Adaptive Modulation and Filter Configuration in ...

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 17, NO. 3, MARCH 2018

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Adaptive Modulation and Filter Configuration in Universal Filtered Multi-Carrier Systems Xiao Chen , Liang Wu , Member, IEEE, Zaichen Zhang , Senior Member, IEEE, Jian Dang, Member, IEEE, and Jiangzhou Wang, Fellow, IEEE

Abstract— Universal filtered multi-carrier (UFMC) is a potential waveform technology, which can efficiently combat different carrier frequency offsets (CFOs) of multiple users. First, adaptive modulation and power allocation are applied for each subcarrier to meet a preset bit error rate (BER) requirement by ignoring CFOs. Then, we prove that different CFOs will cause different interference variances to adjacent users, which results in performance degradation in UFMC systems. To reduce the interference caused by CFOs and improve achievable rate, a novel adaptive filter configuration algorithm is proposed to adaptively design the parameters of the finite impulse response filters. Specifically, the proposed algorithm is available for the UFMC systems, where the user equipments are allocated with different bandwidths. Finally, simulation results show that the proposed adaptive filter configuration algorithm can dramatically eliminate the interference caused by different CFOs, and achieve better BER performance and a higher achievable rate than the conventional scheme. Index Terms— Achievable rate, adaptive filter, bit error rate (BER), carrier frequency offset (CFO), universal filtered multi-carrier (UFMC).

I. I NTRODUCTION

W

ITH the 1000 times higher data rates requirement, the fifth generation (5G) networks are expected to support many application requirements such as Internet of Things (IoT), gigabit wireless connectivity, tactile Internet and machine-type communication (MTC) [1], [2]. Nowadays, orthogonal frequency division multiplexing (OFDM) [3]–[5] is the dominant standard waveform for high-speed wireless communication systems in the fourth generation (4G) and IEEE 802.11 (WiFi) standards. However, some disadvantages of OFDM cannot be overcome [6], [7]. OFDM is not spectrally Manuscript received June 24, 2017; revised October 19, 2017 and November 30, 2017; accepted December 17, 2017. Date of publication December 29, 2017; date of current version March 8, 2018. This work was supported in part by NSFC under Project 61501109, Project 61571105, and Project 61601119, in part by the national key research and development plan under Grant 2016YFB0502202, and in part by the Jiangsu NSF under Project BK20140646. The associate editor coordinating the review of this paper and approving it for publication was E. A. Jorswieck. (Corresponding authors: Liang Wu; Zaichen Zhang.) X. Chen, L. Wu, Z. Zhang, and J. Dang are with the National Mobile Communications Research Laboratory, Southeast University, Nanjing 210096, China (e-mail: [email protected]; [email protected]; [email protected]; [email protected]). J. Wang is with the School of Engineering and Digital Arts, University of Kent, Canterbury CT2 7NT, U.K. (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TWC.2017.2786231

efficient due to its strict requirement of orthogonality and its utilization of the cyclic prefix (CP) [8], [9]. Besides, the high out-of-band (OOB) emission of OFDM will cause adjacentchannel interference (ACI), which makes it impossible to implement the dynamic spectrum access [10]. Most importantly, the strict synchronization requirements of OFDM systems are the crucial drawback for the considered application scenarios for 5G systems. Due to these drawbacks of OFDM, it is of great importance to design new waveforms to meet the requirements of the future communication, such as higher data rates, lower latencies, relaxed time-frequency alignment, flexible resource allocation, etc [1], [8]–[11]. With the waveform contenders for 5G, new waveforms have been proposed including generalized frequency division multiplexing (GFDM), filter bank multi-carrier (FBMC), universal filtered multi-carrier (UFMC), etc. As a promising contender, GFDM was proposed for the air interface of 5G network. GFDM applies a prototype filter to each individual block consisting of a number of subcarriers, and the pulse shaping of the filter is circularly shifted in the timefrequency domain. By employing the prototype filter, GFDM can reduce the OOB emission and apply a feasible resource allocation with strong spectrum fragmentation [12]–[14]. However, with the non-orthogonal subcarriers, both intercarrier interference (ICI) and inter-symbol interference (ISI) exist in the GFDM systems. In order to mitigate the selfinterference, algorithms should be designed for GFDM at the cost of complexity [13], [15]. As one of the new waveforms, FBMC has gained attention and been as an alternative technique to OFDM in recent years [6], [16]. FBMC modulator applies a pulse shape/prototype filter to each subcarrier, which results in a reduced sidelobe and can effectively eliminate the ICI. Without CP, FBMC can achieve an efficient data transmission. Besides, FBMC can be applied to the situation where the users are asynchronous [17]–[20]. However, the filter of FBMC is applied for a subcarrier as a subband so that the frequency response of the filter needs to be narrow, which requires long filter length. In general, the filter length of FBMC is two or three times longer than the symbol length, which causes long ramp up and ramp down areas in short burst transmission. Furthermore, in some multiple-input multiple-output (MIMO) applications, FBMC with filtered multi-tone (FMT) or staggered multitone (SMT) principle is more complex than OFDM and UFMC [6], [7], [16].

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UFMC is a potential waveform technology, considered as a generalization of filtered OFDM and FBMC. As an advanced waveform for future communication systems, UFMC inherits the simplicity of OFDM and collects the advantages of FBMC. Compared to OFDM and FBMC, there are some unique advantages in UFMC systems. Firstly, UFMC with no requirement of CP needs a shorter guard period for filter ramp up and ramp down, which results in a slightly better spectral efficiency than OFDM [21], and UFMC can flexibly use the noncontiguous spectrum resources by assigning different subbands with non-contiguous spectrum. Secondly, the finite impulse response (FIR) filter is one of the key technologies in UFMC systems. With the FIR filter, UFMC can reduce the OOB sidelobe levels and eliminate the inter-block interference (IBI) by filtering a group of adjacent subcarriers as a subband [22]. As a result, UFMC can reduce the impact of time-frequency misalignment which is not available for OFDM but a common issue in 5G [23], [24]. Thirdly, the filter length of UFMC is shorter than that of FBMC, and UFMC is applicable to short burst transmission and delay-sensitive scenario, which is not suitable for FBMC [25], [26]. As OFDM, UFMC can also efficiently combat the frequency-selective fading [21]. With the above advantages, UFMC will be an irreplaceable waveform in future wireless communication systems. In an uplink multi-user system, carrier frequency offset (CFO) is one of the critical issues. The CFO is caused by Doppler effect resulting from the mobility of users and the local oscillator misalignment between the transmitter and receiver [22]–[24], [27]. Therefore, at the base station (BS), there are multiple CFOs due to the different features of multiple users. For OFDM systems, CFO issues have been extensively studied [28]–[31], and techniques have been proposed to estimate CFOs and reduce the ICI due to CFOs. However, the CFOs cannot be eliminated completely by applying the CFO estimation and compensation technologies. On the other hand, the weighted overlap and add based OFDM (WOLAOFDM) is a filtered CP-OFDM, and it is slightly better than UFMC in asynchronous configuration. However, it has been proved that WOLA-OFDM has an inferior side-lobe reduction performance compared to UFMC, which is a very important indicator when considering the interference caused by CFOs [32], [33]. Moreover, in future communications, the millimeter wave (mm-wave) spectrum will be the key resources. As a result, the magnitude of CFOs will increase resulting in a serious effect on the ICI robustness. UFMC systems are more robust than OFDM/WOLAOFDM systems [11], [32], because the FIR filter of UFMC systems can effectively reduce the OOB emissions of the subcarriers. Based on CFOs, filter optimizations have been studied to reduce the OOB emissions in the UFMC systems [23], [24]. However, the Dolph-Chebyshev filter considered in [23] and [24] is not optimal for the UFMC systems due to its high OOB emissions [34]. Besides, the filter length is fixed in [23] and [24], even though, the achievable rate can be improved by optimizing the filter length. Therefore, our objective is not only to design the FIR filter with the reduced OOB emissions to eliminate the interference caused by CFOs, but also to design the FIR filter with its length as short as possible

