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The limitation of permutation polynomial interleavers for turbo codes and a scheme for dithering permutation polynomials
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SHORT COMMUNICATION
The limitation of permutation polynomial interleavers for turbo codes and a scheme for dithering permutation polynomials Jonghoon Ryu a , Lucian Trifina b,∗ , Horia Balta c a
Samsung Electronics, Inc., Suwon, Republic of Korea “Gheorghe Asachi” Technical University, Faculty of Electronics, Telecommunications and Information Technology, Department of Telecommunications, Bd. Carol I, No. 11 A, 700506 Iasi, Romania c University Politehnica of Timisoara, Faculty of Electronics and Telecommunications, Department of Telecommunications, V. Parvan 2, 300223 Timisoara, Romania b
a r t i c l e
i n f o
Article history: Received 5 February 2014 Accepted 16 June 2015 Keywords: Permutation polynomial Quadratic permutation polynomial Turbo codes Interleaver
a b s t r a c t In this letter, partial upper bounds on minimum distance for turbo codes with permutation polynomial (PP) based interleavers over integer rings are derived using the fact that PPs are equivalent to a family of linear permutation polynomials (LPPs). It is shown that upper bounds on minimum distance of turbo codes using higher order PP based interleavers are bounded by a function of the number of equivalent LPPs for PPs. Besides, it is shown that when the constant terms of LPPs are dithered, the resulting dithered LPP interleavers perform better than the quadratic permutation polynomial (QPP) based interleavers used in long term evolution (LTE) standard or than other good QPP or cubic permutation polynomial (CPP) based interleavers given in the literature. © 2015 Elsevier GmbH. All rights reserved.
1. Introduction
2. LPP representation of higher order PPs
Permutation polynomial (PP) based interleavers over integer rings have been widely studied [1–3,6–8]. In particular quadratic permutation polynomial (QPP) based interleavers were emphasized due to their simple implementation [3] as well as excellent performance [1]. In [1], upper bounds on minimum distance of turbo codes with QPP based interleavers are shown. Higher order PP based interleavers have also been investigated for better performance and implementation, in particular for cubic permutation polynomial (CPP) based interleavers [2,3]. However little is known for minimum distance of turbo codes with higher order PP based interleavers. In this letter, the technique shown in [3] is used to decompose higher order PPs into linear permutation polynomials (LPPs) and partial upper bounds on the minimum distance for turbo codes using higher order PP based interleavers are shown. It is also shown that when the constant terms of the LPPs which are equivalent to PPs are dithered, better frame error rate (FER) performance is obtained. For a more succinct writing, in the following, PP based interleavers are denoted as PP.
In this section, previous results on higher order PPs are briefly reviewed and upper bounds on the minimum distance for turbo codes using PPs are shown. Firstly, the equivalence of PPs and a family of LPPs is shown. In the following, a parallel LPP (PLPP) is defined.
∗ Corresponding author. Tel.: +40 232701679. E-mail addresses: [email protected] (J. Ryu), [email protected] (L. Trifina), [email protected] (H. Balta).
Definition 2.1.
p(x) =
[3] Let p(x) be an interleaver such that
⎧ p0 (x) = P1,0 x + P0,0 , mod(x, L) = 0 ⎪ ⎪ ⎪ ⎨ p1 (x) = P1,1 x + P0,1 ,
mod(x, L) = 1
⎪ ⎪ ··· ⎪ ⎩
pL−1 (x) = P1,L−1 x + P0,L−1 ,
mod(x, L) = L − 1,
which can be also represented in the following form,
p(x) =
⎧ p0 (y) = P1,0 · Ly + P0,0 , x = Ly ⎪ ⎪ ⎪ ⎨ p1 (y) = P1,1 · Ly + P0,1 ,
⎪ ⎪ ··· ⎪ ⎩
x = Ly + 1
pL−1 (y) = P1,L−1 · Ly + P0,L−1 ,
x = Ly + (L − 1),
with 1 ≤ L < N, where N is the interleaver length, L|N and 0 ≤ y ≤ N − 1. Then p(x) is called a PLPP (i.e., p(x) consists of L LPPs). L For each l = 0, 1, · · · , L − 1, pl (y) is a LPP and since a LPP can be implemented using only additions and comparisons, a PLPP can also
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Table 1 The least numbers of LPPs for equivalent PLPPs (L) of LTE-QPPs of various lengths N [5]. Lengths range
L=1
L=2
L=3
L=4
L>4
40–512 528–1024 1056–2048 2112–6144
1
55 21 1
3 3 1 1
6 26 48
1 2 4 15
60 32 32 64
77
8
80
22
188
1
be implemented using the same address generation method for a LPP [3]. Although not all PPs are equivalent to PLPPs [3], the following lemma shows that all PPs are equivalent to PLPPs when the interleaver lengths are of the form N = 23 · M, where M is a positive integer.
K
Lemma 2.2. [3] Let f (x) = f xk (mod N) be a PP. Suppose that k=1 k N = 23 · M, with M a positive integer. Then f(x) is equivalent to a PLPP and L ≤ 2M. Lemma 2.2 is obtained by computing the f(x) at each point x = 2My + l and using the modulo operation and the zero polynomials shown in [4,3] to remove quadratic and higher order terms. In particular, a sufficient condition for a QPP to have an equivalent PLPP is given in [3]. By the lemma 2.2, all the LTE-QPPs are equivalent to PLPPs, since their interleaver lengths are multiples of 8. In practice, L’s are relatively small numbers compared to interleaver lengths, as shown in Table 1. For example, let f(x) = 15x + 32x2 (mod256), then f(2y) = 15 · 2y + 32 · (2y)2 = 15 · 2y + 32 · (2y)2 + 256/2 · y + 256/2 · (y)2 = 79 · 2y, f(2y + 1) = 15 · (2y + 1) + 32 · (2y + 1)2 = 15 · (2y + 1) + 32. Note that (N/2) · y + (N/2) · y2 is a zero polynomial for all y. Thus, the equivalent PLPP, with L = 2, of the previous QPP f(x), is:
p(x) =
p0 (x) = 79x, mod(x, 2) = 0 p1 (x) = 15x + 32, mod(x, 2) = 1
Let L = 4, then by using a similar method, the equivalent PLPP of the previous QPP f(x), is:
p(x) =
⎧ ⎪ ⎨ p0 (x) = 15x, mod(x, 4) = 0 p1 (x) = 15x + 32,
mod(x, 4) = 1
p3 (x) = 15x + 32,
mod(x, 4) = 3
⎪ ⎩ p2 (x) = 15x + 128, mod(x, 4) = 2
Since L is relatively small compared to the interleaver length for QPPs in [5], the interleaver/deinterleaver for the PPs can be efficiently generated as shown in [3]. Note that the number of
coefficients of the L LPPs depends only on L, not on the degree of PP. Thus a PP of arbitrary degree can be implemented using L LPPs if it is equivalent to L LPPs. In the following, upper bounds on minimum distance for turbo codes with PPs using Lemma 2.2 are shown. Lemma 2.3. Let the first coefficients of PLPP be equal for all l, i.e., P1,0 = P1,1 = . . . = P1,L−1 = P. Let also m and n be positive integers and L|(m · (2 − 1)), where is the degree of the primitive feedback and monic feedforward polynomials of recursive systematic convolutional codes, which are component codes of a conventional turbo code. If there exists a critical interleaver pattern of size 4 as shown in Fig. 1, the minimum distance of the turbo code with this PLPP interleaver is upper bounded by (m + n) · 2 + 12. Proof. Consider the constituent codewords 1 and 2 generated by the interleaver pattern, both containing two fundamental paths with input sequences of weight 2 as shown in Fig. 1, where there are two error patterns with input sequences of weight 2 at points xi , xi + m · (2 − 1), and xj , xj + m · (2 − 1) respectively. It is easy to check that the weight of codeword generated by the constituent code 1 is 2 · (m · 2−1 + 2). Let us consider the error sequence with an input sequence of weight 2 xi , xi + m · (2 − 1). Since the distance between the two points is m · (2 − 1) and L|(m · (2 − 1)), the two points are on the same ith LPP. Thus, each point is mapped to Pxi + Li and P(xi + m · (2 − 1)) + Li , respectively. Since the input sequences for the constituent codes 1 and 2 are mapped by an interleaver, there is a point in the input for the constituent code 1 that is mapped to the point Pxi + Li + n · (2 − 1) in the input for the constituent code 2. Let us call it xj . Then Pxj + Lj = Pxi + Li + n · (2 − 1). Since the distance between the points xj , xj + m · (2 − 1) is m · (2 − 1) and L|(m · (2 − 1)), the two points are in the same jth LPP. Finally, P(xj + m · (2 − 1)) + Lj = Pxj + Lj + Pm · (2 − 1) = Pxi + Li + (Pm + n) · (2 − 1), which is equal to P(xi + m · (2 − 1)) + Li + n · (2 − 1). Thus, an input sequence of weight 4 exists for PLPP with L LPPs and the weight of the corresponding codeword is 2 · (m · 2−1 + 2) + 2 · (n · 2−1 + 2) + 4 = (m + n) · 2 + 12. 䊐 It should be mentioned that the upper bound in Lemma 2.3 assumes that the first coefficients of PLPP are all equal. This constraint was also imposed for the computation of L for LTE-QPPs in Table 1. In Table 2, upper bounds on the minimum distance for turbo codes with PPs when = 3 are shown. The result in Lemma 2.3 is similar to Tables II and III in [1], however, Lemma 2.3 can also be applied to higher order PPs.
constituent code 1
xi
x i+m(2v−1)
xj
x j+m(2v−1)
interleaver
Pxi +Li
Pxi +Li +n(2 v−1)
P(xi+m(2v−1))+L i P(xi+m(2 v−1))+Li +n (2v−1)
constituent code 2 Fig. 1. Critical interleaver pattern of size 4 (Fig. 3 in [1]).
