FEC Coding with a Rate Adaptive Spre!ad Spectrum OFDM System

FEC Coding with a Rate Adaptive Spre!ad Spectrum OFDM

System

Zhong Ye*, Gary J. Saulnier”, Dennis Lee* , and Michael J. Medleyt “ ECSE Dept, Rensselaer Polytechnic Institute

Troy, New York 12180-3590

fAiT Force Research Laboratory, IFGC Rome, New York 13441-4505

ABSTRACT

This paper investigates the use of a spread spectrum OFDM mod-ulation to provide a variable rate communications link that is tolerant of multipath fading. The system adaptively varies the amount of spreading in response to changing channel conditions in an attempt to optimize throughput while maintaining acceptable error rate performance. System performance results for AWGN and Rayleigh fading channels, both frequency selective and fre-quency non-selective, show that a desired BER performance can be maintained, A trade-off analysis between Forwa,rd Error Cor-rection (F’EC) coding and spreadhg for a fixed bandwidth ex-pansion factor is performed. For convolutional coding the results shown tha,t a low-rate code provides better performimce while, for a Reed-Solomon code, a coding rate of approximately 1/2 is best.

1. INTRODUCTION

Orthogonal frequency division multiplexing (OFDIM) [1, 2] has received considerable attention as a method to efficiently uti-lize channels with non-flat frequency responses and/or non-white noise. In its most common form, a high rate data stream is divided up among the many carriers in the system in a manner which op-timizes the capacity of the overall channel. OFDM can also be used as a spread spectrum modulation (OFDM-SS) [3, 4, 5, 6] wherein spectral spreading is accomplished by putting the same data on all the carriers, producing a spreading factor equal to the number of carriers. At the receiver, the energy from all the carriers is coherently combined to produce the decision variable. Multiple users can be supported in the same channel through Code Division Multiple Access (CDMA) [7].

Conventional OFDM can provide high rate services at the ex-pense of low tolerance to channel impairments su,ch as interfer-ence while OFDM-SS provides interferinterfer-ence immunity at the cost of bandwidth. We have introduced the concept of rate adaptive OFDM (RA-OFDM) [8,9] which combines the advantages of both the conventional OFDM and OFDM-SS systems. When anti-jam capability is needed, the signaling format is adjusted to increase the spreadhlg while, when the channel is clear, spreading is re-duced to increase the throughput. More specifically, in a favorable channel, the system would operate much like the conventional OFDM system where different data bits are transmitted on dif-ferent carriers yielding high throughput. In a severe interference channel, the system would operate like the OFDM-SS system, sacrificing throughput to provide the needed processing gain to overcome the jamming. In most cases, the system would operate between these two extremes, providing sufficient processing gain to mitigate the channel effects while maximizing throughput.

In the following sections, we present the RA-OFDM system which allows a smooth transition of processing gain to track the changing channel conditions. The performance of the system un-der both AWGN and frequency selective and flat fading channels is investigated with special attention to the trade-off between for-ward error correction (FEC) coding and spreading. The paper *This work is partially supported by the Air Force Research Labo-ratory, Air Force Materiel Command, USAF, under agreement number F30602-98-1-0054 and contract number F30602-98-C-0018. Mr. Ye is also partially supported by the GTE Graduate Fellowship.

concludes by providing some insight into the results through the use of asymptotic coding gain results.

2, THE RA-OFDM SYSTEM

With OFDM modulation, variable throughput can be imple-mented by changing the amount of redundancy in the placement of data bits on the carriers. For simplicity, assume initially that the data is not coded. The highest throughput is accomplished by sendhg different data on each of the carriers. This arrange-ment does not provide significant anti-jam capability nor does it produce any frequency dhersity to help mitigate the effects of frequency selective fading. One way to insert a processing gain of 2 is to send the same data on a pair of carriers, preferably spaced apart in frequency to produce frequency diversity for both multipath and anti-jam protection. Similarly, the processing gain can be increased to four by sending the same data on 4 carriers, a processing gain of 8 by sending the same data on 8 carriers, etc. The maximum processing gain available equals the number of carriers in the system.

