Performance Analysis of SC-FDMA and OFDMA in the … · Performance Analysis of SC-FDMA and OFDMA...

Performance Analysis of SC-FDMA and OFDMA

in the Presence of Receiver Phase Noise

Gokul Sridharan∗ and Teng Joon Lim†

∗Edward S. Rogers Sr. Department of Electrical and Computer Engineering

University of Toronto, Canada† Department of Electrical and Computer Engineering, National University of

Singapore, Singapore

Email:[email protected], [email protected]

Abstract

In this paper we study the effect of receiver phase noise on single carrier frequency division multiple

access (SC-FDMA) and orthogonal frequency division multiple access (OFDMA). We show that in both SC-

FDMA and OFDMA, common phase error rotates all the symbols by a certain angle and that the higher order

frequency components of phase noise result in inter-carrier interference, or ICI. We then study the effect of

phase noise on the performance of linear receivers that are often used in practice. In particular, we show that

the amount of ICI affecting the sub-carriers depends closely on the allocation of sub-carriers among different

users and prove that the performance of linear receivers in the presence of receiver phase noise deteriorates

much more in the case of interleaved SC-FDMA than in the case of localized SC-FDMA. We identify the

association of the significant phase noise components with the components of multi-user interference to be

the fundamental reason behind the performance gap between interleaved and localized SC-FDMA.

Index Terms

phase noise, SC-FDMA, inter-carrier interference, linear MMSE receivers.

Manuscript submitted to the IEEE Transactions on Communications on April 10, 2011. The material in this paper was presented

in part at the IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC) 2011, Toronto, ON,

Canada, Sept. 11-14, 2011.

IEEE TRANSACTIONS ON COMMUNICATIONS 2

I. INTRODUCTION

In recent years, many standards have adopted frequency-division schemes such as OFDMA and

SC-FDMA to serve multiple users in a network, both in the uplink and downlink. It is well known

that OFDM has a high peak-to-average power ratio (PAPR) which leads to difficulties in transceiver

design. To address this issue, SC-FDMA has been adopted for uplink transmission in the LTE (3rd

Generation Partnership Project Long Term Evolution, put forth by European Telecommunications

Standards Institute) standard [1]. SC-FDMA is essentially a DFT (discrete Fourier transform) precoded

OFDMA scheme and is known to have a much smaller PAPR than OFDMA [2]. In either scheme,

sub-carriers can be allocated to users in various ways – with channel knowledge at the transmitter of

varying degrees, it is meaningful to solve a resource allocation problem that optimizes some metric

of performance such as outage probability or capacity. In this paper however, we focus on the two

allocation approaches adopted in LTE that does not require channel knowledge of any form at the

transmitter – localized and interleaved. In the former, each user is allocated a set of contiguous

sub-carriers while in the latter, the users are allocated isolated sub-carriers spaced evenly over the

transmission bandwidth. Interleaved SC-FDMA offers some diversity benefits over localized SC-

FDMA. The basic steps involved in SC-FDMA transmission are shown in Fig. 1. The figure also

includes the use of a linear frequency domain MMSE equalizer at the receiver that compensates for

the channel on a per carrier basis. Such a structure is widely used in receiver design because of its

low complexity and ease of implementation. In this paper we focus exclusively on such receivers.

With multi-carrier systems being widely adopted, it is pertinent for us to study their performance

under realistic scenarios where receiver impairments play a critical role. In particular, phase noise

(PHN) is an impairment that needs special attention because unlike other impairments, it changes

substantially over the duration of a multi-carrier symbol and cannot be compensated for in the training

stage. In this paper, we present a detailed analysis on the effect phase noise has on an SC-FDMA

signal and on how it alters the performance of a linear receiver that is oblivious to the presence of

phase noise. The framework that we use lets us generalize our results to OFDMA as well.

PHN arises from imperfections in the frequency synthesizer that result in random fluctuations in

the phase of its output sinusoidal signal. In this paper we assume phase noise to be a first-order

auto-regressive (AR(1)) process as suggested in [3] for the IEEE 802.11g standard. Further details

on such a process are given in Appendix A.

Initial studies on SC-FDMA systems under ideal conditions have shown that SC-FDMA performs

April 22, 2012 DRAFT


as well as OFDMA [4] while using frequency domain MMSE equalization. Recent studies have

also analyzed the performance of SC-FDMA under the influence of different RF impairments such

as frequency offsets, timing offsets, IQ imbalance etc. In [5] a performance comparison between

SC-FDMA and OFDMA is presented in the presence of power amplifier non-linearity, where SC-

FDMA is shown to be better. Performance of SC-FDMA under multi-carrier frequency offsets has

been simulated in [6]. Antilla et al. [7] discuss the effect of receiver IQ imbalance on SC-FDMA

waveforms. Sensitivity of SC-FDMA to large frequency and timing offsets was studied in [8]. Priyanto

et al. [9] show how IQ imbalance, power amplifier non-linearities, as well as phase noise affect the

SNR of the transmitted signal, but do not discuss the effect these non-idealities have on detection at

the receiver. To our knowledge, there has been no detailed discussion on how receiver phase noise

affects an SC-FDMA signal. The effect of PHN on OFDM/OFDMA has been studied extensively

[10]–[15] and can be characterized by the rotation of all the sub-carriers by a certain angle called the

common phase error and the leakage of neighboring sub-carriers resulting in inter-carrier interference

(ICI). We show that both these effects are also seen in the case of SC-FDMA but are characterized

differently.

The performance of linear MMSE receivers in the presence of phase noise and its dependence

on sub-carrier allocation among users is the primary subject of this paper and has so far not been

thoroughly investigated. A surprising result in this context is that although interleaved SC-FDMA

out-performs localized SC-FDMA under ideal conditions by exploiting frequency diversity, in the

presence of phase noise, interleaved SC-FDMA may perform worse. Using insights from the signal

model that we derive, we show how this difference in performance arises and further quantify the

performance loss through a detailed analysis of the SINR (signal to interference and noise ratio)

expression in the two cases.

The paper is organized as follows. In Section II we set up the signal model for a multi-user uplink

transmission using SC-FDMA and focus on linear MMSE receivers used to detect SC-FDMA signals.

In Section III we study how phase noise affects the received signal before and after being processed

by the linear receiver. We then discuss how the SINR of the received signal after receiver processing is

affected by the choice of sub-carrier allocation among different users and establish that localized SC-

FDMA is more robust to PHN than interleaved SC-FDMA. Finally, in Section IV we investigate how

interference resulting because of phase noise depends on various system and phase noise parameters.

In this paper, all vectors are represented in bold lowercase font and all matrices are represented in



bold uppercase font. The notation diag(M) is used to represent the column vector formed using the

diagonal of the matrix M. circ(v) is used to represent the square right circulant matrix formed using

the vector v. E[.] and V ar(.) are used to denote the expectation and variance of a random variable.

II. RECEIVED SIGNAL MODEL

A. The received signal without phase noise

We first consider the detection of an SC-FDMA symbol transmitted over a block fading frequency

selective channel where the channel stays constant over the duration of a whole symbol and there is

no phase noise. We assume that perfect frame synchronization, including carrier frequency recovery

has been established in the training stage. We further assume that current channel conditions have

been estimated during the training phase and that the channel state information is available at the

receiver. The model established here is essential in understanding the the effect of phase noise on a

conventional LMMSE receiver for SC-FDMA.

We consider a wireless network with K users communicating to a base station using SC-FDMA or

OFDMA, with each user being assigned M sub-carriers. Let the total number of sub-carriers be N ,

with N = MK. Each user performs an M-DFT precoding of the symbols, maps the M-DFT output

to its assigned sub-carriers, and then performs multi-carrier modulation with an N-IDFT operation.

