Low Complexity ZF Receiver Design for Multi-User GFDMA Uplink

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Low Complexity ZF Receiver Design for Multi-User GFDMA Uplink Systems Hong Wang, and Rongfang Song

Abstract—This paper presents a low complexity receiver for multi-user uplink transmission using generalized frequency division multiplexing (GFDM). By exploiting the frequency spreading filter in the GFDM, the banded and circulant properties of the frequency transformed modulation matrices of individual users are developed. According to the newly developed properties in the frequency domain, a novel low complexity zero-forcing (ZF) receiver is developed for multi-user GFDM uplink systems in the frequency selective channels. The proposed receiver can be realized by successive discrete Fourier transforms (DFTs) and one-tap equalization, avoiding the huge computation caused by the inverse of the large dimensional equivalent system matrix. Simulation results show that the proposed receiver provides nearly the same symbol error rate as the traditional ZF equalizer with substantially reduced computational complexity. Index Terms—Low complexity, ZF receiver, uplink, GFDM.

I. I NTRODUCTION Since orthogonal frequency division multiplexing (OFDM) is susceptible to timing and frequency synchronization errors and has large out-of-band (OOB) radiation, it is of utmost importance to explore alternative modulation methods to meet the foreseeable application requirements, such as massive machine type communications [1], [2]. Recently, non-orthogonal generalized frequency division multiplexing (GFDM) has been proposed as a promising multicarrier modulation scheme, because it has the advantages of low OOB emission and robustness to asynchronous transmissions [3], [4]. The non-orthogonality of the prototype filter pulse of GFDM causes self-interference [5], [6]. Thus, the GFDM receiver requires a more complicated equalizer than the OFDM system in multipath fading channels. To tackle the self-interference in detection, three effective receivers, namely, match filter (MF), zero-forcing (ZF), and minimum mean square error (MMSE) receivers, were presented in [7], [8]. Since the computational costs of the direct implementation of MF, ZF, and MMSE This work was supported in part by Natural Science Foundation of Jiangsu Province under Grant BK20170910, in part by Open Research Foundation of National Mobile Communications Research Laboratory, Southeast University, under Grant 2018D09, in part by NUPTSF under Grant NY217005 and Grant NY217031, in part by Natural Science Foundation of Jiangsu Higher Education Institutions under Grant 16KJB510035, and in part by Open Research Fund of Jiangsu Engineering Research Center of Communication and Network Technology of NJUPT. H. Wang is with School of Communication and Information Engineering, Nanjing University of Posts and Telecommunications, Nanjing 210003, China, and also with National Mobile Communications Research Laboratory, Southeast University, Nanjing 210096, China (e-mail: [email protected]). R. Song is with School of Communication and Information Engineering, Nanjing University of Posts and Telecommunications, Nanjing 210003, China, and also with Jiangsu Engineering Research Center of Communication and Network Technology, Nanjing University of Posts and Telecommunications, Nanjing 210003, China (e-mail:[email protected]).

