Low-Complexity Equalization for TDS-OFDM Systems

IEEE TRANSACTIONS ON BROADCASTING,VOL.51,NO.3,SEPTEMBER2005401 Low-Complexity Equalization for TDS-OFDM Systems

Over Doubly Selective Channels

Jian Fu,Chang-Yong Pan,Zhi-Xing Yang,and Lin Yang

Abstract—Time variation of a multipath channel leads to in-terchannel interference(ICI)in orthogonal frequency-pision multiplexing(OFDM)systems.It results in the performance degradation,therefore,limits the achievable throughput.Some methods have been proposed to suppress ICI,unfortunately,they are either computationally complex or at the price of spectral ef?ciency.In this paper,a low-complexity equalization method for time-domain synchronous OFDM(TDS-OFDM)systems is proposed under the assumption that the channel impulse response (CIR)varies in a linear fashion within a block period.The ratio-nale behind our method is to use a?nite power series expansion for the inverse of the equalization matrix.This method provides a desired tradeoff between the performance and the processing com-plexity.Theoretical analysis and simulation results demonstrate that the proposed method can effectively mitigate ICI caused by the channel variations with low computational complexity.

Index Terms—Doubly selective channel,equalization,inter-channel interference(ICI),TDS-OFDM.

I.I NTRODUCTION

O RTHOGONAL frequency-pision multiplexing(OFDM) has emerged as a popular technique to combat with the inter-symbol interference(ISI)over the frequency selective fading channels.The impact of ISI can be removed by inserting the guard interval[1].OFDM has been extensively used in broadcasting domain,such as the Digital Video Broadcasting for Terrestrial Television(DVB-T)proposed as the European digital television standard[2],and the Terrestrial Digital Mul-timedia/Television Broadcasting(DMB-T)[3]proposed by Tsinghua University as a candidate for the digital television terrestrial broadcasting(DTTB)standard for the People’s Republic of China.

When the OFDM block duration is much smaller than the channel coherent time,i.e.,relatively mild Doppler,the channel can be approximated as constant over an OFDM block.This assumption simpli?es the equalizer to a one-tap?lter in the frequency-domain.However,wireless terminals are expected to operate at high levels of mobility and capacity.The channel variations within an OFDM block due to high Doppler will destroy the subcarrier orthogonality and result in interchannel interference(ICI),causing severe performance degradation

Manuscript received December8,2004;revised May13,2005.This work was supported in part by the China National Science Foundation under Grant 50177001and60372007,the Ministry of Information Industry Foundation under Grant2002291.

J.Fu,C.-Y.Pan,and Z.-X.Yang are with the State Key Lab.on Microwave and Digital Communications,Electronics Engineering Department,Tsinghua University,100084Beijing,China(e-mail:fuj02@b3d4e8260066f5335a812110). L.Yang is with Legend Silicon Corp.,Fremont,CA94539USA(e-mail: lyang@b3d4e8260066f5335a812110).

Digital Object Identi?er10.1109/TBC.2005.852249[4]–[8].The channel fading under this circumstance becomes doubly(time and frequency)selective[4]–[6].

There has been a body of literature devoted to mitigate the ICI due to the channel variations.In[4],the authors reduce the com-plexity of linear minimum mean-squared error(MMSE)equal-ization by omitting the small ICI coef?cients within the equal-ization matrix.In[5],matched-?lter,least square,as well as MMSE equalizers incorporated with the decision feedback are presented.The authors of[6]combine methods of[4]and[5], and then derive a recursive algorithm to calculate the equalizer coef?cients.In[7],the authors construct a bank of LTV?lters to maximize the power ratio of the signal to the ICI-plus-noise in the multiple-antenna scenario.In[8],low-complexity equaliza-tion and channel acquisition schemes are presented based on a ?nite Taylor series expansion for the time-varying channel fre-quency response.All these approaches in[5]–[8]

take computational complexity

with equal to the OFDM symbol length,becoming impractical

when is large.

In this paper,a novel equalization method

with com-plexity is proposed for the time-domain synchronous OFDM (TDS-OFDM)system over a doubly selective channel.Time variations of a channel response are assumed linear during an OFDM block.We use a?nite power series expansion for the inverse of the equalization matrix as a satisfactory compromise between the system performance and the processing complexity. The effectiveness of this scheme is demonstrated by both anal-ysis and simulation.This paper is organized as follows.In Sec-tion II,doubly selective channel statistics as well as the system model of TDS-OFDM are described.In Section III,the low-complexity equalization method is proposed and theoretically analyzed.Before the?nal conclusions in Section V,results and discussions are presented in Section IV.

