Two-Dimensional Interpolation-Assisted Channel Prediction for OFDM Systems

648 IEEE TRANSACTIONS ON BROADCASTING, VOL. 59, NO. 4, DECEMBER 2013

Two-Dimensional Interpolation-Assisted ChannelPrediction for OFDM Systems

Hsuan-Chang Lee, Member, IEEE, Jih-Ching Chiu, Member, IEEE, and Ken-Huang Lin, Member, IEEE

Abstract—Channel prediction can provide up-to-date informa-tion about the state of a channel in OFDM systems at the expenseof increasing the complexity of wireless systems. Therefore,this paper proposes a novel approach for optimizing the shapeand intervals between samples in two-dimensions (2-D) foran orthogonal frequency-division multiplexing (OFDM) systemto reduce the complexity of predicting the channel frequencyresponse (CFR). Furthermore, the proposed prediction schemehas the flexibility to integrate with an adaptive algorithm whichdoes not require any statistical prior knowledge and is able totrack nonstationary channel statistics. The number of multipliersthat are required to predict the CFR is reduced by predicting theCFRs at particular subcarriers and incorporating interpolationmethods. This paper also deals with the proposal of a 2-D curvefitting (CF) interpolator to be used for predicting the CFR thatallow improving the accuracy of the CFR prediction. Simulationand analysis results demonstrate that the 2-D CF interpolatorcombined with the proposed scheme has lower complexity thanthe other approaches to predicting CFR, without compromisingthe mean square error (MSE) or bit error ratio, even in a noisyand fast-fading channel environment.

Index Terms—Channel prediction, 2-D interpolation, curvefitting, DVB-T

I. Introduction

ORTHOGONAL frequency-division multiplexing(OFDM) is an efficient modulation scheme for

high-data-rate digital communications that can provide ahigh spectral efficiency in wireless transmission channels[1], and has been utilized commercially in terrestrial digitalvideo broadcasting (DVB-T), long-term evolution (LTE)systems, wireless local area networks (IEEE 802.11a),and the IEEE 802.16 work group. OFDM systems use anadaptive modulation and coding scheme (AMC) to evaluatethe maximum performance of an interactive OFDM systemwith the channel frequency response (CFR) [2]. Channelestimation is an extensively used method for obtaining CFRsin OFDM systems [3]–[5]. In [3], a channel estimationmethod based on a time-domain threshold was presented toreduce the noise in OFDM systems. An analytical model ofinterference for channel estimation [4] has been presented to

Manuscript received April 15, 2013; revised August 7, 2013; acceptedSeptember 19, 2013. Date of current version December 10, 2013.

The authors are with the Electrical Engineering Department, National SunYat-sen University, Kaohsiung 804, Taiwan (e-mail: hc−[email protected];[email protected]; [email protected]).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TBC.2013.2283097

deal with the imperfect superimposed sequence cancellationat the receiver. The block timing error statistics of theOFDM time-synchronizer has been utilized to improveminimum mean square error (MMSE) channel estimation[5]. Attempts have also been made to reduce the complexityof channel estimation [6], using multiple adaptive filters toexploit the frequency correlation of the channel. However,in high-mobility environments, where the Doppler frequencyis high and the CFR changes rapidly, previous techniquesfor obtaining the CFR cannot be used by the techniquessuch as delay-free detection or AMC because of processingand propagation delays, which cause significant performancedegradation. For example, in [1], a channel modeling withthe maximum Doppler spread of 80 Hz and an SNR of 30 dBin OFDM systems, the normalized mean square error (MSE)of the channel estimation is approximately 0.1.Channelprediction is an effective means of overcoming this delay.

Channel prediction algorithms for flat-fading channels havebeen extensively studied in the past several years. The ESPRITalgorithm [1] and the sub-space based root-MUSIC method [8]have been used to predict the power spectrum that is associatedwith the fading process. An adaptive long-range predictionmethod [9] has been applied to reduce feedback for OFDMsystems, and the MMSE method [10] has been utilized topredict the fading CFR from numerous past observations.

However, predicting the CFR in each subcarrier not onlyrequires the observation of several past CFRs to generatethe observation vector but also involves the computationallycomplex determination the coefficient function. To reduce thenumber of parameters in an observation vector, the samplingrate for observing the CFRs is determined and the samplingpattern is designed to have a diamond shape in the predictionof CFRs. The size of the matrix of the coefficient function ofCFR prediction depends on the number of the parameters ofan observation vector and is closely related to the complexityof the prediction. Accordingly, simplifying the observationvector of the CFR predictor reduces the size of the coef-ficient matrix of the prediction, reducing the complexity ofthe prediction.

The developed prediction scheme also predicts the CFRsat particular subcarriers and applies interpolation methods toreduce the number of the predicted CFRs, further reducing thecomputational complexity of the CFR prediction. Additionally,the proposed prediction scheme has the flexibility to involvewith an adaptive algorithm for implementation. The adaptiveversion of our prediction schemes has low complexity and

0018-9316 c© 2013 IEEE

LEE et al.: TWO-DIMENSIONAL INTERPOLATION-ASSISTED CHANNEL PREDICTION FOR OFDM SYSTEMS 649

Fig. 1. Block diagram of a channel prediction method.

