# OFDM Synchronization Errors and Effects of Phase Noise on WLAN Transceivers

Authors: Mourad Melliti; Salem Hasnaoui; Ridha Bouallegue
Journal: Journal of Computer Systems, Networks, and Communications
 (2008)

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## Abstract

This work aims at frequency synchronization in OFDM IEEE 802.11g Transceivers. The degradation of performance results from the detrimental effects introduced by frequency offset and phase noise. First, we present a robust method to detect and synchronize the OFDM signal with severe noise and interference corruption based on calculating the accumulated phase difference for each subcarrier. Then, we present a simple architecture to implement the proposed methods. These methods are simulated and analyzed by computer. Finally, we propose many algorithms to correct the common phase error (CPE) of the phase noise and track well its variation.


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## Body

## 1. Introduction

Orthogonal frequency
division multiplexing (OFDM) has been gaining popularity in a variety of
digital communication systems, such as wireless LAN 802.11a and 802.11g. The
imperfect synchronization between the tuner's local oscillation in transmitter
and in receiver brings not only frequency offset but also phase noise. Many
algorithms have been put forward for estimating and correcting common phase
error (CPE). This paper proposes a novel adaptive algorithm for correcting CPE.
The simulation results show that the algorithm is of good performance in CPE
estimation.After a brief introduction to OFDM
systems in Section2, this paper will be concerned in defining the general
problem of improper frequency synchronization. In Sections 3 and 4, we will
explain why frequency offsets have detrimental effects on the received signal's
spectrum. We will quantify the offset with respect to the carrier frequency
spacing and will consider the two cases where the offset is either an integer
multiple of the carrier spacing or not. Depending on this condition, we will see
how the orthogonality among the carriers may be destroyed. We will then turn to
study the effects of phase noise. We will characterize phase noise by
describing the 2 types of phase error that it induces, namely, common phase
error (CPE) and intercarrier interference (ICI). The description will fully
explain the modification induced by each type on the received signal's
spectrum. In Section 7, we will briefly discuss the common approaches to remove
the effects of CPE.


## 2. OFDM Systems

The main reasons OFDM was adopted in the
wireless local area network (WLAN) standards IEEE 802.11a [1] and g [2] are its
high spectral efficiency and ability to deal with frequency-selective fading
and narrowband interference. However, the main disadvantage that might result
is the induced intersymbol interference (ISI). OFDM addresses this problem within
the more general framework of multicarrier modulation. The bit-stream is
divided into substreams that are each sent with a lower data rate on an
individual subchannel.As in all digital communication systems,
OFDM requires proper synchronization between the transmitter and the receiver.
Both time and frequency synchronization influence the performance of OFDM
systems. The subdivision of the initial symbol into N parallel symbols results
in a longer symbol duration. This will cause the system to be less sensitive to
timing offsets. However, frequency offsets may heavily degrade the system's
performance [1].


## 3. Implementation of QAM Transceiver

By implementing a QAM-based transceiver,
the OFDM signal is formed by first converting the frequency samples into time
samples through the use of the inverse Fourier transform represented before by
the IFFT [3].The output time samples take the
following form:(1)Sn=1N∑k=0N−1Skej2πnk/N,n=0,1,…,N−1.These generated parallel time samples
are then converted to a serial stream through a parallel-to-serial converter.
The last step required before transmitting the stream over the channel is to
add a cyclic prefix code. Indeed, one of the major properties of the continuous
Fourier transform is the duality between convolution in time domain and
multiplication in frequency domain. This property does not completely extend to
the discrete case. For it to be correctly applied, one should carry circular
convolution instead of the more common linear convolution in the time domain.
To this end, we assume that the discrete channel impulse response has lengthk.
Adding the cyclic prefix corresponds to appending the last k symbols of each
block [s0,s1⋯sN] to the beginning of the
same block. We get the following block: [sN−k,sN−k−1⋯s0,s1⋯sN]. This operation preserves the
desired property of multiplication in the frequency domain. The resulting
sequence is passed through a digital-to-analog converter and then up converted
to a higher carrier frequency. The signal is then sent through the channel.On the receiver end, noise introduced by
the channel will be superimposed on the received signal. The signal is first
down converted to base-band and then passed through a lowpass filter to remove
the effects of high-frequency terms. The time samples are retrieved by passing
the resulting signal through an analog-to-digital converter. The cyclic prefix
is first removed from each stream and then the stream is converted to a parallel
stream through a serial-to-parallel converter. The DFT is then performed to
recover the original coefficients. These are then converted to a serial stream
to form the symbol that was originally sent.As can be clearly seen from Figure1,
the required overall bandwidth of an OFDM system is equal to NB/2. In this
sense, OFDM systems make better use of the available system bandwidth.Effects of frequency shift on OFDM spectrum.
(a)(b)


