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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 63, NO. 11, NOVEMBER 2015 4501

Widely Linear Estimation for Space-Time-CodedGFDM in Low-Latency Applications

Maximilian Matthé, Luciano Leonel Mendes, Nicola Michailow, Dan Zhang, and Gerhard Fettweis, Fellow, IEEE

Abstract—This paper presents a solution for achieving trans-mit diversity with generalized frequency division multiplexing(GFDM). Compared to previous works, the proposed solutionsignificantly improves symbol error rate (SER) performance andlatency, where both aspects are crucial for future 5G cellularnetworks. It is shown that widely linear estimation at the re-ceiver side can jointly equalize and demodulate the space-timeencoded GFDM signal. Moreover, maximum ratio combining canfurther increase the SER performance with multiple receive anten-nas. SER performance is evaluated in Rayleigh fading multipathchannels.

Index Terms—Estimation, modulation, transmit diversity,GFDM, tactile internet.

I. INTRODUCTION

DURING the past decades, the evolution of mobile commu-nication networks has increased their capacity in terms of

number of users and throughput. However, only higher through-put will not be enough to address future challenges foreseenfor the fifth generation (5G) of mobile networks. Low latencyhas been pointed out as an important requirement to triggernew services, such as the Tactile Internet [1], [2]. Besideslow latency, low out-of-band (OOB) emissions, fragmentedspectrum allocation and relaxed requirements for synchronicitywill also be important in 5G systems [3].

Orthogonal Frequency Division Multiplexing (OFDM) hasproven to be a robust and reliable modulation scheme for highdata rate communication systems. The circularity introduced bya cyclic prefix (CP), which needs to be longer than the channeldelay profile, allows for frequency domain equalization with asingle tap per subcarrier and increases the system performanceover fading multipath channels. Due to orthogonality of the

Manuscript received April 13, 2015; revised July 7, 2015 and August 7,2015; accepted August 7, 2015. Date of publication August 13, 2015; dateof current version November 13, 2015. This work was supported in part byCNPq-Brasil and Finep/Funttel/Radiocommunication Reference Center underGrant No. 01.14.0231.00 hosted by Inatel and was performed in the frameworkof the FP7 project ICT-619555 “RESCUE,” which is partly funded by theEuropean Union. The associate editor coordinating the review of this paper andapproving it for publication was S. Gezici.

M. Matthé, N. Michailow, D. Zhang, and G. Fettweis are with the VodafoneChair Mobile Communication Systems, Technische Universität, 01062Dresden, Germany (e-mail: [email protected]; [email protected]; [email protected]; [email protected]).

L. L. Mendes is with the Telecommunication Engineering Department,Intituto Nacional de Telecomunicações, 37540-000 Sta. Rita do Sapucai-MG,Brazil (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TCOMM.2015.2468228

subcarriers, OFDM can be easily integrated with space-timecoding (STC) [4] to obtain transmit diversity over time-variantfrequency-selective channels.

Nevertheless, OFDM has some drawbacks that might hamperits usage in the 5G physical layer (PHY). OFDM exhibitshigh OOB emission, restricted to −35 dBc, which demandsfor a channel filtering to meet the emission masks imposedby regulation agencies. This is a significant disadvantage in aflexible or fragmented spectrum allocation scenario [3]. Reduc-ing the time duration of the OFDM frame to the low-latencyrequirements of 5G also poses a challenge. Due to the CPoverhead, the size of the channel impulse response does notallow the use of short OFDM symbols efficiently. Moreover,short time domain OFDM symbols produce wide subcarriers inthe frequency domain, which are subject to frequency selectiv-ity per subcarrier. In such conditions, the attractive frequency-domain equalization (FDE) cannot be easily implemented sinceOFDM has a resolution of only one tap per subcarrier in thefrequency-domain [5].

Several new waveforms are being proposed to address therequirements of 5G networks [3], where Generalized FrequencyDivision Multiplexing (GFDM) [6] is one waveform candidate.In this technique, a block of data symbols is transmitted persubcarrier and each subcarrier is circularly convolved with apulse shaping filter. A CP can be added to the GFDM frameto provide low-complexity FDE. The self-interference through-out the data symbols, caused by using non-orthogonal, well-localized pulse shaping filters, can be mitigated by linear oriterative approaches at the receiver side. GFDM can achieve lowOOB emission [7] and it can reduce the latency on the PHY [8].Because each GFDM subcarrier is represented by multiplefrequency samples, FDE is possible even when the channel isfrequency selective per subcarrier [6]. The shortening of GFDMsubsymbols is not problematic because only a single CP isadded for the entire frame. These properties make GFDM apromising candidate for the 5G PHY layer.

