(Nshortened/Npunctured)/[R/(1−R)] greater than one. The qnormalized value depends on the particular matrix design and the code rate, R. When the preservation of the original code rate is required, the normalized shortening to puncturing ratio shall be set to one (qnormalized=1).
- The amount of puncturing is limited to a certain value, which depends on the particular parity check matrix design.
s′=floor{s. (L/96)},
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US20190181884A1 (en) * | 2016-08-08 | 2019-06-13 | Nokia Technologies Oy | Inter-Block Modifications to Generate Sub-Matrix of Rate Compatible Parity Check Matrix |
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US8024641B2 (en) | 2011-09-20 |
USRE49225E1 (en) | 2022-09-27 |
US7917829B2 (en) | 2011-03-29 |
US8301975B2 (en) | 2012-10-30 |
WO2006039801A1 (en) | 2006-04-20 |
US20120117440A1 (en) | 2012-05-10 |
US20090259915A1 (en) | 2009-10-15 |
US20130013983A1 (en) | 2013-01-10 |
US8583980B2 (en) | 2013-11-12 |
US8291289B2 (en) | 2012-10-16 |
US8099646B2 (en) | 2012-01-17 |
US20070094580A1 (en) | 2007-04-26 |
US20110107177A1 (en) | 2011-05-05 |
US20100211847A1 (en) | 2010-08-19 |
US7996746B2 (en) | 2011-08-09 |
US7747934B2 (en) | 2010-06-29 |
USRE48212E1 (en) | 2020-09-15 |
US20100251065A1 (en) | 2010-09-30 |
US20110307755A1 (en) | 2011-12-15 |
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