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Phase-Noise Mitigation in OFDM by Best MatchTrajectories

Senay Negusse, Per Zetterberg, Peter Händel

Abstract—This paper proposes a novel approach to phase-noise compensation. The basic idea is to approximate the phase-noise statistics by a finite number of realizations, i.e., a phase-noise codebook. The receiver then uses an augmented receivedsignal model, where the codebook index is estimated along withother parameters. The realization of the basic idea dependsonthe details of the air interface, the phase-noise statistics, thepropagation scenario and the computational constraints. In thispaper, we will focus on a MQAM-OFDM system with pilot sub-carriers within each OFDM symbol. The channel is frequencyselective, fading and unknown. A decision-feedback methodis employed to further enhance performance of the system.Simulation results are shown for uncoded and coded systemsto illustrate the performance of the algorithm, which is alsocompared with previously employed methods. Our simulationsshow that for a 16-QAM coded OFDM system over a frequencyselective Rayleigh fading channel affected by phase noise withroot-mean-square (RMS) of 14.4 degrees per OFDM symbol, theproposed algorithm is 1.5dB from the ideal phase-noise freecaseat a BER of 10−4. The performance of the best reference schemeis 2.5dB from the ideal case at BER of10−4. The proposed schemeis also computationally attractive.

Index Terms—OFDM, phase-noise, BER and SNRs, codebook,fading channel, channel estimation, pilot sub-carrier.

I. I NTRODUCTION

Orthogonal frequency division multiplexing (OFDM) isadopted by most of the current and future telecommunica-tion standards for high-rate data transmission, particularly inwireless communication systems. Its resilience to multipathchannel fading, the spectral efficiency it provides as well as thesimplicity of the equalization, has enabled OFDM to remainthe most popular modulation scheme. However, OFDM isknown to be sensitive to various hardware imperfections, theDirty-RF effect [1, 2], originating in the transceiver hardware.

A. Previous Works

The phase-noise (PHN) phenomenon and its underlyingeffects on various OFDM systems were studied extensivelyin [2–8]. The effect of PHN on OFDM systems is classifiedinto two components. The common phase error (CPE), whichrotates all the sub-carriers in one OFDM symbol by a commonphase distortion, and the inter-carrier interference (ICI), whicharises due to the loss of orthogonality between each sub-carrier. Frequency domain approaches to PHN estimation andcompensation mainly deal with the CPE and ICI components

S. Negusse (negusse@kth.se), P. Zetterberg (perz@kth.se), and P. Händel(ph@kth.se), are with the Department of Signal Processing,ACCESS Lin-naeus Center, KTH Royal Institute of Technology, 100 44 Stockholm, Sweden.

separately, while time domain approaches attempt to com-pensate for both jointly. It has been shown that a significantimprovement in performance can already be achieved withCPE correction only by treating the ICI term as an additiveGaussian noise [9–11]. However, CPE correction only is notalways sufficient for high rate transmission, therefore ICIcompensation is necessary.

Most ICI compensation techniques employ decision-directed feedback (DD-FB) for frequency domain PHN esti-mation [5, 12–16]. A vast majority of works assume a knownchannel frequency response [5, 12, 15–19]. For unknownchannel, joint channel and PHN estimation is considered in[13] where an ICI reduction scheme over a Rayleigh fadingchannel is presented in which the PHN process within anOFDM symbol is modeled as a power series. Although themethod presented showed a significant bit error rate (BER)improvement, the cost in computation is large. A less complexmethod is presented in [14] where the estimate of channel andCPE at pilot subcarriers are interpolated to obtain channelfrequency response and CPE followed by a DD-FB loop toestimate ICI components.

A non-iterative compensation scheme in which the CPEbetween consecutive OFDM symbols are interpolated linearlyto estimate the time varying PHN is presented in [17, 18].In [20], a method of suppressing the ICI is presented bylinearly combining the cyclic-prefix and the correspondingOFDM samples. The linear coefficients are obtained such thatthe ICI power is minimized. Other methods using the Bayesianframework for joint channel and PHN estimation have beenproposed in [21–24]. Soft-input maximum a posteriori andextended Kalman smoother are proposed in [21]. In [22–24], Monte-Carlo methods are employed to approximate theposterior probability distribution of the unknown quantitiesfor PHN tracking and channel estimation. Although theyare shown to provide good BER performance, Monte-Carlomethods, such as particle filters, are known for their numericalcomplexity. The main interest in the paper lies on PHNcompensation over unknown channels using pilot sub-carriers.However, the proposed method also uses DD-FB loop forimproved performance. For the known channel the proposedmethod is benchmarked against [5]. For known and unknownfrequency selective Rayleigh fading channels the methods in[13, 14] are also used as benchmarks. The problem setting inwhich the methods presented in [5], [13, 14] and are employedis identical to the setting the proposed algorithm.

2

B. Contributions

In this paper, we employ a simple and novel scheme forPHN cancellation which is shown to have superior perfor-mance in terms of BER compared to previously proposedmethods. The proposed technique can be applied to anyassumed distribution of the PHN process,θ(n). Additionally,no approximation based on the magnitude of the PHN is takeninto account, e.g.ejθ(n) ≃ 1 + j sin(θ(n)), as is common insome previous works.

A codebook ofK vectors representing a set of trajectorieswhich aim to closely match with the PHN realization is used.An uncountable set of possible PHN realizations is representedby a codebook withK quantized trajectories, which are storedat the receiver.1 The complexity of the algorithm is thereforedependent on the number of trajectories in the codebook,K.For a moderately small sized codebook, it is shown that theimpact of the PHN is significantly reduced (e.g. withK = 27,there is 60% reduction in terms of the effective PHN meansquare error (MSE), see Section III-C).

In this paper, we focus on a solution where the tra-jectory that minimizes the Euclidean distance between theconstellation of the received symbols at the pilot positionsand the constellation of the known pilot symbols is chosen.Additionally, a DD-FB technique is employed such that boththe estimate of data symbols from the channel decoder andpilot symbols are used to compute the Euclidean distance. Theproposed technique can also be used for combined channelestimation and PHN compensation schemes, in which MMSEbased channel estimation is employed for each of the vectorsin the codebook. The channel estimate corresponding to thetrajectory in the codebook which best approximates the PHNrealization is chosen. We will show by the simulation that,for AWGN and fading channels with a known frequencyresponse, the proposed scheme outperforms previously pro-posed PHN compensation techniques. In this scenario, theproposed method does not employ DD-FB while the referenceschemes require previously detected symbols for ICI estima-tion. Simulation results are also shown for unknown channelfrequency response, in which the proposed method employs aDD-FB technique for combined channel estimation and PHNcompensation. It is shown that the method presented providesimproved performance compared to previously proposed meth-ods. Additional results based on a PHN process modelled asOrnstein-Steinbeck is also shown to illustrate the applicabilityof the proposed method for a PHN model other than Weinerprocess.

The paper is organized as follows. Sec. II introduces thesystem model and parameters while Sec. III presents design ofthe codebook for Weiner PHN along with some performanceanalysis. In Sec. IV, details of implementation of the code-book for PHN compensation together with channel estimation.Simulation results are shown in Sec. V and a computationalanalysis of the proposed algorithm as well as some of the

1The technique is similar to the design of vector quantization codebookwhere a set ofn vectors from somem-dimensional space is efficientlyrepresented by a codebook with a smaller set of vectors fromm-dimensionalspace [25].

reference methods is given in Sec. VI. We end with concludingremarks in VII.