The performance of an uncoded RA-OFDM system (i.e., spreading only) has been investigated in [8, 9]. Performance of the system under AWGN, frequency selective and frequency flat fadin,g has shown that the system can maintain a desired BER through the proper control of the processing gain. In this pa-per we will investigate the pa-performance of the RA-OFDM system using FEC codes. An important issue is how to divide the band-width expansion between spreading and coding. For example, if the bandwidth is going to be expanded by a factor of 8, should no cclding and a processing gain of 8 be used or a rate 1/2 code and ii processing gain of 4 or a rate 1/4 code and a processing gain of 2? This trade-off will be examined for both block codes and convolutional codes.

This trade-off between spreading and coding in the RA-OFDM system will be studled both analytically and through simulation. In all cases, perfect carrier and symbol synchronization is assumed between the transmitter and receiver. In the simulation, the RA-OFDM modulator with variable processing gain and a number of di Eerent FEC coders, several channel models, and the rate adaptive receiver are implemented and used to perform Monte Carlcl simulations. The equalizer coefficients are derived directly from the known channel response rather than through the use of an aclaptive algorithm. The equalizer coefficients are the complex conjugate of the channel response for the corresponding frequency band. As a result, the simulations provide an estimate the per-formance that would be provided by an ideal equalizer and does not include any degradation due to imperfect equalizer channel estimation or tracking.

The channels considered are the AWGN channel and frequency selective and frequency-nonselective (flat) Rayleigh fading chan-nels. The flat Rayleigh fading channel is implemented in the sim-ulation by filtering a complex Gaussian using the method from Jakes [10]. The frequency selective channel is implemented by summing multiple rays spaced at sample intervals, T.,each having the same power. Each ray has independent flat Rayleigh fading with the same Doppler rate. The resulting signal has a uniform intensity profile and a total delay spread equal to the number of rays in the channel times the sampling rate.

For analysis purposes, the slow flat Rayleigh fading channel is modeled by an A/f-state Markov chain where each state corre-sponds to a particular level of fading or, equivalently, a particu-lar received SNR. This concept of a finite-state Markov channel emerged from the early work of Gilbert [11] and Elliott [12]. Wang extended the concept of the two-state Gilbert-Elliott channel into the finite-state Markov channel [13]. The channel model is shown in Figure 1, where S~ corresponds to the ith channel state and

P12 99 ““” P1,1 P2,2

P

M.lM ---x Q SM .~ PhI,M-1 p M,M

Figure 1. ill-state Markov chain model for a R;ayleigh fading channel.

there are M possible states. The channel is in state S; when the received SNR, ~, satisfies ‘yi-1 ~ T < ~i for some cOnStants 7+1 and Ti.

Unless otherwise specified, the RA-OFDM signal consists of 64 carriers, the FFT in the receiver has 64 bins andl perfect sym-bol synchronization is assumed, i.e. the blocks of the data pro-cessed by the FFT are time aligned with the symbol intervals of the non-delayed path. Similarly, all processing is performed at baseband, effectively assuming perfect carrier sy nchronizat ion. Random data is sent by the transmitter and binary signaling is used.

The goal of the RA-OFDM system is to adapt the format of the transmissions to compensate for a wide variety of channel condl-tions. As the quality of the channel changes, the RA-OFDM system can respond by changing any of a number of parameters, including the amount of processing gain, the code type and/or rate and the type and/or depth of the interleaving. Any adaptiv-ity, however, comes with a cost, either in the form of additional signaling overhead or increased transceiver complexity. As a re-sult, our goal is to develop a combination of signaling parame-ters that can provide acceptable performance over a broad range of conditions with a minimum of adaptation cost. The system, therefore, may not be strictly optimal for any particular channel but should provide acceptable performance while making efficient use of channel resources.

A mobile packet radio system can expect to experience many different channel conditions, including AWGN, flat fadkg and frequency selective fading. The fading channels can be changing slowly or rapidly. In cases where the channel is static or changing slowly, the RA-OFDM system will track the channel variations, changing the signaling parameters to maintain the best link for the current conditions. When channel variations are too rapid to be tracked, such as in fast fading, the signaling parameters will adjust to the statistics of the channel rather than tracking instantaneous variations. Under these conditions, interleaving is necessary to break up the channel correlation bebween symbols. The definition of what rate of channel fluctuations is “too rapid” to track depends on how a transmitter obtains information about the channel from itself to a receiver and, in particular, how current that information can be. In any event, unlike conventional non-adaptive schemes, the interleaving depth will not have to become excessive when the fading becomes very slow because tracking the channel and adapting the signal to it will become the primary mitigation technique.

