Design And Analysis Of Turbo Codes On Rayleigh Fading Channels

160 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 16, NO. 2, FEBRUARY 1998

Design and Analysis of TurboCodes on Rayleigh Fading Channels

Eric K. Hall and Stephen G. Wilson,Member, IEEE

Abstract—The performance and design of turbo codes usingcoherent BPSK signaling on the Rayleigh fading channel is con-sidered. In low signal-to-noise regions, performance analysis usessimulations of typical turbo coding systems. For higher signal-to-noise regions beyond simulation capabilities, an average upperbound is used in which the average is over all possible interleavingschemes. Fully interleaved and exponentially correlated Rayleighchannels are explored. Furthermore, the design issues relevantto turbo codes are examined for the correlated fading channel.Turbo interleaver design criteria are developed and architecturalmodifications are proposed for improved performance.

Index Terms—Codes, concatenated coding, fading channels,interleaved coding, Rayleigh channels.

I. INTRODUCTION

FOR wireless applications on fading channels, channelcoding is an important tool for improving communications

reliability. Turbo codes, introduced in [1], have been shown toperform near the capacity limit on the additive white Gaussiannoise (AWGN) channel. As a powerful coding technique,turbo codes offer great promise for improving the reliabilityof communications over wireless channels where fading isproblematic. To date, only limited attention has been givento the performance of turbo codes on fading channels [2], [3].In this work, we explore both the performance and designof turbo codes for fully interleaved channels and correlatedRayleigh slow-fading channels.

The organization of the paper includes a brief overview ofturbo codes followed by a discussion of the channel model. Wethen proceed to discuss the turbo code average upper boundingtechnique along with an examination of two-codeword prob-ability bounds on correlated and independent Rayleigh fadingchannels. Simulation results are then presented for typicalturbo schemes, followed by the results and conclusions fromour simulations and the average bound. We conclude with anexamination of the design of turbo codes on correlated fadingchannels.

II. SYSTEM MODEL

Turbo codes, introduced in [1], are, in essence, parallelconcatenated convolutional codes (PCCC). The turbo en-

Manuscript received October 1996; revised April 25, 1997. This work wassponsored by the National Science Foundation under Grant NCR-9415996and NASA/LeRC under Contract NAG3-1948. This paper was presented inpart at CISS’96, Princeton, NJ, March 1996 and the 6th Mini-Conference onCommunications in conjunction with IEEE GLOBECOM’96, London, U.K.,November 1996

The authors are with the Department of Electrical Engineering, Universityof Virginia, Charlottesville, VA 22903 USA.

Publisher Item Identifier S 0733-8716(98)00164-4.

coder is constructed from simple recursive systematic convo-lutional (RSC) encoders connected in parallel and separatedby interleavers. The decoder uses an iterative, suboptimal,soft-decoding rule where each constituent RSC is decodedseparately. The constituent decoders then participate in sharingof bit-likelihood information in an iterative fashion. Theconstituent decoders traditionally use the BCJR algorithm [4],which is a MAP symbol decoding algorithm for block andconvolutional codes. While the global turbo decoder is not ML,it has been shown to perform within 0.7 dB of the Shannonlimit on the AWGN channel for bit-error rates (BER’s) of 10and message lengths of 65 536 [1].

In this paper, we consider coherent BPSK signaling over anondispersive Rayleigh slow-fading channel. With appropriatesampling, the discrete representation of this channel is

where is an integer symbol index, is a BPSKsymbol amplitude ( ), and is an i.i.d. AWGN com-ponent with zero mean and power spectral density . Thefading amplitude is modeled with a Rayleigh pdf,

for . With sufficient channel interleaving(fully interleaved), the ’s are independent. Without sufficientchannel interleaving, we adopted an exponentially correlatedchannel model as in [5] and [6]. In this model, the continuous-time, autocorrelation function is given bywhere is the Doppler bandwidth and is the lag parameter.