II. SYSTEM MODEL AND PROPOSEDAPPROACH

We consider an OFDM transmission system over a fadingtime varying channel withL multipath taps employing QAMmodulation technique. The OFDM system is assumed to haveFFT size ofN of which Np are pilot subcarriers and with acyclic prefix of lengthNcp such thatNcp ≥ L. To obtain them-th OFDM symbol of durationT , a stream of data bits isdivided toN−Np groups ofM -bits which are mapped to2M -QAM complex symbols,Sm(k). Assuming perfect frequencyand timing synchronization, the baseband representation of thereceived time domain signal,ym(n), is given by

ym(n) = ejθm(n)

( L−1∑

ℓ=0

hm(ℓ)sm(n− ℓ)

)+ wm(n) (1)

wheresm(n) is theN point inverse FFT (IFFT) of the complextransmitted symbol,Sm(k), hm(i) is the fading complexchannel pulse response withL propagation paths,wm(n) isa zero mean complex circular Gaussian channel noise withvarianceσ2

w; andθm(n) is the PHN sample at the receiver attime indexn of them-th OFDM symbol. The received signalis sampled at a frequencyfs = (N +Ncp)/T , whereT is theOFDM symbol duration.

After removing the cyclic prefix, an FFT operation isapplied to the received signal such that thek-th subcarrieris given by

Ym(k) =1√N

N−1∑

n=0

L−1∑

ℓ=0

hm(ℓ)sm(n− ℓ)ej(θm(n)−2πkn)

+1√N

N−1∑

n=0

wm(n)e−j2πkn

= Sm(k)Hm(k)Am(0)︸ ︷︷ ︸CPE

+

N−1∑

l=0l 6=k

Sm(l)Hm(l)Am(l − k)

︸ ︷︷ ︸ICI

+Wm(k)(2)

whereAm(k) andWm(k) are respectively the FFT ofejθm(n)

and that of the additive complex Gaussian channel noisew(n).Hm(k) is the instantaneous frequency response of the multi-path channel which is assumed to be constant within oneOFDM symbol. Figure 1 shows the block diagram of thesystem model. Equation (2) can further be written in matrixform as

Ym = AmSdmHm +Wm (3)

whereSdm = diag(Sm(0), .., Sm(N − 1)) is anN ×N matrixrepresenting diagonalization of the symbol vectorSm; and theN × N matrix Am is a unitary circulant matrix containingDFT coefficients ofejθm(n) such thatAm(k, l) = Am(l− k).Hm andWm are vectors of the channel frequency responseand the complex Gaussian noise respectively. Estimation oftherealization of the PHN process,θm = [θm(0), · · · θm(N−1)]T ,would enable compensation of CPE as well as ICI.

3

IFFT

sm(0)

sm(1)

sm(N − 1)

wm(n)

exp(jθm(n))

(sm ∗ hm)(l)

ym(0)

ym(1)

ym(N − 1)

Fading Channelhm(l)

sm(n)P/S

inputbit stream QAM

ModulationSm(k)

Sm(N − 1)

Sm(1)

Sm(0)

QAMDemodulation

Ym(k)

Ym(0)

Ym(1)

Ym(N − 1)

FFT

S/P

P/S

ym(n)

S/P

receivedbit stream

Figure 1: Block diagram of the OFDM system.

A. Approximate Model

We propose a method where the PHN realization at them-thOFDM symbol is approximated by one of a set ofK vectors,φkKk=1, whereφk = [φk(0), φk(1), · · · , φk(N−1)]T plus aninitial phase offset,ψm ∈ [0, 2π). Associated with each vector,φk, is a corresponding unitary circulant matrix containing DFTcoefficients ofejφk(l)N−1

l=0 denoted byA. The signal modelin (3) is then modified as

Ym = ejψmAkmSdmHm +Wm (4)

which is an approximation of (3), where the PHN DFTcoefficient matrix,Am, is now replaced byejψmAkm , whichis a function of an index,km, that takes on a value between1andK with equal probability. The framework can be extendedto unequal probabilities but this is left for future work. Therandom variableψm, which is assumed independent ofkm,is uniformly distributed and models the initial phase of thesymbol. Bothkm and ψm are independent from symbol tosymbol. The design of the codebookφkKk=1 is yet to bediscussed in a later section. It should be noted that thisapproximate model is only used for the purpose of deriving thealgorithm. Data from the original model in (3) is used whentesting and validating the proposed algorithm by simulation.

B. Cost Function

Any estimator derived in the absence of PHN can be writtenas

Ω = argminΩ

C(Y1, . . . , Yq), (5)

where Ω is the estimated parameter,C is some form ofoptimization criterion andq is the number of OFDM symbols.It should be noted thatAkm and its inverseA−1

kmare unitary

circulant matrices. As a result,Akm is invertible andA−1km

Wm

is statistically identical toWm. We may therefore generalize

the criterion (5) as

Ω = arg minΩ,k1,··· ,kq,ψ1,...,ψq

C(e−jψ1A−1k1

Y1, . . . , e−jψqA−1

kqYq).

(6)for the model (4). This includes, for instance, maximum-likelihood estimators where now the estimator in (6) is alsothe maximum likelihood estimator for the model (4). Anexample of a use of the proposed new estimator would bein an OFDM system with an equalizer and Viterbi decoderand a pilot subcarrier. In such a system, the optimizationfunction could be the sum path metric. In this case, (6) impliesthat the entire receiver is run for all possible hypotheses ofk1 . . . kq (of which there areKq), and selects the one with thelowest path metric (in each symbol optimizing overψ). Thisimplementation is generally not realistic. A practical criterionis hard decoding of the M-QAM symbols of OFDM symbols,where the criterion function separates into a term for eachsymbol individually. In this case, the criterion function is theEuclidean distance between estimates of the received signalpoint, Sm, at the output of the equalizer and the nearest pointin the M-QAM constellation. Here, it is realistic to sequencethrough a quite large sizeK. Selection of the codebooks is adesign problem still to be considered.

For the case where the Wiener PHN model is used, asystematic codebook design is presented in the next subsec-tion. For other PHN models different codebook designs mayperform better. However, the design for Wiener PHN maywork well for other PHN models as well. This is illustratedin Section V-B5 where results based on a PHN modelled asOrnstein-Uhlenbeck process is presented.

III. C ODEBOOK DESIGN FORWEINER PHN

The model introduced in [3] and commonly employed inliterature, where the PHN process is described by a Wienerprocess, is used. Under the Weiner model, the PHN sample at

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Time sample in an OFDM symbol

Pha

seE

rror

(ra

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θ(n)θ(n)

Figure 2: A single Wiener PHN realization (blue) withσ2ǫ = 2π(0.01/N)

whereN = 64 and its approximation according to (8) forJ = 8.

