WINDOW DESIGN FOR NON-ORTHOGONAL INTERFERENCE REDUCTION IN OFDM RECEIVERS

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1 WINDOW DESIGN FOR NON-ORTHOGONAL INTERFERENCE REDUCTION IN OFDM RECEIVERS Patric Nicel, Wolfgang Gerstacer, Christof Jonietz, Ger Kilian, Albert Heuberger, an Wolfgang Koch Institute for Mobile Counications, University of Erlangen Nureberg, Cauerstr. 7, 958 Erlangen, Gerany, Eail: {nicel, gersta, jonietz, Counications Departent, Fraunhofer IIS, A Wolfsantel 33, 958 Erlangen, Gerany, Eail: {il, heb}@iis.fraunhofer.e ABSTRACT For conventional orthogonal frequency-ivision ultiplexing OFDM systes, the guar interval is isregare in the receiver an rectangular shape winows are use iplicitly ue to the iscrete Fourier transfor DFT. We consier a Nyquist-shape winow in the receiver, so that the transit signal oes not nee to be oifie, an optiize the winow shape in orer to suppress interference fro non-orthogonal single carrier interferer signals. For the winow coefficients, a close for solution is erive. This type of interference usually occurs in OFDM raio systes lie Digital Raio Moniale DRM that are operating in low frequency bans. Disturbance is given for instance by haronics of switche-oe power supplies.. INTRODUCTION Usually, orthogonal frequency-ivision ultiplexing OFDM systes apply a guar interval as pre- an/ or postfix in orer to obtain flat-faing subcarriers for low coplexity equalization. Diversity of the channel is retaine by interleaving an joint coing over all subchannels. In the Digital Raio Moniale DRM syste [], ultilevel coing MLC is eploye. For ecoing, e.g. iterative approaches are use [], that nee inforation on the signal to noise ratio SNR of each OFDM sybol in orer to guarantee high perforance of the transission syste. The guar interval is ignore in conventional OFDM receivers, as it ay be affecte by intersybol interference ISI, an the iscrete Fourier transfor DFT is use for eoulation of the subchannel sybols. As a part of the guar interval ay not be affecte by ISI ue to a shorter ipulse response of the channel, this unconsue part ay be use in orer to apply a ifferent winow than the iplicitly ue to the DFT use rectangular winow. Different winowing techniques have been publishe in [3 5]. To retain the orthogonality of the OFDM subcarriers, Nyquist winows have been introuce an optiize in these publications in orer to axiize the subcarrier SNR after eoulation an to itigate intercarrier interference ICI occurring ue to iproper frequency synchronization of the receiver. For this scenario, a close for solution for the optiu winow coefficients is erive in [4, 5]. A polynoial-shape winow is propose in [6], for which coefficients are eterine by nuerical optiization. In this paper, non-orthogonal interference fro sinusoial signals with frequencies unequal the OFDM signal subcarrier frequencies is consiere. In our approach, an optiu winow in the sense of iniu interference leaing [3] into ajacent OFDM subcarriers is use. The winow has to be optiize to the current interference scenario. As the interference ay not be copletely reove by winowing, an since, ue to the coe properties, it is better to reuce oerate interference on the less isturbe subcarriers, than strong interference on nearly lost subcarriers, with respect to the optiu bit error rate perforance, a traeoff between ain-lobe with an sie-lobe heights of the interference on the subcarriers is eterine. An aaptive approach that iniizes the overall interference on the OFDM subcarriers is presente in [7] for igital subscriber line DSL systes, which has the tenency to focus ainly on the stronger interfere carriers with the isavantage, that subcarriers with oerate an low interference ay eteriorate. This causes a non-optial ecoing perforance, as usually on the ore interfere carriers not very uch gain can be obtaine. In contrast