Sum Product Decoding Algorithm Computer Science Essay

Published Date: 02 Nov 2017

Abstract

The Multiple Input Multiple Output MIMO technique improves the capacity of the wireless link when operating in a dense multipath scattering environment through the use of multiple antennas in both transmitter and receiver. Wavelet-based MIMO- Orthogonal frequency Division Multiplexing MIMIO-OFDM system can better combat narrowband interferences and more robust to Intercarrier Interference ICI than traditional FFT filters.

A Low Density Parity Check LDPC coded Discrete Wavelet Packet Transform DWPT-MIMO-OFDM system has been proposed and to improve the bit error rate over that of the traditional OFDM which based on Fast Fourier Transform FFT with different models of channels.

The system uses sum product decoding algorithm. The computer simulation tests which implemented using matlab 2012a, show that the BER of the DWT based MC-CDMA has been improved by ……………. dB compared with FFT based MC-CDMA.

Keywords: MIMO-OFDM, , Mallat' algorithm, PCCC, BER, OFDM, MAX-LOG-MAP.

حسام عبد الدائم محمد

[email protected]

مدرس \جامعة بغداد\كليه الهندسة\قسم الهندسة الالكترونية والاتصالات

الخلاصة:

هذا العمل يقدم محاكاة ل Parallel Concatenation Convolution Coding PCCC مع رمز الناقل المتعدد الوصول (MC-CDMA) عبر قناه الخفوت متعدد المسار multipath fading channel مع مقارنة لحالة المعلومات الغير المشفرة و المعلومات التي تستخدم Serial Concatenation Convolution Coding SCCC . تقنية فك الشفرة المستخدمة في المحاكاة كانت فك الشفرة المتكررة حيث تعطي اقصى كفاءة مع ان هناك تحسين بالاست تكرارات . تم استخدام تضمين phase shift keying مع M = 2, 4, 8 and 16 مع مزج تقسيمات التردد المتعامده OFDM. نموذج القناة المتستخدم هو نفسه المخصص 3GPP مع مواصفات تقنية TS 25.101 v2.10 وعرض نطاق قناة قدره 5MHz.

من خلال العمل لوحظ ان هناك تحسين بالاداء بالنسبة للPCCC نسبة الى SCCC و المعلومات غير المشفرة بدلالة ال Signal to Noise Ratio (SNR) بمقدار عدة dBs كما موضح بالجدول رقم (2).

الكلمات الرئيسية: MIMO-OFDM, Mallat' algorithm, PCCC, BER , OFDM, MAX-LOG-MAP.

1-Introduction

The combination of Multiple Input Multiple Output (MIMO) with the OFDM modulation is considered very good solution for the robustness and higher data rates in the next generation wireless LAN. The system in such a case is capable of supporting high bit rates in wireless communication and achieving attractive diversity gain. MIMO-OFDM for the systems in IEEE 802.11n is what distinguishes this WLAN among the others from the same family of IEEE 802.11 and it is considered a very good improvement. MIMO system considers more robustness and offers higher gain in comparison to the single antenna case, and also in MIMO system we can transmit different signals in the same time over different antennas. Using antenna diversity which is very effective in the case of scattering environment, having diversity of receive antennas give us the option to combine, select and switch in order to improve the quality of our received signal[32][44]. In an MIMO system, N signals are transmitted by N antennas at the same time using the same bandwidth and effective processing at the receiver side based on the M received signals by M different antennas, is able to distinguish the different transmitted signals [Abdul-Latef 2]. MIMO techniques can basically be split into two groups: space time coding (STC) [Alamouti] and space division multiplexing (SDM) [Abdul-Latef, Zelst, Wolniansky]. Space time block coding used in this paper to transmit multiple copies of a data stream across a number of antennas and to exploit the various received versions of data to improve reliability of data transfer [Kala]. STC increases the performance of the communication system by coding over the different transmitter branches; whereas SDM achieves a higher throughput by transmitting independent data streams on different transmit branches simultaneously and at the same carrier frequency [Zelst]. Since increasing the bit rates is one of the goals, focusing on SDM algorithms is done in this paper [Abdul-Latef].

MIMO-OFDM systems achieving spectral efficiency and increased throughput. by increasing the number of unique spatial paths between the transmitter and receiver, MIMO-OFDM systems simply exploit the spatial dimension. However, to ensure that the channel matrix is invertible, MIMO systems require an environment rich in multipath [California].

A potential application of the MIMO principle is the next-generation wireless local area network (WLAN). The current WLAN standers IEEE 802.11a [Abdul-Latef, Prasad] are based on orthogonal frequency division multiplexing (OFDM) [Abdul-Latef, Lawerey, Wang].