0

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PSfrag replacements

x1−x1 x2 xr−x2−xr

Ri

R1

Ri−1 Ri+1

RQ· · ·· · ·

Figure 3: Dividing the Gaussian sample space intoQ regions of equal areas,whereQ is an odd integer.

them-th OFDM symbol is given by

θm(n) =

mN+n∑

i=0

ε(i) = ψm +

n∑

l=0

ε(l) (7)

for n = 0, 1, . . . , N − 1, whereε(l) is a Gaussian randomvariable with zero mean and varianceσ2

ε = 2πβT/N (inradians2), andβT denotes the rate at which the PHN variancegrows in one OFDM symbol. The objective is to construct acodebook containing a finite set of vectors which approximatethe trajectories of possible Wiener PHN realizations withinone OFDM symbol. The subscriptm in θm(n) andψm(n) isdropped for notational simplicity. Let theN samples of theOFDM symbol be divided intoJ segments withL = N/Jsamples per segment. Thenθ(n) is defined as a process whichis constant in each segment. The value of the constants,λj ,in each segment are set as the sample average ofθ(n) inthe corresponding segment. The PHN process is thereforeapproximated byθ(n) given by

θ(n+ jL) = λj =1

L

L−1∑

ℓ=0

θ(ℓ + jL)

=1

L

L−1∑

ℓ=0

jL+ℓ∑

i=0

ǫ(i) + ψ

(8)

for n = 0, · · · , L−1 andj = 0, · · · , J−1. This is illustrated inFigure 2 forσ2

ǫ = 2π(0.01/N),N = 64 andJ = 8, where thesamples within each segment are approximated by the samplemean of the elements in the segment. The process,θ(n), issimilar to a Weiner process where each step,Xj = λj+1−λj,occurring everyL samples, is given by

Xj =1

L

L−1∑

ℓ=0

(θ((j + 1)L+ ℓ)− θ(jL+ ℓ)

)

=1

L

L−1∑

ℓ=0

ℓ+(j+1)L∑

i=ℓ+jL+1

ǫ(i)

(9)

where the second equality follows from (8). The increments,Xj , are identically distributed Gaussian random variables withzero mean and varianceσ2

x given by

σ2x = E[X2] =

1

L2

L−1∑

ℓ=0

ℓ+(j+1)L∑

i=ℓ+jL+1

L−1∑

v=0

v+(j+1)L∑

k=v+jL+1

E[ǫ(i)ǫ(k)]

=(2L2 + 1)

3Lσ2ǫ

(10)

where the auto-correlationE[ǫ(ℓ)ǫ(k)] = σ2ǫ δ(k−ℓ) as well as

arithmetic and geometric series are used to obtain the secondequality. Note that forL = 1, that is when the number ofsegments is equal to the number of samples,J = N , thenσ2x = σ2

ǫ . The approximated Weiner PHN process can be re-written in terms ofXj as

θ(jL+ n) = λj−1 +Xj + ψ =

j∑

ℓ=0

Xℓ + ψ (11)

whereλ−1 = 0 andX0 = 0 for j = 0.Since the Gaussian sampling space is unbounded, the ran-

dom variableX is instead represented by a set of quantizedsamples,xiQi=1, which representQ regions,RiQi=1, inthe Gaussian pdf with zero mean and varianceσ2

x. Thisimplies that the increments at the consecutive segments,j = 1, · · · , J − 1, are defined by a setxiQi=1 which defineQ possible trajectories. That is, the second segment,j = 2,hasQ possible increments which setQ possible trajectories.In the next segment, there will beQ possible increments oneach trajectory from the previous segment such that there areQj−1 trajectories at thej-th segment. Therefore, at theJ-thsegment, there will beK = QJ−1 trajectories representingquantized paths of the random walk process each of whichare set as a vector entry in the codebook,φKk=1. The numberof code vectors,K, is therefore determined by the predefinedset of quantization regions,Q, and the number of segmentsthe symbol is divided into, i.e.J .

A. Defining Quantization Regions

The quantization regions can be defined as a set of regions,RiQi=1, which divides the Gaussian sample space intoQregions. For ease of development, the sample space is dividedinto a set of regions with equal areas (i.e. equiprobable). The

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Time sample in an OFDM symbol

Pha

seE

rror

(ra

dia

ns)

PHN processSet of trajectories

Figure 4: A single Wiener PHN realization (blue) withσ2ǫ = 2π(0.01/N)

whereN = 64 and a finite set ofK = 2187 trajectories,* forJ = 8 andQ = 3, which corresponds to possible trajectories that the PHN realizationmight take for equiprobable quantization.

regions are defined as

P

[X ∈ R1

]= · · · = P

[X ∈ RQ

]=

1

Q(12)

whereP [X ∈ Ri] implies the probability that the randomvariableX is in the regionRi. For the case whenQ is odd,the sample space is partitioned byQ+ 1 data points denotedby −∞, x−r, . . . , x−1, x1, . . . , xr,∞, as shown in Figure3, wherex−ℓ = −xℓ, ℓ = 1, . . . , r and r = (Q − 1)/2. Theregions at the tails of the Gaussian pdf are given byR1 =[−∞, x−r[ andRQ = [xr,∞[, while the regions in the middleare bounded by[xℓ, xℓ+1[. The bounding points,xℓ are givenby

xℓ =√2σ2

xerf−1

(2ℓ− 1

Q

), ℓ = 1, · · · , Q− 1

2(13)

where erf−1(·) is the inverse error function. On the other hand,whenQ is even, the sample space is partitioned by2r+1 datapoints denoted by−∞, x−r, · · · , x−1, x0, x1, · · · , xr,∞where x0 = 0 and x−ℓ = −xℓ for ℓ = 0, · · · , r wherer = Q/2. These points are given by

xℓ =√2σ2

xerf−1

(2ℓ

Q

), ℓ = 0, · · · , Q

2. (14)

The plot in Figure 4 is shown as an example to illustrate theprocess for a set of equiprobable quantization regions whereQ = 3 andJ = 8 while Figure 5 shows the trajectory that isclosest to the PHN realization. The regions can also be definedas a set which divides the Gaussian sample space uniformly.This implies that the codebook trajectory would have unequalprobability. The development of this case is left for futurework.

B. Quantization Points

Given the quantization regions,R = R1, . . . , RQ, thequantization points,x1, . . . , xQ, which represents each re-gion is defined as the mean point within the region. That is,

xi = E

[X |X ∈ Ri

]=

∫RixfX(x)dx∫

RifX(x)dx

(15)

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Pha

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(ra

dia

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PHN process

Selected trajectory

Selected trajectory (codebook with uniform quantization)

PHN process

Selected trajectory

Figure 5: The trajectory that best matches the PHN realization.

for i = 1, . . . , Q, whereE[·] is the expectation operator andf(x) is the Gaussian distribution function. It can be shown that(15) provides the set which results in a minimum squared errorquantization of the random variableX with a pdf f(x), [25].Moreover, the expected value of quantization errorqerror =x − x is zero, i.e.E[x] = E[x]. For the Gaussian pdf withzeros mean and varianceσ2

x = (2L2+1)/3Lσ2ǫ , (15) is given

by

xi =−σx

0.5√2π

(exp(−0.5x2i+1/σ

2x)− exp(−0.5x2i /σ

2x)

erf(xi+1/√2σx)− erf(xi/

√2σx)

).

(16)It can be seen that the codebook will contain an all zeros entrywhenQ is odd. This is attractive, since it means no correctionwill be done if the receiver would have no phase noise and theSNR is high. The plot in Figure 4 employs (13) forQ = 3 todefine the quantization regions and (16) to obtain the quantizedpoint for each region.