If we let the M × 1 symbol vector of the kth user be denoted as dk, the N × 1 time domain transmit

vector xk(after cyclic prefix removal) corresponding to the kth user can be written as

xk = FHNTkFMdk, (1)

where Tk is an N ×M sub-carrier mapping matrix that maps the length-M data vector to the M

sub-carriers allocated to the kth user, FN is the N × N DFT matrix with the (l,m)th entry given

by (FN)lm = (1/√N)e−(2πj(l−1)(m−1)/N) and FM is the M ×M DFT matrix defined similarly. The

structure of Tk and some properties associated with operations that involve these mapping matrices

are discussed in Appendix B. We assume that the average symbol energy of the ith symbol of the

kth user (dki) is equal to (σ2d)ki. We let the discrete time impulse response of the channel between

the kth user and the base station be given by gk. The frequency domain channel response is given by

hk = FNgk. Further, we define the channel correlation matrix corresponding to the kth user as Rk,

with Rk = E[hkhHk ]. We make use of the channel correlation matrix in the SINR analysis presented

in Section III. Letting Hk represent diag(hk), the N × 1 received signal vector yk corresponding to



the kth user at the output of the N-DFT front end, while ignoring noise, is given by

yk = HkTkFMdk. (2)

The received signal vector corresponding to the signals received from all the K users after the N -DFT

operation at the receiver is given by

y =K∑k=1

yk + n =K∑k=1

HkTkFMdk + n (3)

where n is a i.i.d Gaussian noise vector with variance σ2n.

It is important to note here that if we were to replace FM in (3) with the M×M identity matrix IM ,

then the transmission scheme is equivalent to OFDMA with sub-carriers being allocated according to

the mapping matrices Tk. Thus, the analysis in this paper naturally extends to OFDMA if we replace

FM by the identity matrix IM .

B. MMSE Equalization

We focus on the detection of the data stream corresponding to the kth user. When an MMSE

equalizer is used, the channel corresponding to a sub-carrier allocated to the kth user is equalized

in the frequency domain by multiplying the received signal corresponding to that sub-carrier by(hkp)∗

|hkp|2+σ2n/(σ

2d)kp

, where p is the sub-carrier index and hkp is the channel coefficient of the kth user on

the pth sub-carrier. For the kth user, we define a diagonal equalization matrix Hk with entries along

the diagonal given by (hkp)∗

|hkp|2+σ2n/(σ

2d)kp

for p = 1 to N .

After the N -DFT operation at the receiver, the receiver aggregates the sub-carriers corresponding

to the kth user. This operation is equivalent to pre-multiplying the received signal vector y with

TTk . After this operation, the receiver equalizes the channel using the M ×M equalization matrix–

TTk HkTk, where the multiplication by the sub-carrier mapping matrices picks the channel equalization

parameters corresponding to the M sub-carriers allocated to the kth user. Thus, we have

(TTk HkTk)TT

k y =(TTk HkTk)(TT

k HkTk)FMdk +

p=K∑p=1,p 6=k

(TTk HkTk)(TT

k HpTp)FMdp +(TTk HkTk)TT

k n (4)

=(TTk HkTk)(TT

k HkTk)FMdk + (TTk HkTk)TT

k n (5)

=(TTk HkHkTk)FMdk + (TT

k HkTk)TTk n (6)



We have used Property B.6 to simplify (4) to (5) and Property B.5 to simplify (5) to (6). Defining

Sk to be the diagonal matrix with entries σ2n/(σ

2d)kp

|hkp|2+σ2n/(σ

2d)kp

for p = 1 to N , we get

(TTk HkTk)TT

k y =FMdk − (TTk SkTk)FMdk + (TT

k HkTk)TTk n. (7)

Finally, we pass the signal in (7) through an M -IDFT block to obtain

FHM(TTk HkTk)TT

k y =dk − FM(TTk SkTk)FMdk + FM(TT

k HkTk)TTk n. (8)

As noted in [8], we see that even under ideal settings (i.e. no transceiver non-idealities like frequency

offset, phase noise etc.), while using a linear MMSE receiver, the signal after receiver processing is

affected by interference because FM(TTk SkTk)FM is not a diagonal matrix. The second term in the

above expression is termed self interference, where symbols corresponding to the kth user interfere

with each other. Note that in the case of OFDMA, replacing FM with IM makes FM(TTk SkTk)FM a

diagonal matrix and hence there is no self interference under ideal settings.

Fig. 2 compares the performance of OFDMA and SC-FDMA while using a linear receiver under

ideal settings. The plot was generated by simulating a system with 8 users equally sharing a total

of 256 sub-carriers. All the users were assumed to use 64-QAM constellation and for simplicity all

users were assumed to be transmitting with the same power. A 10 tap frequency selective channel

with an exponential power delay profile, was independently generated for every user. The tap gain

(power) of the ith tap was assumed to be −3(i−1) dB, i = 1, 2, . . . , 10, with the channel being further

normalized to have unit energy. The sub-carrier spacing was set to 50 kHz. In the case of localized

SC-FDMA, sets of 32 contiguous sub-carriers were allocated to each user and for interleaved sub-

carrier allocation, users were allocated 32 sub-carriers spaced 8 sub-carriers apart. It can be seen from

the figure that while localized OFDMA, interleaved OFDMA and localized SC-FDMA have almost

identical performance, interleaved SC-FDMA outperforms the other three by virtue of frequency

diversity.

Fig. 2 also plots the performance of SC-FDMA and OFDMA in the presence of phase noise.

For the simulation, phase noise characteristics at the receiver were set to 3 RMS value and 10

kHz loop bandwidth (see Appendix A for details). It is seen from the figure that interleaved SC-

FDMA/OFDMA performs much worse than localized SC-FDMA in spite of outperforming the latter

under ideal settings. This suggests that the overall interference due to phase noise is dependent on the

sub-carrier allocation scheme, with interleaved sub-carrier allocation being more adversely affected.



III. DETECTION IN THE PRESENCE OF PHASE NOISE

A. Received Signal Model

Let θ be the phase noise sequence affecting an SC-FDMA symbol. Let p represent [ejθ1 , . . . , ejθN ]T

and let Q be the matrix given by FNdiag(p)FHN . Using the property that the eigen vectors of a

circulant matrix are the columns of the unitary DFT matrix, it is easy to see that Q is a circulant

matrix and its rows are circular shifts of the frequency domain components of phase noise given by1√N

FHNp = [c0 c1 c2 . . . cN−1]T . In the presence of receiver phase noise, the received signal after the

N -DFT front end can be written as

y = QK∑k=1

HkTkFMdk + n. (9)

Aggregating the sub-carriers corresponding to the kth user by multiplying with TTk gives

TTk y = TT

k QK∑p=1

HpTpFMdp + TTk n. (10)

In Appendix C, we further manipulate this expression to obtain

TTk y =c0(TT

k HkTk)FMdk+(TTk (Q− c0I)Tk)(TT

k HkTk)FMdk

+

p=K∑p=1,p 6=k

(TTk QTp)(TT

p HpTp)FMdp + TTk n. (11)

In (11), the first term is the same as the received signal in the absence of phase noise except for

rotation by common phase error c0. The second term represents the self-interference (interference

within a user’s data stream) and the third term represents the multi-user interference, i.e., interference

resulting from data streams corresponding to other users. An analysis of the received signal at this

stage does not reveal the role played by sub-carrier allocation. Given the circular symmetry of the

matrix Q, all the sub-carriers are equally affected by phase noise irrespective of the sub-carrier

allocation scheme used. The key difference arises after the channel equalization step and is studied

next.

