Generalized Frequency Division Multiplexing: Analysis of An Alternative Multi-Carrier Technique for Next Generation Cellular Systems

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1 Generalized Frequency Division Multiplexing: Analysis o An Alternative Multi-Carrier Technique or Next Generation Cellular Systems Michailow, Nicola; Gaspar, Ivan; Krone, Stean; Lentmaier, Michael; Fettweis, Gerhard Published in: International Symposium on Wireless Communication Systems (ISWCS), 212 DOI: 1.119/ISWCS Link to publication Citation or published version (APA): Michailow, N., Gaspar, I., Krone, S., Lentmaier, M., & Fettweis, G. (212). Generalized Frequency Division Multiplexing: Analysis o An Alternative Multi-Carrier Technique or Next Generation Cellular Systems. In International Symposium on Wireless Communication Systems (ISWCS), 212 (pp ). IEEE--Institute o Electrical and Electronics Engineers Inc.. DOI: 1.119/ISWCS General rights Copyright and moral rights or the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition o accessing publications that users recognise and abide by the legal requirements associated with these rights. Users may download and print one copy o any publication rom the public portal or the purpose o private study or research. You may not urther distribute the material or use it or any proit-making activity or commercial gain You may reely distribute the URL identiying the publication in the public portal Take down policy I you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. L UNDUNI VERS I TY PO Box L und Download date: 18. Nov. 218

2 Generalized Frequency Division Multiplexing: An Alternative Multi-Carrier Technique or Next Generation Cellular Systems Nicola Michailow, Ivan Gaspar, Stean Krone, Michael Lentmaier, Gerhard Fettweis Vodaone Chair Mobile Communications Systems, Technische Universität Dresden, 169 Dresden, Germany {nicola.michailow ivan.gaspar stean.krone michael.lentmaier Abstract Generalized requency division multiplexing (GFDM) is a new concept that can be seen as a generalization o traditional OFDM. The scheme is based on the iltered multi-carrier approach and can oer an increased lexibility, which will play a signiicant role in uture cellular applications. In this paper we present the beneits o the pulse shaped carriers in GFDM. We show that based on the FFT/IFFT algorithm, the scheme can be implemented with reasonable computational eort. Further, to be able to relate the results to the recent LTE standard, we present a suitable set o parameters or GFDM. I. INTRODUCTION In the last couple o years, the popularity o smartphones has grown tremendously and as a consequence to growing demand, mobile internet has become an aordable service or many people. Along with increasing data rates and improved coverage, this trend enables novel applications o wireless cellular systems that had not been easible a ew years back. Among those, the Internet o Things (IoT) is particularly prominent. The idea o an IoT is based on the prediction that in a couple o years, the internet will not only be used by people, but it will also constitute an inrastructure or the interaction o all kinds o machines and devices rom an extremely broad ield o application. Assuming each individual will own several IoT enabled devices, the next generation o cellular systems will be aced with a magnitude o larger number o subscribers. And this will introduce a large variety o new requirements, e.g. regarding mobility, data rates, latency, energy eiciency with respect to low-cost battery driven devices and quality o service. Another approach that tries to satisy the above requirements in a spectrally lexible way is cognitive radio (CR). One particular goal there is to use dynamic spectrum access to exploit spectrum resources, which although they are assigned to a certain service, remain unused at a given time in a given location. This area o application calls or the ability to transmit narrow-band signals with low out-o-band radiation that can be scattered across a large requency range. Many recent wireless standards rely on the orthogonal requency division multiplexing (OFDM) scheme because o various advantages. Like all multi-carrier systems, OFDM beneits rom dividing a high data rate stream into several parallel, low data rate streams that are transmitted on dierent subcarriers, which allows to exploit requency diversity. In combination with a cyclic preix (CP), the scheme enables to consider the individual subcarriers as requency lat and thus enables an easy single-tap equalization. Further, the orthogonality between