to obtain a higher achievable rate in diverse application scenarios. In this paper, at first, without considering CFOs, the adaptive modulation and power allocation are applied for each subcarrier in an uplink multi-user UFMC system to meet a preset bit error rate (BER) requirement. Then, we prove that different CFOs will result in different interference variances for adjacent subbands and severely affect the performance of the UFMC systems. Finally, with the motivations of combatting the CFOs and improving the achievable rate, an adaptive filter configuration algorithm is proposed for different users with different CFOs. The FIR filter based on the weighted Chebyshev approximation is employed in the proposed scheme, and all the parameters of the filter are designed accordingly. The passband and stopband parameters of the filter can be designed according to the bandwidths of different users and the guard interval between adjacent users. The passband ripple of the filter can be designed within 3 dB to guarantee a good system performance. The stopband ripple of the filter is the main parameter, which is designed adaptively according to the different CFOs of different users with a BER target. The filter length can be determined according to the four parameters mentioned above. The proposed adaptive filter configuration algorithm can effectively improve the BER performance and obtain better achievable rate performance. The major contributions of this paper are summarized as follows: (1) In an uplink multi-user UFMC systems, the interference caused by CFOs is analyzed theoretically. It is proved that the interference caused by CFOs has a severe impact on BER performance. (2) A novel adaptive filter configuration algorithm is proposed for the multi-user system, where different users have different CFOs. In the proposed scheme, all parameters of the FIR filter based on the weighted Chebyshev approximation are designed accordingly. The filter length is designed adaptively to improve the achievable rate of the UFMC systems. It is shown that the proposed scheme is especially applicable for the UFMC systems that the user equipments (UEs) are allocated with different bandwidths. (3) Simulation results show that the proposed adaptive filter algorithm can effectively eliminate the interference caused by CFOs, resulting in better BER performance and higher achievable rates than the conventional UFMC schemes. The rest of this paper is organized as follows. In Section II, we describe the system model of the uplink multi-user UFMC system. Section III presents the adaptive modulation and power allocation design scheme. Section IV investigates the impact of CFOs through the theoretical analysis. Section V proposes an adaptive filter configuration algorithm in the UFMC system. In Section VI, we show the simulation results of the proposed adaptive filter configuration algorithm. Section VII concludes this paper. Notations: Vectors and matrices are written in boldface with matrices in capitals. AT , A H and A−1 indicate the transpose, ˜ conjugate transpose and inverse of A, respectively. Let B denotes the corresponding frequency-domain matrix of a timedomain matrix B. [a]k donates the kth element of a.

CHEN et al.: ADAPTIVE MODULATION AND FILTER CONFIGURATION IN UFMC SYSTEMS

II. S YSTEM M ODEL OF UFMC Consider an uplink multi-user UFMC system, as shown in Fig. 1. In this system, K subbands are allocated to K UEs, respectively. The channel is assumed to be frequency-selective and slow fading. The i th (i = 1, · · · , K ) subband contains n i subcarriers for UEi , which indicates that the bandwidth of each subband can be different due to different requirements of users. Besides, it is assumed that the subcarriers within one subband are  Kcontiguous, and the total number of subcarriers is n i ). At the transmitter of UEi , the frequencyN (N ≥ i=1 domain signal s˜i = [˜si (0), s˜i (1), · · · , s˜i (n i − 1)]T is first multiplied by the N-point inverse discrete Fourier transform (IDFT) matrix Vi , where Vi ∈ C N×ni is a partial Fourier matrix consisting of n i columns corresponding to the subband position of UEi . Then, the output of IDFT block will be filtered by a FIR filter Fi with the filter length equal to L i . Fi ∈ C(N+L i −1)×N is a Toeplitz matrix performing the linear convolution to produce the (N + L i − 1)-length transmitted signal xi = Fi Vi s˜i .

(1)

The CFO matrix of UEi in the time domain is ⎡ 1 0 ··· 0 2π ε ⎢ −j N i ··· 0 ⎢0 e Di (εi , k) = ⎢ . . .. . ⎢. .. .. . ⎣. 0

0

···

e− j

2π εi N

expressed as ⎤ ⎥ ⎥ ⎥, ⎥ ⎦

(2)

k

where εi denotes the normalized frequency offset of UEi . Therefore, the received time-domain signal at the BS can be derived as K y= Di Hi xi + ni , (3) i=1

C(N+L i +L ch −2)×(N+L i −1)

is the Toeplitz chanwhere Hi ∈ nel matrix of UEi , L ch is the channel length, Di ∈ C(N+L i +L ch −2)×(N+L i +L ch −2) , and ni ∈ C(N+L i +L ch −2)×1 is the complex additive white Gaussian noise (AWGN) vector whose entries are with zero mean and variance σn2 . At the receiver of BS, the received signal which is padded with zeros has the CFO compensation for each UE. At last, the timedomain signal is transformed to the frequency domain by applying 2N-point fast Fourier transform (FFT). Note that the even subcarriers are utilized to estimate the symbols for all UEs [22]. III. A DAPTIVE M ODULATION AND P OWER A LLOCATION D ESIGN W ITHOUT CFO S Without considering CFOs, adaptive modulation and power allocation of conventional OFDM system [3]–[5] can be employed in the UFMC system. In this section, the subcarrier adaptive modulation and power allocation design are presented to increase the data rate and power efficiency for each subcarrier. For UEi , the received signal in the frequency domain can be expressed as ˜ i P˜ i s˜i + w ˜ i, r˜ i = H

(4)