receivers are high, a couple of low complexity receivers for GFDM were investigated in [9]–[11]. In [9], a low complexity MF receiver was proposed by exploiting interference cancellation in sparse frenquency domain. However, the selfinterference cannot be removed completely. An algorithm with reduced computations to calculate the coefficients of MMSE receiver was reported in [10]. However, the authors in [11] argued that the separation of the calculations of receiver coefficients and signal detections is inefficient in the total computations. By applying the block circulant property of the Gram matrix of the modulation matrix, an efficient implementation for MF and ZF receiver in multipath fading channels was developed in [11]. However, the solutions in the aforementioned works are only applicable to a single user using all the subcarriers but not to the uplink with multiple users. Generalized frequency division multiple access (GFDMA) has the advantage of scheduling multiple users at the same time with flexible subcarrier assignments [12], [13]. However, the receiver schemes developed in the literature cannot be extended to GFDMA uplink systems because the uplink multipath fading channels from users to the receiving node and the numbers of subcarriers adopted by users may be very different. It is worth mentioning that it is questionable to extend single user GFDM receiver to multi-user GFDMA systems straightforwardly in [15] except the case that all channel matrices are same. In this paper, based on the block circulant and banded structure of the frequency transformed modulation matrix, a novel low complexity ZF receiver is proposed in multipath fading channels. The received signals can be detected with the aid of discrete Fourier transform (DFT) operations and onetap equalization. Simulation results show that the proposed scheme provides nearly the same symbol error rate (SER) as the traditional ZF receiver with much lower computational complexity. To the best of our knowledge, this novel receiver scheme is firstly developed in multi-user GFDMA uplink scenarios. The remainder of this paper is organized as follows. Section II describes GFDM signals and the traditional ZF receiver, respectively. In Section III, new design of receiver with low complexity is developed. Simulation results are presented in Section IV. Section V gives the conclusion. Notations: Throughout this paper, uppercase and lowercase boldface letters denote matrices and vectors, respectively. X−1 and XH stand for the inverse and Hermitian transpose of matrix X, respectively. x(i), X[i] and [X]r,c denote the ith element of vector x, ith column of matrix X, and (r, c)th element

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of matrix X, respectively. |B| denotes the cardinality of set B. CM ×N is the set of complex matrices with M rows and N columns. IN and 0M ×N are N ×N identity matrix and M ×N zero matrix, respectively. Q(τ ) is the DFT matrix with}the { √ [ ] . (r, c)th element Q(τ ) r,c = 1/τ exp −j 2π(r−1)(c−1) τ II. GFDM T RANSMISSIONS AND THE T RADITIONAL ZF R ECEIVER A. GFDM Uplink Transmissions A GFDM block with N subcarriers and M time slots is considered. K users are considered for uplink transmissions; the kth user has |Bk | subcarriers with subcarrier index set Bk = {k1 , ..., k|Bk | }. The subcarrier index sets {Bk , k = 1, ..., K} are orthogonal with contiguous subcarriers and have a guard subcarrier between two adjacent sets. A similar model has also been adopted in other literatures, such as [12]. In order to suppress the out-of-band emissions, a frequency spreading filter is used to spread the GFDM signal [16]. For GFDM modulation using a prototype filter sequence {p[t], t = 0, 1, . . . , N M − 1}, the transmitted symbol of user k at time t can be expressed as xk [t] =

−1 ∑ M ∑

pn,m [t]sk,n,m , t = 0, ..., N M − 1,

B. Traditional ZF Receiver In order to detect user signals, a traditional ZF receiver is discussed in this subsection. Expressing (5) in a compact form, we have y = Heq s + n,

(6)

where Heq = [H1 P1 , ..., HK PK ] is the equivalent system matrix, and s = [sT1 , ..., sTK ]T . At the receiver, an equalizer R is used to decorrelate the received signal and its output is ˆs = RHeq s + Rn.

(7)

A ZF equalizer is considered in this paper because of its easy implementation and near optimal performance at high SNR. The ZF equalizer for (7), referred to as traditional ZF equalizer, is −1 H R = (HH Heq . eq Heq )

(8)

The multiplication and inverse of the large dimensional matrices in (8) cause a heavy computation burden. To facilitate the design of a low complexity ZF receiver, we develop two propositions in the next section.

(1)

n∈Bk m=0

III. L OW C OMPLEXITY R ECEIVER D ESIGN

where sk,n,m denotes the data of user k transmitted on the nth subcarrier in the mth time slot, pn,m [t] is the coefficient associated with sk,n,m for output time t given as pn,m [t] = p[(t − mN )N M ] exp{j2πtn/N },

(2)

where (·)N M is the congruent modulo operation with modulus N M . Then, the GFDM signal of user k is represented as

In this section, based on the derived properties of modulation matrix in the frequency domain, a ZF receiver with low complexity is designed. Furthermore, the computational complexity analysis of the traditional ZF and proposed ZF receivers is presented.