the subcarriers enables an eicient, low-complexity transmitter and receiver implementation based on the ast Fourier transorm (FFT) algorithm. However, the scheme also exhibits some disadvantageous properties that make it unable to address several o the previously mentioned requirements. With its strong out-o-band radiation, OFDM can present a non-negligible intererence in overlay systems applications, which makes additional ilters necessary in order to meet a desired spectral mask. OFDM is also very sensitive in terms o carrier requency oset, which requires sophisticated synchronization mechanisms to guarantee that the orthogonally is not aected. Lastly, the cyclic preix approach constitutes a necessary overhead that can reduce the overall energy eiciency o the system. Also, depending on the application, the scheme suers rom high peak-to-average power ratio due to the superposition o many subcarriers, which can increase the requirements to ampliiers. Thus, novel transmission schemes are researched. Several concepts that have emerged in this area during the past years are based on the approach o iltered multi-carrer transmission, which has been known even beore OFDM gained popularity [1], [2]. Among those, generalized requency division multiplexing (GFDM) [3], [4] is a new concept or lexible multi-carrier transmission that introduces additional degrees o reedom when compared to traditional OFDM. In GFDM, the outo-band radiation o the transmit signal is controlled by an adjustable pulse shaping ilter that is applied to the individual subcarriers. Further, a two-dimensional data structure is introduced to group data symbols across several subcarriers and time slots to blocks. The size o the blocks is a variable parameter and allows to implement long ilters or to reduce the total number o subcarriers. The processing o these blocks is done based on tail-biting digital ilters that preserve circular properties across time and requency domain. Similar to OFDM, in GFDM a cyclic preix can be used to combat ISI in a multipath channel. Filter bank multi-carrier (FBMC) [5], [6] is another technique that can provide strong side lobe suppression o the transmit signal, which is dierent rom GFDM. There, the

3 pulse shaping ilter is implemented with the help o a polyphase network. Further, oset QAM modulation is utilized to avoid intercarrier-intererence (ICI) between neighboring subcarriers. The scheme discards the concept o cyclic preix (CP) and relies on a per-subcarrier equalization to combat intersymbol-intererence (ISI). The goal o this paper is on the one hand to extend previous work on GFDM by a low-complexity transmitter model that is suited or a hardware implementation and on the other hand to provide a comparison with the LTE standard. The rest o this paper is organized as ollows: In Section II we discuss the implications o high out o band radiation and recapitulate two ways o looking at the GFDM transmitter, that are known rom previous work. In Section III, a new model suited or low complexity implementation is derived. Section IV deals with the comparison o computational expense among the dierent GFDM models and OFDM and urther a set o reerence parameters suitable or the comparison o GFDM and OFDM is presented. Finally, conclusions are drawn in Section V. II. BACKGROUND Out-o-band radiation is an important issue or any kind o cellular communication system as spectrum resources are subject to strict government regulations. In OFDM based systems, where each subcarrier is shaped with a rectangular pulse in time, the irst side lobes o the corresponding requency domain sin() pulse decay airly slowy. On the one hand, this makes it necessary to introduce additional ilters in order to satisy a certain spectral mask. On the other hand, it makes it diicult to access vacant resources within a system s bandwidth in an opportunistic ashion without adaptive iltering. This iltering can cause ISI which requires a longer CP, otherwise the ISI will eventually cause ICI when detected with a conventional OFDM receiver and degrade the perormance. These two aspects shall serve as the main motivation to introduce additional signal processing eorts to the transmitter and receiver o a wireless system, in order to improve out-o-band radiation properties. In GFDM, each subcarrier individually is shaped with a ilter and as can be seen in Fig. 1, depending on the system parameters this allows to signiicantly improve the spectral properties. And as ISI /ICI are a systematic part o GFDM [4], it is urther expected that the requirements towards synchronization