Transmit diversity is a key feature for the next generationof mobile communication systems [9]. The STC proposedby Alamouti is a simple solution to achieve this property.In previous works [6], time-reversal space-time coding (TR-STC) is shown as a feasible solution for GFDM, since theblock structure of the signal in time domain allows the en-coding of the waveform samples instead of the data symbols.This approach achieves full diversity gain and can be usedas a frequency division multiple access scheme if a singleguard subcarrier is used between multiple users [10]. However,TR-STC-GFDM requires two GFDM frames to build the

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4502 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 63, NO. 11, NOVEMBER 2015

codeword. Although this approach presents a good performancein terms of symbol error rate (SER), it is clearly not optimal forlow-latency applications. To reduce latency, STC can be carriedout on the data symbols within a single block. In previousworks [11] a linear receiver was used to decode this space-timecode, leading to significant SER performance degradation dueto remaining channel intersymbol interference (ISI).

The contribution of this paper is the application of widelylinear estimation (WLE) [12] to minimize the effects of theISI introduced by the multipath channel to the STC-GFDMSER performance. This way, both low latency and good SERperformance can be achieved, however at the cost of increasedcomplexity at the receiver when compared to linear approaches[11]. Moreover, an approach for using maximum ratio com-bining (MRC) with multiple receive antennas is described,which further improves the SER performance. Furthermore, thecomplexity of the proposed algorithm is derived and shownto be linear with the number of subcarries. The SER perfor-mance of the proposed scheme is compared with STC-OFDMand conventional STC-GFDM over fading multipath channelsbased on the Extended Vehicular A (EVA) [13] model. TheWLE-STC-GFDM proposed in this paper is a promising so-lution for applications where low latency and high robustnessare needed and computing power is available. One examplescenario is vehicle-to-vehicle communication [14], where theterminals are not power-limited and the short time window toestablish communication between fast moving objects througha double dispersive channel can be a challenge.

Note that the present proposal is not limited to low-latencyapplications. Instead, the latency properties of the system de-pend on the frame structure of the GFDM signal. However,compared to the proposal in [10], only one GFDM block is nec-essary to achieve full transmit diversity, which is in particularsuitable for low-latency applications.

The remainder of this paper is organized as follows:Section II presents the principles of GFDM. Section III showshow STC can be applied and a linear demodulator is shown,while Section IV derives the WLE for STC-GFDM and anal-yses its complexity. Section V provides the SER performanceanalysis over frequency-selective time-variant channels and,finally, Section VI concludes this paper.

II. GFDM BACKGROUND

In a GFDM block, N = MK data symbols are transmitted onK subcarriers. Each subcarrier is divided into M subsymbols.The complex-valued data symbols for one block are arrangedin a matrix

D = (u0 u1 . . . uM−1). (1)

There, um = (u0,m u1,m . . . uK−1,m)T contains the K data sym-bols that are transmitted in the mth subsymbol. Each symboluk,m is transmitted on a waveform gk,m[n], which is derivedfrom a prototype pulse shaping filter g[n]. The prototype filteris circularly shifted to the kth subcarrier and mth subsymbol,such that

gk,m[n] = g [(n − mK) mod N] exp

(j2π

k

Kn

), (2)

Fig. 1. Illustration of ISI and ICI within a GFDM block for K = 5, M = 5.Data symbols are shown as overlapping diamonds, explaining ISI and ICIbetween transmitted data. Note that in addition to the ISI and ICI due to transmitfiltering, a multipath channel introduces more ISI, which can be removed byfrequency domain equalization (FDE). For example, a multipath channel wouldcause dk,m−2 to interfere into the resource of dk,m.

where n = 0, 1, . . . , N − 1 is the time-index. Usually, g[n] isa raised cosine (RC) filter with rolloff α [6]. Accordingly, thetransmit signal x[n] is given by

x[n] =K−1∑k=0

M−1∑m=0

uk,mgk,m[n]. (3)

Eq. (3) can be formulated to an equivalent matrix expression

x = Ad, (4)

where the vector x = (x[n])T contains the samples of x[n]. d =vec(DT) holds the rows of D transposed and stacked on topof each other. A contains gk,m = (gk,m[n])T as its (kM + m)thcolumn.1 The above model can be extended to the case whenonly Kon subcarriers are switched on with Kon < K. In this case,the columns corresponding to the switched off subcarriers areremoved from A. Furthermore, then matrix D is of dimensionKon × M.

