Direct vs. Two-Step Approach for Unique Word Generation in UW-OFDM
aa r X i v : . [ c s . I T ] S e p Direct vs. Two-Step Approach for Unique WordGeneration in UW-OFDM
Alexander Onic, Mario HuemerInstitute of Networked and Embedded Systems, Alpen-Adria-Universit¨at Klagenfurt, Austria { alexander.onic, mario.huemer } @uni-klu.ac.at Abstract — Unique word OFDM is a novel techniquefor constructing OFDM symbols, that has many ad-vantages over cyclic prefix OFDM. In this paper weinvestigate two different approaches for the generationof an OFDM symbol containing a unique word inits time domain representation. The two-step and thedirect approach seem very similar at first sight, butactually produce completely different OFDM symbols.Also the overall system’s bit error ratio differs signif-icantly for the two approaches. We will prove thesepropositions analytically, and we will give simulationresults for further illustration.
Index Terms — OFDM, Unique word
I. I
NTRODUCTION C YCLICITY of an OFDM symbol is a necessarycondition that needs to be fulfilled in orderto be able to perform OFDM transmission in amultipath environment. Traditionally a cyclic prefix(CP) is used to guarantee the cyclicity.While this method is well examined and un-derstood, there is another possibility to ensure thecyclicity. If a unique word (UW) of length T GI ischosen in advance and introduced at the end of eachOFDM symbol, cyclicity appears, too.Different to the CP, which is copied and addedafter IDFT (Inverse Discrete Fourier Transform), theUW is part of the IDFT output, and therefore alsopart of the DFT interval T DFT . Figure 1 sketches thestructure of CP- and UW-OFDM symbols. UW T GI Data 1 UW T DFT
Data 2 UW T DFT
Data 3 UW T DFT
CP1 Data 1 CP1 T GI T DFT
CP2 Data 2 CP2 T GI T DFT
CP3 Data 3 CP3 T GI T DFT
Fig. 1. OFDM symbol structure using cyclic prefixes andunique words
Note again, that the guard interval in figure 1containing the cyclic prefixes are copies of a part of the payload data and hence random. In contrast theUW is deterministic and known in advance, whichallows additional processing.Summing up, we want to stress some pointsregarding UW-OFDM: • Cyclicity is also ensured as in CP-OFDM. • The UW can be used for synchronization andestimation tasks [1] • Improved performance regarding bit error ratiocompared to CP-OFDM in frequency selectiveenvironments [2]. • Almost no loss in bandwidth efficiency [2].We denote vectors in bold lower case a , frequencydomain vectors additionally with a tilde ˜a and ma-trices in bold upper case A . The operations a T and a H indicate the matrix transpose resp. conjugatetranspose.In section II we introduce the concepts of UW-OFDM and present two different approaches for UWgeneration, that come into mind naturally. We willexamine the OFDM symbol energies resulting fromboth approaches in section III and show numericalexamples proving these results in section IV. Sec-tion V concludes our work.II. U NIQUE W ORD
OFDMThe generation of the final OFDM symbol inUW-OFDM differs a lot from CP-OFDM. For onesymbol in CP-OFDM the data symbols ˜x d ∈ C N d × are loaded onto the subcarriers. Then zero subcar-riers are inserted at the band edges and the DCposition which we describe by a matrix operation ˜x = B˜x d with ˜x ∈ C N × and B ∈ { , } N × N d .This frequency domain vector is then transformedinto time domain via the IDFT operation, which wedenote by a matrix operation x = F − N ˜x utilizingthe N -point DFT matrix F N with the element of the m -th row and the n -th column [ F N ] m,n = e − j πN mn ,where m, n = 0 , , , . . . , N − . The guard intervalis then formed by copying the last values to the front.In UW-OFDM the content of the guard interval isknown in advance and part of the IDFT operation.n order to obtain a predefined sequence at the last N u positions we have to spend at least this amountof freedom on the input side of the IDFT. Thus wedefine some carriers as redundant carriers, which cannot be used for data transmission, but have to beloaded with appropriate values ˜x r ∈ C N r × to yieldthe UW at the output.A transmit symbol can be described by x = F − N BP (cid:20) ˜x d ˜x r (cid:21) , (1)where P is a permutation matrix P ∈{ , } ( N d + N r ) × ( N d + N r ) , that changes the positionsof the data and redundant values in an optimumway. Note that B is now B ∈ { , } N × ( N d + N r ) . Asshown in [2] and [3] the choice of P is crucial forobtaining low energy contributions on the redundantsubcarriers, but its particular design is of no furtherrelevance for the investigations in this paper.Still we haven’t had a look at the resulting timedomain vector x and the symbols on the redundantcarriers ˜x r . Here two approaches can be taken intoaccount and will be explained in detail. A. Two-step approach