can be relaxed. In the rest o this paper, we will introduce a concept or an eicient implementation o GFDM that allows to achieve this strong out-o-band attenuation with reasonable computational complexity and memory requirements. A. Transmitter Model Consider a system according to [3] that is modeled in baseband. Let a set o complex valued data symbols d k [m], k =... K 1, m =... M 1 be given, which are distributed across K active subcarriers and M active time slots. Each subcarrier on its own is pulse shaped with a transmit ilter g Tx [n] and modulated with a subcarrier center power spectral density in db OFDM GFDM M on = 15 GFDM M on = 13 GFDM M on = subcarrier index Fig. 1. Power spectral density (PSD) o OFDM and GFDM (K = 12, N = 248 and examplary root raised cosine pulse with roll-o a =.25) kn j2π requency e N. Each symbol is sampled N K times, leading to a total o MN samples per subcarrier, which is necessary in order to satisy the Nyquist criterion. The transmit signal x[n] = M 1 K 1 m= k= kn j2π d k [m] g Tx [n mn]e N, (1) is obtained through superposition o the iltered data symbols o all subcarriers and time slots. The ilter g Tx [n] is circular with periodicity n mod MN, thus the tail-biting technique is applied at the transmitter. From (1), a linear mapping o a vector d = {d[l]} KM, l = l(k, m) containing KM data symbols to a vector x = {x[n]} NM containing NM transmit samples according to x = Ad (2) can be derived, where A denotes an NM KM modulation matrix. This representation allows to easily apply standard receiver methods to the GFDM system [4]. The structure o A is shown in Fig. 2. From the absolute value o the modulation matrix in Fig. 2(a), it can be seen that there is a repeating pattern that results in a block diagonal structure or all possible phase responses. By closely looking at the individual columns o the irst phase response in Fig. 2(d), it becomes evident that the matrix also contains the responses o the pulse shaping ilter or all possible subcarriers. This leads to an excellent model or studying the nature o GFDM. III. A LOW COMPLEXITY TRANSMITTER IMPLEMENTATION FOR GFDM From a hardware perspective, a straightorward implementation o the models (1) and (2) may turn out not very suitable.

4 t A i,j j i (a) Absolute value o {A} i,j (a) Upsampling in time and requency domain (b) Pulse shaping ilter in time and requency domain t t.2 (c) Filtering in requency domain A i,3 Fig. 3. Subcarrier processing in time and requency domain A i,1 A i, time index j (d) Real and imaginary part o {A} i,1, {A} i,2 and {A} i,3 k = k = 1 k = 2 Fig. 4. Subcarrier superposition in requency domain Fig. 2. Structure o the modulation matrix A or a system with K = 16 subcarriers and M = 9 time slots. By assessing just the number o complex valued multiplications that are necessary to produce x[n], the two approaches result in a number C GFDM,Σ = C GFDM,A = NKM 2. But there is a big potential or savings, when reormulating the GFDM transmitter in a ashion that is similar to the well known IFFT/FFT approach that is used in OFDM. To be able to do that, the transmit signal rom (1) shall be rewritten as x[n] = x k [n], where k x k [n] = [(d k [m]δ[n mn]) g Tx [n]] e j2π k N n (3) is the transmit signal o the kth subcarrier. Note that here g Tx [n] constitutes one complete period o g Tx [n] and thus the circular convolution denoted by is perormed with respect to n and with periodicity NM. So the modulation o an individual subcarrier in (3) can be broken down to the convolution o a Dirac pulse train d k [m]δ[n mn] with a ilter response g Tx [n] and a subsequent multiplication with a complex valued oscillation e j2π k N n. Carrying over this operation to requency domain, it can be equally written as x k [n] = IDFT NM (DFT NM ( d k [m]δ[n mn] ) DFT NM ( g Tx [n] ) DFT NM ( e j2π k N n )), where DFT NM ( ) is the NM-point discrete Fourier transorm and IDFT NM ( ) denotes the corresponding inverse operation. Now the let side o the product, DFT NM ( d k [m]δ[n mn] ) can be interpreted as capturing N periods o the M points periodic sequence DFT M ( d k [m] ), which contains all necessary inormation. Thus the result can be equally produced by copying the values o the M point DFT instead o actually perorming arithmetic operations necessary or an N M point DFT. This concept is illustrated in Fig. 3(a) with N = 2, where three data symbols in time domain, represented by the black dots, produce the same number o points in requency domain. And adding zero samples between the data symbols then results in a repetition o the sequence in requency domain. As the DFT is an operation with periodic inputs and periodic outputs, urther computational savings can be harvested when the periodicity o the time domain signal is maintained during the iltering operation, i.e. tail-biting circular ilters are used (4)