C. MSE Analysis

In this section, the MSE for the designed codebook isanalysed. The objective is to provide a good indication ofthe performance of the designed codebook for a givenJ andQ. The MSE between a random PHN realization and theapproximate model introduced is given by

MSE= Eθ(n)[mink,ψ

∑

n

|θ(n)− ψ − φk(n)|2] (17)

which is difficult to evaluate analytically. Therefore the MSE isevaluated by simulation. However, an approximate expressionfor the MSE, derived in Appendix A, is given by

MSE=(N + J)(N − J)

6Jσ2ǫ + L(J − 1)σ2

q (18)

with σq given by (36).Table I shows the MSE values obtained by Monte-Carlo

simulations (denoted MSEs) normalized by the MSE for theCPE correction only (i.e. MSE forJ = 1). The number ofPHN realizations employed were 5000 for every codebookwith various combination ofJ and Q resulting in K =QJ−1. The approximated theoretical MSEa values computedaccording to (18) are also given. It can be seen that theanalytical and simulation results are similar indicating that

6

Remove CP FFT QAMDemodulation

Π−1 ChannelDecoding

ChannelEncoder

ΠQAM

ModulationCodebook

exp(jθm(n))

φ0, · · · , φK−1

A−1

k

ym(n)Channel Estimation/Equalization and

Selection of the mostlikely trajectory

Estimation using pilots at zeroth iteration

DD-FB i’th iteration

sink

Figure 6: Block diagram of the coded OFDM receiver with PHN compensation and channel estimation.

Table I: Approximated MSE forN = 64 as is given by (18) and simulated MSE denoted by MSEs for various values ofQ andJ . The values are normalizedby the MSE value for CPE correction only given by MSE= (N − 1)(N + 1)σ2

ǫ /6.

Q = 2 Q = 3 Q = 4 Q = 5 Q = 6K MSEs MSEa K MSEs MSEa K MSEs MSEa K MSEs MSEa K MSEs MSEa

J = 1 1 0.9967 1 1 0.9997 1 1 0.9998 1 1 1.0021 1 1 1.0043 1J = 2 1 0.7353 0.7387 2 0.6095 0.6247 3 0.5674 0.5765 4 0.5432 0.5519 5 0.5302 0.5376J = 4 8 0.4479 0.4309 27 0.3488 0.3446 64 0.3076 0.3082 125 0.2878 0.2895 216 0.2770 0.2785J = 5 16 0.4839 0.3607 81 0.2931 0.2855 256 0.2553 0.2538 625 0.2370 0.2375 1296 0.2274 0.2279J = 8 128 0.2631 0.2330 2187 0.1905 0.1813 16384 0.1601 0.1595 78125 0.1487 0.1482 279936 0.1441 0.1416

the reasoning used in the analytical derivation in AppendixA is correct. The MSE analysis assumes that the optimumcode-book entry is chosen. In order to account also for theeffect of noise and decision feedback errors, simulations of thefull receiver for various codebook sizes are performed in Sec.V. These results confirm that the performance improves withincreasing codebook size. The choice of codebook size willultimately depend on the computational cost and performancerequirements for the application at hand. However, it is notedthat complexity grows rapidly withQ and J , thus moderatenumbers such as e.g.Q = 2, J = 5 or Q = 3 and J = 4seems to provide a good compromise.

IV. I MPLEMENTATION ON OFDM SYSTEM

A. Known Channel Response

The received signal vector after removing cyclic prefix,ym = [ym(0), · · · , ym(N − 1)]T , is multiplied by each oftheK trajectoriese−jφk such that a new set ofK de-rotatedOFDM signals is obtained at the receiver

ykm = [e−jφk(0)ym(0), · · · , e−jφk(N−1)ym(N − 1)]T . (19)

An FFT operation on each de-rotated signal vector provides

Ykm = A−1

k Ym

= A−1k (AmSdmHm +Wm),

(20)

where theN×N circulant matrixA−1k is a matrix of DFT co-

efficients ofe−jφk(n). Frequency domain channel equalizationis applied onYk

m, and the criterion by which the trajectorythat best approximates the PHN realization is given by

k∗ = argmink

P−1∑

i=0

∣∣∣∣Skm(ℓi)− Sm(ℓi)

∣∣∣∣2

. (21)

where ℓi ∈ (ℓ0, . . . , ℓP−1) denotes pilot subcarriers,Skm(ℓi)is an element of the estimated symbol vector for thek-thtrajectory given by

Skm = η−1k diag(Hm)−1Yk

m (22)

with complex termηk that corrects for the effective CPE whichincludes the offsetψm at each symbol, the DC level of thecodebook vector,φk, and the accumulated quantization error.It is given by

ηk =

∑P−1i=0 S∗

m(ℓi)Ykm(ℓi)/H

km(ℓi)∑P−1

i=0 |Sm(ℓi)|2. (23)

In a DD-FB loop in which the output of the channel decodercan be exploited, non-pilot symbols can also be used such thatthe criterion in (23) takes into account the decoded symbolsas well

k∗ = argmink

N−1∑

l=0

∣∣∣∣Skm(l)− Sm(l)

∣∣∣∣2

. (24)

for ηk given by

ηk =

∑N−1l=0 S∗

m(l)Y km(l)/Hkm(l)∑N−1

i=0 |Sm(l)|2. (25)

The symbol sent to the decoder is then given by the indexcorresponding toSk

∗

m .

B. Combined Channel Estimation and ICI Cancellation

Since the channel is not always known at the receiver, itneeds to be estimated. Assuming the channel remains station-ary within the time period of an OFDM symbol, for a code-book of sizeK, there will beK received signal candidates,Ykm, which are given by (20) where the channel response,

Hm(k), is considered to be unknown. The MMSE estimate

7

of the channel frequency response vector,Hkm, employing the

k-th trajectory of the codebook is then given by [26]

Hkm = E[Hm(Hm)H ]

(E[Hm(Hm)H ] +

σ2w

EsIN

)−1

HkLS,m

(26)where E[Hm(Hm)H ] is the autocorrelation matrix of thechannel frequency response for the given statistical modelofthe channel andHk

LS,m is the least squares estimate ofHkm

given by

HkLS,m = S−1

m Ykm =

[Y km(0)

Sm(0), · · · , Y

km(N − 1)

Sm(N − 1)

]T. (27)

However, since not all transmitted symbols are known atthe receiver,Np pilot symbols, fitted evenly among theNsubcarriers, are employed to obtain a least squares estimateof the channel frequency response at the pilot positions. Forthe sake of convenience, the LS estimate at pilot positions aredenoted by the vector,Hk,p

LS,m. Therefore, the MMSE estimateof the channel frequency response, fork-th trajectory in thecodebook, is initially obtained based on pilot symbols

Hkm = E[Hm(Hp

m)H ]

