Received March 5, 2018, accepted March 28, 2018, date of publication April 5, 2018, date of current version April 25, 2018.

Digital Object Identifier 10.1109/ACCESS.2018.2823002

Iterative Frequency Domain Equalizationfor MIMO-GFDM SystemsJIE ZHONG1, (Member, IEEE), GAOJIE CHEN 2, (Member, IEEE),JUQUAN MAO3, (Student Member, IEEE), SHUPING DANG 4, (Student Member, IEEE),AND PEI XIAO3, (Senior Member, IEEE)1Department of Information Science and Electronic Engineering, Zhejiang University, Hangzhou 310027, China2Department of Engineering, University of Leicester, Leicester LE1 7RH, U.K.3Institute for Communication Systems, Home of the 5G Innovation Centre, University of Surrey, Guildford GU2 7XH, U.K.4Department of Engineering Science, University of Oxford, Oxford OX1 2JD, U.K.

Corresponding author: Gaojie Chen (gaojie.chen@leicester.ac.uk)

This work was supported by the Engineering and Physical Sciences Research Council through the New Air Interface Techniques for FutureMassive Machine Communications Project under Grant EP/P03456X/1.

ABSTRACT This paper proposes a new iterative frequency domain equalization (FDE) algorithm formultiple-input multiple-output (MIMO)-frequency division multiplexing (GFDM) systems. This newFDE scheme is capable of enhancing the system fidelity by considering the complete frequency-domain sec-ond order description of the received signal. In addition, a new nulling filter design is also proposed forMIMO-GFDM systems to remove the residual interference, which further improves the system fidelitycompared with the traditional scheme. Simulation results are presented to verify the effectiveness andefficiency of the proposed FDE algorithm.

INDEX TERMS FDE, MIMO-GFDM systems, uplink transmission, nulling filter, 5G networks.

I. INTRODUCTIONTo satisfy the communications of emerging applications, e.g.,Internet of Things (IoT), ultra high-definition multimediaand virtual reality (VR), fifth generation (5G) wireless com-munications are faced with more stringent requirements interms of the transmission rate, system capacity, reliabilityand latency [1]–[3]. On the other hand, the improvement onwireless communication systems is restricted by a seriesof deleterious factors in the radio propagation environ-ment, e.g., inter-symbol interference (ISI), multi-user inter-ference (MUI), propagation attenuation and multi-pathfading [4]. These deleterious factors result in a variety oftechnological difficulties in the design of reliable wirelesscommunication systems. One intuitive approach to cope withthese design difficulties is to employ multi-antenna commu-nication paradigms, which have received considerable atten-tion in both academia and industry due to the obvious perfor-mance advantages of network capacity and quality of service(QoS) [5]. By mounting multiple antennas at transmittingand receiving terminals, a multi-input multi-output (MIMO)system is hereby constructed [6], which has been proved to beable to yield a series of performance improvements comparedto single-input single-output (SISO) systems [7].

Besides, the generalized Frequency Division Multiplex-ing (GFDM) was proposed for the 5G air interface, becauseof its low system complexity, flexible spectrum coordination,and low peak-to-average-power ratio (PAPR), which inducesa higher power efficiency compared to OFDM [8]. The flexi-bility of GFDM facilitates its collaboration with single carrierfrequency domain equalization (SC-FDE) and filtered bankmulti carrier (FBMC) [9]. GFDM is based on the modulationin a per-subcarrier manner, in which each subcarrier is modu-lated individually and independently with multiple symbols.Then, the subcarrier is filtered by a prototype filter, whichis circularly shifted in the time and frequency domains. Thisfiltering process is aimed at reducing the out-of-band (OOB)remains, and thereby facilitating the fragmented spectrumand dynamic spectrum allocations without causing severeinterference to the GFDM system per se and/or other users.On the other hand, both ISI and inter-carrier interference (ICI)among subcarriers might be rendered by such a filteringprocess. Fortunately, we can rely on well-designed receivingtechniques tomitigate the interference, e.g., thematched filterreceiver with the iterative interference cancellation, whichhas been found effective to achieve a better performancethan OFDM for numerous applications [10]. More details of

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J. Zhong et al.: Iterative FDE for MIMO-GFDM Systems

modulation and multiple access schemes as well as themotivation of the use of well-designed receiving techniquesfor next generation networks can be found in [11]. Forcompleteness purposes, interesting readers might also referto [12] and [13] for more details of frequency-domainreception.

