Iterative Frequency Domain Equalization for MIMO-GFDM Systems Jie Zhong, Member, IEEE, Gaojie Chen, Member, IEEE, Juquan Mao, Student Member, IEEE, Shuping Dang, Student Member, IEEE, and Pei Xiao, Senior Member, IEEE
Abstract—This paper proposes a new iterative frequency domain equalization (FDE) algorithm for multiple-input multipleoutput (MIMO)-frequency division multiplexing (GFDM) systems. This new FDE scheme is capable of enhancing the system fidelity by considering the complete frequency-domain second order description of the received signal. In addition, a new nulling filter design is also proposed for MIMO-GFDM systems to remove the residual interference, which further improves the system fidelity compared to the traditional scheme. Simulation results are presented to verify the effectiveness and efficiency of the proposed FDE algorithm. Index Terms—FDE, MIMO-GFDM systems, uplink transmission, nulling filter, 5G networks.
I. I NTRODUCTION To satisfy the communications of emerging applications, e.g., Internet of Things (IoT), ultra high-definition multimedia and virtual reality (VR), fifth generation (5G) wireless communications are faced with more stringent requirements in terms of the transmission rate, system capacity, reliability and latency [1]–[3]. On the other hand, the improvement on wireless communication systems is restricted by a series of deleterious factors in the radio propagation environment, e.g., inter-symbol interference (ISI), multi-user interference (MUI), propagation attenuation and multi-path fading [4]. These deleterious factors result in a variety of technological difficulties in the design of reliable wireless communication systems. One intuitive approach to cope with these design difficulties is to employ multi-antenna communication paradigms, which have received considerable attention in both academia and industry due to the obvious performance advantages of network capacity and quality of service (QoS) [5]. By mounting multiple antennas at transmitting and receiving terminals, a multi-input multioutput (MIMO) system is hereby constructed [6], which has been proved to be able to yield a series of performance improvements compared to single-input single-output (SISO) systems [7]. J. Zhong is with Department of Information Science and Electronic Engineering (ISEE), Zhejiang University, Hangzhou, Zhejiang, China, Email:
[email protected]. G. Chen is with the Department of Engineering, University of Leicester, Leicester, UK, Emails:
[email protected]. S. Dang is with the Department of Engineering Science, University of Oxford, Oxford, UK, Emails:
[email protected]. J. Mao and P. Xiao are with the Institute for Communication Systems (ICS), home of the 5G Innovation Centre, University of Surrey, Guildford, Surrey, UK, Emails: {juquan.mao, p.xiao}@surrey.ac.uk. This work was supported by the Engineering and Physical Sciences Research Council (EPSRC) through the New Air Interface Techniques for Future Massive Machine Communications project under Grant EP/P03456X/1.
Besides, the generalized Frequency Division Multiplexing (GFDM) was proposed for the 5G air interface, because of its low system complexity, flexible spectrum coordination, and low peak-to-average-power ratio (PAPR), which induces a higher power efficiency compared to OFDM [8]. The flexibility of GFDM facilitates its collaboration with single carrier frequency domain equalization (SC-FDE) and filtered bank multi carrier (FBMC) [9]. GFDM is based on the modulation in a per-subcarrier manner, in which each subcarrier is modulated individually and independently with multiple symbols. Then, the subcarrier is filtered by a prototype filter, which is circularly shifted in the time and frequency domains. This filtering process is aimed at reducing the out-of-band (OOB) remains, and thereby facilitating the fragmented spectrum and dynamic spectrum allocations without causing severe interference 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 filtering process. Fortunately, we can rely on well-designed receiving techniques to mitigate the interference, e.g., the matched filter receiver with the iterative interference cancellation, which has been found effective to achieve a better performance than OFDM for numerous applications [10]. More details of modulation and multiple access schemes as well as the motivation of the use of well-designed receiving techniques for next generation networks can be found in [11]. For completeness purposes, interesting readers might also refer to [12], [13] for more details of frequency-domain reception. In terms of the spatial domain, the spatial division multiplexing (SDM) systems can achieve high data transmission rate by transmitting and receiving multiple substreams in parallel through a MIMO architecture. However, the complexity raised by the optimal detection for SDM in the MIMO architecture is very high, which grows exponentially with the number of antennas at the transmitter and receiver as well as the signal constellation size. The Vertical Bell Labs Layered Space-Time (V-BLAST) detection technique can be regarded as a satisfactory solution to the performance-complexity tradeoff in detection [14], while suffers from the error propagation problem inherent in its decision feedback process. To address this issue and optimize the performance of V-BLAST detection process, in [15], the co-antenna interference (CAI) components have been replicated by the log-likelihood ratio (LLR) of the interference and the soft replica subtracted from the received composite signal vector. As a consequence, the performance has been improved by repeating this iterative process. Meanwhile, for the case where massive number of