3. SYSTEM PERFORMANCE 3.1. Convolut ionally coded RA-OFDM Syntem

Convolutional codes, especially when coupled with soft decision decoding, can provide large coding gains. These codes can be

used in packet-based systems by appending known tail bits or using “tail-biting” techniques [14, 15]. Here we will investigate the performance of the RA-OFDM system when convolutional codes are combined with the spreading.

If we treat the spreading as an inner repetitional code and the convolutional code as an outer code, we can analyze the perfor-mance of this concatenated coder using the standard coding state diagram and transfer function method from [16]. This approach allows us to determine the upper bound on bit error probability when decoding is performed using the Viterbi algorithm. For soft decision decoding in AWGN, such an upper bound for the bit error probability is [17]

(

)

~fRc7b cW(D, N)

pb<Q

~m

e

~N

k=,,

D=e-@c

(1)

where ~b is used to denote Eb/NO, df is the free distance of the code, Rc is the code rate and T (D, N) is the generating function for the code. Note that the product ~bR. appears in the bound which accounts for the fact that the effective SNR will vary with coding rate because to total bandwidth expansion is being held constant and additional coding results in less spreading. Table 1 lists the generating functions for several codes [18].

_ ,- _-— , 340D’ + . . . ...) k — ‘37Db +....\ 16

E ~~

Rc ‘-IN.1 1/2 Dl”(2 + 22D~ + 60D2 + 14~~4 + ~ 1/3 D’G(l + 24D’ + 113D- +28 114 D’2(2+10D’ +10 D’+8D’~ln D4~ . . ..’..~ I 9.?

Table 1. Generating functions for several convolutional codes. Figure 2 shows the upper bounds for 128-state codes with rates 1/2, 1/3 and 1/4. The bandwidth expansion factor, i.e. the processing gain divided by R., is held constant for all the curves, though its actual value does not ailect the results. It is clear that

‘“-r

-N,

,,,

,y

,

“.tl

10,5 1 15 2 25 3 3,5 4 45 5

Eb/NO in dB

Figure 2. Performance bounds for 128-state convolutional codes in AWGN with a fixed bandwidth expansion factor.

a lowering the rate code results in better BER performance. This result is expected due to the fact that free distance increases as the code rate decreases. However, the incremental gain attained with a reduction in code rate decreases as the code rate gets lower, meaning that the added cost of a very low rate code may not be justified.

Wlhenconsidering flat fading, we will investigate both fast fad-ing, where an interleaver can be used to remove the channel corre-lation between symbol intervals, and slow fading, where full inter-leaving is not possible. With fast fading it is practical to use an

interleaver that is sufficiently deep such that the de-interleaved symbols can be considered independent from one symbol inter-val to the next. Mui derived a performance bound for this case in [18] and Hagenauer further tightened the bound in [19]. Using soft decision decoding and given the perfect knowledge of the fad-ing magnitude, the BER of the coded system is upper bounded by

“+$4+;..,)”

(2) where K is number of information bits in a (N,K) code. Figure 3 shows the performance bounds and shows that, if ideal

interleav-cessing gain andjor code rate changes to track the channel con-ditions. Two strategies were compared: strategy “A” uses fixed rate 1/2 convolutional code and adapts the processing gain from 1 to 8 in steps of 2 and strategy ‘(B” uses no processing gain and adapts the coding rate from 1/2 to 1/16 in the steps of 2. A 4-state Markov chain is used to model the fading process and (3) is used to estimate p, in each state. Figure 4 shows the perfor-mance for both approaches and indicates that the fixed code rate and variable processing gain has a slight advantage at high values of ~,/~iJ whereas the variable code rate system is better at low values of E,/NO. 0.1 Strategy A -+-Strategy B -+---I ,e.(), ~ 10 15 20 25 30 Eb/NO in dB

Figure 4. BER performance for two adaptation strategies in slow Rayleigh fading (~~1’ = 5 * 10-5).