In terms of the spatial domain, the spatial division multi-plexing (SDM) systems can achieve high data transmissionrate by transmitting and receiving multiple substreams inparallel through a MIMO architecture. However, the com-plexity raised by the optimal detection for SDM in theMIMOarchitecture is very high, which grows exponentially withthe number of antennas at the transmitter and receiver aswell as the signal constellation size. The Vertical Bell LabsLayered Space-Time (V-BLAST) detection technique canbe regarded as a satisfactory solution to the performance-complexity trade-off in detection [14], while suffers from theerror propagation problem inherent in its decision feedbackprocess. To address this issue and optimize the performanceof V-BLAST detection process, in [15], the co-antenna inter-ference (CAI) components have been replicated by the log-likelihood ratio (LLR) of the interference and the soft replicasubtracted from the received composite signal vector. As aconsequence, the performance has been improved by repeat-ing this iterative process. Meanwhile, for the case where mas-sive number of users are considered, e.g., applications of IoTand super dense networks [16], [17], the under-determinedMIMO systems should be taken into consideration [18].

Based on these considerations, it motivates us to devise anovel iterative detection algorithm, which takes advantage ofthe non-circular nature of the residual interference after sub-tracting the CAI by their soft symbol estimates, and the per-formance ofMIMO-GFDM systems can be enhanced accord-ingly. In addition to this iterative detection algorithm, we alsopropose a new nulling fitter design based on a modified errordetection criterion, which is capable of removing the residualinterference. To be clear, we summarize the contributions ofthis paper as follows.

1) We propose a novel iterative detection algorithm forMIMO-GFDM systems based on the non-circular char-acteristic of residual interference.

2) We propose a novel nulling fitter design incorporatingwith the proposed iterative detection algorithm.

3) We derive the explicit form of the fitter output of theproposed algorithm.

4) We carry out a series of numerical simulations to con-firm the superiority of our proposed algorithm overother conventional algorithms.

The rest of this paper is organized as follows. We presentthe model of MIMO-GFDM systems in Section II. Then,the iterative FDE algorithm is proposed and detailed inSection III. After that, we provide numerical results to verifythe effectiveness and efficiency of the proposed algorithm inSection IV. Finally, Section V concludes this paper.Notations: we use upper bold-face letters to represent

matrices and vectors. The (n, k)th element of a matrix A

is represented by [A]n,k and the nth element of a vector bis denoted by [b]n. Superscripts (·)H, (·)T , (·)∗ denote theHermitian transpose, transpose, and conjugate, respectively.

II. SYSTEM MODELIn this paper, we mainly focus on the uplink transmissionscenario, in which a cellular multiple access system has nRreceive antennas at the BS and a single transmit antenna atthe ith user terminal, i ∈ {1, 2, · · · ,KT }, where KT is thenumber of users. Meanwhile. it is assumed that a multi-userMIMO system with K (K < KT ) users is employed and eachuser is served at an exclusive time slot, such that K = nR.The system model for a GFDM-based MIMO transmitter andreceiver is shown in Fig. 1. At the transmitter, the informationbits intended to be transmitted are first encoded into a codedbit sequence with a longer length, which is then mappedinto constellation symbols. A set of N symbols is modulatedon one subcarrier aggregated with multi-carriers, which arealso called clusters in Fig. 1. In each subcarrier, a cyclicprefix (CP) is inserted to combat ISI and a data block ishereby constructed. Here, we assume the length of the CP islonger than the impulse response of the channel. The insertionof CP also helps converting linear convolution of the channeland transmitted data sequence into the circular convolution.Subsequently, the data block is interpolated and filtered by aroot raised cosine (RRC) filter, and all subcarriers are shiftedto their own spectrum chunks and summed up for the radiotransmission. As a result, this transmitting band consists ofall occupied spectrum chunks and appears to be a consec-utive M -carrier spectrum. At the receiver, each subcarrieris collected from the whole band, filtered by a RRC filterto eliminate adjacent interference. This filtering process isafter CP removal,N -point frequency domain equalization anddown-conversion. After that, we can apply a MIMO FDEto process the frequency-domain signals, and the design ofthe MIMO FDE is illustrated in Fig. 1 (b). For the receiverdesign, we assume that the linear minimum mean squarederror (MMSE) detection scheme is employed at the receiverfor simplicity. Also, the MMSE detection scheme can pro-vide a satisfactory solution to the trade-off between noiseenhancement and multi-stream interference mitigation [5].By such a receiver design, the output of each subcarrieris combined with a symbol stream, which is then fed toa soft de-mapper to generate estimates of transmitted bits.Finally, the transmitted information can be decoded from thestream.