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˜ i ∈ Cni ×ni is the channel matrix whose diagonal where H element h i, j ( j = 1, · · · , n i ) denotes the channel frequency response of the j th subcarrier, the power allocation  matrix √ √ P˜ i = di ag( p1 , · · · , pni ) has a constraint as nj i=1 p j = Pi with the total transmit power Pi for UEi , s˜i ∈ Cni ×1 is the transmitted signal vector of UEi with E{˜si s˜iH } = Ini , and w ˜ i ∈ Cni ×1 is the noise vector, and each element of w ˜ i has zero mean and variance n i σn2 . According to [35], the maximum data rate of UEi with n i subcarriers can be derived as Ri = max {k j }

s.t.

ni

kj

j =1 ni

p j ≤ Pi

j =1

SNRt h,k ≤ SNRr,k j ,

(5)

where M j = 2k j (k j = 1, · · · , 6) is the modulation order of the j th subcarrier by employing the M-quadrature amplitude modulation (M-QAM), SNRr,k j is the received signal-tonoise-ratio (SNR) of the j th subcarrier with k j th modulation scheme, and SNRt h,k is the SNR threshold for a preset BER target BERt . Specifically, the SNR threshold can be expressed as [35], [36] ⎧   1 k ⎪ ⎨−6 5 · 2 − 4 ln (5 BERt ) k = 1, 3, 5 SNRt h,k = (6) ⎪  ⎩ 2 k − 3 2 −1 ln (5 BERt ) k = 2, 4, 6, where SNRt h,1 < · · · < SNRt h,6 . According to [35], the power allocation can be derived directly, and the modulation order M j can be chosen for the j th subcarrier of UEi adaptively in the UFMC system. IV. T HE T HEORETICAL A NALYSIS OF CFO S This section reports a theoretical analysis for the impact of CFOs. It is assumed that the adaptive modulation and power allocation have already been employed for each subcarrier in the UFMC system. After applying CFO compensation at the BS, the signal corresponding to UEi is expressed as −1 yi = D−1 i y = Di Di Hi xi + zi + ni ,

(7)

where D−1 is the CFO compensation matrix for UEi , and zi i is the interference caused by CFOs from other UEs to UEi and can be written as zi =

K

D−1 i Dk H k x k .

(8)

k=1,k =i

In Appendix A, it is shown that the corresponding frequency-domain form of (7) is derived by applying FFT ˜ i s˜i + z˜ i + n˜ i , y˜ i = H

(9)

where n˜ i ∈ Cni ×1 and z˜ i ∈ Cni ×1 are vectors which are the frequency-domain representations of ni and zi , respectively.

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Fig. 1.

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 17, NO. 3, MARCH 2018

System model of the uplink multi-user UFMC.

approximated as [˜zi ] j ≈ [a˜ i ] j + [b˜ i ] j ,

(11)

where [a˜ i ] j and [b˜ i ] j are the interference from the left-side UEi−1 and the right-side UEi+1 to the j th subcarrier in UEi ’s subband, respectively. Based on (8), (10) and (11), the received interference signal from the UEi−1 to the j th subcarrier in UEi ’s subband can be derived as [a˜ i (εi , εi−1 )] j n i−1   ˜ −1 D ˜ i−1 H ˜ i−1 s˜i−1 ]q I [( j − εi ) − (q − εi−1 )] . = [D i q=1

Fig. 2.

FIR filter frequency response [37].

(12) The detailed derivation is given in Appendix B. ˜ −1 for UEi , we only After applying the CFO compensation D i need to consider the CFO deviation εi−1,i = εi−1 − εi between UEi−1 and UEi . As shown in Appendix C, the interference caused by the CFO deviation from UEi−1 in (12) can be rewritten as

Fig. 3.

[a˜ i (εi−1,i )] j n i−1   = h i−1,q+εi−1,i s˜i−1 (q)I [ j + εi−1,i − q] .

The subcarrier distribution for adjacent three UEs.

Note that the frequency response at the normalized frequency f can be regarded as that of a low-pass linear phase FIR filter with [37] I( f ) =

sin Nπ f . N sin π f

(10)

The frequency response of the FIR filter in (10) is plotted in Fig. 2. As can be seen from Fig. 2, the peak value of I ( f ) is decreasing from the center frequency f = 0 to the two sides. The subcarrier distribution of adjacent UEs (UEi−1 , UEi and UEi+1 ) is shown in Fig. 3, where UEi−1 and UEi+1 are located at the left side and right side of UEi , respectively. Considering the attenuation characteristic of the frequency response shown in Fig. 2, the main interference to UEi is caused by the CFOs of the adjacent UEs (i.e., UEi−1 and UEi+1 ), while the interference from other UEs can be ignored due to the far subcarrier distance from UEi . As a result, the frequency-domain interference to the j th subcarrier within the subband of UEi can be

(13)

q=1

Similarly, the interference from the UEi+1 to the j th subcarrier in UEi ’s subband can be derived as [b˜ i (εi+1,i )] j n i+1   h i+1,q+εi+1,i s˜i+1 (q)I [q + εi+1,i − j ] , =

(14)

q=1

where εi+1,i = εi+1 − εi is the CFO deviation between UEi+1 and UEi . Therefore, the signal-to-interference-plus-noise-ratio (SINR) of the j th subcarrier of UEi can be derived as SINRi, j (εi−1,i , εi+1,i )

 2   ˜  E [Hi s˜i ] j 

=

 . 2   2      ˜ E [a˜ i (εi−1,i )] j + [bi (εi+1,i )] j  + E [n˜ i ] j (15)

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where ω p is the passband cutoff frequency, and ωs is the stopband cutoff frequency. The two parameters ω p and ωs can be easily designed based on the bandwidth requirements. The frequency response and error curve of the filter are shown in Fig. 4, where δ is the passband ripple and α is the stopband ripple. From Fig. 4, it is observed that the filter characteristic is equiripple in both the passband and the stopband. With the parameters ω p , ωs , δ and α, the filter length L can be approximated as [39] √  −20 log10 δα − 13   L≈ + 1. (19) 14.6 ωs − ω p /2π B. Design of Filter Parameters

Fig. 4.

Frequency response and error curve of the low-pass filter [39].

From (13)-(15), it indicates that the SINR of each subcarrier of UEi is directly affected by the CFOs (εi−1 , εi and εi+1 ) of three adjacent UEs. Next, we use zero-forcing (ZF) detection to evaluate the BER performance. The BER of the j th subcarrier with M-QAM scheme is approximated as [38]   1 BERi, j ≈ exp −ψ(M j )SINRi, j (εi−1,i , εi+1,i ) , (16) 5 where ψ(M) is defined as  1.5/(M − 1) square M-QAM ψ(M) = . (17) 6/(5M − 4) rectangular M-QAM From (13)-(16), we see that the BER of each subcarrier is directly affected by the adjacent CFOs (εi−1 , εi and εi+1 ). Besides, different BER degradations will be produced due to the different interference variances caused by different CFOs. Therefore, an adaptive filter configuration scheme can be adopted to eliminate the interference and improve the BER performance for different levels of CFOs. V. P ROPOSED A DAPTIVE F ILTER C ONFIGURATION S CHEME In this section, an adaptive filter configuration algorithm is proposed for the uplink multi-user UFMC system with CFOs. A. Introduction of Filter Parameters In the proposed scheme, the FIR filter based on the weighted Chebyshev approximation is employed, since it can provide accurate edges of passband and stopband. With accurate edges, it is possible to design the filter bandwidth according to the bandwidth of different subband for each UE. Considering a low-pass filter designed by the weighted Chebyshev approximation, the desired frequency response can be expressed as [39]  1 0 ≤ ω ≤ ωp jω (18) H (e ) = 0 ωs ≤ ω ≤ π,