For Rayleigh channels, the turbo decoder must be modifiedto incorporate the appropriate channel statistics. In the MAPalgorithm, this corresponds to formulating the transition metric( ’s). For a fully interleaved channel and known fadingamplitudes (side information, SI), the transition metric from[7] is given as

(1)

The probability is conditionally Gaussian,.

For the fully interleaved channel without side information(NSI), the transition metric is given as

(2)

0733–8716/98$10.00 1998 IEEE

HALL AND WILSON: TURBO CODES ON RAYLEIGH FADING CHANNELS 161

To formulate , we use the law of total probabilityand to write

(3)

In [8], it is noted that the evaluation of the integral in (3) hasno known closed form. To avoid this problem, it is proposedin [8] to assume that is Gaussian in the region ofmost probable . From this assumption, (3) is approximatedas

(4)

(5)

The term is not a function of the conditioning ( ),therefore making its computation unnecessary in the MAPalgorithm [9]. It should also be noted that for the Rayleighchannel with average energy of 1, .

III. PERFORMANCE BOUNDING

The ability to evaluate turbo codes in regions of highsignal-to-noise requires lengthy simulations or an analyticbounding technique. In [10] and [11], an average upper boundis developed for turbo codes. It is shown that this bound is veryuseful in determining the “error floor” as well as understandingthe impact of constituent encoder choice and block size onperformance for the AWGN channel. Here, we apply thisbound to the Rayleigh fading channel.

A. Derivation of the Average Upper Bound

Consider the traditional union upper bound for the MLdecoding of an block code. Without loss of generality,we assume that the all-zeros codeword was sent, and we writethe upper bound on the probability of word error as

(6)

Here, is the number of codewords with Hamming weightand is the probability of incorrectly decoding to

a codeword with weight . For a turbo code with a fixedinterleaver, the construction of requires an exhaustivesearch. Due to complexity issues involved in this search,[10] and [11] propose an average upper bound constructedby averaging over all possible interleavers. The result of thisaveraging can be thought of as the traditional union upperbound, but with anaverage weight distribution. As in [11],the average weight distribution can be written as

(7)

where is the number of input words with Hammingweight and is the probability that an input word

with Hamming weight produces a codeword with Hammingweight . Substituting into (6), the average upper bound forword and bit error can be expressed as

(8)

and

(9)

In (8) and (9), is an expectation with respect tothe distribution . This average upper bound is attractivebecause relatively simple schemes exist for computingfrom the state transition matrix of the RSC [10], [11]. With

, the performance of turbo codes can be studied onvarious statistical channels by formulating the two-codewordprobability for the channel of interest and using (8) or(9).

B. for Fully Interleaved Channels with SI

For the average upper bound, exact two-codeword probabili-ties or tight upper bounds are required. On the fully interleavedchannel with SI, the exact probability of incorrectly decodinga codeword into a codeword which differs from inbit positions indexed by is

(10)

Here, is the tail integral of a standard Gaussian densitywith zero mean and unit variance defined as

To compute the average word error probability, we mustaverage over the channel gains. The resultis a multidimensional integral given as

(11)

If the fading amplitudes are independent, the indexes of thediffering bit positions are of no importance—only the weightof the incorrect codeword matters. Therefore, we can formulatethe two-codeword probability in terms of only the Hammingdistance of the codewords as

(12)


and

For the average upper bound, we need an exact representa-tion of or a tight upper bound. The exact evaluation of(12) is very difficult. To solve this problem, we examine fouroptions. The first option is to simplify (12) to a form that canbe evaluated via numerical integration. The other three optionsavoid the problem of numerical integration by seeking closedform upper bounds for .

Option 1 (Exact): In [12], it is noted that can beexpressed in the alternative form given by

(13)

Substituting into

(14)

Since all the fades are independent, the-dimensional integralfor reduces to a product of integrals over each. Fur-thermore, these integrals have closed-form solutions allowingus to write

(15)

While (15) has no known closed-form solution, it can beevaluated via numerical integration over the single variable.For the average upper bound, we may wish to avoid numericalintegration for every value of and so we nextexamine three bounding options.

