Effect of Carrier Frequency Offset on Channel Capacity in Multi User OFDM-FDMA Systems

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1 Effect of Carrier Frequency Offset on Channe Capacity in Muti User OFDM-FDMA Systems Martin Stemick and Hermann Rohing Abstract Orthogona Frequency Diision Mutipexing (OFDM is a ery robust transmission procedure in mutipath and frequency seectie radio channes. A Frequency Diision Mutipe Access (FDMA resource aocation technique offers the opportunity of a detaied ink adaptation scheme. The combination of these transmission- and mutipe access techniques in OFDM-FDMA is an idea and ery strong candidate for the downink of future 4G mobie communication systems. This technica combination offers high ce capacities by expoiting the inherent muti user diersity effect of the system. To appy OFDM-FDMA in the upink the time and carrier synchronization accuracy becomes ery important. on-idea synchronization of the user signas to the carrier frequency of the base station eads to intercarrier interferences (. In this paper an anaytica mode for the consideration in the upink of a muti user OFDM-FDMA based system is deried. The impact of the carrier frequency offset (CFO on the performance of a ceuar muti user system with respect to different subcarrier aocation schemes is anayzed. T Index Terms OFDM-FDMA Upink Synchronization. I. ITRODUCTIO he fourth generation (4G of mobie communication systems wi hae to offer hundreds of Mbit/s of data rate both in the downink and in the upink direction in order to meet the demands and requirements of future mutimedia appications. This can be fufied if the utiized communication system has a high degree of fexibiity and adaptiity. OFDM based transmission systems in combination with mutipe access schemes were found to be ery promising to proide broadband communication schemes and high system fexibiity simutaneousy [] []. Among a other mutipe access schemes the OFDM- FDMA approach permits a ery high degree of adaptiity in frequency seectie radio channes combined with a moderate technica system compexity []. Howeer in order to proide high data rates not ony in the downink but aso in the upink a number of additiona technica chaenges has to be met. One of these chaenges is the synchronization of a mobie terminas (MT to the base Manuscript receied March 6 7. This work was supported by the German Research Counci (DFG under Grant RO 76/4-. Martin Stemick is with the Institute of Teecommunications Hamburg Uniersity of Technoogy Hamburg 73 Germany (phone: +49-( ; fax: +49-( ; e-mai: stemick@et.tu-harburg.de. Hermann Rohing is Head of the Institute of Teecommunications Hamburg Uniersity of Technoogy Hamburg 73 Germany (e-mai: rohing@tu-harburg.de. station (BS inside a singe ce. An OFDM-FDMA based system diides the aaiabe bandwidth into a set of orthogona but spectray oerapping. Such a system wi be especiay sensitie to carrier frequency offsets (CFO. In the upink eery MT aocates an excusie set of which are superimposed with a other user signas at the BS to an oera OFDM symbo. If eery MT proides an indiidua carrier frequency offset strong intercarrier interferences ( are obsered at the BS [3] [4]. Therefore a precise synchronization of a MTs to the BS is ita for such a radio system. This carrier synchronization scheme can e.g. be done in such a way that in a setup phase the BS measures the CFO for each MT indiiduay. The measured CFO are then fed back to the MTs in order to adjust their oca osciators. The actua measurement of the CFO can be done with dedicated signas or bindy cf. [5] [6] [7] and references therein. Howeer such a procedure needs arge a measurement oerhead and its performance wi strongy depend on measurement accuracies. Therefore in most systems a certain amount of CFO has to be toerated. The objectie of this paper is therefore to inestigate the genera impact of CFO to the performance of muti user OFDM-FDMA based systems. The resuting does not ony depend on the amount of CFO introduced from different users but aso on the empoyed subcarrier aocation scheme. In Section II the considered system mode is introduced foowed by an anaytica mode for the CFO depending in Section III. As wi be seen the effect of can be described by a Gaussian noise signa added to the wanted signas of a users [7] [8]. In Section IV the dependency between the utiized subcarrier aocation scheme and the ee of noise for different users is shown. Section V introduces muti user scenarios to eauate the impact of CFO onto the channe capacity. Section VI finay gies simuation resuts on how much CFO decreases the performance of the introduced OFDM-FDMA based muti user scenarios. II. SYSTEM MODEL The upink of a mobie communication system based on the OFDM-FDMA transmission and aocation technique is considered in this paper. The system consists of a singe ce with a centra BS and u mobie terminas (MT. The oera bandwidth B is diided into. Each MT aocates c / u excusiey to fufi its transmission demands. In a first step it is assumed that a MT hae the same distance to the BS in order to aoid any