In this subsection, the designed for each UE in the 1) Filter Parameter 1 — width n i , the passband ω p,i as

adaptive filter parameters are uplink UFMC system. Passband ω p : With the bandof UEi ’s filter can be designed

ω p,i =

ni 2π. N

(20)

Hence, the passband of each UE with different bandwidth can be designed accordingly. 2) Filter Parameter 2 — Stopband ωs : With the guard interval n g between adjacent subbands, the transition band ω = ωs − ω p can be determined. Using the relationship among ω p , ωs and ω, the stopband ωs,i of UEi ’s filter can be derived as ωs,i =

ni + ng 2π. N

(21)

3) Filter Parameter 3 — Passband Ripple δ: In order to ensure a good system performance, the passband ripple should be set to be δ ≤ 1.5 dB so that the fluctuation can be kept within 3 dB [40] for all subcarriers of each UE. 4) Filter Parameter 4 — Stopband Ripple α: The stopband ripple α (a.k.a. attenuation coefficient) of the filter will be designed based on the interference caused by CFOs. When the j th subcarrier is the first/last subcarrier in the i th subband, it is at or within the 3-dB bandwidth of the filter since the passband ripple δ ≤ 1.5 dB. Besides, Fig. 2 indicates that the peak value of the frequency response is decreasing from center frequency to the two sides. In other words, the first/last subcarrier of UEi is the closest to UEi−1 /UEi+1 , and suffers the heaviest interference from adjacent UEs than other subcarriers of UEi . Based on the filter design principle, if the subcarrier with the heaviest interference can reach the BER target, then all other subcarriers will satisfy the BER requirement. Therefore, to guarantee the BER performance, both the first and the last subcarriers should be taken into account. As mentioned in Section IV, for the first subcarrier of UEi , the main interference is caused by the CFO of UEi−1 which is the closest UE, while the interference from other UEs is ignorable due to the far subcarrier distance. Considering the attenuation characteristic of UEi−1 ’s filter, the SINR of the

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Based on (22) and (26), the filter attenuation coefficient αi−1 of UEi−1 should satisfy   2 2    h i,1 σsi ,1 1 αi−1 ≤  −(N + L ch −1)σn2 .  [a˜ i (εi−1,i )]12 4SNRt h,k1

first subcarrier of UEi can be approximated as SINRi,1 (αi−1 )

 2   ˜  E [H s ˜ ] i i 1   ≈  2  E [a˜ i (εi−1,i )]1 · αi−1  + E [n˜ i ]1 2

(a)

≈

1 2 2 4 h i,1 σsi,1 ,   2 2 [a ˜ i (εi−1,i )]1  + (N + L ch − 1)σn2 αi−1

(27) (22)

where the 41 in the numerator is the filter gain at the 3-dB bandwidth, αi−1 is the filter attenuation coefficient of UEi−1 , and σs2i,1 = p1 is the power allocated to the first subcarrier of UEi . The approximation in (a) is due to N L i . By filtering the subcarriers with a suitable stopband ripple αi−1 , it will make sense to meet the BER requirement by reducing the interference from the OOB emission of adjacent UE. Similarly, for the last subcarrier of UEi , the main interference is caused by the CFO of UEi+1 , and the interference from other UEs can be ignored. The SINR of the last subcarrier of UEi can be derived as SINRi,ni (αi+1 )

 2   ˜  E [H i s˜ i ]n i  ≈  2   2   ˜  E [bi (εi+1,i )]ni · αi+1  + E [n˜ i ]ni  ≈

1 2 2 4 h i,n i σsi,n i  2  2  b αi+1 [ ˜ i (εi+1,i )]ni  + (N

+ L ch − 1)σn2

,

(23)

where αi+1 is the filter attenuation coefficient of UEi+1 . According to (13)-(14), the powers of the interference signals a˜ i and b˜ i are expressed as   [a˜ i (εi−1,i )]1 2 n i−1   = h 2i−1,q+εi−1,i σs2i−1,q I 2 [1 + εi−1,i − q] , (24) q=1

 2  ˜  [bi (εi+1,i )]ni  

n i+1

=

 h 2i+1,q+εi+1,i σs2i+1,q I 2 [q + εi+1,i − n i ] . (25)

q=1

(24) and (25) indicate that the interference caused by CFOs is related to the CFO deviations, the channel coefficients, the number of subcarriers and the signal power of the adjacent UEs. To achieve the BER target, the SINR of the first subcarrier should satisfy SINRi,1 (αi−1 ) ≥ SNR t h,k1 ,

(26)

where SNRt h,k1 is the SNR threshold in (6) with the modulation order M1 = 2k1 .

Similarly, the filter attenuation coefficient αi+1 of UEi+1 should satisfy   2    h i,ni σs2i ,ni 1  2 αi+1 ≤  −(N + L ch −1)σn . 2 ˜  4SNRt h,kni [bi (εi+1,i )]ni (28) Based on the two attenuation coefficients αi−1 and αi+1 , the stopband ripple α of the filter can be designed for the adjacent UEs (UEi−1 and UEi+1 ) to eliminate the interference to UEi and reach the BER target. Similarly, the stopband ripple αi of UEi ’s filter can be designed based on the interference to UEi−1 and UEi+1 . To eliminate the interference to two adjacent UEs, there exist two bounds for αi derived from UEi−1 and UEi+1 . The tighter bound for αi should be selected to combat the interference from UEi to both UEi−1 and UEi+1 . Note that all the filter parameters of all UEs can be designed at the BS to reduce the computational complexity of each UE. Each UE will generate the adaptive filter based on the information feedback from the BS. C. Summary of the Adaptive Filter Configuration Algorithm The main steps of the adaptive filter configuration algorithm are summarized as follows: Algorithm 1 Steps of the Adaptive Filter Configuration Algorithm 1) Solve adaptive modulation and power allocation problems in (5) to initialize transmit signal. 2) With the CFOs of the adjacent UEs, calculate the interference caused by UEi−1 and UEi+1 for UEi , and obtain the SINR of UEi from (22)-(25). 3) Set a BER target and get the corresponding SNR threshold. The attenuation coefficient α of the adjacent UE (UEi−1 / UEi+1 ) is designed adaptively as in (27)-(28), which is based on SINR(α) ≥ SNRt h to achieve the BER target. 4) With the attenuation coefficient, the passband, the stopband and the passband ripple, the filters of UEi−1 and UEi+1 can be derived adaptively according to (19)-(21). 5) The filter of each UE is generated, and UE transmits the modulated signal through the adaptive filter block to combat the interference caused by CFOs. With the designed attenuation coefficient αi and other relevant parameters, the filter length L i of UEi can be calculated from (19) accordingly. In the time domain, the frame lengths of all UEs are the same in the multi-user UFMC system.