Fig. 2. The locations of pilot subcarriers.

does not require any statistical prior knowledge and is able totrack nonstationary channel and noise statistics.

To improve the accuracy for predicting the CFR, a 2-DCF interpolator is also developed. In a 2-D CF interpolator, apower function polynomial with a minimum-variance unbiasedmethod is developed to fit the variation of the CFR. Powerfunction in the proposed interpolation is time or frequency-invariant and does not need to be recalculated to fitting thecurve of CFR, further lowering the computational complexity.

The rest of this paper is organized as follows. Section IIintroduces the system model. Section III discusses the predic-tor of the OFDM system and presents a method to reduce thecomputational complexity of predicting the CFR in OFDMsystems. To improve the accuracy for predicting the CFR,a novel interpolation method that has a lower complexitythan other interpolation methods is presented in Section IV.Section V presents numerically analyzes the performance ofthe various channel-prediction schemes. Section VI drawsconclusions and offers recommendations for future research.

II. System Model

Figure 1 presents a point-to-point OFDM system with Ksubcarriers over a frequency-selective channel. At the transmit-ter, the nth OFDM symbol sequence {Sn,0, Sn,1, . . . , Sn,K−1} issubject to the inverse discrete Fourier transform (DFT). EachOFDM symbol is then transmitted after a cyclic prefix (CP)has been appended to it. After the CP has been removed, theDFT is applied to the time-domain signal at the receiver toobtain the baseband signal at the k∈{0,1, . . . , K-1} subcarrier,which is expressed as, [11]

Yn,k = Sn,kHn,k + Wn,kk = 0, 1, 2, · · · K − 1 (1)

where Hn,k and Sn,k are the CFR and signal that aretransmitted at the k-th subcarrier, respectively, and Wn,k isa zero-mean and complex additive white Gaussian noise(AWGN) with variance σ2

W . For ease of expression, thereceived signals are collected and K × 1 vectors Yn =[

Yn,0 ... Yn,K−1]T

, Hn =[

Hn,0 ... Hn,K−1]T

andWn =

[Wn,0 ... Wn,K−1

]T, and the diagonal K × K ma-

trix Sn = diag{

Sn,0 ... Sn,K−1}

, are defined. From (1),the vector of the received signals is,

Yn = SnHn + Wn (2)

In OFDM systems, such as DVB-T, the pilot subcarriers aredistributed in both time and frequency, as presented in Fig. 2.The black dots represent the pilot subcarriers, and the whitedots represent the data subcarriers. In this study, Dt and Df

are the pilot intervals, given as a number of symbols in thetime and frequency domains, respectively. The CFR H ′′

n,k at apilot subcarrier is obtained as

H ′′n,k =

Yn,k

Sn,k

= Hn,k + Wn,k, withWn,k =Wn,k

Sn,k

(3)

In this paper the nth OFDM symbol is assumed to bethe current received OFDM symbol, and the (n + p)th OFDMsymbols for p ≤ 0 and p > 0 are the previous and future re-ceived OFDM symbol, respectively. The previous CFRs in theprevious OFDM symbol are obtained by channel estimation.The CFRs in the future OFDM symbol are determined bychannel prediction. The next section will discuss particularmethods of predicting the CFR in OFDM systems.

III. Reducing Complexity of Channel Predictions

For a wide-sense-stationary channel, the strong correlationsamong CFRs effectively exist not only in coherence bandwidthbut also in coherence time, as described by Schafhuber [10].Therefore, the auto-correlation function of Hn,k can bedefined as

E{Hδ+n,γ+kH

∗n,k

}= rf [δ, γ] (4)

where δ ≤2/3Kνmax; γ ≤ K /L; νmax represents the maximumnormalized Doppler shift, and L represents the propagationdelay. This auto-correlation function is adopted by channelpredictors such as the MMSE algorithm.

For clearly realizing the proposed scheme, this paper depictsa logical flowchart to clearly explain the presented algorithm,as follows.


A. OFDM CFR Prediction Algorithm

To predict the CFRs, the CFRs in previousOFDM symbols are used. The observation vectorthat is used in the predictors is given by H′′

n =[H ′′

n,0 · · · H ′′n−M+1,0 · · · H ′′

n,K−1 · · · H ′′n−M+1,K−1

]Twhich has MK parameters. Here, H ′′

n,k is the CFR whosevalue is estimated either by (3) at the pilot subcarrier orby channel estimation at the data subcarrier. The channelprediction that is based on the MK observed parameters canbe performed as described in [12]

Hn+p,k = θHn+p,kH′′

n (5)

where p > 0 is the prediction horizon, and θn+n,k represents theprediction coefficients that are associated with the kth subcar-rier on the n + pth OFDM symbol. The optimal Wiener filtercoefficient θn+n,kthat minimizes the MSE E[|(Hn,k − Hn,k)|2]can be obtained by applying knowledge of the auto-correlationfunction in the time and frequency domains in (4).

