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20 IEEE TRANSACTIONS ON BROADCASTING, VOL. 47, NO. 1, MARCH 2001
Analytical Modeling and Simulation of Phase Noise
Interference in OFDM-Based Digital Television
Terrestrial Broadcasting Systems
Mohamed S. El-Tanany, Yiyan Wu, Fellow, IEEE, and László Házy
Abstract—This paper deals with the problem of modeling of
phase noise in OFDM systems and the impact it may have on the
bit error rate performance of such systems subject to a number
of system variables and to a number of channel conditions which
may be encountered when such systems are deployed for certain
applications such as high speed wireless LANs and Digital Televi-
sion Terrestrial Broadcasting (DTTB). The phase noise processes,
the sources of which are the transmitter’s and receiver’s local
oscillator, are modeled using what is believed to be commercially
realizable phase noise masks. Such masks represent the long-time
averaged power spectral densities of the local oscillator output
signal.
Index Terms—COFDM, digital television, DTTB, MQAM,
OFDM, phase noise.
I. INTRODUCTION
O
FDM provides an effective method to mitigate inter-
symbol interference (ISI) in wideband signalling over
multipath radio channels. The main idea is to send the data in
parallel over a number of narrowband flat subchannels (see
Fig. 1). This is efficiently achieved by using a set of overlapped
orthogonal signals to partition the channel. A transceiver can
be realized using a number of coherent QAM modems which
are equally spaced in the frequency domain and which can
be implemented using the IDFT on the transmitter end and
the DFT on the receiving end [1]–[7]. Due to the fact that
the intercarrier spacing in OFDM is relatively small, OFDM
transceivers are somewhat more sensitive to phase noise by
comparison to single carrier transceivers. It is the purpose of
this paper to examine the impact of L.O. phase noise on the
BER performance of OFDM signals over both AWGN and
frequency selective channels.
The paper first discusses the relationship between the contin-
uous time, continuous frequency L.O. phase noise model and
the discrete time, discrete frequency process that is seen by the
OFDM system. An analysis of the OFDM receiver is presented
to assess the impact of the phase noise on the decision vari-
ables at the receiver (which are the received signal samples cor-
rupted by noise, just before making hard decisions regarding
Manuscript received January 31, 2001; revised March 14, 2001. This work
was supported by the Communications Research Centre,Ottawa, Canada, under
Contract 67-CRC-5-2612.
M. S. El-Tanany and L. Házy are with the Department of Systems and Com-
puter Engineering, Carleton University, Ontario, K1S 5B6.
Y. Wu is with the Communications Research Centre, Shirley Hay, Ottawa,
Ontario.
Publisher Item Identifier S 0018-9316(01)04269-X.
Fig. 1. Using OFDM to mitigate ISI.
the received data). It is then shown that the effect of phase
noise on the decision variables is composed of two compo-
nents: a common component which affects all data symbols
equally and as such causes a sometimes visible rotation of the
signal constellation, and a second component which is more
like Gaussian noise and thus affects the received data points
in a somewhat random manner. This representation in terms of
common and foreigncomponents hasbeen pointed out in the lit-
erature [8]–[10]. What we introduce herethat is different, is that
the temporal variations of the rotational component and its de-
pendence on the frequency spacing between the system carriers
play an important role in determining the symbol-error rate per-
formance of the OFDM system, particularly at higher operating
SNR conditions. Taking the temporal variations of the rotational
component of the phase noises into account, we then proceeded
to derive analytical expressions for the average probability of
errorfor 64-QAMOFDM.Theresulting formulas areinaclosed
form which includes several integrals over the Gaussian prob-
ability curve. The analytical results are then used to quantify
the impact of certain phase noise masks on the average BER
performance under different SNR conditions and also subject
to variations of the OFDM frame length. Computer simulation
is used to treat the problem over channels with arbitrary multi-
path profiles and also, to investigate the impact of phase noise
on channel estimation and channel equalization.The simulation
model requires a user specified phase noise mask as an input. It
alsorequires the user to identify system parameterssuch as sam-
pling frequency, OFDM frame length and the size of the signal
constellation.
