parallel combinatory ofdm signaling

558 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 47, NO. 4, APRIL 1999

Parallel Combinatory OFDM SignalingPal K. Frenger,Student Member, IEEE,and N. Arne B. Svensson,Senior Member, IEEE

Abstract—In this paper, a parallel combinatory–orthogonal fre-quency division multiplexing (PC–OFDM) system is proposed andanalyzed. The proposed system selects at each symbol interval asubset of the available subcarriers, and the selected subcarriersare modulated by points from an M -PSK signal constellation.PC–OFDM systems can be designed to have lower peak-to-average power ratio (PAPR), higher bandwidth efficiency, andlower bit error probability on Gaussian channels compared toordinary OFDM systems. A bit mapping procedure using theJohnson association, together with a position algorithm for thePSK symbols, is proposed. Good analytical approximations of theBER for PC–OFDM systems are derived for AWGN and Riceanfading channels, and extensive simulation results are presented.

Index Terms—AWGN channel, bandwidth efficiency, bit map-ping, M -PSK, multicarrier modulation, OFDM, orthogonal sig-naling, parallel combinatory signaling, peak-to-average powerratio (PAPR), Ricean channel.

I. INTRODUCTION

ORTHOGONAL frequency division multiplexing(OFDM) systems are currently being proposed and

tested for many applications, including high definitiontelevision (HDTV) [2], [3], cellular mobile telephony [4], anddigital audio broadcasting (DAB) [5]. One of the drawbacks ofOFDM systems, however, is the large peak-to-average powerratio (PAPR) that causes problems since linear amplifiersneed to be used to avoid cross talk interference betweensubchannels [6], [7]. Different methods of reducing the PAPRfor OFDM systems are proposed. Clipping of the transmittersignal before amplification, to reduce the peak power, isinvestigated in [8], and various block coding schemes areproposed in [9]–[11]. Block coding for the purpose of reducingthe PAPR will, however, increase the required bandwidth tomaintain the same data rate, and clipping will increase the biterror probability for a given signal-to-noise ratio.

In this paper, a method of reducing the PAPR, withoutreducing the bandwidth efficiency and without increasing thebit error probability, is proposed for OFDM systems using

-PSK modulation. An OFDM system with subcarriersusing -PSK can transmit different waveforms duringone signal interval, and the PAPR for such a system growslinearly with the number of carriers . The method proposedhere is based on expanding the-PSK signal constellation

Paper approved by B. Vucetic, the Editor for Modulation of the IEEECommunications Society. Manuscript received August 13, 1997; revised July3, 1998. This work was supported in part by the Swedish Research Councilfor Engineering Sciences. This paper was presented in part at PIMRC-96,Taipei, Taiwan, October 15–18, 1996.

The authors are with the Communication Systems Group, Departmentof Signals and Systems, Chalmers University of Technology, SE-412 96Goteborg, Sweden (e-mail: [email protected]).

Publisher Item Identifier S 0090-6778(99)03306-1.

with one extra, zero amplitude, point. From this larger signalconstellation, containing different waveforms, asubset of waveforms with lower PAPR may be chosen. Ifchosen properly, this new signal constellation will have atleast the same bandwidth efficiency, and lower bit errorprobability, when compared to the original OFDM system.Parallel combinatory signaling was previously proposed forspread spectrum systems [12], [13] as a method for increasingbandwidth efficiency. In this paper, we use similar princi-ples applied on OFDM systems. The parallel combinatoryOFDM (PC–OFDM) systems proposed in this paper representa generalized class of OFDM systems.

Section II describes the OFDM system and the channel mod-els used in this paper. In Section III, the proposed PC–OFDMsystem is introduced. In Section IV, the bandwidth efficiencyof PC–OFDM systems is evaluated and compared to ordinaryOFDM systems. A new bit-mapping procedure for PC–OFDMsystems is proposed in Section V. Expressions for the bit errorprobability are derived in Section VI. Numerical results, bothsimulated and analytical, are presented in Section VII. Thepaper is then concluded in Section VIII.

II. SYSTEM AND CHANNEL MODELS

OFDM systems may be described in many different ways.We assume, for simplicity, that rectangular pulse shaping isemployed. The th transmitted OFDM symbol can then beexpressed in complex base-band notation as

otherwise(1)

where is the sum of the symbol time and the cyclicprefix length , and is the total number of carriers. Thetransmitted data symbol on subcarrierat time is denoted

, and is the imaginary unit. The cyclic prefix is insertedby the transmitter in order to remove the intersymbol inter-ference (ISI) and interchannel interference (ICI) that wouldotherwise cause degradation to the system performance [14].The transmitted signal may be generated using the inverse fastFourier transform (IFFT) in the transmitter, and the bank ofreceiver filters may be implemented using the ordinary FFTin the receiver [15].