Figure 3. Performance for 256-state convolutional codes with var-ious code rates under Rayleigh fading.

ing is achievable, a lower rate code is advantageous. Once again, however, we see that the gains become smaller as the code rate decreases.

In considering the performance over an uninterleaved slowly fading channel, the channel is assumed to be partiidly correlated from one symbol interval to the next. The results in [20, 21, 22] show that the performance of a coded system under these condi-tions is poor and additional coding may further deteriorate the performance. Interleaving helps, but due to time delay and stor-age constraints, only limited interleaving is possible. When fading is sufficiently slow, the fading magnitude, A, can be treated as a constant over a packet duration, where A is Rayleigh dktributed. An obvious approach to bounding the code performance on a slow fading channel is to apply the union bound procedure to get a bound on the conditional bit error rate, PblA, which can then be integrated over the probability density of A to obtain a bound on the unconditional bit error probabilityy, Pb. As pointed out by Mui in [18], this bound is tight only under weak fading condi-tions because the conditional bound is loose for small values of A where the received Eb/NO is low. The high error rates under these conditions dominate the final result.

To avoid this problem, we generate a table of estimated PblA values by simulating the system in AWGN. If the simulated perfor-mance for a particuhw code in AWGN is given by pb = g(E,/~0 ) where g(. ) is a numerically tabulated function, then. the predicted error probabilityy is given by

/

w

Pb = g(A2Eb/iVo)f(A)dA, (3)

o

where f(A) is the probability density function of A. This ap-proach was extended to the RA-OFDM system where the

pro-With frequency selective fading, the channel can be modeled analytically by decomposing the signal bandwidth into L equal sub-bands, each having a bandwidth equal the coherence band-width of the channel. Each one of these sub-bands is assumed to fade non-selectively and independently. In short, path diver-sity is exchanged for frequency diverdiver-sity of the same order. The ideal equalizer in the RA-OFDM receiver performs maximum ra-tio combining during the respreading process by multiplying each carrier by the complex conjugate of the channel response for the corresponding frequency band and summing the products. This process implements a frequency-domain version of the RAKE re-ceiver. Schramm has published a tight upper bound on the bit error probability of a convolutionally encoded spread spectrum system over a frequency selective Rayleigh fading channel in [23]. This bound was derived for a time domain RAKE receiver but can be applied to the equivalent frequency domain structure in the RA-OFDM receiver. Optimum decoding with the use of chan-nel state information is assumed and the resulting bound on BER at high SNR is lm p,<—

x

r(czL – 1/2) 1 2fiK (4) d=dmin c’ r’(dL) (E&)d’ ‘

where I’(.) is the gamma function, K is the number of information bits in a (N,K) code and L is the number of multipath rays. The key result is that combining coding with multipath propagation in a spread spectrum system results in an effective total diversity that is the product of the channel diversity, L, and the diversity d due to the code at a specific Hamming distance.

Using this result, we can draw some conclusions about the trade-off between coding and spreading for the selective fading situation. When there is channel diversity to be exploited, i.e. the processing gain is less than L, increasing the processing gain has almost the same effect as reducing the code rate because dmi~ in-creases almost linearly as the coding rate dein-creases [16]. Once the processing gain reaches L, dedicating more resources to spreading

will not create any additional diversity and it is generally better to assign the remaining resources to coding.

In considering the results for all the channels, it is generally advantageous from a performance perspective to devote band-width to coding rather than spreading when convolutional coding is used. From a practical viewpoint, however, the adaptation of code rate over a large dynamic range can be prohibitively costly and is generally avoided. With this in mind, the performance results indicate that the use of a fied code rate on the order of 1/2 or 1/4 and variable processing gain may provide a good compromise between performance and complexity.

3.2. Reed-Solomon coded RA-OFDM System

Now we will consider the trade-off between spreading and coding for a RS block code using hard-decision, error-correction decod-ing. The underlying modulation is antipodal signaling and multi-ple bits are grouped together to form RS symbols. Again, we will consider the performance in AWGN, flat fading and frequency-selective fading channels. Unlike the study of convolutional codes, the performance will be evaluated using simulation results rather than performance bounds.