(E[Hp

m(Hpm)H ] +

σ2w

EsINp

)−1

Hk,pLS,m

(28)where (·)p denotes a vector whose elements are positionedat pilot subcarriers. A set ofK channel frequency responseestimates,Hk

mKk=1, corresponding to each trajectory in thecodebook is available at the receiver. Equation (24) is usedtodetermine the trajectory which closely matches the PHN real-ization, and the corresponding channel estimate is then used todemodulate and obtain rough decisions on the data symbols.After channel decoding, a decision feedback technique is thenemployed to obtain (27) based on rough decisions on thesymbols containing pilot and decision feedback symbols,Sm,which is then used in (26) to compute an MMSE estimate ofHkm with improved accuracy.For deeply faded channels, estimation accuracy during

symbolm can be improved by including previously decodedsymbols, i.e.,Sm−1,Sm−2, · · · ,Sm−D+1, in the estimationvector. Therefore, taking into accountD previously decodedOFDM symbols, the MMSE estimator in (28) becomes

Hkm = E[Hm(H)H ]

(E[H(H)H ] +

σ2w

EsIDN

)−1

HLS (29)

where H = [Hm,Hm−1, · · · ,Hm−D+1]T and HLS =

[HkmLS,m, H

km−1

LS,m−1, · · · , Hkm−D+1

LS,m−D+1]T . On the initial run of the

m-th OFDM symbol, symbols at pilot subcarriers are usedin order to compute (28), after which the criterion in (21) isused to choose the best match index,k∗, for the trajectorywhich closely matches the PHN realization,φk

∗

, as well asthe corresponding channel estimate,Hk∗

m . Pilot and decisionfeedback symbols can then be used in the next iteration toobtain a better estimate,Hk

m using (26) or (29) if usingL previous OFDM symbols and (24) to obtaink∗. Jointchannel equalization and PHN compensation is employed onthe received signalYm such that the symbols sent to thedecoder are given by

Algorithm 1 The proposed receiver algorithm with no DD-FBfor uncoded system and known channel

1: procedure2: Receivedm-th OFDM symbol3: for k = 0 : K − 1 do4: Y

k

m = A−1

kYm

5: Computeηk in (23) using pilot symbols6: ComputeF (k) =

∑P−1

i=0|Sk

m(ℓi)− Sm(ℓi)|2

7: end for8: Computek∗ = argmink F (k) in (21) to obtaink∗

9: return Sk∗

m

10: end procedure

Sk∗

m = η−1k∗ diag(Hk∗

m )−1Yk∗

m (30)

wherek∗ denotes the index of the most likely trajectory inthe codebook obtained from (24). Figure 6 shows the OFDMreceiver with the proposed PHN compensation and channelestimation scheme. An outline of the proposed algorithm fora known channel response without DD-FB loop is presented inAlgorithm 1. In Algorithm 2, the algorithm for the combinedchannel frequency response estimation and PHN compensationscheme is presented for a coded frame ofM OFDM symbols.

Algorithm 2 The proposed algorithm for channel estimationand PHN compensation with DD-FB and channel coding

1: procedure2: Received frame3: do Iteration on DD-FB loop4: do for each OFDM symbol in the frame5: for k = 0 : K − 1 do6: Y

k

m = A−1

kYm

7: if Zeroth iterationthen8: Channel estimation using pilots, (28)9: Computeηk in (23) using pilot symbols

10: Detected symbol for eachk using (30)11: ComputeF (k) =

∑P−1

i=0|Sk

m(ℓi)−Sm(ℓi)|

2

12: else13: Channel estimation employing (26)14: Computeηk using (25)15: ComputeF (k) =

∑N−1

i=0|Sk

m(ℓi)−Sm(ℓi)|

2

16: end if17: end for18: Computek∗ = argmink F (k) in (21) (zeroth iter-

ation) or (24) to obtaink∗

19: SelectSk∗

m

20: while End of frame21: Decode frame and deinterleave22: Interleave, channel coding and modulation frame23: DD-FB frameS = [S0, · · · ,Sm, · · · ,SM−1]24: while End of iteration25: end procedure

V. SIMULATION RESULTS

In this section, simulation results are given to show theBER performance of the proposed technique under WienerPHN. Two sets of simulation results are presented where inthe first set, the performance of the algorithm in Algorithm1 for an uncoded system is presented under AWGN channel.

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[13], 3 DD-FB iterations (64-QAM)

Proposed method (64-QAM), no DD-FB

AWGN channel no PHN (64-QAM)

[13], 3 DD-FB iterations (16-QAM)

Proposed method (16-QAM), no DD-FB

AWGN channel no PHN (16-QAM)

Figure 7: BER performance of proposed algorithm compared to[13] for16-QAM and 64-QAM uncoded OFDM under AWGN channel and PHN.Corresponding performance plots for an ideal case (i.e. AWGN channel noPHN) is also given.BT = 0.01.

In the second set, performance of the algorithm in Algorithm2 is shown for a Rayleigh fading channel. Simulation resultsare presented for 16-QAM and 64-QAM modulation schemeswith N = 64, Ncp = 16 andNp = 8 pilot subcarriers. Thecodebook size for the results presented isK = 27, in whichQ = 3 and J = 4, which from the analysis in Sec. III-Cis a good compromise between complexity and performance.Perfect timing and frequency synchronization is assumed atthe receiver.

A. Uncoded System, AWGN Channel

Considering the AWGN channel, the proposed Algorithm 1was compared with the PHN compensation method presentedin [13], employing 3 iterations in the DD-FB. In the proposedmethod no DD-FB iterations are employed. The rate of growthof the PHN varianceβT is set to0.01, and 16-QAM OFDMand 64-QAM OFDM were considered without channel coding.The result is given in Figure 7. For the 64-QAM system(dashed line), the proposed method is2.5dB from the idealPHN free case at a BER of10−2, while the method in [13]is 9dB from the ideal case, i.e., an improvement of6.5dB. Inthe 16-QAM system (solid line), the proposed method is2dBfrom the ideal PHN free case at a BER of10−3, while themethod in [13] is3dB from the ideal case.

B. Coded System with Fading Channel

A 10 tap Rayleigh fading channel with parameters specifiedin Table.II is used. Moreover, a 1/2 rate convolutional codewith a constraint length of 7 is used on the input bit stream,which is then passed on to a bit interleaver over 20 OFDMsymbols. A Viterbi decoder is used at the receiver. The PHNvariance growth rate,βT , is set to0.01 for 16-QAM OFDMsuch that the RMS in degrees is180

√2πβT/π = 14.40; while

for 64-QAM, βT is set to0.005. Higher order modulationschemes such as 64-QAM are known to be affected particu-larly worse by PHN. The simulated channel, also employed in

Table II: summary of the Rayleigh channel parameters

Carrier Frequency,fc 5 GHzSampling rate,fs = 1

Ts25 MHz

rms delay spread,τrmsTs

120nsSpeed,v 7 km/hChannel taps,L 10

[13], is given for a normalized amplitude by [27]

Hm(k) =

L−1∑

i=0

φ(τi)ej(ϕi+2πfD

i m+2πkτiN

) (31)

whereαi, ϕi, fDi andτi are respectively the amplitude, phase,Doppler frequency and time delay of thei-th propagation path.The time correlation for the given channel model is definedas

Rt(m− n) = E[Hm(k)H∗n(k)]. (32)

Assuming uniformly distributed angles of arrival at the mobilestation, the correlation follows the well known Jakes model[28] such thatRt(m − n) = J0(2πf

Dmax(m − n)), where

J0(·) denotes the zeroth order Bessel function of the firstkind, andfDmax = (vfc/c)T is the maximal Doppler frequencynormalized by the OFDM symbol duration,T . Assuming anexponentially decaying power delay profile with normalizedRMS delay spreadτrms, [29], the correlation between differentfrequency bins is given by

Rf (k − l) = E[Hm(k)H∗m(l)]

=1− e−L(1/τrms+j2π(k−l))

(1− e−L/τrms)(1 + j2π(k − l)τrms)

(33)

which satisfiesE[|Hm(k)|2] = 1.1) Channel fully known at the Receiver:Assuming that the

frequency response of the channel is known at the receiver,the performance of the proposed algorithm is presented andcompared with previously proposed techniques in [5] and [13].