Please cite this article in press as: Ryu J, et al. The limitation of permutation polynomial interleavers for turbo codes and a scheme for dithering permutation polynomials. Int J Electron Commun (AEÜ) (2015), http://dx.doi.org/10.1016/j.aeue.2015.06.007
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J. Ryu et al. / Int. J. Electron. Commun. (AEÜ) xxx (2015) xxx–xxx Table 2 Upper bounds (UBs) for the minimum distance of turbo codes using PP based interleavers. Generator matrix of recursive systematic convolutional codes is as in the LTE turbo code [5]. L
m
n
UB
L
m
n
UB
1 or 7 2 or 14 3 or 21 2 or 14
1 2 3 2
1 1 1 2
28 36 44 44
4 or 28 5 or 35 6 or 42 8 or 56
4 5 6 8
1 1 1 1
52 60 68 84
The results in Table 2 show that the minimum distance of turbo codes with PPs is upper bounded by a function of L. Thus, for higher order PPs, L should also be considered in searching for good interleavers when the PPs are implemented using LPPs.
3
length is essentially composed of LPPs and the set of LPPs set a limit on the minimum distance of the turbo code with PP [3]. In other direction, the constant terms of LPPs which are in decomposition of PPs can be dithered, i.e., a PP is represented using the equivalent LPPs and the constant terms of the LPPs are dithered. PPs are highly structured so that the regularity of PP can lower the error performance. By dithering PPs, the edge-effect can be avoided. Good dithered LPPs for some lengths were searched. For lengths smaller or equal than 560, L = 2, and for the others, L = 4. Dithered LPP interleavers in the paper were not found in purely random way, but using the rules from [8] or [9], supplemented with simulations. The goal was to find interleavers better than LTE-QPPs or other good QPPs or CPPs. 3.1. Comparison between dithered LPPs and good QPPs or CPPs from the point of view of the minimum distance and simulated FER at high SNR
3. Dithered LPPs Increasing the order of PPs may give better FER performance. However, it also increases the number of computations and required storage for the coefficients. QPPs, CPPs and higher order PPs can be decomposed into PLPPs. In the previous section, it was shown that the minimum distance of turbo codes with PPs is upper bounded by a function of the number of LPPs (L). If CPPs or higher order PPs’ performance is good, they can be efficiently implemented using small number of parallel LPPs. But if not, the performance of turbo codes with PPs may not necessarily improve as the order of polynomial increases. The reason is that any PP with practical
In Tables 3 and 4, the simulation results are shown in terms of FER at high SNR values for LTE-QPPs and for the found dithered LPPs. For simulation, the maximum a posteriori (MAP) algorithm with an iteration stopping criterion based on the maximum absolute value of logarithm likelihood ratio (LLR) was used. The threshold of LLR was set to 15 and the channel model was considered with additive white Gaussian noise (AWGN). At least 70 erroneous frames were simulated for each interleaver at SNR given in Tables 3 and 4. For all reported lengths the FER of dithered LPPs is lower than that for
Table 3 Minimum distances (dmin ), corresponding multiplicities of codewords (Ndmin ) and FER at high SNR for LTE-QPP [5] and dithered LPP interleavers for lengths N ≤ 448. Generator matrix of recursive systematic convolutional codes and trellis termination are as in the LTE turbo code [5]. N
SNR [dB]
(x) (LTE-QPP [5])
dmin /Ndmin
FER (×10−6 )
(x) (Dithered LPP)
dmin /Ndmin
FER (×10−6 )
40 48 56 64 72 88 96 112 128 136 152 200 224 280 320 448
5 5 5 5 5 5 4 4 3.5 3.5 3.5 3 2.75 2.5 2.5 2.5
3x + 10x2 7x + 12x2 19x + 42x2 7x + 16x2 7x + 18x2 5x + 22x2 11 + 24x2 41 + 84x2 15 + 32x2 9 + 34x2 9x + 38x2 13x + 50x2 27x + 56x2 103x + 210x2 21x + 120x2 29x + 168x2
11/1 13/1 13/1 12/1 15/1 15/1 16/2 17/2 16/1 16/3 15/1 20/1 22/99 21/1 20/1 22/105
11.7724 2.6327 3.4949 1.2373 0.3567 2.9358 2.1871 0.8782 3.3163 5.8068 4.7129 1.6036 10.3331 21.1729 3.0100 17.9973
33x, 33x + 28 19x, 19x + 6 9x, 9x + 18 53x 23x 31x, 31x + 80 61x 15x, 15x + 54 49x 87x 63x, 63x + 102 13x, 13x + 146 185x, 185x + 72 87x, 87x + 274 49x, 49x + 210 15x, 15x + 152
13/1 15/1 16/2 17/1 19/6 16/1 19/1 19/1 19/1 20/3 19/1 23/1 23/1 27/3 23/1 25/1
2.5534 0.9787 0.3466 0.1716 0.1520 1.4006 1.2937 0.3567 1.2232 1.4123 0.5288 0.7663 1.4753 2.3912 1.6460 1.1631
Table 4 Minimum distances (dmin ), corresponding multiplicities of codewords (Ndmin ) and FER at high SNR for LTE-QPP [5] and dithered LPP interleavers for lengths N ≥ 560. Generator matrix of recursive systematic convolutional codes and trellis termination are as in the LTE turbo code [5]. For some lengths, in the fourth and seventh columns, in parentheses, 107 ×TUB(FER) values are given for AWGN channel and distance spectrum with maximum distance given in the second column, in parentheses. N
SNR [dB] (dmax )
(x) (LTE-QPP [5])
dmin /Ndmin (107 × TUB(FER))
560
2.2 (39) 1.66
227x + 420x2
29/1 (5.7142) 27/1
848
1.58 (39)
239x + 106x2
29/2 (3.4249)
6.6300
992
1.6
65x + 124x2
22/2
7.8000
1.56 (31)
55x + 84x2
30/322 (148.78)
784
1008
25x + 98x2
FER (×10−6 ) 2.2400 11.0600
11.6600
(x) (Dithered LPP)
dmin /Ndmin (107 × TUB(FER))
FER (×10−6 )
367x, 367x + 484 19x, 19x, 19x + 552, 19x + 488 19x, 19x, 19x + 327, 19x + 761 19x, 19x, 19x + 624, 19x + 568 19x, 19x, 19x + 563, 19x + 149
25/1 (6.1668) 32/1
1.8923
28/1 (3.3580)
5.6668
33/1
5.7500
27/1 (4.0043)
6.7000
3.7500
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Table 5 Minimum distances (dmin ), corresponding multiplicities of codewords (Ndmin ) and FER at high SNR for optimum dmin QPP [6] and dithered LPP interleavers. Generator matrix of recursive systematic convolutional codes and trellis termination are as in the LTE turbo code [5]. For some lengths, in the fourth and seventh columns, in parentheses, 107 ×TUB(FER) values are given for AWGN channel and distance spectrum with maximum distance given in the second column, in parentheses. N
SNR [dB] (dmax )
(x) (Optimum dmin QPP [6])
dmin /Ndmin (107 × TUB(FER))
40 48 56 64 72 88 96 112 128 136 152 200 224 280 320
5 5 5 5 5 5 4 4 3.5 3.5 3.5 3 2.75 2.5 2.5 (35) 2.5 (35)
13x + 10x2 19x + 12x2 27x + 14x2 21x + 8x2 7x + 18x2 5x + 22x2 7 + 24x2 Not given in [6] 37 + 96x2 9 + 102x2 9x + 38x2 13x + 150x2 61x + 168x2 Not given in [6] 99x + 240x2
12/1 12/3 14/21 14/1 15/1 15/1 16/2
448
55x + 336x2
17/1 14/1 15/1 20/1 21/1 23/1 (1.591) 28/1 (0.166)
For the three previously mentioned lengths with exceptions, in Table 4, the distance spectra computed for turbo codes with corresponding interleavers and the truncated upper bounds of FER for AWGN channel, denoted TUB(FER), are presented. TUB(FER) for AWGN channel [11] is given by numdist
Nd (i) · erfc(d(i) · Rc · SNR),
(1)
i=1
where numdist is the number of distances in the distance spectrum of the turbo code, d(i) is the ith distance in the distance spectrum (d(1) = dmin ), Nd (i) is the corresponding codeword multiplicity (Nd (1) = Ndmin ), Rc is the coding rate of the turbo code, SNR is the signal-to-noise ratio and erfc(·) is the complementary error function, given by 2 erfc(x) = √ ·
∞
2
e−t dt.
dmin /Ndmin (107 × TUB(FER))
FER (×10−6 )
1.4693
33x, 33x + 28 19x, 19x + 6 9x, 9x + 18 53x 23x 31x, 31x + 80 61x 15x, 15x + 54 49x 87x 63x, 63x + 102 13x, 13x + 146 185x, 185x + 72 87x, 87x + 274 49x, 49x + 210
2.5534 0.9787 0.3466 0.1716 0.1520 1.4006 1.2937 0.3567 1.2232 1.4123 0.5288 0.7663 1.4753 2.3912 1.6460
3.2486
15x, 15x + 152
13/1 15/1 16/2 17/1 19/6 16/1 19/1 19/1 19/1 20/3 19/1 23/1 23/1 27/3 23/1 (1.221) 25/1 (0.525)
3.6714 6.3377 4.7129 1.6139 32.8682
1. The dithered LPP has a smaller minimum distance and lower simulated FER than the QPP/CPP. 2. The dithered LPP has an equal minimum distance with higher or equal multiplicity and lower simulated FER than QPP/CPP. 3. The dithered LPP has a higher minimum distance and higher simulated FER than the QPP/CPP. 4. The dithered LPP has a equal minimum distance with smaller or equal multiplicity and higher simulated FER than the QPP/CPP.
