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Extended GFDM Framework: OTFS and GFDM Comparison

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Extended GFDM Framework: OTFS and GFDM Comparison Extended GFDM Framework: OTFS and GFDM Comparison Ahmad Nimr, Marwa Chafii, Maximilian Matthe,´ Gerhard Fettweis Vodafone Chair Mobile Communication Systems, Technische Universitat¨ Dresden, Germany ffirst name.last [email protected] Abstract—Orthogonal time frequency space modulation Orthogonal time frequency space modulation (OTFS) modu- (OTFS) has been recently proposed to achieve time and frequency lation technique has been recently proposed in [6] to deal with diversity, especially in linear time-variant (LTV) channels with high mobility scenarios. In this approach, the data symbols large Doppler frequencies. The idea is based on the precoding of the data symbols using symplectic finite Fourier transform are spread in the time and frequency domains using sym- (SFFT) then transmitting them using orthogonal frequency plectic finite Fourier transform (SFFT)-based precoding. As division multiplexing (OFDM) waveform. Consequently, the de- a consequence, high diversity gain is achieved. The spread modulator and channel equalization can be coupled in one data symbols are then transmitted with OFDM waveform. processing step. As a distinguished feature, the demodulated data Compared to OFDM, a very significant gain in terms of symbols have roughly equal gain independent of the channel selectivity. On the other hand, generalized frequency division frame error rate (FER) is shown in high mobility case [6]. multiplexing (GFDM) modulation also employs the spreading Furthermore, the processing is of relatively low cost, where over the time and frequency domains using circular filtering. the channel equalization and demodulation are coupled at the Accordingly, the data symbols are implicitly precoded in a similar receiver. Interestingly, all the estimated symbols have the same way as applying SFFT in OTFS. In this paper, we present signal-to-noise ratio (SNR) at the output of the equalizer. an extended representation of GFDM which shows that OTFS can be processed as a GFDM signal with simple permutation. This feature can not be achieved with OFDM unless power Nevertheless, this permutation is the key factor behind the out- allocation techniques are applied. However, this requires the standing performance of OTFS in LTV channels, as demonstrated channel knowledge at the transmitter, which is infeasible under in this work. Furthermore, the representation of OTFS in the higher Doppler frequencies. GFDM framework provides an efficient implementation, that Because GFDM is based on circular filtering in the time and has been intensively investigated for GFDM, and facilitates the understanding of the OTFS distinct features. frequency domains, the data symbols are implicitly spread in Index Terms—GFDM, OTFS a similar way as in OTFS. This motivates us to investigate the structure of GFDM and reveal a relation between both systems. As we show in this paper, the OTFS samples can be generated I. INTRODUCTION from a permutation of GFDM signal, which exchanges the In the contention between 5G waveform candidates, several role of the number of subcarriers in OTFS to be the number modulation techniques were proposed to attain the require- of subsymbols in the GFDM terminology and vice versa. ments for different use cases. Some of the waveforms focus This observation is relevant for two reasons; first, a slight on providing very low out-of-band (OOB) emission, e.g. filter modification of a real-time flexible implementation of GFDM bank multicarrier (FBMC) [1], which allows asynchronous transceiver [7], enables the processing of OTFS. Second, the multiple access. Other designs concern about the implemen- properties of OTFS can be clearly derived from the structure of tation complexity, which leads to several proposals based on GFDM. For example, we show that the coupled equalizer and orthogonal frequency division multiplexing (OFDM), such as demodulator are equivalent to circular filtering of the received windowed-OFDM [2], filtered-OFDM [3]. Furthermore, gene- OTFS signal using a GFDM receive filter computed from the ralized frequency division multiplexing (GFDM) [4] was first transmit pulse and the estimated channel. proposed as an alternative to OFDM to improve the spectral The remainder of the paper is organized as follows: in efficiency by reducing the cyclic prefix (CP) overhead of