## 4. Frequency Synchronization

As in all digital communication modems,
OFDM requires proper synchronization between the transmitter and the receiver.
Both time and frequency synchronizations influence the performance of OFDM systems.
The subdivision of the initial symbol intoN parallel symbols results in a
longer symbol duration. This will cause the system to be less sensitive to timing
offsets. However, frequency offsets may heavily degrade the system's
performance. To see this, we first derive the spectrum of an OFDM symbol from
its time domain representation. The time domain version of the symbol is the
one transmitted over the channel. We represent the transmitted OFDM symbol as
follows:  (2)S(t)=∑k=0N−1Skej2π(f0+kΔf)trect(tNTs),Sk denotes
the complex amplitude of the kth carrier, Δf is the carrier spacing, and f0 is the frequency of the 0th carrier.
The multiplication by the window function is due to the fact that each OFDM
symbol is time limited to a duration Ts.We then compute the Fourier transform of
the aforementioned time signal to get(3)S(f)=∑k=0N−1Sksinc(NTsf)*δ(f−(f0+kΔf)).Now, we note that the total bandwidth is
equal toNΔf and so the symbol time Ts is equal to 1/NΔf.
Replacing NTs by 1/Δf, we get (4)S(f)=∑k=0N−1Sksinc(f−(f0+kΔf)Δf).It might happen that on the
demodulator's side, the carriers get shifted by an amountδf.
The spacing between the carriers is still equal to Δf.
This can be seen from Figure 2.ICI effects on OFDM signal.
(a)(b)(c)(d)In this case, we have 3 subcarriers and
the symbols on the 3 different curves correspond to the different contributions
of each of the subcarriers to the sum of the sampling points. In Figure1, the
configuration is optimal since no offset is present between the receiver and
the transmitter. However, in Figure 2, an offset of δf is introduced and causes the different
subcarriers to interfere.The amount of shift might either be an
integer multiple of the subcarrier spacingΔf,
in which case, the configuration would be similar to Figure 1. However, the
received data symbols are now in the wrong position and so this will induce a bit-error
rate of 0.5 [1]. However, the orthogonality among the subcarriers is still
maintained. Indeed, the spectrum would now become  (5)S(f)=∑k=0N−1Sksinc(f+CΔf−(f0+kΔf)Δf),  where C is an integer constant.The shift does not affect the location of the
points of intersection of the different sinc waveforms that constitute the spectrum of the OFDM symbol. Orthogonality is
thus preserved. If the shift is not an integer multiple of the carrier spacing,
then we end up with interference among the different subcarriers. This
interference may be quantified in the following way:(6)Il=∑k=0,k≠1N−1Sksinc(δf+(l−k)ΔfΔf).It represents the amount of contribution other
carriers have on the waveform associated
with carrierl. So, Il is obtained by
evaluating S(f)
at the frequency of interest, namely, the
one associated with subcarrier l. Note that if the shift is zero or an
integer multiple of the carrier-spacing, the aforementioned interference term
would evaluate to zero. A nonzero value for Il can be
interpreted as a loss of orthogonality of the basis functions because this
would suggest that the projection of the waveform associated with carrier l has
nonzero components in the direction of the waveforms associated with the other
carriers. In this sense, ICI is also referred to as the effect of loss of
orthogonality. We will see this effect more in details in Section 5 in the
context of phase noise.