II.TDS-OFDM-B ASED DTTB S YSTEM

A.Doubly Selective Channel Statistics

A doubly selective channel can be characterized with the im-pulse

response

(is the sample interval

and,

with as the maximum delay).Due to the motion of the

vehicle,’s are wide-sense stationary (WSS)narrowband complex Gaussian process,which are in-dependent for different path.We assume

that has the same normalized correlation

function for all.Therefore, the correlation function

of for different sampling time and path

is

(1)

0018-9316/$20.00?2005IEEE

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Fig.1.Signal frame format for TDS-OFDM.

where the superscript denotes the complex conjugate

and is the average power of the -th path.The channel frequency

response at

time

and -th subcarrier

is (2)

Hence,the correlation function of the frequency response for

different sampling time and subcarrier

is

(3)

where is the total average power of the channel impulse response (CIR)de ?ned

as

(4)(5)

In (3),the correlation function

of is expressed as

the multiplication of a time-domain

correlation and a fre-quency-domain

correlation

.is determined by the ve-hicle speed or the Doppler frequency,

while depends on the multipath delay spread [9].From Jakes ’model

[10],

(6)

where is the zeroth-order Bessel function of the ?rst kind

and

is the Doppler frequency.B.System Model of TDS-OFDM

Fig.1shows the Frame architecture of TDS-OFDM.The Signal Frame consists of two parts,the Frame Head and the Frame Body.The Frame Head,as the guard interval,is com-posed of a Pre-amble,a pseudo-random number (PN)Sequence,and a Post-amble.The Pre-amble and Post-amble are cyclical extensions of the PN Sequence.A baseband TDS-OFDM system is shown in Fig.2without encoder,interleaver,de-inter-leaver,and decoder.Here we assume the ideal

synchronization.

Fig.2.

Baseband TDS-OFDM system.(a)Transmitter,(b)receiver.

At the transmitter side,each 3780of the symbols are grouped into one OFDM Frame Body.Time-domain signal is obtained through the inverse fast Fourier transform (IFFT)[11]block.PN Sequence with Pre-and Post-ambles,is inserted ahead of signal for the channel estimation.At the receiver side,the samples are fed in parallel to the channel estimator.A PN sequence,the same as the Frame Head,is generated by the local PN-sequence-generator to facilitate the channel estimation.The transmitted data can then be recovered by our proposed equalization method.

After removing the interference of the Frame Head to the Frame Body with the estimated CIR,TDS-OFDM signal is equivalent to the zero padded OFDM (ZP-OFDM)signal.By using the overlap and add (OLA)method,ZP-OFDM signal now becomes the cyclic pre ?xed OFDM (CP-OFDM)signal [12].Note that the step of removing Frame Head and OLA method is done with the imperfectly estimated CIR,causing cancellation errors after performing the above two-step oper-ation.These errors could be considered as another source of noise,making the overall noise level at the receiver slightly higher.Fortunately,this noise increase at the receiver can be ignored because (1)the duration of Frame Head is relatively short compared to that of Frame Body and (2)the precision of channel estimation is quite good for most signal-to-noise ratio (SNR)and Doppler range of our concern.

Let the Frame Body vector of an arbitrary OFDM block at the

transmitter

be

in frequency-domain,then the time-domain vector will

be

.At the

receiver,

and is the Frame Body vector before and after Frame Head removing as well as OLA operation,respec-tively.When taking into account of the time variation of the channel within each OFDM block,we have the output

vector

from the FFT in Fig.2as

follows

(7)