Option 2 (Bound): To avoid numerical integration in (15),consider upper bounding the integrand as

(16)

This bound will be tight for large values of and willallow us to upper bound with the closed-form expression

(17)

where

(18)

Option 3 (Bound): Another option for upper boundingon the fully interleaved fading channel with SI is

to upper bound the function in the integrand of (12). Forthis approach, we have two possibilities. First, consider the

-function bound

(19)

With (19), we bound the function of (12) as

(20)

Substituting this bound into (12), we observe a product ofintegrals, each having closed-form solutions, simplifying thebound to

(21)

Option 4 (Bound): As an alternative to (19), consider the-function bound

(22)

From this bound, the function of (12) can be bounded as

(23)

Substituting this bound into (12) and evaluating the integrals,we upper bound as

(24)

In Fig. 1, the four options for the evaluation of the two-codeword probability are illustrated for . The slope of thecurves in this figure is determined by the diversity order which,for independent fading, corresponds to the codeword distance

. For example, Fig. 1 shows a reduction in oftwo orders of magnitude from 10–14 dB. In thesemilog plot, the slope of the curve is

slope

Comparing the bound options to the exact curve, in regionsof low SNR, Option 4 is the tighter of the three bounds.However, in this region, note the weakness of Option 2. Inregions of high SNR, the roles are somewhat reversed, withOption 2 being almost exact, while Options 3 and 4 differfrom the exact by a scale factor.


Fig. 1. Comparison ofP2(d) expressions for the fully interleaved Rayleigh channel with SI andd = 5.

C. for Fully Interleaved Channels with NSI

For the fully interleaved channel with NSI, we use the bounddeveloped by Hagenauer in [8]. This bound is based on thesimplified decoding metric (5), and is given by

(25)

where and .

D. for Correlated Channels with SI

For the exponentially correlated fading channel with SI, wewill appeal to bounds developed in [5] and [6] for . In[6], it is noted that tight bounds on require knowl-edge of thepositionsof differing symbols in the codewords

and . This bound is given as

(26)

In this expression, is the Hamming distance between thetwo codewords, and indicates the time spacing betweendiffering code symbols with duration . For example,(100101) compared to the all-zeros codeword has and

and . Note that there will be ’s.For the union bound, we can loosen the bound in (26) as

in [6] and [5] by making the pessimistic assumption that alldiffering symbols are adjacent ( , ). With this

assumption, the two-codeword probability can be bounded byan expression which is a function ofand given as

(27)

IV. RESULTS

A. Simulation Results

For the low signal-to-noise region, analytic evaluation ofturbo codes has proven very difficult. Therefore, we examineperformance based on simulations. Simulations will considerrate-1/3 turbo schemes using the 16-state RSC with generator(21/37) . This scheme is often denoted (1, 21/37, 21/37) due toits construction via parallel concatenation. The 16-state RSChas 6, one less than the optimum free distance codewhich has 7. However, this is the encoder used inoriginal paper of turbo codes [1].

For the fully interleaved channel, we have plotted simulationdata for rate-1/3 turbo schemes with different block sizes inFigs. 2 and 3. Here, we are considering input frames of length

420, 5000, and 50 000 bits. In each plot, a capacity limitis shown for reference (see the Appendix for a derivation ofthese limits). In all simulations, the turbo decoder uses theBJCR algorithm with modifications found in [7]. The resultsfor 420 are shown after eight iterations, while both

5000 and 50 000 are shown after 15 iterations.It should be noted the simulations without side informationwere done using the optimum metric rather than the simplifiedmetric shown previously [13]. In all cases, a single turbointerleaver is used with the interleaver fixed for all simulatedframes. For 420, a helical interleaver is used whichhas been shown to be effective on the AWGN channel [14].


Fig. 2. Simulations on the fully interleaved Rayleigh channel with side information (SI).

Fig. 3. Simulations on the fully interleaved Rayleigh channel without side information (NSI).

