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A Study of the Concatenated Reed Solomon – Convolutional Coding Performance used in WiMAX Sean Stamplecoskie Defence R&D Canada Ottawa TECHNICAL MEMORANDUM DRDC Ottawa TM 2006-026 January 2006

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A Study of the Concatenated Reed Solomon – Convolutional Coding Performance used in WiMAX

Sean Stamplecoskie

Defence R&D Canada √ Ottawa TECHNICAL MEMORANDUM

DRDC Ottawa TM 2006-026 January 2006

A Study of the Concatenated Reed Solomon – Convolutional Coding Performance used in WiMAX

Sean Stamplecoskie

Defence R&D Canada – Ottawa Technical Memorandum DRDC Ottawa TM 2006-026 January 2006

DRDC Ottawa TM 2006-026

Principal Author

“original signed by”

Sean Stamplecoskie

Defence Scientist

Approved by

“original signed by”

Bill Katsube

Section Head - CNEW

Approved for release by

“original signed by”

Bert Bridgewater

A/Chief Scientist

© Her Majesty the Queen as represented by the Minister of National Defence, 2006

© Sa Majesté la Reine, représentée par le ministre de la Défense nationale, 2006

DRDC Ottawa TM 2006-026 i

Abstract

WiMAX is the next generation of wireless internet solutions. WiMAX is designed to provide the “last mile” solution, bringing wireless internet to our homes and businesses.

Increased Quality of Service is obtained through the use of multiple air interfaces, adaptive modulation and coding to provide the optimum performance to various users under various signal strength conditions and various application requirements.

This paper is a performance study of WiMAX’s coding performance using the concatenated Reed Solomon-Convolutional coder. The implementation uses an OFDM air interface, the modulation format selected is QPSK and the channel models used were AWGN and a frequency selective channel.

Résumé

WiMAX représente la prochaine génération de solutions sans fil pour Internet. WiMAX est conçu pour fournir la solution du "dernier mille", en apportant l'Internet sans fil à nos maisons et entreprises.

Un service d’une plus grande qualité est obtenu grâce à l'utilisation d’ interfaces multiples d'air, de la modulation adaptative ainsi que du codage adaptif pour fournir une exécution optimale aux utilisateurs dans des conditions et les demandes et la qualité du signal à la réception peuvent varier.

Ce rapport étudie la performance de l'exécution du codage de WiMAX avec le codeur convulution à concaténation Reed Solomon.L'exécution emploie une interface d'air (MROF) Multiplexage par Répartition Orthogonale de la Fréquence et la modulation (QPSK) Modulation par Déplacement de Phase en Quadrature les modèles de canal utilisés incluent un canal avec (BBGA) Bruit Blanc Gaussien Additif et un canal sélectif de fréquence.

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Executive summary

A Study of the Concatenated Reed Solomon – Convolutional Coding Performance used in WiMAX

Stamplecoskie, S.D.; DRDC Ottawa TM 2006-026; Defence R&D Canada – Ottawa; January 2006.

The next wireless protocol to expand wireless capabilities is WirelessMAN (Metropolitan Area Network) 802.16, and its subset WiMAX. WiMAX is designed to provide the “last mile” solution, bringing wireless internet to our homes and businesses.

The standard uses multiple air interfaces, adaptive modulation and coding to provide the optimum performance to various users under various signal strength conditions and various application requirements.

This paper is a performance study of WiMAX’s coding performance in the standard implementation. This implementation uses an OFDM air interface, the standard RS-CC FEC. The modulation format selected is QPSK and the channel models used were AWGN and a frequency selective channel.

Considerable coding gain was realised as the Reed Solomon and Convolutional coding schemes compliment each other. The Viterbi decoder demonstrates considerable coding gain; however it is susceptible to burst errors. The Reed Solomon decoder is proficient at correcting burst errors and recovered many of the decoding errors output from the Viterbi decoder. The concatenated coding strategy demonstrated greater performance than either coder on its own.

Two primary impairments affecting WiMAX are large peak to average ratio and doppler effects. Future investigation should be focussed on reducing the effects of these impairments. Additionally, channel estimation and equalization must be studied to combat the severe degradation caused by frequency selective channels.

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Sommaire

A Study of the Concatenated Reed Solomon – Convolutional Coding Performance used in WiMAX

Stamplecoskie, S.D.; DRDC Ottawa TM 2006-026; R & D pour la défense Canada – Ottawa; January 2006.

Le prochain protocole sans fil pour augmenter des possibilités sans fil est WirelessMAN (réseau métropolitain) 802.16, et son sous-ensemble WiMAX. WiMAX est conçu pour fournir la solution du "dernier mille"en apportant l'Internet sans fil à nos maisons et entreprises.

Ce protocole emploie les interfaces multiples d'air modulation adaptative et codage pour fournir l'exécution optima à de divers utilisateurs dans de diverses conditions de force de signal et diverses conditions d'application.

Cet ouvrage étudie la performance du codage WiMAX pour une implémentation standard. Cette dernière utilise une interface d'air MROF , l’interface standard FEC RS-CC. Le format de modulation choisi est QPSK et les modèles de canal utilisés incluent un canal BBAG et un canal sélectif de fréquence.

Un gain considérable de codage a été réalisé étant donné que les codes Reed Solomon et les codes convolutifs se complètent. Le décodeur Viterbi démontre aussi un gain considérable de codage quoi qu’il demeure susceptible à des erreurs éclat. Le décodeur Reed Solomon permet la correction des erreurs éclatées et la récupération de plusieurs erreurs de décodage du décodeur de Viterbi. La stratégie de codage avec concaténation a démontré une performance supérieure à celle de l'un ou l'autre des codeurs lorsque ils sont utilisés seul.

Les deux principaux points faibles de WiMAX sont rapport de crête et les effets de Doppler moyens. La recherche future devrait viser la réductionces effets. De plus, l'évaluation et l'égalization du canal doivent être étudiées afin de combattre la dégradation sérieuse provoquée par les canaux sélectifs de fréquence.