2 shadowing and path oss effects inside the used signa mode. The considered system mode is depicted in Fig.. The assumed radio channe mode is to eauate soey the effects of. It is assumed that there exists some means of synchronization inside the oera system in such a way that the MT are abe to perform a coarse synchronization based on OFDM symbo ee anaysis. Therefore a perfect positioning of the FFT window for a MT and therefore zero interference between consecutie OFDM symbos (ISI can be assumed. Aso samping offsets between MT and BS are assumed to be negigibe. Fig.. Considered system mode III. AALYTICAL MODEL FOR CFO In OFDM based systems the transmitted time signa x n of a user can be described by the foowing equation: kn j π xn Xk e ( k Where X k is the compex aued moduation symbo of user on subcarrier k. In the foowing k is the subcarrier index on the transmitter (MT side. To mode the mutipe access procedure based on the OFDM-FDMA transmission technique the moduation symbos X k of a user show non-zero aues on the aocated and zero aues esewhere. The transmit time signas x n of eery MT wi pass through indiidua mutipath radio channes and show indiidua CFO. A user signas are then combined and superimposed at the BS to an oera receie signa. The oera receied time signa y n is u jπ kn yn X kz k( n e + ηn ( k with η n as the white Gaussian noise component at the receier and Zk ( n as the combined channe fading and CFO infuence to the signa of user. At the receier side perfect positioning of the FFT window and no signa samping offset is assumed so Z ( n can be expressed as k n f jπ δ k ( k Z n H e (3 where H k represents the channe transfer factor for user on subcarrier k and δ f describes the CFO of user normaized to the subcarrier spacing Δ f. In the foowing we wi consider the interference situation in the receier (BS for a specific user ς on a subcarrier assigned to this user where is the subcarrier index in the receier. After cacuating the FFT in the receier and combining ( and (3 the receied symbo Y on subcarrier can be described by ( n δ f u nk j π j π k k + n k Y X H e e (4 Equation (4 can be spit up into the wanted user moduation symbo X ς of user ς who excusiey transmits on subcarrier and noise on from the remaining k {[ ] \ } see (5. Y α ς H ς X ς + + (5 The noise can be spit into the indiidua contributions from each user (incuding user ς inside the ce. An anaytica expression for is gien in (6. Besides the channe infuence H ς there is an additiona oss α ς on moduation symbo X ς due to the indiidua CFO of user ς on subcarrier see (7. n( δ f + k j π k k n k k X H e (6 α ς n n δ fς j π e (7 The moduation symbos are assumed to be statisticay independent random ariabes from a predefined moduation * aphabet i.e. E{ Xk X i j} for i k j. Additionay the transmit power for the moduation E X hods. symbos is normaized such that { k } Therefore according to the centra imit theorem the noise can be modeed as a zero-mean Gaussian distributed noise signa with ariance { } ( E cf. [7] [8]. The anaytica expression is as foows: ( H k e k n d k k k H ( n d jπ f k [ δ + ] ( π ( δ f + k ( π( δ f + k sin k for Further from (7 the power oss L ς due to CFO of user ς aocated to subcarrier can be cacuated see (9. { ας } δ fς jπ n ς n ( d π + cos d d L E e (9 δ fς To gie an impression what happens if mutipe users transmit to the BS with certain CFOs δ f a simpe exampe with two users is depicted in Fig.. This exampe shows the reception of two users at the BS where the integer samping points of the are shown on the x-axis. At the BS user shows a frequency shift of δ f. This eads to a oss of signa power on the samping points of user (cf. (9 which is marked with a square on one exempary subcarrier. On the same subcarrier sef-interference of user can be (8

3 user user obsered which is marked with trianges. Additionay user introduces to the signas of user (cf. (8 which is highighted with circuar markers for one exempary samping point. A this can be summarized for the genera case using (5 (8 and (9 which yieds SR u ς Hς L ( + ( ( Equation ( describes the oera SR on subcarrier. This SR depends on the CFO and channe situation of user ς who is aocated to and aso on disturbances by from a users. Additionay the Gaussian noise power at the receier is represented by ( { } E. The ariance of the oera interference noise on subcarrier is therefore Rx Amp. äf u ( (. ( user user Fig.. Exampe for two users at BS where user exhibits CFO f/ Äf IV. EVALUATIO OF After proiding the anaytica background in the preious paragraph now the quantitatie impact of CFO-induced can be shown for genera OFDM-FDMA systems. One important factor in this context is the subcarrier aocation scheme that is used in the upink. To increase the performance of OFDM-FDMA adaptie subcarrier aocation schemes which expoit the mutiuser diersity inside the system can be used [9] []. The two aocation schemes considered here are either bockwise- or