to this, our approach efines a region, where interference parts are not inclue in the optiization. Other solutions ealing with the sinusoial interference proble [8,9] use fixe winows lie Hanning, Haing an raise-cosine winows with ifferent paraeter sets. In ifference to e.g. [8], no oifications at the transitter sie are neee for our solution. The paper is structure as follows. The syste oel is introuce in Section, an the erivation of an optiu Nyquist-shape winow for the interference scenario is given in Section 3. Finally, perforance is presente in ters of siulation results an analyze in Section 4.. SYSTEM MODEL We aopt the transission oel of [5]. For the generation of the OFDM signal, a -point inverse DFT IDFT is applie. A ν enotes the coplex aplitue of the transit sybol on subcarrier ν, where A ν is taen fro the sae zero-ean signal set for all subcarriers ν. The subcarrier spacing is given by f sub / T, where T enotes the oulation interval. A guar interval of D g D pre + D post saples in total is ae with D pre saples as cyclic prefix an D post saples as cyclic postfix, respectively. The iscrete-tie coplex baseban transit We prefer the syetric representation of the guar interval for clarification /6/$. 6 IEEE. SPAWC6

2 sequence for ties D pre < + D post is given by s ν π +j A ν e ν. We efine the average transit signal power as σs E{ s }, where E{ } enotes the expectation operator. In ifference to [5], we assue perfect synchronization of the carrier frequency an ay therefore neglect ICI coponents. The consiere signal is now isturbe by an interferer signal, which is given by z π +j Z e γ, with coplex aplitue value Z. The interferer frequency is given by f z an the noralize frequency is γ f z / f sub f z T. The average interference power is σz E{ z } Z/. The receive signal for the interfere aitive white Gaussian noise AWGN channel is given by r s + z + n, 3 with variance σn E{ n } of the noise saples n. Applying the winow w to the receive signal, an perforing a DFT, the OFDM eoulator elivers the output Y π j w r e. 4 Assuing D g, we set w <, an can siplify 4 as Y D u + π j w +n r +n e. n 5 }{{} y This expression now represents the cascae operations of winowing an DFT. Instea of r, now y is processe by the DFT. Therefore, winowing can be viewe an ipleente as preprocessing operation of existing OFDM receivers. 3. OPTIMUM WINDOW SHAPE In orer to aintain the orthogonality of the OFDM subcarrier signals, the Nyquist conition in tie oain has to be fulfille for the winow function [4, 5]: w +i C, <. 6 i Here, C represents a constant factor, which is arbitrarily set to C. We obtain Y S + Z + N, 7 where S A, 8 Z π j w z e, an 9 Dw Dw Dw c g g w Dw Dw Dw Fig.. Decoposition of the Nyquist winow w into the su of a rectangular function c an an optiize part g g for D w. N Dw Dw Dw +Dw +Dw +Dw π j w n e. For the assue white noise an orthogonal DFT basis functions, we obtain uncorrelate, zero-ean noise ters in frequency oain of variance σ N E { N } σ n w. We efine the rectangular winow coponent of conventional OFDM receivers as { for < c otherwise. In orer to fulfill the Nyquist conition 6 iplicitly for our winow function w, we efine w c + g g, 3 where g represent the paraeters to be optiize. As in general the largest part of the guar interval is ipaire by ISI ue to the elay sprea of the channel, only the reaining part is available for winowing. We assue here, that at least D w saples are usable for winowing, therefore D w aitional saples are use on either sie of the guar interval as epicte by Fig. an g < D w D w. Now we consier the interference Z fro 9 an insert an 3: Z c + g g e j π γ. 4 The nuber of aitional winowing saples is efine as a ultiple of for easier representation. The elay sprea of the channel can be eterine by the algorith of [], so that the winow length can be atche properly to the channel conitions.