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To guarantee the same guard period, the filters of different UEs are padded with zero such that the lengths equal to the maximum filter length L max = max {L 1 , · · · , L K }. Therefore, the achievable rate of UEi can be expressed as n i j =1 k j Ci = (bit/s/Hz), (29) (N + L max − 1)Ts · Bi where Ts is the sampling period, Bi is the bandwidth of the i th subband. VI. S IMULATION R ESULTS In this section, we evaluate the performance of the proposed adaptive filter configuration scheme in the uplink multi-user UFMC system via simulations under different parameter settings. In all simulations, the channels are modeled as multipath Rayleigh fading with complex normal distribution CN (0, 1), and the channel length is L ch = 8. All simulation results are obtained by averaging over 1000 channel realizations, where 1000 symbols are transmitted per channel realization. The IDFT size is N = 128. For comparison of the proposed scheme and the conventional scheme, we first consider the situation that all the UEs have the same bandwidth, and then the case of different bandwidths is shown. In the simulations, 4 subbands each consisting of 16 subcarriers are allocated to 4 UEs, respectively. More specifically, 12 subcarriers in each subband are used to carry information, and the other 4 subcarriers are used as guard interval between adjacent subbands. The subbands are allocated to UE1 , UE2 , UE3 , and UE4 in sequence. The adaptive modulation and power allocation are designed based on the channel matrix without considering CFOs or filters, and they are employed for the UFMC system in all simulations. The BER target is set as BERt = 10−3 for both adaptive modulation design and adaptive filter design. A. Performance of the Adaptive Modulation and Power Allocation With No CFOs or Filters This subsection shows the effect of the adaptive modulation and power allocation in the uplink multi-user system without considering CFOs or filters. Fig. 5 shows the simulated and the theoretical BER performance in (16) of all UEs employing the adaptive modulation and power allocation without considering CFOs or filters. It can be seen from Fig. 5 that, with the adaptive modulation and power allocation, the BER target 10−3 can be satisfied in the entire SNR region. Moreover, the gap between the theoretical BER and the simulated BER is within 2.5 dB due to the approximation error. Similar results are reported in [38]. For comparison, Fig. 5 also depicts the simulated BER performance of all UEs employing 16-QAM without the power allocation design. It can be seen that the simulated BER performance with 16-QAM is worse than that of the adaptive scheme in the entire SNR region, and it cannot satisfy the BER target 10−3 in the low SNR region. Therefore, adaptive modulation is required, when a BER target is set. Fig. 6 illustrates the average modulation order performance of the 4 UEs employing the adaptive modulation and power

Fig. 5. BER comparison of the schemes with different modulation strategies, when the CFOs or filters are not considered.

Fig. 6. The average modulation order of each UE employing the adaptive modulation and power allocation without considering CFOs or filters.

allocation without considering CFOs or filters. In Fig. 6, different UEs have the same average modulation order for a specific SNR. It is clear that, when the SNR increases, the average modulation orders of the 4 UEs are increasing accordingly. The reason is that the adaptive modulation and power allocation are designed to maximize the achievable rate with a BER target, which results in the higher modulation order with the higher SNR. B. Effect of CFOs in the UFMC System With No Filters In this subsection, we show the effect of CFOs on the performance of UEs in the UFMC system without filters. In this case, the UFMC system is equivalent to the OFDM system. Fig. 7 depicts the CFO effect on UE2 ’s BER performance with 3 UEs (n 1 = n 2 = n 3 = 12) for different SNR values, and the CFOs are set to be ε1 = ε3 = 0 and ε2 ∈ [−0.5, 0.5]. In Fig. 7, the simulated BER performance becomes worse from center ε2 = 0 to two sides for all SNR values, which indicates that the increasing CFO leads to the worse BER performance. In addition, when the SNR increases, the BER performance becomes worse except for ε2 = 0. The reason is that the

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Fig. 7. CFO’s effect on UE2 ’s BER performance in the UFMC system (n 1 = n 2 = n 3 = 12) with CFOs as ε1 = ε3 = 0 and ε2 ∈ [−0.5, 0.5] but without filters.

Fig. 8. Comparison of BER performance among each UE (n 1 = n 2 = n 3 = n 4 = 12) in the UFMC system with CFOs as ε1 = −0.2, ε2 = −0.1, ε3 = 0.1, and ε4 = 0.2 but without filter.

interference caused by CFO increases with the increasing SNR in (24)-(25), which leads to a BER degradation. On the contrary, when the CFO of UE2 equals 0, UE2 ’s noise is caused by the Gaussian noise which resulting in a better BER when the SNR increases as shown in Fig. 5. It is inferred that the BER performance with high SNR has a higher sensitivity to CFO than that with low SNR. To show the effect of CFOs in the multi-user system, the normalized CFOs of the 4 UEs are assumed to be ε1 = −0.2, ε2 = −0.1, ε3 = 0.1, and ε4 = 0.2, respectively. In this case, the CFO deviation ε1,2 = −0.1 that will cause interference from UE1 to UE2 is used to design the attenuation coefficient α1 of UE1 ’s filter. Similarly, ε2,1 = 0.1 with ε2,3 = −0.2, ε3,2 = 0.2 with ε3,4 = −0.1, and ε4,3 = 0.1 are used to design α2 , α3 , and α4 , respectively. Fig. 8 illustrates the simulated BER performance of the UFMC system with CFOs where no filter is applied in each subband. By comparing Fig. 5 with Fig. 8, we see that the BER performance of the UFMC system is greatly degraded by the presence of CFOs. Specifically, the BER performance

Fig. 9. CFO effect on UE2 ’s BER performance in the UFMC system (n 1 = n 2 = n 3 ) with the CFOs (ε1 = ε3 = 0 and ε2 ∈ [−0.5, 0.5]) and the proposed adaptive filters.