To be specific, we introduce the notations and nomencla-tures in terms of an arbitrary subcarrier for a single useras follows, which are highly realted to the proposed FEDalgorithm described in the next section and evolved from theformer system proposed in [19]. The roll-off factor α of theRRC filter leads to additional spectrum occupation of α × Ncarriers. However, the in-band N -point carriers dominate theperformance, and we simplify the derivation based on them.We denote DFM = IK ⊗ FM and FM is the M × M Fouriermatrix with the element [FM ]m,k = exp(−j 2πM (m−1)(k−1)),

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J. Zhong et al.: Iterative FDE for MIMO-GFDM Systems

FIGURE 1. The system model of MIMO-GFDM, where (a) denotes the transmitter structure and (b) denotes the receiverstructure. (a) Transmitter structure. (b) Receiver structure.

where k,m ∈ {1, · · · ,M} denote the numbers of samples andfrequency tones, respectively. Here⊗ denotes the Kroneckerproduct, and IK is a K ×K identity matrix. D−1FM correspondsto the KM × KM inverse Fourier matrix, which can bedetermined by IK ⊗ F−1M and F−1M denotes a M ×M inverseFourier matrix with the element [F−1M ]m,k = 1

M exp(j 2πM (m−1)(k − 1)). Similarly, DFN and D−1FN are defined as DFM

and D−1FM with a difference in the matrix size. Meanwhile,zn and z−1n represent the subcarrier mapping and demap-ping matrices, respectively, which represent the N allocatedsubcarriers in the M carriers, so that the dimension of znis M × N .In the receiving side, the received signal processed by the

RF module becomes r = HD−1FM (IK⊗

zn)GDFN s + w,where s = [sT1 , · · · , s

TK ]

T∈ CKN×1 denotes the data

sequences transmitted by all K users, and si ∈ CN×1, i ∈{1, · · · ,K }, denotes the transmitted data block for the ith userwith E[sisiH ] = IN ; w ∈ CMnR×1 denotes a circularly sym-metric complex Gaussian noise vector with zero mean andcovariance matrix N0I ∈ RMnR×MnR , i.e., w ∼ CN (0,N0I);H is a nRM × KM channel matrix. The RRC filter matrixG ∈ CKN×KN is a block diagonal matrix, in which the ithsub-matrix is expressed as Gi =diag{gi,1, gi,2, · · · , gi,N } ∈CN×N and gi,n (i ∈ {1, 2, · · · ,K }) is the RRC filter’sresponse in the frequency domain for the ith user on thenth subcarrier.

III. FREQUENCY DOMAIN EQUALIZATIONWith the MIMO FDE, the time-domain output signal can bewritten as [19]

z = D−1FNAHG(IK ⊗z−1n )DFM r

= D−1FNAHG(IK ⊗z−1n )DFM (HD−1FM

× (IK ⊗zn)GDFN s+ w)

= D−1FNAHG(HGDFN s+ w) = D−1FN z, (1)

where A denotes a KN × KN equalization matrix and H =(IK ⊗ z−1n )DFM HD−1FM (IK ⊗zn) ∈ CKN×KN ; w ∈ CnRN×1

denotes a circularly symmetric complex Gaussian noise vec-tor with zero mean and covariance matrix N0I ∈ RnRN×nRN .The received signal after the RRC filtering can be expressedas

r = GHGs+ Gw = GHGDFN s+ Gw, (2)

where s = DFN s represents the transmitted signal in thefrequency domain. By applying a classic linearMMSE equal-izer [20], we can then perform operations of a FDE matrix Aon r to yield the equalized signal z = AHr, where A canbe generated by the classic approach by minimizing the costfunction

e = E[‖z− s‖2] = E[‖AHr− s‖2], (3)

which leads to the FDE matrix A = R−1r T r [20], andthe autocorrelation and cross correlation matrices can be