3 This forula can be split up into two parts, Z Z Z, with Z Z D u c e j π γ, an g g e j π γ. With, the first part can be expresse as Z D u + e j π γ Z j π γ j π γ e j π γ+ sin πγ sin π γ, 5 an for the secon part we obtain with Z Z κ Z D w g e j π γ κ D w g e j π γ + j π γ e j π γ κ jπγ e j π γ κ, 6 where Z an the efinition of g was use in the last step. With the assuption, that the interferer frequency is nown 3, we now consier the interference eission on the less isturbe subchannels. For this, we efine an interval aroun the noralize interferer frequency, given by the set M {M,..., M }, within which the interference is isregare. Here, we choose M γ N +.5 an M γ + N.5, where enotes the rouning operation to the nearest integer value an N represents the nuber of isregare frequency bins aroun the noralize interferer frequency. In the case that M < or M, the interval has to be split in orer to account for the cyclic property of the DFT. Outsie this winow, the total signal to interference plus noise ratio SINR is axiize in orer to iprove ecoing perforance. As the signal power is fixe, we now iniize the average interference plus noise power. We assue that all subcarriers are occupie, so that only the relative position of the interferer frequency to the subcarrier frequencies is relevant in further consierations. We consier the total SINR in the for: σ A using the average sybol power σ A ρ tot σ Z + σn, 7 E{ A }, an the 3 This assuption is only neee for winow optiization. average subchannel interference power σ Z σz N, 8 with / M σz { E Z } E { Z + Z } σz π ej γ+ sin πγ sin π γ + D w κ D w jπγ e j π γ κ. 9 In the following, [ ] i an [ ] i,l enote the i-th eleent of a colun vector an the i, l-th eleent of a atrix, with the first eleents inexe by an,, respectively. T, H an stan for the transpose, Heritian transpose an conjugation operation on atrices an vectors, respectively. The I L atrix of zeros is efine as I L, the I L atrix of ones as I L, an the L L ientity atrix as I L L. Defining the iagonal atrix S : { for i {,..., } \ M [S ] i,i elsewhere, the colun vector e : [e ] i σ z e j π γ+i sin πγ sin π γ i, an the colun vector e : e B g, with the D w atrix B: [B] i,l σ z jπγ e j π γ il Dw, 3 an the D w colun vector g: [g] l g l Dw, 4 we can express 8 as σ Z e + e H S H S e + e N }{{} S e + B g H S e + B g. 5 N With an 3, we efine w + C g, 6 with [ T D w T D ] T, u T D w an 7 C [ I T D w D w T D ] T, u D w D w IT D w D w 8 an rewrite as σn σ n w H w σ n + C g H + C g. 9 Now, ρ tot is axiize with respect to the vector g: g opt arg ax ρ tot. 3 g

4 log σ Z /σ z [B] Rectangular winow Optiize winow N Optiize winow N Fig.. Noralize interference variance on subcarriers ajacent to the interferer frequency for noralize interferer frequency γ 5.5. Using the Wirtinger erivative [], we obtain: ρ tot g σa g σ Z + σn σ A σ Z σ + σ Z N g + σ N g σ S σ + σ Z N B H S e + B g + σ nc H + C g. N The solution for 3 is foun by setting ρ tot g BH S e + B g opt N σ nc H + C g opt. The optiu vector is obtaine as g opt B H S B + σ n N C H C 3 3 D u B H S e + σ n N C H. 33 For the special case, that the noise power can be neglecte, g opt B H S B B H S e PERFORMANCE COMPARISON In the following, we analyze the properties of the propose receive winowing for interference suppression an show the perforance of ifferent winow functions in a practical syste. We copare the aaptive winow accoring to 34 to other fixe winows lie the rectangular winow, the raisecosine winow, an the trapezoial winow for ifferent paraeters N. As exaple, we consier an OFDM syste w w Fig. 3. Winow functions for noralize interferer frequency γ 5.5 legen of Fig. vali. with 56, D g 64 an D w 8. As a worst case scenario, we use γ 5.5, so that the interferer frequency is locate in the ile between two ajacent subcarriers. For this case, the noralize interference variance on subcarriers ajacent to the interferer frequency is shown in Fig.. The use winow functions are epicte in Fig. 3, where in the lower part only the transition interval [ 8,..., 7] is shown. The optiization of the winow function is one for N an N 5, respectively. It can be observe, that with increasing value of N, the ain lobe of the interference aroun γ gets broaer, but at the sae tie, the sie lobes are ore an ore suppresse. Therefore, for practical applications, a trae off has to be foun epenent on the worst interference scenario which is anageable. Being esigne for such a scenario, the winow ight be fixe accoring to the esire target bit error rate BER. In Fig. 4, we investigate a DRM transission [] with DRM robustness oe B, spectru occupancy an protection level overall coe rate R all.5, 64-QAM MLC oulation, long interleaving over two ifferent channels as exaple. The figures show the BER versus E S /N E S : average receive sybol energy, N : noise power spectral ensity at the receive antenna. Here, a carrier to interference ratio CIR of log CIR 6 B is consiere, for