Normally OFDM is implemented using IFFT and FFT’s in order to multiplex the signals together and decode the signal at the receiver respectively [Khaizuran]. To increase the delay spread of the channel so that it becomes larger than the channel impulse response, the system adds cyclic prefixes (CP) before transmitting the signal. This is done to minimize inter-symbol interference (ISI). However, this is done at the expense of reducing the spectral containment of the channels [Khaizuran]. To decrease the BW waste [4] which results from adding cyclic prefix, wavelet based OFDM is utilized. Since wavelet based OFDM does not use cyclic prefix, the spectral containment of the channels is better.

The spectral containment of the channels is better in WT based OFDM since it does not use cyclic prefix to deal with delay spreads of the channel. This due to the overlapping nature of wavelet properties [Rashmi]. Due to use of wavelet transform, the transmission power is reduced. The wavelet transform is discrete both in time as well as scale as compared to other transforms such as Fourier transform.

The increasing demand of higher rates and robust transmission in modern communication systems, have motivated the search for a suitable error correction schemes allowing high performance and errors near the theoretical limit (Shannon limit) [Ammar].

In this paper FFT-MIMO-OFDM is replaced by LDPC coded DWT-MIMO-OFDM using STBC technique in order to further reduce the level of interference and increase spectral efficiency. The performance of both FFT-MIMO-OFDM and DMWT- MIMO-OFDM systems was compared over the AWGN and fading channel models. The results show that the BER greatly reduced by adopting the suggested system.

2-LDPC Coding

The advantages and the performance of LDPC codes make it one of the best options among other correcting codes, in which these codes are capable of the fulfillment of these requirements and have been proposed for the next generation wireless standards. Now these codes are widely applied in the current wireless networks, such as WiFi (IEEE 802.11n), WiMAX (IEEE802.16e) and DVB-S2 standards. Also they are an important option for forward error correction in fourth generation (4G) wireless communication systems[Ammar] .

LDPC codes are linear block codes specified by a very sparse (containing mostly 0’s and only a small number of 1’s) random parity-check matrix, but are not systematic. The parity-check matrix of an LDPC is an M × N matrix A , where M is the number of parity bits, and N is the transmitted block length (N = K + M , with K as the source block length). The matrix A is specified by a fixed column weight j and a fixed row weight k = j N /M (in the MacKay’s and Neal’s codes k is as uniform as possible [MacKay 1999, Aqiel 2011, Husam 2012], and code rate R = K /N. LDPC codes can be decoded using probability propagation algorithm known as the sum-product or belief propagation algorithm [Kschischang 2001, Husam 2012], which is represented by a factor graph [Tanner Graph, Aqiel 2012] that contains two types of nodes: the "bit nodes" corresponding to a column of the parity-check matrix, which also corresponds to a bit in codeword and the "check nodes" corresponding to a row of the parity-check matrix, which represents a parity-check equation.

2-1-SUM-PRODUCT DECODING ALGORITHM

The decoding problem is to find the most probable vector x such that Ax mod 2 = 0 , with the likelihood of x given by xnΠnfn, where fn0 = 1 − fn1 and f1n = 1/(1 + exp(−2yn / σ2 )) for AWGN channel or f1n = (yn / σ2 ) exp[−y2n / 2σ2 ] for Rayleigh channel, and yn, σ2 represent the received bit and noise variance, respectively. We denote the set of bits, n, that participate in check m as N (m) ≡ {n : Amn = 1}, where Amn represents the element of the mth row and nth column in the parity-check matrix. Similarly, we define the set of checks m in which bit n participates as M(n ) ≡ {m : Amn = 1}. We denote a set N (m) with bit n excluded as N (m) \ n. The algorithm has two alternating parts, in which quantities qmn and rmn associated with each non-zero element in the matrix A are iteratively update. The quantity x qmn is meant to be the probability that bit n of x is x, given the information obtained via checks other than check m. The quantity rmn is meant to be the probability of check m being satisfied if bit n of is x considered fixed at x and the other bits have a separable distribution given by the probabilities {qmn' : n ' ∈ N (m) \ n }. The aposteriori probabilities for a bit are calculated by gathering all the extrinsic information from the check nodes which connected to it, which can be obtained by the iterative sum-product procedure described in details in [Luis 2006, Aqiel 2011, Aqiel 2012].