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Table of contents

Abstract ............................................................................................................................................ i Résumé ............................................................................................................................................. i Executive summary ........................................................................................................................ iii Sommaire......................................................................................................................................... v Table of contents ........................................................................................................................... vii List of figures ................................................................................................................................. ix List of tables ................................................................................................................................... xi 1. Introduction............................................................................................................................... 1 2. Orthogonal Frequency Division Multiplexing.......................................................................... 2 3. IEEE 802.16 WirelessMAN Standard ...................................................................................... 5 4. WiMAX Specification .............................................................................................................. 7

4.1 Randomizer ................................................................................................................... 7 4.2 Forward Error Correction .............................................................................................. 8

4.2.1 Reed-Solomon Encoder .................................................................................. 8 4.2.2 Convolutional Encoder.................................................................................... 9

4.3 Interleaver.................................................................................................................... 10 4.4 Symbol Mapping ......................................................................................................... 10 4.5 Serial to Parallel Converter ......................................................................................... 10 4.6 OFDM Modulator........................................................................................................ 11 4.7 Cyclic Prefix Insertion................................................................................................. 11

5. WiMAX Implementation Under Test ..................................................................................... 12 5.1 Input Message.............................................................................................................. 12 5.2 Forward Error Correction ............................................................................................ 12

5.2.1 Reed Solomon Encoding............................................................................... 12 5.2.2 Convolutional Encoding................................................................................ 13

5.3 Interleaving.................................................................................................................. 14 5.4 Symbol Mapping ......................................................................................................... 14 5.5 Serial to Parallel Converter ......................................................................................... 15 5.6 OFDM Modulator........................................................................................................ 15 5.7 Cyclic Prefix Insertion................................................................................................. 15 5.8 Channel Model ............................................................................................................ 16

5.8.1 AWGN Channel ............................................................................................ 16 5.8.2 Frequency Selective Channel ........................................................................ 16

6. Simulation Procedures ............................................................................................................ 17 7. Results..................................................................................................................................... 18

7.1 Simulated WiMAX Overhead Losses ......................................................................... 18 7.2 Performance WiMAX Coding Components................................................................ 19

viii DRDC Ottawa TM 2006-026

7.2.1 Performance of the Convolutional Coder...................................................... 19 7.2.2 Performance of the Reed-Solomon Coder .................................................... 20 7.2.3 Concatenated RS-CC Coder.......................................................................... 21

7.3 Comparison of Coded QPSK vs. Coded WiMAX over an AWGN Channel .............. 22 7.4 Comparison of Coded QPSK vs. Coded WiMAX over a Frequency Selective

Channel........................................................................................................................ 24 8. Discussion............................................................................................................................... 25 9. Conclusion .............................................................................................................................. 26 References ..................................................................................................................................... 27 List of symbols/abbreviations/acronyms/initialisms ..................................................................... 29

DRDC Ottawa TM 2006-026 ix

List of figures

Figure 1. Single Carrier Wideband vs OFDM................................................................................ 2 Figure 2. Single Carrier vs. OFDM subjected to Frequency Selective Channel Impairments. ...... 3 Figure 3. Insertion of the Cyclic Prefix. ......................................................................................... 3 Figure 4. Relationship between cell radii, signal strength and modulation format. ....................... 5 Figure 5. Physical Layer of the WiMAX wireless broadband channel. ......................................... 7 Figure 6. The WiMAX rate 1/2, convolutional encoder................................................................. 9 Figure 7. Frequency Spectrum of WiMAX transmission............................................................. 11 Figure 8. WiMAX interleaver implemented for this performance study. .................................... 14 Figure 9. QPSK symbol map........................................................................................................ 14 Figure 10. Magnitude and Phase response of the frequency selective channel. ........................... 16 Figure 11. Flowchart of the simulation processing chain............................................................. 17 Figure 12. Performance losses due to cyclic prefix insertion and unused OFDM carriers. ......... 19 Figure 13. Performance of the WiMAX convolutional coder. ...................................................... 20 Figure 14. Performance of the WiMAX Reed-Solomon Coder. .................................................. 21 Figure 15. Performance of the concatenated RS-CC coder. .......................................................... 22 Figure 16. Comparison of QPSK vs. WiMAX performance with and without coding. ............... 23 Figure 17. Simulated performance of QPSK and OFDM over a frequency selective channel. ... 24

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List of tables

Table 1. Coding formats supported by WiMAX. ........................................................................... 8 Table 2: Coding Rates, free distances and puncturing patterns specified for WiMAX’s rate

1/2 constraint length 7 convolutional encoder. ........................................................... 10

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1. Introduction

Wireless LAN (Large Area Network) 802.11 has given computer users increased flexibility and mobility via an effective wireless internet connection. The success of this technology is apparent by the increasing number of “hot spots” available in our libraries, schools, malls and places of business. However, the usefulness of Wireless LAN is limited by its short communication range.

The next wireless protocol to expand wireless capabilities is WirelessMAN (Metropolitan Area Network) 802.16, and its subset WiMAX. WiMAX is designed to provide the “last mile” solution, bringing wireless internet to our homes and businesses.

WiMAX was designed with flexibility in mind. The standard uses multiple air interfaces, adaptive modulation and coding to provide the optimum performance to various users under various signal strength conditions and various application requirements.

The air interfaces used by WiMAX are Single Carrier (SC) and Orthogonal Frequency Division Multiplexing (OFDM). OFDM uses a 256 point FFT application.

The modulations supported by WiMAX are BPSK, QPSK, 4-QAM, 16-QAM, 64-QAM and 265QAM.

The following Forward Error Correction (FEC) strategies are supported: Reed-Solomon (RS), Block Convolutional Coding (BCC), concatenated Reed-Solomon-convolutional coding (RS-CC), Block Turbo Coding (BTC), Convolutional Turbo Coding (CTC) and Low Density Parity Check Coding (LDPC).