II. S
YSTEM DESCRIPTION AND MODEL
A functional block diagram of an OFDM system is shown in
Fig. 2. The incoming data is fist applied to a baseband
-ary
0018–9316/01$10.00 © 2001 IEEE
EL-TANANY et al.: ANALYTICAL MODELING AND SIMULATION OF PHASE NOISE INTERFERENCE 21
Fig. 2. Block diagram of an OFDM transmitter.
Fig. 3. Block diagram of an OFDM receiver.
QAM modulator which maps each binary bits into
one of the
constellation points. The -QAM points coming
out of the baseband modulator are then grouped into frames,
each containing
complex constellation points. Each frame is
applied to an inverse DFT processor which outputs
-complex
transform coefficients. A circular prefix of length
is then
appended to the
complex transform coefficients to form a
transmitted frame which is
points long. The transmitted
frame is then applied to a serial-to-parallel converter and then
applied to an IQ modulator to translate the spectral content of
the signal to some UHF or microwave frequency band. The IQ
modulation is accomplished by multiplying the complex enve-
lope of the signal with the output of a local oscillator. This step
isoften accomplished at a convenientIF frequency and the mod-
ulated signal is then upconverted using a higher frequency local
oscillator. For our purpose, it is sufficient to consider one local
oscillator as indicated in, Fig. 2.
The local oscillator is not perfect. Its output is usually de-
graded due to many factors, including short term frequencydrift
that may in part be caused by temperature variations. The short
term frequency drifts manifest themselves as phase noise which
has traditionally been characterized in terms of its power spec-
tral density.
A functional block diagram of a simplified OFDM receiver
is depicted in Fig. 3. The received signal, usually corrupted by
additive noise and channel distortion, is first applied to a low
noise microwavefront-end where it is amplified and perhaps fil-
tered to suppress unwanted interference. The received signal is
then downconverted to an IF frequency and applied to an I&Q
demodulator which brings the signal down to baseband in the
form in-phase and quadrature components. These in turn are ap-
plied to a A/D converter which outputs complex baseband sam-
ples at a rate of one sample per received symbol. The complex
samples are then grouped into received frames which contain
points each. Assuming that the frame synchronization
is working, the received frames are first reduced to
points
each by removing the circular prefix, and then are applied to an
-point DFT processor. The received frame is also used to esti-
mate the frequency response of the channel. The DFT output is
then equalized to generate
-decision variables which may be
used to recover the data either based on threshold comparison or
applied to a sequential estimation procedure such as the Viterbi
algorithm.
The important block in Fig. 3 is the local oscillator, which
likethe transmitter local oscillator may haveits ownphase noise
which will degrade the quality of the received signal and the
overall BER performance. For analog TV (ATV) applications
the Tx local oscillator is of much better spectral purity since
this exists only in thebase station and as such it does not have to
be very economical. The Rx local oscillator signal on the other
hand is provided with a cheap commercial TV tuner which ex-
hibits high levelsof phase noise. It is for this reason that we will
concentrate on the Rx LO phase noise without explicit mention
of the Tx LO phase noise. It should however be mentioned that
the analysis we will develop can still be applied to cases where
the phase noise is introduced by a combination of both local
oscillators.
III. P
HASE NOISE ANALYSIS
Let represent tile analog (i.e., continuous time and fre-
quency) phase noise process of the local oscillator which will
be assumed Gaussian with zero mean and power spectral den-
sity specified by some phase noise mask. As such,
can be
written as follows:
(1)
where
is the impulse response of a low-pass linear filter
whose frequency response is given by:
scale factor (2)
where
is the phase noise mask of the local oscillator and
is a white Gaussian noise process with power spectral
22 IEEE TRANSACTIONS ON BROADCASTING, VOL. 47, NO. 1, MARCH 2001
density . By definition, the autocorrelation function of the
phase noise process is given by:
(3)
which can also be written in terms of
as follows:
(4)
We are interested in the autocorrelation function of the sam-
pled phase-noise process. This is defined by:
(5)
where
(6)
and therefore
(7)
which can be rewritten as
(8)
From (8) and (3) and (4) the discrete autocorrelation function
can be rewritten in terms of the continuous phase noise power
spectral density as follows:
(9)
In order to simplify the analysis that follows we make the
assumptions:
• the communication channel is additive white Gaussian
• the channel frequency response is flat
• the phase noise variance is very small compared to unity,
in which case the assumption that
can
be invoked.
