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Massive MIMO Downlink based on Single CarrierFrequency Domain Processing
Zahra Mokhtari, Student Member, IEEE, Maryam Sabbaghian, Member, IEEE, Rui Dinis, Senior Member, IEEE.
Abstract—In this paper, we investigate the suitability of singlecarrier frequency domain processing (SC-FDP) as the downlinktransmission scheme in a massive multiple input multiple output(MIMO) system. By deriving the sum-rate of the SC-FDPmassive MIMO system theoretically, we show that this methodobtains a sum-rate similar to that of orthogonal frequencydivision multiplexing (OFDM) massive MIMO. We also derivethe theoretical sum-rate of both SC-FDP and OFDM in a non-synchronized massive MIMO scenario and show that the formeris significantly larger than the latter. Moreover, we theoreticallyanalyze the sum-rate of both systems in the presence of poweramplifier non-linearity. All the sum-rates are derived for bothzero forcing (ZF) and matched filter (MF) precoding schemes.The results show that the effect of power amplifier non-linearityon the sum-rate of both systems is similar when the number ofusers is large. We also compare SC-FDP with OFDM from thepeak to average power ratio (PAPR) and complexity viewpoints.Although the PAPR of SC-FDP signals is lower than that ofOFDM signals, for MIMO systems this difference decreases aswe increase the number of users, which means that the techniquescan have similar PAPR in massive MIMO systems. The overallcomplexities are similar for SC-FDP and OFDM. Due to thementioned facts, we can conclude that SC-FDP is a promisingtransmission scheme for the downlink of the massive MIMOsystem in the presence of carrier frequency offset (CFO) andpower amplifier non-linearities.
Keywords: Single Carrier, Massive MIMO, Precoding, CFO,PAPR.
I. INTRODUCTION
The appearance of smart phones accelerated the increasingdemand for high rate wireless communications. This trend hasmotivated a great amount of research to develop schemes toameliorate the spectral efficiency and accommodate high rateusers more efficiently [1]. One of the promising solutions tocope with this problem is exploiting massive multiple inputmultiple output (MIMO) systems. This method enhances thespectral and power efficiency significantly [2], [3]. However,to get this technology to the efficient implementation stage,new challenges such as pilot contamination in time divisionduplex (TDD) mode, training and feedback overhead in fre-quency division duplex (FFD) mode, hardware impairmentsand appropriate air interface need to be encountered [4]- [8].
One of the critical issues is to examine the suitability ofdifferent modulation schemes to be adopted in massive MIMOsystems [8]. One of the well-known modulation schemes
Z. Mokhtari and M. Sabbaghian are with the School of Electrical andComputer Engineering, University of Tehran, Tehran, Iran, {z.mokhtari,[email protected]}. M. Sabbaghian is the corresponding author. R. Dinisis with Faculdade de Cincias e Tecnologia, Universidade Nova de Lisboa(FCT-UNL) and Instituto de Telecomunicacoes {[email protected]}
for wireless communications is orthogonal frequency divisionmultiplexing (OFDM). This method converts the frequencyselective wireless channel into flat fading channels over eachsub-carrier. Thus, the system can be analyzed in a flat fad-ing scenario, which has been the subject of many papersin massive MIMO [5], [9]- [14]. In [5], the fundamentalproblem of pilot contamination in the downlink of a multi-cell massive MIMO system has been analyzed and a multi-cell precoding scheme based on minimum mean square error(MMSE) criterion has been developed to mitigate the pilotcontamination. In [9], the authors consider the downlink (DL)of a single cell scenario and compare the spectral and energyefficiency of precoding schemes based on matched filter (MF)and zero forcing (ZF) criteria. The effect of phase noise in theuplink of a massive MIMO system with zero forcing equalizerhas been studied in [10]. The authors in [11] have consideredboth uplink and downlink of a single cell massive MIMOsystem and proposed an MMSE based pilot reuse algorithmto reduce the pilot overhead. In [12], the authors analyze howthe optimal number of scheduled users depends on the numberof BS antennas and other system parameters in a multi-cellmassive MIMO systems. Efficient power and training durationallocations have also been studied in [13] and [14]. We wouldlike to highlight that all the enumerated papers consider aflat fading scenario or, equivalently, an OFDM transmissionscheme.
OFDM is a special case of generalized multi-carrier (GMC)systems [15]. GMC air interfaces are based on utilizing fastFourier transform (FFT) and inverse FFT (IFFT) as efficienttools to deal with the channel frequency selectivity. They alsoallow flexible allocation of time/frequency resources in bothsingle user and multi-user scenarios. Another special caseof GMC, which is used as uplink transmission scheme infourth generation cellular communication systems, is blockwise single carrier scheme [16]. This method equalizes thereceived signal in the frequency domain, thus, it is known assingle carrier frequency domain equalizer (SC-FDE). Any ofthe GMC methods can be used in a massive MIMO framework.It is noteworthy that in the massive MIMO it is usuallyassumed that the base station (BS) has full channel knowledgewhile the user equipments only know the channel statistics[17], [18]. Thus, the main signal processing load is carried bythe BS. This means that in the uplink we have to equalize thesignal and in the downlink we have to precode it while bothequalization and precoding are performed in the BS. Thesetwo processes in the block-wise single carrier systems, areperformed in the frequency domain. Therefore, in this paperwe refer to this system as single carrier frequency domain
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processing (SC-FDP). We used the term processing to includeboth equalization and precoding.
In this paper, we focus on the downlink transmission in amassive MIMO system. We consider SC-FDP as a candidateair interface and investigate its favorable and unfavorableattributes. The research on SC-FDP, unlike OFDM, is verylimited in the context of massive MIMO and is confinedto [19], [20]. Contrastingly to our work, [19] considers theuplink transmission of a single cell massive MIMO system. In[19], a low complexity iterative detection algorithm for singlecarrier frequency division multiple access (SC-FDMA) schemehas been developed. The proposed algorithm combines a fre-quency domain MMSE equalization with parallel interferencecancellation. As in our paper, [20] considers the downlinktransmission and the BER and peak to average power ratio(PAPR) of the orthogonal frequency division multiple access(OFDMA) and SC-FDMA schemes are compared. However,this comparison is performed via simulations and there is noanalytical derivation of sum-rate.