A popular implementation of the air interface is the OFDM, 256-point FFT. Channel coding is performed in three steps, randomization, FEC and interleaving. The OFDM interface must implement concatenated Reed Solomon-Convolutional Coding (RS-CC). This is a RS outer code with a CC inner code. BTC and CTC are implemented as options. The modulation formats supported by the OFDM air interface are BPSK, QPSK, 16-QAM and 64-QAM.

This paper is a performance study of WiMAX’s OFDM air interface, using the standard RS-CC FEC and QPSK modulation over an AWGN channel and a frequency selective channel.

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2. Orthogonal Frequency Division Multiplexing

“Orthogonal Frequency Division Multiplexing (OFDM) is a promising technique for achieving high data rate and combating multipath fading in wireless communications”[1]. OFDM combines two communications concepts: multi-carrier modulation (MCM); and orthogonal frequency shift keying (FSK).

MCM attains wideband data-rates by dividing a wideband data stream into parallel, lower rate data streams and transmitting them over multiple narrowband carriers. The benefits of this strategy are longer symbol intervals and narrowband channel characteristics.

Orthogonal FSK enables each OFDM carrier to be orthogonal to all other carriers transmitted. This orthogonality is achieved by separating the carriers by an integer multiple of the symbol duration of the narrowband data stream.

OFDM is an effective means of communicating in non line-of-sight (NLOS) environments due to its long symbol interval, which is able to operate within conditions of large delay spreads, common in NLOS environments [2]. A comparison of a single carrier wideband communication channel and an OFDM system is depicted in Figure 1:

Figure 1. Single Carrier Wideband vs OFDM.

As indicated by the figure, single carrier wideband communications utilize longer symbol intervals over lower bandwidth communication channels. Also apparent is the benefits of OFDM in frequency selective environments. Since signal degradation is present on portions of the band rather than the entire band and over narrower channels, the equalization of the received signal is considerably less complex. This comparison is illustrated in Figure 2.

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Figure 2. Single Carrier vs. OFDM subjected to Frequency Selective Channel Impairments.

The effects of inter-symbol-interference (ISI) are mitigated via the use of longer symbol duration, as well as the use of a cyclic prefix (CP). The cyclic prefix is a copy of the tail of the symbol period that is appended to the start of the message as a guard interval. The CP is illustrated in Figure 3.

Figure 3. Insertion of the Cyclic Prefix.

This guard interval serves many purposes in addition to reducing ISI. The CP aids in timing recovery and mitigation of inter-carrier-interference (ICI). One negative aspect of adding a CP to a message is a loss in Eb/No since this portion of the symbol does not contain any new information and is discarded by the receiver. The Eb/No loss is calculated as:

⎟⎟⎠

⎞⎜⎜⎝

⎛×=

s

b

o

b

TTLossN

E

10

log10 Equation 1

4 DRDC Ottawa TM 2006-026

The OFDM modulation technique has many benefits for operating in NLOS environment; however, it is not perfect. OFDM’s long symbol interval is vulnerable to doppler effects experienced in mobile environments. Additionally, the large peak to average ratio poses significant problems due to demanding linear amplifier requirements. As a result, considerable effort has been targeted at mitigating doppler effects and decreasing the cost of high performance linear amplifiers.

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3. IEEE 802.16 WirelessMAN Standard

OFDM has been a recent focus of wireless access system development due to its ability to transmit high data rate information over LOS and NLOS communication channels. The first OFDM implementation with significant success was the IEEE 802.11 wireless LAN. The next standard to adopt this modulation format is 802.16, which places OFDM as a key enabler for the standard.

IEEE 802.16 is an international standard for local and metropolitan area networks. 802.16 defines the air interface for fixed broadband wireless access systems [3]. 802.16 promises high-speed portable internet service as a “last mile” solution. The 802.16 standard proposes coverage areas of up to 50 km in radius.

In addition to OFDM, IEEE 802.16 uses adaptive modulation as a means of increasing quality of service (QoS). Adaptive modulation ensures high performance with the maximum data rate sustainable. The modulation formats supported are listed in increasing complexity:

• binary phase shift keying (BPSK), • quadrature phase shift keying (QPSK), • 16-quadrature amplitude modulation (16-QAM), and • 64-quadrature amplitude modulation (64-QAM).

As the complexity of the modulation increases, a higher data rate is supported; however, the increase in data rate is attained at the expense of lower performance in the presence of noise. To ensure the best QoS, the base station monitors the noise level of the communication channel and the signal strength of the mobile device. Based on the signal strength and noise level present on the channel, the portable device is instructed to use the highest data rate possible while maintaining a predefined level of performance. As the signal strength is related to the transmitter-receiver separation distance, the closer the devices are, the higher the data rate supported2. The relationship between the modulation formats used is illustrated in the following diagram:

Figure 4. Relationship between cell radii, signal strength and modulation format.

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In addition to adaptive modulation, 802.16 supports adaptive coding. The coding formats supported are variations of Reed Solomon, concatenated Reed Solomon-Convolutional, Block Turbo Coding, Convolutional Turbo Coding and Low Density Parity Check Coding.

The modulation formats supported are single carrier, OFDM 256-carrier and Orthogonal Frequency Division Multiple Access (OFDMA) a 1024-carrier OFDM variant.

Many other signal processing structures are supported by 802.16. Randomization is supported on all channels to protect against the transmission of unmodulated carriers. Power control is supported to maintain sufficient signal strength in dynamic signal environments. Automatic Repeat Request (ARQ) is supported to correct errors not corrected by FEC. Subchannelization is supported to increase the power in designated bands, increasing range of transmission and avoiding frequency bands of severe degradation. One final option supported by 802.16 is space time coding which provides a form of time diversity to combat fading channels.

802.16 is a highly adaptive communication standard proposed for high data rate wireless access systems. The question asked when implementing such a flexible and all encompassing standard is where to begin. This is where WiMAX comes in.

