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Generalized Frequency Division Multiplexing in a Gabor Transform Setting Maximilian Matthé, Luciano Leonel Mendes and Gerhard Fettweis
Abstract—This letter shows the equivalence of the recently proposed Generalized Frequency Division Multiplexing (GFDM) communications scheme with a finite discrete critically sampled Gabor Expansion and Transform. GFDM is described with the terminology of Gabor analysis and the Balian-Low theorem is applied to prove the non-existence of zero-forcing receivers for certain configurations, having strong impact on the system performance. An efficient algorithm for calculation of specific GFDM receiver filters is derived and numerical examples confirm the theoretical results. Index Terms—Gabor transform, multicarrier modulation, GFDM
I. I NTRODUCTION New waveforms are being proposed for the future mobile communication physical (PHY) layer [1] and Filter Bank Multicarrier (FBMC) [2] is a main candidate due to its advantageous properties of, among others, spectral shaping and robustness against time and frequency offsets. However, its long filter tails required to filter each subcarrier hinder burst transmissions for low latency applications. GFDM [3] is another waveform candidate for 5G networks. Among its proposed advantages are low out of band radiation, robustness against time and frequency offsets and flexibility to accommodate a variety of channels and applications. Its block based structure enables burst transmission for low latency applications [3]. Gabor analysis and time-frequency analysis originated in 1947 when Dennis Gabor published his Theory of Communication, where proposed to transmit an arbitrary signal as the linear combination of time- and frequency-shifted Gaussian impulses. Since then, mathematicians and engineers have put research effort into the study when such an expansion is possible and which properties arise from it. A detailed mathematical introduction to the foundations of time-frequency analysis is given in e.g. [4], while a broad overview of current topics in Gabor analysis is given in e.g. [5]. The main contribution of this paper is to show that GFDM transmission and linear reception are equivalent to a critically sampled finite discrete Gabor expansion and transform, Manuscript received May 26, 2014; revised June 12, 2104; accepted June 14, 2014. Date of publication: ... Maximilian Matthé and Gerhard Fettweis are with Vodafone Chair Mobile Communications Systems, TU Dresden, Germany. e-mail: {maximilian.matthe,fettweis}@ifn.et.tu-dresden.de Luciano Leonel Mendes is with Instituto Nacional de Telecomunicações (Inatel), Sta. Rita do Sapucai, MG, Brazil. e-mail:
[email protected] This work has been performed in the framework of the FP7 project ICT619555 RESCUE which is partly funded by the European Union. The authors would like to thank CNPq - Brasil for partially funding the work presented in this paper.
kn
d0 [m] d~
ej2π K ↑K
~g[n]
+
S/P dK−1 [m]
↑K
D/A
~g[n]
Fig. 1. Block diagram of the GFDM transmitter.
respectively, which offers the possibility to apply the vast mathematical background of Gabor analysis to the study of GFDM. Two main results that follow from the application of Gabor analysis to GFDM are analyzed. Primarily, Gabor analysis brings theoretic reasoning to the existence and performance of certain receiver configurations. Secondly, although efficient algorithms for the practical implementation of GFDM exist [3], still a concern is the derivation of employed filter coefficients, which requires a huge matrix inversion. Here, the Gabor transform structure of GFDM can be exploited to circumvent the matrix inversion requirement and provide a fast algorithm for calculation of GFDM receiver filters. The remainder of this work is structured as follows: Sec. II describes GFDM from the conventional point of view in terms of a filtered multicarrier system. In Sec. III the discrete Gabor transform is introduced. Sec. IV puts GFDM into the notation of Gabor analysis and consequences resulting from Gabor theory are pointed out. Numerical examples are provided in Sec. V and conclusion is drawn in Sec. VI. II. GFDM PRINCIPLES The block-based communication scheme GFDM is described in [3] as an innovative modulation technique suitable for the air interface of 5G networks. The block diagram of a GFDM transmitter is shown in Fig. 1. In a GFDM block with length of N = M K samples, M complex valued subsymbols are transmitted on each of the K subcarriers. The M data symbols dk [m], m = 0, 1, . . . , M − 1 on the kth subcarrier are upsampled by factor K and filtered with a circular convolution with the transmitter filter g[n]. The signal is upconverted to the frequency of the kth subcarrier to yield the transmit signal xk [n] of the kth subcarrier by ! M −1 X 0 0 xk [n] = g[n ] ~ dk [m]δ[n − mK] exp j2π kn , K m=0
n
(1)
where ~ denotes circular convolution with respect to the block length N and n0 is the convolution index variable. Expressing the convolution explicitly and superpositioning all xk [n], the
2
kn
e−j2π K
~h∗ [−n]
d0 [m]
↓K
A/D
P/S ~h∗ [−n]
↓K
dˆ
dK−1 [m]
Fig. 2. Block diagram of the GFDM receiver.