## 5. Effects of Phase Noise

Signals are usually transmitted in the
radio frequency range; but this is not compatible with the range of frequencies
over which modulators/demodulators operate. Indeed, they operate at a much
lower frequency usually referred to as the intermediate frequency. Converting
the RF frequency to IF is thus crucial for proper demodulation of the signal.
This down-conversion is performed by local oscillators which turn out to have
the detrimental effect of introducing noise that will be superimposed on the
received signal [4].The introduced noise is caused by random
variations of the phase about the steady-state sinusoidal waveform. In what
follows, we assume that the transmitted OFDM signal has the same form as
mentioned before, that is,(7)s(t)=∑k=0N−1Skej2π(f0+kΔf)t.In ideal conditions (if the local
oscillator does not introduce noise), the received signal would be of the
following form:(8)r(t)=∑k=0N−1HkSkej2π(f0+kΔf)t,  where Hk denotes the complex frequency response of the channel in the kth subchannel.On the other hand, if the local
oscillator is not ideal, we need to take into account the effect of noise. The
latter is modeled as a complex random process and denoted byejφ(t). We get the following received signal: (9)r(t)=∑k=0N−1HkSkej2π(f0+kΔf)t=∑k=0N−1Rkej2π(f0+kΔf)tejϕ(t),   where Rk=HkSk.At the receiver, we project the received
signal over each one of the possible carrier waveforms to determine the
demodulated carrier amplitudes. This is translated in the following:(10)Al=1Ts∫0Tsr(t)e−j2π(f0+lΔf)tdt=1Ts∫0Tsejϕ(t)∑k=0N−1Rkej2π(k−l)Δftdt,   where Ts denotes an OFDM symbol time and is
equal to Ts=1/NΔf and N is equal to the number of different subcarriers.
In a practical design, the afore-mentioned integral would be replaced by a
summation [5], but the authors decided to use the integral because it makes the
derivation closer to intuition even though nothing would change if they replace
it by a summation.The afore-mentioned expression denotes
the projection of the received signal on the waveform associated with the
carrier:(11)2πf1=2πf0+2πlΔf.The perturbations introduced by the
random processφ(t) are assumed to be relatively small so that the
following approximation is justified: (12)ejϕ(t)≈1+jϕ(t).Now, we can writeAl in the
following way: (13)1T∫0Ts(1+jϕ(t))∑k=0N−1Rkej2π(k−l)Δftdt=1Ts∑k=0N−1Rk∫0Tsej2π(k−l)Δftdt+1Ts∑k=0N−1Rk∫0Tsjϕ(t)ej2π(k−l)Δftdt.   The second term will be
denoted by Yl.This separation of the two terms is
justified by the fact that the first part is what we would have received in the
absence of noise while the second term includes the effect of noise.Now, we write(14)1Ts∑k=0N−1Rk∫0Tsjφ(t)ej2π(k−l)Δftdt   in the following way: (15)1Ts∑k=0,k≠lN−1Rk∫0Tsjϕ(t)ej2π(k−l)Δftdt+RlTs∫0Tsjϕ(t)dt.   The second term of the afore-mentioned
summation represents the contribution of the lth carrier to the
summation and is the one that we expect since we are projecting the received
signal on the waveform associated with the lth carrier. It evaluates to jϕ0Rl, where φ0 is a real constant that results from the integration.We note that the constantφ0 is independent of l and so it is the same for all symbols of the signal
constellation. We refer to this error as the CPE, which is directly proportional
to the received amplitude Rl of carrier l. We conclude that the physical meaning
of this contribution is a rotation of the signal constellation by an angle
equal to (16)ϕ0=1Ts∫0Tsjϕ(t)dt.We now characterize the other term that
contributes to the previous summation, namely,(17)1Ts∑k=0,k≠lN−1Rk∫0Tsjϕ(t)ej2π(k−l)Δftdt.This
term represents the remainingN−1 contributions of the other carriers. This
contribution consists in first shifting the spectrum of φ(t) by 2 2π(k−l)Δf and then scaling it by the received amplitude
of the corresponding carrier. This amplitude is complex and thus this noise
term will also be complex. It is commonly referred to as ICI or loss of orthogonality