(x) (Dithered LPP)
5.3434 3.8845 7.1716 1.0681 0.3567 2.9358 1.7141
LTE-QPPs. In Tables 3 and 4, the minimum distances (dmin ) and the corresponding multiplicities of codewords (Ndmin ) for LTE-QPPs and for dithered LPPs are also given. The post-interleaver flushing trellis termination was used, as in LTE turbo code [5]. To find the minimum distances or the distance spectra of turbo codes the method in [10] was used. From Tables 3 and 4 it is observed that minimum distances for dithered LPPs are higher than those for LTE-QPPs for all lengths, excepting three of them (namely 560, 848 and 1008). This shows that the dithering constant term of a PLPP may increase the minimum distance of turbo codes with a such interleaver. Further, exceptions have been considered when:
TUB(FER) = 0.5 ·
FER (×10−6 )
(2)
x
The maximum distance in each distance spectrum is given in parentheses in the second column of Table 4. The 107 ×TUB(FER) values are given in parentheses in the fourth and seventh columns, respectively. It is seen that TUB(FER) for LTE-QPPs is greater than
1.1631
or very close to TUB(FER) for dithered LPPs. This explains the difference or similarity in simulated FER performance. In [1] the minimum distances for all LTE-QPPs are given, but dual trellis termination [12] was considered. With LTE trellis termination the minimum distance decreases considerably compared to dual termination. In [6] QPPs with optimum minimum distance for a large number of lengths N between 32 and 512 and for N = 1024 were obtained. Because in [6] also dual termination was used, the minimum distances and the corresponding multiplicities of codewords for QPPs from Table I in [6], with lengths that are also in Table 3, were computed, when LTE trellis termination is used. It is mentioned that QPPs with optimum minimum distances for dual termination do not necessarily have optimum minimum distances for LTE trellis termination. The results are given in Table 5 along with those for dithered LPPs. Simulated FER values for QPPs in [6] with previously mentioned lengths are also reported. Excepting length 448, for all other lengths in Table 3 our dithered LPPs lead to higher or equal minimum distances compared to those of QPPs from [6] with LTE trellis termination. In addition, simulated FER for dithered LPPs is lower than FER for QPPs from [6] for all reported lengths, excepting the length of 320 for which FER values are very close to each other. For these two lengths with exceptions (320 and 448) the 107 ×TUB(FER) values and the maximum distance in the distance spectra as in Table 4, are given. For the length of 320 the TUB(FER) values are very close to each other, as shown by the simulated FER values. For interleavers of length 448, we have assumed that besides the distance spectrum other parameters influence FER performance. It should be mentioned that a high minimum distance is not sufficient for a good performance of a turbo code. In [2,7,8] few methods for finding good QPP or CPP based interleavers are given. These methods also take into account other parameters of QPP/CPP based interleavers (such as D parameter or spread and refined nonlinearity degree). In Tables 6 and 7, the our dithered LPPs are compared with the QPPs from [2] or [7,8] and the CPPs from [2], for some lengths, considering the simulated FER and the minimum distance. It is observed that for all reported lengths dithered LPPs lead to higher or equal minimum distances (excepting CPP of length 40 and QPPs of lengths 320, 560, 848 and 1008, respectively). The dithered LPPs also lead to lower simulated FER than QPPs or CPPs, excepting few cases (CPPs of lengths 48 and 88 and QPP of length 320). For these lengths with exceptions, the 107 ×TUB(FER) values were computed, as shown in Tables 4 and 5. For lengths 448, 560
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Table 6 Minimum distances (dmin ), corresponding multiplicities of codewords (Ndmin ) and FER at high SNR for QPP from [2] or [8], CPP from [2] and dithered LPP interleavers for lengths N ≤ 200. Generator matrix of recursive systematic convolutional codes and trellis termination are as in the LTE turbo code [5]. For some lengths, in the fourth and seventh columns, in parentheses, 107 ×TUB(FER) values are given for AWGN channel and distance spectrum with maximum distance given in the second column, in parentheses. N 40 48 56 64 72 88 96 112 128 136 152 200
SNR [dB] (dmax )
(x) (QPP from [2] or [8] and CPP from [2])
dmin /Ndmin (107 × TUB(FER))
FER (×10−6 )
(x) (Dithered LPP)
dmin / Ndmin (107 × TUB(FER))
FER (×10−6 )
5 5 (37) 5 5 (39) 5 5 5 5 5 5 5 5 (40) 4 4 4 4 3.5 3.5 3.5 3.5 3.5 3 3
13x + 10x2 [2] 3x + 8x2 + 16x3 7x + 36x2 [2] 5x + 6x2 + 12x3 3x + 42x2 [2] 5x + 14x2 + 42x3 7x + 16x2 [2] 5x + 24x2 + 48x3 5x + 54x2 [8] 7x + 0x2 + 4x3 5x + 22x2 [2] 27x + 22x2 + 66x3 13x + 72x2 [2] 9x + 12x2 + 56x3 41x + 28x2 [2] 41x + 0x2 + 28x3 17 + 32x2 [2] 19 + 102x2 [2] 19x + 0x2 + 34x3 59x + 38x2 [2] 59x + 0x2 + 114x3 13x + 150x2 [2] 3x + 0x2 + 80x3
12/1 15/3 (4.44) 15/2 15/3 (1.59) 13/1 14/23 12/1 15/1 16/2 16/2 15/1 15/1 (0.325) 16/1 15/1 17/2 17/2 18/1 16/1 16/1 16/1 16/1 20/1 20/1
5.3434 4.5677 2.6038 0.9258 3.4457 2.6896 1.2373 0.3527 0.3063 0.6231 2.9358 0.1576 1.5356 1.7407 0.8507 0.7587 1.5770 4.0654 4.1556 3.5498 4.6401 1.6139 2.2218
33x, 33x + 28 19x, 19x + 6 9x, 9x + 18 53x
13/1 (7.553) 15/1 (2.039) 16/2
2.5534
17/1
0.1716
23x
19/6
0.1520
31x, 31x + 80 61x
16/1 (0.104) 19/1
1.4006
15x, 15x + 54 49x 87x
19/1
0.3567
19/1 20/3
1.2232 1.4123
63x, 63x + 102 13x, 13x + 146
19/1
0.5288
23/1
0.7663
and 1008, the dithered LPPs have smaller TUB(FER) values and also lower simulated FER. For lengths 48, 320 and 848, TUB(FER) values are close to each other as simulated FER values. As for length of 448 in Table 5, we have assumed that other parameters besides the distance spectrum influence FER performance, for interleavers of lengths 40 and 88. 3.2. Simulation results To see the FER behaviour of dithered LPPs in a wider range of SNR, the simulation results for several lengths (N = 40, N = 64 and N = 448) are given below. In Fig. 2, FER curves for interleavers of length N = 40 are given. The simulated interleavers are LTE-QPP, optimum dmin QPP from [6] (which is the same as QPP with largest
0.9787 0.3466
1.2937
spread and best distance spectrum from [2]), CPP with largest spread and best distance spectrum from [2] and dithered LPP. It is observed that at high SNR, dithered LPP has the best performance, CPP from [2] has performance closer to dithered LPP, excepting for SNR=5 dB, LTE-QPP has the weakest performance and QPP from [6] or [2] has a medium performance. In Fig. 3, FER curves for interleavers of length N = 64 are given. Simulated interleavers are LTE-QPP (which is the same as QPP with largest spread and best distance spectrum from [2]), optimum dmin QPP from [6], CPP with largest spread and best distance spectrum from [2] and dithered LPP. Again at high SNR, dithered LPP has the best performance. LTE-QPP and QPP from [6] have the weakest performance and CPP from [2] have performance closer to dithered LPP one, excepting for SNR = 5 dB.