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4. WiMAX Specification

WiMAX is an implementation of a subset of the 802.16 standard for broadband communication over Line of Sight (LOS) (50 km) and NLOS (8 km) propagation environments. WiMAX utilizes OFDM and considerable FEC as key enablers to achieve broadband rates over great distances.

The WiMAX physical layer is illustrated in Figure 5:

Figure 5. Physical Layer of the WiMAX wireless broadband channel.

The following subsections describe the WiMAX Physical Layer in greater detail.

4.1 Randomizer

The randomizer is pseudo-random code generator that takes the input message and adds the pseudo-random code to the input message using modulo-2 addition. The effect of this process is a scrambled message entering the signal processing chain. Scrambling is performed to prevent the transmission of an unmodulated carrier and has no effect on the coding gain of the system. After scrambling, a 0x00 byte is appended to the message sequence. The purpose of this 0x00 byte is to ensure the convolutional encoder in the FEC finishes each message encoding in state 0.

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4.2 Forward Error Correction

The second element in the signal processing chain is the FEC encoder. WiMAX has chosen the concatenated RS-CC encoder for error correction and detection. The RS-CC is an outer RS encoder with an inner convolutional encoder. The encoder pairs support various overall coding rates by shortening and puncturing the data sequences. The overall coding rate of the RS-CC encoder for the mobile station must be configured to 1/2 [1] when acquiring service with the network. The coder configurations supported by WiMAX are described in Table 1:

Modulation Uncoded Block Size

(Bytes)

Coded Block Size

(Bytes)

Overall Coding Rate

RS Code

(N’, K’, T’)

Convolutional

Coding Rate

BPSK 12 24 1/2 (12,12,0) 1/2

QPSK 24 48 1/2 (32,24,4) 2/3

QPSK 36 48 3/4 (40,36,2) 5/6

16-QAM 48 96 1/2 (64,48,8) 2/3

16-QAM 72 96 3/4 (80,72,4) 5/6

64-QAM 96 144 2/3 (108,96,6) 3/4

64-QAM 108 144 3/4 (120,198,6) 5/6

Table 1. Coding formats supported by WiMAX.

4.2.1 Reed-Solomon Encoder

Reed Solomon codes are Maximum Distance Separable (MDS) which means they achieve the maximum possible minimum distance (dmin) possible for the FEC with the specified parameters N,K. Having the greatest possible minimum distance means that the code can correct the maximum number of errors for the specified parameters.

The RS encoder is an N = 255, K = 239, T= 8 encoder with elements taken from GF(28) = GF(256). Therefore, each symbol is a byte. The generator polynomial for the encoder is:

))...()()()(()( 123210 −+++++= Txxxxxxg ααααα , where α = 02HEX

Equation 2

and the primitive polynomial used to create GF(256) is:

DRDC Ottawa TM 2006-026 9

1)( 2348 ++++= xxxxxp Equation 3

by multiplying the components of g(x) we get,

1202252194

318241695147619179183976

10161111021210913107141041512016)(

ααα

ααααααα

αααααα

+++

+++++++

++++++=

xxxxxxxxx

xxxxxxxxg Equation 4

This encoder supports all code rates and error correction requirements stipulated by the WiMAX specifications. To achieve the desired coding rate K’ and error correcting capability T’, the code is both shortened and punctured. The code is shortened by appending K-K’ zero bytes to the start of the input message. After encoding, these K-K’ zero bytes are discarded. The code is punctured to obtain an error-correcting capability T’ by discarding the last 2(T-T’) parity bytes.

4.2.2 Convolutional Encoder

A rate 1/2, constraint length 7, binary convolutional code is used as the inner encoder of the concatenated RS-CC encoder. The output connections for the two outputs are 171OCT and 133OCT as defined in the 802.16 standard. Figure 6 illustrates the convolutional encoder used in WiMAX devices.

Figure 6. The WiMAX rate 1/2, convolutional encoder

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To accommodate the various overall coding rates defined in the WiMAX specifications, the output of the convolutional encoder is punctured [1]. The convolutional coding rates, free distances dfree and puncturing patterns supported by WiMAX are illustrated in the following table; where output 1 is represented by an X and output 2 is represented by a Y:

Code Rates

Rate 1/2 2/3 3/4 5/6

dfree 10 6 5 4

Punctured Output Pattern

X1Y1 X1Y1Y2 X1Y1Y2X3 X1Y1Y2X3Y4X5

Table 2: Coding Rates, free distances and puncturing patterns specified for WiMAX’s rate 1/2 constraint length 7 convolutional encoder.

As previously stated, the convolutional encoder always ends in the all zero state since the input message sequence is padded with a 0x00 byte.

The encoder configuration is the optimum configuration for a constraint length 7, rate 1/2 convolutional encoder [4]. This configuration provides the maximum free distance (dfree) of 10.

4.3 Interleaver

The data is interleaved immediately after encoding. The purpose of the interleaver is to protect the convolutional encoder from burst errors by spreading the burst across the entire received codeword. The size of the interleaver is scaleable to accommodate the various modulation formats and their corresponding codeword sizes.

4.4 Symbol Mapping

Symbol mapping is the modulation of a data sequence using a modulation format. The modulation formats supported are BPSK, QPSK, 16 QAM and 64 QAM (optional).

4.5 Serial to Parallel Converter

The data sequence is a serial data stream and the OFDM modulator inputs many data streams in parallel. The serial to parallel converter takes a serial data stream and breaks it into blocks; which are input into the OFDM modulator in parallel.