For block sizes greater that 1000, it has been observed thatrandomly generated interleavers generally perform better thandeterministic interleaver designs [15]. Therefore, for5000 and 50 000, the fixed interleaver is generatedrandomly and used without optimization.

With SI, it can be observed that for 50 000, the perfor-mance of is within 0.7 dB of the capacity limit on the Rayleighchannel. However, even for 420, the performance ismuch better than uncoded BPSK which achieves BERat 44 dB. Therefore, these rate-1/3 turbo codes are

capable of coding gains exceeding 40 dB. Without channelside information, the performance degrades approximately 0.8dB, consistent with the corresponding capacity limits (see theAppendix). Furthermore, the performance for 50 000remains within 0.7 dB of the capacity limit for BER .

In Figs. 4 and 5, the performance of the same turbo schemeswith 420 and 5000 is shown for various fadingbandwidths. It can be observed that performance deterioratesrapidly as decreases (fading process slows). It should alsobe noted that for large blocks, the penalty for decreased fading


Fig. 4. Simulations on the exponentially correlated Rayleigh channel withk = 420 and SI.

Fig. 5. Simulations on the exponentially correlated Rayleigh channel withk = 5000 and SI.

bandwidth is less severe. For a BER and0.01, the 420 scheme suffers roughly a 4-dB penaltyversus fully interleaved performance, while 5000 is onlypenalized 2 dB. Similar results can be observed for0.001, implying that large blocks contribute greater diversityto the system.

B. Bound Results

We now evaluate the average upper bounding techniquefor various channel models and various turbo schemes

(Figs. 6–12). It should be noted that the tightness of thesebounds to actual simulation data from a specific interleavingscheme is questionable for several reasons. These includethe union bound, the averaging over all interleavers, and thebounding of . This fact is illustrated in Fig. 6. It shouldalso be noted that the true performance of turbo codes doesnot “diverge” at low SNR, as indicated by the bounds. In fact,the change in slope of the BER bound curves around 10isnot an effect of the channel, but rather an artifact of the unionbound attributable to overcounting [11].


Fig. 6. Simulation and average bound for (1, 21/37, 21/37),K = 420 (SI and NSI).

Fig. 7. Average upper bound results on the fully interleaved Rayleigh channel with SI using different two-codeword probabilities and the (1, 5/7, 5/7)turbo scheme withK = 10.

In Fig. 7, a plot for small block length illustrates the effectsof the different bounds outlined previously. Motivatedby this figure, the bound results in the remainder of thiswork will use the bound given in (24). This bound offers thebest compromise between performance in regions of low SNRwhile not requiring numerical integration.

In Figs. 8 and 10, the effects of increased block length canbe observed. In [10], it is noted that the interleaver gain forincreased block length is proportional to for all RSC’s.

Therefore, for an increase in block length from 100 to 1000,the performance increase is 1/10, which can be observed in thefigures. The slopes of BER bounds will eventually convergeto the minimum distance of the ensemble of codes. Therefore,the effects of using encoders with better free distance can beseen in Figs. 9–11. This effect is most evident is regions ofhigh SNR. In regions of low SNR where our bounds are lessuseful, the actual weight spectrum becomes more important ininfluencing performance.


Fig. 8. Average bounds for the fully interleaved Rayleigh channel withK = 100 and SI.

Fig. 9. Average bounds for the fully interleaved Rayleigh channel with 16-state RSC’s and SI.

TABLE IENERGY DEGRADATION DUE TO CORRELATED

FADING AT HIGH SNR FOR K = 100

For the exponentially correlated channel, Fig. 12 shows theaverage bound results for 100 and the (1, 21/37, 21/37)turbo code. As in the simulations, performance degrades asthe correlation of the channel increases ( decreases). In

fact, this degradation can be predicted if we return to the two-codeword probability. Approximating (27) by dropping the lasttwo terms, the term can be viewed as anenergy degradation factormultiplying . Therefore, ifwe assume that the minimum distance dominates, as is thecase at high SNR, we can approximate the difference relativeto the fully interleaved channel. These differences are shownin Table I as well as Fig. 12.