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given by

Rr = E[rrH] = GHGGHHHGH+ N0I (4)

and

T r = E[rsH] = GHG. (5)

A. CONVENTIONAL ITERATIVE DETECTION ALGORITHM:THE BENCHMARKDenote the symbol vector

s =[s1, . . . , sn−1, sn, sn+1, . . . , sNK

]T (6)

Now, let us focus on the decoding of symbol sn. By apply-ing the classic iterative interference cancellation method, thereceived signal vector can be written as [21]

rn = r− GHGDFN sn = GHGDFN [s− sn]+ Gw, (7)

where rn is the interference-canceled version of r, and

sn =[s1, . . . , sn−1, 0, sn+1, . . . , sNK

]T, (8)

which contains the soft estimates of the interfering symbolss from the previous iteration. One should note that (7) rep-resents a decision-oriented iterative scheme, by which thedetection procedure at the pth iteration utilizes the symbolestimates from the (p− 1)th iteration. As a result of this iter-ative procedure, the performance can be improved, becausesymbols are more accurately estimated and the interferencecancellation works better. With a slight abuse of notation,we do not specifically distinguish the iteration index in thispaper, as no ambiguity arises.

In order to further mitigate the effects of residual interfer-ence on rn, we can apply an instantaneous linear filter to rnto obtain zn = wH

n rn, where the vector of filter coefficientswn ∈ CNK×1 is determined by minimizing

en = E{|wHn rn − sn|

2} (9)

by the MMSE criterion. Mathematically, this process can beexpressed by

wn = [GHGDFNVnDHFNG

HHHGH+ N0I]−1

× (GHGDFN )n, (10)

where (GHGDFN )n is the nth column of the matrixGHGDFN ; the matrix Vn ∈ RNK×1 is formed by

Vn = diag{var(s1), var(sn−1), σ 2s , var(sn+1), var(sNK )]},

(11)

where σ 2s = E[|sj|2] and var(sj) = E[|sj − sj|2]. More details

and a complete description of this classic iterative algorithmcan be found in [21]–[24].

B. PROPOSED ITERATIVE DETECTION ALGORITHMAs can be seen from the description in the previous subsec-tion, an obvious drawback of the classic iterative detectionalgorithm is the error propagation caused by incorrect estima-tions, which can be alleviated by applying the widely linearprocessing (WLP) [25], [26]. TheWLP exploits the completesecond-order statistics of the received signals, and not onlyprocesses rn, but also its conjugated version r∗n to yield thefiltered output, i.e.,

zn = anrn + bnr∗n = 0Hn yn (12)

where 0n =[an bn

]H and yn =[rTn (r∗n)

T ]T . The filter0n can be designed by minimizing the MSE E{|en|2}, whereen = zn − sn = 0H

n yn − sn. According to the orthogonalityprinciple E[yne

∗n] = E[yn(0

Hn yn − sn)H] = 0, we can have

the solution:

0n = (E[ynyHn ])−1E[yns

∗n] = 9

−1yy 9ys, (13)

where 9yy and 9ys are explicitly expressed in (14) and (15)at the top of the next page.For ASK modulations, we can simply have Vn = Vn.

For M -PSK, M -QAM modulations, the matrix Vn can becalculated by

Vn = E{[sn − sn][sn − sn]T }= diag{31, . . . , 3n−1, 0,3n+1, . . . , 3N }. (16)

Denoting a complex M -PSK or M -QAM symbol sp =sp,I + jsp,Q, and sp = sp,I + jsp,Q, where sp = E[sp], thepth diagonal element of Vn can be determined by

3p = E[(sp − sp)2] = E[s2p]− (sp)2

= E[s2p,I + 2jsp,I sp,Q − s2p,Q]− (sp,I )2 − 2jsp,I sp,Q

+ (sp,Q)2

= (sp,Q)2 − (sp,I )2. (17)

Also, the soft estimate si in (8) and the variance var(si) in (11)can be given by [27]

si = E{si} =M∑m=1

smPr (si = sm);

var(si) = E[|si|2]− |E{si}|2, (18)

where E[|si|2] =∑M

m=1 |xm|2 Pr (si = xm).