users are considered, e.g., applications of IoT and super dense networks [16], [17], the under-determined MIMO systems should be taken into consideration [18]. Based on these considerations, it motivates us to devise a novel iterative detection algorithm, which takes advantage of the non-circular nature of the residual interference after subtracting the CAI by their soft symbol estimates, and the performance of MIMO-GFDM systems can be enhanced accordingly. In addition to this iterative detection algorithm, we also propose a new nulling fitter design based on a modified error detection criterion, which is capable of removing the residual interference. To be clear, we summarize the contributions of this paper as follows. 1) We propose a novel iterative detection algorithm for MIMO-GFDM systems based on the non-circular characteristic of residual interference. 2) We propose a novel nulling fitter design incorporating with the proposed iterative detection algorithm. 3) We derive the explicit form of the fitter output of the proposed algorithm. 4) We carry out a series of numerical simulations to confirm the superiority of our proposed algorithm over other conventional algorithms. The rest of this paper is organized as follows. We present the model of MIMO-GFDM systems in Section II. Then, the iterative FDE algorithm is proposed and detailed in Section III. After that, we provide numerical results to verify the effectiveness and efficiency of the proposed algorithm in Section 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 b is denoted by [b]n . Superscripts (·)H , (·)T , (·)∗ denote the Hermitian transpose, transpose, and conjugate, respectively. II. S YSTEM M ODEL In this paper, we mainly focus on the uplink transmission scenario, in which a cellular multiple access system has nR receive antennas at the BS and a single transmit antenna at the ith user terminal, i ∈ {1, 2, · · · , KT }, where KT is the number of users. Meanwhile. it is assumed that a multiuser MIMO system with K (K < KT ) users is employed and each user is served at an exclusive time slot, such that K = nR . The system model for a GFDM-based MIMO transmitter and receiver is shown in Fig. 1. At the transmitter, the information bits intended to be transmitted are first encoded into a coded bit sequence with a longer length, which is then mapped into constellation symbols. A set of N symbols is modulated on one subcarrier aggregated with multi-carriers, which are also called clusters in Fig. 1. In each subcarrier, a cyclic prefix (CP) is inserted to combat ISI and a data block is hereby constructed. Here, we assume the length of the CP is longer than the impulse response of the channel. The insertion of CP also helps converting linear convolution of the channel and transmitted data sequence into the circular convolution. Subsequently, the data block is interpolated and filtered by a root raised cosine (RRC) filter,
and all subcarriers are shifted to their own spectrum chunks and summed up for the radio transmission. As a result, this transmitting band consists of all occupied spectrum chunks and appears to be a consecutive M -carrier spectrum. At the receiver, each subcarrier is collected from the whole band, filtered by a RRC filter to eliminate adjacent interference. This filtering process is after CP removal, N -point frequency domain equalization and down-conversion. After that, we can apply a MIMO FDE to process the frequency-domain signals, and the design of the MIMO FDE is illustrated in Fig. 1 (b). For the receiver design, we assume that the linear minimum mean squared error (MMSE) detection scheme is employed at the receiver for simplicity. Also, the MMSE detection scheme can provide a satisfactory solution to the trade-off between noise enhancement and multi-stream interference mitigation [5]. By such a receiver design, the output of each subcarrier is combined with a symbol stream, which is then fed to a soft de-mapper to generate estimates of transmitted bits. Finally, the transmitted information can be decoded from the stream. To be specific, we introduce the notations and nomenclatures in terms of an arbitrary subcarrier for a single user as follows, which are highly realted to the proposed FED algorithm