In this paper, we analytically examine the suitability ofthe SC-FDP method for downlink transmission of massiveMIMO systems in the presence of hardware impairments. Theconsidered hardware impairment includes the instabilities ofthe oscillators in the form of carrier frequency offset (CFO)and the nonlinearities of the power amplifiers. We analyticallyderive the achievable sum-rate of a SC-FDP massive MIMOsystem in the presence and absence of CFO. To put this anal-ysis into perspective, we compare the achievable rate in bothscenarios with the corresponding values of OFDM massiveMIMO systems. Although the sum-rate of a fully synchronizedOFDM massive MIMO system has been already derived inother papers, the literature lacks such analysis in the presenceof CFO. Therefore, we derived the sum-rate of an OFDMmassive MIMO system hit by CFO. Through our analysis, weshow that the SC-FDP and OFDM obtain similar sum-rates ina synchronized case. In a non-synchronized system, however,we show analytically that the interference term generated dueto CFO does not vanish in OFDM as the number of antennasgoes to infinity while this is not the case for SC-FDP systemswhen the users know only the channel statistics. We havealso derived the sum-rate of these two systems in presence ofpower amplifier non-linearity for systems with large number ofusers which is usually the case in the massive MIMO systems.Our analysis show that for small number of users the SC-FDP massive MIMO system has lower PAPR and out-of-bandradiation compared to the OFDM massive MIMO systems.However, for large number of users the derived sum-rate,PAPR and out-of-band radiation are identical for both systems.Furthermore, as presented in our analysis in the presence ofpower amplifier non-linearity, the sum-rate of both systemsdoes not increase unlimitedly as the number of BS antennastends to infinity. We can summarize the contributions of thepaper as follows:
• Deriving the sum-rate of the SC-FDP in a synchronizedmassive MIMO system.
• Deriving the sum-rate of the SC-FDP in a non-synchronized massive MIMO system.
• Deriving the sum-rate of the OFDM in a non-synchronized massive MIMO system.
• Deriving the sum-rate of the SC-FDP in the presence ofpower amplifier non-linearity.
• Deriving the sum-rate of the OFDM in the presence ofpower amplifier non-linearity.
• Comparing the PAPR and out-of-band radiation of theSC-FDP and OFDM massive MIMO systems.
All the sum-rate derivations are done for both MF and ZFprecoding schemes. Overall, we conclude that in a massiveMIMO scenario SC-FDP outperforms OFDM when hardwareimpairments such as CFO and power amplifier non-linearityare taken into account. This is of crucial importance speciallyin distributed massive MIMO systems where the base stationantennas are geographically distributed over the cell [21]- [23].In this framework, each antenna is impelled to have its own RFchain including the oscillator and power amplifier. Distributedmassive MIMO is more expensive than a system where mas-sive number of antennas are located in a single array. To makethe system economically worthwhile, we have to utilize low-cost oscillators and power amplifiers. Inexpensive oscillatorsare imprecise and suffer from frequency instabilities. Cheappower amplifiers’s characteristic, likewise, are nonlinear whichintroduces in-band distortion and spectral splatter resulting inperformance degradation of in-band or adjacent users.
The rest of the paper is organized as follows. Section IIdescribes the system model of SC-FDP massive MIMO in thedownlink. In section III achievable sum-rates are derived inthe presence and absence of CFO and in presence of poweramplifier non-linearity. Section IV compares SC-FDP fromdifferent points of view with OFDM. Section V presents theperformance results and section VI concludes the paper.
II. SYSTEM MODEL
We consider the downlink in a single cell scenario wherethe BS is equipped with M antennas and there are K singleantenna users. All users exploit the same time and frequencyresources, simultaneously. The channel coefficient of the l-th path of the frequency selective channel between the m-thantenna of the BS and k-th user is denoted by hm,k,l which isa zero-mean complex Gaussian random variable with varianceof σ2
k,l. We assume the channel coefficients are independentand normalized such that
∑L−1l=0 σ2
k,l = 1 for all users whereL is the number of paths. The n-th coefficient of the channelfrequency response which is obtained as
Hm,k,n =
L−1∑l=0
hm,k,le− j2πnlN (1)
is a zero-mean complex Gaussian random variable with vari-ance of 1 and N is the data block length. The BS has perfectchannel state information and the user equipment has thechannel statistics only. In the rest of the paper, vectors andmatrices are denoted by boldface letters. The vectors in thetime domain are denoted by bold small letters and those inthe frequency domain are denoted by bold capital letters.
In the rest of this section, we explain the details of thetransmitted signal in the downlink of a massive MIMO system
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with SC-FDP modulation. We present the q-th symbol of thedata block which should be transmitted from the BS to the k-th user by xk,q where q = 0, ..., N − 1. The data symbols areassumed to be independent and identically distributed randomvariables with zero mean and variance P . By taking an N -point FFT of this vector, we convert the data block into thefrequency domain and present the n-th component of thefrequency domain vector by Xk,n. Then, the n-th data symbolsof the corresponding blocks of all K users are precoded in thefrequency domain using the M ×K precoding matrix W n asfollows
Y n = W nXn, (2)
where Y n = [Y1,n, Y2,n, ..., YM,n]T , Xn =[X1,n, X2,n, ..., XK,n]T , n = 0, ..., N − 1 and Ym,n isthe n-th symbol of data block of the m-th BS antenna infrequency domain. It is worth mentioning that the matrix W n
is associated with each user’s channel frequency response onthe n-th sub-carrier. Thus, we need N precoding matrices toprecode the whole data block of all users. After the precodingin the frequency domain, the vector obtained for each BSantenna is converted to time domain by an N point IFFT asfollows
ym,i =1
N
N−1∑n=0
Ym,nej2πniN
=1
N
N−1∑n=0
(K∑k=1
wm,k,nXk,n
)ej2πniN
(3)
where ym,i is the i-th component of the vector transmittedfrom the m-th antenna and wm,k,n is element in the m-th rowand k-th column of W n. The generated time domain vectoris preceded by cyclic prefix and transmitted over the channel.
The i-th component of received vector at user u, afterremoving the CP, can be expressed as
xu,i =1
N
M∑m=1
N−1∑n=0
Hm,u,nYm,nej2πniN + ηu,i
=1
N
M∑m=1
N−1∑n=0
K∑k=1
N−1∑q=0
wm,k,nHm,u,nxk,qej2πn(i−q)
N + ηu,i
(4)
where ηu,i is the i-th element of the zero-mean complex whiteGaussian noise with variance of σ2 added to the u-th usersignal. Fig. 1 shows the block diagram of SC-FDP massiveMIMO system in the downlink.
III. ACHIEVABLE SUM-RATE
In this section, we derive the achievable sum-rate in thedownlink of a SC-FDP massive MIMO system taking advan-tage of Theorem 1 in [24]. We consider three scenarios whichare presented in the following three sub-sections. In the firstscenario, the system is fully synchronized and there is nopower amplifier non-linearity and in the second and third onethe system suffers from frequency instabilities in the form ofa fixed CFO and power amplifier non-linearity, respectively.