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4.6 OFDM Modulator

The OFDM modulation is completed by performing an Inverse Discrete Fourier Transform (IDFT) [5]. The IDFT operation interprets its parallel inputs as the frequency spectrum components of a signal and converts the data to a time domain signal for transmission. Since, WiMAX supports 256 OFDM, there are N = 256 spectral components considered in the IDFT. The IDFT operation is defined as:

1,...,1,0,2exp][1][1

0

−=⎟⎠⎞

⎜⎝⎛= ∑

=

NnknN

jkXN

nxN

k

π Equation 5

The demodulation of the received signal is completed by performing the inverse operation, the Discrete Fourier Transform (DFT). The DFT converts a time domain signal into its spectral components. The DFT operation is defined as:

1,...,1,0,2exp][1][1

0

−=⎟⎠⎞

⎜⎝⎛ −= ∑

=

NkknN

jnxN

kXN

n

π Equation 6

WiMAX uses 256 OFDM. Of the 256 carriers, 56 are used as Guard Carriers, 8 as Pilot Carriers and 192 as data carriers. The Guard Carriers are the 28 lowest frequencies, 27 highest frequencies and 1 DC subcarrier. The 8 subcarriers are evenly spaced amongst the data carriers at locations 40, 65, 90, 115, 143, 166, 191 and 216. The frequency spectrum of the OFDM waveform is illustrated in figure 7.

Figure 7. Frequency Spectrum of WiMAX transmission

4.7 Cyclic Prefix Insertion

The insertion of a CP is performed by copying a portion of the tail of the OFDM signal and appending it to the start of the OFDM signal. The amount of CP is variable and the values (G) supported by WiMAX are 1/4, 1/8, 1/16 and 1/32 [1]. The rates represent the portion of the transmitted bit that is appended to the start of the transmitted symbol. The CP time is calculated as:

bg TGT •= Equation 7

Where Tb is the duration of the transmitted bit, and Tg is the duration of the transmitted CP.

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5. WiMAX Implementation Under Test

WiMAX is a flexible standard used with various modes of operation. The specific mode tested in this report is the rate 1/2 RS-CC FEC which is mandatory when accessing all WiMAX networks. The modulation format chosen is QPSK and OFDM-256 is used as the final modulation stage. The randomizer is omitted from this study, as its purpose is to protect against the transmission of unmodulated carriers and does not affect the performance of the physical layer. The individual WiMAX signal processing elements are discussed in greater detail in the following subsections.

5.1 Input Message

As illustrated in figure 7, the OFDM modulator uses 192 carriers. Each carrier is QPSK modulated. Thus, each OFDM symbol requires 2x192 = 384 bits after FEC. The overall rate of the FEC is 1/2, therefore the number of input bits required at the input to the signal processing chain, for one OFDM symbol is 384/2 = 192 bits. As a result, the input bits are chosen as a random vector of binary bits of length 192.

5.2 Forward Error Correction

The rate 1/2 concatenated RS-CC coder was implemented as two separate coding blocks. The individual coding blocks are described in greater detail in the following subsections.

5.2.1 Reed Solomon Encoding

The RS encoder operates over GF(q) = GF(256) = GF(28); therefore, each symbol consists of one byte (8 bits). The encoder is an N = 255, K = 239 and T = 8 systematic RS encoder. As indicated in Table 1, the RS encoder is shortened and punctured to an N’ = 32, K’ = 24 and T’ = 4 RS code.

The shortening is performed by appending 239 – 32 = 207, 0x00 bytes to the start of the 32 byte message sequence prior to encoding. After encoding, these 207, 0x00 bytes are discarded.

The code is punctured to T’ = 4 by discarding the 2(T-T’) = 8 tailing parity bits. The parity bits are then appended to the start of the message bits before interleaving.

The decoding of a RS encoded codeword is performed in five steps. First, the parity bits are taken from the start of the codeword and placed at the end. Second, 207 0x00 bytes are inserted to fill in the shortened byte positions. Third, a codeword is created by substituting the eight punctured parity bytes with all zero bytes. Fourth, the codeword is decoded using a RS decoder. Fifth, the 207, 0x00 bytes are discarded leaving the original transmitted message.

The rate of the RS code (R) is 3/4 and the dmin is 2T’+1 = 9. The asymptotic coding gain for this RS coding scheme using hard decision decoding is [6]:

( ) dBRd 283.5943

21log10

21log10 10min10 =⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=⎟

⎠⎞

⎜⎝⎛ Equation 8

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A modification of the Matlab Reed Solomon encoder/decoder was required to realize the true performance in the presence of puncturing. When a RS code is punctured, the error correction capability is:

TeT =⎟⎠⎞

⎜⎝⎛+

2' Equation 9

Where T’ represents the error correcting capability of the punctured code, e represents the number of erasure symbols and T represents the error correction capability of the code prior to puncturing. However, Matlab’s rsdec interprets erasures as errors. As a result, the 8 punctured bits use up the full error correction capability of the code. To simulate the T’= 4 error capability of the code, 4 punctured bits are sent directly to the decoder erasure positions. With this modification, the error correction capability of the code is 4, representing the capability of the RS punctured decoder with knowledge of erasure positions.

5.2.2 Convolutional Encoding

The convolutional encoder is depicted in Figure 6. Table 1 indicates that the convolutional coding rate required for an overall coding rate 1/2 and QPSK modulation is R = 2/3. From Table 2 it is observed that this convolutional encoder has a free distance, dfree= 6 and the puncturing format is X1Y1Y2.

The convolutional code is decoded using a Viterbi decoder. The Viterbi decoder uses soft decision decoding. A one is interpreted as a one and a zero is interpreted as a zero. The punctured bits are substituted with the value 0.5. The trace back depth of the Viterbi decoder is 100 bits. The Viterbi decoder starts and ends each decoding process in the all zero state.

The input bits are obtained from a QPSK demodulator using hard decision decoding, but the Viterbi decoder uses soft decision decoding. The asymptotic coding gain for this convolutional code is between:

( ) dBRd free 010.3632

21log10

21log10 1010 =⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=⎟

⎠⎞

⎜⎝⎛ Hard Decision Equation 10

and

( ) dBRd free 020.6632log10log10 1010 =⎟

⎠⎞

⎜⎝⎛= Soft Decision Equation 11

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5.3 Interleaving

The interleaver is a matrix of size 12x32. The 384 input bits are read in as rows and read out serially as columns. This process separates adjacent bits by 32 bit positions, protecting the convolutional encoder from burst errors.

The interleaving strategy is illustrated in the following diagram.