θn+p,k = Q−1n+p,kχn+p,k (6a)

with

χn+p,k=

⎡⎢⎢⎣

rf [p, −k], . . ., rf [p + M − 1, −k], . . .,rf [p, 1 − k], . . ., rf [p + M − 1, 1 − k], . . .,

· · ·rf [p, K − k − 1], . . ., rf [p + M − 1, K − k − 1]

⎤⎥⎥⎦

T

(6b)

Qn+p,k=

⎡⎢⎣

σ2H+σ2

Wn+p,k· · · rf [M−1, K−k−1]

.... . .

...r∗f [M−1t , K − k − 1] · · · σ2

H+σ2Wn+p,k

⎤⎥⎦

(6c)where χn+p,k is a MK × 1 vector and Qn+p,k is aMK × MKmatrix.θn+n,k should be frequently updated innon-stationary scenarios. Therefore, the implementation wouldrequire a real-time inversion of a MK × MK matrix Qn+p,k

and the computational complexity of predicting the CFR ineach subcarrier is O(M3K3).

B. Downsampling Channel Prediction and 2-D Interpolation

The size of the matrix in (6c) depends on the number ofthe parameters of an observation vector and is closely relatedto the complexity of the prediction. Accordingly, simplifyingthe observation vector of the CFR predictor will reduce thesize of the coefficient matrix of the prediction, reducingthe complexity of the prediction. To reduce the number ofparameters, the observation vector is obtained using the downsampling approach [13], with the restriction that the samplingrate must exceed double the minimum rate to prevent aliasing.The sampling rate should therefore satisfy the inequality

1

DtTO

≥2�h ⇒ 1

2�hTO

≥Dt (7)

where T o is the minimum sampling rate of the OFDMsystem, and �h is the maximum rate of variation of the CFRs.The one-dimensional (1-D) sampling theory that is specified as

Fig. 3. The locations of the observation CFRs.

(7) can be directly utilized to reduce the number of parametersof the observation vector. However, unlike in the conventional1-D sampling method, the application of the sampling theoremto the 2-D sampling space can further reduce the number of theparameters of an observation vector. To minimize the MSE ofthe CFR estimation, the optimal sampling arrangement on the2-D time-frequency grid should have the shape of a diamond[15]. The proposed method arranges the observed subcarriersare in a diamond shape, as presented by � in Fig. 3. Dt is thesampling distance of the observation vector in time domain.Hence, the index of the observation vector (M-1) is multiple ofDt . Since the arrangement is diamond-shaped in 2-D, similarsituations exist in every Df subcarriers in an OFDM symbol,and Df is multiple of Dt to substantially reduce the samplingdistance from Df to Dt . In this paper, Df is set to triple Dt

to fit the location of pilots for easily obtaining the observedCFRs. Therefore, the scheme of predicting the CFRs can bereformulated as

H ′n+p,k = qH

n+p,kHn (8a)

with

Hn =

[H ′′

n,0 H ′′n−Dt,0 · · · H ′′

n−M+1,0H ′′

n−Dt+1,(Df /Dt )H ′′

n−2Dt+1,(Df /Dt )· · ·

· · · H ′′n,K−1 H ′′

n−Dt,K−1 · · · H ′′n−M+1,K−1

]T(8b)

where the observation vector Hn is M‘ × 1, and M ′ is MK/Df .The simplified coefficient functions qn+p,k are now

qn+p,k = R−1n+p,kgn+p,k (9a)

with

gn+p,k =

⎡⎢⎢⎣

rf [p, −k], rf [p + Dt, −k], . . .rf [p, Df

/Dt − k], rf [p + Dt, Df

/Dt − k], . . .

· · ·rf [p, K − k − 1], rf [p + Dt, K − k − 1], . . .

⎤⎥⎥⎦

T

(9b)


Fig. 4. The locations of the predicted CFRs in sereval times prediction.

Rn+p,k=

⎡⎢⎢⎢⎣

σ2H+σ2

Wn+p,krf [Dt, −k]

r∗f [Dt, −k] σ2

H + σ2Wn+p,k

......

r∗f [M−Dt, K−k−1] r∗

f [M−2Dt, K−k−1]· · · rf [M−Dt, K−k−1]· · · rf [M − 2Dt, K − k − 1]. . .

...· · · σ2

H+σ2Wn+p,k

⎤⎥⎥⎥⎦

(9c)where gn+p,k is a 1 × MK /Df vector and Rn+p,k is anMK /Df × MK /Df matrix. As stated above, the number ofparameters of the observation vector is reduced from MK inH′′

n to MK /Df in Hn. Therefore, the size of the matrix inthe coefficient functions is reduced from MK × MK in (6c) toMK /Df × MK /Df in (9c), and the complexity of the filter thatperforms matrix inversion is greatly reduced from O((MK)3)to O((MK /Df )3).