Figure 1: Block diagram of SC-FDP and OFDM Massive MIMOsystems in the downlink
A. Synchronized Massive MIMO with no Power AmplifierNon-linearity
As explained earlier, the user knowledge of the channel isconfined to its statistics. Thus, to satisfy the conditions ofTheorem 1 in [24] in order to derive the ergodic achievablesum-rate we can rewrite (4) as [18]
xu,i =1
N
M∑m=1
N−1∑n=0
E{wm,u,nHm,u,n}xu,i︸ ︷︷ ︸Desired signal
+ξu,i (5)
where E{.} is the expectation and ξu,i is the effective noisedefined as
ξu,i =1
N
M∑m=1
N−1∑n=0
wm,u,nHm,u,nxu,i
− 1
N
M∑m=1
N−1∑n=0
E{wm,u,nHm,u,n}xu,i
+1
N
N−1∑q=1q 6=i
M∑m=1
N−1∑n=0
wm,u,nHm,u,nxu,qej2πn(i−q)
N
+1
N
K∑k=1k 6=u
N−1∑q=0
M∑m=1
N−1∑n=0
wm,k,nHm,u,nxk,qej2πn(i−q)
N + ηu,i.
(6)
Considering (5) and (6), we can obtain the SINR for the u-thuser as follows
SINRu,i =
E
{∣∣∣ 1N
∑Mm=1
∑N−1n=0 E{wm,u,nHm,u,n}xu,i
∣∣∣2}Pξu,i
,
(7)
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where Pξu,i is the power of the effective noise at user u andis defined as
Pξu,i = PVar
(1
N
N−1∑n=0
M∑m=1
wm,u,nHm,u,n
)
+ P
N−1∑q=0q 6=i
E
∣∣∣∣∣ 1
N
N−1∑n=0
M∑m=1
wm,u,nHm,u,nej2π(i−q)n
N
∣∣∣∣∣2
+ P
K∑k=1k 6=u
N−1∑q=0
E
∣∣∣∣∣ 1
N
N−1∑n=0
M∑m=1
wm,k,nHm,u,nej2π(i−q)n
N
∣∣∣∣∣2
+ σ2
(8)
If we use MF precoding, the precoder coefficient in the m-throw and u-th column of W n can be expressed as
wm,u,n =H∗m,u,n√
K∑Mm=1 |Hm,u,n|2
'H∗m,u,n√
KME{|Hm,u,n|2}=H∗m,u,n√KM
,
(9)
where H∗m,u,n is the complex conjugate of Hm,u,n and theapproximation in the above equation is due to the fact that1M
∑Mm=1 |Hm,u,n|2 can be approximated by E{|Hm,u,n|2}
for large M . Substituting (9) in (7) and noting thatE{|Hm,k,n|4} = 2 for k = 1, ...,K, m = 1, ...,M andn = 0, ..., N − 1, the SINR of the u-th user is simplifiedto
SINRMFu,i =
M
K +K σ2
P
, (10)
Since in (5), E{wm,u,nHm,u,n} is known and xu,i and the ef-fective noise can be assumed uncorrelated [18], by consideringthe Gaussian distribution for data symbols, we can obtain theachievable sum-rate of users as [24, Theorem 1]
RMFSC−FDP = K log2
(1 +
M
K +K σ2
P
), (11)
For the ZF precoding, the precoding matrix W n is defined as
W n =HH
n (HnHHn )−1√
E{trac(HHn (HnH
Hn )−1)}
=
√M −KK
HHn (HnH
Hn )−1,
(12)
where Hn is a K × M matrix whose k-th row and m-thcolumn element is Hm,k,n. In this case since HnW n =√
M−KK IK×K , the i-th received symbol at user u is
xu,i =
√M −KK
xu,i + nu,i. (13)
So the SINR of the received signal at user u is
SINRZFu,i =P (M −K)
Kσ2, (14)
and the achievable sum-rate of users is
RZFSC−FDP = K log2
(1 +
P (M −K)
Kσ2
). (15)
B. Non-Synchronized Massive MIMO
In this sub-section, we investigate the achievable sum-ratein the downlink of the SC-FDP massive MIMO system whenthe system is hit by carrier frequency offset. Let us present thenormalized CFO of the u-th user by εu. The i-th componentof the received signal of this user can be expressed as
xu,i =1
N
M∑m=1
N−1∑n=0
K∑k=1
N−1∑q=0
wm,k,nHm,u,nxk,qej2πn(i−q)
N ej2πεuiN
+ ηu,i.(16)
The rotation of the received signal caused by CFO results inSINR and BER degradation. Since the users know the channelstatistics, the received signal can be written as
xu,i =1
N
N−1∑n=0
M∑m=1
E{wm,u,nHm,u,n}xu,i︸ ︷︷ ︸Desired signal
+ζu,i, (17)
where ζu,i, the effective noise at xu,i, is defined as
ζu,i =1
N
M∑m=1
N−1∑n=0
wm,u,nHm,u,nej2πεuiN xu,i
− 1
N
N−1∑n=0
M∑m=1
E{wm,u,nHm,u,n}xu,i
+1
N
N−1∑q=0q 6=i
M∑m=1
N−1∑n=0
wm,u,nHm,u,nej2π(ni−nq+εui)
N xu,q
+1
N
K∑k=1k 6=u
N−1∑q=0
M∑m=1
N−1∑n=0
wm,k,nHm,u,nej2π(ni−nq+εui)
N xk,q
+ ηu,i.(18)
The power of ξu,i is
Pξu,i = PE
{∣∣∣∣∣ 1
N
M∑m=1
N−1∑n=0
wm,u,nHm,u,nej2πεuiN
− 1
N
N−1∑n=0
M∑m=1
E{wm,u,nHm,u,n}
∣∣∣∣∣2
+ P
N−1∑q=0q 6=i
E
∣∣∣∣∣ 1
N
N−1∑n=0
M∑m=1
wm,u,nHm,u,nej2π(i−q)n
N
∣∣∣∣∣2
+ P
K∑k=1k 6=u
N−1∑q=0
E
∣∣∣∣∣ 1
N
N−1∑n=0
M∑m=1
wm,k,nHm,u,nej2π(i−q)n
N
∣∣∣∣∣2+ σ2.
(19)
Henceforth, the SINR for the i-th sample of the u-th user isas follows
SINRu,i =E∣∣∣ 1N
∑N−1n=0
∑Mm=1E{wm,u,nHm,u,n}xu,i
∣∣∣2Pξu,i
.
(20)
5
For the MF and ZF precoders, the above SINR is simplifiedto (21) and (22), respectively.
SINRMFu,i =
1
2− 2 cos( 2πεuiN ) + K
M + σ2KPM
, (21)
SINRZFu,i =1
(cos( 2πεuiN )− 1)2 + sin2( 2πεui
N ) + σ2KP (M−K)
.