Subject to the above, the input to the DFT block during one
OFDM frame interval may be written as:
to (10)
where
are the IDFT coefficients of the QAM symbol
sequence
and is the sampling period. With the approx-
imating assumptions made above,
can be rewritten:
(11)
and the output of the DFT block can therefore be written as
(12)
By examining (12) it becomes clear that the phase noise
disturbs the decision variables in two ways:
• The first bracketed component to the right implies a phase
rotation in the amount
for all the received data
symbols in the current OFDM frame. This is a common
rotation and it results in a rotation of the entire signal con-
stellation. We will use
to denote this common rotation
factor, that is
(13)
• The second bracketed component depends on the data
symbols within the current frame. Assuming that the
frame is relatively large and that the data points are
statistically independent, it can then be argued that the
result is a Gaussian process which is likely to affect the
decision variables in the same way additive Gaussian
noise does. This second component has been referred to
in the literature as a foreign component, but here it will
be referred to as a dispersive component due to the way it
affects the received signal constellation. We will use
to denote this second term, that is
(14)
The common component is linearly related to the sampled
phase noise process and as a result,
is Gaussian with a mean
and variance given by:
(15)
(16)
(17)
The foreign dispersive component is also Gaussian with zero
mean and variance which is evaluated next. The variance of the
sum of the rotational and the dispersive components is equal
to the sum of the variances of both components. As such, the
variance of the dispersive component can be easily obtained if
we know the variance of the sum. This is given by:
(18)
In other words,
(19)
EL-TANANY et al.: ANALYTICAL MODELING AND SIMULATION OF PHASE NOISE INTERFERENCE 23
where
(20)
and which can be rewritten as
(21)
Therefore, we get
(22)
and
(23)
where
is the average signal power. We conclude that the
variance of the sum of the two noise components normalized to
the signal power is equal to the area under the phase noise mask.
The variance of the dispersive component is therefore given by
.
We now summarize the important points in this section. The
decision variables which are represented by the output of the
DFT block in the receiver block diagram of Fig. 3 are corrupted
by the phase noise process in two different ways:
First, the entire signal constellation is rotated by an amount
which is a zero mean Gaussian process with variance given
by
(24)
where
represents the OFDM frame length,
represents the sampling frequency and
is the power spectral density of the local oscillator’s
phase noise process.
It should be noted here that
is in reality a band-limited
process with a bandwidth less than half of the sampling rate.
Therefore, the integration limits in (24) can be changed to the
range
. Assuming that is held constant, the
variance
is a monotonically decreasing function of .It
reaches a maximum for
and approaches zero as ap-
proaches infinity. This implies that OFDM systems which use
shorter frame lengths will experience larger constellation rota-
tions while systems with verylargeframes exhibit much smaller
rotation.
Fig. 4. Analytical results showing the symbol error rate as a function of
SNR/bit.
The decision variables are also corrupted by a dispersive
noise component that is essentially Gaussian with zero mean
and variance
(25)
We notice that
is a monotonically increasing function of
with a minimum of zero as (i.e., for a single carrier
system). If left uncompensated, the rotational component can
severely impact the BER performance of the OFDM system.
The impact of this component on the average BER performance
of
-QAM OFDM for and is analyzed in the
following section.
IV. P
ROBABILITY OF ERROR ANALYSIS
The conditional BER performance (conditioned on the
rotational component) for 16-QAM OFDM and for 64-QAM
OFDM for a given value of the rotational noise component
is derived in the Appendix and it can be written as follows:
(26)
where
(27)
24 IEEE TRANSACTIONS ON BROADCASTING, VOL. 47, NO. 1, MARCH 2001
Fig. 5. Carrier recovery processing steps.
(28)
Therefore,
(29)
The average symbol error probability is obtained by aver-
aging the, conditional probability of (26) over the distribution
of the rotational component. This component is Gaussian with
zero mean and variance given by (24).