intereaed aocation. A bockwise aocation means that a singe user aocates ony bocks of adjacent see Fig. 3. The bocksize can ary between one and c according to the system requirements. In the intereaed scheme each user assigns a set of equidistanty spaced. The subcarrier sets are positioned in an intereaed way oer the system bandwidth which is depicted in Fig. 4. This is a specia case of a bockwise aocation with bocksize where the of an indiidua user hae the maximum mutua distance. In the foowing the ee of interference power ( is eauated for these different aocation schemes for the simpe case of two users inside the system. user user user - u Fig. 3. Bockwise subcarrier aocation user u - Fig. 4. Intereaed subcarrier aocation user u - Two exampes are considered. In both exampes a tota bandwidth of 56 is assumed which is equay shared between the users. At the BS a CFO of δ f. is obsered for user whie user is perfecty synchronized. Then the ee of ( is cacuated (cf. (8( on each subcarrier. The db-aues of ( in Fig. 5 and Fig. 6 are reated to the transmit power { } E X. In the first exampe bockwise aocation with a bocksize of 8 is considered so ony two bocks are present in the system see Fig. 5. Because user is not we synchronized some is obsered on a. This is constanty high for user because his aocated interfere with each other. Ony at the border to user the decreases because there is no interference from user to user due to his perfect synchronization. A which wi be obsered by user stems from user and is steadiy decreasing for which are far away from the interfering carriers. Most to user is we beow -3dB. A different situation can be seen in Fig. 6. There the bocksize is decreased to 6 and the bocks are aocated in an intereaed manner. The situation for user worsened compared to Fig. 5 because now eery bock of user experiences heay interference from both sides. ó [db] user user Fig. 5. for two users with bockwise aocation; 8 for each user bocksize 8 user with δ f. user with δ f.

4 ó [db] user user Fig. 6. for two users with bockwise aocation 8 for each user bocksize 6 user with δ f. user with δ f As a resut the interference ee of user is aways higher than -3 db. On the other hand user profits from the changed subcarrier aocation. At the border- of his aocated bocks he sti obseres a ow interference ee. But now the number of bocks has increased and therefore the number of border- aso has increased so that the oera performance of user is improed. If the bocksize in Fig. 6 is further reduced to one a purey intereaed aocation scheme is obtained. Then the interference situation for the two users is conersed: ow user sees the owest interference since the immediate neighbors of his are the perfecty synchronized carriers of user. This effect can be shown by a simpe anaysis: In an intereaed scheme with two users and bocksize the subcarrier aocation pattern is ery reguar. Therefore if the at the borders of the bandwidth are negected the interference situation is amost the same for a of a specific user. For further simpification we assume that the strongest interference to a subcarrier stems from its two neighboring. Using (8 and the aboe prerequisites the interference power ( for a of user can be cacuated as foows: ( δ f + ( sin( π( δ f+ ( ( π ( δ f To cacuate the interference ( for user the CFO in ( has to be set to δ f. Due to our assumptions ( is independent of. Since in our exampe δ f hods equation ( yieds (. This means user sees no interference amost as assumed aboe. Of course user wi experience from its own more far away but these hae minor infuence and are negected here. On the other hand eery subcarrier of user now obseres heay interference from user. This can be seen by substituting δ f with δ f in (. Since δ f. hods ( wi take a aue of -7dB. So the situation from Fig. 5 and Fig. 6 where user saw the highest and user the owest interference is competey conersed. V. CFO AD SYSTEM CAPACITY After discussing the infuence of the subcarrier aocation scheme on the interference ee for indiidua users it is now of interest to eauate which effect the CFO wi hae on the oera performance of a muti user system. The system mode introduced in section II wi be considered in the foowing. Inside the system 6 users wi equay share a bandwidth of 56. The channe between the MTs and the BS wi be with CFOinduced as an additiona disturbance. Intereaed and bockwise aocation wi be empoyed. As a performance criterion the theoretica channe capacity C of each user wi be measured based on C bits og ( SR OFDM-Sym.. (3 + of user The -power is assumed to be (.. To get significant aerage resuts for eery user the aocation of a users to the subcarrier bocks is done in a random manner. In the simuations two scenarios were considered: In the first scenario a MTs except one are perfecty synchronized to the BS. The capacity of the bady synchronized user is eauated for arious aocation schemes and CFOs. In this way the sef inficted interference of a user can be eauated in dependency on the aocation