[6]. This is due to the fact that the orthogonal property of the basis
functions that was used to produce the OFDM signal is no longer preserved.
Indeed, the projection of the received signal on the waveform associated with
the lth subcarrier now contains a new term which is completely random.
Because of this randomness, it cannot be corrected.More generally, we defineφm to be equal to (18)ϕm=1Ts∫0Tsϕ(t)ejmΔftdt.The noise effect can then be represented
by the following matrix:(19)[Y0Y1Y2:YN−2YN−1]=j[φ0φ1φ2⋯φN−2φN−1φ−1φ0φ1⋯φN−3φN−2φ−2φ−1φ0⋯φN−4φN−3⋮⋮⋮⋮⋮⋮φ−(N−2)φ−(N−3)φ−(N−4)⋯φ0φ1φ−(N−1)φ−(N−2)φ−(N−3)⋯φ−1φ0]×[R0R1R2⋮RN−2RN−1].  The diagonal of the afore-mentioned matrix is
constant because, as was explained before, this constitutes the CPE, which is
the same for all carriers. The off-diagonal elements are the weighting
coefficients that multiply each of the complex amplitudes of the received
signals. The nomenclature intercarrier interference is thus clear from the
structure of the afore-mentioned matrix.φ0 can be considered to be the output of a filter
with the following impulse response: (20)h(t)=rect(t/Ts)Ts.   The impulse response is
zero except for a rectangle of height 1/Ts for a duration of Ts seconds.Taking the Fourier transform ofh(t), we get (21)H(f)=sinc(fΔf).It is very common to assume that the
random process is a zero-mean Gaussian process. From this, we conclude that the
random variablesφm are also Gaussian zero-mean. To fully
characterize these random variables, we need to compute their variances.Forφ0,
we find that its power spectral density is given by |Φ(f)H(f)|2, where Φ(f) denotes the power spectral density of φ(t).
Its variance may then be computed as follows: (22)σ02=∫−∞∞|Φ(f)H(f)|2df=∫−∞∞sinc2(fΔf)|Φ(f)|2df.More generally,φm is a zero-mean Gaussian random variable with
variance equal to (23)σ02=∫−∞∞sinc2(fΔf)|Φ(f−mΔf)|2df.In fact, CPE can be accounted for as
will be described in Section7. The component we are left with is ICI.We rewrite it for ease of reference:(24)1Ts∑k=0,k≠lN−1Rk∫0Tsjϕ(t)ej2π(k−l)Δftdt.Using the afore-described filter, this
can be rewritten as(25)1Ts∑k=0,k≠lN−1Rk∫−∞∞jΦ(f−(k−l)Δf)sinc(fΔf)df.We make the change of variableu=f−(k−l)Δf and get (26)ICI=1Ts∑k=0,k≠lN−1Rk∫−∞∞jΦ(u)sinc(uΔf+k−l)du.We assume that thekth subcarrier
has average power = 2. We can then compute the power of the ICI term and find it
to be equal to (27)∑k=0,k≠lN−1σk2∫−∞∞|Φ(u)|2sinc2(uΔf+k−l)du.From the afore-mentioned expression, we
can see that if the power spectral density ofφ(t) is decreasing with frequency, then the major
contribution to ICI will be that of subcarriers that are near the subcarrier of
interest. We can also say that subcarriers near the center of the frequency
band will suffer more from
ICI than subcarriers at the edge of the band because contribution from adjacent
subcarriers will be higher in the former case. In [5], it is stated that
subcarriers near the center of the frequency band will be subject to more
interference than subcarriers at the band edge by up to a factor of two.Another important observation is that
interference increases as the spacing between the subcarriers decreases.From the power of the ICI, we can
compute the noise-to-signal ratio. For a subcarrier near the middle of the
band, assuming equal power on all subcarriers, the noise-to-signal ratio is
given in [2] as (28)ICI2σl2=2∑k=0N/2−1σk2∫−∞∞|Φ(u)|2sinc2(uΔf+k−l)du.We have seen that the effect of ICI
increases as the relative spacing of the subcarriers decreases. So in this case, ICI effect may
be considered small as compared to that of CPE. However, ICI is still present
because the symbols in the constellation tend to have the shape of a cloud
which depicts the random perturbation introduced by ICI. However, we can
appreciate more the effect of CPE because it can be seen that the constellation
has undergone a rotation by a constant angle. CPE can be corrected as will be
briefly discussed in Section7 and so after removing its effect.