Table 7 Minimum distances (dmin ), corresponding multiplicities of codewords (Ndmin ) and FER at high SNR for QPP from [2] or [7,8] and dithered LPP interleavers for lengths N ≥ 224. Generator matrix of recursive systematic convolutional codes and trellis termination are as in the LTE turbo code [5]. For some lengths, in the fourth and seventh columns, in parentheses, 107 ×TUB(FER) values are given for AWGN channel and distance spectrum with maximum distance given in the second column, in parentheses. N
SNR [dB] (dmax )
(x) (QPP from [2] or [7,8])
dmin /Ndmin (107 × TUB(FER))
FER (×10−6 )
(x) (Dithered LPP)
dmin /Ndmin (107 × TUB(FER))
FER (×10−6 )
224
2.75
22/98
10.6936
1.4753
2.5
22/127
32.2522
27/3
2.3912
320
2.5 (35) 2.5 (35) 2.2 (39) 1.66
185x, 185x + 72 87x, 87x + 274 49x, 49x + 210 15x, 15x + 152 367x, 367x + 484 19x, 19x, 19x + 552, 19x + 488 19x, 19x, 19x + 327, 19x + 761 19x, 19x, 19x + 624, 19x + 568 19x, 19x, 19x + 563, 19x + 149
23/1
280
27x + 168x2 [2] 17x + 210x2 [2] 21x + 80x2 [2] 139x + 112x2 [7,8] 37x + 420x2 [8] 25x + 294x2 [8]
23/1 (1.221) 25/1 (0.525) 25/1 (6.1668) 32/1
1.6460
28/1 (3.3580)
5.6668
33/1
5.7500
27/1 (4.0043)
6.7000
448 560 784
25/1 (0.912) 25/1 (3.285) 29/1 (8.3184) 29/1
848
1.58 (39)
185x + 318x2 [8]
29/1 (3.4930)
992
1.6
65x + 124x2 [8]
22/2
1.56 (31)
55x + 588x2 [8]
29/1 (149.53)
1008
1.2546 1.8168 6.1013 17.2973
13.0017
7.8000
17.4220
1.1631 1.8923 3.7500
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10
10
LTE−QPP ( 29x+168x2 (mod 448) ) Optimum dmin−QPP [6] ( 55x+336x2 (mod 448) )
2
LTE−QPP ( 3x+10x (mod 40) ) Optimum d −QPP [6] min
2
−1
−1
or QPP from [2] ( 13x+10x (mod 40) ) CPP from [2] ( 3x+8x2+16x3 (mod 40) ) DLPP (P1,0=P1,1=33, P0,0=0, P0,1=28, N=40)
10
−2
−2
10
FER
FER
10
−3
10
−3
10
−4
−4
10
−5
10
10
−5
10
−6
10
−6
10
0
2
QPP from [7] ( 139x+112x (mod 448) ) DLPP (P1,0=P1,1=15, P0,0=0, P0,1=152, N=448)
10
1
2
3
4
5
0
0.5
1
1.5
2
2.5
SNR [dB]
SNR [dB] Fig. 2. FER curves for interleavers of length N = 40. Generator matrix of recursive systematic convolutional codes and trellis termination are as in the LTE turbo code [5]. DLPP stands for dithered LPP.
Fig. 4. FER curves for interleavers of length N = 448. Generator matrix of recursive systematic convolutional codes and trellis termination are as in the LTE turbo code [5]. DLPP stands for dithered LPP.
Finally, in Fig. 4, FER curves for interleavers of length N = 448 are given. Simulated interleavers are LTE-QPP, optimum dmin QPP from [6], QPP with largest spread and best distance spectrum from [7,8] (in [2] only QPP/CPP interleavers up to length of 352 are given) and dithered LPP. It is observed that at high SNR, dithered LPP and QPP from [7,8] have the best performance. LTE-QPP has weakest performance and QPP from [6] has a medium performance among the simulated interleavers, although it has the biggest minimum distance.
51). It is shown in this letter that for every PP (not just QPPs), the upper bound on the minimum distance is 52 when L = 4. Although the PPs that achieve the upper bound 52, have not been identified, this shows that the upper bound will increase only with 1 or 2 for any degree of polynomials when L = 4, i.e., no large difference in terms of minimum distance. This partly explains why the L of good QPPs is larger than 4 when the interleaver length is large. By increasing the order of the PPs, better PPs may be found as shown in the example, although there may not be large difference in minimum distance when L = 4. However, PPs have the maximum contention-free property which enables flexible hardware design. Our search in finding good interleavers may be constrained in a pool of PPs. Since all the practical PPs are equivalent to PLPPs, if our search is constrained in finding good PPs with small L, they can be generated in an efficient way. Thus, if a maximum contention free property is desired, PPs may be used and if better error performance is desired, dithered LPPs may be used. When L was chosen properly, both PPs and dithered LPPs can be generated in a same way. The approaches for using PPs or dithered LPPs in turbo codes are given below:
3.3. The usefulness of the partial upper bound and approaches for using PPs or dithered LPPs in turbo codes The upper bound in this letter shows some interesting points, specifically, for L = 4. In [1], it is shown that the upper bound on the minimum distance of turbo codes with QPP based interleavers, when QPPs have a quadratic inverse, is 50 (and for some QPPs with non-QPP inverse,
0
10
1. If there exist CPPs or higher order PPs with performance better than QPPs, figure out if the number of LPPs is small. 1-1) If there exist CPPs or higher order PPs, then they can be decomposed into a family of LPPs. 1-2) If there is no such CPPs or higher order PPs, then dithered LPPs can be used. 2. Instead of storing the coefficients of PPs, the ones of PLPPs can be stored, for both cases 1-1 and 1-2.
2
LTE−QPP or QPP from [2] ( 7x+16x (mod 64) ) Optimum dmin−QPP [6] ( 21x+8x2 (mod 64) )
−1
10
2
3
CPP from [2] ( 5x+24x +48x (mod 64) ) DLPP (P1,0=P1,1=53, P0,0=0, P0,1=0, N=64)
−2
10
−3
FER
10
−4
10
In this way, algebraic interleavers for turbo codes with good performance, less storage and computation are available.
−5
10
4. Conclusions −6
10
−7
10
0
1
2
3
4
5
SNR [dB] Fig. 3. FER curves for interleavers of length N = 64. Generator matrix of recursive systematic convolutional codes and trellis termination are as in the LTE turbo code [5]. DLPP stands for dithered LPP.
Using the fact that PPs over integer rings are decomposed into LPPs, it was shown that the upper bound on minimum distance for turbo codes with PPs is a function of the number of LPPs. This shows that 2 or 4 LPPs can be used for short interleaver lengths, but for medium to long lengths, 4 or 8 LPPs can be used. Dithered LPPs have been also proposed, where only the constant terms of equivalent LPPs for a PP are dithered. It was shown that for many short to
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medium lengths, dithered LPPs outperform in terms of simulated FER at high SNR, LTE-QPPs and other good QPPs, such those from [6] (when LTE trellis termination is used), [2] or [7,8] or CPPs from [2]. An analysis of reported interleavers in terms of the minimum distance or the truncated upper bound of FER was done. This shows that dithered LPPs lead, in most of the cases, to higher minimum distances or smaller TUB(FER) values. The usefulness of the upper bound derived in the paper was shown and approaches for using PPs or dithered LPPs in turbo codes were given. Acknowledgments The work of Horia Balta was supported by a grant of the Romanian Ministry of Education, CNCS – UEFISCDI, project number PN-II-RU-PD-2012-3-0122. The authors would like to thank the anonymous reviewers for their helpful comments and suggestions that improved the paper. References [1] Rosnes E. On the minimum distance of turbo codes with quadratic permutation polynomial interleavers. IEEE Trans Inform Theory 2012;58(July (7)):4781–95.
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[2] Trifina L, Tarniceriu D. Analysis of cubic permutation polynomials for turbo codes. Wirel Pers Commun 2013;69(March (1)):1–22. [3] Ryu J. Permutation polynomials of higher degrees for turbo code interleavers. IEICE Trans Commun 2012;E95-B(December (12)):3760–2. [4] Singmaster D. On polynomial functions (mod m). J Number Theory 1974;6(October (5)):345–52. [5] 3GPP TS 36.212 v8.3.0 (2008–05) E-UTRA: Multiplexing and channel coding (Release 8). [6] Rosnes E, Takeshita O. Optimum distance quadratic permutation polynomialbased interleavers for turbo codes. In: IEEE international symposium on information theory (ISIT 2006). 2006. p. 1988–92. [7] Trifina L, Tarniceriu D, Munteanu V. Improved QPP interleavers for LTE standard. In: IEEE international symposium on signals, circuits and systems (ISSCS 2011). 2011. p. 403–6. [8] Trifina L, Tarniceriu D. Improved method for searching interleavers from a certain set using Garello’s method with applications for the LTE standard. Ann Telecommun 2014;69(June (5–6)):251–72. [9] Balta H, Kovaci M, Nafornita M, Balta M. Multi-binary turbo-code design based on convergence of iterative turbo-decoding process. In: 5th European conference on circuits and systems for communications (ECCSC’10). 2010. p. 240–3. [10] Garello R, Pierleoni P, Benedetto S. Computing the free distance of turbo codes and serially concatenated codes with interleavers: algorithms and applications. IEEE J Sel Areas Commun 2001;19(May (5)):800–12. [11] Proakis JG. Digital communications. 3rd ed. New York: McGraw-Hill; 1995. [12] Guinand P, Lodge J. Trellis termination for turbo encoders. In: Proc. 17th biennial symp. commun. 1994. p. 389–92.