The PC–OFDM systems in this paper will use-PSK sig-nal constellations extended with an additional zero amplitudepoint, as in Fig. 1. From an -PSK constellation, we thusconstruct an ( )-ary amplitude and phase shift keying(APSK) constellation. In the next section, we introduce a

0090–6778/99$10.00 1999 IEEE

FRENGER AND SVENSSON: PARALLEL COMBINATORY OFDM SIGNALING 559

(a) (b)

(c) (d)

Fig. 1. Signal spaces of the (M + 1)-ary amplitude and phase shift keying,(APSK), constellations used in this paper: (a) 3-APSK, (b) 5-APSK, (c)9-APSK, and (d) 17-APSK.

constraint in the choice of data symbols that aretransmitted in parallel at time index, using a combinatorialapproach.

The channel models used in this paper for evaluating thebit error rate (BER) of the proposed PC–OFDM system willbe additive white Gaussian noise (AWGN) and independentlyRicean fading channel models [16].

III. PARALLEL COMBINATORY SIGNALING

The PAPR for an OFDM system using -PSK equalsdB, and for an OFDM system using -

APSK the PAPR is dB. If we choosethe set of data points such that there are alwayspointswith zero amplitude, and points with nonzero amplitude,where , we obtain a system with a PAPR of

dB, which is lower than for the previous twosystems. We call this new system a parallel combinatory (PC–)OFDM system using out of carriers. In the transmitter,we first choose which subcarriers to be zero and nonzero,respectively. Then the nonzero subcarriers are modulatedby points from an ordinary -ary PSK constellation. Thuswe map bits to a subset of carriers to use andbits to the phase of the selected subcarriers. We defineasthe row vector containing the parallel combinatory bits, i.e.,the bits determining which subcarriers to be nonzero. Furtherwe define as the row vector containing the PSK bits, i.e.,the bits determining the phases of the selected subcarriers, andfinally is the concatenation of the PC and thePSK bits. Please note that we have skipped the time subscriptindex and only study one single PC–OFDM symbol. Choosing

out of carriers can be done in different ways.

The number of bits per PC–OFDM symbol, , is therefore1

(2)

where is the largest integer smaller than, or equal to,and denotes the base-2 logarithm of.

After the FFT in the receiver, we use a maximum likelihood(ML) detector, finding for each subcarrier the point in the( )-APSK constellation closest to the received value onthat subcarrier. If the number of subcarriers whose receiveddata symbols are closest to a nonzero signal constellation point(denoted ) equals , the received word is accepted andthe bits are decoded. Otherwise, if , ( is apositive integer), there are too many nonzero subcarriers, andthe nonzero subcarriers that have the smallest amplitude areset to zero. If the number of nonzero subcarriers is less than

, e.g., , we find among the subcarriers set tozero the subcarriers closest to any of the nonzero pointsof the signal constellation, and decode these subcarriers to thecorresponding points. After this correction, the received bitsmay be decoded. This receiver procedure allows us to fullyutilize the coding gain provided by the parallel combinatorybit mapping.

IV. BANDWIDTH EFFICIENCY

The bandwidth of any OFDM system depends on thenumber of subcarriers and the type of pulse shaping employed.Assuming that the number of subcarriers is relativelylarge, we define the bandwidth of the OFDM system as

. We also assume that the length of the cyclicprefix is negligible compared to the useful symbol duration

, and therefore define the bit rate as .These assumptions will not affect the comparison between thePC–OFDM systems and the ordinary OFDM systems, sincethe excess bandwidth and the length of the cyclic prefix willbe the same in these systems.

The bandwidth efficiency of the PC–OFDM system maynow be defined as . In Fig. 2 we plot the bandwidthefficiency versus the ratio of carriers used, , for

2, 4, 8, and 16. The black and white circles representordinary OFDM systems using -PSK and -APSK,respectively. Triangles pointing down represent PC–OFDMsystems with the same bandwidth efficiency as an ordinaryOFDM system using -PSK, and triangles pointing up repre-sent PC–OFDM systems with maximum bandwidth efficiencyfor a given . We see that the PC–OFDM systems arecapable of achieving almost the same bandwidth efficiencyas the corresponding OFDM systems using -APSK.Furthermore, we note that a PC–OFDM system using BPSK( ) should use fewer carriers than a system using 16-PSK ( ) in order to obtain the maximal bandwidthefficiency. This is obvious since when deciding not to use onesubcarrier for PSK modulation, by always choosing one ofthe data symbols equal to zero, we lose one bit in the BPSKcase and four bits in the 16-PSK case. However, the gain in

1We will assume in this paper thatM is a power of two, and thuslog2M

is an integer.


Fig. 2. Bandwidth efficiency of the PC–OFDM systems forM = 2, 4, 8,and 16. The black and white circles represent ordinary OFDM systems usingM -PSK and(M+1)-APSK, respectively. Triangles pointing down representPC–OFDM systems with the same bandwidth efficiency as an ordinary OFDMsystem usingM -PSK and triangles pointing up represent PC–OFDM systemswith maximum bandwidth efficiency for a givenM .

the number of parallel combinatory bits, is in both cases, thesame.