The two-step approach aims on generating a zeroword at the position of the unique word first. In asecond step we add the unique word in time domain: x ′ = (cid:20) x p (cid:21) = F − N BP (cid:20) ˜x d ˜x r (cid:21) (2) x = x ′ + (cid:20) u (cid:21) , (3)with the unique word x u ∈ C N u × and the payload x p ∈ C ( N − N u ) × .Splitting up these matrix operations into appropri-ately sized sub-matrices F − N BP = (cid:20) M M M M (cid:21) , (4)we can extract the zero word generation and solvefor the redundant subcarrier symbols = M ˜x d + M ˜x r (5) ˜x r = − M − M ˜x d = T˜x d (6)by the matrix T = − M − M .Since we define the number of redundant subcar-riers as N r = N u , M is quadratic with permutedVandermonde structure and invertible. With the result for the redundant subcarrier sym-bols we can find the following expression for thefrequency domain vector ˜x that is fed into the IDFT: ˜x = BP (cid:20) ˜x d ˜x r (cid:21) = BP (cid:20) ˜x d T˜x d (cid:21) = BP (cid:20) IT (cid:21) ˜x d (7)We let G = P (cid:20) IT (cid:21) to get the simple expression ˜x = BG˜x d . (8)Following this approach, the final transmit vectoris found by computing BG˜x d , changing to timedomain by applying the IDFT and finally adding theUW, as initially intended.Note, that with this approach the UW spectrumis added to the result of (2). So the UW exertsinfluence on every subcarrier in general, dependingon the choice of the UW. B. Direct approach
In contrast to the two-step approach, the uniqueword can also be generated directly at the output ofthe IDFT: (cid:20) x p x u (cid:21) = F − N BP (cid:20) ˜x d ˜x r (cid:21) . (9)Following the derivations of the two-step ap-proach we get x u = M ˜x d + M ˜x r ˜x r = M − x u − M − M ˜x d = M − x u + T˜x d (10)and finally obtain ˜x = BP (cid:18)(cid:20) IT (cid:21) ˜x d + (cid:20) − (cid:21) x u (cid:19) = BG˜x d + BP (cid:20) − (cid:21) x u . (11)After transformation of ˜x into time domain thesignal is ready to be sent.While in the two-step approach the UW is ableto influence all subcarriers, here this is not possible.The only impact of the UW in frequency domain canbe seen in (10), which is on the redundant carriers,regardless of the actual UW spectrum. . Receiver design The channel propagation of one OFDM symbolcan be modeled with the cyclic channel convolutionmatrix H ∈ C N × N and additive white Gaussiannoise n ∈ C N × as r = Hx + n .Transforming the received vector into frequencydomain and removing the zero carriers using B T ,we get the disturbed vector of data and redundantsubcarrier symbols as ˜y = B T F N r = B T F N Hx + B T F N n . (12)As presented in [2] an LMMSE estimator can bederived that extracts the data part by b ˜x d = f W e H − (cid:16) ˜y − e HB T ˜x u (cid:17) (13)using the spectral influence of the UW ˜x u , describedlater in this section, and a Wiener smoother f W = G H (cid:18) GG H + N σ n σ d (cid:16) e H H e H (cid:17) − (cid:19) − . (14)This suggests the following decoding procedure:1) Transform received symbol into frequency do-main and discard zero carriers.2) Subtract spectrum influence of the UW, afterpassing the channel e HB T ˜x u .3) Apply zero-forcing equalization e H − .4) Apply Wiener smoothing f W .This procedure is the same for the two-step andthe direct approach, with the only distinction in thedefinition of ˜x u in (13). For the two-step approachthis is simply the unique word in frequency domain,added in (3): ˜x u = F N (cid:20) u (cid:21) (15)For the direct approach this derives from (11): ˜x u = F N BP (cid:20) − (cid:21) x u (16)The remaining procedure does not depend on theused UW generation method.III. S YMBOL ENERGIES IN
UW-OFDMAlthough the two presented approaches how togenerate the unique word for the OFDM symbolseem almost identical, the impact on the symbolenergies is tremendous, as we will show in thissection, based on [3].