In [5], a Fourier series representation is used to approximatethe PHN realization at them-th OFDM symbol, therebyaiming to suppress all the influence of the PHN. CPE com-pensation and symbol estimates are initially obtained usingpilot subcarriers, after which a DD-FB loop is employedfor MMSE estimation of the vector of Fourier coefficientsof the PHN realization which is subsequently used for ICIcompensation. Similarly, in [13], a two stage cancellationtechnique is employed in which the CPE is corrected, followedby ICI cancellation using decision feedback symbols. The ICIcancellation technique in [13] is derived based on power seriesexpansion of the PHN over an OFDM symbol.

The results in our approach were obtained using Algorithm2, but where step 8 and 12 are omitted since the channel isalready known. The results are shown by the solid curves inFigure 8 and Figure 9 for 16-QAM and 64-QAM respectively.

First, in Figure 8 (solid line), it can be seen that theperformance of the proposed algorithm without employingDD-FB is 0.5dB from the ideal case (no PHN) for a BER of10−4. On the other hand, for 64-QAM, Figure 9 shows that the

9

performance of the proposed algorithm is2dB from the idealplot for a BER of10−4. In both Figures, the proposed methodshows performance gain in BER compared to the previouslyproposed methods in [5], which employs a single iteration onthe DD-FB, and the method proposed in [13], which is shownfor 2 and 3 DD-FB iterations respectively for 16-QAM and64-QAM. It should be noted that no DD-FB loop is performedin the proposed approach.

2) Unknown Channel Response:For the case where thechannel is unknown, joint channel estimation and PHN com-pensation is employed according to Algorithm 2. Performanceof the techniques in [13] and [14] is also presented forcomparison.

In [14], FFT interpolation is employed using the leastsquares estimates of the joint channel frequency response andCPE at pilot positions to obtain estimates over all subcarriers.The method also proposes estimating the CPE by averagingthe phase displacements on pilot subcarriers which is usedto remove its influence from the estimated channel frequencyresponse. Previously decoded symbols are then employedfor least squares (LS) estimation of the vector of Fouriercoefficients of the PHN realization for ICI compensation usingprevious channel and symbol estimates. Three sets of resultsare presented for the method in [14]. The first two resultsconsider estimation of only one and three Fourier coefficientsof the PHN realization closest to the DC for ICI compensationthus requiring less computational complexity. The third resultrequires inverting a64 × 64 Toeplitz matrix to estimate theentire vector of Fourier coefficients of the PHN realizationforICI compensation.

The proposed method as well as the method in [13] useD = 3 past OFDM symbols as described in (24) to enhanceperformance of the channel estimation. Performance plots forthe unknown channel case are shown with the dashed curve inFigure 8 and Figure 9 for 16-QAM and 64-QAM respectively.In both Figure 8 and Figure 9, the performance of the proposedtechnique shows improved performance, in BER, over thepreviously proposed methods in [13], [14].

In Figure 8, the proposed method is1.5dB from the idealcase, (no PHN and known channel response) at a BER of10−4

employing only 2 iterations on the DD-FB loop. On the otherhand, the method in [13], shown for 8 iterations, is2.5dB fromthe ideal case while the method in [14] is far from the idealcase.

For the 64-QAM system, Figure 9 shows that the proposedmethod provides improved performance compared to [13] andthe method in [14]. With 10 iterations, the proposed methodsis3dB from the ideal case at a BER of10−3, while the methodin [13] is 5dB from the ideal case for the same number ofiterations.

3) Performance with respect to size ofK: In Figure 10,simulation results are presented showing the BER performanceof the proposed algorithm for various codebook sizeK andfor βT = 0.01 and βT = 0.05. The presented result doesnot employ any DD-FB loop. A 16-QAM OFDM system withthe same description as in Sec. V-B is considered assumingknown and unknown Rayleigh fading channel. It can be seenthat the BER improves aK grows. However, the relative

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Proposed method, without DD-FB loop (known channel)

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ICI-cancellation Method in [5] (known channel)

Proposed method, 2 DD-FB iterations with channel estimation

Proposed method without DD-FB with channel estimation

[13], 8 DD-FB iterations with channel estimation

[13], 2 DD-FB iterations with channel estimation

[13], without DD-FB with channel estimation

[14], full matrix

[14], partial matrix 3

[14], partial matrix 1

Figure 8: BER performance of the proposed algorithm compared to [13] and[14] for 16-QAM OFDM. Performance plot for the ideal case (i.e. perfectchannel knowledge and no PHN) is also given. ICI cancellation technique in[5] is also shown for perfectly known channel.BT = 0.01

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Proposed method, 10 DD-FB iterations with channel estimation

Proposed method, 3 DD-FB iterations with channel estimation

[13], 10 DD-FB iterations with channel estimation

[13], 3 DD-FB iterations with channel estimation

[14], full matrix

Figure 9: BER performance of proposed algorithm compared to[13] and [14]for 64-QAM OFDM. Performance plot for the ideal case (i.e. perfect channelknowledge and no PN) is also given. ICI-cancellation technique in [5] is alsoshown for perfectly known channel.BT = 0.005

improvement asK grows beyondK = 27 is very small.For βT = 0.01, BER improves only slightly at high SNRin both cases where the channel is assumed known and wherethe channel is estimated asK increases from 27. The relativeimprovement in BER asK increases is more noticeable forβT = 0.05, even though it is still small.

4) Sensitivity to input parameters:The plots in Figure11 show the sensitivity of the proposed algorithm and thealgorithm in [13] to the estimated channel and PHN parametersrelative to the true parameters. The rate at which the PHNvariance grows is set toBT = 0.03 and the normalizedmaximal Doppler frequency is set tofDmax = 0.03. All otherRayleigh channel parameters are kept the same as in Table II.Moreover, the size of the codebook wasK = 27 with Q = 3

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K = 27, Q = 3, J = 4

K = 625, Q = 5, J = 5

K = 2187, Q = 3, J = 8

K = 1, Q = 1, J = 1

K = 4, Q = 5, J = 2

K = 16, Q = 2, J = 5

with channel estimation

known channel

βT = 0.01 (red)

βT = 0.05 (blue)

Figure 10: BER performance of the proposed algorithm on 16-QAM codedOFDM for various number of codebook size.

0 0.005 0.01 0.03 0.05 0.07 0.09

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βT = 0

βT = 0.01

βT = 0.03

βT = 0.05

βT = 0.07

Proposed method

Method in [13]Method in [14]

fDmax

Figure 11: BER performance of the proposed algorithm (blackline) and themethod in [13] (red line) as a function of the input doppler frequencyfD

max

for various input values ofβT in a 16-QAM OFDM on a Rayleigh channelwhere the truefD

max = 0.03, βT = 0.03 and SNR= 20dB.

andJ = 4 and SNR= 20dB. Both the proposed method andthe method in [13] use the past two symbols together with thecurrent symbolD = 3 for channel estimation.