transmit signal is given by x[n] =
K−1 −1 XM X
kn
dkm g0m [n]ej2π K =
k=0 m=0
K−1 −1 XM X
dkm gkm [n],
1) Matched Filter (MF) Receiver: The MF receiver matrix is given by B(hMF ) = AH g where Ag = A(g). It maximizes the signal-to-noise ratio per subcarrier, but introduces selfinterference due to the possible non-orthogonality of the transmit filter. 2) Zero-Forcing (ZF) Receiver: The ZF matrix is given by B(hZF ) = A−1 g . It cancels out all self-interference at the cost of increasing the noise power. Furthermore, there are cases when A(g) becomes singular and no ZF receiver exists. 3) Linear Minimum-mean-square-error (MMSE) Receiver: The matrix for the linear MMSE receiver is given by 2 −1 H B(hMMSE ) = (AH Ag g Ag + σw I)
k=0 m=0
(2) kn
where gkm [n] = g[(n − mK) mod N ]ej2π K are circularly time-frequency shifted versions of the prototype transmit filter g[n] and dkm = dk [m]. Note that, as a GFDM corner case, (2) reduces to the IDFT expression of OFDM if M = 1 and g[n] = √1K . On the other hand, GFDM reduces to SC-FDE (Single-Carrier Frequency Domain Equalization), when K = 1 and g[n] = δ[n] [3]. Equation (2) can be reformulated into a single matrix equation [3] by ~ (x[n])> = A(g)d, (3) n
>
where (·) denotes transpose, A(g) is a N × N matrix containing gkm [n] from (2) as its (k + mK)th column and d~ is a vector containing dkm at the corresponding rows. Note that A(g) is a linear mapping according to αA(g) + βA(h) = A(αg + βh),
(4)
where g, h are arbitrary windows and α, β ∈ C. Assuming an additive white gaussian noise (AWGN) channel the received signal is given by y[n] = x[n] + w[n]
(5)
2 . σw
where w[n] is AWGN with variance The block diagram of the linear GFDM receiver is shown in Fig. 2. Each subcarrier is shifted to the DC subcarrier, circularly convolved with a specific receiver filter h∗ [−n] and downsampled to the symbol rate. At the kth subcarrier the mth received subsymbol dˆk [m] is therefore given by 0 dˆk [m] = y[n0 ] ~ (h∗ [−n0 ] exp(−j2π kn ) . (6) K
where is the variance of the AWGN and I is the identity matrix. It balances self-interference and noise-enhancement and outperforms both MF and ZF receivers at the cost of an increased complexity due to the necessary estimation of the noise variance. III. T HE DISCRETE G ABOR E XPANSION AND T RANSFORM In its traditional form the Gabor expansion expresses a timecontinuous signal x(t) in terms of a linear combination of time-frequency shifts of a prototype window g(t) [6]. A finite discrete version of Gabor analysis is obtained by periodization and sampling of the time-continuous signal, as is described in e.g. [7]. The resulting signal x[n] is constrained to be discrete in time and periodic with the period of N samples or, equivalently, of finite extent of N samples. x[n] is expanded from T F functions, which are circular time-frequency shifts uf t [n] of a prototype window u[n] by x[n] =
where B(h) = (A(h))H is the N × N matrix that contains h∗km as its (k + mK)th row and (·)H denotes conjugate transpose. From (3) three standard linear receiver types are readily available [3]:
af t (x)uf t [n]
(10)
with uf t [n] = u[(n − m∆T ) mod N ] exp(j2π ∆F N f n)
(11)
where N = ∆T T = ∆F F ; F, T, ∆F, ∆T ∈ N+ equals the length of the discrete transform and ∆T and ∆F N describe the step size for time- and frequency-shifts, respectively. The corresponding discrete Gabor transform with a window ν[n] is defined by [7] af t (x) =
