Modified Ω' Metric For QPP Interleavers Depending On SNR

Lucian Trifina1, Lucian Ghercă1, Bogdan Lupu1, Ana-Mirela Rotopănescu 1

1 Faculty of Electronics, Telecommunications and Information Technologies, Technical University “Gh. Asachi” Iasi, Bd. Carol I no. 11, 700506, Romania , luciant@etc.tuiasi.ro

Abstract— In this paper a modified Ω' metric, depending on signal to noise ratio (SNR), is proposed. This metric is more suitable for choosing of quadratic permutation polynomial (QPP) interleavers, because it is closer to the upper bound of frame error rate (FER). The minimizing of this metric leads to the better choice of the polynomial from the error rate point of view, between an interleaver with maximum-spread and one with the best classical Ω' metric. The simulations for two lengths of interleaver and two component codes confirm this metric.

I. INTRODUCTION The polynomial interleavers are the most recent interleavers

from literature having the following advantages [1]: excellent performance, completely algebraic structure and efficient implementation (high speed and little memory requirements). Among these the QPP interleavers are simplest.

A QPP interleaver of length L is defined in [1]:

π(x)=(q0+q1x+ q2x2) mod L, x= 0, 1L − (1)

where q1, q2 are chosen so that the quadratic polynomial from (1) to be a permutation polynomial, q0 having the effect of a cyclic shift of the permutation elements.

In [1] QPP interleavers are found that are designed based on certain parameters, specific for the interleavers only, which do not depend on the global turbo code. The parameters of QPP interleavers used in the paper are: the nonlinearity degree, the refined nonlinearity degree, the degree of shift invariance, the spread factor and Ω’ metric.

The nonlinearity degree of a QPP interleaver is achieved by measuring the number of distinct orbits (a set of points) of the action of an isometry group on the interleaver code, that is, on the points (x, π(x)). It is demonstrated that the nonlinearity degree of a QPP interleaver is given by the relation:

ζ=L/gcd(2q2,L) (2)

“gcd” meaning „greatest common divisor”. The refined nonlinearity degree ζ' is given by the number of

distinct elements of set q2x2 (mod L), x=0,1, ..., ζ-1. Takeshita states in [1] that the first spectral line with high

multiplicity in distances spectrum of a turbo code with a QPP interleaver is very close to the degree of shift invariance ε, defined as the size of the orbits. The relation between ε and ζ is the following:

ζ =L/ε (3)

Spread factor (D parameter) is defined by relation:

D= ( ) ,

min ,L

L i ji ji j

p pδ≠

∈

, (4)

where ( ),L i jp pδ is Lee metric between the points

( )( ),ip i iπ= and ( )( ),jp j jπ= :

( ) ( ) ( ),L i j L Lp p i j i jδ π π= − + − . (5)

The notation Li j− means:

( )( ) ( )( ) min mod , modLi j i j L j i L− = − − . (6) In [1] are given the quadratic polynomials which have the

largest spread (D parameter) and also which maximize Ω’ metric defined as:

Ω’= ζ’·ln(D), (7)

The simulations results for different length of interleavers

show that in some cases better results are obtained with interleavers which have maximum D parameter maximum and in other cases with interleavers which have maximum Ω’ metric. From this fact a problem arises: how can one find the better interleaver by a single combined metric?

The paper is organized such as follows: in section II a modified Ω’ metric, that depends on SNR, is proposed and the founded QPP are given, in section III are given the simulations results and in section IV the conclusions.

II. THE MODIFIED METRIC Takeshita [1] explains the proposal of Ω’ metric by the fact

that it realizes a compromise between the code words multiplicity, controlled by the refined nonlinearity degree (ζ') and the free effective distance (the minimum Hamming distance between code words generated on information sequences of two weights), controlled by D parameter. The use of the

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logarithm is explained by fact that the minimum distance of a turbo code doesn’t increase more than the logarithm of the interleaver length [2], and the D parameter increases by some degree with the length.

