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.
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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
[14] T. C. W. Schenk, R. W. van der Hofstad, E. R. Fledderus, and P. F. M. Smulders, “Distribution of the ICI term in phase noise
impaired OFDM systems,” in IEEE Transactions on Wireless Commununications, vol. 6, no. 4, April 2007, pp. 1488–1500.
[15] V. Syrjala, M. Valkama, Y. Zou, N. N. Tchamov, and J. Rinne, “On OFDM link performance under receiver phase noise with
arbitrary spectral shape,” in IEEE Wireless Communications and Networking Conference 2011, March 2011.
[16] G. Sridharan and T. J. Lim, “Blind estimation of common phase error in OFDM and OFDMA,” in 2010 IEEE Global
Telecommunications Conference, December 2010.
[17] K. Nikitopoulos and A. Polydoros, “Phase-impairment effects and compensation algorithms for OFDM systems,” IEEE Trans.
on Commun., vol. 53, no. 4, pp. 698 –707, April 2005.
[18] D. D. Lin and T. J. Lim, “The variational inference approach to joint data detection and phase noise estimation in OFDM,” IEEE
Transactions on Signal Processing, vol. 55, no. 5, pp. 1862–1874, May 2007.
[19] D. D. Lin, R. Pacheco, T. J. Lim, and D. Hatzinakos, “Joint estimation of channel response, frequency offset, and phase noise
in OFDM,” IEEE Transactions on Signal Processing, vol. 54, no. 9, pp. 3542–3554, September 2006.
April 22, 2012 DRAFT
IEEE TRANSACTIONS ON COMMUNICATIONS 21
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.
April 22, 2012 DRAFT
IEEE TRANSACTIONS ON COMMUNICATIONS 22
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
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.
April 22, 2012 DRAFT
IEEE TRANSACTIONS ON COMMUNICATIONS 24
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.
April 22, 2012 DRAFT
IEEE TRANSACTIONS ON COMMUNICATIONS 25
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.
April 22, 2012 DRAFT
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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
April 22, 2012 DRAFT
IEEE TRANSACTIONS ON COMMUNICATIONS 27
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
April 22, 2012 DRAFT
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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.
April 22, 2012 DRAFT
IEEE TRANSACTIONS ON COMMUNICATIONS 29
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
April 22, 2012 DRAFT
IEEE TRANSACTIONS ON COMMUNICATIONS 30
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.
April 22, 2012 DRAFT
IEEE TRANSACTIONS ON COMMUNICATIONS 31
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
April 22, 2012 DRAFT
IEEE TRANSACTIONS ON COMMUNICATIONS 32
? 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
April 22, 2012 DRAFT
IEEE TRANSACTIONS ON COMMUNICATIONS 33
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.