Section II we provide a short overview of the conventional OFDM. However, due to the flexibility in tuning the different GFDM. Section III is dedicated to the advanced representation parameters, such as the number of subcarriers and subsymbols, of GFDM and extended framework. The relation between the the prototype pulse shape and the active subsymbol set, GFDM conventional GFDM and OTFS is discussed in Section IV. can be reconfigured to meet different requirements [5]. This Numerical comparisons through simulation are introduced in flexibility inspires the design and implementation of unified Section V. Finally, Section VI concludes the paper. multicarrier framework based on the GFDM model. II. GFDM OVERVIEW The work presented in this paper has been performed in the framework In the GFDM modulation, a time-frequency resource block of the ORCA project [https://www.orca-project.eu/].This project has received funding from the Eropean Union’s Horizon 2020 research and innovation of duration T and bandwidth B is used to convey a message programme under grant agreement No 732174. of maximum N data symbols. For that, the frequency band is divided into K equally spaced subcarriers with subcarrier where a~ = FPQa, unvecQ×P fag denotes the inverse of B spacing ∆f = K , and the time duration is divided into M vectorization operation, FP is the P -point discrete Fourier T ij −j2π P equally spaced subsymbols with subsymbol spacing Tsub = M transform (DFT) matrix, defined by [FP ](i;j) = e . The ~ such that ∆fTsub = 1. Each pair of subcarrier-subsymbol (a) ¯(a) matrix ZP;Q, ZQ;P are known as the discrete Zak transform (k; m) is used to modulate one data symbol dk;m using a (DZT) [8] of a and a~, respectively. pulse shape gk;m(t), and thus, N = MK. The pulse shapes are generated by a shift in the time and frequency domains of a periodic prototype pulse shape gT (t), in addition to windowing to confine the block within the duration T , thus j2πk∆ft gk;m(t) = wT (t)gT (t − mTsub)e ; (1) where wT (t) is a rectangular window of duration T . Accor- dingly, one GFDM block in the time domain is expressed as K=2−1 M−1 X X j2πk∆ft Fig. 1: Visualization of (8) and (9). x(t) = wT (t) dk;mgT (t − mTsub)e : (2) k=−K=2 m=0 A. Time domain representation based on this representation, GFDM can be seen as generali- The GFDM block equation (3) can be reformulated by zation of OFDM, where gT (t) = 1 and M = 1. The discrete- time representation can be achieved by using a sampling representing n with two indexes q = 0; ··· ;K − 1 and p = 0 ··· ;M − 1, such that n = q + pK. Thereby, frequency Fs = B. The samples of one GFDM block can be represented in a vector x 2 N×1 such that [4], M−1 K−1 C X X k j2π K q K−1 M−1 [x](q+pK) = dk;mg[< q + pK − mK >N ]e k n X X j2π K n m=0 k=0 [x](n) = x( B ) = dk;mg[< n − mK >N ]e : (3) k=0 m=0 Using the notations in (8), then Here, < : >N denotes the modulo-N operator. The correspon- M−1 K−1 h (x) i h (g) i k X X j2π K q ding representation in the frequency domain is given by VM;K = VM;K [D](k;m) e (p;q) (<p−m> ;q) m=0 M k=0 K−1 M−1 m M−1 X X −j2π M n [x~](n) = dk;mg~[< n − kM >N ]e : (4) X h (g) i h T H i = VM;K D FK : k=0 m=0 (<p−m> ;q) (m;q) m=0 M The GFDM demodulator applies circular filtering on the The second line defines a circular convolution between the q- received signal y[n] with a receiver pulse γ[n] such that, (g) T H th column of VM;K and the q-th column of D FK , which N−1 X ∗ −j2π k n can be expressed in the frequency domain with M-DFT as d^k;m = y[n]γ [< n − mK >N ]e K : (5) h (x) i h (g) i h i n=0 T H FM VM;K = FM VM;K · FM D FK : (10) (p;q) (q;q) (p;q) In the conventional matrix representation, the GFDM block N×N Using the notation in (9), we get can be expressed using a matrix A 2 C as 1 h i j2π k n (x) H (g) T H x = Ad; [A] = g[< n − mK > ]e K ; (6) VM;K = FM KZM;K FM D FK : (11) (n;k+mK) N MK Here, denotes the element-wise multiplication operator. where d = vec fDg with [D](k;m) = dk;m. Moreover, the demodulation matrix B 2 CN×N has a similar structure and B. Frequency domain representation is generated from the receiver pulse γ as Following similar derivation on x~ defined in (4), we get j2π k n H [B] = γ[< n − mK >N ]e K ; d^ = B y: (7) K−1 M−1 (n;k+mK) h (x~) i h (g~) i m X X −j2π M p VK;M = VK;M [D](k;m) e : (q;p) (<q−k> ;p) III. GFDM ALTERNATIVE REPRESENTATION k=0 K m=0 1 h i Hence, V (x~) = F KZ¯(g~) F H DF : (12) K;M K K K;M K M In this section, we introduce an alternative matrix model C.
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