3-MIMO-OFDM

MIMO-OFDM for the systems in IEEE 802.11n is what distinguishes this WLAN among the others from the same family of IEEE 802.11 and it is considered a very good improvement. MIMO system considers more robustness and offers higher gain in comparison to the single antenna case, and also in MIMO system one can transmit different signals in the same time over different antennas [Ammar].

The general transceiver structure of MIMO-OFDM is presented in Figure (1). The system consists of N transmit antennas and M receive antennas. The transmit antennas transmit independent data (say A1, A2, …, AN) simultaneously and in the same frequency band. A MIMO decoder uses M ≥ N antennas at the receiver side. The received MIMO-OFDM symbol of the n:th subcarrier and the m:th OFDM symbol of the i:th receive antenna after FFT can be written as [Helka]

where Aj[n,m] is the transmitted data symbol on n:th carrier and m:th OFDM symbol, Wi[n,m] is the additive noise contributions. and Hi,j[n,m] is the channel coefficient in the frequency domain between the j:th transmit antenna and the i:th receive antenna. The channel matrix H is an NxM matix corresponding to the n:th subcarrier and m:th OFDM symbol.

The received data symbols of all antennas can be found as in below:

Where

are the Nx1 and Mx1 vectors of the transmitted and received data symbols. The transmitted data symbols can be obtained by solving equation (8) which is called MIMO-OFDM equalization.

This equalization works well in case of small noise and no ISI or ICI.

Wavelet Packet Transform and Wavelet Packet Based MIMO-OFDM system

Let the data stream at the transmitter is first modulated and STBC coded. Here the modulation schemes adopted may be the BPSK, QPSK, and M-QAM. The modulated output symbols are defined as be X = (x[1], x[2], …., x[N]). This stream is converted into parallel sequences Sk and then modulation is done using M-array inverse wavelet packet transform (IWPT). Figures (1) shows the wavelet packet based MCM transmitter operating Mallat’s fast algorithm [Giovanni, Haleh]. The transmitted signal Y, consists of successive K symbols, as the sum of M amplitude modulated waveforms by φk as:

where Y = (y[1], y[2], …., y[n],…, y[N]), is the transmitted signal. SK = (sο [k], s1 [k],….Sm [k], …, SN [k]), is the output of serial to parallel converter ,and φk,

is the waveforms matrix in which φm[n] represent the WPT-MIMO-OFMM basis function that are mutually Orthogonal [jamine] to reduce the symbol errors [Haleh] .

In the wavelet packet scheme, one limits his analysis to subcarrier waveforms defined through a set of FIR filters, and implemented by Mallat’s fast algorithm [halleh, Mallat] with less complexity for wireless communication.

The Mallat algorithm implemented by quadrature mirror filter pair (QMF) that consists of the scaling filter gand dilatation filter h. the knowledge of the scaling filter and wavelet tree depth is sufficient to design the wavelet transform. The scaling filter gand dilatation filter h, and the corresponding reversed filters g(-n) and h(-n), are used to form a wavelet packet tree. These filters satisfy following conditions [Hongbing, Kenneth, Haitham, Giovanni, Haleh aqiel]:

where M is the length of the filters.

The carrier waveforms are obtained by iteratively filtering the signal into high and low frequency components. The waveforms φm[m] are derived by J successive iterations as the following recursive equations:

Where j is the iteration index, , and m the waveform index . Y is the transmitted signal through the channel H, with L multi-paths, H = (h[0], h[1], …,h[l], …, h[L-1]). At the output of the channel, the received signal can be calculated as:

where R = (r[1], r[2], …,r[n], …, r[N]), is the received signal, and W=(w[1], w[2], …, w[n], …, w[N]) is additive white Gaussian noise (AWGN).

6- Proposed System

The proposed system block diagram shown in figure (2). This model shows the system of OFDM based on FFT transform and DWPT transform systems. At the transmitter the incoming data stream is first encoded using the LDPC code. Then it is modulated using QAM modulation scheme with M = 2, 4, 8, and 64 and converted into parallel sequences. Then, the parallel sequences are modulated with M-array inverse wavelet packet transform (IWPT) which baesd on Mallat’s fast algorithm. Figures (3), show the discrete wavelet packet transform based MCM transceiver operating.

As in the FFT-OFDM system model, the input serial data stream is converted into parallel word size required for transmission. The data to be transmitted on each carrier is modulated into a MPSK format. An Inverse Wavelet Packet Transform (IWPT) is used to find the corresponding time domain waveform. The IWPT requires two groups of data input, the first part is called approximation and the second group called details (the length of approximation part equal to the length of details part). In the proposed system the input parallel BPSK data represents approximation part, while zeros are inserted as a details part. The length of output signal from the IWPT stage is equal to double the length of parallel BPSK data. This output data is converted to serial vector, then, this vector is convoluted with the channel selected in the work.