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SHORT COMMUNICATION
The limitation of permutation polynomial interleavers for turbo codes and a scheme for dithering permutation polynomials Jonghoon Ryu a , Lucian Trifina b,∗ , Horia Balta c a
Samsung Electronics, Inc., Suwon, Republic of Korea “Gheorghe Asachi” Technical University, Faculty of Electronics, Telecommunications and Information Technology, Department of Telecommunications, Bd. Carol I, No. 11 A, 700506 Iasi, Romania c University Politehnica of Timisoara, Faculty of Electronics and Telecommunications, Department of Telecommunications, V. Parvan 2, 300223 Timisoara, Romania b
a r t i c l e
i n f o
Article history: Received 5 February 2014 Accepted 16 June 2015 Keywords: Permutation polynomial Quadratic permutation polynomial Turbo codes Interleaver
a b s t r a c t In this letter, partial upper bounds on minimum distance for turbo codes with permutation polynomial (PP) based interleavers over integer rings are derived using the fact that PPs are equivalent to a family of linear permutation polynomials (LPPs). It is shown that upper bounds on minimum distance of turbo codes using higher order PP based interleavers are bounded by a function of the number of equivalent LPPs for PPs. Besides, it is shown that when the constant terms of LPPs are dithered, the resulting dithered LPP interleavers perform better than the quadratic permutation polynomial (QPP) based interleavers used in long term evolution (LTE) standard or than other good QPP or cubic permutation polynomial (CPP) based interleavers given in the literature. © 2015 Elsevier GmbH. All rights reserved.
1. Introduction
2. LPP representation of higher order PPs
Permutation polynomial (PP) based interleavers over integer rings have been widely studied [1–3,6–8]. In particular quadratic permutation polynomial (QPP) based interleavers were emphasized due to their simple implementation [3] as well as excellent performance [1]. In [1], upper bounds on minimum distance of turbo codes with QPP based interleavers are shown. Higher order PP based interleavers have also been investigated for better performance and implementation, in particular for cubic permutation polynomial (CPP) based interleavers [2,3]. However little is known for minimum distance of turbo codes with higher order PP based interleavers. In this letter, the technique shown in [3] is used to decompose higher order PPs into linear permutation polynomials (LPPs) and partial upper bounds on the minimum distance for turbo codes using higher order PP based interleavers are shown. It is also shown that when the constant terms of the LPPs which are equivalent to PPs are dithered, better frame error rate (FER) performance is obtained. For a more succinct writing, in the following, PP based interleavers are denoted as PP.
In this section, previous results on higher order PPs are briefly reviewed and upper bounds on the minimum distance for turbo codes using PPs are shown. Firstly, the equivalence of PPs and a family of LPPs is shown. In the following, a parallel LPP (PLPP) is defined.
∗ Corresponding author. Tel.: +40 232701679. E-mail addresses: [email protected] (J. Ryu), [email protected] (L. Trifina), [email protected] (H. Balta).
Definition 2.1.
p(x) =
[3] Let p(x) be an interleaver such that
⎧ p0 (x) = P1,0 x + P0,0 , mod(x, L) = 0 ⎪ ⎪ ⎪ ⎨ p1 (x) = P1,1 x + P0,1 ,
mod(x, L) = 1
⎪ ⎪ ··· ⎪ ⎩
pL−1 (x) = P1,L−1 x + P0,L−1 ,
mod(x, L) = L − 1,
which can be also represented in the following form,
p(x) =
⎧ p0 (y) = P1,0 · Ly + P0,0 , x = Ly ⎪ ⎪ ⎪ ⎨ p1 (y) = P1,1 · Ly + P0,1 ,
⎪ ⎪ ··· ⎪ ⎩
x = Ly + 1
pL−1 (y) = P1,L−1 · Ly + P0,L−1 ,
x = Ly + (L − 1),
with 1 ≤ L < N, where N is the interleaver length, L|N and 0 ≤ y ≤ N − 1. Then p(x) is called a PLPP (i.e., p(x) consists of L LPPs). L For each l = 0, 1, · · · , L − 1, pl (y) is a LPP and since a LPP can be implemented using only additions and comparisons, a PLPP can also
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2
Table 1 The least numbers of LPPs for equivalent PLPPs (L) of LTE-QPPs of various lengths N [5]. Lengths range
L=1
L=2
L=3
L=4
L>4
40–512 528–1024 1056–2048 2112–6144
1
55 21 1
3 3 1 1
6 26 48
1 2 4 15
60 32 32 64
77
8
80
22
188
1
be implemented using the same address generation method for a LPP [3]. Although not all PPs are equivalent to PLPPs [3], the following lemma shows that all PPs are equivalent to PLPPs when the interleaver lengths are of the form N = 23 · M, where M is a positive integer.
K
Lemma 2.2. [3] Let f (x) = f xk (mod N) be a PP. Suppose that k=1 k N = 23 · M, with M a positive integer. Then f(x) is equivalent to a PLPP and L ≤ 2M. Lemma 2.2 is obtained by computing the f(x) at each point x = 2My + l and using the modulo operation and the zero polynomials shown in [4,3] to remove quadratic and higher order terms. In particular, a sufficient condition for a QPP to have an equivalent PLPP is given in [3]. By the lemma 2.2, all the LTE-QPPs are equivalent to PLPPs, since their interleaver lengths are multiples of 8. In practice, L’s are relatively small numbers compared to interleaver lengths, as shown in Table 1. For example, let f(x) = 15x + 32x2 (mod256), then f(2y) = 15 · 2y + 32 · (2y)2 = 15 · 2y + 32 · (2y)2 + 256/2 · y + 256/2 · (y)2 = 79 · 2y, f(2y + 1) = 15 · (2y + 1) + 32 · (2y + 1)2 = 15 · (2y + 1) + 32. Note that (N/2) · y + (N/2) · y2 is a zero polynomial for all y. Thus, the equivalent PLPP, with L = 2, of the previous QPP f(x), is:
p(x) =
p0 (x) = 79x, mod(x, 2) = 0 p1 (x) = 15x + 32, mod(x, 2) = 1
Let L = 4, then by using a similar method, the equivalent PLPP of the previous QPP f(x), is:
p(x) =
⎧ ⎪ ⎨ p0 (x) = 15x, mod(x, 4) = 0 p1 (x) = 15x + 32,
mod(x, 4) = 1
p3 (x) = 15x + 32,
mod(x, 4) = 3
⎪ ⎩ p2 (x) = 15x + 128, mod(x, 4) = 2
Since L is relatively small compared to the interleaver length for QPPs in [5], the interleaver/deinterleaver for the PPs can be efficiently generated as shown in [3]. Note that the number of
coefficients of the L LPPs depends only on L, not on the degree of PP. Thus a PP of arbitrary degree can be implemented using L LPPs if it is equivalent to L LPPs. In the following, upper bounds on minimum distance for turbo codes with PPs using Lemma 2.2 are shown. Lemma 2.3. Let the first coefficients of PLPP be equal for all l, i.e., P1,0 = P1,1 = . . . = P1,L−1 = P. Let also m and n be positive integers and L|(m · (2 − 1)), where is the degree of the primitive feedback and monic feedforward polynomials of recursive systematic convolutional codes, which are component codes of a conventional turbo code. If there exists a critical interleaver pattern of size 4 as shown in Fig. 1, the minimum distance of the turbo code with this PLPP interleaver is upper bounded by (m + n) · 2 + 12. Proof. Consider the constituent codewords 1 and 2 generated by the interleaver pattern, both containing two fundamental paths with input sequences of weight 2 as shown in Fig. 1, where there are two error patterns with input sequences of weight 2 at points xi , xi + m · (2 − 1), and xj , xj + m · (2 − 1) respectively. It is easy to check that the weight of codeword generated by the constituent code 1 is 2 · (m · 2−1 + 2). Let us consider the error sequence with an input sequence of weight 2 xi , xi + m · (2 − 1). Since the distance between the two points is m · (2 − 1) and L|(m · (2 − 1)), the two points are on the same ith LPP. Thus, each point is mapped to Pxi + Li and P(xi + m · (2 − 1)) + Li , respectively. Since the input sequences for the constituent codes 1 and 2 are mapped by an interleaver, there is a point in the input for the constituent code 1 that is mapped to the point Pxi + Li + n · (2 − 1) in the input for the constituent code 2. Let us call it xj . Then Pxj + Lj = Pxi + Li + n · (2 − 1). Since the distance between the points xj , xj + m · (2 − 1) is m · (2 − 1) and L|(m · (2 − 1)), the two points are in the same jth LPP. Finally, P(xj + m · (2 − 1)) + Lj = Pxj + Lj + Pm · (2 − 1) = Pxi + Li + (Pm + n) · (2 − 1), which is equal to P(xi + m · (2 − 1)) + Li + n · (2 − 1). Thus, an input sequence of weight 4 exists for PLPP with L LPPs and the weight of the corresponding codeword is 2 · (m · 2−1 + 2) + 2 · (n · 2−1 + 2) + 4 = (m + n) · 2 + 12. 䊐 It should be mentioned that the upper bound in Lemma 2.3 assumes that the first coefficients of PLPP are all equal. This constraint was also imposed for the computation of L for LTE-QPPs in Table 1. In Table 2, upper bounds on the minimum distance for turbo codes with PPs when = 3 are shown. The result in Lemma 2.3 is similar to Tables II and III in [1], however, Lemma 2.3 can also be applied to higher order PPs.
constituent code 1
xi
x i+m(2v−1)
xj
x j+m(2v−1)
interleaver
Pxi +Li
Pxi +Li +n(2 v−1)
P(xi+m(2v−1))+L i P(xi+m(2 v−1))+Li +n (2v−1)
constituent code 2 Fig. 1. Critical interleaver pattern of size 4 (Fig. 3 in [1]).