A. Symbol energy for the two-step approach
Using the expectation operator E {·} , Parseval’stheorem, which allows us to derive the symbol en-ergy in frequency domain and omittint the matrices BP , that do not change the energy budget, the meanenergy of an OFDM symbol, when averaging overall possible data vectors, is given by E x = E n x H x o = 1 N E n ˜x H ˜x o + x H u x u = 1 N E (cid:26)(cid:2) ˜x H d ˜x H r (cid:3) (cid:20) ˜x d ˜x r (cid:21)(cid:27) + x H u x u = 1 N E n ˜x H d ˜x d o| {z } E d + 1 N E n ˜x H r ˜x r o| {z } E r + x H u x u | {z } E u . (17)The data symbols are assumed to be uncorrelatedand from a QAM alphabet with zero mean andvariance σ d which provides E d = N d σ d N .The trace operation tr ( · ) , which is the sum of themain diagonal elements of a matrix, and (6) help usto rewrite the energy of the redundant carriers as E r = 1 N E n ˜x H r ˜x r o = 1 N E n tr (cid:16) ˜x r ˜x H r (cid:17)o = 1 N tr (cid:16) E n ˜x r ˜x H r o(cid:17) = 1 N tr (cid:16) E n T˜x d ˜x H d T H o(cid:17) = 1 N tr (cid:16) T E n ˜x d ˜x H d o T H (cid:17) = σ d N tr (cid:16) TT H (cid:17) . (18)The amount of energy needed for the redundantsymbols depends on T , which is only influenced bythe number and positions of the redundant carriers. B. Symbol energy for the direct approach
For the direct approach we start the derivation infrequency domain: E x = 1 N E n ˜x H ˜x o = 1 N E n ˜x H d ˜x d o + 1 N E n ˜x H r ˜x r o = N d σ d N + 1 N E n ˜x H r ˜x r o . (19)e use now (10) to further get E n ˜x H r ˜x r o = E n tr (cid:16) ˜x r ˜x H r (cid:17)o = tr (cid:16) E n ˜x r ˜x H r o(cid:17) = tr (cid:16) E n(cid:0) T˜x d + M − x u (cid:1) (cid:0) T˜x d + M − x u (cid:1) H o(cid:17) = tr (cid:16) E n T˜x d ˜x H d T H + M − x u ˜x H d T H + T˜x d x H u ( M H ) − + M − x u x H u ( M H ) − o(cid:17) = σ d tr (cid:16) TT H (cid:17) + tr (cid:16) M − x u x H u ( M H ) − (cid:17) (20)and finally obtain E x = N d σ d N | {z } E d + σ d N tr (cid:16) TT H (cid:17)| {z } E r + 1 N x H u ( M H ) − M − x u | {z } E u . (21)We identify the same expressions for E d and E r as in the two step approach. The diference is in E u which is the only term that depends on the actualchoice of the unique word.In [3] it is shown that the inequality x H u x u ≤ N x H u M − H M − x u (22)holds in any case. Hence the OFDM symbol from thetwo-step approach always needs equal or less energythan the symbol generated by the direct approach.We emphasize the difference between the energyof the unique word x H u x u and the energy effected bythe generation of the unique word E u . The latter iscontained in x u and ˜x p in the time domain symbol,or in ˜x r only when looking at the frequency domainrepresentation. We will call the difference betweenthe left and the right hand side excess energy , whichis only present in x p or ˜x r .IV. N UMERICAL EXAMPLES
With (22) it is obvious that the energy of anOFDM symbol generated by the two-step approachis always equal or lower than the energy of a symbolgenerated by the direct approach. We also assumethe excess energy of the direct approach is wastedand does not contribute to transmission reliability interms of bit error ratio (BER), since this energy isconcentrated on the redundant carriers only.In this section we want to give some insightto the dimension of this issue for practical setups.Therefore we consider three common unique wordsequences, compare their energy consumption in thetwo-step and the direct approach and finally showtheir impact on the BER.The BER curve of a CP-OFDM system accord-ing to IEEE 802.11a [4] will be included as a reference. We apply the same parameters for UW-OFDM wherever possible, i.e. N = 64 , the lengthof the guard interval will be the unique word length N u = N r = 16 = N r and 12 zero carriers will beincluded at the band edges and the DC carrier. IEEE802.11a also includes