described in the next section and evolved from the former system proposed in [19]. The roll-off factor α of the RRC filter leads to additional spectrum occupation of α × N carriers. However, the in-band N -point carriers dominate the performance, and we simplify the derivation based on them. We denote DFM = IK ⊗ FM and FM is the M × M Fourier matrix with the element [FM ]m,k = exp(−j 2π M (m − 1)(k − 1)), where k, m ∈ {1, · · · , M } denote the numbers of samples and frequency tones, respectively. Here ⊗ denotes the Kronecker product, and IK is a K × K identity matrix. D−1 FM corresponds to the KM × KM inverse Fourier −1 matrix, which can be determined by IK ⊗ F−1 M and FM denotes a M × M inverse Fourier matrix with the element 1 2π [F−1 M ]m,k = M exp(j M (m − 1)(k − 1)). Similarly, DFN −1 and DFN are defined as DFM and D−1 FM with a difference represent the in the matrix size. Meanwhile, zn and z−1 n subcarrier mapping and demapping matrices, respectively, which represent the N allocated subcarriers in the M carriers, so that the dimension of zn is M × N . In the receiving side, the received signal processed by the ˜ −1 (IK N zn )GDF ˜s + w, ˜ RF module becomes ˜r = HD N FM ˜ = [˜ ˜TK ]T ∈ CKN ×1 denotes the data where s sT1 , · · · , s ˜i ∈ CN ×1 , sequences transmitted by all K users, and s i ∈ {1, · · · , K}, denotes the transmitted data block for the ith ˜ ∈ CM nR ×1 denotes a circularly user with E[s˜i s˜i H ] = I N ; w symmetric complex Gaussian noise vector with zero mean and ˜ ∼ CN (0, N0 I); covariance matrix N0 I ∈ RM nR ×M nR , i.e., w ˜ is a nR M × KM channel matrix. The RRC filter matrix H G ∈ CKN ×KN is a block diagonal matrix, in which the ith sub-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’s response in the frequency domain for the ith user on the nth subcarrier.
(a) Transmitter structure
(b) Receiver structure Fig. 1. The system model of MIMO-GFDM, where (a) denotes the transmitter structure and (b) denotes the receiver structure.
III. F REQUENCY D OMAIN E QUALIZATION
and T r = E[rsH ] = GHG.
With the MIMO FDE, the time-domain output signal can be written as [19] ˜= z =
H D−1 FN A G(IK H D−1 FN A G(IK
⊗ ⊗
z−1 r n )DFM ˜ −1 ˜ −1 zn )DFM (HD FM
˜ × (IK ⊗ zn )GDFN ˜s + w) =
H ˜ D−1 FN A G(HGDFN s
+ w) =
(1)
D−1 FN z,
where A denotes a KN × KN equalization matrix and KN ×KN ˜ −1 H = (IK ⊗ z−1 ;w∈ n )DFM HDFM (IK ⊗ zn ) ∈ C nR N ×1 C denotes a circularly symmetric complex Gaussian noise vector with zero mean and covariance matrix N0 I ∈ RnR N ×nR N . The received signal after the RRC filtering can be expressed as ˜ + Gw, r = GHGs + Gw = GHGD FN s
(2)
˜ represents the transmitted signal in the where s = D FN s frequency domain. By applying a classic linear MMSE equalizer [20], we can then perform operations of a FDE matrix A on r to yield the equalized signal z = AH r, where A can be generated by the classic approach by minimizing the cost function e = E[kz − sk2 ] = E[kAH r − sk2 ],
(3)
R−1 r Tr
which leads to the FDE matrix A = [20], and the autocorrelation and cross correlation matrices can be given by Rr = E[rr H ] = GHGGH H H GH + N0 I
(4)
(5)
A. Conventional Iterative Detection Algorithm: the Benchmark Denote the symbol vector T s = s1 , . . . , sn−1 , sn , sn+1 , . . . , sN K
(6)
Now, let us focus on the decoding of symbol sn . By applying the classic iterative interference cancellation method, the received signal vector can be written as [21] ¯n = GHGD FN [˜ ¯n ] + Gw, r n = r − GHGD FN s s−s (7) where r n is the interference-canceled version of r, and T ¯n = s¯1 , . . . , s¯n−1 , 0, s¯n+1 , . . . , s¯N K s ,
(8)
˜ which contains the soft estimates of the interfering symbols s from the previous iteration. One should note that (7) represents a decision-oriented iterative scheme, by which the detection procedure at the pth iteration utilizes the symbol estimates from the (p − 1)th iteration. As a result of this iterative procedure, the performance can be improved, because symbols are more accurately estimated and the interference cancellation works better. With a slight abuse of notation, we do not specifically distinguish the iteration index in this paper, as no ambiguity arises. In order to further mitigate the effects of residual interference on r n , we can apply an instantaneous linear filter to r n
to obtain zn = wH n r n , where the vector of filter coefficients wn ∈ CN K×1 is determined by minimizing 2 en = E{|wH n r n − sn | }
(9)
by the MMSE criterion. Mathematically, this process can be expressed by wn =[GHGD FN V
H H H H n D FN G H G
Also, the soft estimate s¯i in (8) and the variance var(si ) in (11) can be given by [27]
+ N0 I]
−1
× (GHGD FN )n ,
(10)
where (GHGD FN )n is the nth column of the matrix GHGD FN ; the matrix V n ∈ RN K×1 is formed by
s¯i = E{si } =
sm Pr (si = m=1 E[|si |2 ] − |E{si }|2 ,
σs2
2
2
where = E[|sj | ] and var(sj ) = E[|sj − s¯j | ]. More details and a complete description of this classic iterative algorithm can be found in [21]–[24].