A CP with length NCP is added to the transmitted signal toavoid interference between subsequent GFDM blocks. Sinceonly one CP is needed for all subsymbols, the CP overhead ofGFDM is significantly reduced compared to OFDM [6].

On the receiver side, assuming the channel impulse responseshorter than the CP, the received signal after the CP removal isgiven by

r = HAd + w, (5)

where H is the circulant channel matrix and w is the additivewhite Gaussian noise (AWGN) vector with variance σ 2

w. Themodulation process introduces intercarrier interference (ICI)and ISI between the transmitted data symbols, as shown inFig. 1 and the multipath channel further increases the ISI. Theoccuring interference needs to be combated at the receiver inorder to provide acceptable SER performance. For example, thetransmitted data symbols can be estimated by a linear operationusing a zero-forcing receiver (ZFR) or minimum mean squareerror receiver (MMSER) [6]. Considering ZFR, the estimateddata symbols are given by

d = BZFH−1r, (6)

1Throughout the paper we assume zero-based indexing for matrices.

MATTHÉ et al.: WIDELY LINEAR ESTIMATION FOR SPACE-TIME-CODED GFDM IN LOW-LATENCY APPLICATIONS 4503

Fig. 2. Block diagram of the proposed STC-WLE-transceiver.

where

BZF = A+ = (AHA)−1

AH (7)

is the demodulation matrix, containing the ZF filters γk,m[n].Note that H is diagonalized by the Fourier transform and henceFDE can be efficiently performed.

The MMSER jointly performs channel equalization and de-modulation. Its demodulation matrix is given by

BMMSE = (Rw + AHHHHA)−1

AHHH, (8)

where (·)H denotes conjugate transpose and Rw is the noisecovariance matrix. The estimated symbols are given by

d = BMMSEr. (9)

The MMSER reduces the noise enhancement for low signalto noise ratios (SNR) and it can mitigate the self-generatedinterference introduced in the modulation process when non-orthogonal waveforms are employed [15]. However, it is notunbiased and hence a rescaling operation before constellationdetection is necessary.

III. SPACE-TIME CODE FOR GFDM

Transmit diversity is an important feature for mobile com-munication systems to provide robustness against frequencyand time fading. Alamouti has proposed a simple and effectiveSTC to obtain diversity in single carrier systems by usingtwo transmit antennas without reduction of the overall datarate compared to single-antenna transmission [4]. Alamouti’sSTC scheme is easily integrated with orthogonal multicarriersystems [16], but it is a challenge to apply this technique tonon-orthogonal systems [17].

In order to keep the overall latency on the PHY low, noadditional delay should be introduced by the STC. Hence, inthe proposed solution, STC is carried out within a single GFDMblock by space-time encoding the data symbols transmitted bytwo subsequent subsymbols on each subcarrier. This approachrequires an even number M of subsymbols in each GFDM blockto be space-time encoded. However, in [18] it was shown thatGFDM requires an odd number of subsymbols when commonpulse shaping filters are used. Therefore, the first subsymbolin the GFDM block is not encoded but left empty for pilots orsynchronization signaling. The corresponding loss in spectralefficiency is compensated by the more efficient use of theCP in most useful applications. In fact, given a number of

subcarriers K, STC-GFDM with one empty subsymbol will bemore frequency efficient than STC-OFDM if

M >K + 2NCP

NCP. (10)

Note that pilots and synchronization resources for OFDM arenot considered here, which would further reduce its spectralefficiency.

The block diagram of the proposed STC transmitter is de-picted in the left part of Fig. 2. Let Ds contain all but the firstcolumn of D and let

D(1)s = [u1 u2 . . . uM−2 uM−1

]= Ds

D(2)s = [−u∗

2 u∗1 . . . − u∗

M−1 u∗M−2

]= D∗

s Ps

(11)

be the space-time encoded data to be transmitted by the twotransmit antennas. There, Ps is given by

Ps = I M−12

⊗[

0 1−1 0

], (12)

where In is the n × n identity matrix and ⊗ denotes the matrixKronecker product. Let As be a shortened A, where the Kcolumns relating to the first subsymbol are discarded.

