Extended GFDM Framework: OTFS and GFDM Comparison
Ahmad Nimr, Marwa Chafii, Maximilian Matthe,´ Gerhard Fettweis Vodafone Chair Mobile Communication Systems, Technische Universitat¨ Dresden, Germany {first name.last name}@ifn.et.tu-dresden.de
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 {a} 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 ∈ 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) (
2) Windowing: the spread data matrix Ds is element- IV. OTFS INTHE GFDM FRAMEWORK wise multiplied with a transmitter windowing matrix A. OTFS overview W ∈ K×M , which is generated based on the prototype tx C In the OTFS modulation [6], a packet burst of duration pulse shape: X = Wtx Ds. The entries of Wtx depends T = NoTo and bandwidth B = Mo∆f o is used to transmit on the implementation domain, No×Mo the data symbols Do ∈ C . Although OTFS is originally T (TD) (g) (FD) (g˜) (14) designed in the delay-Doppler domain, it can be simply Wtx = KZM,K , Wtx = KZK,M . introduced in the time-frequency representation with three-step 3) Transformation: the matrix X can be seen as frequency processing. First, the data symbols are spread with the inverse domain blocks of length M in rows, or as time domain 1 of SFFT to generate the spread data matrix Ds defined by symbols of length K in columns. Thus, a final M-DFT or o 1 K-IDFT is applied to obtain the block samples, as follows D = F H D F ∈ No×Mo . so No o Mo C (18) No (x) 1 H T (x˜) VM,K = FM X , VK,M = FK X . (15) M No×Mo Then, a transmit window Wtx ∈ C can be applied, 4) Allocation: the final vector is achieved by allocating the X = W D . (19) generated samples to the corresponding indexes. Specifically, o tx so
n (x) T o n (x˜) T o Finally, the time-domain signal is generated with a multicarrier x = vec VM,K , x˜ = vec VK,M . (16) modulation of Mo subcarriers and No subsymbols as Additionally, N-IDFT is required for the frequency domain M o −1 1 H 2 No−1 implementation so the time domain signal is x = N FN x˜. X X j2πm∆f o(t−nTo) The demodulator performs the inverse steps. First, the matrix s(t) = [Xo](n,m) gtx(t − nTo)e . Y ∈ K×M is constructed from the received signal y , Mo n=0 C eq m=− 2 h iT (TD) (yeq) (FD) 1 H (y˜eq) (17) Despite of its general representation, it is implicitly conside- Y = FM VM,K , Y = K FK VK,M . red, as stated in the works related to OTFS, e.g. [10], that W Y Then a receive window rx is applied to followed by gtx(t) is a rectangular pulse of duration To and To∆f o = 1. ˆ 1 H despreading, so that D = M FK (Wrx Y ) FM , which is Therefore, the discrete OTFS block is given by an alternative representation of the circular demodulation in Mo−1 No−1 (5). Table I summarizes all the processing steps. X X T j2π ml [s] = X g [< l−nM > ]e Mo . (l) o (m,n) tx o (NoMo) m=0 n=0 D. Extended flexibility of GFDM This shows that the OTFS block is composed of No In addition to the main configuration parameters K, M and consecutive OFDM symbols of length Mo. Here, the (xo) 1 H T the prototype pulse shape, further degrees of freedom can be CP is not considered. Let VM ,N = FM Xo , with n o o o Mo o exploited to extend the flexibility of GFDM modem, namely: x = vec {V (xo) }T , then o Mo,No • Flexible spreading: the spreading can be altered to enable n o s = vec V (xo) . (20) or disable the spreading matrices, as in [9]. Mo,No • Flexible transformation: the DFT or IDFT transformation B. OTFS relation to GFDM can be turned on or off to get more options. • Flexible allocation: the conventional allocation preserves Comparing the OTFS samples (20) with the GFDM samples the spectral characteristics of the GFDM signal. However, (15), we reveal that the OTFS samples can be generated the mapping can be customized by changing the dis- using the first three steps of the GFDM modulator with the tribution of the samples in the transmitted signal. This parameters M = Mo, K = No. The difference between both extended flexibility is exploited in the next section to blocks is in the allocation step. Actually, the OTFS block is a generate the OTFS signal. 1The subscript o is added to [6] to distinguish the OTFS terminology. TABLE I: The four steps of conventional GFDM modulator.