1 2 3 4 5 6 . . . 32 33 34 35 36 37 38 . . . 64 65 66 67 68 69 70 . . . 96

: : :

353 354 355 356 357 358 . . . 384

Figure 8. WiMAX interleaver implemented for this performance study.

5.4 Symbol Mapping The symbol mapping in this implementation is QPSK modulation. The data is read into the QPSK

modulator as a block of 384 bits. The bits are grouped into bit pairs b0 and b1. Each Pair of bits corresponds to a single complex valued QPSK symbol. The symbol mapping is illustrated in Figure 9.

Figure 9. QPSK symbol map.

1,2,3,4,5,6, . . . , 384

IN

1,33,65,97 . . . , 384

OUT

DRDC Ottawa TM 2006-026 15

The QPSK demodulator can be viewed as two orthogonal BPSK demodulators [4]. Therefore, in the presence of Additive White Gaussian Noise (AWGN), the performance of the QPSK receiver is:

⎟⎟⎠

⎞⎜⎜⎝

⎛=

o

be N

EQP 2 Equation 12

5.5 Serial to Parallel Converter

The serial to parallel conversion process is accomplished by grouping the QPSK symbols into groups of 192 symbols and sending this block of symbols to the OFDM modulator.

5.6 OFDM Modulator

The OFDM modulator inputs the 192 QPSK symbols and places them onto the data carriers as illustrated in Figure 7. The remaining bit positions are filled with data values (b0=1,b1=0), which corresponds to a BPSK modulated pilot. The modulator performs an IDFT of length 256 on the input data, completing the OFDM modulation.

The demodulation of the OFDM signal is performed by performing the IDFT on the received data and extracting the data from the relevant bit positions.

Assuming all 256 carriers are used, the 256-OFDM performance is identical to an equivalent wideband QPSK modulation format operating at the same bit rate as the 256-OFDM.

However, only 192 carriers are used by WiMAX and this reduction in relevant signal information causes a performance decrease equivalent to:

⎟⎠⎞

⎜⎝⎛

256192log10 10 = -1.25 dB Equation 13

The 1.25 dB performance loss calculated for WiMAX is due to the overhead induced by implementing guard carriers and subcarriers. The guard carriers are used to enable the signal to naturally decay, while maintaining the “brick wall” FFT shape. The subcarriers provide the same functionality, when the subchannelization option is selected by the mobile station.

5.7 Cyclic Prefix Insertion

Since the CP is immediately discarded after timing recovery, it does not contribute any new information to the code. Since this evaluation assumes coherent reception, the CP is not added to the transmitted signal. However, there is a loss associated with the addition of the CP [1]. This loss is caused by the increase in noise observed at the input to the receiver. Relevant signal information is measured over the bit period Tb, but the noise is present over the symbol period Ts = Tb+Tg. The ratio of CP time simulated in this evaluation is 1/8 and Tg=G·Tb. Therefore, the theoretical decrease in performance is:

dBTTP

s

be 51.0

98log10log10 1010 −=⎟

⎠⎞

⎜⎝⎛=⎟⎟

⎞⎜⎜⎝

⎛=∆ Equation 14

16 DRDC Ottawa TM 2006-026

The theoretical loss associated with the insertion of the CP is 0.52 dB. Together, with the loss associated with the unused OFDM carriers, the overhead loss inherent to WiMAX is -1.76 dB.

5.8 Channel Model

Two channel models were tested in this evaluation. The first is the standard AWGN channel and the second is a frequency selective channel.

5.8.1 AWGN Channel

The AWGN channel is the standard noise channel. The noise has 0 dB gain and a variance of N0/2. The OFDM data consists of 256 complex samples of the time domain signal. A 256-sample complex-noise vector is added to the input signal and received by the OFDM demodulator.

5.8.2 Frequency Selective Channel

The frequency selective channel is a variant of the AWGN channel from section 5.8.1. The transmitted data is first filtered through a frequency selective channel and then impaired by an AWGN signal at the receiver. The gain of the filter is 0dB. The magnitude and phase of the frequency selective filter used is illustrated in Figure 10.

Figure 10. Magnitude and Phase response of the frequency selective channel.

DRDC Ottawa TM 2006-026 17

6. Simulation Procedures

The simulation evaluated the performance of WiMAX for probability of bit error (Pb) values greater than 1x10-5. The simulation was initialized with an Eb/No = 0 dB. The Eb/No value was incremented by 1 dB when a predefined number of errors were measured or a predefined number of message frames were sent. The maximum number of errors was a variable, ranging from 500 to 1000. The maximum number of message frames was a variable ranged from 1000 to 3000.

The simulation processing chain is illustrated in the following flowchart:

Figure 11. Flowchart of the simulation processing chain.

18 DRDC Ottawa TM 2006-026

7. Results

All simulations were performed using Matlab 6.5.

7.1 Simulated WiMAX Overhead Losses

The CP loss and the combined CP plus under-utilized OFDM carrier loss was simulated in MATLAB. The theoretical QPSK performance was used as a benchmark. This performance was calculated using Equation 12. The comparable OFDM transmission is the 256-OFDM transmission with all carriers used. The performance is very similar since each OFDM channel can be interpreted as an individual QPSK channel. Figure 12 indicates the performance of QPSK and WiMAX, as well as the overhead losses inherent to WiMAX.

The theoretical and simulated losses are displayed in the following figure.

DRDC Ottawa TM 2006-026 19

Figure 12. Performance losses due to cyclic prefix insertion and unused OFDM carriers.

The insertion of the guard band was simulated by increasing the noise level by 1/G. The theoretical value was calculated with the same noise increase. As calculated in Section 5.7, a performance decrease of 0.51dB is realized as a result of the guard interval insertion.

The suboptimal utilization of 192 of the total 256 OFDM carriers causes an additional overhead penalty. In section 5.6 this decrease in performance was found to be 1.25 dB. This is comparable to the simulated results in Figure 12.