We also see that for a PC–OFDM system with ,we can employ one-fourth, and for one half, of theavailable subcarriers simultaneously and still obtain the samebandwidth efficiency as the corresponding ordinary OFDMsystems using -PSK. Thus we can obtain approximately a6 and a 3 dB reduction of the PAPR for BPSK and QPSK,respectively. If, for example, , we can transmit thesame number of bits per second per Hz as an ordinary OFDMsystem with -PSK by using 9, 18, 25, and 29 for

2, 4, 8, and 16, respectively.Thus the PC–OFDM systems can obtain a higher bandwidth

efficiency than ordinary OFDM systems using-PSK, andthe maximal bandwidth efficiencies are only slightly less thanthose of the -APSK systems. In addition to this, thecoding gain, as we will see in Section VII, can decrease thebit error rate as well.

V. BIT MAPPING PROCEDURE

The problem of mapping the bits to a PC–OFDM symboldeserves some attention. First we need to find out whichsubcarriers to use for -PSK signaling, and which to setto zero. Several solutions to this problem are possible. Theobvious one is to generate a lookup table with entries.The entries may consist of vectors of length, where a “zero”in one position denotes that the corresponding subcarriershould be set to zero, and a “one” that it should be usedfor -PSK signaling. To reduce the storage space neededfor this approach, we may produce a smaller lookup tablecontaining only entries that when shifted cyclically produceall entries. However, these solutions will fail whenis large due to the exponential increase in storage needed. Wemay use a constant weight code instead, preferably with high

code rate to avoid losing data rate. Constant weight codes are,however, nonlinear to their nature (the all zero word is not acodeword) and are only available for very specific lengths andweights. Thus we may use a specific constant weight code ifits weight and lengths happen to coincide with the and

we want to use. To the best of our knowledge, there is noknown general family of constant weight codes for arbitrarylengths and weights with high rates, which are easy to decode.We have therefore chosen a different solution that is describedin the sequel.

A. Selecting the Subcarriers

The problem of generating a lookup table consisting of allpossible ways to choose out of may be formulatedrecursively. The computational complexity to find out thecontents of a certain row in that lookup table grows linearwith so there is no need to store the table. Letdenote this table. The first subscript index denotes the weight(number of ones), and the second denotes the length of eachrow. We note that may be written as

(3)

where denotes a column vector of length

containing only ones, and is a vector of lengthcontaining only zeros. The association scheme expressed in(3) is known as the Johnson association scheme [17]. We nowsee that if we are interested in finding the contents of rowin , we simply compare to to find out if the firstposition equals 0 or 1 and the other positions follow by therecursive form of . We only need to take care of thetrivial problems of finding (containing only ones) and

(containing only zeros) to stop the recursion. Obviouslythis mapping works both ways, i.e., given the contents of acertain row we can calculate the row index. We may easilyreformulate the problem if we prefer an iterative solutioninstead. For mapping of the bits to a row index , wehave chosen the natural binary code (NBC). Other possiblemappings such as the gray code (GC), the sign-magnitudecode (SMC), and the twos-complement code (TCC), as definedin [18], have been investigated. These mappings give noimprovements compared to the NBC in terms of minimizingthe average number of errors in the parallel combinatory bitswhen erroneously detecting a row of with Hammingdistance 2 from the transmitted one.

B. Positions of the PSK Symbols

If we are not careful in how to map the remaining bitson the selected subcarriers, we will have error events of highprobability where almost all bits are received incorrectly.The reason for this is best shown by the following example.

Suppose we use , , and togetherwith the bit mapping of the PC bits as given in Table I. Themapping of pairs of PSK bits to data symbols follows the Graycode according to .This system transmits 3 bits in the choice of which subcarriers


Fig. 3. The proposed bit mapping procedure.

TABLE IMAPPING OF THE PARALLEL COMBINATORY BITS, bbbpc

are nonzero, and 14 bits in the received phase of the selectedsubcarriers. (Note that we can transmit one extra bit comparedto a system using QPSK modulation of all 8 subcarriers.)Assuming that [0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1,0, 0, 1, 1], we will produce the vector [ 1, 1, 1, 1,

1, 1, 1, 0] as input to the OFDM modulator. The first threebits of select the subcarriers 1 to 7, and the following 14bits select the phases of these carriers. Suppose that due tochannel disturbances we detect the vector [0, 1, 1, 1,

1, 1, 1, 1] as received, where we erroneously have chosenthe last subcarrier instead of the first. Following the demappingprocedure, we then decode the bit vector [1, 1, 1, 0, 0,1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0] where every single bit is wrong.Even though six of the subcarriers transmitting the PSK bitswere correctly detected, a shift occurred such that the first twoPSK bits were decoded from (the erroneously detected) firstsubcarrier. The PSK bits decoded from the second subcarrierthus occur in the third and fourth positions of , which iswrong.