B. Proposed Iterative Detection Algorithm As can be seen from the description in the previous subsection, an obvious drawback of the classic iterative detection algorithm is the error propagation caused by incorrect estimations, which can be alleviated by applying the widely linear processing (WLP) [25], [26]. The WLP exploits the complete second-order statistics of the received signals, and not only processes r n , but also its conjugated version r ∗n to yield the filtered output, i.e., zn = an r n + bn r ∗n = ΓH (12) n yn H T T where Γn = an bn and y n = r n (r ∗n )T . The filter Γn can be designed by minimizing the MSE E{|en |2 }, where en = zn − sn = ΓH n y n − sn . According to the orthogonality H principle E[y n e∗n ] = E[y n (ΓH n y n − sn ) ] = 0, we can have the solution: −1 Γn = (E[y n y H E[y n s∗n ] = Ψ−1 yy Ψys , n ])
(13)
where Ψyy and Ψys are explicitly expressed in (14) and (15) at the top of the next page. For ASK modulations, we can simply have V˜ n = V n . For M -PSK, M -QAM modulations, the matrix V˜ n can be calculated by ¯n ][sn − s ¯ n ]T } V˜ n = E{[sn − s = diag{Λ1 , . . . , Λn−1 , 0, Λn+1 , . . . , ΛN }. Denoting a complex sp,I + jsp,Q , and s¯p = pth diagonal element of
(16)
p=1
where 1 λ(bpi ) [1 + bpi tanh( )]. (20) 2 2 In what follows, we employ systems applying 4ASK and QPSK as examples to demonstrate how the log-likelihood ratio λ(bpi ) can be derived by the proposed iterative detection algorithm, because each symbol sn is associated with two bits b0n and b1n . Although we fix the constellation size to four in this paper for illustration purposes, the proposed iterative algorithm can be easily extended to systems with an arbitrary constellation size. 1) Proposed iterative detection algorithm for systems applying ASK: For systems applying ASK, the nulling filter’s output can be written as Pr (bpi ) =
zn = µn sn + ηn ,
M -PSK or M -QAM symbol sp = s¯p,I + j¯ sp,Q , where s¯p = E[sp ], the V˜ n can be determined by
= E[s2p,I + 2jsp,I sp,Q − s2p,Q ] − (¯ sp,I )2 − 2j¯ sp,I s¯p,Q 2
= (¯ sp,Q )2 − (¯ sp,I )2 . (17)
(21)
where the randomness of noise and residual interference can be characterized by a Gaussian random variable ηn . Subsequently, we can derive the LLR values for b0n and b1n for systems applying 4ASK based on the assumption that the interference-plus-noise term ηn at the output of a nulling filter is a non-circular random variable [28]. The parameters µn and Nη can therefore be derived by H ∗ µn = E{zn s∗n } = ΓH n E[y n sn ] = Γn C yd (GHGD FN )n = ΓH ∗ , n (GHGD FN )n
Nη = E{|ηn |2 } = E{|zn − µn sn |2 } = E{|zn |2 } − |µn |2 = µ∗n − |µn |2 .
(22)
The above equations hold since zn = ΓH n y n and Γn = C −1 yy C yd . Therefore, we can further derive H H E{|zn |2 } = E{ΓH n y n y n Γn } = Γn C yy Γn H −1 ∗ = CH yd C yy C yy Γn = C yd Γn = µn .