V. V
ARIANCE OF THE ROTATIONAL COMPONENT
The phase noise process of commercial tuners is often de-
scribed in terms of a phase noise mask which is piecewiselinear
andhas units of dBc/Hz.We willuse
todenote such a mask.
Phase noise masks can be fully described in terms of the coordi-
nates of their breakpoints. Using these coordinates and the fact
thatthemask ispiecewiselinear we can compute the phase noise
spectrum in absolute units as follows:
(30)
(31)
(32)
(33)
(34)
The average probability of symbol errors for a 64-QAM
signal has been calculated according to the procedure outlined
above with the results shown in Fig. 7. These results are based
on a phase noise mask with break-points at 1 kHz and 200 kHz.
The average power of the phase noise process is about 29 dB
below the carrier. Fig. 7 suggests that:
• The effect of phase noise on SER performance is negli-
gible for relatively low SNR (below 12 dB).
• The performance degradation increases rapidly as the
SNR/bit increases beyond 25 dB. This is due to the
presence of an irreducible error floor. The impact on the
8k and 2k frame sizes is about the same since no attempt
has been made to compensate for phase noise in either
case. The SNR degradation due to phase noise, at an error
rate of
, is about 3 dB.
VI. C
ARRIER RECOVERY
Carrier recovery refers to the estimation of, and compensa-
tion for the phase noise sequence
across an OFDM frame. This task can be accomplished with
the aid of at least one CR (carrier recovery) pilot tone. The de-
sign of the CR pilot differs from the channel estimation pilots
in that a frequency guard interval is mandatory on both sides of
the CR pilot to make possible the extraction of the prominent
phase noise sidebands with as little interference from the data
symbols as possible. The selection of the frequency guard in-
terval is obviously dependent on the shape of the phase noise
spectral mask.
For the sake of simplicity, we shall assume that the carrier
recoveryis achieved using an ideal bandpass filter with its pass-
band centered on the CR pilot-tone frequency. This filtering op-
eration is to be performed in the frequency domain using the
FFT and IFFT. The frequency guard interval normalized to the
OFDM tone spacing will be denoted by
, and the CR filter
bandwidth normalized to the OFDM tone spacing is denoted
.
The CR pilot tone normalized frequency is taken as
.
The processing steps involved in the carrier phase recovery
are illustrated in Fig. 5. The window function
is a rectan-
gular window which is equal to unity for
and for
, and is zero otherwise.
Theoutputofthe carrier recoverysubsystem in Fig. 5 consists
of three components:
1) A desired component which is directly dependent on the
phase noise sidebands within a bandwidth
. This com-
ponent is of the form
(35)
with a variance equal to
(36)
2) An interfering component caused by interference from
the information bearing OFDM tones as a result of tone
spreading that is created by the phase noise process. The
EL-TANANY et al.: ANALYTICAL MODELING AND SIMULATION OF PHASE NOISE INTERFERENCE 25
variance of this information related interference is given
by
(37)
where
(38)
3) An interfering component caused by the additive thermal
noise in the system and which has a variance of the form
(39)
where SNR
is the average signal-to-noise ratio per tone
and
is the average symbol power.
The normalized mean-square error of the recovered carrier
can therefore be written as
NMSE
(40)
The first factor in the numerator of (40), namely
is recognized as the power of the
phase noise process falling outside a bandwidth equal to the
frequency guard interval and as such, it decreases monotoni-
cally with
. This component also increases with the number
of points in an OFDM frame (i.e., the FFT size
). The second
term in the numerator accounts for the additive white Gaussian
noise disturbance to the recovered phase noise sequence. The
denominator of the second factor in (40) represents the phase
noise power (desired component) measured within the CR filter
bandwidth.
As an example, let us consider the phase noise mask shown
in Fig. 6. Assume that this mask represents the LO phase noise
in an OFDM system which utilizes a
-point FFT. From (40)
abovewecalculatedtheCRnormalized MSE as afunctionofthe
CR filter bandwidth relative to the OFDM signal tone spacing,
with the frequency guard interval as a parameter. The result is a
family of curves as illustrated in Figs. 7–9.
From Figs. 7 and 9 we notice that under high SNR conditions
the normalized MSE has an irreducible value caused by the first
term in the numerator of (40). This value increases as the fre-
quency guard interval is reduced. The curves also suggest that
there is an optimum value for the CR filter bandwidth.