N
where hkm [n] = h[(n − mK) mod N ] exp(j2π kn K ) is the prototype receiver filter h[n] circularly shifted to the corresponding subcarrier and subsymbol. Again, (7) reduces to the DFT expression of the OFDM receiver, when M = 1 and h[n] = √1K . The reception of all symbols at the same time can be combined into a single matrix equation ˆ d~ = B(h)~y = B(h)A(g)d~ + B(h)w, ~ (8)
F −1 T −1 X X f =0 t=0
n=mK
which is expressed with the standard scalar product in C by dˆkm = hy[n], hkm [n]iCN (7)
(9)
2 σw
N −1 X n=0
x[n]νf∗t [n] = hx[n], νf t [n]iCN
(12)
with νf t [n] = ν[(n − m∆T ) mod N ] exp(j2π ∆F N f n)
(13)
i.e. the scalar product of the signal x[n] with the corresponding time-frequency shift νf t [n] of the window ν[n]. A main concern in classical Gabor analysis is, if for a given u[n] the expansion coefficients af t (x) in (10) exist for every x[n]. Furthermore, what is the corresponding ν[n] to calculate them? Wexler and Raz [7] proved that the coefficients for (10) are calculated by (12) if and only if N −1 X n=0
u−f,−t [n]ν ∗ [n] =
N δ0f δ0t TF
(14)
is fulfilled for every 0 ≤ t < ∆F, 0 ≤ f < ∆T , which is known as the prominent Wexler-Raz duality condition. The corresponding ν[n] is called the dual window to u[n]. The choice of the step sizes ∆T and ∆F significantly influences the existence and uniqueness of ν[n]. Dual windows exist only if ∆T ∆F ≤ N , where equality is referred to as critical sampling. In the oversampled case ∆T ∆F < N multiple dual windows may exist for a given u[n], whereas in the critically sampled case, existing dual windows are unique. IV. GFDM IN A G ABOR T RANSFORM S ETTING When setting ∆T = K and ∆F = M with M K = N a critically sampled Gabor transform and expansion pair is obtained. Furthermore, when the transmit filter g[n] of GFDM is identified with the Gabor window u[n], the transmitted data dkm with the Gabor expansion coefficients akm , and the receiver filter with some h[n] it is obvious that GFDM transmission is a critically sampled Gabor expansion with window g[n] and GFDM reception is a critically sampled Gabor transform with window h[n]. The receiver filters for MF, ZF, and MMSE receivers can therefore be described in terms of a Gabor transform window. 1) MF Receiver: The receiver matrix of the MF GFDM receiver is given by B(hMF ) = AH g . Therefore the window of the corresponding Gabor transform is equal to hMF = g. 2) ZF Receiver: The ZF receiver matrix is given by B(hZF ) = (Ag )−1 . Accordingly, when looking at the first row of B(hZF )Ag = I we have N −1 X
gkm [n](hZF [n])∗ = δ0k δ0m
(15)
n=0
for 0 ≤ k < K, 0 ≤ m < M . This is exactly the Wexler-Raz duality condition from (14) when identifying g−k,−m [n] = gK−k,M −m [n]. Hence, the Gabor window of the ZF receiver is exactly the window that is dual to g[n], denoted by hZF = D(g). The noise enhancement factor (NEF) of the ZF receiver is given by ξhZF = khZF k2 . 3) MMSE Receiver: Equation (9) can be rewritten as follows, when AH g is included into the matrix inversion 2 H −1 2 B(hMMSE ) = [Ag + σw (A−1 = [A(g) + σw A(hZF )]−1 g ) ] (16)
and with linearity (4) it simplifies to 2 ZF −1 B(hMMSE ) = [A(g + σw h )] .
(17)
Accordingly, the window for the Gabor transform of the MMSE receiver is equal to 2 hMMSE = D(g + σw D(g)).