The a priori probability of each symbol Pr (si) can bewritten as

Pr (si) =log2M∏p=1

Pr (bpi ), (19)

where

Pr (bpi ) =

12[1+ bpi tanh(

λ(bpi )

2)]. (20)

In what follows, we employ systems applying 4ASK andQPSK as examples to demonstrate how the log-likelihoodratio λ(bpi ) can be derived by the proposed iterative detection

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9yy = E{ynyHn } = E

{[rnr∗n

] [rHn rTn

]}=

[GHGDFNVnDH

FNGHHHGH

+ σ 2n I GHGDFN VnDT

FNGT HT GT

G∗H∗G∗D∗FN V∗

nDHFNH

HGH G∗H∗G∗D∗FNVnDTFNG

T HT GT+ σ 2

n I

](14)

9ys = E{yns∗n} = E

{[rns∗nr∗ns∗n

]}=

[(GHGDFN )n(GHGDFN )

∗n

]for M -ASK

[(GHGDFN )n

0

]for M -PSK, M -QAM

(15)

algorithm, because each symbol sn is associated with twobits b0n and b

1n. Although we fix the constellation size to four

in this paper for illustration purposes, the proposed iterativealgorithm can be easily extended to systems with an arbitraryconstellation size.

1) PROPOSED ITERATIVE DETECTION ALGORITHM FORSYSTEMS APPLYING ASKFor systems applying ASK, the nulling filter’s output can bewritten as

zn = µnsn + ηn, (21)

where the randomness of noise and residual interference canbe characterized by a Gaussian random variable ηn.Subsequently, we can derive the LLR values for b0n and b

1n

for systems applying 4ASK based on the assumption that theinterference-plus-noise term ηn at the output of a nulling filteris a non-circular random variable [28]. The parametersµn andNη can therefore be derived by

µn = E{zns∗n} = 0Hn E[yns

∗n] = 0

Hn Cyd

= 0Hn

[(GHGDFN )n(GHGDFN )

∗n

],

Nη = E{|ηn|2} = E{|zn − µnsn|2}

= E{|zn|2} − |µn|2 = µ∗n − |µn|2. (22)

The above equations hold since zn = 0Hn yn and 0n =

C−1yy Cyd . Therefore, we can further derive

E{|zn|2} = E{0Hn yny

Hn 0n} = 0

Hn Cyy0n

= CHydC−1yy Cyy0n = CH

yd0n = µ∗n. (23)

Note that, throughout the derivations of the proposed schemepresented above, we consider the non-circular nature of ηn,and employ the relation Nη = E[η2n] 6= 0.Furthermore we can determine Nη by

Nη = E[η2n] = E[(zn − µnsn)2] = E{z2n} − |µn|2

= E{0Hn yny

Tn 0∗n} − |µn|

2= 0H

n Cyy0∗n − |µn|

2. (24)

The above equation holds since0Hn y = yTn 0

∗n. Following this

rationale, we can further have

E{z2n} = E{0Hn yny

Tn 0∗n} = 0

Hn E{yny

Tn }0

∗n = 0

Hn Cyy0

∗n,

(25)

where Cyy is explicitly given in (26) at the bottom of the nextpage. Let us denote zn = zn,I + jzn,Q, µn = µn,I + jµn,Q,and ηn = ηn,I + jηn,Q. As a consequence, the filter’s outputzn = µnsn + ηn can be reformed by[

zn,Izn,Q

]︸ ︷︷ ︸

zn

=

[µn,I snµn,Qsn

]︸ ︷︷ ︸

sn

+

[ηn,Iηn,Q

]︸ ︷︷ ︸ηn

(27)

Meanwhile, since the probability distribution of a com-plex random variable or vector can be characterized bythe joint distribution of its real and imaginary part, wehave

f (zn|sn) = f (zn|sn)

=1

2π√det σ n

exp(−12(zn − sn)Hσ−1n (zn − sn)

),

(28)

where the covariance matrix of the Gaussian noise is σ n =

E[ηnηHn ]. Define the mapping matrix as J = 1

√2

[1 j1 −j

],

which is a unitary matrix as JJH = JHJ = I , and J−1 =JH [29]. We can then derive

Jσ nJH = JE[ηnηHn ]JH = E[(Jηn)(Jηn)

H]

=12E[εnεHn ] =

12φn, (29)

where εn =[ηnη∗n

]and

φn = E[εnεHn ] = E{[ηnη∗n

] [η∗n ηn

]}= E

{[ηnη∗n ηnηn

η∗nη∗n η∗nηn

]}=

[Nη NηN ∗η Nη

](30)

From (29), it can be observed that σ n = 12J

HφnJ , and σ−1n =

2JHφ−1n J . The probability density function (PDF) in (28) canthus be reformed by

f (zn|sn) =1

2π√det σ n

exp[−(zn − sn)H

× JHφ−1n J(zn − sn)]. (31)

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TABLE 1. Complexity for estimating one subcarrier by a single iteration.