Accordingly, the signal x(i) transmitted by the ith antennabefore CP addition is given by

x(i) = Asvec(

D(i)s

T)

. (13)

Let FN and FHN denote the N-point unitary discrete Fourier

transform (DFT) and its inverse operation. The received signalat the jth receive antenna after removing the CP and moving tofrequency domain is given by

r(j) =[H

(1,j)H

(2,j)P] [ds

d∗s

]+ w(j). (14)

There, H(i,j) = FNH(i,j)As, H(i,j) denotes the circulant channel

matrix from the ith transmitting to the jth receiving antenna,ds = vec(DT

s ), P = IK ⊗ PTs and w(j) is the frequency domain

AWGN at the jth receiving antenna.Linear space-time decoding can be used to recover the trans-

mitted information, however it suffers from the ISI betweenGFDM subsymbols, introduced by the multipath channel, cf.Fig. 1, because no FDE can be carried out prior to Alamouticombining. Instead, the received time domain GFDM blocksare first demodulated with the ZFR yielding

d(j) = BZFFHN r(j). (15)


Note that FHN is only introduced for notation purposes and not

necessary in real implementations, as the signal is available intime domain anyway. In order to achieve diversity gain, d(j) isrestructured into a K × (M − 1) matrix with columns u(j)

m whichare combined according to [11]

um =∑J

j=1 H(1,j)∗K u(j)

m + H(2,j)K u(j)∗

m+1∑Jj=1

∣∣∣H(1,j)K

∣∣∣2 +∣∣∣H(2,j)

K

∣∣∣2 m odd,

um =∑J

j=1 H(1,j)∗K u(j)

m − H(2,j)K u(j)∗

m−1∑Jj=1

∣∣∣H(1,j)K

∣∣∣2 +∣∣∣H(2,j)

K

∣∣∣2 m even,

(16)

where J is the number of receive antennas, H(i,j)K is the K-point

Fourier transform of the channel impulse response between theith transmit and jth receive antenna and operations are to beunderstood element-wise.

Note that the channel-imposed ISI between the subsymbolstransmitted on each subcarrier is not removed by combiningthe STC after the GFDM demodulation. Thus, the estimatedsubsymbols um obtained by (16) have residual ISI that canseverely reduce the system performance depending on thechannel impulse response, leading to an error floor in the SERperformance curve [11].

IV. WIDELY LINEAR EQUALIZATION FOR STC-GFDM

In order to improve on the STC-GFDM performance, themultipath channel ISI needs to be equalized. This can be ac-complished by jointly demodulating, equalizing and combiningthe received signal with the help of a widely linear estimator,where the block diagram of the processing chain is givenin Fig. 2. Widely linear estimation outperforms conventionallinear estimation when either the estimand or the measurementis an improper process [12]. The following section shows thatthe received signal is improper and the widely linear estimatorfor STC-GFDM is derived. Subsequently, the structure of theresulting equation system is analyzed and a low-complexitysolution is explained.

A. Derivation of Widely Linear Estimators

Mapping an i.i.d. bit sequence onto a rotationally invariantconstellation with unit average symbol energy provides

E[dsdH

s

] = IK(M−1) E[dsdT

s

] = 0K(M−1), (17)

where 0n is a n × n null matrix. Further, assuming the channelimpulse responses between the ith transmit and jth receiveantennas are invariant during the transmission of one GFDMblock, which is reasonable for short block durations, the auto-correlation �(j) of r(j) is given by

�(j) = E[r(j)r(j)H

](18)

=[H

(1,j)H

(2,j)P] [ H(1,j)H

PHH(2,j)H

]+ σ 2

wIMK . (19)

Similarly, the pseudoautocorrelation C(j) is given by

C(j) = E[r(j)r(j)T

](20)

=[H

(1,j)H

(2,j)P] [PTH(2,j)T

H(1,j)T

]. (21)

Note that PPH = PPT = I. Since C(j) �= 0, r(j) is an improper(non-circular) process and WLE of ds can improve the esti-mation performance. Compared to a linear estimator, a widelylinear estimator jointly processes the received signal and itsconjugate to estimate the transmitted data by

d(j)s =

[U(j)

V(j)

]H [ r(j)

r(j)∗]

. (22)

The filter coefficients U(j) and V(j) are chosen to minimize themean squared error (MSE) between ds and d

(j)s and are solutions

to the linear system [12][�(j) C(j)

C(j)∗ �(j)∗]

︸︷︷︸F

[U(j)

V(j)

]=[

�(j)

�(j)∗]