Fig. 8 presents the resultsfor an additive noise SNR of 30 dB,
which is approximately the value needed for an error rate of
assuming 64-QAM modulation is used. This set of curves
Fig. 6. An example phase noise mask (used in many computations to follow).
Fig. 7. Normalized MSE under high SNR conditions, for a 8192 FFT system.
Fig. 8. Normalized MSE for a SNR 30 dB, for a 8192 FFT system.
exhibits a similar trend as observed in Figs. 7 and 9, except
that now the sensitivity of the normalized MSE to the frequency
guard interval is reduced. It is worthy noting that the numerical
results shown above are based on the assumption that the CR
pilot power is equal to 1.5% of the OFDM signal power.
26 IEEE TRANSACTIONS ON BROADCASTING, VOL. 47, NO. 1, MARCH 2001
Fig. 9. Normalized MSE under high SNR conditions, for a 2048 FFT system.
Fig. 10. OFDM frame format.
In summary, the analysis given in this section can be used to
identify benchmarks for the necessary frequency guard interval
required around the CR pilot, and also to identify a benchmark
for the carrier recovery filter bandwidth.
VII. S
IMULATION
In order to quantify the effect of phase noise on OFDM over
channels with frequency selective responses we had to resort to
simulation. Our model assumes
-QAM modulation. No for-
ward error correction is assumed in the model. The transmitted
OFDM frame structure is shown in Fig. 10.
Several equally spaced pilots are used for channel estimation
purposes and a carrier recovery pilot is inserted in the middle
of the frame with a guard band on both sides. The number of
pilots used and their relative power levels (compared to the data
subscribers) are user selectable. The OFDM frame size is also
selectable, but we are going to discuss results for frame sizes of
8k and 2k carriers only.
Fig. 11 illustrates the frequency domain picture of the trans-
mitted signal. The larger peaks represent channel estimation pi-
lots, and the carrier recovery estimation pilot.
Fig. 12 is a functional diagram for the main processing func-
tions of the OFDM receiver used for simulation.
The received signal is first processed to estimate the received
phase noise process using the carrier phase pilot. The carrier re-
covery bandwidth was found to be an important parameter and
as a result it is user selectable. The baseband signal is multiplied
by the complex conjugate of the estimated phase noise vector.
Fig. 11. Illustration of the power spectral density of an OFDM frame.
The compensated signal is also processed to extract an estimate
of the frequency response of the channel based on the channel
estimation pilots. The estimated channel response is used to
equalize the phase-compensated signal frame. Automatic gain
control is performed and finally the baseband signal is applied
to a slicer to recover the received data symbols.
A. Results
Fig. 13 shows the symbol error rate as a function of the
symbol energy to noise density ratio. These results are based
on the simulation of an 8k OFDM system. The phase noise
mask used here is similar to the one shown in Fig. 6 except that
the break points are at 2 kHz and 100 kHz. From Fig. 13 it is
clear that:
• The SER performance degradation due to channel
estimation errors is negligible (in this case the CE pilots
are relatively high at 10 dB above the average energy per
information symbol).
• Performance degradation at SER
, as caused by
phase noise, is about 3 dB relative to the performance of
an ideal system.
• Phase estimation and compensation using the CR algo-
rithm shown in Fig. 5 results in an SER performance that
is within 0.6 dB from ideal.
Fig. 14 shows the symbol error rate as a function of the
symbol energy to noise density ratio. These results are based on
the simulation of an 8k OFDM system The phase noise mask
used here is identical to that shown in figure. From Fig. 14 it
is clear that:
• The SER performance degradation due to channel estima-
tion errors is slightly higher by comparison to the case of
Fig. 13.
• Performance degradation at SER
, as caused by
phase noise, is about 3 dB relative to the performance of
an ideal system. This is identical to the value observed in
Fig. 13.
• Phase estimation and compensation using the CR algo-
rithm shown in Fig. 5 results in an SER performance that
is within 1 dB from ideal. This is higher than the 0.6 dB
observed in Fig. 13.