(18)
A. Calculation of the ZF Receiver Window A main concern for the application of the GFDM ZF and MMSE receivers is that it requires the numerically intensive inversion of Ag of size KM × KM . Since current communication systems can have thousands of subcarriers and subsymbols (e.g. LTE TTI: K = 2048, M = 15) this matrix may even hardly fit into memory of current hardware.
|(Zg)( 12 , f )|
3
1 0.5 0
M =5 M =6 M = 31
0
α = 0.2 α = 0.7
0.2
0.4
0.6
0.8
1
f
Fig. 3. Modulus of the CTZT of a RC with rolloff 0.7 and 0.2 and the sampling positions for K = 128 and different M . For M = 6 the DZT contains a zero, for M = 5 the zero is not sampled and the values of the DZT are high. With M = 31 also small values are contained in the DZT, which increases the norm of the ZF receiver filter and, hence, its NEF.
Dual windows, and hence ZF and MMSE receiver filters, can be efficiently calculated with the help of the discrete ZakTransform (DZT) [8] without requiring the inversion of A(g). The DZT is derived from the continuous-time Zak-Transform (CTZT) [9] X (Zg)(t, f ) = g(t + l)e−j2πlf (19) l∈Z
by sampling the time and frequency components [8] according to (Z (K,M ) g)[n, k] = (Zg)
n k K, M
=
M −1 X
kl
n g( K + l)e−j2π M .
l=0
(20) This discretization assumes that g(t) is sampled with an 1 interval K and is made periodic with the period M [8]. For the critical sampling case, as in GFDM, the relation [10] (Z (K,M ) γ)[n, k] =
1 K(Z (K,M ) g)∗ [n, k]
(21)
between the DZT of the Gabor window g[n] and its dual γ[n] holds. Accordingly the ZF and MMSE receiver windows can be calculated with numerical efficient algorithms [10]. B. Singularity of Ag In previous works [3] it has been mentioned without reasoning that Ag becomes singular with certain configurations of M , K, and g. In these cases no ZF receiver exists and MMSE and MF receiver perform worse in terms of bit error rate. In this paper, Gabor analysis is employed to develop the theoretical background to the question of singularity of Ag . As is evident from (21) the dual window γ[n] will not exist if the DZT of g[n] has zeros [8]. Hence, Ag will be singular in these cases. The Balian-Low theorem (BLT) states that, at critical sampling, functions that are well localized in time and frequency cannot constitute a Gabor frame and, accordingly, their CTZT has at least one zero [4]. In previous works [3] GFDM mostly employs time-frequency well localized raised cosine (RC) or root-raised cosine transmit filters. By the nature of the CTZT, 1 1 these real symmetric windows g(t) fulfill (Zg) , 2 2 = 0 [9]. M Consequently, (Z (K,M ) g) K , = 0 when K, M are even 2 2 and no dual window and ZF receiver exist. Hence, only odd M or K should be considered for practical implementations. However, even though the ZF receiver exists for odd K or odd M , the DZT of g[n] contains smaller values when K or
α = 0.2 α = 0.7
4
10−1 SER
ξgZF [dB]
4
2
10−2 10−3
0 0
20
40
60
10−4
M
MMSE ZF MF FBMC
10
M = 7, α2
M = 8, α1
MMSE ZF
M = 7, α1
14 Es /N0
(a) Noise enhancement factor.
18
10
M = 31, α1 M = 7, α1
14 Es /N0
18
(b) Symbol error rate performance.
Fig. 4. (a) NEF for K = 128, RC filter with different rolloffs α. Only odd M are depicted since no ZF receiver exists for even M . (b) Symbol error rates under AWGN. All systems are simulated with K = 128 and use a RC with α1 = 0.2 or α2 = 0.7 and 16-QAM modulation.