Λp = E[(sp − s¯p )2 ] = E[s2p ] − (¯ sp )2 + (¯ sp,Q )
sm );
var(si ) = (18) PM where E[|si |2 ] = m=1 |xm |2 Pr (si = xm ). The a priori probability of each symbol Pr (si ) can be written as log2 M Y Pr (si ) = Pr (bpi ), (19)
V n = diag{var(s1 ), var(sn−1 ), σs2 , var(sn+1 ), var(sN K )]},
(11)
M X
(23)
Note that, throughout the derivations of the proposed scheme presented above, we consider the non-circular nature of ηn , ˜η = E[ηn2 ] 6= 0. and employ the relation N ˜η by Furthermore we can determine N ˜η = E[η 2 ] = E[(zn − µn sn )2 ] = E{z 2 } − |µn |2 N n n H˜ ∗ T ∗ 2 2 = E{ΓH y y Γ } − |µ | = Γ C n yy Γn − |µn | . (24) n n n n n T ∗ The above equation holds since ΓH n y = y n Γn . Following this
Ψyy =
E{y n y H n}
=E
rn H rn r ∗n
r Tn
Ψys = E{y n s∗n } = E
r n s∗n r ∗n s∗n
=
# " (GHGD F N )n ∗ (GHGD FN )n
for M -ASK
" # (GHGD FN )n 0
for M -PSK, M -QAM
H ∗ H˜ ∗ T ∗ T E{zn2 } = E{ΓH n y n y n Γn } = Γn E{y n y n }Γn = Γn C yy Γn , (25)
˜ yy is explicitly given in (26) at the top of the next where C page. 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 output zn = µn sn + ηn can be reformed by µn,I sn ηn,I zn,I = + (27) µn,Q sn ηn,Q zn,Q | {z } | {z } | {z } sn
ηn
Meanwhile, since the probability distribution of a complex random variable or vector can be characterized by the joint distribution of its real and imaginary part, we have
f (zn |b0n = 1) f (zn |b0n = 0) exp[−(z n − s+ )H J H φ−1 n J (z n − s+ )] ≈ ln exp[−(z n − s− )H J H φ−1 n J (z n − s− )] H H −1 = (z n − s− ) J φn J (z n − s− )
λ(b0n ) = ln
− (z n − s+ )H J H φ−1 n J (z n − s+ ),
where the covariance matrix of the Gaussian noiseis σ n = 1 j 1 H √ E[η n η n ]. Define the mapping matrix as J = 2 , 1 −j H H −1 which is a unitary matrix as J J = J J = I, and J = J H [29]. We can then derive H J σ n J H = J E[η n η H = E[(J η n )(J η n )H ] n ]J 1 1 = E[n H (29) n ] = φn , 2 2 η where n = n∗ and ηn ηn ∗ η η φn = E[n H ] = E n n n ηn∗ ˜η Nη N ηn ηn∗ ηn ηn =E = ˜η∗ Nη (30) ηn∗ ηn∗ ηn∗ ηn N
From (29), it can be observed that σ n = 21 J H φn J , and σ −1 = 2J H φ−1 n n J . The probability density function (PDF) in (28) can thus be reformed by 1 √ exp[−(z n − sn )H 2π det σ n × J H φ−1 n J (z n − sn )].
(32)
where s+ and s− denote the estimated signal vector determined 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 corresponds to 1, and the real part of the symbols s1 , s2 corresponds to 0. Then, it is straightforward to derive λ(b1n ) in a similar manner: f (zn |b1n = 1) f (zn |b1n = 0) ˜− )H J H φ−1 ˜− ) ≈ (z n − s n J (z n − s
f (zn |sn ) = f (z n |sn ) 1 1 H −1 √ exp − (z n − sn ) σ n (z n − sn ) , = 2 2π det σ n (28)
f (zn |sn ) =
(15)
Therefore, the LLR value of b0n can be determined by
rationale, we can further have
zn
GHGD FN V˜ n D TFN GT H T GT G∗ H ∗ G∗ D ∗FN V n D TFN GT H T GT + σn2 I (14)
H H H 2 GHGD FN V n D H FN G H G + σn I = ∗ ∗ ∗ ∗ ∗ H H H G H G D FN V˜ n D FN H G
λ(b1n ) = ln
˜+ ), ˜+ )H J H φ−1 − (z n − s n J (z n − s
(33)