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here and is the frequency-domain Gaussian noise vector and with each and the -point FFT matrix, respectively. element are expressed as follows (8) elsewhere
(9) is the CIR during the In the above two questions, current (or -th) Frame Body interval at path and instant , , for i.e., , where and denotes the total Frame and the Frame Head length, respectively, that is, . Furthermore, (10) with the ICI term of [13]. III. EQUALIZATION A. Proposed Equalization Method We approximate the time variation of each path with a linear model that has a constant slope over an entire OFDM block, i.e., can be expressed as [4], [8] the (11) where , is the average of for path over the current ( -th) OFDM block . Let , -point FFT of and , respectively. Denote (15) with as as the spectrum norm of the matrix , de?ned
Fig. 3.
Diagram of the proposed equalization method for TDS-OFDM system.
We further change the expression of (12) into , and is the identity matrix, where which yields (14) The power series representation for the matrix inversion [14] shows that if , then
(16) is the standard norm of and the maximum is taken where over all the vectors of the same dimension as . Discarding the high-order components of the series, we have a -th order approximation as follows (17) Then, by substituting (17) into (14), we obtain the ICI-free data as follows, which is the key to the implementation of our method with very low complexity.
and be the
where is the diagonal matrix with diagonal vector . Substituting (11) into (9), we have (12) Therefore, the data after equalization can be expressed as (13) However, the direct implementation of requires the com, essentially impractical for a large . To solve plexity of this problem, we propose a novel equalization method with complexity based on a ?nite power series expansion for the inverse of the equalization matrix .
(18) Computational Complexity: Fig. 3 shows the implementation diagram of the proposed equalization method in (18) for TDS-OFDM systems. The computation cost of FFT or IFFT ; and are both diagonal matrices, therefore, only is complex multiplications are needed to calculate . In addition, is also diagonal, hence, the total complexity . In of our method according to (18) is and the overall computapractice, is much smaller than tional complexity of our method is .

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Channel Estimation: In order to construct the matrix equaand . In the tion in (12), it is necessary to estimate TDS-OFDM systems, each Signal Frame has a known PN Sequence inserted as the Frame Head. This PN sequence is used for the channel sounding in the time domain, allowing simple and . The Frame Head yet ef?cient estimation on duration is relatively small compared to the whole block length, which means the channel characteristics keep approximately constant for the Frame Head duration. By convolving the received PN sounding sequence with the known local PN, we can derive the estimated CIR at Frame Head [15], [16], denoted as . The estimation of and can be then easily obtained through the following linear operations
, which As mentioned earlier, the condition for (15) is may not always hold in the presence of Doppler effect. However, this happens very rarely for the Doppler frequency within the range of our concern and in worst case, results in the bit error rate (BER) of 1 for this OFDM block. As the probability of this scenario is extremely low, its contribution to the system BER performance is negligible and it doesn’t matter at all for being 1 or any large yet ?nite number. To facilitate our analysis by 1 just for without losing generality, we upper bound converge no matter what T is. It the purpose of making is seen that
(23) (19) (20) (24) where is the estimated CIR at current (or -th) Frame Head and is the estimated CIR at next (or -th) Frame Head. Residual ICI: To con?rm the ef?ciency of the proposed equalization method given in (18), we evaluate the residual average signal-to-interference ratio (SIR) after equalization, which is given by [5] (21) To simplify the derivation, we denote , and Equation (23) can be then rewritten as , . And therefore
(25) The auto and cross correlation coef?cients of following expressions and have the
(26) With as the estimated CIR at the Frame Head, can be obtained by various schemes. They are: the direct scheme (DS), our proposed scheme (PS), the conventional scheme (CS) [15] and the average scheme (AS) [16]. The direct scheme is based on (14), while our method with different parameter is based on (18). Note the AS is a special case of the PS with , and the conventional scheme assumes block invariance and as the estimated CIR for the current ( -th) OFDM uses block. Therefore, with the subscript of denotes the estimation methods described above, we have , , and , where is the -point FFT of . B. Approximation Error Analysis of the Proposed Equalization Method To evaluate the approximation error of the proposed method to the one with direct implementation of matrix inversion, we de?ne the -th order relative error as
(27)
(28) It is obvious that is linear combination of a Gaussian is also a Gaussian disdistributed vector, hence, tributed vector. Therefore, from (28), we have the observations that and are statistically independent [17]. In [18], channel coef?cients are assumed to be identical if within the same coherence subspace while independent if from one subspace to another. The coherent subspace is determined by both the corand a given threshold . For example, relation coef?cient and are approximately independent if while identical provided . According to the above observations and de?nitions, we obtain the following approximation and are independent are identical if if , (29) .
(22)
and