(3)

A. New Properties of Modulation Matrix in the Frequency Domain

where the transmitted signal vector sk = [sTk,1 , ..., sTk,|Bk | ]T with sk,n = [sk,kn ,0 , ..., sk,kn ,M −1 ]T , and Pk is expressed as

The matrix G in the frequency domain can be expressed as

x k = P k sk ,

Pk = [Ek1 G, Ek2 G, ..., Ek|Bk | G], (4) [ ] N M ×M where G = g, g((N )) , ..., g(((M −1)N )) ∈ C with T g = [p[0], p[1], ..., p[N M − 1]] , g((mN )) is a circulant shift version of g with mN elements downwards, and Ei = 2πi(N M −1) diag{1, exp{j 2πi }}. N }, ..., exp{j N The transmission channel from user k to the receiver is a multipath channel with channel impulse response (CIR) vector [hk,0 , ..., hk,L ]T , where L is the order of the CIR, and {hk,l } are independent complex Gaussian random variables (r.v.s) with zero mean. In this paper, it is assumed that the CIR remains unchanged in one GFDM block. A cyclic prefix (CP) is inserted to the GFDM block at the transmitter and removed at the receiver. At the receiver, after the removal of the CP, the received signal can be expressed as ∑K y= Hk Pk sk + n, (5) k=1

where Hk is a circulant CIR matrix from user k to the receiver with the first column [hk,0 , ..., hk,L , 0, ..., 0]T ∈ CN M ×1 , n is the noise vector whose entries are independent and identically distributed (i.i.d.) complex Gaussian r.v.s with zero mean and variance σv2 .

˜ = Q(N M ) G, G

(9)

˜ is the DFT of the prototype Accordingly, the first column of G filter g given as g ˜ = Q(N M ) g. ˜ and the frequency transformed The properties of the matrix G ˜ modulation matrix Pk are respectively described in the following two propositions. ˜ can be partitioned into N subProposition 1. The matrix G T ˜ ˜ ˜ T ]T with G ˜ 1 and blocks as G = [G1 , 0M ×M , ..., 0M ×M , G N ˜ GN expressed as √ ˜ 1 = M Γ1 Q(M ) , (10) G √ (M ) ˜ GN = M ΓN Q , (11) respectively, where Γ1 = diag{˜ g(0), ..., g ˜(M − 1)}, ΓN = diag{˜ g(N M − M ), ..., g ˜(N M − 1)}. Proof: The practical prototype filters, such as raised cosine (RC) and root RC with roll-off factor α ≤ 1 [17], have at most 2M nonzero frequency components. In this paper, the prototype filter is a RC function. Since these nonzero

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frequency components are located at the frequency indices of [0, M − 1] and [(N − 1)M, N M − 1], we have g ˜(n) = 0, for M ≤ n < (N − 1)M.

(12)

B. New Receiver Design Procedures 1) Transform the received signal into frequency domain: Since Hk is a circulant square matrix, it can be diagonalized by the discrete Fourier transform matrix Q(N M ) as

Using the cyclic shift property of DFT, we have △

g ˜l = Q

(N M )

g((lN )) = g ˜ ⊙ wl ,

Q(N M ) Hk Q(N M )H = Λk , (13)

2πl(N M −1) T where wl = [exp{−j 2πl0 }] , and ⊙ M }, ..., exp{−j M is the Hadamard product operator. Substituting (13) into (9), we have

˜ = [˜ G g, g ˜1 , ..., g ˜M −1 ] = diag{˜ g}W,

(14)

where the (r, c)th element of W ∈ CN M ×M is [W]r,c = exp{−j 2π(r−1)(c−1) }. Based on (12) and (14), Proposition 1 M is proved. Next, we study the property of the frequency transformed modulation matrix of user k. Define the modulation matrix of user k in the frequency domain as ˜ k = Q(N M ) Pk (4) P = [Q(N M ) Ek1 G, ..., Q(N M ) Ek|Bk | G]. (15) Proposition 2. For the subcarrier assignment scheme men˜ k can be expressed as tioned in Section II, the matrix P ˜ k = [0|B |M ×(k −1)M , DTk , 0|B |M ×(N −|B |−k )M ]T , (16) P 1 1 k k k (|Bk |+1)M ×|Bk |M

where the tall matrix Dk ∈ C  ˜ GN 0  ˜1 G ˜N  G   .. Dk =  0 .   . . ..  .. 0 ···

··· .. . .. .