5 domain conversion requency domain processing domain conversion D W M R (L) Γ P (k) W H NM d d 1... FFT IFFT x upsampl. ilter upconv. M 1 M 1 LM 1 LM 1 NM 1 NM 1 Fig. 5. Low complexity GFDM implementation model in the process [7]. In that case the (circular) convolution o the data sequence and pulse shaping ilter rom (3) turns into a regular multiplication in requency domain in (4). Also, since the aim o the pulse shaping is to keep out-o-band radiation minimal, the utilized pulse may turn out to be sparse in requency domain, i.e. many o the coeicients can be zero and thus multiplications do not need to be carried out. Consequently, in general the ilter pulse spans over 1 L N subcarriers in requency domain. For the root-raised cosine (RRC) typically L = 2, which again saves operations as outlined in Fig. 3(c). ( ) Lastly, the DFT o a sinusoid DFT NM e j2π k N n corresponds to δ ( k N ) in requency domain and convolution with a Dirac results in a shit. Consequently, the subcarrier upconversion can be implemented by shiting the samples in requency domain according to Fig. 4. The modiications listed above lead to a GFDM transmitter model as depicted in Fig. 5. A. Matrix model Consider a data matrix D that contains M K complex valued data symbols d k [m], where d k is the kth column o D and denotes the data transmitted on the kth subcarrier. First, an M point DFT is perormed on each vector d k, which can be expressed mathematically with a Fourier matrix W M = 1 M { w k,n } M M, (k 1)(n 1) wk,n j2π = e N. (5) Sequentially, each o the transormed vectors W M d k undergoes three stages o processing in requency domain. First, the samples o the vector are reproduced L times according to R (L) W M d k, by multiplying with a matrix R (L) = {I M, I M,..., I M } T, which is a concatenation o L identity matrices I M o size M M. Next, the pulse shaping ilter Γ is applied through multiplication according to ΓR (L) W M d k, where Γ is a matrix that contains W LM g on its diagonal and zeros otherwise and g = {g[l]} LM contains the time samples o the ilter pulse. In the last stage, the kth subcarrier s signal X k is created by moving the vector to the position o the corresponding subcarrier with a permutation matrix P (k), such that X k = P (k) ΓR (L) W M d k. Therein P (1) = {I LM LM LM... } T, P (2) = { LM I LM LM... } T and so on, with LM being an LM LM all zero matrix. Finally, all subcarrier signals are superpositioned. The transmit signal is then produced with an NM point IDFT according to x = WNM H P (k) ΓR (L) W M d k. (6) k Note that the processing chain in Fig. 5 can be divided into three general parts. Initially, the data matrix D, in which each row corresponds to a time slot and each column corresponds to a subcarrier, is given in time-requency domain. By applying the M point DFT along each column, the data is converted to the requency-requency domain, where all the processing takes place. Finally, the signal is transormed back to timetime domain by the NM point IDFT, which is the domain necessary or transmission. A. Complexity Analysis IV. RESULTS Assuming that an M point DFT can implemented with the FFT algorithm at the expense o M log 2 M complex valued multiplications, the processing o (6) requires K times M log 2 M multiplications or the M point FFTs o K subcarriers, K times LM multiplications or the iltering o K subcarriers, NM log 2 NM multiplications or the NM point IFFT. The operations related to R (L) and P (k) can be realized by means o pointer/memory operations and are thus not counted. This leads to an implementation eort o C GFDM,FFT = KM log 2 M + KLM + MN log 2 MN = MN log 2 N + (K + N)M log }{{} 2 M + KLM }{{} OFDM complexity GFDM overhead (7) or GFDM, while generating the OFDM transmit signal o the same amount o data is at the cost o C OFDM = MN log 2 N. Altough not analyzed in detail here, an implementation according to (6) also gives savings in the memory consumption, because the processing is perormed on vectors and does not require storing the NM KM modulation matrix A rom (2). A comparison o the implementation complexity o the dierent transmitter approaches in terms o complex valued multiplications is given in Fig. 6. An OFDM signal can be