Therefore, the LLR value of b0n can be determined by

λ(b0n) = lnf (zn|b0n = 1)f (zn|b0n = 0)

≈ lnexp[−(zn − s+)HJHφ−1n J(zn − s+)]

exp[−(zn − s−)HJHφ−1n J(zn − s−)]

= (zn − s−)HJHφ−1n J(zn − s−)

− (zn − s+)HJHφ−1n J(zn − s+), (32)

where s+ and s− denote the estimated signal vector deter-mined by max{f (zn|s3), f (zn|s4)} and max{f (zn|s1), f (zn|s2)},in which the real part of the symbols s3, s4 correspondsto 1, and the real part of the symbols s1, s2 correspondsto 0. Then, it is straightforward to derive λ(b1n) in a similarmanner:

λ(b1n) = lnf (zn|b1n = 1)f (zn|b1n = 0)

≈ (zn − s−)HJHφ−1n J(zn − s−)

− (zn − s+)HJHφ−1n J(zn − s+), (33)

where s+ and s− denote the estimated signal vector deter-mined by max{f (zn|s2), f (zn|s4)} and max{f (zn|s1), f (zn|s3)},in which the imaginary part of the symbols s2, s4 correspondsto 1, and the imaginary part of the symbols s1, s3 correspondsto 0.

Finally, referring to (18) and (20), we can convert LLRs tothe soft symbol estimate sn and var(sn), which are utilized forinterference cancellation in the next iteration.

2) PROPOSED ITERATIVE DETECTION ALGORITHM FORSYSTEMS APPLYING QPSK/M-QAMFor systems applying QPSK/M-QAM, the nulling filter’soutput can be expressed as

zn = µnsn + νns∗n + ηn, (34)

and the parameters µn, νn,Nη and Nη are determined by

µn = E{zns∗n}

= 0Hn E[yns

∗n] = 0

Hn Cyd = 0

Hn

[(GHGDFN )n

0

], (35)

νn = E{znsn}

= 0Hn E[ynsn] = 0

Hn Cyd = 0

Hn

[0

(GHGDFN )∗n,

], (36)

Nη = E[|ηn|2] = E[|zn − µnsn − νns∗n|2]

= E{|zn|2} − |µn|2 − |νn|2 = µ∗n − |µn|2− |νn|

2 (37)

and

Nη = E[η2] = E[(zn − µnsn − νns∗n)2] = E{z2n} − 2µnνn

= E{0Hn yny

Tn 0∗n} − 2µnνn = 0H

n Cyy0∗n − 2µnνn.

(38)

Let us denote zn = zn,I+jzn,Q, sn = sn,I+jsn,Q,µn = µn,I+jµn,Q, νn = νn,I + jνn,Q, and ηn = ηn,I + jηn,Q for simplicity.The filtered output zn = µnsn + νns∗n + ηn is reformed by[

zn,Izn,Q

]︸ ︷︷ ︸

zn

=

[(µn,I + νn,I )sn,I + (νn,Q − µn,Q)sn,Q(µn,Q + νn,Q)sn,I + (µn,I − νn,I )sn,Q

]︸ ︷︷ ︸

sn

+

[ηn,Iηn,Q

]︸ ︷︷ ︸ηn

.

(39)

Then, we can similarly refer to (32) and (33) to calculate theLLR value of b0n and b

1n for systems applying QPSK. Hence,

by converting the LLR to the complex soft symbols estimatedaccording to (18) and (20), the interference can be canceledin the next iteration.

For clarity, we present the complexity comparison amongdifferent detection algorithms in Table 1, where X = N ×nR;nR = K for simplicity; |X | denotes the constellation size.