0

˜1 G 0

˜N G ˜1 G

0 .. .

is given as      .   

The (r, c)th element of Q Eki Q [ ] Q(N M ) Eki Q(N M )H =(Q

(N M )

T

r,c (N M )

˜ k. Hk Pk = Q(N M )H Λk P

k=1

where n ˜ = Q(N M ) n. Unlike single user GFDM systems, the transformed channels Λk ’s (in the frequency domain) cannot be equalized simultaneously in multi-user GFDMA systems ′ ′ because Λ−1 k Λk ̸= IN M for k ̸= k. 2) Channel equalization for each user: Based on the struc˜ k in (16), the received signal corresponding to user ture of P k can be extracted from y ˜ as [ ]T y ˜k = y ˜(k1 M − M + 1), ..., y ˜(k|Bk | M ) (23) ˜ = Λk Dk sk + n ˜k , (24) ˜ k is a submatrix of Λk expressed as where Λ ˜ k = diag{[Λk ](k −1)M +1,(k −1)M +1 , ..., [Λk ]k M,k M }. Λ 1 1 |Bk | |Bk |

˜ −1 y y ¯k = Λ k ˜k

(25)

= Dk sk + n ¯k ,

(26)

˜ −1 n where n ¯k = Λ k ˜k . 3) Circulant construction: Cutting off the last M elements from the vector y ¯k and adding them to the first M elements of y ¯k yield a new vector as (1)

(2)

zk = y ¯k + y ¯k ,

is expressed as

(27)

where (1)

T

y ¯k = [¯ yk (1), ..., y ¯k (|Bk |M )] , [ ]T (2) y ¯k = y ¯k (|Bk |M + 1), ..., y ¯k (|Bk |M + M ), 01×(|Bk |M −M ) .

∗

[r]) Eki (Q [c]) } { ∑N M −1 i M −c) exp −j 2πn(r−k NM

= N1M n=0 { 1, c = (r − ki M )N M , r = 1, ..., N M = , 0, otherwise

(21)

Substituting (21) into (5), the received signal in the frequency domain can be expressed as ∑K ˜ k sk + n y ˜ = Q(N M ) y = Λk P ˜, (22)

(17)

˜ for ki ∈ Bk . (18) Q(N M ) Eki G = Q(N M ) Eki Q(N M )H G, (N M )H

where Λk is∑the diagonal matrix with the nth diagonal element L [Λk ]n,n = l=0 hk,l exp{−j 2πnl N M }. Accordingly, we have

Using one-tap equalization, the equalizer output is given as

Proof: Firstly, the ith block in (15) can be expressed as

(N M )

(20)

Substituting (26) into (27) gives ˜ k sk + vk , zk = D

(19)

where Q(N M ) [r] denotes the rth column of Q(N M ) . Thus, Q(N M ) Eki Q(N M )H is a circular shift matrix to shift the elements of a column vector downwards by ki M elements. ˜ in Proposition 1, the Then, based on the expression of G ˜ expression of Pk in (16) can be obtained. Based on two propositions above, we will propose a new ZF receiver design with low complexity in the next subsection.

(28)

where the ith entry of noise vector vk is { ¯ k (i) + n ¯ k (|Bk |M + i), i = 1, ..., M n vk (i) = . (29) ¯ k (i), n otherwise ˜ k is a block Based on the structure of Dk in (17), D circulant equivalent channel matrix with the first block column ˜T ,G ˜ T1 , 0M ×(|B |−2)M ]T for user k. [G N

k

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4) Equalizing the new block circulant matrix: Recalling the ˜ k , we can get block circulant property of the matrix D ˜ TH k Dk Tk = diag{Bk,0 , ..., Bk,|Bk |−1 }

(30)

with Tk = Q(|Bk |) ⊗ IM , Bk,i

(31)