B. Effect on Linear Receivers

We consider the use of a linear MMSE receiver, as described in Section II-B, on an SC-FDMA

signal corrupted by PHN. Using (11), the signal after channel equalization and the final M-IDFT



operation can be written as

dk =FHM(TTk HkTk)TT

k y

=c0dk + c0FM(TTk SkTk)FMdk + FHM(TT

k HkTk)(TTk (Q− c0I)Tk)(TT

k HkTk)FMdk

+

p=K∑p=1,p 6=k

FHM(TTk HkTk)(TT

k QTp)(TTp HpTp)FMdp + FHM(TT

k HkTk)TTk n. (12)

The first term represents rotation by CPE, as before. The second term in (12) is self interference

resulting from MMSE equalization as seen in (8). The third and fourth terms are self interference and

multi-user interference due to phase noise. From the third term, the self interference SIki affecting

the ith symbol of the kth user can be written as

(SI)ki =M∑j=1

M∑m=1

M∑l=1

[f ∗ji(TT

k HkTk)jj(TTk (Q− c0I)Tk)jm(TT

k HkTk)mmfmp(dk)l]

(13)

=M∑j=1

M∑m=1

M∑l=1

f ∗ji(ˆHk)jj((Q− c0I))jm(Hk)mmfml(dk)l, (14)

where we have denoted a matrix of the form TTk MTk as M and fmp is the (m, p)th component of

FM . Assuming each of the terms in the above summation to be independent of each other, we can

compute the variance of SIki to be

V ar(SIki) ≈E

[M∑j=1

M∑m=1

M∑l=1

∣∣∣f ∗ji( ˆHk)jj(Q− c0I)jm(Hk)mmfml(dk)l∣∣∣2]

=1

M2

M∑j=1

M∑m=1

M∑l=1

E[|( ˆHk)jj(Hk)mm|2

]E[|((Q− c0I))jm|2

]E[|(dk)l|2

]

=1

M2

M∑j=1

M∑m=1

M∑l=1

E[|( ˆHk)jj(Hk)mm|2

]E[|(Q− c0I)jm|2

](σ2

d)kl

≈ 1

M2

M∑j=1

M∑m=1

M∑l=1

Eh1(Tk,Rk, j,m, σ2n/(σ

2d)kj)Ec(Tk,Tk,Φ, j,m)(σ2

d)kl, (15)

where the function Eh1(Tk,Rk, j,m, σ2n/(σ

2d)kj) computes the second moment of ( ˆHk)jj(Hk)mm

and Ec(Tk,Tk,Φ, j,m) computes the variance of the term (Q)jm. The function Eh1(.), given in

Appendix D, computes the expression given in (37) with α = (TTk RkTk)jm, where Rk is the channel

correlation matrix and σ2 = σ2n/(σ

2d)kj . The function Ec(.) outputs the (j,m)th entry in the matrix

TTk circ

(diag( 1

NFHΦF)

)Tk where circ

(diag( 1

NFHΦF)

)is the correlation matrix of the frequency

components of phase noise.



Using similar approximations, the variance of the multi-user interference MUIki (fourth term in

(12)) affecting the ith symbol of the kth user can be written as

V ar(MUIki) ≈K∑p=1p 6=k

M∑j=1

M∑m=1

M∑l=1

E

[∣∣∣f ∗ji(TTk HkTk)jj(TT

k (Q)Tp)jm(TTp HkTp)mmfml(dp)l

∣∣∣2]

=K∑p=1p 6=k

M∑j=1

M∑m=1

M∑l=1

E[|( ˆHk)jj(Hp)mm|2

]E[|(TT

k QTp)jm|2]E[|(dp)l|2

]

≈ 1

M2

K∑p=1p 6=k

M∑j=1

M∑m=1

M∑l=1

Eh2

(σ2n/(σ

2d)kj)Ec(Tk,Tp,Φ, j,m)(σ2

d)pl

=1

M2

K∑p=1p 6=k

M∑j=1

M∑m=1

M∑l=1

Eh2

(σ2n/(σ

2d)kj)Ec(Tk,Tp,Φ, j,m)(σ2

d)pl, (16)

where the function Eh2(.) computes the expression in (37) with α set to zero. Setting α to zero is

valid here because channel coefficients from two different users are independent of each other.

We use equations (15) and (16) to compute the SINR at the ith sub-carrier of the kth user. We

define SINR to be

SINRki =V ar

((c0FHM(TT

k HkHkTk)FMdk)i)

V ar(SIki) + V ar(MUIki) + V ar((FM ˆHkTT

k n)i) . (17)

Note that in (17) we have included the common phase error to be part of the signal component. This

is justified as the effect of CPE does not cause interference amongst different sub-carriers and its

effect is only to rotate the signal component by a certain angle. Further, there are many algorithms

to effectively estimate common phase error and compensate for it [16], [17]. To compute (17), we

approximate HkHk to be I so that the signal variance is given by (σ2d)ki and the variance of the noise

term (FM ˆHkTTk n)i is approximated to be Eh2(σ2

n/(σ2d)ki)σ

2n. Since (15) and (16) are independent of

the index i, the SINR expression computed above is valid for any sub-carrier allocated to the kth user

whenever all sub-carriers are transmitted with the same power i.e., (σ2d)ki is the same for all i.

Fig. 3 compares the theoretical and empirical SINRs for interleaved and localized SC-FDMA. The

system settings and phase noise parameters are the same as those used to generate Fig. 2. Since the

system settings and sub-carrier allocation (localized/interleaved) are symmetric across all users in our

simulations, the theoretical estimate of SINR is valid for any user, i.e, SINR for all data streams are

identical.



It can be observed from Fig. 3 that the theoretical estimate for SINR is a very good approximation

to the empirical SINR for both interleaved and localized SC-FDMA. The observed SINR in the case

of interleaved SC-FDMA is much worse than that for localized SC-FDMA. This drop in SINR in

turn results in the performance gap that emerges between interleaved and localized SC-FDMA in Fig.

2. A seen in Fig. 4, the difference in SINRs arises primarily because of the difference in the variance

of multi-user interference for the two scenarios.

C. Difference between interleaved and localized SC-FDMA

Consider a generic term that contributes towards SI as given in (14):

tSI = f ∗ji (ˆHk)jj︸︷︷︸hj

((Q− c0I))jm︸︷︷︸cjm

(Hk)mm︸︷︷︸hm

fml(dk)l. (18)

Terms that contribute towards MUI (tMUI) also have a similar form, except that hj and hm correspond

to terms from the channel equalization matrix and the frequency domain channel matrix of two

different users and hence are independent of each other.

We are interested in the relative behavior of the variance of tSI and tMUI as a function of the

various parameters involved. Since f ∗ji and fml are entries from a unitary DFT matrix, they have unit

magnitude and do not affect the variance. Further, so as to distill out the effects of power control

across different sub-carriers, we assume all users to be using constellations of the same energy. This

allows us to focus on the product (hjhm)cjm and make the following observations:

• In the case of SI, the terms hj and hm can be strongly correlated. For correlation factor α =

E[hjh∗m], equation (37) gives the variance of the product hjhm. In general, this variance is a

decreasing function of the correlation factor. When the two components are independent as in

the case of MUI, at high SNR, the variance of the product hihj can become very large as the

distribution of |hjhm|2 ≈ |hm/hj|2 resembles a Fisher-Snedecor distribution, whose mean is

unbounded. Thus it is reasonable to assume that the variance E[|hjhm|2] is higher in the case of

MUI when compared to SI. Note that this holds for both interleaved and localized sub-carrier

allocation.