Next, the developed scheme reduces the number of recal-culations of the coefficient function to predict the CFRs ofall subcarriers in the n + pth OFDM symbol. As mentionedabove, the optimal pattern for estimating the CFR in OFDMsystems has a diamond shape. However, in prediction mode,constructing the sampling pattern with a diamond shape isdifficult, since the CFRs after the n + pth OFDM symbol areunavailable, as shown in Fig. 4. Hereafter, the nth predictionrefers to the attempt to predict the CFRs of all subcarriers inthe n + pth OFDM symbol. To overcome the aforementionedissue, the presented scheme predicts only those the CFRs atsubcarriers (|n + p + Dt-1| mod Dt) = ([k mod Df ] × Dt/Df ) inthe n + Dt + p-1th OFDM symbol, as represented as black dotsin Fig. 4. The relevant formula is

H ′n+Dt+p−1,k=

⎧⎨⎩qH

n+Dt+p−1,kHnk ∈( |n + p + Dt − 1| mod Dt

=[k mod Df

]Dt

/Df

)0k= otherwise

(10)The CFRs at some subcarriers in the n + Dt + p-2th, n + Dt + p-3th, and n + pth OFDM symbol are represented as •, •, and •

in Fig. 4, respectively. These are obtained in the n-1th, n-2th,and n-3th predictions. After the CFRs at the black dots havebeen obtained, the CFRs between � and black dots can beobtained using an interpolation method in the time domain.The same procedures are performed for the other subcarriers.For example, after the CFRs of • have been obtained, the CFRsbetween • and � are obtained using the interpolation methodin the time domain. Now, the obtained CFRs are regarded asthe virtual pilots, and all of the CFRs in the n + pth OFDMsymbol can now be determined by frequency interpolationusing these virtual pilots.

C. Adaptive Filter and 2-D Interpolation

In this paper, the MMSE algorithm is integrated into theproposed prediction scheme for suitable analysis. Neverthe-less, the proposed prediction scheme has the flexibility toincorporate with an adaptive algorithm, such as the recursiveleast squares (RLS) algorithm [12], which the predictionerror is minimized for implementation. Rewriting the relevantformula of the predicted CFRs in (10) as

H ′n+Dt+p−1,k =

⎧⎨⎩αH

n+Dt+p−1,kHnk ∈( |n+p+Dt−1| mod Dt

=k mod Df

/Dt

)0k= otherwise

(11)In the RLS algorithm, the coefficient function αn,k is calcu-

lated until minimizing the error en,k

en,k =n∑

n′=n−Dt−p+1

λn−n′ ∣∣Hn′,k − αHn′,kHn′−Dt−p+1

∣∣2 (12)

where λ is a forgetting factor. Stable operation of the RLSalgorithm requires 0 < λ ≤1. The selection of λ determines theconvergence speed, the excess MSE, and the ability to tracknonstationary channel statistics. We obtained good results withλ=0.99. However, the exact value is rather uncritical. Theupdated equation for αn,k

αn,k = αn−Dt,k + �n−Dt−p+1ε∗n,k (13)

where εn,k = Hn,k − αHn,kHn−Dt−p+1 and �n is the RLS gain

vector that is given by

�n = (PnHn)/(

λ + HTn PnHn

),n≥Dt (14a)

with

Pn =1

λ

(I − �nHT

n

)Pn−Dt

,n≥Dt (14b)

The RLS recursion is initialized as

�n = PnHn =1

+∣∣HT

n Hn

∣∣Hn,n < Dt (15a)

and

Pn =( I + HnHT

n

)−1=

1

[I − HT

n Hn

+∣∣HT

n Hn

∣∣]

,n < Dt

(15b)where the stabilization factor is in the range 0 < < < 1and is chosen 0.05 in this paper. In contrast with the MMSE


algorithm, the RLS method requires no prior statistical knowl-edge, and can be used track non-stationary channel andnoise statistics. Furthermore, the computational complexity ofpredicting the CFR in each subcarrier is O((MK /Df )2) plusthe complexity of the interpolation.

IV. Channel Prediction Assisted by Interpolation

Methods

This section presents two typical methods for interpolatingall CFRs using the virtual pilots in Section III-B. A novelinterpolation method is also developed to provide excellentperformance with lower implementation complexity than othersuch methods.

A. Channel Prediction Assisted by MMSE Interpolation

The optimal MMSE method has been widely studied [5].Combined with our presented CFR prediction in (10), theWiener filter interpolates the CFRs in the time domain, asfollows.

Htnp,k = H′

np+Dt,k(RtHH)−1rt

HH (16)

with

H′np+Dt−1,k =

[H ′

np+Dt−1,k H ′np−1,k .... H ′

np−(Mt−2)×Dt−1,k

](17a)

RtHH =

⎡⎢⎢⎢⎣

rf [0, 0] rf [Dt, 0]r∗f [Dt, 0] rf [0, 0]

......

r∗f [(Mt − 1)Dt, 0] r∗

f [(Mt − 2)Dt, 0]· · · rf [(Mt − 1)Dt, 0]· · · rf [(Mt − 2)Dt, 0]. . .