Fig. 15 shows the symbol error rate as a function of the
symbol energy to noise density ratio. These results are based on
the simulation of an 2k OPDM system. The phase noise mask
used here is identical to the one used in Fig. 6. From Fig. 15 it
is clear that:
• The SER performance degradation due to channel estima-
tion errors is lower than the degradation shown in the 8k
OFDM case of Fig. 14.
EL-TANANY et al.: ANALYTICAL MODELING AND SIMULATION OF PHASE NOISE INTERFERENCE 27
Fig. 12. Functional diagram of the simulated OFDM receiver.
Fig. 13. SER performance of an 8192-tone OFDM system in the presence of
phase noise over flat channels. This system uses 128 CE pilots at 6 dB relative
to the corner points of the 64-QAM constellation. The phase noise mask break
points are at 2 kHz and 100 kHz. The average phase noise power is
29 dB
relative to the carrier.
• Performance degradation at SER as caused by
phase noise, is about 3 dB relative to the performance of
an ideal system. This is identical to the value observed in
Figs. 14 and 13.
• Phase estimation, and compensation using the CR algo-
rithm shown in Fig. 5 results in an SER performance that
is within 0.5 dB from ideal. This is about half the degra-
dation suffered by the 8k OFDM system of Fig. 14.
Fig. 16 shows the symbol error rate as a function of the
symbol energy to noise density ratio. These results are based on
the simulation of an 8k OFDM system. The phase noise mask
used here is similar to the one used in Fig. 6. From Fig. 16 it
is clear that:
• The SER performance degradation due to channel estima-
tion errors is relativelyhigh by comparison to the degrada-
tion shown in the 8k OFDM case of Fig. 14. This increase
is due to the use of 3 dB less power in the CE pilots.
• Performance degradation at SER
, as caused by
phase noise, is about 2.2 dB relative to the performance of
an ideal system. This is lower than the value observed in
Figs. 14 and 13. This reduction is due to the fact that the
CE pilots an now lower and hence, they interfere less with
the information symbols.
Fig. 14. SER performance of an 8192-tone OFDM system in the presence of
phase noise over flat channels. This system uses 128 CE pilots at 6 dB relative
to the corner points of the 64-QAM constellation. The phase noise mask break
points are at 1 kHz and 200 kHz. The average phase noise power is
29 dB
relative to the carrier. The circles show the ideal performance curve.
• Phase estimation and compensation using the CR algo-
rithm shown in Fig. 5 results in an SER performance that
is within 0.7 dB from ideal. This is about the same perfor-
mance as shown in Fig. 14.
Fig. 17 shows the symbol error rate as a function of the
symbol energy to noise density ratio. These results are based on
the simulation of an 8k OFDM system. The phase noise mask
used here is similar to the one used in Fig. 6. From Fig. 17 it
is clear that:
• The SER performance degradation due to channel estima-
tion errors is relatively high by comparison to the degrada-
tion shown in the 2k OFDM case of Fig. 15. This increase
is due to the use of a 3 dB less power in the CE pilots.
• Performance degradation at SER
, as caused by
phase noise, is about 3 dB relative to the performance of
an ideal system.
• Phase estimation and compensation using the CR algo-
rithm shown in Fig. 5 results in an SER performance that
is within 0.7 dB from ideal. This is slightly worse com-
pared to the performance shown in Fig. 15.
Fig. 18 shows the symbol error rate as a function of the
symbol energy to noise density ratio over a frequency selective
channel. These results are based on the simulation of an
28 IEEE TRANSACTIONS ON BROADCASTING, VOL. 47, NO. 1, MARCH 2001
Fig. 15. SER performance of a 2048-tone OFDM system in the presence of
phase noise over flat channels. This system uses 32 CE pilots at 6 dB relative
to the comer points of the 64-QAM constellation. The phase noise mask break
points are at 1 kHz end 200 kHz. The average phase noise power is
29 dB
relative to the carrier. The circles show the ideal performance curve.
Fig. 16. SER performance of an 8192-tone OFDM system in the presence of
phase noise over flat channels. This system uses 128 CE pilots at 3 dB relative
to the corner points of the 64-QAM constellation. The phase noise mask break
points are at 1 kHz and 200 kHz. The average phase noise power is
29 dB
relative to the carrier. The circles show the ideal performance curve.