(22)
Since
E{xu,iζ∗u,i} = PE{ 1
N
M∑m=1
N−1∑n=0
wm,u,nHm,u,n}(ej2πεuiN − 1
),
we can conclude that the desired signal and the effective noiseare uncorrelated for small εu. Thus, regarding Theorem 1 in[24], the achievable sum-rate for the SC-FDP massive MIMOsystem with MF and ZF precoder for small CFO is as (23)and (24), respectively.
RMFCFO
SC−FDP =1
N
K∑u=1
N−1∑i=0
log
(1 +
1
2− 2 cos( 2πεuiN
) + KM
+ σ2KPM
),
(23)
RZFCFO
SC−FDP =
1
N
K∑u=1
N−1∑i=0
log
(1 +
1
(cos( 2πεuiN
)− 1)2 + sin2( 2πεuiN
) + σ2KP (M−K)
)(24)
C. Massive MIMO with Non-linear Power Amplifier
In this subsection, we investigate the achievable sum-ratein the downlink of the SC-FDP massive MIMO system withnon-linear power amplifier. We first investigate the distributionof the power amplifier input signal. The i-th sample of thepower amplifier input signal of the m-th antenna in the SC-FDP massive MIMO system is
ym,i =1
N
K∑k=1
N−1∑n=0
N−1∑q=0
wm,k,nxk,qej2πn(i−q)
N . (25)
Let us define
ςk =
N−1∑n=0
N−1∑q=0
wm,k,nxk,qej2πn(i−q)
N .
The parameter ςk is not necessarily Gaussian. Since wm,k,nand wm,k′,n and the two random variables xk,q and xk′,qare independent for k 6= k′, then ςk’s are also independentfor different users. Thus, in general the distribution of ym,iis not Gaussian. However, for large K the ςk’s are inde-pendent and identically distributed and we can approximatethe distribution of ym,i as Gaussian due to the central limittheorem. Fortunately, in the massive MIMO systems, K isusually large and the Gaussian approximation is valid. Thus,we use the Bussgang Theorem to derive the sum-rate of thissystem. According to the Bussgang Theorem, the non-linearpower amplifier output signal is [25]
ym,i =α
N
N−1∑n=0
K∑k=1
wm,k,nXk,nej2πniN + ηapm,i, (26)
where α is the attenuation factor and ηapm,i is the distortionsignal modeled by a zero mean Gaussian process with varianceσ2ap which is independent of the signal. Consequently, the i-th
sample of the received signal of u-th user is
xu,i =α
N
M∑m=1
N−1∑n=0
E{wm,u,nHm,u,n}xu,i︸ ︷︷ ︸Desired signal
+ξu,i, (27)
where ξu,i is the effective noise defined as
ξu,i =α
N
M∑m=1
N−1∑n=0
wm,u,nHm,u,nxu,i
− α
N
M∑m=1
N−1∑n=0
E{wm,u,nHm,u,n}xu,i
+α
N
N−1∑q=1,q 6=i
M∑m=1
N−1∑n=0
wm,u,nHm,u,nxu,qej2πn(i−q)
N
+α
N
K∑k=1,k 6=u
N−1∑q=0
M∑m=1
N−1∑n=0
wm,k,nHm,u,nxk,qej2πn(i−q)
N
+1
N
M∑m=1
N−1∑n=0
Hm,u,nNapm,ne
j2πniN + ηu,i.
(28)
In the above equation, Napm,n is the distortion noise of the m-
th antenna on the n-th sub-carrier. Considering (27) and (28),we can obtain the SINR for the u-th user as follows
SINRu,i =
E
{∣∣∣ αN ∑Mm=1
∑N−1n=0 E{wm,u,nHm,u,n}xu,i
∣∣∣2}Pξu,i
,
(29)
where Pξu,i is the power of the effective noise for the u-thuser defined as
Pξu,i = PVar
(1
N
N−1∑n=0
M∑m=1
wm,u,nHm,u,n
)
+ P
N−1∑q=0q 6=i
E
∣∣∣∣∣ 1
N
N−1∑n=0
M∑m=1
wm,u,nHm,u,nej2π(i−q)n
N
∣∣∣∣∣2
+ P
K∑k=1k 6=u
N−1∑q=0
E
∣∣∣∣∣ 1
N
N−1∑n=0
M∑m=1
wm,k,nHm,u,nej2π(i−q)n
N
∣∣∣∣∣2
+Mσ2ap + σ2.
(30)
For the MF precoding, the SINR is simplified to
SINRMFu,i =
M
K +Mσ2ap
Kα2P + σ2 K
Pα2
. (31)
Using Theorem 1 in [24], the achievable sum-rate is
RMFAP
SC−FDP = K log2
(1 +
M
K +Mσ2ap
Kα2P + σ2 K
Pα2
).
(32)
If we use the ZF precoder, it yields the following SINR
SINRZFu,i =Pα2(M −K)
MKσ2ap +Kσ2
, (33)
6
whose corresponding achievable sum-rate is
RZFAP
SC−FDP = K log2
(1 +
Pα2(M −K)
MKσ2ap +Kσ2
). (34)
IV. COMPARISON WITH THE OFDM MASSIVE MIMO
In this section, we compare different aspects of the SC-FDP massive MIMO and OFDM massive MIMO system.The system model of OFDM massive MIMO is similar tothat of SC-FDP massive MIMO, except that the FFT blocksare moved from the BS to the user terminals. Fig. 1 showsthe block diagram of SC-FDP and OFDM massive MIMOsystems.
A. Achievable Sum-Rate
The following sub-sections discuss the achievable sum-rateof an OFDM massive MIMO system in three scenarios. In thefirst scenario the system is synchronized and there is no poweramplifier non-linearity. In the second and third scenarios thesystem suffers from CFO and power amplifier non-linearity,respectively. The achievable sum-rate of the first scenario isalready obtained in the literature, however, the second andthird scenarios have not been investigated in the literature.
1) Synchronized Massive MIMO with no Power Ampli-fier Non-linearity: Using the same notation of the previ-ous section, in an OFDM massive MIMO system, Xk =[Xk,0, ..., Xk,N−1] is the data block to be sent to the k-thuser where in Xk,n, n denotes the subcarrier number. Then-th data symbols of all users are precoded simultaneouslyusing the matrix Wn as explained in (2). The precoded blockassociated with each BS antenna is then passed through anN -point IFFT and preceded by CP. At the user terminal, afterremoving the CP, the signal goes through an N -point FFTblock. Thus, the n-th symbol received by the u-th user is
Xu,n =
M∑m=1
K∑k=1
Hm,u,nwm,k,nXk,n +Nu,n
=
M∑m=1
E{Hm,u,nwm,u,n}Xu,n + Ξu,n,
(35)
where the subscripts denote the n-th sub-carrier of the u-thuser. The parameter Nu,n is the zero mean additive Gaussiannoise whose variance is σ2 and Ξu,n is the effective noisedefined as
Ξu,n =M∑m=1
Hm,u,nwm,u,nXu,n −M∑m=1
E{Hm,u,nwm,u,n}Xu,n
+
M∑m=1
K∑k=1,k 6=u
Hm,u,nwm,k,nXk,n +Nu,n.