...· · · rf [0, 0]

⎤⎥⎥⎥⎦

(17b)

rtHH =

[rf [0, 0], . . ., rf [(Mt − 1)Dt, 0]

]T(17c)

where np is the n + pth OFDM symbol index; H′np+Dt−1,k is

the observed vector of interpolation whose size is 1 × M t; andRt

HHand rtHH are the auto-correlation matrix and the cross-

correlation vector in the time domain, respectively. The pre-dicted CFR is considered at the kth

p subcarrier in the nthp OFDM

symbol. Similarly, the Wiener filter for the interpolation in thefrequency domain is given by,

Hnp,kp= Ht

np,k(RfHH)−1rf

HH (18)

with

Htnp,k =

[Ht

np,k Htnp,k+Df

.... Htnp,k+(Mf −1)×Df

](19a)

RfHH =

⎡⎢⎢⎢⎣

rf [τn, 0] · · · rf [τn, (Mf − 1)Df ]r∗f [τn, Df ] · · · rf [(τn, Mf − 2)Df ]

.... . .

...r∗f [τn, (Mf − 1)Df ] · · · rf [τn, 0]

⎤⎥⎥⎥⎦

(19b)

rfHH =

[rf [τn, k − kp]. . ., rf [τn, k − kp + (Mf − 1)Df ]

]T(19c)

where Htnp,k is the interpolation vector whose size is 1 × Mf ;

and RfHH and rf

HH are the auto-correlation matrix and thecross-correlation vector in the frequency domain, respectively.The complexities of the MMSE coefficients in time andfrequency interpolations are O(M3

t ) and O(M3f ), respectively.

B. Channel Prediction Assisted by Lowpass Sinc Interpolationwith Kaiser Window (LSIKW)

The lowpass sinc filter has been studied [18] for use infrequency offset and carrier phase estimation. They can beused for pilot symbol-assisted modulation in fading channels.When the CFRs are predicted using (10), lowpass filters inboth time and frequency domains can be utilized to reconstructHnp,kp

. This lowpass filter requires a sinc interpolator ofinfinite length in either the time or the frequency domain.Simply truncating the sinc interpolator to a finite length caninduce large passband and stopband fluctuations. Accordingly,the sharpness of the transition band is controlled by settingthe interpolator length. One investigation [14] found that theimprovement in the performance is provided by the flexibilityof the shape of the Kaiser window. This design can beextended to combine the LSIKW with the channel-predictionscheme. At position [np, kp], the predicted CFR is expressedas,

Hnp,kp=

MtDt∑u=−MtDt

Mf Df∑v=−Mf Df

Wsk[u, v]H ′np−u,kp−v (20)

where M t and Mf are the length of the sinc interpolator inthe time and frequency domain, respectively. H ′

n,k is given by(10). Wsk [u,v] is the interpolation filter,

Wsk[u, v] =

χ [u − n, v − k] DtDf

HtHf× sin c

(u−nHt

)sin c

(v−kHf

) (21a)

with

Dt≤Ht≤ 12fD

Df ≤Hf ≤ 12 f

(21b)

where fD is the normalized maximum Doppler frequency and�f is the subcarrier spacing. Parameters Ht and Hf determinethe lowpass filter cutoff frequencies in the time and frequencydomains, respectively. The Kaiser window function χ[u, v] is

χ[u, v] =

I0

(βt

√1 −

(uαt

)2)

I0 (βt)

I0

(βf

√1 −

(vαf

)2)

I0(βf

) (22)


where αt = M tDt , αf = Mf Df , and the window size is2 M tDt + 1 by 2 Mf Df + 1. The shape factors βt and βf arefunctions of the passband/stopband peak fluctuations. LargerH t and Hf result in lower cutoff frequencies, so causing moreinterpolation errors to lead to error floors. Additionally, higherorders of interpolation M t and Mf must be required to satisfythe need for a lower transition band.

C. Channel Prediction Assisted by CF Interpolation

The CF of a (c-1)th-order polynomial reduces AWGN andapproaches the a curve of (c-1)th-order at the location ofinterest in [15]. Power function polynomial depends on therelationship between the location of interest and the locationswhere the observations are made. This time or frequency-invariant property is suitable to develop an efficient interpola-tion method for combining with the proposed CFR predictionschemes in Section III. To proceed, CFRs are predicted using(10) or (11), and the vector of interpolation in the time domainHnp,k can be obtained and fitted by a (c-1)th-order polynomial,as follows.

Hnp,k =

{θtAt

np+Dt−1,kk mod Df = 00k= otherwise

(23)

with

Hnp,k =[

Hnp,k Hnp+1,k · · · Hnp+Dt−1,k

]T(24a)

Atnp+Dt−1,k =

[H ′

np+Dt−1,k H ′np−1,k

· · · H ′np−(Mt−2)Dt−1,k

]T (24b)

θt = Ft

(BT

t Bt

)−1BT

t (24c)

and

Ft =

⎡⎢⎣

1 1 · · · 1c−1

......

. . ....

1 Dt · · · Dc−1t

⎤⎥⎦ (24d)

Bt =

⎡⎢⎢⎢⎣

1 Dt · · · Dc−1t

1 0 · · · 0c−1

......

. . ....