M increase due to the finer sampling grid. Hence, the DZT of hZF [n] contains growing values, constituting an increased noise enhancement. Additionally, filters with wider rolloff contain lower CTZT values in overall. Fig. 3 illustrates these results. From a more general point of view, every transmit filter whose DZT contains values close to zero shows reduced performance due to an increased noise enhancement, which can lead to a design criteria for GFDM filters. Although the non-symmetric Xia filters [11] can be used as prototype for GFDM [3], simple calculation shows that filter (ZgXia ) 14 , 12 = 0 which again leads to singularity problems. Additionally, in [3] the use of a periodized version of the sincfunction as the transmit filter is suggested. Due to its bad timelocalization, this orthogonal filter does not suffer from the BLT and its CTZT has constant modulus. Hence, the NEF does not increase with a finer sampling, but the bad time-localization can make the system more vulnerable to timing-offsets. V. N UMERICAL E XAMPLES Fig. 4(a) shows the NEF of a GFDM system with K = 128 and varying odd M , using a RC filter with two different rolloffs. No NEF can be given for even M due to the nonexistence of the ZF receiver. Evidently, for both filters, the noise enhancement increases with M . From a conventional point of view, with higher M and rolloff the ZF receiver needs to cancel more ISI and ICI which is bought for noise amplification. From a Gabor analysis point of view, with a wider opening or denser sampling of the CTZT of the transmitter filter, the DZT contains values that approach zero (cf. Fig. 3), constituting larger values in the DZT of the dual window, which increases the dual window’s norm and, therefore, increases the NEF. To underline the obtained theoretic results, Fig. 4(b) shows simulated symbol error rates (SER) of different receiver types under AWGN for GFDM systems with different parameter combinations. For comparison, the performance of FBMC over AWGN channel is additionally provided. Notice that the effect of FBMC filter tails and the cyclic prefix in GFDM has not been taken into account, i.e., the same spectral efficiency has been considered for both cases. For even M no ZF receiver exists and MF and MMSE also perform poorly, whereas for odd M the ZF receiver exists and MF and MMSE perform better. When the noise enhancement is small (small M , small rolloff), the system is nearly orthogonal and ZF detection
performs close to FBMC and optimal linear MMSE detection. With higher noise enhancement, orthogonality is lost and performance degrades in general but the MMSE receiver outperforms the ZF receiver. VI. C ONCLUSION This letter has bridged the current understanding of GFDM as a novel multicarrier transmission scheme and the mathematical theory of Gabor analysis. It was shown that GFDM is equivalent to a critically sampled Gabor expansion and analysis pair. Gabor analysis was used to efficiently calculate the ZF and MMSE receiver filters without requiring a matrix inversion, making it possible to provide receiver filters for large systems. Insight into the singularity conditions of GFDM was obtained by investigating the zeros of the DZT of the transmitter filter. It was proven the fact that certain parameter configurations lead to singular systems which eventually influences system design and practical implementations. The obtained theoretical results were confirmed by numerical examples. R EFERENCES [1] G. Wunder et al., “5GNOW: non-orthogonal, asynchronous waveforms for future mobile applications,” IEEE Communications Magazine, vol. 52, no. 2, pp. 97–105, Feb. 2014. [2] Y. Medjahdi et al., “Performance analysis in the downlink of asynchronous OFDM/FBMC based multi-cellular networks,” IEEE Transactions on Wireless Communications, vol. 10, no. 8, pp. 2630–2639, Aug. 2011. [3] Author A, Author B, Author C, “Generalized Frequency Division Multiplexing for 5th Generation Cellular Networks,” Submitted to IEEE Transactions on Communications, 2014. [4] K. Gröchenig, Foundations of Time-Frequency Analysis. Birkhäuser, 2001. [5] H. G. Feichtinger and T. Strohmer, Eds., Advances in Gabor Analysis (Applied and Numerical Harmonic Analysis). Birkhäuser, 2002. [6] M. J. Bastiaans, “Gabor’s signal expansion and the Zak transform.” Applied optics, vol. 33, no. 23, pp. 5241–55, Aug. 1994. [7] J. Wexler and S. Raz, “Discrete Gabor expansions,” Signal Processing, vol. 21, no. 3, pp. 207–220, Nov. 1990. [8] H. Bolcskei and F. Hlawatsch, “Discrete Zak transforms, polyphase transforms, and applications,” IEEE Transactions on Signal Processing, vol. 45, no. 4, pp. 851–866, Apr. 1997. [9] A. J. E. M. Janssen, “The Zak transform : a signal transform for sampled time-continuous signals.” Philips Journal of Research, vol. 43, no. 1, pp. 23–69, 1988. [10] T. Strohmer, “Numerical Algorithms for discrete Gabor Expansions,” in Gabor Analysis and Algorithms, H. Feichtinger and T. Strohmer, Eds. Birkhäuser, 1998, ch. 8, pp. 267–294. [11] X. Xia, “A family of pulse-shaping filters with ISI-free matched and unmatched filter properties,” Communications, IEEE Transactions on, vol. 45, no. 10, pp. 1157–1158, 1997.