• For phase noise modeled as an AR(1) process, the power in the higher order frequency terms

drops off exponentially (Appendix A, eq. (??)) and only the first few terms in the frequency

domain representation of phase noise are significant. Consequently, since the matrix Q is a

circulant matrix whose rows are right circular shifts of the frequency domain components, the

significant components of phase noise lie in the vicinity of the main diagonal.



In the case of SI in localized SC-FDMA, the matrix Q−c0I (equal to (TTk (Q−c0I)Tk)) is a M×M

sub-matrix formed by contiguous rows and columns of Q− c0I and is centered around the main

diagonal. This is easy to see from the structure of Tk for localized SC-FDMA as described in

Appendix B. This observation is illustrated in (19) below, where such sub-matrices are enclosed

in solid lines (in this example, N = 9, M = 3). It is also easy to see that such a matrix contains

most of the significant frequency components of phase noise (c1, c2, cN−1, cN−2 etc). Reasoning

along similar lines, in the case of MUI, we see that matrices (TTk (Q−c0I)Tp) with k 6= p consist

of a subset of contiguous rows and columns of (Q − c0I) whose indices do not overlap and

therefore consist only of the insignificant components of PHN.

We hence see that in the case of SI in localized SC-FDMA, cjm is an element from (TTk (Q −

c0I)Tk) and is likely to have a much higher variance (power) than a corresponding term associated

with MUI that comes from a matrix of the form (TTk (Q− c0I)Tp) with k 6= p.

Q− coI =

0 c1 c2 c3 c4 c5 c6 c7 c8

c8 0 c1 c2 c3 c4 c5 c6 c7

c7 c8 0 c1 c2 c3 c4 c5 c6

c6 c7 c8 0 c1 c2 c3 c4 c5

c5 c6 c7 c8 0 c1 c2 c3 c4

c4 c5 c6 c7 c8 0 c1 c2 c3

c3 c4 c5 c6 c7 c8 0 c1 c2

c2 c3 c4 c5 c6 c7 c8 0 c1

c1 c2 c3 c4 c5 c6 c7 c8 0

(19)

In the case of interleaved SC-FDMA, (TTk (Q − c0I)Tk) is a sub-matrix formed from rows and

columns spaced equally apart. Elements in such a matrix are shown enclosed in dotted lines in

(19). Sub-matrices of the form (TTk (Q− c0I)Tp) with k 6= p also have a similar structure and it

is seen that the dominant phase noise components get spread out among all such sub-matrices.

Thus, unlike localized SC-FDMA, not all significant phase noise components cjm are associated

with SI.

Putting the two observations together, we see that in the case of localized SC-FDMA, the association

of the dominant phase noise frequency components with SI prevents further amplification of MUI

terms where the product hjhm already has a large variance. In the case of interleaved SC-FDMA,

dominant phase noise components can get associated with MUI terms where the product hjhm is

formed of independent components and result in higher MUI variance.



In short, the association of significant phase noise components with terms that contribute to multi-

user interference is the fundamental reason behind the performance gap between interleaved and

localized SC-FDMA.

Fig. 4 shows the empirical and theoretical estimates of self interference and multi-user interference

for localized and interleaved SC-FDMA. We can clearly see that while self interference for the two

cases is comparable, multi-user interference is much higher in the case of interleaved SC-FDMA.

At high SNR, the total interference seen in the case of interleaved SC-FDMA is higher than the

interference seen in the case of localized SC-FDMA by a factor of about 4, which explains the

difference in the SINRs seen in Fig. 3.

As further evidence that our hypothesis is indeed true, we test our hypothesis under the following

two circumstances. Since the hypothesis hinges on correlation between channel coefficients and phase

noise energy concentration in the lower order frequency components, removal of either property

must render the effects on interleaved and localized SC-FDMA equivalent. To check this, rather

than generating channel coefficients from a multi-tap channel, we directly generate independent

channel coefficients corresponding to different sub-carriers while making no changes to the phase

noise characteristics. Consequently, localized SC-FDMA does not have the advantage of correlated

channel coefficients being associated with dominant phase noise frequency components and sub-

carrier mapping must not impact system performance. This is verified through simulations in Fig. 5,

where it is seen that interleaved and localized SC-FDMA have very similar performance under such

circumstances.

In the second test, rather than modeling phase noise as a correlated process, we model it as an

uncorrelated process while retaining the original channel model. As a result, the energy in the phase

noise process is equally spread amongst all the frequency components. Using the same reasoning as

before, we expect interleaved and localized SC-FDMA to perform identically. Simulation results in

Fig. 5 under such a scenario further validate our hypothesis.

IV. DEPENDENCE ON SYSTEM AND PHASE NOISE PARAMETERS

So far we have established all results for a particular choice of system (8 users, 256 tones)

parameters and phase noise parameters (3 RMS, 10 kHz bandwidth). In this section we see how SI

and MUI vary as as a function of these parameters. Specifically, we see how SI and MUI vary as a

function of (a) number of users being served over one SC-FDMA symbol and (b) loop bandwidth



of the phase noise process. Since varying phase noise RMS scales both SI and MUI by the same

amount and their relative strengths remain the same, we do not vary phase noise RMS value.

In Fig. 6, we compare the performance of interleaved/localized SC-FDMA as a function of the

number of users scheduled simultaneously. We make the following observations:

1) For both interleaved and localized SC-FDMA, MUI increases with number of users N , while

SI decreases with number of users. This is expected as the number of sub-carriers assigned to

each user decreases as the number of users increase.

2) In the case of interleaved SC-FDMA, MUI is significantly greater (by almost two orders) than

SI, while the two are comparable in the case of localized SC-FDMA for a wide range of N .

In Fig. 7, we compare the performance of interleaved/localized SC-FDMA as a function of the

loop bandwidth of the phase noise process. We make the following observations:

1) As before, for interleaved SC-FDMA, MUI is significantly greater than SI for a wide range of

Ωo.

2) For interleaved and localized SC-FDMA, MUI as well as SI increase with Ωo. Since increasing

the loop bandwidth results in more energy being distributed in the higher order components of

the phase noise process, the increase in MUI with Ωo is not surprising. The above reasoning

also requires that SI decrease with Ωo, however, this is not the case with SI.The increase in SI

with Ωo is primarily a consequence of energy redistribution of the phase noise process from the

CPE (zeroth order component) to the higher order components. As can be seen from Fig. 8, the

fraction of energy in CPE drops from almost 80% for Ωo = 5 kHz to 25% when Ωo = 50 kHz

(see Appendix A for more details). This leads to a significant increase in the energy associated

with the frequency components of phase noise that are associated with the SI terms (c1, c2 etc.).

We thus see that the observations we made about MUI and SI in the previous section continue to

hold for a wide range of system and phase noise parameters. These results provide useful thumb rules

to account for the effect of phase noise while cumulatively assessing the impact of various receiver

non-idealities. This work also brings to light the importance of evaluating the impact of receiver non-

idealities after all the signal processing steps at the receiver and not immediately after the N-DFT

front end as is most often done.

V. CONCLUSION

In this paper we developed a framework to analyze SC-FDMA signals affected by receiver phase

noise. This framework enabled us to characterize the interference arising before and after receiver



processing while using the linear MMSE receiver. This characterization helped us in computing a

theoretical estimate of SINR for interleaved and localized SC-FDMA systems affected by phase

noise. Using this analysis we provided an explanation for the performance gap between interleaved

and localized SC-FDMA in the presence of phase noise. We further investigated the variation of SI

and MUI as a function of system parameters and phase noise parameters.