To avoid this problem, we need to place the PSK symbolsso that an erroneous decision of subcarriers results in as fewerrors as possible in the PSK modulated bits. Thus, oncethe PC bits have decided which subcarriers to use for PSKmodulation, we need to decide where (on which subcarriers)to place the PSK symbols. Let denote the th row of

. We now assign position numbers, ranging from 1to , to the nonzero coordinates of. This is denoted bya new vector which is constructed using Algorithm 1. Inshort, the idea of the algorithm is to transmit (if possible) the

th PSK symbol on a subcarrier with an indexas small aspossible such that , where . Thepresented algorithm is not known to be optimal. However, anexhaustive computer search for better mapping procedures hasbeen performed for all cases where , and we have foundseveral equally good, but none better. The criterion used was

TABLE IIMAPPING OF THE PARALLEL COMBINATORY BITS,

bbbpc, AND POSITIONING OF THE PSK SYMBOLS

the average number of PSK symbol errors provided that twocarriers are erroneously switched. The bit mapping procedureis summarized in Fig. 3.

Algorithm 1 Positioning of the PSK Symbols

Let denote the th position of the vector .Let denote the vectorof PSK symbols.Let .for to do

if thenif is not already assignedto a subcarrierthen

Assign to subcarrier ,e.g., set .

end ifend if

end forif any PSK symbols are not yet assigned toa subcarrierthen

assign the remaining PSK symbols to anysubcarriers that are not already employed.

end if

We now continue with our example to see what will happen.The positions of the PSK symbols are now chosen accordingto Table II. If we receive the same vector as before,[0, 1, 1, 1, 1, 1, 1, 1], we will now decode it to the bitvector [1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1,1] with five erroneous bits and twelve bits correct. We cannotexpect the parallel combinatory bits to be correct nor the bits


corresponding to the subcarriers we did not choose, but all theother bits were in this example received correctly, and this isa significant improvement.

VI. BIT ERROR PROBABILITY

A. Gaussian Channel

We are now ready to derive the bit error performance of thePC–OFDM system on an AWGN channel. Assume that thetransmitted word at time index is

(4)

where denotes the transmitted symbol of subcarrier, andthe apostrophe denotes vector transpose. Thus the firstsubcarriers are used for transmitting-PSK symbols. Thetime subscript index is dropped for convenience. Further-more, we assume that (where is the real-valuedand positive transmitted symbol) for all , meaningthat the phases of all the nonzero transmitted symbols are zero.At the receiver, we obtain the vector where

(5)

is a vector of independent and identically distributed (i.i.d.)Gaussian complex random variables with variances andzero mean. The received values on the subcarriers transmittinga nonzero data symbol are denoted by, and on subcarrierstransmitting a zero amplitude data symbol, by, that is,

(6)

We define the real and imaginary parts of and asand . We further

define the probability density, and distribution functions ofGaussian random variables with zero mean and varianceas and

, respectively. The real and imaginary partsof are independent, and consequently we obtain their jointprobability density function as .The random variable will lie within one of the ML-decision regions associated with an -PSK constellation.Thus, may be erroneously detected to the correspondingsignal point in that ML-decision region, or correctly detectedas a zero-amplitude point. Because of the rotational symmetryof the ML-decision regions, we may assume that the closestnonzero signal point is . The probability of detecting asthe signal point will depend only on the value of the randomvariable along the real axis in this region. This random variableis denoted and its probability density function is found to be

(7)

For the case , we get the simpler expression; . The distribution function of is denoted

. The largest of therandom variables in the set is denoted and itsprobability density function is [19]

(8)

We now turn our attention to the random variables ,. Since the real and imaginary parts of

are independent, we have .The probability density function, and distribution function,of along the real axis are and

, respectively. The smallest one of

the random variables is denoted , and itsdistribution function is [19]

(9)

We will obtain errors in the parallel combinatory bits if. This will occur with the probability

(10)

Equation (10) may be numerically evaluated.Using (10) we get an approximation of the bit error proba-

bility, that is valid for , as

(11)

where

erfc (12)

and . Thus, with probability , wechoose the wrong subset of subcarriers. When this occurs, weassume that the error probability of the PC bits is 1/2.Furthermore, we assume that two PSK symbols are lost, sotwo of the PSK symbols have a bit error probability of1/2 and the remaining PSK symbols have the bit errorprobability . We may also, with probability ,choose the correct subcarrier subset. In this case, the PC bitsare received correctly and the PSK bits have the errorprobability . Averaging over these events we obtainthe expression in (11). For the OFDM systems using-PSKmodulation, we may use (12) as an approximation of the biterror rate for high , and for the ( )-APSK systemswe will simulate the symbol error probability, , anduse

(13)

to obtain an upper and lower bound on the bit error rate.