As the results indicate, the simulated and theoretical losses inherent to CP insertion and unused OFDM carriers simulated for the theoretical losses match very closely. The theoretical 1.76 dB decrease in performance is realized. For the remainder of the WiMAX simulations, the overhead will be included to give the most accurate performance estimate.

7.2 Performance WiMAX Coding Components

7.2.1 Performance of the Convolutional Coder

As calculated in section 5.2.2, the convolutional coder is expected to have a hard decision asymptotic coding gain of 3.01 dB and a soft decision asymptotic coding gain of 6.02 dB. Since a hard decision data slicer was used in conjunction with a soft decision Viterbi decoder, the coding gain is expected to be greater than the better performer, the soft decision decoder.

Additionally, the performance of the coding scheme is lower bounded by [6]:

⎟⎟⎠

⎞⎜⎜⎝

⎛ ⋅=

⎟⎟⎠

⎞⎜⎜⎝

⎛••

⋅⋅=

⎟⎟⎠

⎞⎜⎜⎝

⎛≥

o

b

o

b

o

bfreeb

NE

Q

NE

Q

NEd

Qk

p

8241

98

25619262

241

21

Equation 15

Where 192/256 is the loss due to suboptimal FFT usage and 8/9 represents the loss due to guard insertion. The performance of the convolutional coder is illustrated in Figure 13.

For the following upper bound [7] was used to predict the performance of the convolutional coding scheme:

)/)(2/( obfree

free

NEdRdb eBP ⋅−≈ Equation 16

20 DRDC Ottawa TM 2006-026

Where Bd_free is the weight of the information bits on the dfree1 path. Since dfree = 10 is the path associated with a single one as the input, Bd_free = 1. The value of dfree is 6 after puncturing the code to a rate R=2/3.

Figure 13. Performance of the WiMAX convolutional coder.

As Figure 13 illustrates, the performance bound of Equation 15 is a very weak lower bound. A more accurate means of determining the error performance of large constraint length convolutional codes is through simulation [7], especially under low signal to noise ratios.

The simulated data is constrained by the upper bound as expected and the bound appears to be tighter than that of the lower bound. The simulated data falls between the upper and lower bounds for the AWGN channel.

7.2.2 Performance of the Reed-Solomon Coder

The Reed-Solomon coding scheme has a rate 3/4, dfree 9, k = 24 and n = 32. The lower bound of this RS code is governed by the following equation [4,8]:

( )( ) ( ) iNM

iM

N

ti

Nik

k

b PPiN

P −

+=

−⎟⎟⎠

⎞⎜⎜⎝

⎛−

≥ ∑ 1112

21

1

Equation 17

DRDC Ottawa TM 2006-026 21

( )( ) ( ) iiN

tiib PPiP −

+=

−⎟⎟⎠

⎞⎜⎜⎝

⎛−

≥ ∑ 3244

1

3224

23

1321

122

The simulated performance of the WiMAX Reed Solomon coder, theoretical lower bound performance and hard decision asymptotic coding gain calculated in section 5.2.1 are illustrated in Figure 14.

Figure 14. Performance of the WiMAX Reed-Solomon Coder.

As Figure 14 indicates, the lower bound is a good approximation of the performance of the Reed Solomon coding scheme. The greater the Eb/No, the closer the simulated output will be to the lower bound.

7.2.3 Concatenated RS-CC Coder

The RS-CC concatenated coder is a unique configuration of complimentary coding schemes. The convolutional coder is capable of correcting many errors when they are randomly distributed. When the decoder enters an error state, a burst of errors is passed to the RS decoder. The strength of the RS decoder is in decoding burst errors. This particular decoder is capable of correcting four symbols which are in error. The simulated performance of the concatenated code and its components is illustrated in Figure 15.

22 DRDC Ottawa TM 2006-026

Figure 15. Performance of the concatenated RS-CC coder.

Figure 15 demonstrates the performance of the concatenated RS-CC coding scheme. The simulation was run until the performance of the RS-CC coder produced a bit error rate less than 10^-6. The concatenated coder provides a slight increase in performance over the convolutional encoder up to Eb/No = 5 dB. After this point, the RS decoder corrected all errors passed from the Viterbi decoder. Since the next value for Eb/No is 0 = 10-inf, Matlab does not plot this value and the sharp drop in probability of bit error is not as apparent.

7.3 Comparison of Coded QPSK vs. Coded WiMAX over an AWGN Channel

The following is a comparison of the RS-CC coding scheme over a wideband QPSK channel and the WiMAX 256-OFDM implementation. Figure 16 depicts the simulated performance of uncoded and concatenated RS-CC coded QPSK and OFDM.

DRDC Ottawa TM 2006-026 23

Figure 16. Comparison of QPSK vs. WiMAX performance with and without coding.

As Figure 16 illustrates, a considerable performance gain can be achieved through the use of concatenated RS-CC coding. The doubling of the bit rate caused by the coding is a small price to pay for the substantial coding gain.

For a probability of bit error of 1x10-4 the simulated results indicate a coding gain of 5 dB. The simulated results indicate the rate of increase in performance relative to the decrease in Eb/No is much greater for the coded results than for the uncoded results. As a result, one can expect greater coding gains for higher Eb/No levels.

It is not fair to compare the QPSK directly to the WiMAX results as the WiMAX results include overhead losses and the QPSK data does not. Assuming similar overhead, the QPSK performance would shift to the right and produce results similar to WiMAX. This is apparent when viewing the relative changes between the two modulation schemes, both exhibit the same degree of coding gain.

24 DRDC Ottawa TM 2006-026

7.4 Comparison of Coded QPSK vs. Coded WiMAX over a Frequency Selective Channel

The following is a comparison of the RS-CC coding scheme over a wideband QPSK channel and the WiMAX 256-OFDM implementation. Figure 17 depicts the simulated performance of uncoded and concatenated RS-CC coded QPSK and OFDM over the frequency selective channel described in section 5.8.2.

Figure 17. Simulated performance of QPSK and OFDM over a frequency selective channel.