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J. Ryu et al. / Int. J. Electron. Commun. (AEÜ) xxx (2015) xxx–xxx Table 2 Upper bounds (UBs) for the minimum distance of turbo codes using PP based interleavers. Generator matrix of recursive systematic convolutional codes is as in the LTE turbo code [5]. L
m
n
UB
L
m
n
UB
1 or 7 2 or 14 3 or 21 2 or 14
1 2 3 2
1 1 1 2
28 36 44 44
4 or 28 5 or 35 6 or 42 8 or 56
4 5 6 8
1 1 1 1
52 60 68 84
The results in Table 2 show that the minimum distance of turbo codes with PPs is upper bounded by a function of L. Thus, for higher order PPs, L should also be considered in searching for good interleavers when the PPs are implemented using LPPs.
3
length is essentially composed of LPPs and the set of LPPs set a limit on the minimum distance of the turbo code with PP [3]. In other direction, the constant terms of LPPs which are in decomposition of PPs can be dithered, i.e., a PP is represented using the equivalent LPPs and the constant terms of the LPPs are dithered. PPs are highly structured so that the regularity of PP can lower the error performance. By dithering PPs, the edge-effect can be avoided. Good dithered LPPs for some lengths were searched. For lengths smaller or equal than 560, L = 2, and for the others, L = 4. Dithered LPP interleavers in the paper were not found in purely random way, but using the rules from [8] or [9], supplemented with simulations. The goal was to find interleavers better than LTE-QPPs or other good QPPs or CPPs. 3.1. Comparison between dithered LPPs and good QPPs or CPPs from the point of view of the minimum distance and simulated FER at high SNR
3. Dithered LPPs Increasing the order of PPs may give better FER performance. However, it also increases the number of computations and required storage for the coefficients. QPPs, CPPs and higher order PPs can be decomposed into PLPPs. In the previous section, it was shown that the minimum distance of turbo codes with PPs is upper bounded by a function of the number of LPPs (L). If CPPs or higher order PPs’ performance is good, they can be efficiently implemented using small number of parallel LPPs. But if not, the performance of turbo codes with PPs may not necessarily improve as the order of polynomial increases. The reason is that any PP with practical
In Tables 3 and 4, the simulation results are shown in terms of FER at high SNR values for LTE-QPPs and for the found dithered LPPs. For simulation, the maximum a posteriori (MAP) algorithm with an iteration stopping criterion based on the maximum absolute value of logarithm likelihood ratio (LLR) was used. The threshold of LLR was set to 15 and the channel model was considered with additive white Gaussian noise (AWGN). At least 70 erroneous frames were simulated for each interleaver at SNR given in Tables 3 and 4. For all reported lengths the FER of dithered LPPs is lower than that for
Table 3 Minimum distances (dmin ), corresponding multiplicities of codewords (Ndmin ) and FER at high SNR for LTE-QPP [5] and dithered LPP interleavers for lengths N ≤ 448. Generator matrix of recursive systematic convolutional codes and trellis termination are as in the LTE turbo code [5]. N
SNR [dB]
(x) (LTE-QPP [5])
dmin /Ndmin
FER (×10−6 )
(x) (Dithered LPP)
dmin /Ndmin
FER (×10−6 )
40 48 56 64 72 88 96 112 128 136 152 200 224 280 320 448
5 5 5 5 5 5 4 4 3.5 3.5 3.5 3 2.75 2.5 2.5 2.5
3x + 10x2 7x + 12x2 19x + 42x2 7x + 16x2 7x + 18x2 5x + 22x2 11 + 24x2 41 + 84x2 15 + 32x2 9 + 34x2 9x + 38x2 13x + 50x2 27x + 56x2 103x + 210x2 21x + 120x2 29x + 168x2
11/1 13/1 13/1 12/1 15/1 15/1 16/2 17/2 16/1 16/3 15/1 20/1 22/99 21/1 20/1 22/105
11.7724 2.6327 3.4949 1.2373 0.3567 2.9358 2.1871 0.8782 3.3163 5.8068 4.7129 1.6036 10.3331 21.1729 3.0100 17.9973
33x, 33x + 28 19x, 19x + 6 9x, 9x + 18 53x 23x 31x, 31x + 80 61x 15x, 15x + 54 49x 87x 63x, 63x + 102 13x, 13x + 146 185x, 185x + 72 87x, 87x + 274 49x, 49x + 210 15x, 15x + 152
13/1 15/1 16/2 17/1 19/6 16/1 19/1 19/1 19/1 20/3 19/1 23/1 23/1 27/3 23/1 25/1
2.5534 0.9787 0.3466 0.1716 0.1520 1.4006 1.2937 0.3567 1.2232 1.4123 0.5288 0.7663 1.4753 2.3912 1.6460 1.1631
Table 4 Minimum distances (dmin ), corresponding multiplicities of codewords (Ndmin ) and FER at high SNR for LTE-QPP [5] and dithered LPP interleavers for lengths N ≥ 560. Generator matrix of recursive systematic convolutional codes and trellis termination are as in the LTE turbo code [5]. For some lengths, in the fourth and seventh columns, in parentheses, 107 ×TUB(FER) values are given for AWGN channel and distance spectrum with maximum distance given in the second column, in parentheses. N
SNR [dB] (dmax )
(x) (LTE-QPP [5])
dmin /Ndmin (107 × TUB(FER))
560
2.2 (39) 1.66
227x + 420x2
29/1 (5.7142) 27/1
848
1.58 (39)
239x + 106x2
29/2 (3.4249)
6.6300
992
1.6
65x + 124x2
22/2
7.8000
1.56 (31)
55x + 84x2
30/322 (148.78)
784
1008
25x + 98x2
FER (×10−6 ) 2.2400 11.0600
11.6600
(x) (Dithered LPP)
dmin /Ndmin (107 × TUB(FER))
FER (×10−6 )
367x, 367x + 484 19x, 19x, 19x + 552, 19x + 488 19x, 19x, 19x + 327, 19x + 761 19x, 19x, 19x + 624, 19x + 568 19x, 19x, 19x + 563, 19x + 149
25/1 (6.1668) 32/1
1.8923
28/1 (3.3580)
5.6668
33/1
5.7500
27/1 (4.0043)
6.7000
3.7500
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Table 5 Minimum distances (dmin ), corresponding multiplicities of codewords (Ndmin ) and FER at high SNR for optimum dmin QPP [6] and dithered LPP interleavers. Generator matrix of recursive systematic convolutional codes and trellis termination are as in the LTE turbo code [5]. For some lengths, in the fourth and seventh columns, in parentheses, 107 ×TUB(FER) values are given for AWGN channel and distance spectrum with maximum distance given in the second column, in parentheses. N
SNR [dB] (dmax )
(x) (Optimum dmin QPP [6])
dmin /Ndmin (107 × TUB(FER))
40 48 56 64 72 88 96 112 128 136 152 200 224 280 320
5 5 5 5 5 5 4 4 3.5 3.5 3.5 3 2.75 2.5 2.5 (35) 2.5 (35)
13x + 10x2 19x + 12x2 27x + 14x2 21x + 8x2 7x + 18x2 5x + 22x2 7 + 24x2 Not given in [6] 37 + 96x2 9 + 102x2 9x + 38x2 13x + 150x2 61x + 168x2 Not given in [6] 99x + 240x2
12/1 12/3 14/21 14/1 15/1 15/1 16/2
448
55x + 336x2
17/1 14/1 15/1 20/1 21/1 23/1 (1.591) 28/1 (0.166)
For the three previously mentioned lengths with exceptions, in Table 4, the distance spectra computed for turbo codes with corresponding interleavers and the truncated upper bounds of FER for AWGN channel, denoted TUB(FER), are presented. TUB(FER) for AWGN channel [11] is given by numdist
Nd (i) · erfc(d(i) · Rc · SNR),
(1)
i=1
where numdist is the number of distances in the distance spectrum of the turbo code, d(i) is the ith distance in the distance spectrum (d(1) = dmin ), Nd (i) is the corresponding codeword multiplicity (Nd (1) = Ndmin ), Rc is the coding rate of the turbo code, SNR is the signal-to-noise ratio and erfc(·) is the complementary error function, given by 2 erfc(x) = √ ·
∞
2
e−t dt.
dmin /Ndmin (107 × TUB(FER))
FER (×10−6 )
1.4693
33x, 33x + 28 19x, 19x + 6 9x, 9x + 18 53x 23x 31x, 31x + 80 61x 15x, 15x + 54 49x 87x 63x, 63x + 102 13x, 13x + 146 185x, 185x + 72 87x, 87x + 274 49x, 49x + 210
2.5534 0.9787 0.3466 0.1716 0.1520 1.4006 1.2937 0.3567 1.2232 1.4123 0.5288 0.7663 1.4753 2.3912 1.6460
3.2486
15x, 15x + 152
13/1 15/1 16/2 17/1 19/6 16/1 19/1 19/1 19/1 20/3 19/1 23/1 23/1 27/3 23/1 (1.221) 25/1 (0.525)
3.6714 6.3377 4.7129 1.6139 32.8682
1. The dithered LPP has a smaller minimum distance and lower simulated FER than the QPP/CPP. 2. The dithered LPP has an equal minimum distance with higher or equal multiplicity and lower simulated FER than QPP/CPP. 3. The dithered LPP has a higher minimum distance and higher simulated FER than the QPP/CPP. 4. The dithered LPP has a equal minimum distance with smaller or equal multiplicity and higher simulated FER than the QPP/CPP.
