PERFORMANCE COMPARISON OF PROPOSED INTERLEAVER WITH DIFFERENT TYPES FOR PARALLEL TURBO CODE *

Interleaver type is important in parallel turbo code system. It’s responsible of improve code performance against additive white Gaussian noise AWGN. In this paper, introduce a new interleaver with its mathematical model. Study its performance as comparison with different interleavers, including random interleaver, QPP quadratic primitive polynomial interleaver as standard of LTE, chaotic interleaver and finally 2-D interleaver for burst error environment, Study the performance as simulation. The system takes in consideration also different modulation schemes including QPSK, 16-QAM and 64QAM. Result comparison between different types summarized in performance BER curves and Tables.


Introduction
Interleaver one of the most important component that construct Turbo code, it plays major role in enhance overall Turbo code performance against burst errors. in communication applications interleaver design effects system performance, many researches interest in interleaver design and focus on its performance against burst errors, complexity design and time consuming in both interleave and deinterleave data.Interleaver applied in many communication system applications, for examples we could see in wireless half-duplex [1] interleaver gain important for receive diversity schemes of distributed Turbo codes in such subject interleaver design established to be important factor in DTC distributed Turbo code performance [1].
In [2] introduce interleaver design that provide facilities in high throughput Turbo decoding which is required for next generation wireless systems.Interleaver in wireless communication [3] applied in division multiple access IDMA with multiple users in wireless communication systems, the chip interleaver here represent the only means of user separation.Interleaver continue take its location in more applications even for OFDM system used to improve BER by using random interleaver [4].Also for forward error correcting codes FEC the Block interleaver used as main burst error treatment in OFDM based WIMAX (IEEE 802.16d) system [5].As see, interleaver still present in modern communication system applications and here not about to list all its application but just a brief look where its applied, for coding with FEC and Turbo codes and also brings attention in [6] to estimate Interleaver period for Reed-Muller coded signals .In this paper a study of different interleavers with parallel turbo code system are introduce, paper divided as sections illustrate each used interleaver apart, finally introduce the proposed interleaver in separated section.More information on used system parameters where highlight with considered different modulation schemes.The simulation results listed at the end of paper as comparison between different interleaver types and different modulation schemes ware used in system simulation.

System Transmitter
The considered system encoder takes parallel turbo code system as encoder rate 1/3, with two identical convolutional encoders of maximum free distance of d free = 5, generated polynomials (5, 7) defined in octal system number and constraint length of 3. The interleaver located as shown in Figure 1 in between to convolutional encoder.This interleaver type will changed as case study.Then followed by modulation process takes one of QPSK, 16-QAM or 64-QAM type.The choice of modulation scheme is done also according case study.

System Receiver
At receiver side a reciprocal of transmitter operation done.It starts by demodulation process and then iterative decoder of parallel turbo code system with 6 iterations.

Random Interleaver
It represents the simplest type of interleaver, which permutes the input vector randomly according initial seed.It may give different output sequence at each run time since it randomly rearranges the input elements sequence depends on initial seed.So reciprocal operation means restore the original data sequence.This process should be use the same initial seed used in random interleaving process.The restore data sequence called deinterleaver.So deinterleaver processes actually represent the inverse of interleaver process [7].

QPP Quadratic Primitive Polynomial
Quadratic permutation polynomial interleaver represents the standard interleaver used with parallel turbo code encoder system for LTE long term evolution.Its permutation operation depends on mathematical formula describes the new sequence of permutated data.The QPP interleaver formula in standard of LTE support frame length of 188 different values, from smallest of 40 to 6144 as largest value.The QPP formula: Where k is input frame length, f 1 and f 2 are standards constants depends on k length summarized in reference [2], "i" is the index or sequence position of original location and finally P(i) gives the new position of permuted data.The deinterleaver process at receiver side do reciprocal operation by tracing permuted data frame and restore it to its original location.