APPENDIX A

PHASE NOISE CHARACTERISTICS

The phase noise model used in this paper is exactly the same as in [18]. Phase noise is modeled

as a first order auto-regressive (AR(1)) process and is characterized by two parameters — 1/Ts, the

sampling rate and Ωo, the one sided 3-dB bandwidth of the phase locked loop, also known as the loop

bandwidth. σθ is the RMS value of the PHN process in radians, a typical value being around 3. In

general, the 3-dB bandwidth of an oscillator tends to be in the range of tens of kHz. We denote the

covariance matrix of a length-N sequence of PHN by Φ. Common phase error (CPE) is the sample

mean θ of a length-N sequence of phase noise θ = θ1, θ2, . . . , θN given by θ = 1N

∑Nk=1 θ[k]. It

can be shown that this is a zero mean Gaussian random variable with variance σ2θ

= 1TΦ1/N2 [19].

The frequency domain representation of the phase noise vector is given by 1√N

FHejθ. We let the

vector c = [c0, c1 . . . cN−1] denote the frequency domain components. Note that c0 is representative

of the common phase error i.e., 1N

∑k=Nk=1 e

jθ[k] ≈ 1 + 1N

∑θk (under small angle assumption). The

covariance matrix of the frequency domain components is given by 1N

FHNΦFN . Since the phase noise

PSD tapers off rapidly beyond the loop bandwidth, most of the energy (σ2θ ) of the phase noise process

is contained in the first few lower order frequency components i.e. c1, cN−1, c2, cN−2 etc.

An important aspect of any phase noise process is the split in power between CPE and higher order

components. Using the covariance matrix Φ, in Fig. 8 we see that as the sub-carrier spacing increases,

CPE variance grows larger. We also see that increasing loop bandwidth decreases CPE variance. In

relative terms, if a high percentage of energy is concentrated in CPE, effects of CPE dominate over

that of higher order components and vice versa.

APPENDIX B

PROPERTIES OF THE SUB-CARRIER MAPPING MATRIX Tk

The N ×M matrix Tk maps M data symbols of the kth user to M out of the N available sub-

carriers. Assuming the number of users to be K, we have MK = N . Let Uk represent the set of



indices of the sub-carriers allocated to the kth user. Suppose the ith symbol in the data vector dk is

mapped to the jth sub-carrier with j ∈ Uk then, Tk(i,j) = 1, and is zero otherwise. Of the N rows of

Tk, N−M rows are all zeros and the remaining M rows consist of the M unit vectors of the standard

basis of an M-dimensional space. As a result, TkTTk is an N ×N diagonal matrix with the jth entry

along the diagonal being one if the jth row of Tk is non-zero and is zero otherwise. Further, the

columns of Tk form a subset of M the unit vectors of the standard basis of an N -dimensional space.

As a result, TTk Tk is the M ×M identity matrix. We denote the concatenated matrix [T1T2 . . .TK ]

as T. Since a sub-carrier is assigned exclusively to a user, T is a permutation of the identity matrix

and is orthogonal i.e. TTT = I. We observe the following properties of a mapping matrix Tk.

Property B.1: Given an N ×N matrix R, TTk R is an M ×N matrix consisting of a subset of M

rows from R whose indices are in Uk.

Proof: Since every row of TTk is a unit vector from the standard basis of an N -dimensional space

and no two rows are identical, every row in TTk picks a row from R. Specifically, if TT

k(i,j) = 1, then

the ith row in the resulting matrix is simply the jth row of R.

Property B.2: Given an N ×N matrix R, RTk is an N ×M matrix consisting of a subset of M

columns from R whose indices are in Uk.

Proof: This follows from Property B.1 once we note that RTk is (TTk RT )T .

Property B.3: Given an N × N matrix R, TTk RTk is an M ×M sub-matrix of R formed by M

columns and M rows of R, sharing the same set of indices Uk.

Proof: This property follows immediately from the Property B.1 and Property B.2.

Property B.4: Suppose R is an N ×N diagonal matrix then TTk RTk is a M ×M diagonal matrix

with entries picked from diag(R) such that the indices of the chosen entries belong to Uk.

Proof: This property follows from Property B.3.

Property B.5: If R and S are two N×N diagonal matrices, then TTk RTkTT

k STk is equal to TTk RSTk.

Proof: We first note that TkTTk is a diagonal matrix. Since R and S are also diagonal matrices,

we can change the order of the product. Hence, we have

(TTk RTk)(TT

k STk) = TTk (TkTT

k )RSTk = (TTk Tk)(TT

k RSTk) = TTk RSTk, (20)

where the last equality follows from the fact that (TTk Tk) is the M ×M identity matrix.

Property B.6: If R is an N ×N diagonal matrix, then, TTk RTp = 0 for p 6= k.



Proof: Using Property B.1 and Property B.2, we note that the set of indices of the rows and

columns that form the resulting matrix belong to the sets Uk and Up respectively. Since users do not

share sub-carriers, Uk ∩ Up = ∅ and hence no chosen row and column share the same index. Since

all the non-zero entries in R lie along the diagonal, the result follows.

Property B.7: If R is an N ×N invertible diagonal matrix, then, (TTk RTk)

−1 = TTk R−1Tk.

Proof: We first note that TTk RTk is an M ×M diagonal matrix. Specifically, if the ith row in TT

k

has a non-zero entry in the jth location, then the (i, i)th element in TTk RTk is given by R(j,j). Since

the resulting matrix is a diagonal matrix, inverting TTk RTk is equivalent to inverting the individual

components along the diagonal of the resulting matrix. Since these individual components are entries

along the diagonal in R, we first invert R and then compute the product TTk R−1Tk.

All the above properties are true for any general sub-carrier allocation. We now focus on properties

of permutation matrices specific to localized and interleaved sub-carrier mapping. In the case of

localized SC-FDMA, the first user is allocated the first M sub-carriers, thus T1 (1:M, 1:M) = I. Similarly,

for the kth user, Tk (1+(kM):(k+1)M, 1:M) = I.

Property B.8: For the case of localized sub-carrier allocation, if R is an N ×N circulant matrix,

then TTk RTk = R(1+(kM):(k+1)M, 1+(kM):(k+1)M) = R(1:M, 1:M) and is independent of k.

Proof: This property follows from Property B.3 and the observation that all M×M sub-matrices

of R formed from a set of M contiguous rows and columns that share the same set of indices are

equivalent.

For interleaved sub-carrier mapping, the data symbols corresponding to the kth user are mapped

to sub-carriers with indices k,M + k, 2M + k, . . .KM + k. As a result, the matrices Tk are row

shifted copies of each other.

Property B.9: For the case of interleaved sub-carrier allocation, if R is an N ×N circulant matrix,

then TTk RTk is independent of k.

Proof: Let the first row of R be denoted as v. Then R(i,j) = v((j−i) mod N)+1. Now, any entry in the

sub-matrix TTk RTk belongs to a row in R whose index is given by pM+k for some p ∈ 0, 1, . . . (N−

1) and a column in R whose index is given by qM+k for some q ∈ 0, 1, . . . (N−1). We now note

that such an element in R is given by R(pM+k,qM+k) = v((qM+k−pM−k) mod N)+1 = v((qM−pM) mod N)+1,

which is independent of k.