IV. SIMULATIONSIn this section, we compare the error performance of differentiterative detection algorithms by applying them to 4× 4 and4 × 3 uncoded and coded MIMO-GFDM systems. Specifi-cally, a normalized six-path equal-power fading channel withunit average channel gain is adopted for simulations. Thefading coefficient of each path is modeled as an independentidentically distributed (i.i.d.) complex Gaussian variable. Formodulation schemes, we adopt 4ASK and QPSK, so thateither the data transmission rate or the spectrum efficiencycan be the same. Simulation parameters are listed in Table 2

Cyy = E{ynyTn } = E

{[rnr∗n

] [rTn rHn

]}=

[Crr Crr

C∗rr C∗

rr

]

=

[GHGDFN VnDT

FNGT HT GT GHGDFNVnDH

FNGHHHGH

+ σ 2n I

G∗H∗G∗D∗FNVnDTFNG

T HT GT+ σ 2

n I G∗H∗G∗D∗FN V∗

nDHFNG

HHHGH

](26)

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FIGURE 2. Performance comparison between the proposed and conventional iterative algorithms of uncoded MIMO-GFDM systems withdifferent number of iterations: (a) uncoded 4ASK 4 × 4 MIMO-GFDM systems; (b) uncoded QPSK/16QAM 4 × 4 MIMO-GFDM systems;(c) uncoded 4ASK 4 × 3 MIMO-GFDM systems; (d) uncoded QPSK/16QAM 4 × 3 MIMO-GFDM systems.

TABLE 2. Simulation parameters.

and the simulation results shown in this section are producedby averaging over at least 50,000 random trials.

Fig. 2 shows the BER performance with differentnumbers of iterations for 4 × 4 and 4 × 3 uncoded

MIMO-GFDM systems applying different modulationschemes (i.e. the conventional and proposed iterative detec-tion algorithms). As the iterative process goes on, the BERsdecrease for all three modulations (4ASK, QPSK and16QAM). Our simulations show that it takes approximatelythree iterations for the aforementioned algorithms to reachsteady states, and more iterations do not yield noticeableperformance improvement. For this reason, we carry out therest of simulations by fixing the number of iterations to three.

In Fig.3, the BER performance is simulated for two itera-tive algorithms in uncoded MIMO-GFDM systems applyingdifferent modulation schemes. Fig.3a and Fig.3b show 4 ×4 and 4× 3MIMO cases respectively. Through the numericalresults, we have the following observations:

• The proposed algorithm performs better in under-determined 4× 3 than in 4× 4 uncodedMIMO systems.

• When 4ASK is employed as the modulation scheme,the proposed iterative algorithm significantly

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FIGURE 3. Performance comparison between the proposed and conventional iterative algorithms for the 4ASK/QPSK/16QAM modulated 4 × 4and 4 × 3 MIMO-GFDM systems, given the number of iterations to be three. (a) Uncoded 4 × 4 3rd iteration. (b) Uncoded 4 × 3 3rd iteration

FIGURE 4. Performance comparison between the proposed and conventional iterative algorithms for the 4ASK/QPSK/16QAM modulated 4 × 4and 4 × 3 coded MIMO-GFDM systems, given the number of iterations to be three. (a) Coded 4 × 4 3rd iteration. (b) Coded 4 × 3 3rd iteration.

outperforms its conventional counterpart, especially inthe moderate and high SNR region in both 4 × 4 and4 × 3 MIMO-GFDM systems.

• When QPSK is employed as the modulation scheme,there is still a gap between two iterative algorithms, butnot as significant as that of the above cases.

• When 16QAM is employed as the modulation scheme,the curve regarding the proposed algorithm is similarto that of the conventional algorithm in 4 × 4 MIMOsystems, while there exists a obvious performance gapin 4 × 3 MIMO systems.

Fig. 4 demonstrates the performance comparisonof 4ASK/QPSK/16QAM modulated 4 × 4 and 4 × 3 codedMIMO-GFDM systems. The employed channel code is aconvolution code with a rate of 1/3, generator polynomial

(25,33,37) and the constraint length of five. The resultssuggest that the proposed algorithm has a comparable per-formance with the conventional counterpart in 4 × 4 MIMOsystems with an exception when 16QAM is applied, and aminor gap exists for this exceptional case. While in under-determined 4 × 3 MIMO systems, substantial enhancementyielded by the proposed algorithm is achieved for all threemodulation schemes.