cannot achieve the BER target 10−3 in either low or high SNR region. It is also seen that, when the SNR increases, the BER decreases first and then increases in Fig. 8. This mainly results from the fact that, in the low SNR region, the Gaussian noise is the dominant impairment; while in the high SNR region, the main interference is the inter-subband interference caused by the CFOs. Therefore, when SNR increases, the interference from other UEs increases proportionally in (24)-(25), which leads to a BER degradation in the high SNR region. Furthermore, it can be observed that the BER relationship satisfies BER1 ≈ BER4 ≤ BER2 ≈ BER3 in Fig. 8. The reason is that UE1 (UE4 ) has only one adjacent UE, and the main interference of UE1 (UE4 ) is caused by only one CFO deviation ε2,1 = 0.1 (ε3,4 = −0.1). However, the main interference of UE2 (UE3 ) is caused by two CFO deviations ε1,2 = −0.1 and ε3,2 = 0.2 (ε2,3 = −0.2 and ε4,3 = 0.1) from two adjacent UEs. As a result, UE1 (UE4 ) suffers from smaller interference resulting in better BER performance than UE2 (UE3 ). The simulated results prove that, first, the interference caused by CFOs will directly affect the BER performance; second, different CFOs will cause different interference variances, which are consistent with the results in (24)-(25). C. Performance of the Proposed Adaptive Filter in the UFMC System With CFOs In this subsection, we investigate the proposed adaptive filter configuration scheme in the UFMC system. Fig. 9 shows the CFO effect on UE2 ’s BER performance when the proposed adaptive filter is employed. The simulation scenario is same as that in Fig. 7. Considering the achievable rate requirement, the adaptive filter length is limited to be smaller than 2L ch . By comparing Fig. 7 with Fig. 9, we observe a significant performance improvement achieved by employing the adaptive filter. To show the range of validity of the designed filter, Fig. 10 depicts the simulated BER performance of the UFMC system with the filters optimized for the given CFOs but applied to other CFOs. The given CFOs are set to be ε1 = −0.1, ε2 = 0

CHEN et al.: ADAPTIVE MODULATION AND FILTER CONFIGURATION IN UFMC SYSTEMS

Fig. 10. UE2 ’s BER performance in the UFMC system (n 1 = n 2 = n 3 = 12) employing the optimized filters for the given CFOs (ε1 = 0.1, ε3 = −0.1, and ε2 = 0) but with the CFOs as ε1 = 0.1, ε3 = −0.1, and ε2 ∈ [−0.5, 0.5].

Fig. 11. Comparison of BER performance among each UE (n 1 = n 2 = n 3 = n 4 = 12) in the UFMC system with CFOs (ε1 = −0.2, ε2 = −0.1, ε3 = 0.1, and ε4 = 0.2) and the proposed adaptive filters.

and ε3 = 0.1 as an example. With the adaptive filter designed for the given CFOs, the CFO of UE2 is assumed to change from -0.5 to 0.5. It is seen from Fig. 10 that the optimized filters can effectively combat the CFOs in the low SNR region (e.g., 5 dB and 10 dB). However, in the region of 15 dB ∼ 25 dB, the given filters cannot eliminate the interference caused by CFOs when |ε2 | ≥ 0.3. That is to say, for the low SNR, the given filters can combat the CFOs in the wide simulated CFO range, while the SNR increases, the valid range gets narrower. In Fig. 11 and Fig. 12, the adaptive filter configuration algorithm is employed in the UFMC system to make the comparison with the results in Fig. 8. We assume that the filter length L i (i = 1, 2, 3, 4) has a constraint as L i < 2L ch to ensure the achievable rate. Fig. 11 shows that, with the proposed adaptive filter configuration strategy, the simulated BER performance of all UEs has been improved significantly compared with the results in Fig. 8. As expected, the BER target 10−3 has been attained in the entire SNR region for all UEs. In the low SNR region,

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Fig. 12. Comparison of adaptive filter length among each UE (n 1 = n 2 = n 3 = n 4 = 12) in the UFMC system with CFOs (ε1 = −0.2, ε2 = −0.1, ε3 = 0.1, and ε4 = 0.2) and the proposed adaptive filters.

Fig. 13. Comparison of sum rates achieved by the proposed adaptive filter scheme and the conventional fixed-length filter schemes with L f 1 = 16 and L f 2 = 32 in the UFMC system with CFOs (ε1 = −0.2, ε2 = −0.1, ε3 = 0.1, and ε4 = 0.2).

the BER performance of all UEs is similar due to the Gaussian noise is the dominant impairment. In the high SNR region, UE1 and UE4 have better BER performance than UE2 and UE3 for the same reason as that in Fig. 8. It is suggested that the system performance has a higher sensitivity to CFOs with higher SNR, and the FIR filter is necessary. Fig. 12 depicts the average filter lengths of the proposed adaptive filters for all UEs in the UFMC system. In the low SNR region, the filter lengths are close to zero for the reason that the interference caused by CFOs is negligible resulting in a relaxing requirement of filter. In the high SNR region, the lengths of adaptive filters increase when the SNR increases, since the interference caused by CFOs increases with SNR. Besides, the filter length of different UE is different in Fig. 12. The filter lengths of UE1 and UE4 are shorter due to the smaller interference caused by CFOs than that of UE2 and UE3 . From (27)-(28), the attenuation coefficients α is the decreasing function of the power of interference (˜ai and b˜ i ) caused by CFOs. Hence, the attenuation coefficient decreases

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D. Proposed Adaptive Filter Employed in the UFMC System With Different Bandwidths

Fig. 14. Comparison of BER performance among UEs with different bandwidths (n 1 = 8, n 2 = 10, n 3 = 12, n 4 = 14) in the UFMC system with CFOs (ε1 = −0.2, ε2 = −0.1, ε3 = 0.1, and ε4 = 0.2) and the proposed adaptive filters.

In this subsection, we study the performance of the proposed adaptive filter scheme in the UFMC system with different bandwidths for different UEs. In the following, the numbers of subcarriers allocated to the 4 UEs are set to be n 1 = 8, n 2 = 10, n 3 = 12, and n 4 = 14, respectively. Fig. 14 and Fig. 15 show the simulated BER performance and average filter lengths of the proposed adaptive filter scheme, respectively. As can be seen from Fig. 14, the BER of all the 4 UEs satisfies the BER target 10−3 in the entire SNR region. Besides, it is interesting to observe that the BER performance gap between UE1 and UE4 is larger than that in Fig. 11. The reason is that, the interference from UE3 to UE4 is larger than that from UE2 to UE1 as n 3 > n 2 . By comparing Fig. 12 with Fig. 15, we see little variation in the filter length. Overall, the results in Fig. 14 and Fig. 15 indicates that the proposed adaptive filter is suitable for the UFMC system with different bandwidths. VII. C ONCLUSION

Fig. 15. Comparison of adaptive filter length among UEs with different bandwidths (n 1 = 8, n 2 = 10, n 3 = 12, n 4 = 14) in the UFMC system with CFOs (ε1 = −0.2, ε2 = −0.1, ε3 = 0.1, and ε4 = 0.2) and the proposed adaptive filters.

with the increase of SNR, and the decreasing attenuation coefficient gives rise to the increasing filter length as shown in (19). Fig. 13 illustrates the sum rate performance of the 4 UEs achieved by the proposed adaptive filter and the conventional fixed-length filters. In the conventional schemes, the filter lengths are set to be L f1 = 16 and L f2 = 32 as in [22] and [24]. It can be seen from Fig. 13 that the sum rates are increasing with SNR due to the increased modulation order. Furthermore, the proposed adaptive filter has a remarkable rate improvement in the entire SNR region. This is due to the fact that the filter length of the proposed adaptive filter is shorter than that of the conventional fixed-length filters. The result is consistent with (29) that the achievable rate is a decreasing function of filter length L. Besides, the performance gain of the proposed adaptive filter scheme will greatly increase when the number of UEs increases.