B. Flat Ricean Fading Channel

On a flat Ricean fading channel, all subcarriers will havethe same attenuation and we can write the received vector attime index as , where isa complex Gaussian distributed scalar random variable witha Ricean distributed amplitude. The statistics of may bedefined by the ratio of the power in the direct path () andthe Rayleigh fading indirect path (). For this purpose weintroduce the notation , where is the variance


Fig. 4. Analytical (solid) and simulated (dotted) BER results as a functionof Eb=N0 in decibels for PC–OFDM systems withNc = 32 andNpc = 4,8, 12, 16, 20, 24, and 28, using BPSK modulation on the selected subcarriers.Also shown are BER results for BPSK (dashed) and upper and lower boundsof the BER for 3-APSK (dash–dotted).

of . To obtain the bit error probability on a fading channel,we average (11) over the probability density function of thereceived amplitude , e.g.,

(14)

where the approximation sign comes from the approximationsmade in (11) and (12). For the Ricean fading channel we have

(15)

where denotes the zeroth order Bessel function of the firstkind.

VII. N UMERICAL RESULTS

A. Bit Error Rate on AWGN Channels

In Figs. 4–7, we show the bit error rate (BER) perfor-mances of the PC–OFDM systems, with BPSK, QPSK, 8-PSK,and 16-PSK modulation of the used subcarriers, respectively.Simulated results for PC–OFDM systems with areshown for 4, 8, 12, 16, 20, 24, and 28 with dottedlines marked with white circles. Using solid lines we showthe corresponding approximated analytical results according to(11). For comparison we also show simulated BER results forOFDM systems using -PSK modulation using dashed lines.Also shown in Figs. 4–7 are dash–dotted lines, correspondingto the upper and lower bounds in (13), of the BER for OFDMsystems using ( )-APSK modulation. All simulationswere performed until at least 1000 bit or symbol errorsoccurred. It should be noted that the results presented in thissection do not take the effects of any nonlinearities in thetransmitter into account.

By comparing the simulated and analytical results for thePC–OFDM systems, we see a close correspondence indicatingthat the approximations made in (11) are tight. We see that thebit error rate decreases when decreases, except for very

Fig. 5. Analytical (solid) and simulated (dotted) BER results for PC–OFDMsystems as a function ofEb=N0 in decibels withNc = 32 andNpc = 4, 8,12, 16, 20, 24, and 28, using QPSK modulation on the selected subcarriers.Also shown are BER results for QPSK (dashed) and upper and lower boundsof the BER for 5-APSK (dash–dotted).

Fig. 6. Analytical (solid) and simulated (dotted) BER results for PC–OFDMsystems as a function ofEb=N0 in decibels withNc = 32 andNpc = 4, 8,12, 16, 20, 24, and 28, using 8-PSK modulation on the selected subcarriers.Also shown are BER results for 8-PSK (dashed) and upper and lower boundsof the BER for 9-APSK (dash–dotted).

high error rates. It is clear that to obtain the lowest possible biterror rate, we should use , where we have a systemvery similar to orthogonal signaling [16]; but looking at Fig. 2we see that such a system has very low bandwidth efficiencyand the comparison is in that respect unfair. Instead we cantrade BER performance against bandwidth efficiency to someextent. This behavior is similar to that of traditional channelencoding.

The BER results for are shown in Fig. 4. Wesee that, compared to an OFDM system using BPSK, thePC–OFDM systems require equal, or lower, to obtainthe BER for . Compared to an OFDM systemusing 3-APSK, the required to obtain the BERis lower, or approximately equal, for the PC–OFDM systemswith .


Fig. 7. Analytical (solid) and simulated (dotted) BER results for PC–OFDMsystems as a function ofEb=N0 in dB with Nc = 32 andNpc = 4, 8, 12,16, 20, 24, and 28, using 16-PSK modulation on the selected subcarriers. Alsoshown are BER results for 16-PSK (dashed) and upper and lower bounds ofthe BER for 17-APSK (dash–dotted).

Fig. 8. Analytical BER results as a function ofEb=N0 in dB for PC–OFDMsystems withNc = 8; 32 and 128. Results are shown forM = 4 (dashedlines),M = 8 (dash–dotted), andM = 16 (solid).Npc = Nc=2 in all cases.

In Fig. 5 we show the BER results for . We seethat lower is required to obtain the BER forPC–OFDM systems with compared to OFDMsystems using QPSK and compared to OFDM systems using5-APSK the required is lower for all values of .

The results for are shown in Fig. 6. The requiredto obtain the BER is lower for the PC–OFDM

systems compared to both the OFDM system using 8-PSK andthe OFDM system with 9-APSK.

Finally we also show in Fig. 7 the BER results for .Also here we see, as expected, that the PC–OFDM systemsrequire lower to obtain the BER .