, (23)

where

�(j) = E[r(j)dH

s

]= H

(1,j)(24)

and

�(j) = E[r(j)dT

s

]= H

(2,j)P. (25)

The solution of (23) is given by

U(j) = S(j)−1(�(j) − C(j)�(j)−1∗

�(j)∗)V(j) = S(j)−1∗ (

�(j)∗ − C(j)∗�(j)−1�(j)

),

(26)

where S(j) = �(j) − C(j)�(j)−1∗C(j)∗ is the Schur complement

of �(j)∗ in F.Since H

(i,j)is a tall matrix, �(j) becomes singular when

σw → 0 and hence a zero forcing (ZF) estimation cannot bedirectly derived. Instead, the system model (14) for the widelylinear estimation problem is reformulated to a linear estimationproblem of double size according to[

r(j)

r(j)∗]

︸︷︷︸r(j)

a

= H(j)eq

[ds

d∗s

]︸︷︷︸

da

+[

w(j)

w(j)∗]

︸︷︷︸w(j)

a

, (27)

where

H(j)eq =

[H

(1,j)H

(2,j)P

H(2,j)∗P H(1,j)∗

]. (28)

The linear minimum mean square error (LMMSE) estimatorfor da in (27) is given by

d(j)a,MMSE = H

(j)Heq

(H

(j)eqH

(j)Heq + σ 2

wI)−1

︸︷︷︸B(j)

MMSE

r(j)a . (29)


Direct calculation shows that H(j)eq H

(j)Heq + σ 2

wI = F from (23)and hence (29) is equivalent to (22) and (23). Writing theLMMSE estimator in (29) to its alternate form [19] results in

d(j)a,MMSE =

(H

(j)Heq H

(j)eq + σ 2

wI)−1

H(j)Heq r(j)

a . (30)

This reformulation allows the derivation the ZF estimator by

d(j)a,ZF =

(H

(j)Heq H

(j)eq

)−1H

(j)Heq r(j)

a = B(j)ZFr(j)

a , (31)

where B(j)ZF = H

(j)+eq is the Moore-Penrose pseudo inverse of

H(j)eq . From the Gauss-Markov theorem [19] it is known that (31)

is the best linear unbiased estimator (BLUE) for da and is hencethe best widely linear unbiased estimator for ds.

Note that d(j)a,(·) =

[d

(j)T

s,(·) d(j)H

s,(·)]T

contains redundant infor-

mation and hence only the upper half of B(j)(·), denoted by B(j)

s,(·),is required in actual processing.

Analogously, widely linear MMSE and ZF estimators canbe derived when J receive antennas are jointly combined. Thesystem model changes to

⎡⎢⎢⎢⎢⎣

r(1)a

r(2)a...

r(J)a

⎤⎥⎥⎥⎥⎦

︸︷︷︸ra

=

⎡⎢⎢⎢⎢⎢⎣

H(1)

eq

H(2)

eq...

H(J)

eq

⎤⎥⎥⎥⎥⎥⎦

︸︷︷︸Heq

da +

⎡⎢⎢⎢⎢⎣

w(1)a

w(2)a...

w(J)a

⎤⎥⎥⎥⎥⎦

︸︷︷︸wa

(32)

and widely linear MMSE and ZF estimators are given by

da,MMSE = HHeq

(HeqH

Heq + σ 2

wI)−1

ra

da,ZF = H+eqra.

(33)

This approach considerably increases the system size and

computational complexity. Alternatively, the d(j)

can be esti-mated separately for every receiving antenna and then com-bined, weighted by the quality of the channels.

In the following the ZF MRC receiver is derived, the com-putation of the MMSE MRC is straightforward but requiresheavier notation. For the ZF receiver we find that

d(j)s,ZF = ds + B(j)

s,ZFw(j)a (34)

and thus the MSE of the estimated data equals

e(j) = diag

(E

[(d(j)

s,ZF − ds

) (d(j)

s,ZF − ds

)H])

(35)

= σ 2wdiag

(B(j)

s,ZFB(j)H

s,ZF

). (36)

The operator diag(·) returns the diagonal of a matrix argumentand returns a diagonal matrix for a vector argument. The

Fig. 3. Sample equivalent channel Heq for M = 9, K = 8 using a RC filterwith α = 1.

estimated data symbols from the J receiving antennas are nowlinearly combined, weighted by their inverse MSE

s(j) =[diag

(e(j))]−1

(37)

according to

ds =⎡⎣ J∑

j=1

s(j)

⎤⎦−1⎛⎝ J∑

j=1

s(j)d(j)s

⎞⎠ . (38)

Note that ds = d(1)

s for J = 1.