(36)
Therefore, the SINR of the u-th user received data over then-th sub-carrier is
SINRu,n =E{|∑Mm=1E{Hm,u,nwm,u,n}Xu,n|2
}PΞu,n
, (37)
where PΞu,n is the power of the effective noise which isdefined as
Ξu,n = PVar{M∑m=1
Hm,u,nwm,u,n}
+ P
K∑k=1,k 6=u
E
{|M∑m=1
Hm,u,nwm,k,n|2}
+ σ2.
(38)
For the MF and ZF precoders, the SINR is simplified to (39)and (40), respectively.
SINRMFu,n =
M
K +K σ2
P
, (39)
SINRZFu,n =P (M −K)
Kσ2. (40)
By considering Theorem 1 of [24] the corresponding sum-ratesare [9]
RMFOFDM = K log2
(1 +
M
K +K σ2
P
), (41)
RZFOFDM = K log2
(1 +
P (M −K)
Kσ2
). (42)
Comparing (11) and (41) and comparing (15) and (42)shows that the achievable sum-rate in the downlink of syn-chronized SC-FDP and synchronized OFDM are identical forboth MF and ZF precoders. This is due to the fact that in thedownlink the users only know the channel statistics.
2) Non-Synchronized Massive MIMO: In the presence ofcarrier frequency offset, the data received on the n-th sub-carrier of the u-th user can be expressed as
Xu,n =
1
N
N−1∑i=0
M∑m=1
N−1∑r=0
K∑k=1
Hm,u,rwm,k,rXk,rej2πi(r−n+εu)
N +Nu,n
=
(1
N
N−1∑i=0
ej2πiεuN
)M∑m=1
K∑k=1
Hm,u,nwm,k,nXk,n
+1
N
N−1∑i=0
M∑m=1
N−1∑r=0r 6=n
K∑k=1
Hm,u,rwm,k,rXk,rej2πi(r−n+εu)
N
︸ ︷︷ ︸Additional interference
+Nu,n,
(43)
Comparing (35) and (43), we can see that in OFDM massiveMIMO the CFO generates an additional interference term dueto users’ interfering adjacent sub-carriers.
Since the users know the channel statistics, we can writeXu,n as
Xu,n =
M∑m=1
E{wm,u,nHm,u,n}Xu,n + Ξu,n, (44)
7
where Ξu,n is the effective noise over the n-th sub-carrier ofthe u-th user which is defined as
Ξu,n =
M∑m=1
wm,u,nHm,u,n(1
N
N−1∑i=0
ej2πεuiN )Xu,n
−M∑m=1
E{wm,u,nHm,u,n}Xu,n
+1
N
N−1∑r=0,r 6=n
N−1∑i=0
M∑m=1
(wm,u,rHm,u,rej2π(r−n+εu)i
N )Xu,r
+1
N
K∑k=1,k 6=u
N−1∑r=0
N−1∑i=0
M∑m=1
(wm,k,rHm,u,rej2π(r−n+εu)i
N )Xk,r
+Nu,n.(45)
The power of Ξu,n will be
PΞu,n = E
{∣∣∣∣∣M∑m=1
wm,u,nHm,u,n(1
N
N−1∑i=0
ej2πεuiN )
−M∑m=1
E{wm,u,nHm,u,n}
∣∣∣∣∣2
+P
N2
N−1∑r=0r 6=n
E
∣∣∣∣∣N−1∑i=0
M∑m=1
(wm,u,rHm,u,rej2π(r−n+εu)i
N )
∣∣∣∣∣2
+P
N2
K∑k=1k 6=u
N−1∑r=0
E
∣∣∣∣∣N−1∑i=0
M∑m=1
(wm,k,rHm,u,rej2π(r−n+εu)i
N )
∣∣∣∣∣2
+ σ2. (46)
Hence, we can derive the SINR for the n-th sub-carrier of theu-th user as follows
SINRu,n =P∣∣∣∑M
m=1E{wm,u,nHm,u,n}∣∣∣2
PΞu,n
. (47)
By using the MF precoding, the SINR is obtained as
SINRMFu,n =
M
K + 2M(
1− 1N
∑N−1i=0 cos( 2πεui
N ))
+ σ2KP
.
(48)
For the ZF precoding, the SINR is calculated as
SINRZFu,n =1
2(
1− 1N
∑N−1i=0 cos( 2πεui
N ))
+ σ2KP (M−K)
.
(49)
Since
E{Xu,iΞ∗u,i} = PE{M∑m=1
wm,u,nHm,u,n}
(1
N
N−1∑i=0
ej2πεuiN − 1
),
we can conclude that the desired signal and the effective noiseare uncorrelated for small εu. Thus, considering Theorem 1in [24], the achievable sum-rate of the users for MF and ZFprecoding for small CFO are
RMFCFO
OFDM =
K∑u=1
1
N
N−1∑n=0
log2
(1 + SINRMF
u,n
), (50)
RZFCFO
OFDM =
K∑u=1
1
N
N−1∑n=0
log2
(1 + SINRZFu,n
). (51)
Therefore, if M tends to infinity, the sum-rate in the non-synchronized OFDM massive MIMO system is limited to
RMFCFO
OFDM∞ = RZFCFO
OFDM∞ =
K∑u=1
1
N
N−1∑n=0
log2
1 +1
2(
1− 1N
∑N−1i=0 cos( 2πεui
N )) .
(52)
This is due to the fact that the interference term generatedby the CFO in (48) and (49) does not vanish as M tendsto infinity. Consequently, the achievable sum-rate does notincrease unlimitedly.To compare this value with that of the non-synchronized SC-FDP massive MIMO system, we note that when M tends toinfinity, (23) and (24) are simplified to
RMFCFO
SC−FDP∞ = RZFCFO
SC−FDP∞ =
1
N
K∑u=1
N−1∑i=0
log
(1 +
1
2(1− cos( 2πεuiN ))
).
(53)
Since 1 − cos( 2πεuiN ) = 0 for i = 0, the sum-rate of the
non-synchronized SC-FDP massive MIMO system increasesunlimitedly as the number of BS antennas tends to infinity.
Comparing (53) and (52) indicates that, unlike the syn-chronized case where the achievable sum-rate of SC-FDP andOFDM are identical, in non-synchronized case where the usersknow the channel statistics there is a fundamental differencebetween the enumerated massive MIMO systems. As can beverified, in SC-FDP by increasing M the achievable sum-rate increases. In OFDM, however, for very large number ofantennas the achievable sum-rate is bounded by (52). This dif-ference is of vital importance specially in distributed massiveMIMO systems where due to having several independent localoscillators we have to use inexpensive ones in which frequencystability is a challenge and suffer further from the frequencyoffset.