(x) (Dithered LPP)
5.3434 3.8845 7.1716 1.0681 0.3567 2.9358 1.7141
LTE-QPPs. In Tables 3 and 4, the minimum distances (dmin ) and the corresponding multiplicities of codewords (Ndmin ) for LTE-QPPs and for dithered LPPs are also given. The post-interleaver flushing trellis termination was used, as in LTE turbo code [5]. To find the minimum distances or the distance spectra of turbo codes the method in [10] was used. From Tables 3 and 4 it is observed that minimum distances for dithered LPPs are higher than those for LTE-QPPs for all lengths, excepting three of them (namely 560, 848 and 1008). This shows that the dithering constant term of a PLPP may increase the minimum distance of turbo codes with a such interleaver. Further, exceptions have been considered when:
TUB(FER) = 0.5 ·
FER (×10−6 )
(2)
x
The maximum distance in each distance spectrum is given in parentheses in the second column of Table 4. The 107 ×TUB(FER) values are given in parentheses in the fourth and seventh columns, respectively. It is seen that TUB(FER) for LTE-QPPs is greater than
1.1631
or very close to TUB(FER) for dithered LPPs. This explains the difference or similarity in simulated FER performance. In [1] the minimum distances for all LTE-QPPs are given, but dual trellis termination [12] was considered. With LTE trellis termination the minimum distance decreases considerably compared to dual termination. In [6] QPPs with optimum minimum distance for a large number of lengths N between 32 and 512 and for N = 1024 were obtained. Because in [6] also dual termination was used, the minimum distances and the corresponding multiplicities of codewords for QPPs from Table I in [6], with lengths that are also in Table 3, were computed, when LTE trellis termination is used. It is mentioned that QPPs with optimum minimum distances for dual termination do not necessarily have optimum minimum distances for LTE trellis termination. The results are given in Table 5 along with those for dithered LPPs. Simulated FER values for QPPs in [6] with previously mentioned lengths are also reported. Excepting length 448, for all other lengths in Table 3 our dithered LPPs lead to higher or equal minimum distances compared to those of QPPs from [6] with LTE trellis termination. In addition, simulated FER for dithered LPPs is lower than FER for QPPs from [6] for all reported lengths, excepting the length of 320 for which FER values are very close to each other. For these two lengths with exceptions (320 and 448) the 107 ×TUB(FER) values and the maximum distance in the distance spectra as in Table 4, are given. For the length of 320 the TUB(FER) values are very close to each other, as shown by the simulated FER values. For interleavers of length 448, we have assumed that besides the distance spectrum other parameters influence FER performance. It should be mentioned that a high minimum distance is not sufficient for a good performance of a turbo code. In [2,7,8] few methods for finding good QPP or CPP based interleavers are given. These methods also take into account other parameters of QPP/CPP based interleavers (such as D parameter or spread and refined nonlinearity degree). In Tables 6 and 7, the our dithered LPPs are compared with the QPPs from [2] or [7,8] and the CPPs from [2], for some lengths, considering the simulated FER and the minimum distance. It is observed that for all reported lengths dithered LPPs lead to higher or equal minimum distances (excepting CPP of length 40 and QPPs of lengths 320, 560, 848 and 1008, respectively). The dithered LPPs also lead to lower simulated FER than QPPs or CPPs, excepting few cases (CPPs of lengths 48 and 88 and QPP of length 320). For these lengths with exceptions, the 107 ×TUB(FER) values were computed, as shown in Tables 4 and 5. For lengths 448, 560
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Table 6 Minimum distances (dmin ), corresponding multiplicities of codewords (Ndmin ) and FER at high SNR for QPP from [2] or [8], CPP from [2] and dithered LPP interleavers for lengths N ≤ 200. Generator matrix of recursive systematic convolutional codes and trellis termination are as in the LTE turbo code [5]. For some lengths, in the fourth and seventh columns, in parentheses, 107 ×TUB(FER) values are given for AWGN channel and distance spectrum with maximum distance given in the second column, in parentheses. N 40 48 56 64 72 88 96 112 128 136 152 200
SNR [dB] (dmax )
(x) (QPP from [2] or [8] and CPP from [2])
dmin /Ndmin (107 × TUB(FER))
FER (×10−6 )
(x) (Dithered LPP)
dmin / Ndmin (107 × TUB(FER))
FER (×10−6 )
5 5 (37) 5 5 (39) 5 5 5 5 5 5 5 5 (40) 4 4 4 4 3.5 3.5 3.5 3.5 3.5 3 3
13x + 10x2 [2] 3x + 8x2 + 16x3 7x + 36x2 [2] 5x + 6x2 + 12x3 3x + 42x2 [2] 5x + 14x2 + 42x3 7x + 16x2 [2] 5x + 24x2 + 48x3 5x + 54x2 [8] 7x + 0x2 + 4x3 5x + 22x2 [2] 27x + 22x2 + 66x3 13x + 72x2 [2] 9x + 12x2 + 56x3 41x + 28x2 [2] 41x + 0x2 + 28x3 17 + 32x2 [2] 19 + 102x2 [2] 19x + 0x2 + 34x3 59x + 38x2 [2] 59x + 0x2 + 114x3 13x + 150x2 [2] 3x + 0x2 + 80x3
12/1 15/3 (4.44) 15/2 15/3 (1.59) 13/1 14/23 12/1 15/1 16/2 16/2 15/1 15/1 (0.325) 16/1 15/1 17/2 17/2 18/1 16/1 16/1 16/1 16/1 20/1 20/1
5.3434 4.5677 2.6038 0.9258 3.4457 2.6896 1.2373 0.3527 0.3063 0.6231 2.9358 0.1576 1.5356 1.7407 0.8507 0.7587 1.5770 4.0654 4.1556 3.5498 4.6401 1.6139 2.2218
33x, 33x + 28 19x, 19x + 6 9x, 9x + 18 53x
13/1 (7.553) 15/1 (2.039) 16/2
2.5534
17/1
0.1716
23x
19/6
0.1520
31x, 31x + 80 61x
16/1 (0.104) 19/1
1.4006
15x, 15x + 54 49x 87x
19/1
0.3567
19/1 20/3
1.2232 1.4123
63x, 63x + 102 13x, 13x + 146
19/1
0.5288
23/1
0.7663
and 1008, the dithered LPPs have smaller TUB(FER) values and also lower simulated FER. For lengths 48, 320 and 848, TUB(FER) values are close to each other as simulated FER values. As for length of 448 in Table 5, we have assumed that other parameters besides the distance spectrum influence FER performance, for interleavers of lengths 40 and 88. 3.2. Simulation results To see the FER behaviour of dithered LPPs in a wider range of SNR, the simulation results for several lengths (N = 40, N = 64 and N = 448) are given below. In Fig. 2, FER curves for interleavers of length N = 40 are given. The simulated interleavers are LTE-QPP, optimum dmin QPP from [6] (which is the same as QPP with largest
0.9787 0.3466
1.2937
spread and best distance spectrum from [2]), CPP with largest spread and best distance spectrum from [2] and dithered LPP. It is observed that at high SNR, dithered LPP has the best performance, CPP from [2] has performance closer to dithered LPP, excepting for SNR=5 dB, LTE-QPP has the weakest performance and QPP from [6] or [2] has a medium performance. In Fig. 3, FER curves for interleavers of length N = 64 are given. Simulated interleavers are LTE-QPP (which is the same as QPP with largest spread and best distance spectrum from [2]), optimum dmin QPP from [6], CPP with largest spread and best distance spectrum from [2] and dithered LPP. Again at high SNR, dithered LPP has the best performance. LTE-QPP and QPP from [6] have the weakest performance and CPP from [2] have performance closer to dithered LPP one, excepting for SNR = 5 dB.