1 −(Mt − 2)×Dt · · · − ((Mt − 2)×Dt)c−1

⎤⎥⎥⎥⎦

(24e)where At

np+Dt−1,k is the vector that is observed of size M t × 1;Bt is an M t × c Vandermonde matrix; Ft is a Dt × c matrix;Hnp,k is an interpolated vector in the time domain of sizeDt × 1, and θt is a Dt × Mt matrix and is time or frequency-invariant. Therefore, the coefficient vector does not have tobe recalculated in the interpolation process to reduce thecomplexity of the interpolation. In the following, the CFmethod is again utilized for interpolation in the frequencydomain, and the CFRs that are obtained from (23) and SectionIII are treated as the observation vector of the interpolation.

All of the CFRs in the nthp OFDM symbol fitted by a (z-1)th-

order polynomial with Hnp,k, which is

Hnp,k = θf Af

np,k (25)

with

Hnp,k =[

Hnp,k Hnp,k+1 · · · Hnp,k+Df −1]T

(26a)

Af

np,k=

⎧⎪⎨⎪⎩

[Hnp,k Hnp,k+Df

· · · Hnp,k+(Mf −1)Df

]Tk ∈ first block[

Hnp,k−(Mf −1)Df· · · Hnp,k−Df

Hnp,k

]Tk ∈ last block[

Hnp,k−DfHnp,k · · · Hnp,k+(Mf −2)Df

]Tother blocks

(26b)

θf = Ff

(BT

f Bf

)−1BT

f (26c)

and

Ff =

⎡⎢⎣

1 1 · · · 1...

.... . .

...1 Df · · · Dz−1

f

⎤⎥⎦ (26d)

Bf =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎡⎢⎣

1 · · · 1

1. . .

...1 · · · ((Nf − 1)Df + 1)z−1

⎤⎥⎦ k is the first taps

⎡⎢⎣

1 · · · (−(Nf − 1)Df + 1)z−1

1. . .

...1 · · · 1

⎤⎥⎦ k is the last taps

⎡⎢⎢⎢⎣

1 Df +11 1...

...1 (Nf −2)Df + 1

· · · (−Df +1)z−1

· · · 1. . .

...· · · ((Nf −2)Df +1)z−1

⎤⎥⎥⎥⎦ k ∈ otherwise

(26e)

where Af

np,k is the observation vector of size Mf × 1.

Equation (18) gives the components of Af

np,k; Hnp,k is theobtained vector which has a size of Df × 1. Bf is an Mf × zVandermonde matrix; Ff is a Df × z matrix, and θf is aDf × Mf matrix. Notably, Ft , Bt , Ff , and Bf are all timeor frequency-invariant and independent of the parameter n

or k. Therefore, the coefficients θt and θf are time andfrequency-invariant, respectively, and are not recalculated inthe interpolation process to reduce the complexity of theinterpolations. Finally, the prediction scheme easily predicts allof the CFRs in the np OFDM symbol. The Appendix analyzesthe MSE performance of the CF interpolation that is combinedwith the proposed prediction scheme.

D. Complexity Analysis

Table I presents the complexity of each approachthat isdiscussed herein, where MMSE + Downsampling Channel Pre-diction (DSCP) is an MMSE interpolator that is combinedwith the channel-prediction scheme that was presented inSection III-B; LSIKW + DSCP is an LSIKW interpolator that


TABLE I

Comparison of Complexities of Predictors

is combined with the channel-prediction scheme that waspresented in Section III-B, and CF + DSCP is a CF interpolatorthat is combined with the channel-prediction scheme that waspresented in Section III-B; CF + Adaptive predictor is a CFinterpolator that is combined with the adaptive predictionmethod that was presented in Section III-C.

For the traditional MMSE predictor [12], the number of theparameters of the observation vectors is MK . Implementationof the traditional MMSE predictor requires a real-time inver-sion of the matrix MK × MK so the computational complexityin each subcarrier is O(M3K3)

In contrast to the traditional predictor, the developed predic-tion scheme, the number of the parameters of the observationvector is MK /Df . Therefore, the complexity of the proposedDSCP is O((MK /Df )3) plus the complexity of the interpola-tion.

In MMSE + DSCP, the MMSE interpolation requires twoWiener filters that involve M t × M t and Mf × Mf ma-trix inversions in the time and frequency domains, re-spectively. Recall that the Wiener filter in the time do-main must be recalculated for every Df subcarriers. There-fore, the complexity of MMSE + DSCP in each subcarrieris O((MK /Df )3) + O(M3

t ) + O(M3f ). Notably, to perform the

MMSE interpolation, only information such as the channelstatistics in both the time and the frequency domains, and thesignal-to-nose ratio of a future OFDM symbol, is required.This information may be unavailable.

The LSIKW interloper depends on the symmetry of the sincfunction, and its complexity is double the number of parame-ters of the observation vector. Therefore, the numbers of pa-rameters of the observation vectors in the time and frequencydomains are M t and Mf , and the complexities of the LSIKWinterloper in the time and frequency domains are 2 M t/Df and2 Mf , respectively. Similarly, the coefficients of the LSIKWinterloper in the time domain must be recalculated every Df

subcarriers. The computational complexity of LSIKW + DSCPin each subcarrier is O((MK /Df )3) + 2 M t/Df + 2 Mf .