In Fig. 8 we show analytical BER results for PC–OFDMsystems with , and . In all cases, we haveused , and results are shown for (dashedlines), (dash–dotted), and (solid). We seethat for , the degradation when compared

Fig. 9. BWE versus requiredEb=N0 to obtain the BER10�6, Analyticalresults according to (11) are shown forM = 2 (solid), M = 4 (dashed),M = 8 (dash–dotted), andM = 16 (dotted). Simulated results for thePC–OFDM systems are shown as circles. Results for OFDM systems usingM -PSK are shown with asterisks and for OFDM systems using (M+1)-APSKwith crosses.

to when is about 1 dB at the BER . Resultsfor (not shown in Fig. 8) show a similar behavior.This is because in (10) increases whenincreases, and this error event dominates in (11) forand 4. For 8 and 16, we see much smaller differencesfor the different values of . This is due to the termin (11) which is more dominating in these cases. In fact, theBER decreaseswhen increases. This is caused by the twoterms and in (11), where the first termincreases and the second decreases when(and thus also

in this case) increases. For , the results will beclose to identical to those for for all .

The results of Fig. 2 (bandwidth efficiency) and Figs. 4–7(BER as a function of ) are summarized in Fig. 9.There we plot the bandwidth efficiency as a function of the

required to obtain the BER for all PC–OFDMsystems, as well as for the OFDM systems with-PSK and( )-APSK. Analytical results are shown for PC–OFDMsystems with , and using solid, dashed,dash–dotted, and dotted lines, respectively. Simulated resultsfor the PC–OFDM systems are shown as white circles. Resultsfor OFDM systems using -PSK are shown with asterisks andlower bounds for OFDM systems using ( )-APSK withcrosses.

Based on the results in Fig. 9, we compare OFDM systemsusing -PSK to PC–OFDM systems with nearly the samebandwidth efficiency in Table III. The first row of Table IIIshows the number of nonzero subcarriers used,, by thePC–OFDM systems when . The second row showsthe bandwidth efficiency [bit/s/Hz] of these PC–OFDM sys-tems. In parentheses, the bandwidth efficiencies of the OFDMsystems using -PSK are shown. The third row then showsthe difference in PAPR of the two systems, e.g.,PAPR PAPR – in dB. We see, for , a6-dB reduction, and for , a 3-dB reduction of the PAPR


TABLE IIIPC–OFDM VERSUS OFDM WITH M -PSK

TABLE IVPC–OFDM VERSUS OFDM WITH (M + 1)-APSK

in favor to the PC–OFDM systems. For and ,the achievable reduction in PAPR is smaller; about 1.2 and 0.6dB, respectively. The difference in , needed to obtainthe BER , between the two systems is denotedand is shown in row 4 of Table III. Here we see coding gainsfor the PC–OFDM systems of slightly more than 1 dB for

, and , and slightly less than 1 dB for .The same comparison is made between PC–OFDM system

with the maximal possible bandwidth efficiency and OFDMsystems using ( )-APSK and the results are shownin Table IV. Looking at row 2, we see that the maximalbandwidth efficiencies of the PC–OFDM systems are closeto the bandwidth efficiencies of the OFDM systems using( )-APSK which are shown in parentheses. The reductionin PAPR is approximately 4, 2, 1, and 0.4 dB for ,and , respectively. The coding gain measured at the BER

is more than 1 dB for , and .We may also compare PC–OFDM systems with

to OFDM systems using 5-APSK and BPSK. Comparingto 5-APSK, we see that the bandwidth efficiencies and therequired to obtain the BER are approximatelythe same if the PC–OFDM system uses 16 subcarriers. Thedifference in PAPR between the two systems is, however,

dB. The same comparisonmay be made between an OFDM system using BPSK and aPC–OFDM system with and . The differencein PAPR is here dB in favorof the PC–OFDM system. Comparing an OFDM system usingBPSK to a PC–OFDM system with and thePC–OFDM system has dBlower PAPR and the required to obtain the BERis approximately 2.5 dB lower for the PC–OFDM system.

B. Results on Flat Ricean Fading Channels

In Fig. 10 we show bit error probability results on Riceanfading channels for PC–OFDM systems using QPSK modu-lation of the selected subcarriers. The number of carriersis 32 and analytical results, according to (14), are shown for

, and . Also shown are simulated bit error

Fig. 10. Analytical bit error probability results on Ricean fading channels forPC–OFDM systems withNc = 32 using QPSK modulation of the selectedsubcarriers.Npc = 32; 16; 8, and 4. = 10 and 0 dB. Also shown aresimulated results forNpc = 32 (white circles) andNpc = 16 (black circles).

Fig. 11. Analytical bit error probability results on Ricean fading channels forPC–OFDM systems withNc = 32 using 8-PSK modulation of the selectedsubcarriers.Npc = 32; 24; 16, and8. = 10 and 0 dB. Also shown aresimulated results forNpc = 32 (white circles) andNpc = 24 (black circles).

probability results for equals 16 (black circles) and 32(white circles). These two systems have approximately thesame bandwidth efficiency. We see that the analytical approxi-mations correspond almost perfectly with the simulations. Theratio of the power in the direct path and the Rayleigh fadingpath is 0 and 10 dB, respectively. When dB, thePC–OFDM with requires an approximately2.5 dB higher than an ordinary OFDM system ( ) toobtain the BER . We thus conclude that the PC–OFDMsystems are more sensitive to channel fading than an ordinaryOFDM system. For low SNR, the probability of selecting thewrong subcarrier subset in the receiver will be large, and thiseffect is dominating on a fading channel.