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This leads to the following for the cumulative density function of (CDF)
TABLE I 6-PATH HILLY TERRAIN CHANNEL PARAMETERS OF SIMULATIONS
TABLE II PARAMETERS OF THE TDS-OFDM SYSTEM
(30) where (31) and is the number of coherent subspaces expressed as (32) represents the count of cases of In the above equation, and is the nearest integer that is less than or equal to . Therefore, the approximate expression for the of is probability density function (PDF)
The upper bound for the expectation of as
(33) can be obtained
Fig. 4. Bound for E [ (Q)] versus f (Hz).
(34) Similarly, the upper bound for the variance of expressed as can be
IV. RESULTS AND DISCUSSIONS In this section, we demonstrate the performance of the proposed equalization method. The channel model for simulation is the COST 207 6-path typical hilly terrain channel [19] with the parameters listed in Table I. The TDS-OFDM system with parameters in Table II is used for simulations to verify the ef?ciency of the proposed equalization scheme. We ?rst give the approximation error analysis results based on . For the given COST 207 channel, (34) and (35) with and , Figs. 4 and 5 show the bounds and versus the Doppler frequency with of different , respectively. They both indicate that the bounds for and decrease with the decrease of yet with the increase of . It clearly illustrates that our proposal of
(35)

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Fig.5.Bound for D [

(Q )]versus f

(Hz).

Fig.6.BER versus SNR (dB);COST 207channel and f =20Hz .

using a ?nite power series for approximation is a viable scheme yet higher Doppler frequency always brings a more severe neg-ative impact on the accuracy of our method.Other interesting observations are:(1)there is a signi ?cant accuracy improvement

using our method

with

over .(2)the improvement becomes much less signi ?cant with the increase

of .These two observations leads to the conclusion that it would be accu-rate enough of

using

to characterize the time varying channel even

with

.Then we simulate the BER performances of the DS,the PS,the CS in [15]and the AS in [16],with the estimated CIR.Figs.6and 7show the BER performances versus SNR under various

schemes in the case

of

(the normalized Doppler frequency is 0.011,corresponding to the receiver velocity of 25to 46km/h in the TV UHF band @470to 862MHz)

and

,respectively.Fig.8is the BER performances of var-ious schemes versus Doppler frequency with a ?xed SNR of 40dB.From all these three ?gures,the DS always has the best

BER Fig.7.BER versus SNR (dB);COST 207channel and f =60Hz

.

Fig.8.BER versus f (Hz);COST 207channel and SNR =40(dB).

performance;our scheme achieves tremendous performance im-provements compared to the CS and AS under different channel situations.The advantage becomes even more obvious with the increase

of ,suggesting excellent capability of our scheme to combat with the Doppler Effect.

Finally,we measure the residual average SIR performance of our equalization method against other schemes and the results are given in Fig.9.It indicates that the performances for the PS and the DS are very close.Our scheme unsurprisingly of-fers a great SIR gain against the CS and AS.For instance,it can provide 22.5dB and 4dB gain in SIR over the CS with nor-malized Doppler frequency of 0.01and 0.2,respectively.These two numbers change to 10and 3dB if comparing our proposal against the AS.

The following observations can be drawn consistently from the analysis and simulation results given in Figs.4–9:(1)Our scheme is superior to both AS and CS because it can track the time variation within an OFDM block using linear interpolation.

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Fig.9.Residual average SIR versus normalized Doppler frequency.

Therefore,our method is less sensitive to the Doppler frequency.(2)With Doppler Frequency not too high,the performance gap between our method and the DS can be further narrowed with the

parameter increased.Hence,our method inherently pro-vides the tradeoff between the performance and processing com-plexity.(3)In most cases of our concern,

smaller ,

(i.e.,

and )is good enough to close the gap to the DS.This really pushes down the complexity effectively.(4)When the Doppler frequency is very high,all schemes will have their per-formances degraded with error-?oor emerging even though our method is hurt less severely.This is mainly because the assump-tion of the linear variation of CIR within a block is not valid any more [4].Therefore,new methods should be studied to address this issue,which could be of our future interest.

V .C ONCLUSION

Based on the ?nite power series expansion under the valid assumption that CIR changes linearly within an OFDM block,

a novel equalization method

with

complexity is proposed to mitigate the ICI in OFDM systems.Its effectiveness has been proven by both theoretical analysis and computer simulation.

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