8k OFDM system. The phase noise mask used here is similar
to the one used in Fig. 6. From Fig. 18, it is clear that:
• Performance degradation at SER
, as caused by
phase noise, is about 5 dB relative to the performance of
art ideal system.
• Phase estimation and compensation using the CR algo-
rithm shown in Fig. 5 results in an SER performance that
is within 3.5 dB from ideal. This is much worse compared
to the performance shown in Fig. 15. This may be ex-
plained by the fact that the ICI due to residual phase noise
is enhanced due to the presence of powerful tones in the
vicinity of weaker tones (the unevenness of tone power is
caused by the frequency selectivity of the channel).
Fig. 19 shows the symbol error rate as a function of the
symbol energy to noise density ratio over a frequency selective
channel. These results are based on the simulation of an
Fig. 17. SER performance of a 2048-tone OFDM system in the presence of
phase noise, over flat channels. This system uses 32 CE pilots at 3 dB relative
to the corner points of the 64-QAM constellation. The phase noise mask break
points are at 1 kHz and 200 kHz. The average phase noise power is
29 dB
relativetothecarrier. The circles showanidealperformancecurve for reference.
Fig. 18. SER performance of an 8192-tone OFDM system in the presence
of phase noise over a frequency selective channel with impulse response
[1zeros(1,7) zeros(1,7) 1/4]. Thissystem uses128CE pilotsat3 dBrelative
to the corner points of the 64-QAM constellation. The phase noise mask break
points are at 1 kHz and 200 kHz. The average phase noise power is
29 dB
relative to the carrier. The solid lines show performance based on the simulated
channel and phase estimates. The circles show an ideal performance curve for
reference.
2k OFDM system. The phase noise mask used here is similar
to the one used in Fig. 6. From Fig. 19 it is clear that:
• Performance degradation at SER
, as caused by
phase noise, is about 5 dB relative to the performance of
an ideal system.
• Phase estimation and compensation using the CR algo-
rithm shown in Fig. 5 results in an SER performance that
is within 1.2 dB from ideal. This is much better compared
to the performance shown in Fig. 18. This may be ex-
plained by the fact that the ICI due to residual phase noise
is less significant due to the fact that the tone spacing is
now 4 times the tone spacing in the case of the 8k OFDM
system, in spite of the unevenness of tone power.
EL-TANANY et al.: ANALYTICAL MODELING AND SIMULATION OF PHASE NOISE INTERFERENCE 29
Fig. 19. SER performance of a 2048-tone OFDM system in the presence of
phase noise over a frequency selective channel with impulse response
. This system uses 32 CE pilots at 3 dB relative to the
corner points of the 64-QAM constellation. The phase noise mask break points
acre at 1 kHz and 200 kHz. The average phase noise power is
29 dB relative
to the carrier. The solid lines show performance based on the simulated channel
and phase estimates. The circles show an ideal performance curve for reference.
VIII. CONCLUSIONS
This paper presented an analytical procedure to quantify the
impact of local oscillator phase noise on the performance of
OFDM systems over additive white Gaussian noise channels.
It also addressed the questions of performance over frequency
selective channels. Based on simulation results, the paper also
discussed the potential performance improvement that can be
gained if the OFDM system employed a means of phase com-
pensation on the receiving end. The impact of completely un-
compensated phase noise is to create an irreducible error floor.
Partial compensation for phase noise is also possible. The cases
studied here indicate significant performance improvement in
terms of SNR for both flat and for frequency selective channels.
A
PPENDIX
DERIVATION OF BER IN THE PRESENCE OF PHASE NOISE
The -QAM signal constellation is rectangular with constel-
lation points represented by
(A.1)
with
for 64-QAM and
for 16-QAM.
The impact of the phase noise process is modeled by multi-
plying the signal points above with the vector
to account
for the rotational component, and by adding a noise component
to account for the dispersive component. This results in a
decision variable with a mean
(A.2)
and a varianceequal to the sum of the variancesof the dispersive
component and the variance of the additive thermal raise com-
ponent. The decision variables can thus be written of the form
(A.3)
where
represents a complex Gaussian process which is equal
to the sum of the additive noise and the dispersive component
that is caused by the local oscillator phase noise process.