Table 7 Minimum distances (dmin ), corresponding multiplicities of codewords (Ndmin ) and FER at high SNR for QPP from [2] or [7,8] and dithered LPP interleavers for lengths N ≥ 224. Generator matrix of recursive systematic convolutional codes and trellis termination are as in the LTE turbo code [5]. For some lengths, in the fourth and seventh columns, in parentheses, 107 ×TUB(FER) values are given for AWGN channel and distance spectrum with maximum distance given in the second column, in parentheses. N
SNR [dB] (dmax )
(x) (QPP from [2] or [7,8])
dmin /Ndmin (107 × TUB(FER))
FER (×10−6 )
(x) (Dithered LPP)
dmin /Ndmin (107 × TUB(FER))
FER (×10−6 )
224
2.75
22/98
10.6936
1.4753
2.5
22/127
32.2522
27/3
2.3912
320
2.5 (35) 2.5 (35) 2.2 (39) 1.66
185x, 185x + 72 87x, 87x + 274 49x, 49x + 210 15x, 15x + 152 367x, 367x + 484 19x, 19x, 19x + 552, 19x + 488 19x, 19x, 19x + 327, 19x + 761 19x, 19x, 19x + 624, 19x + 568 19x, 19x, 19x + 563, 19x + 149
23/1
280
27x + 168x2 [2] 17x + 210x2 [2] 21x + 80x2 [2] 139x + 112x2 [7,8] 37x + 420x2 [8] 25x + 294x2 [8]
23/1 (1.221) 25/1 (0.525) 25/1 (6.1668) 32/1
1.6460
28/1 (3.3580)
5.6668
33/1
5.7500
27/1 (4.0043)
6.7000
448 560 784
25/1 (0.912) 25/1 (3.285) 29/1 (8.3184) 29/1
848
1.58 (39)
185x + 318x2 [8]
29/1 (3.4930)
992
1.6
65x + 124x2 [8]
22/2
1.56 (31)
55x + 588x2 [8]
29/1 (149.53)
1008
1.2546 1.8168 6.1013 17.2973
13.0017
7.8000
17.4220
1.1631 1.8923 3.7500
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0
10
10
LTE−QPP ( 29x+168x2 (mod 448) ) Optimum dmin−QPP [6] ( 55x+336x2 (mod 448) )
2
LTE−QPP ( 3x+10x (mod 40) ) Optimum d −QPP [6] min
2
−1
−1
or QPP from [2] ( 13x+10x (mod 40) ) CPP from [2] ( 3x+8x2+16x3 (mod 40) ) DLPP (P1,0=P1,1=33, P0,0=0, P0,1=28, N=40)
10
−2
−2
10
FER
FER
10
−3
10
−3
10
−4
−4
10
−5
10
10
−5
10
−6
10
−6
10
0
2
QPP from [7] ( 139x+112x (mod 448) ) DLPP (P1,0=P1,1=15, P0,0=0, P0,1=152, N=448)
10
1
2
3
4
5
0
0.5
1
1.5
2
2.5
SNR [dB]
SNR [dB] Fig. 2. FER curves for interleavers of length N = 40. Generator matrix of recursive systematic convolutional codes and trellis termination are as in the LTE turbo code [5]. DLPP stands for dithered LPP.
Fig. 4. FER curves for interleavers of length N = 448. Generator matrix of recursive systematic convolutional codes and trellis termination are as in the LTE turbo code [5]. DLPP stands for dithered LPP.
Finally, in Fig. 4, FER curves for interleavers of length N = 448 are given. Simulated interleavers are LTE-QPP, optimum dmin QPP from [6], QPP with largest spread and best distance spectrum from [7,8] (in [2] only QPP/CPP interleavers up to length of 352 are given) and dithered LPP. It is observed that at high SNR, dithered LPP and QPP from [7,8] have the best performance. LTE-QPP has weakest performance and QPP from [6] has a medium performance among the simulated interleavers, although it has the biggest minimum distance.
51). It is shown in this letter that for every PP (not just QPPs), the upper bound on the minimum distance is 52 when L = 4. Although the PPs that achieve the upper bound 52, have not been identified, this shows that the upper bound will increase only with 1 or 2 for any degree of polynomials when L = 4, i.e., no large difference in terms of minimum distance. This partly explains why the L of good QPPs is larger than 4 when the interleaver length is large. By increasing the order of the PPs, better PPs may be found as shown in the example, although there may not be large difference in minimum distance when L = 4. However, PPs have the maximum contention-free property which enables flexible hardware design. Our search in finding good interleavers may be constrained in a pool of PPs. Since all the practical PPs are equivalent to PLPPs, if our search is constrained in finding good PPs with small L, they can be generated in an efficient way. Thus, if a maximum contention free property is desired, PPs may be used and if better error performance is desired, dithered LPPs may be used. When L was chosen properly, both PPs and dithered LPPs can be generated in a same way. The approaches for using PPs or dithered LPPs in turbo codes are given below:
3.3. The usefulness of the partial upper bound and approaches for using PPs or dithered LPPs in turbo codes The upper bound in this letter shows some interesting points, specifically, for L = 4. In [1], it is shown that the upper bound on the minimum distance of turbo codes with QPP based interleavers, when QPPs have a quadratic inverse, is 50 (and for some QPPs with non-QPP inverse,
0
10
1. If there exist CPPs or higher order PPs with performance better than QPPs, figure out if the number of LPPs is small. 1-1) If there exist CPPs or higher order PPs, then they can be decomposed into a family of LPPs. 1-2) If there is no such CPPs or higher order PPs, then dithered LPPs can be used. 2. Instead of storing the coefficients of PPs, the ones of PLPPs can be stored, for both cases 1-1 and 1-2.
2
LTE−QPP or QPP from [2] ( 7x+16x (mod 64) ) Optimum dmin−QPP [6] ( 21x+8x2 (mod 64) )
−1
10
2
3
CPP from [2] ( 5x+24x +48x (mod 64) ) DLPP (P1,0=P1,1=53, P0,0=0, P0,1=0, N=64)
−2
10
−3
FER
10
−4
10
In this way, algebraic interleavers for turbo codes with good performance, less storage and computation are available.
−5
10
4. Conclusions −6
10
−7
10
0
1
2
3
4
5
SNR [dB] Fig. 3. FER curves for interleavers of length N = 64. Generator matrix of recursive systematic convolutional codes and trellis termination are as in the LTE turbo code [5]. DLPP stands for dithered LPP.
Using the fact that PPs over integer rings are decomposed into LPPs, it was shown that the upper bound on minimum distance for turbo codes with PPs is a function of the number of LPPs. This shows that 2 or 4 LPPs can be used for short interleaver lengths, but for medium to long lengths, 4 or 8 LPPs can be used. Dithered LPPs have been also proposed, where only the constant terms of equivalent LPPs for a PP are dithered. It was shown that for many short to
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medium lengths, dithered LPPs outperform in terms of simulated FER at high SNR, LTE-QPPs and other good QPPs, such those from [6] (when LTE trellis termination is used), [2] or [7,8] or CPPs from [2]. An analysis of reported interleavers in terms of the minimum distance or the truncated upper bound of FER was done. This shows that dithered LPPs lead, in most of the cases, to higher minimum distances or smaller TUB(FER) values. The usefulness of the upper bound derived in the paper was shown and approaches for using PPs or dithered LPPs in turbo codes were given. Acknowledgments The work of Horia Balta was supported by a grant of the Romanian Ministry of Education, CNCS – UEFISCDI, project number PN-II-RU-PD-2012-3-0122. The authors would like to thank the anonymous reviewers for their helpful comments and suggestions that improved the paper. References [1] Rosnes E. On the minimum distance of turbo codes with quadratic permutation polynomial interleavers. IEEE Trans Inform Theory 2012;58(July (7)):4781–95.
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[2] Trifina L, Tarniceriu D. Analysis of cubic permutation polynomials for turbo codes. Wirel Pers Commun 2013;69(March (1)):1–22. [3] Ryu J. Permutation polynomials of higher degrees for turbo code interleavers. IEICE Trans Commun 2012;E95-B(December (12)):3760–2. [4] Singmaster D. On polynomial functions (mod m). J Number Theory 1974;6(October (5)):345–52. [5] 3GPP TS 36.212 v8.3.0 (2008–05) E-UTRA: Multiplexing and channel coding (Release 8). [6] Rosnes E, Takeshita O. Optimum distance quadratic permutation polynomialbased interleavers for turbo codes. In: IEEE international symposium on information theory (ISIT 2006). 2006. p. 1988–92. [7] Trifina L, Tarniceriu D, Munteanu V. Improved QPP interleavers for LTE standard. In: IEEE international symposium on signals, circuits and systems (ISSCS 2011). 2011. p. 403–6. [8] Trifina L, Tarniceriu D. Improved method for searching interleavers from a certain set using Garello’s method with applications for the LTE standard. Ann Telecommun 2014;69(June (5–6)):251–72. [9] Balta H, Kovaci M, Nafornita M, Balta M. Multi-binary turbo-code design based on convergence of iterative turbo-decoding process. In: 5th European conference on circuits and systems for communications (ECCSC’10). 2010. p. 240–3. [10] Garello R, Pierleoni P, Benedetto S. Computing the free distance of turbo codes and serially concatenated codes with interleavers: algorithms and applications. IEEE J Sel Areas Commun 2001;19(May (5)):800–12. [11] Proakis JG. Digital communications. 3rd ed. New York: McGraw-Hill; 1995. [12] Guinand P, Lodge J. Trellis termination for turbo encoders. In: Proc. 17th biennial symp. commun. 1994. p. 389–92.
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