Modulation Demodulation TD implementation FD implementation TD implementation FD implementation 1 H ˆ 1 ˆ H Spread. Ds = K FK DFM D = M FK DsFM W = K{Z(g) }T W = KZ¯(g˜) W generated from W Wind. tx M,K tx K,M rx tx X = Wtx Ds Dˆ s = Wrx Y T (x) 1 H T (x˜) h (yeq)i 1 H (y˜eq) Trans. VM,K = M FM X VK,M = FK X Y = FM VM,K Y = K FK VK,M
n (x) T o n (x˜) T o h (yeq)i h (y˜eq)i Alloc. x = vec {VM,K } x˜ = vec {VK,M } VM,K = [yeq](k+mK) VK,M = [y˜eq](m+kM) (m,k) (k,m)
permutation of the GFDM block xo with square the commu- received GFDM block after removing the CP is: tation matrix P ∈ F V (yo) = H˜ (e) XT + E + V (22) V. NUMERICAL SIMULATION Mo Mo,No o o do o The purpose of this section is first to compare the perfor- ˜ (e) M×K where H ∈ C is the equivalent channel defined by mance of the conventional GFDM2 and OTFS systems genera- Mo−1 L−1 ted from the same parameters. The simulation parameters are h (e)i X X −j2π lp H˜o = h(l, Ncp+m+q(Mo+Ncp))e Mo , listed in Table II. The data bits are encoded with Long Term (p,q) m=0 l=0 Evolution (LTE)-Turbo code of code rate 1/2 and mapped to 16-QAM symbols. The transmit window is generated from Edo denotes the interference terms resulting from Doppler g spread and Vo is the additive noise samples. Furthermore, let a prototype pulse , which is a periodic raised-cosine (RC) with roll-off factor α = 0. The samples in both systems are (e) ˜ (e)T (e) (e) (23) Wtx = Ho Wtx, Xo = Wtx Dso , generated from the GFDM modulator depicted in Fig. 2 using 1 the parameters K = No and M = Mo. The MMSE receiver then V (yo) = F H X(e)T + E + V Mo,No Mo o do o Mo (24) in the case of OTFS is performed using MMSE pulse shape (e) (e) (xo ) generated based on the equivalent window W (23), while = V + Ed + Vo tx Mo,No o the GFDM-MMSE receiver considers the linear model [5] (e) where, xo is an equivalent time-domain GFDM block gene- n˜ (e)o ˜ (e) y˜ = diag h Ad + ˜d + v˜. (29) rated by the window Wtx (15). Furthermore, the relation The demodulated symbols are fed to soft-input decoder. Both y = P T · r = x(e) + + v , (25) o No,Mo o do o modulation techniques are evaluated in the extended vehicular shows that the received OTFS block can be processed with a A channel model (EVA) [12] with the consideration of ideal ready implemented real-time GFDM receiver, e.g. [7], under channel estimation per block as in (37) in the appendix. The additive noise channel. In this case, the minimum mean square SNR is defined by the ratio Es/N0, where Es is the average error (MMSE) receive pulse shape can be efficiently calculated symbol power and N0 is the additive white noise power. (e) h i h i based on the inverse Zak transform of Wtx [11]. Moreover, ˆ 2 ˆ 2 E kd − dk E |dn − dn| due to the circular filtering in the combined equalization NMSE = 2 , SNR[n] = 2 . (30) and demodulation, the symbols at the output achieve roughly E [kdk ] E [|dn] | the same SNR considering the additional interference. The 2In this section, GFDM refers to the conventional GFDM waveform [4]. 0 10 11 10-1 10.5 10 10-2 9.5 10-1 9 10-3 8.5 -4 10 8 10-2 7.5 5 10 15 20 5 10 15 20 500 1000 1500 2000 (a) BER vs. SNR. (b) FER vs. SNR. (c) SNR per symbol. Fig. 3: Performance evaluation of OFDM, GFDM and OTFS with long frames. 100 -1 10 10-1 -1 10-2 10 10-2 -3 10 10-2 10-3 10-4 10-3 10-4 5 10 15 20 5 10 15 20 500 1000 1500 2000 2500 3000 (a) BER vs. SNR. (b) FER vs. SNR. (c) FER vs. Doppler frequency. Fig. 4: Performance evaluation of OFDM, GFDM and OTFS with short frames. TABLE II: Simulation parameters fd = 31.5Hz the FER gain is 3dB at SNR > 14dB. This can OTFS No = 16,Mo = 128 be explained, as illustrated in Fig. 3c, by the almost equal SNR Long-GFDM K = 16,M = 128 per-symbols at the output of the OTFS demodulator unlike in Long-OFDM N = 2048 GFDM and OFDM, where the SNR per symbol experiences Short-GFDM K = 16,M = 8 significant variations. Additionally, these variations are even Short-OFDM N = 128 higher in OFDM such that GFDM achieves slightly lower FER CP length N 32 CP than OFDM. Based on that, the corresponding bit errors in Bandwidth B = Fs 8 MHz OTFS subcarrier spacing ∆fo 62.5 KHz OFDM and GFDM can be burst, which increases the number Modulation and coding 16-QAM, LTE-Turbo (1/2) of frame errors. But in the case of OTFS, it is more likely that an erroneous frame has higher number of bit errors. On the other hand, it can be observed that, with increased Doppler Fig. 3 shows a comparison between OTFS, conventional shift, e.g. fd = 312.5Hz, OTFS maintains smaller FER GFDM and OFDM using long block. A frame is represented in through the exploitation of time and frequency diversity, while a code word corresponding to one GFDM block. It is shown the GFDM and OFDM