APPENDIX C

SC-FDMA RECEIVED SIGNAL IN THE PRESENCE OF PHN

Here we show how we manipulate (10) to get (11). From (10), we have

TTk y =TT

k QK∑p=1

HpTpFMdp + TTk n (21)

=TTk QTTT

K∑p=1

HpTpFMdp + TTk n (22)

=TTk QT

K∑p=1

TTHpTpFMdp + TTk n (23)

=[TTk QT1 TT

k QT2 . . .TTk QTK

] K∑p=1

TTHpTpFMdp+ TTk n (24)

=[TTk QT1 TT

k QT2 . . .TTk QTK

]×

K∑p=1

[(TT

1 HpTpFMdp)T (TT2 HpTpFMdp)T . . . (TT

KHpTpFMdp)T]T

+ TTk n (25)

=i=K∑i=1

p=K∑p=1

(TTk QTi)(TT

i HpTp)FMdp + TTk n (26)

Now, we use Property B.6 to simplify the previous equation to get

TTk y =(TT

k QTk)(TTk HkTk)FMdk +

p=K∑p=1,p 6=k

(TTk QTp)(TT

p HpTp)FMdp + TTk n (27)

=c0(TTk HkTk)FMdk+(TT

k (Q−c0I)Tk)(TTk HkTk)FMdk

+

p=K∑p=1,p 6=k

(TTk QTp)(TT

p HpTp)FMdp + TTk n (28)

APPENDIX D

COMPUTING THE VARIANCE

Let x be an exponential random variable distributed as f(x) = λe−λx, with λ = 1. For such a

random variable,

E

[1

x+ c

]=

∫ ∞0

1

x+ ce−xdx = ec

∫ ∞c

1

ye−ydy = ecΓ(0, c) (29)

where, c > 0 and Γ(.) is the upper incomplete gamma function.



Also,

E

[1

(x+ c)2

]=

∫ ∞0

1

(x+ c)2e−xdx = ec

∫ ∞c

1

y2e−ydy = ecΓ(−1, c). (30)

Let hk and hp be two zero mean circular symmetric Gaussian random variables with unit variance.

Let the correlation between hk and hp be α, i.e., α = E[hph∗k]. We are interested in the mean and

variance of h∗khp|hk|2+σ2 . In order to compute the mean and variance, we first approximate hp as αhk+βg,

where g is a zero mean circular symmetric Gaussian random variable with unit variance and is

independent of hk, and β =√

(1− |α|2). Using this approximation, the expression h∗khp|hk|2+σ2 can be

written as

h∗khp|hk|2 + σ2

= α− ασ2

|hk|2 + σ2+

βh∗kg

|hk|2 + σ2(31)

The mean can be computed to be

E

[h∗khp

|hk|2 + σ2

]=E

[α− ασ2

|hk|2 + σ2+

βh∗kg

|hk|2 + σ2

]=α− ασ2E

[ 1

|hk|2 + σ2

]+ E

[β|hk||hk|2 + σ2

]E[ejθk ]E[g]

=α− ασ2eσ2

Γ(0, σ2). (32)

Denoting γ1 = α, γ2 = − ασ2

|hk|2+σ2 and γ3 = β|hk|ejθkg|hk|2+σ2 , the second moment of h∗khp

|hk|2+σ2 can be written

as the summation

E

[∣∣∣∣ h∗khp|hk|2 + σ2

∣∣∣∣2]

=3∑i=1

3∑j=1

E[γiγ∗j ]. (33)

The term E[|γ2|2] can be written as

E[|γ2|2] = |α|2σ4E

[1

(|hk|2 + σ2)2

]= |α|2σ4eσ

2

Γ(−1, σ2). (34)

The term E[|γ3|2] can be written as

E[|γ3|2] = E

[(|β||hk||g||hk|2 + σ2

)2]

= |β|2E[|g|2]E

[|hk|2

(|hk|2 + σ2)2

]= |β|2

(eσ

2

Γ(0, σ2)− σ2eσ2

Γ(−1, σ2)). (35)



The term E[γ∗1γ2] can be written as

E[γ∗1γ2] = α∗E[− ασ2

|hk|2 + σ2] = −|α|2σ2eσ

2

Γ(0, σ2) (36)

We further note that E[γ∗1γ3], E[γ∗2γ3], E[γ∗3γ1], E[γ∗3γ2] all are equal to zero. Thus the second moment

can be written as

E

[∣∣∣∣ h∗khp|hk|2 + σ2

∣∣∣∣2]

=|α|2 + Γ(0, σ2)[|β|2eσ2 − 2α2σ2eσ

2]

+ Γ(−1, σ2)[|α|2σ4eσ

2 − |β|2σ2eσ2].

(37)

The function Eh1(Tk,Rk, j,m, σ2n/σ

2d) computes the expression given in (37) with α set to

(TTk RkTk)jm, where Rk is the channel correlation matrix and σ2 = σ2

n/σ2d. The function Eh2(σ2

n/σ2d)

computes the expression given in (37) with α set to zero and σ2 = σ2n/σ

2d.

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in OFDM,” IEEE Transactions on Signal Processing, vol. 54, no. 9, pp. 3542–3554, September 2006.



ADC/

RF

DAC/

RF

P to

SN point

IDFT

Sub carrier

Mapping

M point

DFT

S to

P

Symbol

Stream

CHANNEL

N point

DFT

Sub carrier

De−mapping

sub−carrier

equalization

channel based

(ZF/MMSE)

M point

IDFT S to

P

P to

S

StreamSlicer

Symbol

Fig. 1. Block diagram representing the SC-FDMA scheme and the use of a frequency domain MMSE equalizer at the receiver. Note

that N > M , and usually, MN

= K, an integer representing the number of users in the uplink.



18 20 22 24 26 28 30 32 34 3610

−4

10−3

10−2

10−1

Eb/No (in dB)

BE

R

Local OFDMA, no PHNIntrlv. OFDMA, no PHNIntrlv. SCFDMA, no PHNLocal SCFDMA, no PHNLocal OFDMA, PHNLocal SCFDMA, PHNIntrlv. OFDMA, PHNIntrlv. SCFDMA, PHN

Fig. 2. Plot comparing the performance of interleaved and localized OFDMA/SC-FDMA while using MMSE channel equalization

under ideal settings and in the presence of phase noise. 256 sub-carriers were equally shared by 8 users, each using a 64-QAM

constellation. The sub-carrier spacing was 50 kHz. The phase noise characteristics at the receiver were set to 3 RMS value and 10

kHz loop bandwidth



18 20 22 24 26 28 30 32 34 3610

15

20

25

30

35

Eb/No (in dB)

Eb/(N

o+I)

(in d

B)

Intrlv SC−FDMA (th)ideal caseIntrlv SC−FDMA (emp)Local SC−FDMA (emp)Local SC−FDMA (th)

Fig. 3. Plot comparing the theoretical and empirical SINRs in localized and interleaved SC-FDMA while using a linear MMSE

receiver. The x-axis represents the signal to noise ratio(SNR) without taking interference into account while y-axis represents the signal

to noise and interference ratio(SINR). The phase noise parameters were Ωo = 10 kHz and σθ = 3. The sub-carrier spacing was 50

kHz with 8 users equally sharing 256 sub-carriers.

18 20 22 24 26 28 30 32 34 360

1

2

3

4

5

6

7

8x 10−3

Eb/No (in dB)

Nor

mal

ized

Var

ianc

e of

Inte

rfere

nce

Intrlv MUI variance (emp)Intrlv MUI variance (th)Local MUI variance (emp)Local MUI variance (th)Intrlv SI variance (emp)Local SI variance (emp)Local SI variance (th)Intrlv SI variance (th)

Fig. 4. Plot comparing the variance of self interference and multi-user interference due to phase noise in localized and interleaved

SC-FDMA while using a linear MMSE receiver. The phase noise parameters were Ωo = 10 kHz and σθ = 3. The sub-carrier spacing

was 50 kHz with 8 users equally sharing 256 sub-carriers.