In Fig. 11 we show similar results for PC–OFDM systemsusing 8-PSK modulation on the selected subcarriers. By com-


paring the dashed ( ) and solid ( ) linesfor the case when equals 10 dB, we see that the PC–OFDMsystem requires slightly lower to obtain the BER .

VIII. SUMMARY AND CONCLUSIONS

We have shown that PC–OFDM systems can be designed tohave lower PAPR than OFDM systems using-PSK whilemaintaining the bandwidth efficiency. On AWGN channels,the BER for large is lower for most PC–OFDMsystems than for OFDM systems using-PSK. The maximumachievable bandwidth efficiencies for PC–OFDM systems arealmost as high as for OFDM systems using ( )-APSK,but the PAPR and the BER for high are lower for thePC–OFDM systems.

Results on Ricean fading channels show that PC–OFDMsystems are not as robust against fading as ordinary OFDMsystems. However, on fading channels the use of PC–OFDMmay be motivated by its higher bandwidth efficiency andreduced peak-to-average power ratio. The bit error rate will,for most cases, be higher for the PC–OFDM system comparedto an ordinary OFDM system.

OFDM systems are more robust against impulse noise thana single carrier system due to the long symbol interval. Thiscan motivate the use of PC–OFDM on channels with AWGNand impulse noise. As an example of such a channel, wecan mention communications on high-voltage power lines thatsuffer from Gaussian noise due to partial discharges to thesurrounding air and impulse disturbances due to switchingelectrical machinery on and off.

PC–OFDM represents a generalization of the ordinaryOFDM scheme, and for (and ) PC–OFDM isvery similar to orthogonal signaling. It should also be notedthat PC–OFDM only achieves a marginal PAPR improvementcompared to what is theoretically possible [20]. Thus aPC–OFDM system could be followed by some other scheme(i.e., clipping [8] or block-coding [21]) to further reduce thePAPR. Further studies are needed to design such a scheme,as well as to find a suitable error control scheme for thePC–OFDM system.

REFERENCES

[1] P. Frenger and A. Svensson, “A parallel combinatory OFDM system,” inProc. IEEE Int. Symp. Personal, Indoor, Mobile Radio Commun.,Taipei,Taiwan, Oct. 15–18, 1996, pp. 1069–1073.

[2] T. de Couason, R. Monnier, and J. B. Rault, “OFDM for digital TVbroadcasting,”Signal Processing,vol. 39, nos. 1/2, pp. 1–32, Sept. 1994.

[3] H. Sari, G. Karam, and I. Jeanclaude, “Transmission techniques fordigital terrestrial TV broadcasting,”IEEE Commun. Mag.,vol. 33, pp.100–109, Feb. 1995.

[4] C. Reiners and H. Rohling, “Multicarrier transmission technique incellular mobile communications system,” inProc. IEEE Veh. Technol.Conf., Stockholm, Sweden, June 8–10, 1994, pp. 1660–1664.

[5] ETSI ETS 300 401, “Radio broadcasting systems: Digital audio broad-casting (DAB) to mobile, portable and fixed receivers,” Tech. Rep.,Sophia Antipolis, France, 1995.

[6] M. Burgos-Garcıa and F. Perez-Martınez, “Simple procedure for opti-mum linearization of amplifiers in multicarrier applications,”Electron.Lett., vol. 30, no. 2, pp. 114–115, Jan. 1994.

[7] M. Johansson, T. Mattsson, L. Sundstrom, and M. Faulkner, “Lineariza-tion of multi-carrier amplifiers,” inProc. IEEE Veh. Technol. Conf.,Secaucus, NJ, 1993, pp. 684–687.

[8] R. O’Neill and L. B. Lopes, “Performance of amplitude limited multi-tone signals,” inProc. IEEE Veh. Technol. Conf.,Stockholm, Sweden,1994, pp. 1675–1679.

[9] S. J. Shepherd, P. W. J. Van Eetvelt, C. W. Wyatt-Millington, andS. K. Barton, “Simple coding scheme to reduce peak factor in QPSKmulticarrier modulation,”Electron. Lett.,vol. 31, no. 14, pp. 1131–1132,July 1995.

[10] A. E. Jones, T. A. Wilkinson, and S. K. Barton, “Block coding schemefor reduction of peak to mean envelope power ratio of multicarriertransmission schemes,”Electron. Lett.,vol. 30, no. 25, pp. 2098–2099,1994.

[11] D. Wulich, “Reduction of peak to mean ratio of multicarrier modulationusing cyclic coding,”Electron. Lett.,vol. 32, no. 5, pp. 432–433, Feb.1996.

[12] S. Sasaki, H. Kikuchi, H. Watanabe, J. Zhu, and G. Marubayashi,“Performance evaluation of parallel combinatory SSMA systems inRayleigh fading channel,” inProc. IEEE Int. Symp. Spread SpectrumTechniques Appl.,Oulu, Finland, 1994, pp. 198–202.