B. Complexity Analysis

According to (29), the widely linear MMSE and ZF estima-tors for ds require to solve a linear equation system of size 2N.The application of general-purpose solvers for such systemsrequires a computational complexity of cubic order in both thenumber of subcarriers and subsymbols which is prohibitivelycomplex for low-latency applications. In what follows we an-alyze the structure of the equation systems and show that, dueto the sparsity of the transmit filter in the frequency domain,a solution can be found with linear complexity in the numberof subcarriers, where the number of complex operations isconsidered as a figure of merit.

Consider the equivalent channel matrix H(j)eq in (28). When

using a band-limited filter that only has B non-zero coefficientsin the frequency domain, FNAs is a sparse matrix with Bentries per column. For example, when using a RC filter withrolloff α, (FNgk,m)〈n〉N

= 0 when |n − kM| > (1 + α)M2 and

B = (1 + α)M. Accordingly,

FNH(i,j)As = FNH(i,j)FHN FNAs (39)

obeys the same sparsity since the channel matrix is diagonalizedby the DFT. Note that due to periodicity properties of the DFTFNg0,m has contributions around n = 0 and n = N − 1. Since

P only operates within isolated subcarriers, the blocks of H(j)eq

are equally sparse. An illustration is given in Fig. 3.


Let T2U be a 2U × 2U permutation matrix given by

T2U = [e0 eU e1 eU+1 . . . eU−1 e2U−1]T (40)

where ei is a zero column vector of length 2U with 1 at its ithposition. With Non = Kon(M − 1) a permutation is applied to(27) by

T2Nr(j)a︸︷︷︸

r(j)′a

= T2NH(j)eq TT

2Non︸︷︷︸Q(j)

eq

T2Nonda︸︷︷︸d′

a

+w(j)a (41)

where Q(j)eq is a sparse matrix of size 2N × 2Non where each

column only has 2B non-zero elements. Eq. (41) is now solvedfor d′

a by

d′a =

((Q(j)

eq

)HQ(j)

eq + σ 2n I)−1 (

Q(j)eq

)Hr(j)′

a (42)

and subsequently the inverse permutation is applied to acquirethe original da. However, as the present permutations onlydescribe the order of variables in the system, they are notaccounted for complexity. On the other hand, the calculation

of (Q(j)eq )

Hr(j)′

a requires only 2BN complex multiplications.

(Q(j)eq )

HQ(j)

eq is a positive definite hermitian band diagonal ma-trix with periodic boundary conditions with S = 4(M − 1) − 1superdiagonals since only the subsymbols of adjacent subcar-riers overlap in the frequency domain. The periodic boundarycondition exists due to the edge subcarriers, that wrap aroundin the discrete frequency domain. Accordingly, such subcarrierssimultaneously have frequency components at the lowest andhighest frequencies, making them unsuitable for spectral local-ization. Furthermore, similar to OFDM, empty edge carrierscan be used to reduce emission into adjacent systems. But,due to good frequency-localization of GFDM, much fewerempty carriers would be needed for this purpose. Furthermore,without edge carriers, i.e. in case of Kon < K, the periodicboundaries vanish since the last subcarrier does not overlapwith the first one. Then, the band diagonal structure of (42)can be solved with Non((S2 + 1) + 4S) complex multiplications[20, LAPACK zpbsv]. If K = Kon, the periodic boundary con-ditions can be overcome with the application of the Woodburyformula [21] and the solution is accomplished with an extraeffort in the order of O(K · M3) complex multiplications whichis still linear with the number of subcarriers. Hence, by ex-ploiting the structure of the equation system, the computationaleffort can be significantly reduced to O(K · M3) compared toO(K3M3) when general-purpose solvers are employed. In thebeneficial case of empty edge carriers, even the application ofthe Woodbury formula is not necessary at all and complexityreduces to O(K · M2)

V. SER PERFORMANCE ANALYSIS

An approximation for the SER performance over fadingmultipath channels for STC-OFDM [22] and STC-GFDM with