V. CONCLUSIONThis paper analyzed an uplink MIMO-GFDM system with anovel iterative FDE algorithm. The simulation results showedthat when considering the complete second order descriptionof the received signal in the frequency domain, the pro-posed iterative FDE algorithm outperforms its conventional

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counterpart. The performance gain is significant, especiallyfor under-determined MIMO-GFDM systems. This specialfeature makes the proposed FDE algorithm a satisfactorycandidate for supporting under-determined communicationsin next generation networks, in which the number of devicesis expected to be greater than the number of antennas in basestations.

ACKNOWLEDGMENTThe authors would like to thank the editor and the anony-mous reviewers for their constructive comments which helpimprove the quality of this manuscript.

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JIE ZHONG received the B.Eng. and Ph.D.degrees in communication and information sys-tems from Zhejiang University, Hangzhou, China,in 2003 and 2008, respectively. He was a VisitScholar in 5GIC with the University of Surrey,Guildford, U.K. He is currently an Associate Pro-fessor with the College of Information Scienceand Electronic Engineering, Zhejiang University,China. He holds over 10 international patents onwireless communications. His research interests

include multiantenna signal processing, air interface waveform design, radioaccess networks, and massive multiple-input and multiple-output systems.

GAOJIE CHEN (S’09–M’12) received the B.Eng.and B.Ec. degrees in electrical information engi-neering and international economics and tradefrom Northwest University, China, in 2006, andthe M.Sc. (Hons.) and Ph.D. degrees in electricaland electronic engineering from LoughboroughUniversity, Loughborough, U.K., in 2008 and2012, respectively. From 2008 to 2009, he wasa Software Engineering with DTmobile, Beijing,China. From 2012 to 2013, he was a Research

Associate with the School of Electronic, Electrical and Systems Engineering,Loughborough University. He was a Research Fellow with 5GIC, Facultyof Engineering and Physical Sciences, University of Surrey, U.K., from2014 to 2015. Then, he was a Research Associate with the Departmentof Engineering Science, University of Oxford, U.K., from 2015 to 2018.He is currently a Lecturer with the Department of Engineering, Universityof Leicester, U.K. His current research interests include information the-ory, wireless communications, cooperative communications, cognitive radio,secrecy communication, and random geometric networks.

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JUQUAN MAO received the B.Eng. degree incomputer science and technology from QingdaoUniversity, China, in 2003, and the M.Sc. degreein computer science and technology from theBeijing University of Posts and Telecommunica-tions, China. He is currently pursuing the Ph.D.degree with the Institute for Communication Sys-tems, University of Surrey, U.K. He was a Techni-cal Engineer with Huawei Technologies Co., Ltd.,from 2006 to 2012. In 2012, he joined the Depart-

ment of Electrical Engineering, London South BankUniversity, as a Lecturer.Hismain areas of research interest include 5G newwaveforms, physical layernetwork slicing, and non-orthogonal multiple access.

SHUPING DANG (S’13) received the B.Eng.degree (Hons.) in electrical and electronic engi-neering from The University of Manchesterand the B.Eng. degree in electrical engineer-ing and automation from Beijing Jiaotong Uni-versity in 2014 via a joint ’2+2’ dual-degreeprogram. He is currently pursuing the D.Phil.degree with the Department of Engineering Sci-ence, University of Oxford. He was also a Certi-fied LabVIEW Associate Developer by National

Instrument from 2014 to 2016. He serves as a Reviewer for the IEEETRANSACTIONS ON WIRELESS COMMUNICATIONS, the IEEE TRANSACTIONS ON

COMMUNICATIONS, the IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, DIGITAL

SIGNAL PROCESSING, EURASIP Journal on Wireless Communications andNetworking, and a number of leading conferences in communication engi-neering. His current research interests include cooperative communications,wireless signal processing, and 5G communication system design.

PEI XIAO (SM’11) was with Newcastle Uni-versity and Queen’s University Belfast. He alsoheld positions at Nokia Networks, Finland. Heis currently a Professor of wireless communi-cations with the Institute for CommunicationSystems, Home of 5G Innovation Centre (5GIC),University of Surrey. He is also the TechnicalManager of 5GIC, leading the research team aton the new physical layer work area, and coor-dinating/supervising research activities across all

the work areas within 5GIC. He has published extensively in the fields ofcommunication theory and signal processing for wireless communications.

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