four pilot carriers with anoverall energy of / of the whole OFDM symbol.Since in UW-OFDM we aim to use the unique wordfor synchronization tasks, we scale the UW to thesame percentage for comparison reasons.The unique word sequences used for comparisonin this work are:1) The generalized Barker sequence [7] of length padded with zeros to the final length of .2) A CAZAC sequence (constant amplitude, zeroautocorrelation) from [5], as often used forchannel estimation, frequency offset estima-tion and timing synchronisation.3) The length unique word from [6], whichalso has CAZAC properties.The average energy demand per OFDM symbolwhen using these sequences as unique words isshown in figure 2, split in data energy E d , redundantenergy E r and UW generation energy E u . t w o - s t e p 1 ) , d i r e c t ) , d i r e c t ) , d i r e c t e n e r gyp e r O F D M s y m bo l E d E r E u Fig. 2. Symbol energies for different unique words andapproaches
The first bar shows the symbol energy if the zeroword is used as UW. Since all UW energies x H u x u are normalized, the two-step approach yields thesame symbol energy for any UW, according to (17).Thus the second bar represents this case with onlya barely noteable E u bar, being in fact / ofthe whole, topping the zero word energy. The threeremaining bars show the average energies of theOFDM symbols generated by the direct approach forthe chosen UWs. We note the huge amount of excessenergy needed only by using the direct approach forW generation. As stated earlier, the excess energydepends on the particular UW design and on thepositions of the redundant carriers.Figure 3 shows the performance of an uncodedtransmission over an AWGN channel. The bit errorrate is plotted against E b /N . − − − − − − − E b /N [dB] b it e rr o rr a t e CP-OFDM 1), direct -word 2), directtwo-step 3), direct Fig. 3. BER performance using the different UWs and differentsymbol generation approaches
Since the symbols generated by the two-step ap-proach have the same energies, the curves coincideand only one representative is shown. The bit errorperformances of the direct approach transmissionssuffer from the huge excess energy. If we comparethe curve of sequence No. 3), direct vs. two-stepapproach, we note a constant shift of about .This exactly coincides to the corresponding symbolenergies E (3) x E (two-step) x = 9 . .
25 = 7 . , . . (23)From figure 3 it can be seen that the simulatedUW-OFDM system (using the two step approach)shows a small degradation over the reference CP-OFDM system (IEEE 802.11a) in an AWGN envi-ronment. However UW-OFDM shows its potentialin frequency selective environments. Figure 4 showsresults (taken from [3]) for two different indoormultipath scenarios, one featuring deep spectralfades (channel 1), and one representing a nearlyflat fading channel (channel 2). Note that all curvesare generated without using an outer channel code,furthermore the two step approach is applied inUW-OFDM. Especially for channel 1 UW-OFDMsignificantly outperforms CP-OFDM.
10 15 20 25 3010 − − − − − − “channel 2” “channel 1” E b /N [dB] b it e rr o rr a t e CP-OFDMUW-OFDM
Fig. 4. BER performance comparing UW-OFDM with thereference CP-OFDM system in frequency selective scenarios
V. C
ONCLUSION
In this work we had a closer look on two possibil-ities of unique word generation in UW-OFDM andderived analytical expressions for the correspondingOFDM symbol energies. With the aid of three par-ticular example UW sequences, we showed that thesymbol energies of OFDM symbols generated by thedirect approach need much more energy than thosegenerated by the two-step approach. The excessenergy of the direct approach is completely wasted,since it does not improve the transmission reliability.Thus we will exclude the direct approach infurther research. R
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