˜+ and s ˜− denote the estimated signal where s vector determined 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 corresponds to 1, and the imaginary part of the symbols s1 , s3 corresponds to 0. Finally, referring to (18) and (20), we can convert LLRs to the soft symbol estimate s¯n and var(sn ), which are utilized for interference cancellation in the next iteration. 2) Proposed iterative detection algorithm for systems applying QPSK/M-QAM: For systems applying QPSK/M-QAM, the nulling filter’s output can be expressed as zn = µn sn + νn s∗n + ηn ,
(34)
˜η are determined by and the parameters µn , νn , Nη and N µn =
E{zn s∗n }
=
∗ ΓH n E[y n sn ]
=
ΓH n C yd
H˜ νn = E{zn sn } = ΓH n E[y n sn ] = Γn C yd
(GHGD FN )n = , 0 (35) 0 = ΓH , ∗ n (GHGD FN )n , (36) ΓH n
Nη = E[|ηn |2 ] = E[|zn − µn sn − νn s∗n |2 ] (31)
= E{|zn |2 } − |µn |2 − |νn |2 = µ∗n − |µn |2 − |νn |2 (37)
˜ rr C rn T T H ˜ C yy = E{y n y n } = E = rn rn r ∗n C ∗rr GHGD FN V˜ n D TFN GT H T GT = ∗ G H ∗ G∗ D ∗FN V n D TFN GT H T GT + σn2 I
C rr ˜ ∗rr C
H H H 2 GHGD FN V n D H FN G H G + σn I ∗ ∗ ∗ ∗ ∗ H H H H ˜ G H G D FN V n D FN G H G
(26)
TABLE I C OMPLEXITY FOR ESTIMATING ONE SUBCARRIER BY A SINGLE ITERATION Operations
×/÷
+/-
Conventional Proposed
3X 3 + 6X 2 + 2X + 2|X | 3 18X + 16X 2 + 8X + 2|X |2 + 6|X | + 28
3X 3 + 2X 2 + X + |X | + log2 |X | − 2 3 18X + 4X 2 + 6X + |X |2 + 2|X | + log2 |X | + 13
-1
10
10-2
10-2
BER
BER
-1
10
10-3 conv. 1st iter. conv. 2nd iter. conv. 3rd iter. impr. 1st iter. impr. 2nd iter. impr. 3rd iter.
10-4
10-5 0
5
10
16QAM conv. 1st iter. 16QAM conv. 2nd iter. 16QAM conv. 3rd iter. 16QAM impr. 1st iter. 16QAM impr. 2nd iter. 16QAM impr. 3rd iter. QPSK conv. 1st iter. QPSK conv. 2nd iter. QPSK conv. 3rd iter. QPSK impr. 1st iter. QPSK impr. 2nd iter. QPSK impr. 3rd iter.
10-3
10-4
10-5 15
20
25
30
35
0
5
10
Eb/No (dB)
10-1
10-1
10-2
10-2
10-3 conv. 1st iter. conv. 2nd iter. conv. 3rd iter. impr. 1st iter. impr. 2nd iter. impr. 3rd iter.
10-5 0
5
10
10-3
10-5 20
25
30
35
30
35
16QAM conv. 1st iter. 16QAM conv. 2nd iter. 16QAM conv. 3rd iter. 16QAM impr. 1st iter. 16QAM impr. 2nd iter. 16QAM impr. 3rd iter. QPSK conv. 1st iter. QPSK conv. 2nd iter. QPSK conv. 3rd iter. QPSK impr. 1st iter. QPSK impr. 2nd iter. QPSK impr. 3rd iter.
10-4
15
20
(b) Uncoded QPSK/16QAM 4 × 4
BER
BER
(a) Uncoded 4ASK 4 × 4
10-4
15
Eb/No (dB)
25
30
35
Eb/No (dB) (c) Uncoded 4ASK 4 × 3
0
5
10
15
20
25
Eb/No (dB) (d) Uncoded QPSK/16QAM 4 × 3
Fig. 2. Performance comparison between the proposed and conventional iterative algorithms of uncoded MIMO-GFDM systems with different 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.
and ˜η = E[η ] = E[(zn − µn sn − νn s∗n )2 ] = E{zn2 } − 2µn νn N H˜ ∗ T ∗ = E{ΓH n y n y n Γn } − 2µn νn = Γn C yy Γn − 2µn νn . (38) 2
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 = µn sn + νn s∗n + ηn is
-1
10
-1
-2
10
10
-2
BER
BER
10
10-3 16QAM conv. 16QAM impr. QPSK conv. QPSK impr. 4ASK conv. 4ASK impr.
10-4
10-5 0
5
10
10-3 16QAM conv. 16QAM impr. QPSK conv. QPSK impr. 4ASK conv. 4ASK impr.
10-4
10-5 15
20
25
30
35
0
5
10
Eb/No (dB)
15
20
25
30
35
Eb/No (dB)
(a) Uncoded 4 × 4 3rd iteration
(b) Uncoded 4 × 3 3rd iteration
10-1
10-1
10-2
10-2
BER
BER
Fig. 3. Performance comparison between the proposed and conventional iterative algorithms for the 4ASK/QPSK/16QAM modulated 4 × 4 and 4 × 3 MIMO-GFDM systems, given the number of iterations to be three.