links are almost broken. Furthermore, that, the Doppler has less influence on OTFS compared to the Doppler interference increases with the increase of the the GFDM and OFDM. As discussed in the appendix, the is transmit power, which explains the BER floor. because of the accuracy of the equivalent channel estimation, However, the design of OFDM and GFDM can be updated which increases with the decrease of the product P νd, where with respect to the channel mobility conditions. Therefore, fd νd = B and P is the DFT size. In OTFS, P = Mo, while another configuration considering shorter block length of in the long GFDM and OFDM P = KMo, and thus, GFDM GFDM and OFDM is evaluated in Fig. 4. Moreover, the frame and OFDM obviously suffer from higher Doppler interference. length in OTFS, i.e. the code word is shortened accordingly. In This is reflected in the bit error rate (BER) performance as this scenario, GFDM and OFDM attain better performance due shown in Fig. 3a. However, the achieved gain is at the cost of to the lower Doppler interference but less robust than OTFS increased CP overhead in the case of OTFS. Although GFDM against increased Doppler frequencies as illustrated in Fig. 4a. and OFDM achieve similar performance in term of BER for Moreover, due to the spreading technique in OTFS, the short smaller Doppler shift, OTFS significantly outperforms them frame is spread over larger time duration. As a result, the FER in term of FER as depicted in Fig. 3b. For instance, for performance gain of OTFS is appealing as shown in Fig. 4b, especially at high Doppler shift. For instance, the gain is higher [12] 3GPP, “Base Station (BS) radio transmission and reception,” Technical Specification (TS) 36.104, 05 2008, v8.2.0. than 4 dB at fd > 1500Hz an SNR = 18dB, as can bee seen in Fig. 4c. On the other hand, considering additional low-latency APPENDIX and throughput constraints, the conventional GFDM provides Consider a signal x(cp)[n], n = 0 ··· N +N −1 with x(cp)[n] = acceptable performance under moderate mobility conditions. cp x[< n − Ncp >N ] transmitted through LTV channel defined by the delay-Doppler response H(l, ν), where ν is the normalized Doppler VI.CONCLUSION frequency, i.e. ν = f , the received signal can be expressed as In this paper, we have proposed an extended GFDM frame- Fs Z L−1 work that provides unified four-step implementation structure X (cp) j2πnν r[n] = H(l, ν)x [n − l]e dν, (31) in the time and frequency domains. The OTFS modulation, ν l=0 which is a powerful transmission technique in LTV channels, can be processed with the GFDM framework with simple with L ≤ Ncp − 1 is the maximum delay spread. Then we extract the samples y[n] as y[n] = r[N + n], n = 0 ··· N − 1, modification. Namely, it is shown that the OTFS block results cp Z L−1 from a permutation of the corresponding conventional GFDM X j2π[Ncp+n]ν then y[n] = H(l, ν)x[< n − l >N ]e dν block and the received OTFS block can be represented as a ν l=0 (32) GFDM block in additive noise channel. Thus, the received Z N−1 1 X j2π kn j2π[N +n]ν OTFS signal can be fed to the conventional GFDM demodu- = H˜ (k, ν)˜x[k]e N e cp dν. N lator with a reconfigurable receive pulse shape that depends on ν k=0 the channel estimation. As a result, the OTFS equalization and L−1 kl ˜ P −j2π N Here, H(k, ν) = l=0 H(l, ν)e . This corresponds to remo- demodulation are combined in a circular filtering that produces ving the CP and using N-DFT representation for circular convolution, data symbols with equal SNR. In addition, the MMSE-OTFS which is attained due to the CP. The DFT of the received signal is receiver can be implemented with low complexity. The simu- N−1 X −j2π nq lation results show that OTFS outperforms the counterparts y˜[q] = y[n]e N long-GFDM and long-OFDM in terms of BER and FER. n=0 Furthermore, OTFS significantly outperforms the short blocks Z N−1 N−1 1 X j2πN ν X j2π n(k−q+Nν) of GFDM and OFDM in high mobility scenarios, due to the = H˜ (k, ν)˜x[k]e cp e N dν. N ν exploitation of time and frequency diversity. However, GFDM k=0 n=0 can be a reasonable choice for moderate mobility and low- Assuming that the maximum Doppler spread fd satisfies N|νD| 1, fd where νd = , then, latency use cases as it outperforms OFDM in term of FER. Fs In the future work, the extended GFDM framework is used N−1 N−1 X j2π (k−q)n j2πnν X j2πnν to develop alternative forms of OTFS with applications for e N e ≈ e δ(q − k), (33) multiuser scenarios. n=0 n=0 N−1 Z REFERENCES 1 X j2πN ν j2πnν y˜[q] ≈ H˜ (q, ν)˜x[q]e cp e dν [1] B. Farhang-Boroujeny, “OFDM versus filter bank multicarrier,” SPM, N ν n=0 (34) vol. 28, no. 3, pp. 92–112, 2011. 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