At the basis of the metric proposed by Takeshita stands the upper bound of frame error rate (FER) which for the first M distances from distances spectrum of a code is:

( )~

1

0.5M

i i ci

FER N erfc d R SNR=

< ⋅ ⋅ ⋅ ⋅∑ (8)

where id represents the Hamming distance between the code

words and „all-zero” word or the weight of that code word (the turbo code being linear), iN represents the multiplicity of code words of weight id (i.e. the number of such code words), Rc is the coding rate, and the complementary error function is defined bellow:

( ) 22 t

x

erfc x e dtπ

∞−= ⋅ ∫ (9)

being a decreasing function.

It is desired that the frame error rate to be small and therefore it is necessary that the first multiplicities in the distances spectrum to be as small as possible and the first distances to be as high as possible.

Ω’ metric use an approximation of code words multiplicities by the first significantly high multiplicity in the distance spectrum of turbo code and the approximation of the distance by the free effective distance.

Doing a correlation between relations (8) and (7), the following modification on the Ω’ metric is proposed:

Ω’SNR=(L/ζ’)*erfc( ( )3logcR SNR D⋅ ⋅ (10)

From the right side of (8) only one term was retained, the

multiplicity is replaced by the degree of shift invariance, given in (3) and the distance is replaced by the base 3 logarithm of D parameter. The refined nonlinearity degree (ζ') was used because it was shown in [1] that it better describes a QPP interleaver. The base 3 logarithm was used, because in [3] it is shown that the minimum distance of a turbo code doesn’t increase more than the base 3 logarithm of the interleaver length. It is obvious that Ω’SNR metric given in (10), must be minimized (like FER) and not maximized, like classical Ω’ metric. The value of the SNR has to be sufficiently high such as the difference between the performances of various interleavers to be evident. However, as it will be seen, the value of the SNR chosen in (10) is higher than the SNR at which a turbo code can work using an interleaver found by the Ω’SNR metric. For a good performance though, the value of SNR in metric must to be in the “error-floor” region of turbo code.

The exact value of the coding rate Rc depends on the choice of the type of trellis termination. In this paper a post-interleaver trellis termination, with the terminations bits transmitted, was used. In consequence, for a classical turbo code that uses a convolutional component code with m elements of memory, the coding rate is:

3 4cLR

L m=

+ (11)

TABLE I. Ω’SNR METRIC FOR MS-QPP AND Ω’-QPP INTERLEAVERS (EXACTLY CODING RATE)

MS-QPP Ω’-QPP L SNR [dB] π(x) D ζ ΩSNR ζ' Ω’

SNR π(x) D ζ ΩSNR ζ' Ω’SNR

40 5 x+ 10x2 4 2 2.40 2 2.40 x+ 10x2 4 2 2.40 2 2.40 80 4 9x+ 20x2 10 2 2.70 2 2.70 9x+ 20x2 10 2 2.70 2 2.70