In the CF interpolation, power function polynomial istime- and frequency-invariant and can be determined withoutchannel statistics. Neither recalculation in any prediction norsymmetry of the sinc function is required. Therefore, thecomplexities of CF interpolation in the time and frequency

TABLE II

Simulation Parameters

domains are M t/Df and Mf , respectively. The complexity ofthe proposed CF + DSCP is O([MK/Df ]3) + M t/Df + Mf .

In CF + Adaptive predictor, the proposed prediction schemeis incorporated in the RLS algorithm which requires no priorstatistical knowledge, and can be used track non-stationarychannel and noise statistics. Furthermore, the computationalcomplexity of predicting the CFR in each subcarrier isO((MK /Df )2) + M t/Df + Mf which is the lowest complexityany of the mentioned predictors.

V. Simulation and Discussion

In this section, simulations are conducted to evaluatethe proposed scheme. The fundamental OFDM simulationparameters are consistent with the operating mode of theDVB-T system [19]. The receiver that is presented in Fig.1involves demodulation and zero-forcing equalization, givenby Yn,k = Yn,kHn,k.Hn,k is determined using the channel-prediction scheme. The best-performing algorithm of the pre-dictor is MMSE, and the processing delay is assumed to be lessthan four OFDM symbols. Therefore, the prediction horizonp is four. Table II lists the parameters of the OFDM system;the modulation is 64-QAM. The locations of the pilots are aspresented in Fig. 2 where Dt is four and Df is 12. The numberof parameters of the predictor, M, is set to 33, and the ordersof interpolation M t and Mf are eight.

Figures 5(a) and 5(b) present the MSE and bit error rate(BER), respectively, that are associated with the differentchannel-prediction schemes for a vehicle speed of 30 km/h.As expected, the predictor performs best, as presented inFigs. 5(a) and (b). The MMSE + DSCP can perform almostas well as the predictor in the slow time-varying channel. TheLSIKW interpolation results in a great interpolation error indeep fading. Therefore, LSIKW + DSCP has a poorer MSEthan all other schemes, as presented in Fig. 5(a). Based onthe error tolerance in the demodulation decision, the BER ofLSIKW + DSCP is close to those of the other approaches, asshown in Fig. 5(b). Integrating with DSCP and the adaptivepredictor, CF interpolation method can fit the channel curveso the MSE and BER are close to the performance of thepredictor in the slowly time-varying channel.

Figures 6(a) and (b) plot the MSE and BER performancesof various channel-prediction schemes for a vehicle speed of


Fig. 5. (a) MSE of various prediction schemes for a speed of 30 km/hr,(b) BER of various prediction schemes for a speed of 30 km/hr.

300 km/h. CF + DSCP and CF + adaptive prediction methodssuffer from a large error that are caused by the time-varyingchannel, but they almost fit the CFRs. Therefore, the proposedapproaches as well as the traditional MMSE algorithm have theexcellent performance in a high-speed rail speed environment,as presented in Figs. 6(a) and (b).

Figures 7(a) and (b) display the MSE and BER performanceof the different schemes when an airplane is traveling at a highspeed of 1000 km/h. Given a rapidly time-varying channel, thechannel-prediction schemes introduce not only an interpolationerror but also a prediction error. The prediction error is greaterthan the interpolation error in the demodulation decision.However, the proposed prediction scheme is flexible enoughto be combined with various interpolation and predictionmethods for ease of application. Besides, it is observed thatthe prediction method becomes more crucial than interpolationmethod when the Doppler spread is relative large.

Fig. 6. (a) MSE of various prediction schemes for a high train speed,(b) BER of various prediction schemes for a high train speed.

VI. Conclusion

This paper presented an efficient method for challenging theprediction task in an OFDM system. It considered the sam-pling rate of CFRs to achieve low complexity for predictingthe CFR. The proposed scheme involves a newly designedefficient pilot pattern for channel prediction in the OFDMsystem and applies a 2-D interpolation method to minimizethe number of the required multipliers in the predictionprocess. Furthermore, the proposed prediction scheme has theflexibility to integrate with an adaptive algorithm which cantrack nonstationary channel statistics without any statisticalprior knowledge. To improve the accuracy of the predictionof CFR, a novel CF interpolator was applied as a powerfunction polynomial manner to fit the curve of the CFR withexcellent performance. It is simpler than the Wiener filter,which depends on a matrix inverse operation and channelstatistics, and the LSIKW filter, which must be symmetricaland has a sideband for implementation of the sinc function.Combining the proposed scheme with the CF interpolationconsiderably simplified the prediction process. The adaptiveversion of the proposed method reduces the complexity ofpredicting the CFR from O((MK)3) to O((MK /12)2) in DVB-T


Fig. 7. (a) MSE of various prediction schemes for an airplane speed,(b) MSE of various prediction schemes for an airplane speed.

systems. The method also has potential for use in otherwireless OFDM communication systems.