which the conventional receiver ay reach BERs saller than 4. The transissions are consiere along with the presente winow functions. e to the applie ulti-level coing, a steep transition can be observe at log ES /N 6 B for the AWGN channel. The su of interference an noise power is assue to be nown at the receiver for each subchannel, if not, it has to be estiate. For the AWGN channel, a gain of.7. B can be observe for winows ifferent fro the rectangular winow, representing the conventional receiver. The visible error floor is ue to the reaining interference on the subcarriers. A lower error floor can be observe for the presente optiize winows. As an exaple for a tie variant frequency-selective channel, we consier DRM channel No. 5 [], that consists of two equal power Rayleigh faing paths with Doppler sprea Hz each an elays s an 4 s, respectively. For this channel, a larger gain of ore than B can be observe in the transition range of the curves copare to the

5 BER 3 Rectangular winow Optiize winow N Optiize winow N 5 channel 5 channel BER 3 Rectangular winow Opt. win. N, fixe Opt. win. N 5, fixe Opt. win. N 5, ajuste to γ log ES /N [B] Fig. 4. Perforance coparison of ifferent receive winows for DRM transission oe B. DRM channel No. AWGN channel an DRM channel No. 5 with interference, log CIR 6 B. conventional receiver. It can be notice, that the paraeter N shoul be chosen larger than in orer to iprove perforance. A value of N 5 was foun to be optiu by coputer search. For this setting, a soewhat better perforance than for the nown winow functions can be observe, although for these heuristic winow functions alreay goo perforance is obtaine. Fro the presente results, especially the use of fixe winows lie e.g. the trapezoial winow sees reasonable. It turns out, that the optiize winow function is only slightly epenent on γ. Optiization has been carrie out for the entione worst case scenario, as the true interference frequency is generally unnown an the ipact of interference is uch saller for interferer frequencies close to ultiple of f sub. In Fig. 5, BER versus γ is shown for ifferent optiize winows log CIR 6 B, log ES /N B. As alreay state, no perforance ifference between the fixe worst-case winow optiization an the winow optiization ajuste to γ is observable. Using 33 instea of 34 for winow optiization, in aition to the signal an noise power also the interference power has to be nown by the receiver, which is ifficult to achieve in practice. 5. CONCLUSIONS In this paper, a novel receive winow optiization etho is presente, that iproves the robustness of conventional OFDM receivers against non-orthogonal single carrier interferers. For a DRM syste, a gain of ore than B in ters of signal to noise ratio is shown to be achievable in realistic scenarios. The winow ay be aapte to the current strength an location of the interference. 6. REFERENCES [] ETSI Stanar Digital Raio Moniale DRM, ETSI ES 98 V.. 4-6, Syste Specification. [] V. Fischer, A. Kurpiers an F. Kulla: Iprove Multistage Decoing of Multilevel Coes for Digital Raio Fig. 5. Perforance coparison of ifferent receive winows for DRM transission oe B. DRM channel No. 5 with interference, log CIR 6 B, log ES /N B. Moniale DRM, 8th IEEE International Syposiu on Consuer Electronics 4, Unite Kingo, 4. [3] C. Muschalli. Iproving an OFDM reception using an aaptive Nyquist winowing. IEEE Transactions on Consuer Electronics, vol. 4, no. 3, pages 59 69, August 996. [4] S.H. Müller-Weinfurtner. OFDM for wireless counications: Nyquist winowing, pea-power reuction, an synchronization. Reihe Kouniations- un Inforationstechni, Ban 6, Shaer Verlag, Aachen, Dissertation Universität Erlangen-Nürnberg,. [5] S.H. Müller-Weinfurtner. Optiu Nyquist Winowing in OFDM Receivers. IEEE Transactions on Counications, vol. 49, no. 3, pages 47 4, March. [6] R. Song an S.-H. Leung. A Novel OFDM Receiver with Secon Orer Polynoial Nyquist Winow Function. IEEE Counications Letters, vol. 9, no. 5, pages , May 5. [7] J. Refern. Receiver Winow Design for Multicarrier Counication Systes. IEEE Journal on Selecte Areas in Counications, vol., no. 5, pages 9 36, June. [8] M. Valaa, R. Porat, an F. Harris. OFDM transission with receiver winowing for iprove interference rejection. Proceeings of the Thirty-Seventh Asiloar Conference on Signals, Systes an Coputers, Pacific Grove, CA, Noveber 3. [9] D. Zhang, F. Pingyi, an C. Zhigang. Receiver winow esign for narrowban interference suppression in IEEE 8.a syste. Proceeings of the Joint Conference of the th Asia-Pacific Conference on Counications an the 5th International Syposiu on Multi- Diensional Mobile Counications, Beijing, China, August 4. [] M. Oziewicz. On application of MUSIC algorith to tie elay estiation in OFDM channels. IEEE Transactions on Broacasting, vol. 5, no., pages 49 55, June 5. [] S. Hayin. Aaptive Filter Theory. Prentice-Hall, Upper Sale River, New Jersey, Thir Eition, 996. γ