{ } ˜N + G ˜ 1 exp −j 2πi =G |Bk |

(32)

= Σk,i Q(M ) ,

(33)

where Σk,i is a diagonal matrix and can be obtained by substituting (10)-(11) into (32) expressed as } { √ √ 2πi Γ1 , (34) Σk,i = M ΓN + M exp −j |B k| where Γ1 and ΓN are given in the line below (11). Thus, the post-processed signal in (28) can be reexpressed as zk = Tk Σk Uk sk + vk ,

(35)

where Σk = diag{Σk,0 , ..., Σk,|Bk |−1 }, and Uk = diag{Q(M ) , ..., Q(M ) }TH k . Using the ZF equalization, the signal can be detected by −1 H ˆ sk = UH k Σk Tk zk ,

(36)

H where both TH k and Uk can be implemented by a series of DFT (or IDFT) operations, and Σ−1 k is a one-tap equalizer. 5) Implementation summary: The implementation procedures of the proposed scheme can be summarized as: • Step 1: Transform the received signal into the frequency domain by (22); • Step 2: Extract individual user signal by (23) and equalize channel effects by (25); • Step 3: Construct block-circulant structure by (27); • Step 4: Equalize the block circulant matrix in the frequency domain and detect the signal by (36). The block diagram of the proposed receiver is presented in Fig. 1.

Step 1 CP removal

DFT/IDFT

DFT

One-tap equalization

Step 2 Extract signal for each user

IDFT

Step 4

Fig. 1.

One-tap equalization

Circulant construction Step 3

Block diagram of the proposed ZF receiver.

C. Complexity Analysis The computational complexity is composed of the calculation of equalizer coefficients and the detection of the received signals. The complexity analysis is based on the number of complex multiplications (CMs). The complexity of the proposed ZF receiver is shown as follows. In step 1, the

N M -point DFT operations involve N 2 M 2 CMs1 . In step 2, channel equalization requires N M CMs and the calculation of diagonal elements in {Λk } requires N M (L+1) CMs. In Step 4, the multiplications with TH by M times k can be realized ∑K 2 |Bk |-point IDFT operations, which needs M k=1 |Bk | CMs. ∑K −1 The multiplication of Σk involves k=1 |Bk | CMs. Since the calculations of the coefficients in Σ−1 only involve the k prototype filter, they can be pre-computed off-line and saved for use in equalization. The multiplication with UH k takes |Bk | times M -point IDFT ∑ operations and M times |Bk |-point DFT K operations requiring k=1 (M 2 |Bk | + M |Bk |2 ) CMs. Thus, the total CM computations of the proposed ZF receiver are ∑K 2 CPZF = N M (N M + L + 2) + 2M |Bk | + J + M 2 J. k=1 (37) ∑K where J = k=1 |Bk |. The computational complexity of the traditional ZF receiver is analyzed as follows. The construction of Heq needs N M 2 (L + 1)J CMs. The multiplication of 3 2 H HH eq Heq involves N M J /2 CMs. The inverse of Heq Heq 3 3 H includes about M J CMs. The multiplication of Heq on the right hand requires N M 3 J 2 CMs. At last, the detection of the received signal needs N M 2 J CMs. Thus, the total number of CMs of the traditional ZF receiver is CTZF =N M 2 (L + 2)J + 3N M 3 J 2 /2 + M 3 J 3 .

(38)