VII. Appendix

The MSE analysis of CF + DSCP

Following (20), a CFR at the k′th subcarrier in the nthp OFDM

symbol is predicted by

Hnp,k′ = θfk′Af

np,k (27)

where

θfk′ = Ffk′(BT

f Bf

)−1BT

f (1×Nf ) (28a)

Ffk′ =[

1 k1 · · · kz−11

](1×z) (28b)

k1 = k′ mod Df (28c)

The MSE of this prediction scheme is written as

MSE = E{∣∣Hn+p,k′

∣∣2}− 2 (E {Hn+p,k′Hn+p,k′

})+ E

{∣∣Hn+p,k′∣∣2} (29)

To normalize with respect to power, the MSE is rewrittenas

MSE = 1 − 2 (E {Hn+p,k′Hn+p,k′})

+ E{∣∣Hn+p,k′

∣∣2} (30)

The second term is evaluated as

E{Hn+p,k′Hn+p,k′

}= θfk1E

{Hn+p,k′An+p,f

}= θfk′θ′

tE{Hn+p,k′[

A′t,k−(Nf −1)Df

· · · A′t,k−Df

A′t,k

]T} (31)

where θ′tis the coefficient that is interpolated in the time

domain at the k′subcarrier, and does not include the n or pvalue.

θ′t =

⎡⎢⎢⎢⎣

θ′1 0 · · · 0

0 θ′2 · · · 0

......

. . ....

0 0 · · · θ′Nf

⎤⎥⎥⎥⎦ (32)

The second term is calculated as

E{Hn+p,k′Hn+p,k′

}= θfk′θ′

tQHn+p,k

⎡⎢⎣

r∗f [Dt−1, 0]×rf [p+Dt−1, k′−k+(Nf −1)Df ]

...rf [(Nt−2)Dt−1, 0]×rf [p+Dt−1, k′−k]

· · · r∗f [Dt−1, 0]×rf [p+M−1, k′−k+(Nf −1)Df ]

. . ....

· · · rf [(Nt−2)Dt−1, 0]×rf [p+M−1, k′−k]

⎤⎥⎦

T

(33)By an argument similar to that presented above, the third termin (26) can be evaluated as

E{∣∣Hn+p,k′

∣∣2} = θfk′θ′t

⎡⎢⎢⎢⎣

QHp+Dt−1,0

QHp+Dt−1,0

...QH

p+Dt−1,0

⎤⎥⎥⎥⎦

⎡⎢⎢⎢⎣

R0,0 RDt,0

RTDt,0 R0,0...

...RT

(Nt−1)Dt,(Nf −1)DfRT

(Nt−2)Dt,(Nf −1)Df

· · · R(Nt−1)Dt,(Nf −1)Df

· · · R(Nt−2)Dt,(Nf −1)Df

. . ....

· · · R0,0

⎤⎥⎥⎥⎦⎡⎢⎢⎢⎣

QHp+Dt−1,0

QHp+Dt−1,0

...QH

p+Dt−1,0

⎤⎥⎥⎥⎦

T

θ′T θTfk′

(34)

Equations (30) to (34) give the MSE performance of theproposed channel-prediction scheme with CF interpolation.

Acknowledgment

The authors are grateful for many helpful discussions withProf. C.-P. Li and Prof. C.-K. Wen.

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Hsuan-Chang Lee (S’10–M’12) received the M.S. degree from the Instituteof Communications Engineering, National Sun Yat-sen University, Kaohsiung,Taiwan, in 2003. He is currently working toward the Ph.D. degree in theDepartment of Electrical Engineering, National Sun Yat-sen University.

His current research interests include radio-transmission technologies andresource-management schemes for wireless systems.

Jih-Ching Chiu (M’02) received the B.S. and M.S.degrees in electrical engineering from National SunYat-Sen University, Kaohsiung, Taiwan, and Na-tional Cheng-Kung University, Tainan City, Taiwan,in 1984 and 1986, respectively. He received thePh.D. degree in computer science and informationengineering from National Chiao Tung University,Hsinchu City, Taiwan, in 2002. In 1989 he joined theNational Sun Yat-sen University, and is presently anAssociate Professor with the Department of Electri-cal Engineering.

His current research interests include the areas of ILP CPU design, computersystem integration, and embedded system design. Currently, he is involved ina research on new generation processor design and reconfigurable computingbased processor design.

Ken-Huang Lin (S’90–M’93) received the B.S.degree from National Sun Yat-sen University, Kaoh-siung, Taiwan, in 1984, and the M.S. degree fromNational Taiwan University, Taipei, Taiwan, in 1986,both in electrical engineering. In 1987, he pursuedthe Ph.D. and was a Research Assistant with theDepartment of Electrical and Computer Engineeringat the University of Illinois at Urbana-Champaign,Champaign, IL, USA. In 1993, he received thePh.D. degree and joined the Department of ElectricalEngineering, National Sun Yat-sen University and is

currently a professor. Since 2005, he has also been serving as the Director ofIncubation Center, National Sun Yat-sen University. He was a recipient of theYoung Scientist Award in the 25th General Assembly of URSI in 1996, andwas named the Advanced Semiconductor Engineering, Inc. Endowed ChairProfessor in Engineering in 2012.

His current research interests include radio wave propagation, antennas,space science, wireless communication, and electromagnetic compatibility.

Two-Dimensional Interpolation-Assisted Channel Prediction for OFDM Systems

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