scheme. The second scenario assumes that a users except one hae an absoute CFO of δ f. The capacity of the singe perfecty synchronized user is aso eauated for arious aocation schemes and CFOs. This scenario shows the performance oss due to interference inficted from other users. VI. RESULTS In the foowing we wi present some quantitatie resuts for the scenarios introduced in the preious section. Fig. 7 shows the resuts for the first scenario where we obsere the capacity of a non-ideay synched user. In genera the performance of this user decreases with increasing CFO. W..o.g. we wi assume the obsered non-synchronized user to be user. The worst performance figures are obsered for a bockwise aocation with bocksize 6. This means in the considered system mode that eery user gets ony one bock of. The bad performance shown for user stems from his strong sef-interference due to the accumuation of a of his in a singe bock (cf. aso Fig. 5 user. When the bocksize decreases the performance of user improes because aso his sef-interference decreases. For the intereaed aocation there is amost no performance oss because the of user hae maximum distance to each other and therefore obsere ony ow interference. Fig. 8 shows the resuts for the second scenario where ony one user in the system is perfecty synchronized and a others experience a uniform CFO to the BS. We obsere the capacity of the synchronized user (w..o.g.: user. User ony obseres interference noise from other users no sef

5 interference. Therefore aocation schemes with arge bock-sizes are adantageous because then interference is ony obsered at the borders of one bock. In Fig. 8 the triange marked cure refers to the aocation with the argest bocksize and therefore shows the owest capacity oss for user. According to our considerations the performance oss grows worse when the bocksize decreases. Bits per OFDM-Sym Bocksize 6 5 Bocksize 8 Bocksize 4 5 Bocksize Intereaed äf Fig. 7. Capacity of the bady synchronized user for arious aocation schemes. A other users in the system are perfecty synchronized. Bits per OFDM-Sym Bocksize 6 Bocksize 8 Bocksize 4 Bocksize Intereaed shown that the performance oss of indiidua users through depends strongy on the subcarrier aocation scheme. Considering the oera system performance one has to be aware of the fact that changing the aocation scheme structure ony aries the distribution of inside the system but has no effect on the oera power. REFERECES [] A. F. Moisch Wideband Wireess Digita Communication. Upper Sadde Rier J: Prentice Ha st ed.. [] H. Rohing D. Gada R. Gruenheid OFDM: A Fexibe and Adaptie Air Interface for a 4G Mobie Communication System in Proc. Internationa Conference on Teecommunications (ICT Beijing pp [3] A. M. Toneo et. a. Anaysis of the Upink of an Asynchronous Muti-user DMT OFDMA System Impaired by Time Offsets and Muti-path Fading in Proc. IEEE VTS-Fa VTC Boston MA pp [4] L. Schwoerer G. Hetmanczyk Impact of frequency errors in WIGWAM OFDMA upink in Proc. th Int. OFDM-Workshop - InOWo 6 Hamburg 6 pp [5] J. Zhu W. Lee Carrier Frequency Offset Estimation for OFDM Systems With u Subcarriers IEEE Trans. on Vehicuar Technoogy o. 55 no. 5 pp Sep. 6. [6] T. Roman S. Visuri and V. Koiunen Bind Frequency Synchronization in OFDM ia Diagonaity Criterion IEEE Trans. on Sig. Proc. o. 54 no. 8 pp Aug. 6. [7] K. Brüninghaus Ein Beitrag zur Demoduation Synchronisation und Kanaschätzung in OFDM-basierten Funkübertragungssystemen. PhD thesis Technische Uniersität Hamburg-Harburg 3. [8] K. Sathananthan and C. Teambura Probabiity of Error Cacuation of OFDM Systems With Frequency Offset IEEE Trans. Comm. o. 49 no. pp o.. [9] M. Stemick S. Oonbayar H. Rohing PHY-Mode Seection and Mutiuser Diersity in OFDM Transmission Systems Frequenz o. 6 no. - pp. 7- Jan./Feb. 7. [] J. Gross H. Kar F. Fitzek A. Woisz Comparison of heuristic and optima subcarrier assignment agorithms in Proc. Int. Conference on Wireess etworks (ICW 3 Las Vegas V 3 pp äf Fig. 8. Capacity of the perfecty synchronized user for arious aocation schemes. A other users in the system obsere an absoute CFO of δ f. These two exampes depicted in Fig. 7 and Fig. 8 show not ony the dependency of system performance on CFO and subcarrier aocation scheme. It aso becomes obious that in the first exampe a sma bocksize is of adantage whie in the second exampe it is just the other way round. This is because in exampe ony sef-inficted is obsered whie in exampe ony from adjacent users is obsered. The choice of an -optimized subcarrier aocation scheme shoud therefore aso depend on the oera interference situation inside the communication system. For gien CFO the system designer can choose how the is distributed between the users by choosing a suitabe subcarrier aocation scheme. Of course not the absoute amount of inside the system can be infuenced but its distribution. VII. COCLUSIO In this paper the infuence of CFO-induced on a muti user OFDM-FDMA system was eauated. It was

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