Once again, the concatenated RS-CC coding scheme demonstrates a considerable gain in performance in the presence of channel impairments. The frequency selective channel causes severe performance degradation. This is apparent, in the uncoded simulation results. Even with high SNR values, a very high error rate (~1x10-2 for uncoded WiMAX) was measured.

The relative performance gains are similar for both modulation schemes. The coding demonstrates a performance gain of several orders of magnitude for high values of Eb/No. With similar overhead losses, the QPSK modulation scheme is expected to perform very closely to the WiMAX results. This is expected, since both transmission formats have the same underlying modulation scheme, QPSK.

DRDC Ottawa TM 2006-026 25

8. Discussion

As the results indicate, the concatenated RS-CC code is a very robust code. Individually, each coding scheme is simple, and provides moderate coding gain. By combining, them, they compliment each other, producing a powerful, yet computationally moderate coding scheme. The Viterbi algorithm is very powerful, but prone to burst errors. When it enters an error state, the output is a burst of error bits. One of the many strengths of the Reed Solomon decoder is its ability to correct burst errors. As the simulation results indicate, the performance of each individual decoder is reasonable, but combined, they perform impressively.

The relative gains observed in each modulation scheme seem to be similar. For both the AWGN and frequency selective channel, the RS-CC coding performs very well. Even under the severe degradation of a frequency selective channel, the RS-CC coding scheme performs very well.

One of the best methods of increasing the performance of a communication system operating over a frequency selective channel is channel estimation coupled with equalization. This is a difficult problem for wideband systems. However, with OFDM, the problem is simplified. The frequency spectrum can be separated into multiple narrowband channels. For this reason, considerable effort has been directed at channel estimation and equalization for WiMAX. With these performance enhancements, the performance of the frequency selective channel is expected to approach that of the AWGN simulations.

26 DRDC Ottawa TM 2006-026

9. Conclusion

The concatenated RS-CC code demonstrated considerable coding gain when impaired by AWGN and frequency selective channels. The results are greater than either of the two coding formats on their own, with minimal additional complexity. The coding schemes compliment each other. The high performance of the Viterbi decoder produces burst errors, when the decoder fails. The RS-CC performs very well when the bit errors are confined to a small number of symbols. Together they form a very capable code without the complexity associated with single coder strategies with similar error correcting performance.

In addition to the problems associated with peak to average ratio and doppler effects; channel estimation and equalization must be studied to combat the severe degradation caused by frequency selective channels.

DRDC Ottawa TM 2006-026 27

References

[1] Heiskala, Juha and Terry, John; OFDM Wireless LANs: A Theoretical and Practical Guide; SAMS Publishing; Indianapolis, IN, 2002.

[2] WiMAX Technology LOS and NLOS Technology, White Paper 033-100596-001, SR Telecom Inc. Montreal QC, 2004.

[3] IEEE Standard for Local and Metropolitan Area Networks, IEEE Std 802.16-2004, New York, NY, 2004.

[4] Proakis John G.;Digital Communications 4th Edition, The McGraw-Hill Companies Inc.; New York, NY 2001.

[5] Haykin, Simon; Communication Systems 4th Edition, John Wiley & Sons Inc. Hoboken NJ 2001.

[6] Wicker Stephen B; Error Control Systems for Digital Communication and Storage, Prentice Hall, Inc; Upper Saddle River, NJ 1995.

[7] Lin, Shu and Costello, Daniel J. Jr.; Error Control Coding 2nd Edition, Prentice Hall Inc; Upper Saddle River, NJ 2004.

[8] Gulliver, T. Aaron; Performance Evaluation of Reed Solomon Codes, Department of Systems and Computer Engineering, Carleton University, Ottawa ON, 1993.

[9] Rappaport, Theodore S.; Wireless Communications: Principles and Practice 2nd Edition, Prentice Hall Inc; Upper Saddle River, NJ 2002.

[10] Emerging Wireless Technology: Focus On WiMAX, RF Design Magazine, RF Micro Devices Inc, Greensboro, NC, 2005.

28 DRDC Ottawa TM 2006-026

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DRDC Ottawa TM 2006-026 29

List of symbols/abbreviations/acronyms/initialisms

ARQ Automatic Repeat Request AWGN Additive White Gaussian Noise BCC Block Convolutional Coding BTC Block Turbo Coding CP Cyclic Prefix CTC Convolutional Turbo Coding dfree Free Distance dmin Minimum Distance DFT Discrete Fourier Transform DND Department of National Defence Eb Bit Energy FEC Forward Error Correction FSK Frequency Shift Keying G Cyclic Prefix Insertion Factor ICI Inter-Carrier Interference IDFT Inverse Discrete Fourier Transform ISI Inter-Symbol Interference LAN Large Area Network LDPC Low Density Parity Check LOS Line of Sight MAN Metropolitan Area Network MCM Multi-Carrier Modulation MDS Maximum Distance Separable No Noise Level NLOS Non-Line of Sight OFDM Orthogonal Frequency Division Multiplexing OFDMA Orthogonal Frequency Division Multiple Access Pb Probability of Bit Error QoS Quality of Service R&D Research & Development RS Reed Solomon RS-CC Reed Solomon-Convolutional Coding SC Single Carrier Tb Bit Period Ts Symbol Period

30 DRDC Ottawa TM 2006-026

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DRDC Ottawa TM 2006-026 31

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WiMAX is the next generation of wireless internet solutions. WiMAX is designed to provide the “last mile” solution, bringing wireless internet to our homes and businesses.

Increased Quality of Service is obtained through the use of multiple air interfaces, adaptive modulation and coding to provide the optimum performance to various users under various signal strength conditions and various application requirements.

This paper is a performance study of WiMAX’s coding performance using the concatenated Reed Solomon-Convolutional coder. The implementation uses an OFDM air interface, the modulation format selected is QPSK and the channel models used were AWGN and a frequency selective channel.

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WiMAX, OFDM, Reed, Solomon