3) Massive MIMO with Non-linear Power Amplifier: Inthis sub-section, we investigate the achievable sum-rate in thedownlink of the OFDM massive MIMO system with non-linear power amplifier. The i-th sample of the power amplifierinput signal of the m-th antenna is
ym,i =1
N
N−1∑n=0
K∑k=1
wm,k,nXk,nej2πinN . (54)
For large N , the distribution of (54) can be approximated byGaussian. Thus, we use the Bussgang Theorem to derive thesum-rate of this system. According to the Bussgang Theorem,the output signal of the power amplifier of the m-th antennaover the n-th sub-carrier is [25]
Ym,n =α
K∑k=1
wm,k,nXk,n +Napm,n, (55)
where Napm,n is the distortion signal which is independent of
the signal and is modeled by a zero mean Gaussian process
8
with variance of σ2ap. Consequently, the received signal at the
n-th sub-carrier of the u-th user is
Xu,n =α
M∑m=1
E{wm,u,nHm,u,n}Xu,n︸ ︷︷ ︸Desired signal
+Ξu,n, (56)
where Ξu,n is the effective noise defined as
Ξu,n =α
(M∑m=1
wm,u,nHm,u,n −M∑m=1
E{wm,u,nHm,u,n}
)Xu,n
+α
K∑k=1k 6=u
M∑m=1
wm,k,nHm,u,nXk,n +
M∑m=1
Hm,u,nNapm,n +Nu,n.
(57)
Henceforth, we can obtain the SINR of the u-th user as follows
SINRu,n =
E
{∣∣∣α∑Mm=1E{wm,u,nHm,u,n}Xu,i
∣∣∣2}PΞu,i
,
(58)
where PΞu,n is the power of the effective noise defined as
PΞu,n = PVar
(M∑m=1
wm,u,nHm,u,n
)
+ P
K∑k=1k 6=u
E
∣∣∣∣∣M∑m=1
wm,k,nHm,u,n
∣∣∣∣∣2+Mσ2
ap + σ2.
(59)
If we use the MF precoding, we can calculate the SINR asfollows
SINRMFu,n =
M
K +Mσ2ap
Kα2P + σ2 K
Pα2
. (60)
Therefore, by using Theorem 1 of [24], the achievable sum-rate is
RMFAP
OFDM = K log2
(1 +
M
K +Mσ2ap
Kα2P + σ2 K
Pα2
). (61)
For the ZF precoder, the SINR is obtained as
SINRZFu,n =Pα2(M −K)
MKσ2ap +Kσ2
, (62)
and the correspoding achievable sum-rate is
RZFAP
OFDM = K log2
(1 +
Pα2(M −K)
MKσ2ap +Kσ2
). (63)
By comparing (32) and (61) and comparing (34) and (63),we can conclude that the effect of the power amplifier non-linearity on the sum-rate of both systems is similar when thenumber of users is large. As can also be seen, in the presenceof the non-linear power amplifier the sum-rate of both systemsdoes not increase unlimitedly as the number of the BS antennastends to infinity and it is limited to
RMFAP
SC−FDP = RMFAP
OFDM = K log2
(1 +
Pα2
Kσ2ap
), (64)
for both MF and ZF precoders.
−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.40
1
2
3
4Real Part
−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.40
1
2
3
4Imaginary Part
Figure 2: Histogram of the input signal of the power amplifier inSC-FDP for K = 1.
B. PAPR and Out-of-Band Radiation Issues
It is well known that in the absence of precoders, thePAPR of OFDM signals is significantly larger than that ofsingle carrier systems based on the same constellation. Thisnecessitates using expensive RF amplifiers to avoid in-banddistortion and out-of-band radiation. This issue is more criticalin massive MIMO systems where the number of RF amplifiersgrows with M . In the following, we compare the PAPR andout-of-band radiation of OFDM and SC-FDP massive MIMOsystems in the downlink where the system is equipped withprecoder.
The PAPR of the signal transmitted from the m-th antennaof the BS is expressed as
PAPRm =max |ym,i|2
1N
∑N−1i=0 |ym,i|2
.
In the OFDM massive MIMO system, ym,i is defined as (54).We know that for large N , the distribution of (54) can beapproximated by Gaussian. Thus, the CCDF of the PAPR forthe OFDM case can be computed as [26]
Pr(PAPRm > γ0) = 1− (1− e−γ0)2.8N (65)
For SC-FDP massive MIMO ym,i is defined as (25). Wepreviously stated that the distribution of (25) is Gaussian forlarge number of users. So we expect that the SC-FDP andOFDM massive MIMO systems to have the same out-of-bandradiation and PAPR performance for large number of users.For small K, however, we expect that the SC-FDP system tohave a lower PAPR than the OFDM system. Fig. 2 and Fig. 3present the histogram of ym,i in the SC-FDP massive MIMOsystem with K = 1 and K = 10, respectively. These figuresindicate that the distribution of ym,i in SC-FDP for K = 1can also be approximated as Gaussian, but this approximationis more accurate as the number of users increases. So, in thedownlink we expect the PAPR of SC-FDP massive MIMOto be less than that of the OFDM when the number of usersare small. Contrastingly, as the number of users increases, thePAPR of the SC-FDP massive MIMO also increases and tendsto that of the OFDM massive MIMO. The simulation resultsfor the PAPR and out-of-band radiation provided in section Valso confirm this point.
9
−0.6 −0.4 −0.2 0 0.2 0.4 0.60
1
2
3Real Part
−0.6 −0.4 −0.2 0 0.2 0.4 0.60
1
2
3Imaginary Part
Figure 3: Histogram of the input signal of the power amplifier inSC-FDP for K = 10.
C. Complexity
In this section we will analyze the complexity of the trans-mitter and receiver of SC-FDP and OFDM massive MIMOsystems in the downlink. At the transmitter side of SC-FDPmassive MIMO, we have K FFT and M IFFT blocks totransmit the data to frequency domain and back to timedomain, respectively. So in overall these blocks require (K +M)N2 log2N complex multiplications and (K+M)N log2Ncomplex additions. The precoding is done at the transmitter inthe frequency domain with linear precoders. Considering (2),to precode a data block of length N we need NMK complexmultiplications and NM(K−1) complex additions. In OFDM,however, we have M IFFT blocks at the transmitter and a FFTblock at each receiver. So in the OFDM case we have K FFTblocks which are located at the user side and not at the BS.Consistent with SC-FDP, in OFDM the precoding is done atthe transmitter side in the frequency domain.