6 generated with the lowest computational eort. For certain parametrization, i.e. L = 2, with the new model the beneits o pulse shaped subcarriers in GFDM can be exploited at the cost o an increase in complexity by a actor as low as roughly 2. In the impractical case that the pulse shaping ilter spans the complete signal bandwidth, the number o multiplications increases by an order o two magnitudes compared to OFDM. Implementations according to (1) and (2) suer the highest computational load, because they do not beneit rom the log 2 savings o the FFT/IFFT. B. A Case Study or an LTE-like GFDM System The power spectral density (PSD) o OFDM and GFDM is compared in Fig. 1. Therein, the parameters o the OFDM system are chosen such, that they match the speciications o the Long Term Evolution (LTE) standard [8], [9]. For the GFDM system, a comparable set o parameters has been derived in Table I such, that they can serve as a reerence or comparing both concepts in terms o equal sampling time, channel bandwidth and subcarrier bandwidth. Thus the FFT size N and the number o active subcarriers K is also the same or both systems. In GFDM, a block has the Parameter Value Description T s.66 µs sampling time B 2 MHz channel bandwidth B SC 15 khz subcarrier bandwidth N 248 FFT size K 121 active subcarriers M 15 block size M on {15, 13, 11} active time slots ilter RRC pulse shaping ilter shape α.25 ilter roll-o L 2 ilter size in req. domain TABLE I LTE-LIKE PARAMETERS FOR A GFDM SYSTEM duration o M = 15 time slots, which is comparable to a transmission time interval (TTI) o LTE. Further, the parameter M on is introduced, which denotes how many o the time slots are actually illed with data. In total, M M on time slots remain as a guard time to phase the GFDM block in and out. Consequently, the GFDM employs a block structure as depicted in Fig. 5. In the context o this comparison, a rootraised cosine (RRC) ilter with roll-o actor α =.25 is chosen because o the narrow spectrum that it can produce. Note that the GFDM scheme is not restricted to this exemplary pulse. The curves in Fig. 1 show that a GFDM signal with signiicantly stronger out-o-band suppression can be produced. The beneit over OFDM increases with larger guard times at the edges o the GFDM block. A urther improvement is expected rom optimizing the ilter pulse. V. CONCLUSIONS In this paper we motivate the need or a lexible multicarrier communication system that is able to address the expected needs o uture cellular networks. We show that pulse number o complex valued multiplications C Fig. 6. C OFDM C GFDM,FFT,L=2 C GFDM,FFT,L=N C GFDM,A and Σ block size M Comparison o implementation complexity shaped subcarriers can be achieved in GFDM at reasonable computational cost, which is approximately in the same order o magnitude as traditional OFDM. But at the same time, in terms o out o band radiation, GFDM can outperorm OFDM by several orders o magnitude. In order to be able to draw a comparison between both systems, we introduce a set o suitable parameters or GFDM, which relate to the recent LTE standard. The investigation o an optimal ilter pulse shape, that is matched to the properties o GFDM, as well as the design o a low complexity receiver remain interesting topics or urther research. ACKNOWLEDGMENT This work has been perormed in the ramework o the ICT project ICT EXALTED, which is partly unded by the European Union. REFERENCES [1] B. Saltzberg, Perormance o an Eicient Parallel Data Transmission System, IEEE Transactions on Communication Technology, vol. 15, no. 6, pp , December [2] R. W. Chang, High-Speed Multichannel Data Transmission with Bandlimited Orthogonal Signals, Bell Syst. Tech. J., vol. 45, pp , December [3] G. Fettweis, M. Krondor, and S. Bittner, GFDM - Generalized Frequency Division Multiplexing, in Proc. 69th IEEE Vehicular Technology Conerence, VTC Spring 29. [4] N. Michailow, S. Krone, M. Lentmaier, and G. Fettweis, Bit Error Rate Perormance o Generalized Frequency Division Multiplexing, in Proc. 76th IEEE Vehicular Technology Conerence, VTC Fall 212. [5] B. Farhang-Boroujeny, OFDM versus Filter Bank Multicarrier, IEEE Signal Processing Magazine, vol. 28, no. 3, pp , May 211. [6] T. Ihalainen, A. Viholainen, and M. Renors, On Spectrally Eicient Multiplexing in Cognitive Radio Systems, in Proc. 3rd Int. Symp. Wireless Pervasive Computing ISWPC 28. [7] R. Datta, N. Michailow, M. Lentmaier, and G. Fettweis, GFDM Intererence Cancellation or Flexible Cognitive Radio PHY Design, in Proc. 76th IEEE Vehicular Technology Conerence, VTC Fall 212. [8] 3GPP, TS : Physical Channels and Modulation, Tech. Rep. [9] 3GPP, TS 36.14: Base Station Radio Transmission and Reception, Tech. Rep.