## 6. Correction

We have previously mentioned that CPE is
the part of phase noise that can be accounted for and thus can be corrected.The ability to do so stems from the fact
that the effect of CPE is a simple rotation of the signal's constellation by a
fixed angle. Thus, if we can estimate that angle, then correcting for CPE
becomes a simple task as it would only consist in rotating back the
constellation by the same angle.One way to estimate the angle is by
measuring the phase variation of a pilot subcarrier and subtracting that
rotation from all subcarriers. The authors in [7] use an extended Kalman filter
to present an algorithm for pilot-based CPE estimation.Another way would consist in using
differential phase modulation between the subcarriers. This method would,
however, introduce a noise performance penalty when compared to coherent
demodulation. The idea behind this technique is to let one subcarrier have a
fixed phase and to let the information in the other subcarriers be encoded in
their relative phase differences.In [6], the authors propose a new algorithm
for CPE correction.Recall that(29)At=1Ts∑k=0N−1Rk∫0Tsej2π(k−l)Δftdt+1Ts∑k=0,k≠N−1Rk∫0Tsjϕ(t)ej2π(k−l)Δftdt+RlTs∫0Tsjϕ(t)dt,   which can be written as (30)At=1Ts∑k=0N−1Rk∫0Tsej2π(k−l)Δftdt+ICI+CPE.The first term on the left-hand side as
well as ICI are both zero-mean. So, the authors of [6] suggest that we can get
an estimate of the CPE for the lth OFDM symbol by finding the mean of
the phase rotations associated with the subchannels of an OFDM symbol (because
ICI and the first term have zero mean, they will not affect the expectation of Al for all l, and so
we can have an estimate of the CPE by averaging over all the Al).
However, some subchannels may suffer from a low SNR, and thus including their
effect in the CPE estimation will result in undesirable errors. These channels
(for which SNR falls below a certain threshold) are discarded. We will denote
by Nd their number. The Al above the
threshold will then be passed through a frequency equalizer to remove the
effect of the channel. Note that this assumes that we already know the
channel's characteristics. The equalized version of the Al is then passed through a slicer and we perform
an averaging over the difference of the equalized version and the sliced
version of the Al.This can be written as(31)CPEestimate=1Nd∑l,Hl≥threshold(Al,equalized−Al,sliced).More generally, a tracking sequence
occupying OFDM symbolNt (Nt is generally
set to 0) is sent to allow a proper detection of the start of the OFDM frame
and thus a proper removal of the cyclic prefix discussed previously. The above
CPE estimation needs to be carried out for each symbol. CPE variations are
usually very slow and the channel estimation procedure performed by the
frequency equalizer remains the same unless a new tracking sequence is sent. If
the block associated with the training sequence turns to have a high CPE estimate (CPENt),
then it will affect the channel phase estimation as described in [6]. This
needs to be taken into account in the subsequent estimation of the CPE of the
symbols. The authors in [6] use a moving average filter to remove the offset
that might be introduced.The algorithm proposed by the author can be
formulated in the following way.(1)
Select
the subchannels that meet the threshold criteria (SNR above a certain critical
value).(2)
Calculate the mean of the previous
CPE estimates that are in the moving average filter. We call this meanCPEm¯.(3)
UseCPEm¯ to update the phase of the coefficients (Cm−1,l) of the frequency equalizer. This would prevent the occurrence
of the offset previously discussed. The update can be described as follows: (32)Cm,l=Cm−1,l+CPEm¯for0≤l≤N−1.(4)
Get the new estimate for(33)CPEestimate=1Nd∑l,Hl≥threshold(Al,equalized−Al,sliced).(5)
Move the previous value in the moving average filter to calculateCPEm+1¯.


## 7. Conclusion

OFDM is considered as a potential
candidate for next-generation cellular systems. Despite all the advantages it
provides in terms of ISI suppression and efficient bandwidth use, it suffers
from the detrimental effects introduced by frequency offset and phase noise.
The interference among the subcarriers causes the system to loose its orthogonality.
We have seen that ICI places a limit on the carrier spacing because it
increases as the spacing decreases. Local oscillators introduce phase noise
that will be superimposed on the received signal. Many algorithms have been
proposed to correct the CPE part of the phase noise. We have presented one
algorithm based on averaging previous estimates of the CPE to get a better
future estimate. The algorithm tracks well the variations of the CPE even
though it usually varies slowly. On the other hand, ICI has a random structure
and mitigating its effects remains a challenge.



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