IV. S IMULATION R ESULTS In the simulation, a RC filter with roll-off factor α is used. The CIR {hk,l } are independent complex Gaussian r.v.s with zero mean and variance {ηl2 }. The variance ηl2 follows an exponential power profile, i.e., ηl2 = η02 exp{−l/4}, 2 where ∑L η20 is used to normalize the channel power gain with l=0 η0 exp{−l/4} = 1 [14]. Quadrature amplitude modulation (QAM) with a square constellation of size 16 is adopted. Other simulation parameters are shown as follows: The number of users K = 4, the noise power σv2 = −99dBm, the order of the CIR L = 10. Fig. 2 plots the number of CMs required for the traditional ZF and proposed ZF receiver versus block length M for different numbers of total subcarriers. The users are assigned with unequal numbers of subcarriers. The roll-off factor is 0.9. It is shown that the proposed scheme can reduce the complexity substantially. The required computation of the traditional ZF is about 104 − 105 times that of the proposed ZF receiver. The performance of equal subcarrier assignment is similar and omitted for concision. Figs. 3(a) and 3(b) plot the system SERs of the traditional and proposed ZF receivers versus SNR for equal and unequal subcarrier assignment schemes, respectively. It is shown that the SER with small roll-off factor is superior to that with the large one because more inter-carrier interference is leaked into adjacent subcarriers for large roll-off factors. Importantly, it can be observed that the SER performance of the proposed ZF is nearly the same as that of the traditional ZF for both 1 In this paper, we consider the direct DFT operation as a conservative evaluation of complexity. If fast Fourier transform (FFT) is used, the complexity of the proposed scheme can be further reduced.

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0

10

10

12

10

N = 32,64,128,256 bottom to top

−1

10

8

10

SER

Number of CMs

10

10

TZF α = 0.1 PZF α = 0.1 TZF α = 0.5 PZF α = 0.5 TZF α = 0.9 PZF α = 0.9

6

10

−2

10

N = 32,64,128,256 bottom to top

4

10

PZF, N = 32,64,128,256 TZF, N = 32,64,128,256

2

10

0

20

40

60

80

100

−3

Block length M

10

equal and unequal subcarrier assignments. The reason why there is a small gap between the SER performances of the proposed receiver and traditional ZF receiver is that the first M elements in the noise vector are reshaped and the dimension of modulation matrix is reduced in (27). The proposed ZF receiver using the circulant reconstruction has larger noise power than the traditional ZF receiver after equalization. The detailed proof of performance gap is given in Appendix A.

y ¯k = Dk sk + n ¯k , ˜ zk = Dk sk + vk ,

respectively, where the elements in n ¯ k are independent complex Gaussian r.v.s of zero mean and variances {ak,i } with ˜ k ]2 for 1 ≤ i ≤ |Bk |M + M . According to ak,i = σv2 /[Λ i,i

25

30

25

30

10

−1

10

TZF α = 0.1 PZF α = 0.1 TZF α = 0.5 PZF α = 0.5 TZF α = 0.9 PZF α = 0.9

−2

10

−3

10

0

5

10

15

20

SNR (dB)

(b) Unequal subcarrier assignments Fig. 3. SER per subcarrier versus SNR for different roll-off factors with N = 256 and M = 7. Numbers of subcarriers assigned to users of (a) and (b) are [63, 63, 63, 63] and [127, 63, 31, 31], respectively.

(27), the variance of the ith element in vk can be expressed as bk,i =

(40)

20

(a) Equal subcarrier assignments

{

(39)

15

0

A PPENDIX A P ROOF OF P ERFORMANCE G AP Due to the equivalent transforms in Step 1 and 2, the performance of applying zero-forcing equalization to (26) is the same as the traditional ZF receiver, while the zero-forcing equalization to (28) is the proposed ZF receiver. Equations (26) and (28) are given as

10

SNR (dB)

V. C ONCLUSION This paper proposes a new ZF receiver design for multiuser GFDMA uplinks. To avoid the huge computation caused by the inverse of large dimensional equivalent channel matrix, a low complexity receiver with closed-form is developed. By using the banded and block circulant structure of the frequency transformed equivalent channel matrix, the signal detection can be efficiently implemented with the aid of DFT (or FFT) and one-tap equalization operations. Importantly, the proposed scheme can reduce the computational complexity substantially with negligible loss of SER performance in comparing with the traditional ZF receiver.

5

SER

Fig. 2. Number of CMs versus block length for different numbers of total subcarriers. Numbers of subcarriers assigned to users are [N/2 − 1, N/4 − 1, N/8 − 1, N/8 − 1]. PZF: proposed ZF, TZF: traditional ZF.