While these bounds give insights into achievable perfor-mance and how to choose constituent encoders, they tell littleregarding the performance of thebestinterleaving scheme for


Fig. 10. Average bounds for the fully interleaved Rayleigh channel withK = 100 and NSI.

Fig. 11. Average bounds for the fully interleaved Rayleigh channel with 16-state RSC’s and NSI.

a given block size and constituent encoder. In can be saidthat thebestscheme performs better than the ensemble boundbut performance could bemuch better. This is due to thevariation in the achievable minimum distances within the classof interleaving schemes for a given block size and constituentencoder. For example, over the class of all interleavers, the (1,21/37, 21/37) turbo scheme with 420 has a worst caseinterleaver which yields a minimum distance of ten. However,it is known that this scheme using a helical interleaver canachieve a minimum Hamming distance of 22 [16]. Therefore,

in regions of high SNR, actual performance will be muchbetter than the average bound due to the dramatic differencesin the diversity orders. Again, at low SNR, the multipliers onlow-weight events contribute to make the situation less clear.

V. DESIGN FOR CORRELATED FADING

To examine the design of turbo codes on correlated chan-nels, we will consider the union upper bound on block-errorprobabilities for a specific turbo code rather than the ensembleaverage considered above. The union bound for the probability


Fig. 12. Average bounds for the exponentially correlated Rayleigh channel with (1, 21/37, 21/37),K = 100 and SI.

of bit error of an turbo code is given as

(28)

where is the information weight of andis the two-codeword probability. A bound for this probabilityis given in (26). As in [6], we can loosen and simplify thisbound to

(29)

If we substitute (29) into (28), we have

(30)

where the approximation arises from for smalland represents an effective information weight multiplierfor the class of error events of distancedefined as

(31)

The term is referred to as thephrase lengthproduct and can be used as a design parameter [6].

A. Interleaver Design Issues

Using (30) and (31), we can make some statements regard-ing the design of turbo codes in areas of high signal-to-noise.In this region, error events will be dominated by codewordshaving minimum Hamming distance . Therefore, we canrewrite (30) as

(32)

From this approximation and the definition of fromabove, we develop the following design objectives for theturbo interleaving scheme:

1) maximize , referred to as maximizing the diversityorder

2) among the class of interleaving schemes achieving themaximum , minimize .

The first criterion is identical for turbo code design onAWGN channels, and involves creating interleavers that pre-vent short merges in both constituent trellises. However, thetraditional turbo scheme using only one interleaver mighthave more than one interleaver that achieves the maximum

. Therefore, our second criterion states that, within theclass of interleavers achieving maximum , choose the onewhich minimizes . From (31), we observe that thisstatement reduces to maximizing the phrase length productfor sequences achieving . In fact, this product is maxi-mized when the weight is evenly spread throughout the entirecodeword.


Fig. 13. Turbo code fragments.

Fig. 14. Turbo codeword after traditional serialization.

B. Turbo Modifications

In its classical representation, a PCCC turbo scheme havingtwo constituent encoders is generally drawn with only oneinterleaver preceding the second constituent encoder. In thisscheme, codewords are formed by the serialization of thesystematic and parity streams. In this scheme, transmitterlatency is reduced since there is no additional buffering stagefollowing encoding. Receiver latency is also reduced sincethe first constituent decoding can be done as the codewordis received. Despite these benefits in terms of latency, thetraditional serialization procedure will suffer problems oncorrelated fading channels due to clustering of weight withincodewords. This clustering occurs since the event in-volves short merges in both constituent trellises. Figs. 13 and14 illustrate this clustering effect.