128 3.5 15x+ 32x2 16 2 3.58 2 3.58 7x+ 16x2 8 4 3.13 3 4.17 160 3.5 19x+ 40x2 16 2 4.42 2 4.42 9x+ 20x2 10 4 3.23 3 4.30 256 3 15x+ 32x2 16 4 4.42 3 5.89 15x+ 32x2 16 4 4.42 3 5.89 320 3 19x+ 40x2 20 4 4.67 3 6.23 19x+ 40x2 20 4 4.67 3 6.23 400 2.5 17x+ 100x2 20 2 14.72 2 14.72 7x+ 40x2 16 5 6.82 5 6.82 408 2.5 25x+ 102x2 24 2 13.33 2 13.33 25x+ 102x2 24 2 13.33 2 13.33 512 2.5 31x+ 64x2 32 4 6.92 3 9.23 15x+ 32x2 16 8 5.43 4 10.87 640 2.5 39x+80x2 32 4 8.62 3 11.49 19x+ 40x2 20 8 5.85 4 11.69 752 2 31x+ 188x2 32 2 25.80 2 25.80 23x+ 94x2 26 4 14.57 3 19.42 800 2 17x+ 80x2 32 5 10.97 5 10.97 17x+ 80x2 32 5 10.97 5 10.97 1024 2 123x+ 256x2 34 2 33.82 2 33.82 31x+ 64x2 32 8 8.76 4 17.52 1280 2 39x+ 80x2 40 8 9.60 4 19.21 39x+ 80x2 40 8 9.60 4 19.21 1504 1.5 183x+ 376x2 46 2 52.96 2 52.96 23x+ 94x2 26 8 17.89 4 35.77 1600 1.5 49x+ 100x2 50 8 13.48 4 26.97 17x+ 80x2 32 10 13.63 6 22.71 2048 1.25 63x+ 128x2 64 8 17.11 4 34.22 31x+ 64x2 32 16 12.07 7 27.60 2560 1.25 79x+ 160x2 64 8 21.37 4 42.75 39x+ 80x2 40 16 13.48 7 30.82 3200 1.25 79x+ 800x2 80 2 95.76 2 95.76 17x+ 80x2 32 20 15.07 9 33.50 4096 1 173x+ 1024x2 80 2 138.11 2 138.11 31x+ 64x2 32 32 13.30 12 35.46 8192 1 127x+ 256x2 128 16 27.81 7 63.56 31x+ 64x2 32 64 13.30 23 37.00

Searches were performed using the proposed metric (10) up to length of 1024 (for lengths given in [1]) and using the base 3 logarithm of D parameter. The coding rate is computed as in (11) for m=3. The results are not significantly different when m=4.

The founded polynomials are the ones from table I with minimum Ω’SNR metric corresponding to each length. In table I the MS-QPP column contains the interleavers with the largest D parameter, and the Ω’-QPP column contains the interleavers with the largest Ω’ metric. For lengths greater than 1024, due to a high simulation time, Ω’SNR metric was computed only for the MS-QPP and Ω’-QPP interleavers presented in table I, without searching among all QPP interleavers. Table I contains also a ΩSNR metric, where refined nonlinearity degree (ζ') is replaced with simple nonlinearity degree (ζ).

III. SIMULATIONS RESULTS This section gives a comparison between polynomials with

the largest D parameter and polynomials with largest Ω’ metric, from the point of view of bit error rate (BER) and FER performances respectively, in order to verify the proposed metric. For the interleavers with the largest Ω’ metric the corner merit was maximized, as suggested in [1], by the choice of the q0 coefficient. The resulted interleaver is noted Ω’-QPP-MC.

Two lengths of interleavers (160 and 752) was considered and two component codes, the first with 8 states and the second with 16 states, with the generator matrices G=[1, 15/13], and G=[1, 35/23] respectively (octal form). The decoding algorithm is Log-MAP (Logarithmic-Maximum-A-Posteriori) with a stopping criterion of iterations based on absolute value of LLR (Logarithm Likelihood Ratio). The maximum number of iterations is 12 and the threshold of LLR is 10. The simulations used blocks of bits and continued until a given number of errors was accumulated. The given number decreased as the value of SNR increased.

In the figure 1 the simulations results for length of 160 are given. The better performance of MS-QPP interleavers can be observed in both component codes. From the table I can be observed that the Ω’SNR metric for Ω’-QPP-MC interleaver is smaller.

Because the metrics between the two interleavers are very close, in figure 2 it was drawn this metric, for the two interleavers, depending on the SNR. It can be observed that from a certain value of the SNR (which is greater than the one considered in table I) the value of metric for MS-QPP interleaver is smaller than the one for Ω’-QPP-MC interleaver. The conclusion is that when the metrics are very close, the difference between metrics doesn’t reflect the difference between BER (FER) performances especially if the considered SNR value is too small. Also it can be said that ΩSNR metric is not suitable for choosing an interleaver, because the difference between the ΩSNR metrics for MS-QPP and Ω’-QPP interleavers is not reflected in the BER (FER) performances.