18 20 22 24 26 28 30 32 34 36

0.01

0.1

0.005

Eb/No (in dB)

BER

Local SCFDMA Ind. Channel Coeff.

Intrlv SCFDMA Ind. Channel Coeff.

Local SCFDMA Uncorrelated PHN

Intrlv SCFDMA Uncorrelated PHN

Fig. 5. Plot illustrating the performance of interleaved and localized SC-FDMA when all channel coefficients are generated

independently and when phase noise is uncorrelated. 8 users equally shared 256 sub-carriers and phase noise RMS value was set

to 3.

4 8 16 3210

−6

10−5

10−4

10−3

10−2

Number of users

Log

.nor

mal

ized

vari

ance

ofin

terf

eren

ce

MUI var. intrlv. SCFDMA

SI var. intrlv. SCFDMA

MUI var. local. SCFDMA

SI var. local. SCFDMA

Fig. 6. Plot showing the variation in MUI and SI as a function of number of users while using SC-FDMA with 256 sub-carriers.

The phase noise RMS was set to 3 and the loop bandwidth to 10 kHz. The signal to noise ratio was set to 28 dB.



5 kHz 10 kHz 20 kHz 50 kHz10

−6

10−5

10−4

10−3

10−2

10−1

Loop bandwidth

Log

.nor

mal

ized

vari

ance

ofin

terf

eren

ce

MUI var. intrlv. SCFDMA

SI var. intrlv. SCFDMA

MUI var. local. SCFDMA

SI var. local. SCFDMA

Fig. 7. Plot showing the variation in MUI and SI as a function of loop bandwidth of the phase noise process while using SC-FDMA.

The phase noise RMS was set to 3 RMS and 8 users equally shared 256 sub-carriers. The signal to noise ratio was set to 28 dB.

0 50 100 150 200 250 3000

10

20

30

40

50

60

70

80

90

100

sub−carrier spacing in kHz

pe

rce

nta

ge

of

en

erg

y in

CP

E

Ωo=5kHz

Ωo=10 kHz

Ωo=20 kHz

Ωo=50 kHz

Fig. 8. Percentage of phase noise energy concentrated in the CPE term as a function of sub-carrier spacing and loop bandwidth.



RESPONSE TO REVIEWERS

Reviewer 1:

We believe we have not received your complete review and are afraid we haven’t been able to

address all the concerns that you may have had. We hope to address these concerns in the next round

of reviewing.

Abstract can clarify the contrast/similarity between SC-FDMA and OFDMA in case of ICI.

We have expanded the abstract a little bit to provide more details about the paper. In the case of

OFDMA and SC-FDMA there are several terms that contribute towards interference. The interference

in the two scenarios differs in the manner in which the different terms in the overall interference are

related to each other. The difference arises because in order to obtain the signal model for OFDMA,

Fm is replaced with I in the received signal model given in (12). We briefly take note of this at the

end of Section II-A.

An important approximation in our analysis is that individual terms in interference can be assumed

to be independent of each other. Under this assumption, the inter-relationships between the different

terms contributing to interference is neglected and thus the overall structure of ICI becomes quite

similar in OFDMA and SC-FDMA. One subtle observation is that if we were to assume that any

user transmits with equal power on all sub-carriers assigned to him, then all symbols transmitted by

a user are equally affected by interference in the case of SC-FDMA, while in the case of OFDMA,

different symbols see different levels of interference.

The performance of linear receivers in the presence of phase noise is similar for both OFDMA

and SC-FDMA primarily because the structure of interference is the same in both the cases. The

main differences arise between interleaved and localized sub-carrier allocation, which is the primary

focus of our paper.

Reviewer: 2

The submission appears to be an extended version of a paper (with the same title) presented in

PIMRC 2011. The core results of the submission (including figures 1 to 4) seem to be included in



the conference paper. The existence of an earlier conference paper is not indicated in the submission.

I leave it to the editorial board of TCOM to judge whether the submission can be considered for

publication in the journal.

We sincerely apologize on our oversight on this issue. We have now stated clearly in a footnote on

the first page that some parts of this paper were submitted to an earlier conference (PIMRC 2011).

We feel this paper is a much more detailed exposition of the key results presented in our conference

paper in addition to a some new insights. In particular we note that Section III-C now has two new

arguments along with experimental evidence to further back our hypothesis. Further, Section IV

is a completely new addition where we explore the variation in MUI and SI as a function of the

system and phase noise parameters. The conclusions drawn here indicate that for a wide range of

these parameters, the performance difference between interleaved and localized SC-FDMA persists

and thus broadens the scope of our arguments. Appendix A now includes a figure that illustrates the

energy split between CPE and the higher order components of a phase noise process. This is a very

important aspect of the phase noise process that to the best of our knowledge has not been presented

before. Appendix C presents a complete derivation of (11) while Appendix B lists various properties

of the sub-carrier mapping matrices along with proofs.

For all of the above reasons, we feel this is a complete and comprehensive presentation of

our analysis of the performance of SC-FDMA/OFDMA in the presence of phase noise and merits

publication as a full article.

The paper presents a careful and insightful analytical study of the effects of phase noise on

localized and interleaved SC-FDMA systems. Even though the results confirm the intuitive idea that

interleaved SC-FDMA is more sensitive to phase noise (as well as frequency offsets and various other

imperfections), it is useful to have a complete analytical model to quantify these effects.

We believe the critical argument that delineates the performance of interleaved and localized SC-

FDMA is quite non-trivial. Under ideal conditions, interleaved SC-FDMA is expected to outperform

localized SC-FDMA because it is able to better exploit diversity in the frequency selective channel.

However, this proves to not be the case in the presence of non-idealities such as phase noise and this



warrants a thorough investigation to bring to light the interplay between the various factors involved.

The relationship between the correlation among sub-carrier channel coefficients and the distribution

of energy in the higher order phase noise components in determining the overall system performance

has not been explained before.

Also, to the best of our knowledge this is one of the first papers to analyze the performance

subsequent to the linear receiver being used. To this end, we have derived a clear, end to end signal

model through which we have been able to carry out a detailed SINR analysis. It is important to

note that there is essentially no difference (except for sub-carrier mapping) in the signal model

for interleaved and localized SC-FDMA immediately after the IDFT operation at the receiver. The

sub-optimality of the linear MMSE receiver (for per tone channel equalization) for SC-FDMA plays

a key role in our analysis and hence it is important to incorporate this into our signal model. We

believe we have provided a basic framework to better analyze the effects of other non-idealities while

using linear receivers and these results are of potential use to other researchers with such interests.

The numerical examples seem to assume that perfect power control. Presumably differences in

different users received power levels would increase the performance differences.

Our original assumption was that power control will compensate for pathloss and shadowing while

the effect of Rayleigh fading still needs to be taken into account. However, in view of comments from

two reviewers, we have allowed signals on different sub-carriers to have different power levels and

have accordingly modified the SINR analysis. Please note changes to (15) and (16). However, in our

simulations, we have assumed that all users use the same signal strength since the primary focus of

this paper is to explore the effects of phase noise and interleaving in SC-FDMA while using a linear

receiver. Our goal was to find the simplest setting where these effects come to the forefront, with the

intention of zeroing out all other effects and then identifying the sensitivity of our results to various

parameters involved. We felt taking power control into account would significantly complicate the

conclusions drawn, while also restricting it to the specific assumptions made on power control.