In this paper, we have proposed an adaptive filter configuration algorithm to address the problem of the interference caused by CFOs in the uplink multi-user UFMC system. With the adaptive modulation and power allocation, we have proved that the interference caused by CFOs has a direct effect on the system performance, and different CFOs can lead to different interference variances. As a result, an adaptive filter configuration scheme has been proposed to reduce the interference caused by CFOs, which can adaptively design the parameters of the filters aiming to achieve the BER target and improve the achievable rate. Simulation results verify that the interference caused by CFOs results in the degradation of BER performance, while the proposed adaptive filter scheme can dramatically eliminate the interference caused by CFOs and attain an expected BER performance. Besides, the proposed adaptive filter algorithm can effectively improve the achievable rates due to the shorter filter length compared with the conventional fixed-length filter. Moreover, the proposed adaptive filter scheme can also be applied to the UFMC systems with different subband bandwidths. A PPENDIX A D ERIVATION OF (9) As shown in Fig. 1, the received signal yi in (7) is padded with zeros and can be rewritten as  −1          Di 0 Hi 0 xi z ni Di 0 yi = + i + . 0 0 0 0 0 0 0 0 0 2N×2N

2N×2N

2N×2N

2N×1

2N×1

2N×1

(30) Then, the frequency-domain signal by applying a 2N-FFT is given by r˜ i = WiH yi , C2N×2ni

(31)

is a partial Fourier matrix consisting of where Wi ∈ 2n i columns corresponding to the subcarriers position of UEi ,

CHEN et al.: ADAPTIVE MODULATION AND FILTER CONFIGURATION IN UFMC SYSTEMS

which is a submatrix of the Fourier matrix W ∈ C2N×2N . Thus, (31) can be rewritten as r˜ i = {W H yi }2ni ,

i

(b)

= T˜ i x˜ i + g˜ i + f˜i ,

(33)

where the vectors x˜ i , g˜ i and f˜i in (33) are defined as ⎧      ⎪ xi xi ⎪ H H ⎪ x˜ i = W ⎪ = Wi ⎪ ⎪ 0 0 ⎪ ⎪    2ni   ⎪ ⎪ ⎨ zi zi g˜ i = W H = WiH , ⎪ 0 0 ⎪ 2n ⎪ i      ⎪ ⎪ ⎪ ni ni ⎪ H H ⎪ ˜ ⎪ f = W = Wi ⎪ ⎩i 0 0

 i ×2n i

=

Hi 0 0 0 !"

(34)

2n i ×2n i

(35)

and {·}2ni ×2ni donates the matrix consisting of 2n i rows and 2n i columns corresponding to the subcarriers position of UEi . Note that Wi is an orthogonal matrix, and Hi is a Toeplitz matrix. Therefore,  according  to the property of the Toeplitz Hi 0 H ˜ Wi in (35) is a diagonal matrix matrix, Ti = Wi 0 0 with the frequency-domain channel coefficients on the main diagonal [21]. Based on (1) and (34), x˜ i can be further rewritten as       (c) Vi H Fi Vi s˜ i H Vi s˜ i H x˜ i = Wi = Wi = Wi s˜i , (36) 0 0 0 where the filter matrix Fi in (c) can be viewed as an I N matrix while there is no filter. From (36), WiH is the 2N-FFT matrix consisting of 2n i columns, while Vi is the N-IDFT matrix containing n i columns. As a result, the odd subcarriers of x˜ i contain interference from all data symbols, while the even subcarriers carry the useful data symbols [22]. By setting the odd subcarriers equal to zero, x˜ i can be expressed as $ %T x˜ i = 0, s˜i (0), 0, s˜i (1), 0, · · · , s˜i (n i − 1) .

⎛

⎜ ⎜ ⎜ ⎜ ˜i ⎜ = T ⎜ ⎜ ⎪ ⎪ ⎜ ⎪ ⎪ ⎜ ⎪ ⎪ ⎪ ⎝ ⎪ ⎪ ⎩

0 s˜i (0) 0 s˜i (1) 0 .. .

s˜i (n i − 1)

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

⎞

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ + g˜ i + f˜i ⎪ ⎟ ⎪ ⎪ ⎟ ⎪ ⎪ ⎟ ⎪ ⎪ ⎠ ⎪ ⎪ ⎭ ni

(d)

˜ i s˜i + z˜ i + n˜ i , = H

(38)

˜ i ∈ Cni ×ni in (d) is a frequency-domain channel where H matrix by keeping the rows and columns of the even subcarriers from T˜ i . z˜ i ∈ Cni ×1 and n˜ i ∈ Cni ×1 are interference vector and noise vector in the frequency domain which are obtained by keeping the elements of the even subcarriers from g˜ i and f˜i , respectively.

First, from (8), the interference from the UEi−1 to UEi can be written as ai = D−1 i Di−1 Hi−1 xi−1 .

 Wi , #

y˜ i = {˜ri }ni

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

A PPENDIX B D ERIVATION OF (12)

2n i

WiH

carriers as

(32)

where {·}2ni donates the vector consisting of 2n i elements corresponding to the subcarriers position of UEi . By substituting (30) into (32), the frequency-domain signal r˜ i can be rewritten as         xi ni H Hi 0 H zi H r˜ i = W +W +W 0 0 0 0 0 2n i          0 H x z n i i i = WH WW H i +W H +W H 0 0 0 0 0 2n

and T˜ i in (b) is defined as     Hi 0 H ˜ Ti = W W 0 0 2n

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(37)

With (33)-(37), the received frequency-domain signal corresponding to UEi can be derived by keeping the even sub-

(39)

Next, the interference signal ai is padded with zero and transformed to the frequency domain by applying 2N-FFT as shown in Fig. 1, and the output can be expressed as     −1   Hi−1 0 xi−1 Di−1 0 Di 0 r˜ ai = WiH 0 0 0 0 0 0 0      −1   Di−1 0 Hi−1 0 xi−1 Di 0 = WH 0 0 0 0 0 0 0 2n i    −1   0 D D 0 i−1 i = WH W WW H 0 0 0 0     Hi−1 0 xi−1 WW H ·W H 0 0 0 2n i

(e)

=

˜ i−1 T˜ i−1 x˜ i−1 , ˜ −1 C C i

(40)