We note that the real and imaginary parts of
are indepen-
dent Gaussian processes with equal variances. Their joint dis-
tribution is given by
(A.4)
To calculate the probability of error we divide the signal space
intorectangular decision regionswith boundaries at
and
for the inner constellation points. The outer points
at the left, right, top or bottom of the constellation will have
their respective decision regions extending to infinity on at least
one side. With this in mind, we proceed to compute the average
probability of error.
A. The Inner Constellation Points
For the inner constellation points the probability of making a
correct decision is given by
(A.5)
The first integral above can be written as
(A.6)
where
(A.7)
and
(A.8)
Similarly,
(A.9)
where
(A.10)
and
(A.11)
30 IEEE TRANSACTIONS ON BROADCASTING, VOL. 47, NO. 1, MARCH 2001
Therefore, the probability of error for an inner constellation
point is given by
(A.12)
where
can be written in the form
(A.13)
and
(A.14)
B. The Exterior Constellation Points (Left/Right)
For the left/right exterior constellation points it can be shown
that
is given by
(A.15)
where
(A.16)
C. The Exterior Constellation Points (Top/Bottom)
For the top/bottom exterior constellation points it can be
shown that
is given by
(A.17)
where
(A.18)
D. The Diagonal Constellation Points
For the diagonal constellation points it can be shown that
is given by
(A.19)
where
(A.20)
E. Conditional Probability of Error
The BER performance of 16-QAM OFDM and of 64-QAM
OFDM for a given value of the rotational noise component
can now be written as follows:
(A.21)
where
(A.22)
and
(A.23)
Therefore,
(A.24)
R
EFERENCES
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EL-TANANY et al.: ANALYTICAL MODELING AND SIMULATION OF PHASE NOISE INTERFERENCE 31
[7] A. Chini, “Multi Carrier Modulation in Frequency Selective Fading
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Mohammed S. El-Tanany obtained the B.Sc. and M.Sc. in electrical engi-
neering in 1974 and 1978 respectively, both from Cairo University in Giza,
Egypt, and the Ph.D. in electrical engineering from Carleton University, Ot-
tawa, ON, Canada in 1983. He worked with the Advanced Systems division of
Miller Communications in Kanata, Ontario from 1982 to 1985 with principal
involvement in the research and development of digital transmission equipment
for mobile satellite type of applications and also for VHF airborne high-speed
down links. He joined Carleton University in 1985, initially as a research asso-
ciate in thearea ofwireless communications for mobile andindoor communica-
tions. He is currentlya professorwith theDepartment of Systemsand Computer
Engineering where he isactivelyinvolved inseveralresearch programsthat deal
with digital transmission in the PCS and millimeter wave frequency bands, with
emphasis on channelmeasurements, modelingas wellas modulation/coding for
frequency selective fading channels.
Yiyan Wu is a senior research scientist with the Communications Research
Centre, Ottawa, Canada. His research interests include digital video compres-
sion and transmission, high definition television (HDTV), signaland image pro-
cessing, satellite and mobile communications. He is actively involved in the
ATSC technical and standard activities and ITU-R digital television and data
broadcasting studies. He is an IEEE Fellow, an adjunct professor of Carleton
University, Ottawa, Canada, a Member of the IEEE Broadcast Technology So-
ciety Administrative Committee and a member of the ATSC Executive Com-
mittee (representing IEEE).
László Házy obtained a B.Sc. in electronics and telecommunications from the
Polytechnic Institute of Bucharest, Romania in 1992 and a M.Eng. in electrical
engineering from Carleton University, Canada, in 1997. Currently he is a Ph.D.
candidate at Carleton University. Between 1992 and 1995, he was an assistant
professor with the Department of Electronics and Computers at Transilvania
University, Romania, working in the area of radio and microwave communica-
tions. Between 1993 and 1995 he was also involved with the Research Institute
for Computer Technology, Brasov, Romania. His research interests include dig-
ital communications theory and various aspects of wireless communication sys-
tems, and he is currently working on OFDM and multicarrier spread spectrum
systems.