With a single turbo interleaver, this clustering of weight isalways going to be present for sequences that produce lowHamming weight. However, the clustering can be reducedby adding an additional interleaver either before the firstencoder or before the serialization of the systematic sequence(Fig. 15). The extra interleaver should improve performanceby increasing the phrase length product for low output weightsequences, as well as reducing the correlation between adjacentsystematic symbols for the constituent decoders. The system-atic interleaver should operate over the entire input frame,with the design being arbitrary provided that it is sufficientlydifferent from the turbo interleaver. A possible candidate isthe row/column interleaver which reads data into a matrix byrows while reading the data out by columns.

For situations with relaxed constraints on system latency,another possible interleaving scheme is to append a blockinterleaver following the serialization procedure as in Fig. 16.Block interleavers have been shown to perform well in mit-igating fading effects by reducing the effective correlationtime of the channel. The ability of this scheme to reducechannel correlation is directly related to the size of theblock interleaver or the interleaver depth. Unfortunately, theadditional latency for the scheme is proportional to the size

of the interleaver. If received codewords are buffered at thereceiver before decoding, the addition of a block interleaverdesigned to scramble over codewords only increases latencyat the transmitter. For the design of these interleavers, therow/column interleaver again is a strong candidate due tothe fact that it provides uniform spacing between formerlyadjacent symbols.

Figs. 17 and 18 show simulation results illustrating the ben-efits of the alternative interleaving schemes for exponentiallycorrelated channels with SI. In Fig. 17, a (1, 21/37, 21/37)turbo code is used where Here, a 21 20 helicalinterleaver is used and For the block interleavingscheme, a row/column interleaver with dimensions 3536is used. For the systematic interleaving scheme, a 2021row/column interleaver is used. The size of the systematicinterleaver is matched to the input block length ( 420)while the block interleaver is matched to the codeword length( ).

In the simulation, for a BER of 10 , the block andsystematic interleaving schemes offer coding gains of 2 and3 dB, respectively. While these gains do not match the codedperformance on the fully interleaved channel, they have beenobtained with relatively little effort (i.e., latency and extrahardware). If the goal was the performance of the fullyinterleaved channel, the size of the block interleaver couldbe increased.

In Fig. 18, a turbo design with large blockand stronger degree of channel correlation ( 0.001) isconsidered. Here, the turbo interleaver uses the same randomdesign used in the simulations. The block interleaver is a 125

120 row/column design while the systematic interleaver isa 100 50 row/column design. Despite the strong channelcorrelation, the additional interleaving yields coding gains of5 and 7 dB, respectively. The increase in the coding gainsover the previous example is largely related to the size of thesystematic and block interleavers. Again, by increasing thesize of the block interleaver, the fully interleaved performancecan be approached. In both previous examples, it can be notedthat interleaver design is more critical as decreases.

VI. CONCLUSIONS

In this paper, we have shown via simulations that turbocodes are capable of performance very near the capacity limiton fully interleaved fading channels. Furthermore, we haveshown simulation results and bounds to indicate the perfor-mance of turbo systems under conditions of both independentand correlated fading. The bound results give indications ofachievable performance as well as the effects of block lengthand constituent encoder choice. We concluded by proposinginterleaving options as well as structural modifications to im-prove turbo code performance on correlated fading channels.

APPENDIX

With coherent BPSK signaling, the discrete fading channelmodel is given as , where is the channel outputand is an input BPSK signal with energy constraint .


Fig. 15. Systematic interleaving for fading channels.

Fig. 16. Block interleaving for fading channels.

Fig. 17. Simulation results for turbo modifications on the exponentially correlated Rayleigh channel withK = 420,BTs = 0.01, and SI.

The variable is an AWGN component, and is the channelgain with Rayleigh distribution and is independent of.

Channel capacity is defined as the maximum over the inputdistribution of the mutual information between thechannel output and input . For the fading channel,if the fading amplitude is known, the mutual information is

conditioned on this knowledge. For this case, we write thecapacity expression as

(33)

(34)


Fig. 18. Simulation results for turbo modifications on the exponentially correlated Rayleigh channel withK = 5000,BTs = 0.001, and SI.