0

ak,i + ak,|Bk |M +i , 1≤i≤M . ak,i , M < i ≤ |Bk |M

˜ k are respectively rewritten The expressions of Dk and D as  ˜ GN  ˜1  G   Dk =  0   .  .. 0

0 ˜N G .. . ..

. ···

··· .. . .. .

0 .. .

˜1 G 0

˜N G ˜1 G

0

      ∈ C(|Bk |+1)M ×|Bk |M   

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and

 ˜ GN   ˜ ˜ k =  G1 D  .  ..

0 ˜N G .. . ···

0

··· .. . .. . ˜ G1

interference nulling vectors in terms of noise power [18]. Then, we have



˜1 G ..  .   ∈ C|Bk |M ×|Bk |M .  0  ˜N G

(TZF)

ρk,i

˜ k [j] is the jth column of the matrix D ˜ k . Then, after where D the circulant construction in (27), the noise power on the ith data stream of user k is given as (PZF)

=

|fk,i (n)|2 bk,n

n=1 |Bk |M

=

∑

|fk,i (n)|2 ak,n +

n=1

M ∑

|fk,i (n)|2 ak,|Bk |M +n .

n=1

(42) T Lemma 1. For the ith data stream of user k, ¯ fk,i = [fk,i , tTk,i ]T T with tk,i = [fk,i (1), ..., fk,i (M )] is one possible interference nulling vector for (39), i.e., { 1, j = i H ¯ fk,i Dk [j] = . (43) 0, j ̸= i

Proof: For 1 ≤ i ≤ |Bk |M − M , with (41), we have { 1, j = i H H ˜ ¯ fk,i Dk [j] = fk,i Dk [j] = . (44) 0, j ̸= i For |Bk |M − M + 1 ≤ i ≤ |Bk |M , by using (41), we have ˜ N [l] [fk,i (|Bk |M − M + 1), ..., fk,i (|Bk |M )]H G { 1, j = i ˜ 1 [l] = +[fk,i (1), ..., fk,i (M )]H G , 0, j ̸= i

(45)

˜ 1 [l] and G ˜ N [l] are the lth where l = j − (|Bk | − 1)M , G ˜ ˜ columns of G1 and GN , respectively. The result in (45) gives H that ¯ fk,i Dk [j] = 0 for |Bk |M − M + 1 ≤ i ≤ |Bk |M, j ̸= i H ¯ and fk,i Dk [j] = 1 for |Bk |M − M + 1 ≤ i ≤ |Bk |M, j = i. Thus, Lemma 1 is proved. fk,i to (39), the By applying the interference nulling vector ¯ noise power is expressed as |Bk |M +M (null)

ρk,i

=

∑

|¯ fk,i (n)|2 ak,n

n=1 |Bk |M

=

∑

n=1

|fk,i (n)|2 ak,n +

(47)

where ρk,i denotes the noise power on the ith data stream of user k for the traditional ZF receiver. Therefore, we have proved that the proposed ZF receiver using the circulant reconstruction has larger noise power than the traditional ZF receiver. Since the desired signal power per data stream is normalized after equalization, the signal-tointerference-plus-noise ratio per data stream of the traditional ZF is no less than that of proposed ZF receiver, which results in a performance gap between the proposed ZF receiver and the traditional one. ACKNOWLEDGEMENT The authors would like to thank Dr. Shu-Hung Leung from City University of Hong Kong for his helpful comments on the improvement of this paper.

|Bk |M

ρk,i

(PZF)

= ρk,i .

(TZF)

˜k Let fk,i denote the ZF receiving vector for equalizing D of (40) in order to detect the ith data stream of user k. We have { 1, j = i H ˜ fk,i Dk [j] = , (41) 0, j ̸= i

∑

(null)

≤ ρk,i

M ∑

|fk,i (n)|2 ak,|Bk |M +n .

n=1

(46) According to Gauss-Markov theorem, the traditional ZF vector with least square solution is optimal among all the

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2169-3536 (c) 2018 IEEE. Translations and content mining are permitted for academic research only. Personal use is also permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.