It can be seen that the sensitivity of the proposed algorithmas well as the algorithm in [13] to the input values of thenormalized maximal Doppler frequency,fDmax is low exceptfor fDmax = 0. However, for higher value ofD, the BERperformance is expected to be very sensitive to the input valueof fDmax. However, it can be seen that the performance of boththe proposed algorithm and [13] is noticeably sensitive to theinput values ofβT . On the other hand, the method in [14] doesnot depend at all on the channel and PHN statistics and thusthe performance remains unaffected by the estimated inputparameters.

5) PLL Oscillator : The above results have been providedassuming a free running oscillator in which the PHN ismodelled by the Weiner process and based on which thecodebook is derived. In order to demonstrate the applicabilityfor a PHN model other than the Weiner process, we employ theproposed method using the Ornstein-Steinbeck process [30]tomodel the PHN. The Ornstein-Uhlenbeck process is used tomodel the PHN at the output of a voltage controlled oscillator(VCO) in a phase-locked loop (PLL) [5].

In Figure 12, a PHN realization,θ(n), for a free runningoscillator and a PLL is shown. It can be seen that the PHNfrom PLL (Ornstein-Uhlenbeck process) is stable around themean while the PHN from a free running oscillator grows overtime.

A detailed analysis of PLL circuit parameters and the asso-ciated PHN process generated is given in [31]. In Figure 12,the PHN variance of the Weiner process (free running VCO)grows at a rate of(βT )vco = 0.05 while more parameters arerequired to model the Ornstein-Uhlenbeck process. A Chargepump PLL is considered which is set to have a 3dB bandwidthof flp = 20kHz for the low pass filter,fpd = 20kHz for thephase detector andfpll = 100kHz for the PLL itself. The PHNvariance of the VCO and the reference oscillator grow at a rateof βTvco = 0.01 and (βT )ref = 2−9 respectively. Moreover,the center frequency of the VCO isfc = 5GHz while thecenter frequency of the reference oscillator isfref = 100MHzand the sampling frequency isfs = 25MHz. A 16-QAM codedOFDM with the same description as in Sec. V-B is consideredassuming unknown Rayleigh fading channel so that channelestimation is employed along PHN compensation.

In Figure 13, the performances of the proposed method andthe method in [13] is compared for various input of the PHNvariance of the VCO, which is denoted byβT vco, for a freerunning VCO and for PLL at 20dB SNR. Both methods arederived based on the Weiner PHN assumptions. The proposedmethod is run in 2 DD-FB loops while the method in [13]is run in 5 DD-FB loops. The improved in performance forPLL (dashed line) compared to free running VCO (solid line)can be seen in Figure 13 for both the proposed as well asthe method in [13]. It can be seen that the lowest BER forthe proposed method in a PLL is achieved around the inputβT vco = 0.02 which then starts to increase asβT vco > 0.02.The increasing trend in BER can also be seen in free runningVCO whenβT vco > 0.04 for the proposed method. The BERperformance of the method in [13], on the other hand, seemsto be stable around5×10−3 in a PLL and around9×10−3 in afree running VCO for an inputβT vco equal to 0.01 and above.For the range of inputβT vco values displayed, the proposedmethod outperforms [13] in both cases when a PLL and a freerunning VCO is used.

VI. COMPUTATIONAL ANALYSIS

The proposed scheme does incur some computations whichare not part of the reference schemes [13, 14]. These includethe multiplication with the matrixA−1

k in (20), and all theother calculations inside the loop overk in Algorithm 2.It should be noted that the channel estimation in (26), (28)

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Figure 12: PHN samples,θ(n), simulated as an out from a free runningoscillator (Weiner process) and a charge Pump PLL (Ornstein-Uhlenbeckprocess).

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Method in [13], 5 DD-FB iteration, Free running VCO,

Method in [13], 5 DD-FB iteration, PLL

Proposed method, 2 DD-FB iteration, Free running VCO

Proposed method, 2 DD-FB iterations, PLL

Method in [14], Free running VCO

Method in [14], PLL

Figure 13: Comparison of BER performance of the proposed algorithm andthe method in [13] in a free running VCO and PLL as a function ofthe inputβT vco in which the true(βTvco) = 0.05. A 16-QAM coded OFDM and aRayleigh channel where SNR= 20dB used.

and (29) only requires the multiplication of the LS channelestimate with a pre-computed matrix. The multiplication withA−1k is an operation in the frequency domain. More efficient

would be to do the codebook pre-compensation in the timedomain and do a separate FFT for each of theK possiblereceived signals. In this case, the number of complex multi-plications to obtainYk

m for k = 1, ...,K is on the order ofN(1 + K

2 log2N), rather thanN(KN + 12 log2N). A sum-

mary of the number of complex additions and multiplicationsrequired by the most important equations of our algorithm isgiven in Table III.

The method in [13] involves some computations which arenot needed in our algorithm for instance the successive leastsquares estimation of the coefficients of the power series whichapproximates the PHN in addition to anN × N Toeplitz

Table III: Summary of total Number of complex operations required perOFDM symbol and DD-FB iteration excluding the Viterbi and a few othercomputations.

Equation Additions Multiplications Complex inversions(20) KN(N − 1) KN2

−

(27) − KN KN(26) KN(N − 1) KN2

−

(25) (K + 1)(N − 1) K(2N + 1) +N KN + 1(30) − 2KN K(N + 1)(24) KN(N − 1) KN −

matrix inversion. The overall number of complex additions andmultiplications for channel estimation and PHN compensationrequired for one OFDM symbol by the proposed method aswell as the method in [13] is given in Table IV, excluding theViterbi algorithm and without taking into account theN ×NToeplitz matrix inversion for [13]. The parameterΛ in TableIV refers to the order of the polynomial approximating theICI coefficients as presented in [13]. It can be seen that thecomplexity of both the proposed algorithm and the methodin [13] are linearly dependent onK andΛ respectively. ForK = 27, Λ = 6 andN = 64 the proposed algorithm has acomputational advantage over [13]. This is illustrated in Figure14 which compares the measured execution time for variousnumber of DD-FB iterations between the proposed algorithm(Algorithm 2) and the algorithms in [13] for 64-QAM OFDMat 24dB SNR. No DD-FB loop or 0 iteration for the methodin [13] implies channel estimation with CPE compensationwhich reduces the number of computations significantly. TheN × N Toeplitz matrix inversion in [13] has been factoredout of the measured execution time (since we did not takeinto account the Toeplitz structure). Our simulator is a Linuxsystem (Ubuntu 11.04) running on a CPU with Intel-Corei7-2600 and uses the IT++ library in C++ for executingthe algorithms. The results in Figure 14 show that, for thechosen parameterK = 27, the proposed method provides asignificantly better performance-complexity trade-off than themethod in [13].

The DD-FB stage in [5] and [13] are not easily parallelized,while the most computationally demanding operations in theproposed algorithm can obviously be split into up toKparallel processors or dedicated hardware. The algorithmsin[5] and [14] also involve an inverstion of anN×N (Toeplitz)matrix. However, the method in [14] also presents a solutionin which only the ICI coefficients close to the carrier needto be estimated. This requires solving a less complex linearsystem, e.g. based on tridiagonal matrix algorithm whichonly requiresO(N) complex operations. This makes [14]less computationally complex to implement than the proposedmethod and [13]. In addition [14] has the advantage providedby avoiding the requirement of having to know the channeland PHN statistics. These advantages come at the cost ofperformance in BER as shown in Figure 8 and Figure 9.