Thus, as presented in Table I, the overall complexity ofthe SC-FDP and OFDM massive MIMO systems have similarcomputational complexities. Both systems involve M IFFTand K FFT blocks in the downlink. They also use similarlinear frequency domain precoding and detection schemes.However, in the SC-FDP massive MIMO all the computationalload is concentrated in the BS and the computational complex-ity at the user terminal is less than that of the OFDM case. It isworth mentioning that in the multiple access forms of OFDMand SC-FDP, OFDMA and SC-FDMA, there is sub-carrierallocation and deallocation. Thus, we need an additional FFTand IFFT block at the receiver side in the SC-FDMA casetoo. Thus, the complexity of the SC-FDMA massive MIMOis higher than its multi-carrier counterpart.
V. PERFORMANCE RESULTS
In this section, we investigate the efficiency of the SC-FDP, and OFDM massive MIMO systems through comparativesimulations. We consider a massive MIMO system whose datablock length for SC-FDP and OFDM is N = 128 and thedata is modulated as QPSK. We consider multipath channelwith L = 15 paths with uncorrelated Rayleigh fading on thedifferent multipath components and for different antennas. For
Table I: Number of arithmetic operations required for precoding adata block of length N in a system with K users and M BS
antennas and channel impulse response of length L
Number of ComplexMultiplications
Number of ComplexAdditions
SC-FDP (K +M)N2log2 N +
NMK(K + M)N log2 N +NM(K − 1)
OFDM (K +M)N2log2 N +
NMK(K + M)N log2 N +NM(K − 1)
0 100 200 300 400 5000
2
4
6
8
10
12
14
Number of BS Antennas
Rate
per
User
(bits/s
/Hz)
SC−FDP (ZF Precoder)
OFDM (ZF Precoder)
SC−FDP (MF Precoder)
OFDM (MF Precoder)
K=5
K=25
K=5
K=25
Figure 4: Rate per user analysis for fully synchronized SC-FDP andOFDM massive MIMO systems with MF and ZF precoder for
σ2
P= 0.01.
the sake of simplicity, we considered the same exponentialpower delay profile for all links between transmit and receiveantennas and we assume the users to have the same distancefrom the BS. The normalized delay spread of the channel is3.
Fig. 4 and Fig. 5 present the rate per user for a fully synchro-nized massive MIMO system with the considered modulationsversus the number of BS antennas for σ2
P = 0.01 and σ2
P = 1,respectively. The rate is presented for different number ofusers and for both MF and ZF precoders. As expected, byincreasing the number of BS antennas and decreasing thenumber of users, the rate per user increases. Fig. 4 and Fig.5 indicate that at high SNR the rate per user for the ZFprecoder is higher than that of the MF precoder. However,at low SNR for small ρ = M
K , the rate per user for theMF precoder is higher than that of the ZF precoder. Fig. 6demonstrate the sum-rate of the SC-FDP and OFDM massiveMIMO systems with the MF and ZF precoders in the presenceof carrier frequency offset for K = 10 and different numberof BS antennas for σ2
P = 0.01. In this figure the CFOvector of the users is assumed ε = [0.01, 0.02, 0.03, 0.04,0.05, 0.02, 0.03, 0.04, 0.04, 0.05]. This figure confirm that theOFDM massive MIMO is more sensitive to carrier frequencyinstabilities than the SC-FDP massive MIMO system. As canbe seen, in a non-synchronized massive MIMO system, thegap between the performance of the SC-FDP and OFDM islarge in high SNR regime for the ZF precoder and large M .As presented in Fig. 6, the sum-rate of the OFDM massiveMIMO system does not increase unlimitedly and it converges
10
10 20 30 40 50 60 70 80 90 1000
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Number of BS Antennas
Ra
te p
er
Use
r (b
its/s
/Hz)
SC−FDP (ZF Precoder)
OFDM (ZF Precoder)
SC−FDP (MF Precoder)
OFDM (MF Precoder)
K=5
K=15
K=25
Figure 5: Rate per user analysis for fully synchronized SC-FDP andOFDM massive MIMO systems with MF and ZF precoder σ2
P= 1.
0 100 200 300 400 50010
20
30
40
50
60
70
Number of BS Antennas
Sum
−R
ate
(bits/s
/Hz)
SC−FDP (ZF Precoder)
OFDM (M=∞)
OFDM (ZF Precoder)
SC−FDP (MF Precoder)
OFDM (MF Precoder)
Figure 6: Sum-rate of SC-FDP and OFDM massive MIMO systemswith MF and ZF precoder in presence of carrier frequency offset
for K = 10 and σ2
P= 0.01.
to the upper bound derived in (52) as M → ∞. As statedearlier, this is due to the fact that the interference caused bythe CFO in the OFDM massive MIMO does not vanish asM increases. In Fig. 7, the sum-rate of an SC-FDP and anOFDM massive MIMO system with K = 10, M = 500 andσ2
P = 0.01 is presented versus CFO. This figure also confirmsthat OFDM massive MIMO is more sensitive to CFO than SC-FDP massive MIMO system. Fig. 8 shows the sum-rate of theSC-FDP and OFDM massive MIMO systems in the presenceof a non-linear power amplifier for K = 10 and differentnumber of BS antennas and attenuation factor α. In this figure,we have assumed that σ2
ap = 0.01 and σ2
P = 0.01. As can beseen, the sum-rates of both systems are identical and do notincrease unlimitedly as the number of BS antennas tends toinfinity. However, they approach the upper bound derived in(64). Fig. 9 demonstrates the sum-rate of these two systems forthe ZF and MF precoder versus the distortion noise variancewhen K = 10, M = 100, and σ2
P = 0.01. Since the derivedsum-rates for both systems are identical in the presence ofnon-linear power amplifier, we have only showed the curvesfor one system and removed the system type in the legend ofFig. 9. Fig. 8 and Fig. 9 show that the sum-rate of the systemwith the ZF precoder is higher than that of the MF precoder
0 0.02 0.04 0.06 0.08 0.120
40
60
80
100
120
140
Normalized Frequency Offset
Sum
−R
ate
(bits/s
/Hz)
SC−FDP (ZF Precoder)
OFDM (ZF Precoder)
SC−FDP (MF Precoder)
OFDM (MF Precoder)
Figure 7: Sum-rate of SC-FDP and OFDM massive MIMO systemswith MF and ZF precoder versus CFO for K = 10, M = 500 and
σ2
P= 0.01.