˜ i−1 ∈ C2ni ×2ni , T ˜ i−1 ∈ C2ni ×2ni ˜ −1 ∈ C2ni ×2ni , C where C i 2n ×1 i and x˜ i−1 ∈ C in (e) are defined as ⎧       −1 −1 ⎪ 0 0 D D ⎪ i i ˜ −1 = W H ⎪ C W Wi = WiH ⎪ i ⎪ ⎪ 0 0 0 0 ⎪ ⎪ 2n ×2n i i       ⎪ ⎪ ⎪ ⎪ Di−1 0 Di−1 0 ⎪ ˜ ⎪ W Wi C = WH = WiH ⎪ ⎨ i−1 0 0 0 0    2ni ×2ni   . ⎪ Hi−1 0 Hi−1 0 ⎪ H H ⎪ ˜ W W T = W = W ⎪ i−1 i i ⎪ ⎪ 0 0 0 0 ⎪ ⎪ 2n i ×2n i     ⎪ ⎪ ⎪ ⎪ xi−1 xi−1 ⎪ H ⎪ = WiH ⎪ ⎩x˜ i−1 = W 0 0 2n i

(41) Then, the frequency-domain interference signal to the desired n i subcarriers can be derived by keeping the even

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subcarriers as   ˜ i−1 T˜ i−1 x˜ i−1 ˜ −1 C a˜ i = {˜rai }ni = C i

R EFERENCES (f) ni

˜ i−1 H ˜ i−1 s˜i−1 , ˜ −1 D = D i (42)

˜ i−1 ∈ Cni ×ni and H ˜ i−1 ∈ Cni ×ni in ˜ −1 ∈ Cni ×ni , D where D i ( f ) are obtained by keeping the rows and columns of the even ˜ −1 , C ˜ i−1 and T˜ i−1 , respectively. subcarriers from C i Finally, according to (10), the frequency response of the j th subcarrier of UEi can be expressed as I ( j, εi ) = I [ j −εi ] with the CFO (εi ). For UEi−1 , all n i−1 subcarriers have effect on the j th subcarrier of UEi . With the CFO (εi−1 ), the frequency response of the qth subcarrier of UEi−1 at the j th subcarrier of UEi can be expressed as I (q, εi−1 | j, εi ) = I [( j −εi )−(q −εi−1 )], q = 1, · · · , n i−1 . (43) From (42)-(43), for the j th subcarrier of UEi , the interference caused by all n i−1 subcarriers of UEi−1 can be derived as n i−1   ˜ i−1 H ˜ i−1 s˜i−1 ]q I (q, εi−1 | j, εi ) . (44) ˜ −1 D [D [a˜ i ] j = i q=1

Substituting (43) into (44) yields (12). A PPENDIX C D ERIVATION OF (13) ˜ i−1 in (44) can be ˜ −1 D Based on (40)-(42), the product of D i written as  −1      Di−1 0 0 −1 ˜ H Di ˜ Di Di−1 = Wi Wi 0 0 0 0 ni  −1    Di Di−1 0 = WiH , (45) Wi 0 0 n i

D−1 i Di−1 2π ε −j N i

where di ag 1, e

can be expressed according to Di  2π ε − j N i (N+L i +L ch −2) as ,··· ,e

=

D−1 i Di−1  2π(εi−1 −εi ) 2π(εi−1 −εi ) −j −j (N+L i +L ch −2) N N = di ag 1, e ,··· ,e   2π εi−1,i 2π εi−1,i = di ag 1, e− j N , · · · , e− j N (N+L i +L ch −2) . (46) D−1 i Di−1

˜ −1 D ˜ i−1 D i

to the frequency domain, By transforming plays a role in spectrum displacement in the frequency domain. ˜ i−1 ∈ Cni−1 ×ni−1 in (44) is the Furthermore, the matrix H frequency-domain channel matrix of UEi−1 and can be written as ˜ i−1 = di ag(h i−1,1 , h i−1,2 , · · · , h i−1,ni−1 ), H

(47)

where the diagonal element h i−1, j ( j = 1, · · · , n i−1 ) denotes the channel coefficient of the j th subcarrier of UEi−1 . Mul˜ i−1 , the channel coefficient h i−1, j has a ˜ −1 D tiplied by D i spectrum shift (εi−1,i ) yielding h i−1, j +εi−1,i . Thus, from (44)-(47), the interference caused by the CFO deviation between UEi and UEi−1 can be simplified as (13).

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Xiao Chen received the B.S. degree in information science and engineering from the Chien-Shiung Wu Honors College, Southeast University, Nanjing, China, in 2015. She is currently pursuing the Ph.D. degree in electrical and information engineering with Southeast University, Nanjing. Her research interests include massive multiple-input and multiple-output wireless communication systems, advanced waveform techniques, and interference control.

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Liang Wu received the B.S., M.S., and Ph.D. degrees from the School of Information Science and Engineering, Southeast University, Nanjing, China, in 2007, 2010, and 2013, respectively. From 2011 to 2013, he was a visiting student with the School of Electrical Engineering and Computer Science, Oregon State University. He is currently a Lecturer with the National Mobile Communications Research Laboratory, Southeast University, Nanjing. His research interests include ultra-wideband wireless communication system, indoor optical wireless communications, multiple-input and multiple-output wireless communication systems, and orthogonal frequency-division multiplexing.

Zaichen Zhang (M’02–SM’15) was born in Nanjing, China, in 1975. He received the B.S. and M.S. degrees in electrical and information engineering from Southeast University, Nanjing, China, in 1996 and 1999, respectively, and the Ph.D. degree in electrical and electronic engineering from The University of Hong Kong, Hong Kong, in 2002. From 2002 to 2004, he was a Post-Doctoral Fellow with the National Mobile Communications Research Laboratory, Southeast University, China. He joined the School of Information Science and Engineering, Southeast University, China, in 2004, where is currently a Professor. He has published over 150 papers and holds 20 patents. His current research interests include 5G wireless systems, optical wireless communication technologies, and quantum communications.

Jian Dang received the B.S. and Ph.D. degrees from the School of Information Science and Engineering, Southeast University, Nanjing, China, in 2007 and 2013, respectively. From 2010 to 2012, he was a visiting student with the Department of Electrical and Computer Engineering, University of Florida. He is currently a Lecturer with the National Mobile Communications Research Laboratory, Southeast University. His research interests include signal processing in wireless communications, multiple access schemes, non-orthogonal modulation schemes, and visible light communications.

Jiangzhou Wang (F’17) is currently the Head of the School of Engineering and Digital Arts and a Professor of telecommunications with the University of Kent, U.K. He has authored over 200 papers in international journals and conferences in the areas of wireless mobile communications and three books. Prof. Wang is an IET Fellow. He received the Best Paper Award from the 2012 IEEE GLOBECOM. He was the Technical Program Chair of the 2013 IEEE WCNC in Shanghai and the Executive Chair of the 2015 IEEE ICC in London. He serves/served as an Editor for a number of international journals, including the IEEE T RANS ACTIONS ON C OMMUNICATIONS from 1998 to 2013. He was the Guest Editor for the IEEE J OURNAL ON S ELECTED A REAS IN C OMMUNICATIONS and the IEEE Communications Magazine. He is currently an Editor for Science China Information Science. He was an IEEE Distinguished Lecturer from 2013 to 2014.