It would be interesting to see the performance also as a function of phase noise RMS value.



Since the variance of MUI and SI as given in (15) and (16) have a factor of phase noise RMS

value (hidden inside Φ), their relative dependence on the phase noise RMS is the same, i.e., the ratio

of MUI to SI is a constant even as the RMS value is varied. For this reason, the conclusions in the

paper remain unchanged even as phase noise RMS value is varied. We have made a brief mention

of this in the beginning of Section IV.

In the caption of Fig. 6, 8 users is probably a typo.

Thank you for pointing this out, we have fixed it.

The figure captions could be more complete. For example, in Fig. 2, the FFT-size, number of users,

and symbol constellation could be indicated.

We have made the captions more descriptive, giving more information on parameters used to

generate the plot.

Reviewer: 3

The manuscript is generally relatively well written, but is perhaps from time to time a bit more like

report than scientific article. This is also partially matter of presentation and opinion.

We have tightened the language in several parts of the paper. We will appreciate if you could

point to specific sections that you thought could do with better writing.

missing references: the manuscript seems to miss quite a few important references, like those listed

at the end of these review notes ([R1]-[R3]). These should be properly cited in the paper. Also, [R2]

and [R3] contain SINR analyses of OFDMA radio link, in which either (i) more general or arbitrary

spectra shape for the oscillator is allowed or (ii) also the power differences between the neighboring

channels are taking into account. So please compare your work also against these.

Thank you for pointing us to these references, wee have now included these references as well.

We would like to point out that while the focus in these references is on OFDMA, our focus is

almost exclusively on SC-FDMA. Further, for an accurate analysis of SINR it is crucial to take into



consideration the receiver structure being used, and we believe this is one of the first paper to have

incorporated this into the analysis.

Just as in the references, we have now modified our analysis to account for power differences

between sub-carriers. We have however worked under the equal power assumption in our simulations

and discussions so as best illustrate the underlying interactions in the simplest possible setting.

only RX phase noise is considered in the manuscript, why? Please elaborate. This is especially

since the focus is here on cellular mobile uplink in which obviously terminal TX is supposed to

have much lower quality (cheaper) oscillator, and thus perhaps contains much more phase noise than

basestation RX ?

There are two reasons why we did not consider transmitter phase noise in this paper. The first

reason is that in the uplink, we would have had to factor in as many phase noise sequences as there

are users and this brings in a much higher degree of complexity to the overall analysis since we

now have to deal with multiple sets of phase noise process parameters. The second major reason is

that once we factor in transmit phase noise, the transmit signal is no longer cyclic and hence we

can no longer leave the cyclic prefix out of the overall signal model. Thus in addition to the usual

effects of phase noise on the signal, we will also have to factor in the additional penalty due the

signal being acyclic. This calls for a more careful analysis and the current framework cannot handle

this. We hope to address this in our future work.

how sensitive the results are against the assumed oscillator model ? what happens if you try e.g.

with the oscillator models described e.g. in [R3] ? on the other hand, what happens if you use free

running oscillator (brownian motion) in which plain integrator is used to filter white Gaussian noise

(as the PHN generation model) ?

A basic assumption used all through this paper is that the phase noise process is a stationary

process. In this paper, we have modeled phase noise as an AR(1) process, which is a good fit for

phase noise generated from PLL based oscillators. The covariance matrix of such a process is given

in Appendix A and plays a crucial role in our analysis. We can directly extend the results in this

paper to other stationary phase noise models as long as we use the appropriate covariance matrix Φ.



So, yes, the analysis presented in this paper does extend to other stationary phase noise models.

With regard to sensitivity, as long as the overall PSD of the phase noise process looks similar

to that of an AR(1) process, in that it has a cut off frequency beyond which the power decays

exponentially, the conclusions drawn in this paper will hold.

The Wiener process is a non-stationary process whose variance increases over time and is

unbounded. Due to this, large phase shifts are possible unless phase is synchronized at the start of

every OFDM symbol, which is unlikely. Hence, the small-angle approximation is no longer true and

our results do not hold for this scenario. Since communication systems are seldom designed with a

free running oscillator, such a model for phase noise is not likely to arise in a practical receiver design.

since the focus is on uplink, it is not realistic that all users have the same power level. I recommend

that you extend the analysis by taking the possible power differences into account. This would add

the impact of the results. Such SINR analysis with different power levels for adjacent channel signals

is carried out e.g. in [R3], so please compare and cite properly.

Our original assumption was that power control will compensate for pathloss and shadowing while

the effect of Rayleigh fading still needs to be taken into account. However, in view of comments

from two reviewers, we have allowed signals on different sub-carriers to have different power levels

and have accordingly modified the SINR analysis. Please note changes to (15) and (16). However, in

our simulations, we have assumed that all users use the same signal strength since the primary focus

of this paper is to explore the effects of phase noise and interleaving in SC-FDMA while using a

linear receiver. Our goal was to find the simplest setting where these effects come to the forefront so

as to distill out all other effects and then identify the sensitivity of our results to various parameters

involved. We felt taking power control into account would significantly complicate the conclusions

drawn, while also restricting it to the specific assumptions made on power control.

in many experiments, like Fig 2 and Fig 3, it remains unclear whether the results are for only

one given channel realization (per user) or are the results averaged over many independent channel

realizations? If the results are for just one realization, then the validity could be seriously questioned



? please elaborate.

All the results have been averaged over multiple instances of the channel.

also, partially related to above, it remains somewhat unclear what kind of assumptions are made

about the radio channel properties; e.g. correlation across sub-carriers ? please elaborate.

The multipath Rayleigh fading channel used in this paper has 10 taps with an exponential power

delay profile i.e., the power in each tap decays from 0 dB to -27 dB. In addition a normalization

factor was introduced so as to ensure there is no power gain due to the channel and the reported

SINRs are accurate. We have now clearly specified the channel used. (see pg. 6)

overall, I find the presenation style somewhat overwhelming or loose, so the presentation could

be made much more compact. Perhaps a letter type of article? This way also the true contributions

would be more apparent. E.g. Appendix A looks to be just repetition from the literature?

While we have not completely done away with Appendix A, we have significantly shortened it

to only include some new observations and pointed to other references for further details. Fig. 8,

which explores the split in energy of the phase noise process between CPE and the higher order

components has not been presented before, and it is crucial in explaining the results seen in Fig. 7.

We strongly feel there are enough contributions in this paper to merit a full journal article. We

believe the critical argument that delineates the performance of interleaved and localized SC-FDMA

is fairly non-trivial. Under ideal conditions, interleaved SC-FDMA is expected to outperform

localized SC-FDMA because it is able to better exploit the diversity in the frequency selective

channel. However, this proves to not be the case in the presence of non-idealities such as phase

noise and this warrants a thorough investigation to bring to light the interplay between the various

factors involved. The relationship between the correlation among sub-carrier channel coefficients

and the distribution of energy in the higher order phase noise components in determining the overall

system performance has not been explored before.

Also, to the best of our knowledge this is one of the first papers to analyze the performance

subsequent to the linear receiver being used. It is important to note that there is essentially no



difference (except for sub-carrier mapping) in the signal model for interleaved and localized

SC-FDMA immediately after the IDFT operation at the receiver. The sub-optimality of the linear

MMSE receiver (for per tone channel equalization) for SC-FDMA plays a key role in our analysis

and hence it is important to incorporate this into our signal model. We believe we have provided a

solid framework to better analyze the effects of other non-idealities while using linear receivers and

is of potential use to other researchers with such interests.


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