Figure 5 compares the performance when the bandwidth ex-pansion due to coding and spreading is kept fixed at 16. The

le-06 t

\k

t ,,.,, ~ o 2 4 6 8 10 Eb/NOin~B

Figure 5. Performance results in an AWGN channel for various combinations of code rate and spreading for RS codes with a bandwidth expansion factor of 16.

coder block size N = 15 and information symbols/block (K) vary from 1 to 8 to produce various coding rates, resulting in code rates of K/15. The best performance is obtained by the system that uses a moderate rate coder (l?= = 1/2). Also, it is noteworthy that a very low rate coded system (l?. = 1/16) does not produce an acceptable BER performance, most likely because the absence of processing gain results in the decoder operating on a received SNR that is too low to allow the coding gain to be released. To the contrary, inserting some processing gain prior to decoding boosts the SNR seen by the decoder and allows a higher rate code to work effectively. Results obtained for different bandwidth expansion factors show the same trend.

The RS coded RA-OFDM system was simulated using a 4-ray frequency-selective fading channel. In order to allow a fair assess-ment of the frequency diversity utilization, a system with 1024 carriers was used to allow the spread code word to fit in one FFT block. It is desirable to have errors within each code symbol to be correlated, and errors between symbols to be independent [24]. For the RS codes over GF(16), the 4 bits forming a code sym-bol are placed on adjacent carriers while the symsym-bols forming a codeword are separated in frequency. Each RS symbol is further repeated by the amount of the spreading processing gain. The total bandwidth expansion factor was set at 30. Figure 6 shows the system performance and, once again, the use of a relatively

1 0.1 0.01 O.col 0.0001 h-es ... x, “.,., ‘...~ 1,.06 x

I

.- .. 2 4 6 8 10 12 14 16 Eb/NOin dB

Figure 6. Performance of a Reed Solomon coded RA-OFDM spread spectrum system in a 4-ray selective fading channel with f,=50.

high-rate code and spreading provides the best performance. In a slow flat fading channel, the RA-OFDM system will track the channel variations and, therefore, the channel attenuation will be roughly constant over the duration of each packet, making the channel for each packet appear like an AWGN channel. Therefore, it is expected that the above coding/spreading trade-off results for AWGN will apply to this case. When the fading gets so fast that rate adaptation cannot be used to track the channel variations, coding and interleaving is used to maintain performance at an acceptable level. Figure 7 shows the performance under these conditions and, once again, demonstrates that the best resource

Figure 7. Trade-off between block coding and spreading for fast Rayleigh fading channel with a bandwidth expansion factor of 1.6. allocation policy is to use a rate 1/2 coder and dedicate the re-maining bandwidth expansion for spreading.

The results for RS codes are quite different than those for the convolutional codes. For RS codes, using a rate 1/2 coder with spreading is best while, for the convolutional codes, using low ra,te codes is best. Some insight into the behavior of the RS codes can be gained by examining the asymptotic coding gain, G, for these codes which has been derived in [25, 26]. G provides the coding gain, relative an uncoded system, as the SNR goes to infinity and can be a convenient way to compare code performance. For RS codes using hard decision decoding in an AWGN channel, G can

be written as

G= R.(t+l) (5)

where R. and t are the code rate and number of errors corrected, respectively, where t = [~J. Table 2 shows the parameters for the RS codes we used in the simulations. As the table shows,

( ) R. t G– (W) & 3 ~ (15,4) & 5 $-(15,2) & 6 $-(15,1) + 7 # — —

Table 2. Asymptotic coding gain for the RS codes used in the simulations.

lowering the coding rate does not provide more codhg gain for these codes and the largest gain is achieved using a rate 1/2 coder. The use of soft decision decoding with the RS codes would not significantly change this behavior. In this case, the asymptotic coding gain can be expressed as

G = R.(2t+ 1), (6)

resulting in approximately 3 dB more coding gain relative the hard decision case. However, G still favors the use of high-rate codes.

4. DISCUSSION

The results that were presented show that when using FEC coding in the RA-OFDM system, the optimal trade-off between coding and spreading depends on both the channel conditions as well as the type of code used. In general, however, it appears that the use of a fixed code rate of approximately 1/2 with a variable processing gain will provide a good compromise between BER performance and system complexity.

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