VII. C ONCLUSION

Based on a simple codebook table which approximates thephase-noise statistics by a finite number of realizations, anovelPHN compensation approach is introduced. The general ideacan be applied to a wide range of PHN scenarios. Herein,

12

Table IV: Summary of total Number of complex operations required per OFDM symbol excluding the Viterbi and a few other computations.

Method No. of Additions No. of MultiplicationsProposed method (i+ 1)(KN(3N − 2) − k +N − 1) (i+ 1)(K(2N2 + 6N + 1) +N)[13] i(ΛN(N − 1) +N(N − 2)(Λ − 2)(Λ− 1)/2) i(Λ(4N2 + 2N + 1) + Λ(Λ− 1)N(2N − 1)/2)

+(i+ 1)2N(N − 1) +(i+ 1)(2N(N + 1))

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6∗

6∗

7∗

7∗

8∗

8∗

9∗

9∗

10∗

10∗

Method in [13]

Proposed method (Algorithm 2))

No. of iterations

Figure 14: BER performance of Algorithm 2 and the algorithmsin [13] on64-QAM OFDM as a function of averaged execution time per frame (in sec)for K = 27 and Λ = 6 at 24dB SNR using various number of iterations(denoted byi∗) in the DD-FB loop.

we have concentrated on ICI suppression in a convolutionalencoded OFDM system with scattered pilots. The complexityof the algorithm is determined by the size of the codebook.

Using a codebook of moderate size,K = 27, for a 16-QAM uncoded OFDM system in which an AWGN channeland a PHN withBT = 0.01, which is was considered, theproposed algorithm is 2dB from the ideal PHN free case ata BER of 10−3. Considering an identical scenario in a 64-QAM uncoded OFDM, the proposed algorithm is 2.5dB fromthe ideal PHN free case at a BER of10−2. By comparison,the method in [13] is9dB from the ideal case.

For a coded system, the codebook technique is used alongwith DD-FB loop for combined channel estimation and PHNcompensation. For a 16-QAM OFDM system over a Rayleighfading channel withBT = 0.01, the proposed method is1.5dB from the ideal case (PHN free and known channel) ata BER of 10−4 employing 2 iterations on the DD-FB loop;while the method in [13] is2.5dB from the ideal case using8 iterations. For a 64-QAM coded system and PHN withBT = 0.005, the proposed method provides a1dB gain at aBER 10−3 employing 3 iterations in the DD-FB, comparedto the method in [13] which uses 10 iterations. The gainobtained by the proposed method is significant compared tothe method in [14], which uses a pilot interpolation techniquefor channel estimation. Additional results are also presentedfor both 16 and 64 QAM coded systems assuming knownRayleigh channel frequency response at the receiver. Theproposed method is shown to outperform to the reference

schemes without having to employ the DD-FB loop.The performance of the proposed method with respect

to the codebook size is also evaluated showing the relativeimprovement in BER asK increases. However, for the givenexample, the relative improvement is very small asK increasesabove 27. Simulation results are also shown employing PHNfrom a charge pump PLL, modelled as Ornstein-Uhlenbeckprocess, is also presented with the proposed method showingimproved performance compared to the methods in [13] andalso demonstrating the applicability of the proposed algorithmfor a PHN process other than Weiner process.

Therefore, with a moderately small codebook size and lim-ited number of iterations on DD-FB, an improved performanceas well as faster execution time is achieved by the proposedalgorithm.

APPENDIX APROOF OFEQUATION (18)

Let us then first consider the case when the number ofquantization regions,Q, is infinite. In such a case, eachsegment of the selected codebook entry will follow the mean ofsamples of the corresponding segment of the PHN realization.Thus the MSE for infiniteQ is given by

MSE(Q=∞) =

J−1∑

j=0

L−1∑

n=0

Eθ(n)[(θ(n+ jL)− θ(n+ jL))2]

=

J−1∑

j=0

L−1∑

n=0

E[(θ(n+ jL)2

− 2θ(n+ jL)θ(n+ jL) + θ(n+ jL)2]

=

J−1∑

j=0

L−1∑

n=0

E[(θ(n+ jL)2

− 1

L

J−1∑

j=0

L−1∑

n=0

L−1∑

l=0

E[θ(n+ jL)θ(l + jL)]

=(N − J)(N + J)

6Jσ2ǫ

(34)

in which the last equality follows from the covariance of arandom walk process which isE[θ(n + ℓ)θ(n)] = nσ2

ǫ forl ≥ 0.

When Q is finite, an additional error is introduced dueto the quantization of segment averages. As in the designof the codebook, we assume thatψ is set to the averageof the first segment and thus no quantization error occursin the first segment and the PHN realization crosses thefirst sample of each segment of the trajectory. Additionally,by assuming that this error in each segment is identicallydistributed and independent of the difference between the PHN

13

and the codebook trajectory, the MSE due to accumulatedquantization error in each segment isLσ2

q , whereL = N/Jand σq is the standard deviation of quantization error. TheMSE due to the quantization error over the entire trajectoryis therefore, MSE(Q<∞) = (J − 1)Lσ2

q , where(J − 1) is dueto the assumption that the first segment has no quantizationerror. The approximate MSE of the selected trajectory as anestimator of the PHN realization then becomes

MSE=(N + J)(N − J)

6Jσ2ǫ + L(J − 1)σ2

q . (35)

In order to determineσ2q , we assume that the PHN realization

is exactly on one of the codebook entries in the last sampleof the previous segment. In this case, the quantization error ofthe j-th segment average is equal to the quantization error ofthe j-th increment, i.e., no influence of previous increments.The variance,σ2

q , of the quantization error is then given by

σ2q =

Q∑

i=1

E[(X − x)2|X ∈ Ri]P(X ∈ Ri))

=

Q∑

i=1

P(X ∈ Ri)(E[X2|X ∈ Ri]− 2xiE[X |X ∈ Ri] + x2i )

=

Q∑

i=1

P(X ∈ Ri)(E[X2|X ∈ Ri]− x2i )

(36)

where the last equality follows fromE[X |X ∈ Ri] =xi which is given by (16). The conditional ExpectationE[X2|X ∈ Ri] is given by

E

[X2|X ∈ Ri

]=

∫Rix2fX(x)dx

P(X ∈ Ri)(37)

whereP(X ∈ Ri) =∫RifX(x)dx. WhenfX(x) a Gaussian

function∫

Ri

x2fX(x)dx = σ2x/2

(erf(xi+1/

√2σx)− erf(xi/

√2σx)

)

− σx√2π

(xi+1 exp(−0.5x2i+1/σ

2x)− xi exp(−0.5x2i /σ

2x)

)

(38)

and

P(X ∈ Ri) =

∫

Ri

fX(x)dx

= 0.5

(erf(xi+1/

√2σx)− erf(xi/

√2σx)

).

(39)

Equations (38), 39 and (16) are then used on (36) to obtainthe variance,σ2

q , of the quantization error.

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