0 0.5 1 1.5 2 2.5 310

-7

10-6

10-5

10-4

10-3

10-2

10-1

100

SNR [dB]

BE

R (

solid

), F

ER

(da

shed

)

MS-QPP (19x+40x2)mod160Ω'-QPP-MC (115+9x+20x2)mod160

a)

0 0.5 1 1.5 2 2.5 310

-7

10-6

10-5

10-4

10-3

10-2

10-1

100

SNR [dB]

BE

R (

solid

), F

ER

(da

shed

)

MS-QPP (19x+40x2)mod160Ω'-QPP-MC (115+9x+20x2)mod160

b)

Figure 1. BER (FER) curves for interleavers of length L=160 and code with a) 8 states; b) 16 states

3 3.5 4 4.5 5 5.5 6 6.5 70

1

2

3

4

5

6

SNR [dB]

Ω' S

NR m

etric

MS-QPP (19x+40x2)mod160Ω'-QPP-MC (115+9x+20x2)mod160

Figure 2. Ω’SNR metric for interleavers of length L=160

Figure 3 gives the simulations results for length of 752. For

this length the Ω’SNR metric for Ω’-QPP-MC interleaver is smaller than for MS-QPP interleaver. The difference in the BER (FER) performances can be observed from the resulted curves after simulations. In the case of the code with 8 states it can be observed a BER or FER decrease with one measure order. For the code with 16 states the difference can be observed at SNR=1.75 dB.

In [4] it is shown that for length of 512, the performance of the MS-QPP interleaver is better compared to Ω’-QPP-MC one. This fact can also be observed from Ω'SNR metric values for these interleavers.

IV. CONCLUSIONS In this paper a modified Ω’ metric for QPP interleavers was

proposed . The metric depends on a value of the SNR. In order to obtain good performance the value of the SNR in the metric should be bigger than the value of the SNR at which the turbo code can work. This is done to obtain a metric that leads to a good selection of the QPP, with no constrain on the value of the SNR at which the turbo code works in real case. This metric is more suitable for the QPP selection, being closer to the upper bound of the frame error rate.

Minimizing of this metric leads to the better interleaver between the one with largest spread (MS-QPP) and the one with maximum Ω' metric (Ω'-QPP-MC).

It must be noticed that when the metrics of two interleavers are close, the difference between metrics (some tenths) doesn’t necessarily reveals the difference between BER (FER) performances. This fact was shown by simulations for length of 160. When the metrics of two interleavers are significantly different, this difference is also reflected on the performances of turbo codes which use the respective interleavers. The simulations results from section 3 for length of 752 and from [4] for length of 512 show this fact.

REFERENCES [1] Takeshita, Y. Oscar, “Permutation Polynomial Interleavers: An

Algebraic-Geometric Perspective”, IEEE Transactions on Information Theory, Vol. 53, No. 6, pp. 2116-2132, June 2007

[2] M. Breiling, “A logarithmic upper bound on the minimum distance of turbo codes,” IEEE Transactions on Information Theory, Vol. 50, No. 8, pp. 692–1710, Aug. 2004

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.810

-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

SNR [dB]

BE

R (

solid

), F

ER

(da

shed

)

MS-QPP (31x+188x2)mod752Ω'-QPP-MC (619+23x+94x2)mod752

a)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.810

-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

SNR [dB]

BE

R (

solid

), F

ER

(da

shed

)

MS-QPP (31x+188x2)mod752Ω'-QPP-MC (619+23x+94x2)mod752

b)

Figure 3. BER (FER) curves for interleavers of length L=752 and code with a) 8 states; b) 16 states

[3] Perotti, A. and Benedetto, S., “A New Upper Bound on the Minimum

Distance of Turbo Codes”, IEEE Transactions on Information Theory, Vol. 50, No. 12, pp. 2985–2997, December 2004

[4] Trifina, L., Rotopanescu, A.M., Gherca, L. and Lupu, B., “QPP Interleavers with Dispersion Maximization”, Buletinul Stiintific al Universitatii “Politehnica” din Timisoara, Tom 53(67), Fascicola 2, pp. 15-20, 25-26 Sept. 2008