10-3 16QAM conv. 16QAM impr. QPSK conv. QPSK impr. 4ASK conv. 4ASK impr.
10-4
10-5 0
5
10
10-3 16QAM conv. 16QAM impr. QPSK conv. QPSK impr. 4ASK conv. 4ASK impr.
10-4
10-5 15
20
25
30
35
0
Eb/No (dB)
5
10
15
20
25
30
35
Eb/No (dB)
(a) Coded 4 × 4 3rd iteration
(b) Coded 4 × 3 3rd iteration
Fig. 4. Performance comparison between the proposed and conventional iterative algorithms for the 4ASK/QPSK/16QAM modulated 4 × 4 and 4 × 3 coded MIMO-GFDM systems, given the number of iterations to be three.
TABLE II S IMULATION PARAMETERS Parameters
Value
Antennas Mapping Transmitter filter Receiver filter Cyclic prefix Subcarriers Symbols per subcarrier Gap between adjacent carriers Convolutional coding rate Generator polynomial of Conv. code Simulation loops Total simulation symbols
4 TX and 4/3 RX QPSK/4ASK /16QAM RRC with α = 0.1 RRC with α = 0.1 8 symbols 2 12 ∝ × Subcarrier BW 1/3 (25,33,37) 50000 1200000
reformed by zn,I (µn,I + νn,I )sn,I + (νn,Q − µn,Q )sn,Q ηn,I = + . zn,Q (µn,Q + νn,Q )sn,I + (µn,I − νn,I )sn,Q ηn,Q | {z } | {z } | {z } zn
sn
ηn
(39) Then, we can similarly refer to (32) and (33) to calculate the LLR value of b0n and b1n for systems applying QPSK. Hence, by converting the LLR to the complex soft symbols estimated according to (18) and (20), the interference can be canceled in the next iteration.
For clarity, we present the complexity comparison among different detection algorithms in Table I, where X = N × nR ; nR = K for simplicity; |X | denotes the constellation size.
IV. S IMULATIONS In this section, we compare the error performance of different iterative detection algorithms by applying them to 4 × 4 and 4 × 3 uncoded and coded MIMO-GFDM systems. Specifically, a normalized six-path equal-power fading channel with unit average channel gain is adopted for simulations. The fading coefficient of each path is modeled as an independent identically distributed (i.i.d.) complex Gaussian variable. For modulation schemes, we adopt 4ASK and QPSK, so that either the data transmission rate or the spectrum efficiency can be the same. Simulation parameters are listed in Table II and the simulation results shown in this section are produced by averaging over at least 50,000 random trials. Fig. 2 shows the BER performance with different numbers of iterations for 4 × 4 and 4 × 3 uncoded MIMO-GFDM systems applying different modulation schemes (i.e. the conventional and proposed iterative detection algorithms). As the iterative process goes on, the BERs decrease for all three modulations (4ASK, QPSK and 16QAM). Our simulations show that it takes approximately three iterations for the aforementioned algorithms to reach steady states, and more iterations do not yield noticeable performance improvement. For this reason, we carry out the rest of simulations by fixing the number of iterations to three. In Fig.3, the BER performance is simulated for two iterative algorithms in uncoded MIMO-GFDM systems applying different modulation schemes. Fig.3a and Fig.3b show 4 × 4 and 4 × 3 MIMO cases respectively. Through the numerical results, we have the following observations: • •
•
•
The proposed algorithm performs better in underdetermined 4 × 3 than in 4 × 4 uncoded MIMO systems. When 4ASK is employed as the modulation scheme, the proposed iterative algorithm significantly outperforms its conventional counterpart, especially in the moderate and high SNR region in both 4 × 4 and 4 × 3 MIMO-GFDM systems. When QPSK is employed as the modulation scheme, there is still a gap between two iterative algorithms, but not as significant as that of the above cases. When 16QAM is employed as the modulation scheme, the curve regarding the proposed algorithm is similar to that of the conventional algorithm in 4 × 4 MIMO systems, while there exists a obvious performance gap in 4 × 3 MIMO systems.
Fig. 4 demonstrates the performance comparison of 4ASK/QPSK/16QAM modulated 4×4 and 4×3 coded MIMOGFDM systems. The employed channel code is a convolution code with a rate of 1/3, generator polynomial (25,33,37) and the constraint length of five. The results suggest that the proposed algorithm has a comparable performance with the conventional counterpart in 4 × 4 MIMO systems with an exception when 16QAM is applied, and a minor gap exists for this exceptional case. While in under-determined 4 × 3 MIMO systems, substantial enhancement yielded by the proposed algorithm is achieved for all three modulation schemes.