(35)

The steps above are based on the independence ofand ,conditional probability rules, and the law of total probability.

is the mutual information between andconditioned on knowledge of the channel gain.is the expectation over the distribution . Based on theindependence of and , this distribution can be written as

For symmetric channels with a finite input alphabet (i.e.,BPSK), the maximization in the capacity definition is achievedby an equiprobable input distribution

. From the determination ofand , the capacity expression is

(36)

By symmetry, is

(37)

and simplifies to

(38)

In this expression, is the Rayleigh pdf with averagepower of 1 and is the Gaussian pdf with a meanof and variance . The term is defined as

(39)

(40)

If channel side information is not available, the channelcapacity is written

(41)

(42)

(43)

The channel gain emerges in this expression as we examine, which can be written as

Substituting back into and using the simplificationsfor

(44)


Fig. 19. Channel capacity on the fully interleaved Rayleigh channel with coherent BPSK signaling.

TABLE IICAPACITY LIMITS FOR COHERENT BPSK ON

THE INDEPENDENT RAYLEIGH FADING CHANNEL

where

(45)

Equations (38) and (44) can be computed using numericalintegration and the results are plotted in Fig. 19. Notice thatthe lack of SI costs about 1 dB in for codes of rate1/4 to 1/2.

From the noisy channel coding theorem, a code exists thatwill give arbitrarily small error performance provided therate of the code is less than the capacity of the channel

. By equating the code rate and channel capacity, we candetermine the smallest such that arbitrarily small errorperformance is achievable. Through appropriate puncturing ofthe parity sequences, turbo codes of any rate are attainable.However, turbo code rates of 1/2 and 1/3 are most commonlyfound in the literature. Using (38) and (44), the smallest

are shown in Table II for these code rates based onthe following relation:

From the converse to the channel coding theorem, theprobability of error can be lower bounded in regions wherethe code rate is greater than the channel capacity from the

solution to the following equation [17]:

For BPSK signaling with the assumption of equiprobableinputs, and , the true boundary is foundfrom the solution to

(46)

Unfortunately, for fading channels with BPSK signaling,(46) has no closed-form solution due to the complexity ofthe capacity expressions and . Therefore, wesimply show a vertical line to indicate the minimumrequired for zero error performance in Figs. 2 and 3.

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Eric K. Hall received the B.S.E.E. degree fromDuke University, Durham, NC, in 1994 and theM.S.E.E. degree from the University of Virginia,Charlottesville, in 1996.

He spent the summer of 1996 working for Lock-heed Martin Tactical Communications Systems, SaltLake City, UT. Currently, he is working toward thePh.D. degree in electrical engineering at the Uni-versity of Virginia. His research deals with practicaland theoretical aspects of error-control coding.

Mr. Hall is a member of Eta Kappa Nu and theIEEE Communications and Information Theory Societies.

Stephen G. Wilson (S’65–M’68) received theB.S.E.E. degree from Iowa State University,Ames, the M.S.E.E. degree from the Universityof Michigan, Ann Arbor, and the Ph.D. degreein electrical engineering from the University ofWashington, Seattle.

He is currently Professor of Electrical Engineer-ing at the University of Virginia, Charlottesville. Hisresearch interests are in applications of informationtheory and coding to modern communicationsystems, specifically data compression of still

and moving imagery for digital transmission, and digital modulationand coding techniques for satellite channels, wireless networks, spreadspectrum technology, and transmission on time-dispersive channels. Priorto joining the University of Virginia faculty, he was a Staff Engineer forthe Boeing Company, Seattle, WA, engaged in system studies for deep-spacecommunication, satellite air-traffic-control systems, and military spread-spectrum modem development. He also acts as consultant to industrialorganizations in the area of communication system design and analysis anddigital signal processing, and is the author of the graduate-level textDigitalModulation and Coding.

Dr. Wilson is presently Area Editor for Coding Theory and Applicationsof the IEEE TRANSACTIONS ON COMMUNICATIONS.

Documents

Design And Analysis Of Turbo Codes On Rayleigh Fading Channels