Chaotic Interleaver
Chaotic interleaver is derived from Baker map [8].It could be described as method to randomize the bit sequence arranged in two dimensions' square matrix.Chaotic interleaver may be really referred as a concept how to sectaries element in square matrix and rearrange them.its mathematical formula may be illustrated as let B(n 1 , n 2 ….. n k) denotes the discretize map.Where the vector [n 1 ,n 2 ….n k ], represent the sector key.S key .Defining N as the number of data items in one row, the secret key is chosen in way such that each integer n i divides N data items with condition: Let N i = n 1 + n 2 + …… + n i-1 the data items at indices (r, s) is moved to new indices [8]- [13]: To illustrate this equation in steps, the chaotic interleaver process listed in following points [3]: 1.An N×N square matrix is divided into N rectangles of width n i and number of elements N. 2. The elements in each rectangle are rearranged to a row in the permuted rectangle.Rectangles are taken from left to right beginning with upper rectangles then lower ones.3. Inside each rectangle, the scan begins from the bottom left corner towards upper elements.To understand previous steps listed in [8], for example A simple square matrix of 8×8 which mean 64 elements as shown in Figure (3) below at (a), (b) and (c) .The key should be selected by designer to satisfy chaotic interleaver condition, that divide the square matrix to rectangles each of elements equal to N. here in this example N=8.So the selected key = {2, 4, 2}.The rectangle borders are shown in heavy bold lines.In case study an extended to chaotic interleaver including square matrix of 16×16 with elements of 256.
The designed key such that satisfy chaotic interleaver conditions is calculated to be key = {2, 2, 4, 4, 2, 2} with each inside square matrix a rectangle with number of elements equal to N=16.See Figure (4).Then new sequence of input data read row by row to construct chaotic interleaved frame of length 1024 bit.

2D Two Dimensional Prime Interleaver
This type of interleaver works on two-dimension matrix.It really remaps the matrix element sequence in manner depends on mathematical formula set the new positions of interleaved data sequence according calculated row and column vectors.In simulation parameter, 2D prime interleaver is used with matrix of dimension 32×32 with elements of 1024 bit.P row = 3, and P col = 5.

Proposed Interleaver
The proposed interleaver works according predefined step size from designer.The step size takes as reference in interleaving process.Such to insure at most as possible a step distance between two successive elements.For the interleaver input data frame of length N, the interleaving takes also in consideration the input frame length to calculate new position of each element in input data frame.Let examine a simple case.For input frame length of N=10 elements.And simple designed step size of 1. Then the sequence gets jump by step size equal to one, and continue set the new positions, after reaches the position of N=10.It returns to empty position and record the new sequence and so on.See Figure ( 7):   From previous two examples, it's simple to note that for example element 9 change its position from 4 th to 11 th , element 10 change its position from 8 th to 4 th , and new element 11 set at 8 th position.
Review for previous presented four types of interleavers represented by Random, QPP, Chaotic and 2D prime interleaver.Look for interleaver index graph.It's clear the difference between previous presented type and proposed interleaver.See figure (10).List graphs of index of each type of interleaver.The graphs take number of elements of 1024 bit.The graphs show the distribution of new sequences after interleaving.

Simulation Results Discussion
Simulation take parallel turbo code system based on maximum free distance convolutional encoder of generator polynomials in octal defined by (5,7) and constraint length of 3, free distance equal to 6. Frame length of 1024 bit and total input frames of 1000.The proposed interleaver with a step size of 67.The 2D interleaver takes primes 3 for row and 5 for column respectively.QPP interleaver define f 1 of 31 and f 2 of 64 this according standard tables for frame length 1024 bit.Chaotic interleaver designed for 32×32 matrix that produce 1024 interleaved elements.The simulation studies the system performance with three modulation types QPSK, 16-QAM and 64-QAM.For simulation results see figures (12)(13)(14):

Conclusions
In this paper, introduce a new type of interleaver used with parallel turbo code depends on predefined step size.Discuss its operation and put a mathematical model to describe its operation in data interleaving.Also recall four types of interleavers, discuss briefly its operation and the way used by each one to interleave data with examples.The simulation also studies the system performance with different modulation including QPSK, 16-QAM and 64-QAM, simulation results show improvement in BER with using proposed interleaver.It's important to note that long term evolution LTE interleaver represented by QPP need standard fixed values of 188 pair define constants of f 1 and f 2 , while the proposed interleaver not needs such predefined constants with BER performance listed in Table 1 for each case.This leads to reduce interleaver complexity with accepted BER performance as shown in simulation results.
a-Original data sequence from one to 1024.b-Random interleaved data sequence.

Figure 10 .
different interleavers' data sequence pattern comparison.c-QPP interleaved data sequence.d-Chaotic interleaved data sequence.e-2D prime interleaved data sequence.f-Proposed interleaved data sequence.

Figure 12 .
Figure 12.Turbo code system performance, QPSK modulation with different interleaver types.

Figure 13 .
Figure 13.Turbo code system performance, QPSK modulation with different interleaver types enlarged view at curves end.

Figure 15 .
Figure 15.Turbo code system performance, 16-QAM modulation with different interleaver types enlarged view at curves end.

Figure 17 .
Figure 17.Turbo code system performance, 64-QAM modulation with different interleaver types enlarged view at curves end.

Table 1 .
Simulation results comparisons