At the receiver side, the serial data is converted into parallel data by serial to parallel converter. Then, the WPT is used to find the corresponding frequency domain of the parallel data. The length of data out is 2NC. The first half of 2NC output data from WPT stage represents the received signal, and the second half represents details, not used for detection. Single tap frequency domain equalizer is used to overcome channel distortion. Each transmitted subcarrier is then evaluated and converted back into the data word by demodulating the received symbol. Then, the data is decoded to reconstruct the original data.

Therefore, Haar wavelet is used in the design of the proposed model, due its simplicity and better performance.

The parameters and system configuration used in the simulation are summarized in Table (1).

Table (1): The parameters and configurations of the proposed system

Modulation scheme

BPSK

Number of subcarrier of FFT

256 subcarriers

Wavelet level

three

Type of wavelet transform

Haar wavelet

OFDM symbol duration

64*10-6 sec

Required bandwidth

5 MHz

Model of simulated channel

Jacks Model

Number of path

6 paths

Pilot Carriers

8

Doppler frequency

50 Hz

Required BER

10-4

When the number of subcarriers is equal to 128, the number of IDWT points is equal to 256 (128 approximation and 128 details). At the receiver, the length of data out from the DWT is equal to 256 (the first 128 subcarriers represent the received data, while the second 128 subcarriers represent the details).

6.Results

The proposed system illustrated in Figure (2) is the wavelet Packet based MC-CDMA system. An 20 Mbps was transmitted over the system. The modulation schemes are the MPSK with M=2, 4, 16,and 64. The work implied performance comparision of the proposed DWT-MC-CDMA system with that of Fast Fourier Transform FFT MC-CDMA system.

A simulation of the two systems has been made using MATLAB version 2012a. The BER performance of the two systems will also be studied in different channel models which are Additive White Gaussian Noise AWGN and multipath fading channel, with a bandwidth of 5 MHz and Walsh-Hadamard (code 20) has been used with 32 bits

The system uses LDPC code. The LDPC specifications used are irregular [16384] Matrix of rate ½. The decoding algorithm is Sum-Product Decoding Algorithm,which is the soft decision type of message passing. The performance of the LDPC decoding depends upon the number of iteration of the decoder.

6.1.Performance of the Proposed System in AWGN Channel

When the channel was modeled as an Additive White Gaussian Noise AWGN channel. The performance of the system was shown in Figures [6, 7,8,9 and 10]. These figures show the variations of BER versus signal to noise ratio (SNR) (Es/Nͦ signal power to the noise power) for LDPC coded data for both FFT MC-CDMA system and DWPT MC-CDMA system with QAM modulation scheme of M value of 2,4, 16,and 64 respectively. It can be noticed that when M = 64, at SNR of 22 dB the BER is 10-3 DWPT MC-CDMA system, while BER is 10-3 at 26 dB for FFT MC-CDMA system with a gain of 4 dB is obtained by the proposed model.

6.2.Performance of the Proposed System in Multipath Fading Channel

Here, the channel was modeled as a multipath fading channel. Figures [11, 12 and 13] show the performance of LDPC coded FFT MC-CDMA system and LDPC coded DWT MC-CDMA systems with QAM modulation scheme of M value of 2,4, and16 respectively. When M= 16, the BER at 10-3 was achieved at a SNR of 16 dB for DWPT-MC-CDMA system and 20.5 dB for FFT MC-CDMA system with a gain of 4,5 dB is obtained by this model.

7.Conclusion:

The results show that LDPC coded DWPT MC-CDMA gives better results for different channel models including the AWGN and Rayleigh channels over LDPC coded FFT MC-CDMA.

For AWGN channel with LDPC coded DWPT MC-CDMA system, the BER of 10-3 is about 1 dB better than and 2 dB better than LDPC coded FFT MC-CDMA system for BPSK and 6 dB gain over uncoded for QPSK . The gain increases as the order of modulation increase showing superiority for higher data rates. For low SNR the results contain a little difference from both uncoded data and convolution coded data.

For Rayleigh channel, at a BER of 10-3, the performance of the LDPC coded MC-CDMA system is better than that with LDPC coded FFT MC-CDMA system by about 2 and 3 dB for bpsk and 4psk modulation schemes respectively. It’s better by 4.5dB than convolutional coding for 16psk. The number of iterations was set to 10 which represents a low computational complexity comparable to covolutional decoder. For better performance and higher computational complexity, the number of iterations can be increased to 100 as clarified in Fig. 8.