V. C ONCLUSION This paper analyzed an uplink MIMO-GFDM system with a novel iterative FDE algorithm. The simulation results showed that when considering the complete second order description of the received signal in the frequency domain, the proposed iterative FDE algorithm outperforms its conventional counterpart. The performance gain is significant, especially for under-determined MIMO-GFDM systems. This special feature makes the proposed FDE algorithm a satisfactory candidate for supporting under-determined communications in next generation networks, in which the number of devices is expected to be greater than the number of antennas in base stations. ACKNOWLEDGMENT The authors would like to thank the editor and the anonymous reviewers for their constructive comments which help improve the quality of this manuscript. R EFERENCES [1] J. G. Andrews, S. Buzzi, W. Choi, S. V. Hanly, A. Lozano, A. C. K. Soong, and J. C. Zhang, “What will 5G be?” IEEE Journal on Selected Areas in Communications, vol. 32, no. 6, pp. 1065–1082, Jun. 2014. [2] X. Ge, H. Cheng, M. Guizani, and T. Han, “5G wireless backhaul networks: challenges and research advances,” IEEE Network, vol. 28, no. 6, pp. 6–11, Nov. 2014. [3] A. Gupta and R. K. Jha, “A survey of 5G network: architecture and emerging technologies,” IEEE Access, vol. 3, pp. 1206–1232, 2015. [4] F. Khan, “LTE for 4G mobile broadband: air inter technologies and performance,” Cambridge University Press, 2009. [5] A. Paulraj, R. Nabar, and D. Gore, “Introduction to space-time wireless communications,” Cambridge University Press, 2003. [6] W. Jie, J. Zhong, Y. Cai, and M. Zhao, “New detection algorithms based on the jointly gaussian approach and successive interference cancelation for iterative MIMO systems,” International Journal of Communication Systems, vol. 27, no. 10, pp. 1964–1983, Oct. 2012. [7] D. Tse and P. Viswanath, Fundamentals of Wireless Communication, ser. Wiley series in telecommunications. Cambridge University Press, 2005. [8] G. Fettweis, M. Krondorf, and S. Bittner, “GFDM-generalized frequency division multiplexing,” in Proc. 69th IEEE VTC Spring, Barcelona, Spain, Apr. 2009. [9] D. Falconer, S. Ariyavisitakul, A. Benyamin-Seeyar, and B. Eidson, “Frequency domain equalization for single-carrier broadband wireless systems,” IEEE Commun. Mag., vol. 40, no. 4, pp. 58–66, Apr. 2002. [10] R. Datta, N. Michailow, M. Lentmaier, and G. Fettweis, “GFDM interference cancellation for flexible cognitive radio PHY design,” in Proc. 76th IEEE VTC Fall, Qubec City, QC, Canada, Sep. 2012. [11] Y. Cai, Z. Qin, F. Cui, G. Y. Li, and J. A. McCann, “Modulation and multiple access for 5G networks,” IEEE Communications Surveys Tutorials, vol. PP, no. 99, pp. 1–1, 2017. [12] W. Yuan, N. Wu, H. Wang, and J. Kuang, “Variational inferencebased frequency-domain equalization for faster-than-Nyquist signaling in doubly selective channels,” IEEE Signal Processing Letters, vol. 23, no. 9, pp. 1270–1274, Sept. 2016. [13] Q. Shi, N. Wu, X. Ma, and H. Wang, “Frequency-domain joint channel estimation and decoding for faster-than-Nyquist signaling,” IEEE Transactions on Communications, vol. PP, no. 99, pp. 1–1, 2017. [14] G. Golden, G. Foschini, R. Valenzuela, and P. Wolniansky, “Detection algorithm and initial laboratory results using the V-BLAST space-time communication architecture,” IET Electronic Letters, vol. 35, no. 1, pp. 14–16, Jan. 1999. [15] T. Ito, X. Wang, Y. Kakura, M. Madihian, and A. Ushirokawa, “Performance comparison of MF and MMSE combined iterative soft interference canceller and V-BLAST technique in MIMO/OFDM systems,” in Proc. IEEE VTC Fall, Oct. 2003. [16] S. Dang, J. P. Coon, and D. E. Simmons, “Combined bulk and per-tone relay selection in super dense wireless networks,” in Proc. IEEE ICCW, London, UK, Jun. 2015.
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