TABLE ICP EFFICIENCY AND NEF FOR OFDM AND GFDM

QAM modulation and ZFR [6] is given by

pe = 4

(2

μ2 − 1

2μ2

)(1 − ϕ

2

)2J

×

×2J−1∑u=0

(2J − 1 + u

u

)(1 + ϕ

2

)u

, (43)

where μ is the number of bits per symbol of the QAM constel-lation and

ϕ =√√√√ 3RCPσ 2

reqEs

2(2μ − 1)ξN0 + 3RCPσ 2req

Es(44)

is the equivalent average signal to noise ratio. Es and N0 arethe average energy of the QAM constellation and the noisespectral density, respectively. RCP is the rate reduction due toa CP with NCP samples or vacant subsymbols and ξ is the noiseenhancement factor (NEF). These parameters take different val-ues for STC-OFDM and STC-GFDM, as presented in Table I.The equivalent parameter for the multipath Rayleigh channel isgiven by

σ 2req

= σ 2r

I−1∑i=0

h2i , (45)

where h is the channel impulse response with length I and σ 2r

the parameter of the Rayleigh distributed taps.Fig. 4(a) compares the performance of uncoded STC-OFDM,

STC-GFDM and WLE-STC-GFDM with the configurationpresented in Table II where (43) is used as reference for thesimulations results.

Table III presents the non zero taps of the channel delayprofile based on the EVA model [13], assuming the samplingfrequency presented in Table II.

Fig. 4(a) shows that the theoretical performance ofSTC-OFDM and STC-GFDM are similar. STC-GFDM makes abetter use of the CP, but the vacant first subsymbol reduces theoverall efficiency to the same level achieved by STC-OFDM.STC-GFDM has a poor performance when conventional com-bining is used after demodulation due to ISI among the sub-symbols [11], which leads to a clear error floor. The WLEcan remove the ISI during the demodulation and combiningprocess, leading to a significantly improved performance closeto the theoretical curve, but still a slight degradation is present.Fig. 5 shows the ratio between the MSE per subcarrier ofWLE-STC-GFDM and STC-OFDM for random channel re-alizations. Apparently, the ratio is not constant and there are


Fig. 4. Simulated system performance of STC-OFDM and STC-GFDM over mobile channel. (a) Uncoded SER. (b) RS-encoded BER.

TABLE IIGFDM CONFIGURATION

TABLE IIICHANNEL DELAY PROFILE USED FOR THE PERFORMANCE ANALYSIS

Fig. 5. Ratio of MSE per subcarrier between GFDM and OFDM. The ratiowas evaluated for 1000 channel realizations, each one shown in (a) as one line.The histogram in (b) shows the number of occurrences of each ratio, ignoringthe outliers.

channel realizations where WLE-STC-GFDM has a larger MSEthan STC-OFDM which is the reason, why WLE-STC-GFDMdeviates from the theoretical curve. However, this deviation

TABLE IVPARAMETERS OF THE REED SOLOMON CODE

only appears for high SNR where the SER is below 10−3

and can be easily combated with forward error correction.Exemplary, we choose a simple Reed Solomon (RS) code[23], whose parameters from Table IV have been used tofit one GFDM block, such that each GFDM block can bedecoded independently from other GFDM blocks. As can beseen from Fig. 4(b), its limited error correction capability isalready sufficient to combat the performance loss observed inthe uncoded case. This implies the usability of more powerfulchannel codes such as turbo or LDPC codes with short andvariable code-block length [24], [25] that can also satisfy thelow-latency constraints. Given their wide application in today’scommunication systems, they can be more suitable than RScodes due to higher efficiency and simple configuration forother applications.

The simulation results presented in Fig. 4(b) show thatthe channel coding is very effective to address the remainingperformance loss of the WLE-STC-GFDM scheme and the0.3 dB gap between STC-GFDM and STC-OFDM due to themore efficient use of the CP (considering the empty subsymbol)is clearly present for high SNR.

Fig. 4(a) also shows the WLE-STC-GFDM performancewhen two antennas are used in the receiver side, where thesignal on each receiving antenna has been processed separatelyand is then combined by (38). As shown in Fig. 4(a), 2 × 2STC-GFDM does not have the diversity loss observed in the2 × 1 case. A reason is given by (38), where subcarriers withhigh MSE are outweighed by subcarriers with smaller MSE,and hence the influence of bad subcarriers is reduced.


VI. CONCLUSION

This paper has presented an advanced approach on de-modulating a space-time encoded GFDM signal. Space-timeencoding was carried out within a GFDM block in order tokeep the overall system latency. This way transmit diversity canbe achieved without increasing the PHY latency compared to asingle antenna transmission. At the receiver, the GFDM blockis decoded with the help of a widely linear estimator whichprovides significant gains compared to previous works [11].