[13] S. Sasaki, H. Kikuchi, J. Zhu, and G. Marubayashi, “Multiple accessperformance of parallel combinatory spread spectrum communicationsystems in nonfading and Rayleigh fading channels,”IEICE Trans.Commun.,vol. E78-B, no. 8, pp. 1152–1161, Aug. 1995.

[14] J. A. C. Bingham, “Multicarrier modulation for data transmission: Anidea whose time has come,”IEEE Trans. Commun.,vol. 28, pp. 5–14,May 1990.

[15] S. B. Weinstein and P. M. Ebert, “Data transmission by frequencydivision multiplexing using the discrete Fourier transform,”IEEE Trans.Commun.,vol. COM-19, pp. 628–634, Oct. 1971.

[16] J. G. Proakis,Digital Communications,3rd ed. New York: McGraw-Hill, 1995.

[17] F. J. MacWilliams and N. J. A. Sloane,The Theory of Error-CorrectingCodes. Amsterdam, The Netherlands: North-Holland, 1977.

[18] N. S. Jayant and P. Noll,Digital Coding of Waveforms: Principles andApplications to Speech and Video.Englewood Cliffs, NJ: Prentice-Hall, 1984.

[19] A. Papoulis,Probability, Random Variables, and Stochastic Processes,3rd ed. New York: McGraw-Hill, 1991.

[20] S. Shepherd, J. Orriss, and S. Barton, “Asymptotic limits in peakenvelope power reduction by redundant coding in orthogonal frequency-division multiplex modulation,”IEEE Trans. Commun.,vol. 46, pp.5–10, Jan. 1998.

[21] J. A. Davis and J. Jedwab, “Peak-to-mean power control and error cor-rection for OFDM transmission using Golay sequences and Reed–Mullercodes,”Electron. Lett.,vol. 33, no. 4, pp. 267–267, Feb. 1997.

Pal K. Frenger (S’94) was born in Alingsas,Sweden, on October 31, 1968. He received theM.S. (Civilingenjor) degree in electrical engineeringfrom Chalmers University of Technology, Goteborg,Sweden, in 1994, and the Dr.Ing. (Teknisk Licentiat)degree in February 1997. He is currently workingtoward the Ph.D. degree at the CommunicationSystems Group at the Department of Signals andSystems, Chalmers University of Technology.

His main areas of interest are communicationsystems, error control methods, and modulation and

demodulation methods. In particular, he has been working on multicarriermodulation techniques and error control for multiple data rate systems. SinceOctober 1995, he has been involved in the European FRAMES (Future RadioWideband Multiple Access System) project.


N. Arne B. Svensson(S’82–M’84–SM’90) wasborn in Vedakra, Sweden, on October 22, 1955. Hereceived the M.S. (Civilingenjor) degree in electricalengineering from the University of Lund, Sweden,in 1979; and the Dr.Ing. (Teknisk Licentiat) and Dr.Techn. (Teknisk Doktor) degrees from the Depart-ment of Telecommunication Theory, University ofLund, in 1982 and 1984, respectively.

He is currently with the Department of Signal andSystems at the School of Electrical and ComputerEngineering at Chalmers University of Technology,

Goteborg, Sweden, where he was appointed a Professor in CommunicationSystems in April 1993. Before 1985, he held various teaching and researchpositions at the University of Lund. From April 1985 to July 1987, he was aResearch Professor (Docent) at the Department of Telecommunication Theory,University of Lund. In August 1987, he joined the Airborne ElectronicsDivision at Ericsson Radio Systems AB, Molndal, Sweden. After a companyreorganization on January 1988, he became employed by Ericsson RadarElectronics AB, where he first was a member of the New Projects Groupat the Airborne Electronics Division, and then from September 1990 toDecember 1994 a member of the Mobile Telephone Systems Group at theMicrowave Communications Division. His consulting company, BOCOM, isinvolved in studies of error control methods, modulation and demodulationtechniques, spread spectrum and CDMA systems, and computer simulationmethods for communication systems. His current interests include channelcoding and decoding, digital modulation methods, channel estimation, datadetection, multiuser detection, digital satellite systems, CDMA and spreadspectrum system, and personal communication networks. He has published20 journal papers, 4 letters, and more than 80 conference papers.

Dr. Svensson received the IEEE Vehicular Technology Society Paperof the Year Award in 1986, and the Young Scientists Award from theInternational Union of Radio Science, URSI, in 1984. He is an Editor ofthe Wireless Communications Series of IEEE JOURNAL OF SELECTED AREAS IN

COMMUNICATIONS. He is a member of the council of SER (Svenska Elektro-och Dataingenjorers Riksforening), and a member of the Swedish URSIcommittee (SNRV, Svenska Nationalkommitten f¨or Radiovetenskap). He islisted in Who’s Who in the Worldand theEuropean Biographical Directory.

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