0 100 200 300 400 50010
12
14
16
18
20
22
24
26
28
30
Number of BS Antennas
Sum
−R
ate
(bits/s
/Hz)
Upper Bound
SC−FDP (ZF Precoder)
OFDM (ZF Precoder)
SC−FDP (MF Precoder)
OFDM (MF Precoder)
a=0.8
a=0.5
Figure 8: Sum-rate of the SC-FDP and OFDM massive MIMOsystems with MF and ZF precoder in presence of power amplifiernon-linearity with α = 0.5 and α = 0.8 for K = 10, σ
2
P= 0.01
and σ2ap = 0.01.
at high SNR. However, as the variance of the distortion noiseincreases and α decreases, the performance of these precodersget closer.Fig. 10 presents the complementary cumulative distributionfunction (CCDF) of the PAPR of the transmitted signal in theSC-FDP and OFDM massive MIMO systems when M = 50.The out-of-band radiation of these systems for different back-off values are presented for K = 1 and K = 10 in Fig. 11and Fig. 12, respectively. The power amplifier non-linearity ismodeled by the Rapp model. It is worth mentioning that foreach of the considered cases, the PAPR for the ZF and MFprecoders are similar. This statement is also valid for the powerspectrum. Since the corresponding curves are similar whenwe use ZF or MF precoders, the curves presented in Fig. 10,Fig. 11, and Fig. 12 are valid for both ZF and MF precoders.As can be seen, when the number of users is low the SC-FDP massive MIMO system has lower PAPR and consequentlylower out-of-band radiation than those of the OFDM massiveMIMO system. It is noteworthy that by increasing the numberof users, the PAPR and out-of-band radiation of the SC-FDPsystem approaches to those of the OFDM massive MIMO. The
11
0 0.02 0.04 0.06 0.08 0.10
10
20
30
40
50
60
70
80
90
100
Sum
−R
ate
(bits/s
/Hz)
σap
2
ZF Precoder
MF Precoder
Figure 9: Sum-rate of SC-FDP and OFDM massive MIMO systemswith MF and ZF precoder versus σ2
ap for K = 10, M = 500,σ2
P= 0.01 and α = 0.8.
5 6 7 8 9 10 11 12 13
10−4
10−3
10−2
10−1
100
Pr(
PA
PR
>λ)
λ (dB)
SC−FDP (K=1)
OFDM (K=1)
SC−FDP (K=10)
OFDM (K=10)
Figure 10: PAPR analysis for M = 50.
reason resides in the fact that as the number of users increasesin the downlink, the distribution of the precoded signal can bewell approximated by Gaussian. This is similar to the case ofOFDM systems with large block lengths.
Both theoretical and simulation results have shown thesuperiority of massive MIMO systems using SC-FDP over theone using OFDM in the presence of CFO and power amplifiernon-linearity.
VI. CONCLUSION
In this paper, we studied the performance of SC-FDP as amodulation scheme for the downlink of massive MIMO sys-tems, in a single cell scenario. We derived the achievable sum-rates for this system with and without carrier frequency offsetand in presence of power amplifier non-linearity. We then com-pare, analytically and through simulations, the performance ofSC-FDP massive MIMO system with OFDM massive MIMOsystems from different perspectives. Our analysis indicates thatin the absence of CFO, the derived achievable sum-rates ofSC-FDP and OFDM massive MIMO systems are identical.However in the presence of CFO, SC-FDP outperforms OFDMsignificantly, specially in large number of BS antennas. Wealso showed that this difference is due to the effect of the
−0.5 0 0.5−80
−70
−60
−50
−40
−30
−20
−10
Normalized Frequency
Pow
er
Spe
ctr
um
(dB
)
OFDM (BO=5)
SC−FDP (BO=5)
OFDM (BO=7)
SC−FDP (BO=7)
Figure 11: Out-of-band radiation analysis for K = 1 and M = 50.
−0.5 0 0.5−80
−70
−60
−50
−40
−30
−20
−10
Normalized Frequency
Pow
er
Spectr
um
(d
B)
OFDM (BO=5)
SC−FDP (BO=5)
OFDM (BO=7)
SC−FDP (BO=7)
Figure 12: Out-of-band radiation analysis for K = 10 and M = 50.
interference term caused by CFO in OFDM massive MIMOsystem which does not vanish as the number of BS antennastends to infinity. We also show that for large number of users,the achievable sum-rates for OFDM and SC-FDP massiveMIMO systems are identical in the presence of power amplifiernon-linearity. The peak to average power ratio and out-of-bandradiation analysis show the superiority of SC-FDP over OFDMin the massive MIMO systems for small number of users.However, this superiority fades as the number of users grow.
ACKNOWLEDGEMENT
This work was supported in part by Fundacao para a Ciłncia e Tecnologia (FCT) and Instituto deTelecomunicacoes under the project UID/EEA/50008/2013.
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Zahra Mokhtari received the B.Sc. degree from university of Yazd, Yazd,Iran in 2011 and the M.Sc. degree from university of Tehran, Tehran, Iranin 2013, all in electrical engineering. She is currently a Ph.D candidate atuniversity of Tehran. Her research interests are wireless communications, 5Gand Massive MIMO.
Maryam Sabbaghian (S’01-M’08) received the B.Sc. and M.Sc. degreesfrom Sharif University of Technology, Tehran, Iran, in 1999 and 2001,respectively, and the Ph.D. degree from Carleton University, Ottawa, ON,Canada, in 2007, all in electrical engineering. During 2008-2010 she waswith School of Engineering and Applied Sciences at Harvard University as apostdoctoral fellow. In 2010, she joined School of Electrical and ComputerEngineering, University of Tehran, Iran where she is an assistant professor.Dr. Sabbaghian is the recipient of Natural Sciences and Engineering Council(NSERC) of Canada postdoctoral fellowship, Ontario Graduate Scholarship,and NSERC eMPOWER scholarship.
Rui Dinis (S 96, M 00, SM 14)received the Ph.D.degree from Instituto Superior Tcnico (IST), Tech-nical University of Lisbon, Portugal, in 2001 and theHabilitation in Telecommunications from Faculdadede Ciłncias e Tecnologia (FCT), Universidade Novade Lisboa (UNL), in 2010. From 2001 to 2008 hewas a Professor at IST. Currently he is an associatedprofessor at FCT-UNL. During 2003 he was aninvited professor at Carleton University, Ottawa,Canada. He was a researcher at CAPS (Centro deAnlise e Processamento de Sinal), IST, from 1992
to 2005 and a researcher at ISR (Instituto de Sistemas e Robtica) from 2005 to2008. Since 2009 he is a researcher at IT (Instituto de Telecomunicaes). He hasbeen actively involved in several national and international research projectsin the broadband wireless communications area. His research interests includetransmission, estimation and detection techniques. Rui Dinis is editor at IEEETransactions on Communications (Transmission Systems - Frequency-DomainProcessing and Equalization) and IEEE Transactions on Vehicular Technology.He was also a guest editor for Elsevier Physical Communication (Special Issueon Broadband Single-Carrier Transmission Techniques).