The proposed scheme reaches nearly-optimal performancecompared to OFDM, with slight degradations in high SNRregions (SER < 10−3). These deviations can be combated withforward error coding. The scheme can be combined with aMRC approach at the receiver where the symbols from theantennas are linearly combined after WLE has been carriedout per antenna. This way, the matrix inversion complexity forthe WLE is kept constant regardless of the number of receiveantennas. The complexity analysis reveals that processing in thefrequency domain allows the system to be solved with linearcomplexity in the number of subcarriers and is as such suitablefor low-latency implementations.

ACKNOWLEDGMENT

The computations were performed on a computing clusterat the Center for Information Services and High PerformanceComputing (ZIH) at TU Dresden.

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Maximilian Matthé received the Dipl.-Ing degreein electrical engineering from Technical UniversityDresden (TU Dresden), Dresden, Germany, in 2013.He is currently pursuing the Ph.D. in the Voda-fone Chair Mobile Communication Systems at TUDresden. During his studies, he focused on mobilecommunication systems and communication theory.He performed his internship at National InstrumentsDresden and worked on the design and implemen-tation of a measurement site for LTE test UEs. Inhis Diploma Thesis he concentrated on waveform

design for flexible multicarrier transmission systems. His research focuses onthe design and evaluation of MIMO architectures for future cellular networks.

Luciano Leonel Mendes received the B.Sc. andM.Sc. degrees in electrical engineering from Inatel,Brazil, in 2001 and 2003, respectively. In 2007, hereceived the doctor’s degree in electrical engineeringfrom Unicamp, Brazil. Since 2001, he has been aProfessor at Inatel, Brazil, where he has acted asTechnical Manager of the hardware developmentlaboratory from 2006 to 2012. He has coordinatedthe Master Program at Inatel and several researchprojects funded by FAPEMIG, FINEP, and BNDES.He is a post-doc visiting researcher, sponsored by

CNPq-Brasil, at Vodafone Chair Mobile Communications Systems at theTechnical University Dresden since 2013. His main area of research is wirelesscommunication and currently he is working on multicarrier modulations for 5Gnetworks and future mobile communication systems.


Nicola Michailow received the Dipl.-Ing. degree inelectrical engineering with focus on wireless com-munications and information theory from TechnicalUniversity Dresden, Dresden, Germany, in 2010.From 2008 to 2009, he worked in the R&D Depart-ment of Asahi Kasei Corporation, Japan, developingsignal processing algorithms for sensor data anal-ysis. Since 2010, he has been Research Associateat the Vodafone Chair at TU Dresden, pursuing aDr.-Ing. degree. His scientific interests are focused onflexible and non-orthogonal multi-carrier waveforms

for next generation cellular systems. During his time at the Vodafone Chair, hecontributed to the FP7 projects 5GNOW, CREW, QOSMOS, EXALTED, andwas part of the RF Lead User Program with National Instruments.

Dan Zhang received the B.Sc. degree in electri-cal engineering from Zhejiang University, China,in 2004, and the M.Sc. and the Dr.-Ing. degree inelectrical engineering and information technologyfrom RWTH Aachen University, Aachen, Germany,in 2007 and 2013, respectively. Her research interestsare in the area of statistical signal processing, com-munication theory, and optimization theory. Duringher Ph.D. studies, she focused on iterative receiverdesigns. Since September 2014, she has worked asa Postdoctoral Researcher at the Vodafone Chair

Mobile Communications Systems at Technical University Dresden, Dresden,Germany. She has worked on various national and international researchprojects with publication in journals, conference proceedings, and workshops.Her current research focus is on the transceiver design of the waveform GFDM(GFDM) with the objective of providing a new air interface for 5G networks.

Gerhard Fettweis (F’09) received the Ph.D. degreefrom RWTH Aachen, Germany, in 1990. Thereafterhe was at IBM Research and then at TCSI Inc.,USA. Since 1994, he has been Vodafone Chair Pro-fessor at Technical University Dresden, Dresden,Germany, with his main research interest on wirelesstransmission and chip design. He is an HonoraryDoctorate at Technical University Tampere. As arepeat entrepreneur he has co-founded 11 startupsso far. He has setup funded projects in size of closeto EUR 1/2 billion, notably he runs the German

Science Foundation’s CRC HAEC and COE cfAED. He is actively involvedin organizing IEEE conferences, and has been TPC Chair of ICC 2009 andTTM 2012, General Chair of VTC Spring 2013, and DATE 2014.

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