Performance Analysis of Equalized OFDM
Systems in Rayleigh Fading
Ming-Xian Chang, and Yu T. Su, Member, IEEE
Abstract—Channel estimation is usually needed to compensate
for the amplitude and phase distortions associated with a received orthogonal frequency-division multiplexing (OFDM) waveform. This paper presents a systematic approach for analyzing the bit-error probability (BEP) of equalized OFDM signals in Rayleigh fading. Closed-form expressions for BEP performance of various signal constellations [phase-shift keying (PSK), differ-ential phase-shift keying (DPSSK), quaternary phase-shift keying (QPSK)] are provided for receivers that use a linear pilot-assisted channel estimate. We also derive the optimal linear channel esti-mates that yield the minimum BEP and show that some previous known results are special cases of our general formulae. The results obtained here can be applied to evaluate the performance of equalized single-carrier narrowband systems as well.
Index Terms—Error analysis, frequency division multiplexing,
gain control.
I. INTRODUCTION
O
RTHOGONAL frequency-division multiplexing(OFDM) is a promising candidate technique for high speed transmissions in a frequency-selective fading envi-ronment [1]. By converting a wideband signal into an array of properly-spaced narrowband signals for parallel transmission, each narrowband OFDM signal suffers from frequency-flat fading and, thus, needs only a one-tap equalizer to compensate for the corresponding multiplicative channel distortion.
One popular method to estimate the multiplicative channel response (CR) is to insert pilot symbols among transmitted data symbols [2]–[9]. Though many pilot-assisted channel estima-tion algorithms have been suggested, there still lacks a gen-eral analysis for the associated bit-error probability (BEP) per-formance. This paper presents a systematic approach for eval-uating the BEP performance of OFDM receivers in Rayleigh fading when a linear pilot-assisted channel estimate is used. These BEP expressions are functions of the average bit signal energy to noise level ratio (SNR), and some corre-lation coefficients that depend on the true channel statistic and the estimation method used. Our derivations are based on the
Manuscript received September 18, 2001; revised November 20, 2001; ac-cepted March 31, 2002. The editor coordinating the review of this paper and approving it for publication is Y.-C. Liang. This work was supported in part by the MOE Program of Excellence under Grant 89-E-FA06-2-4 and National Sci-ence Council of Taiwan under Grant NSC86-2221-E-009-058. Part of this work was presented at the IEEE VTC’2000, May 2000, Tokyo, Japan.
M.-X. Chang was with the Department of Communication Engineering, Na-tional Chiao Tung University, Hsinchu 30056, Taiwan. He is now with the De-partment of Electrical Engineering, National Cheng Kung University, Tainan, Taiwan (e-mail: mxc@ncku.edu.tw).
Y. T. Su is with the Department of Communication Engineering, National Chiao Tung University, Hsinchu 30056, Taiwan (e-mail: ytsu@cc.nctu.edu.tw).
Digital Object Identifier 10.1109/TWC.2002.804181
facts that Rayleigh fading is resulted from a zero-mean com-plex Gaussian CR process and a linear pilot-assisted channel estimate is a linear function of the true CRs at pilot symbol loca-tions. Using our general result, we are able to derive the optimal linear channel estimate that provides the best BEP performance. The rest of this paper is organized as follows. Section II pro-vides a mathematical model of an OFDM system in Rayleigh fading and gives a brief descriptions of some CR estimation al-gorithms. Section III contains our BEP analysis for phase-shift keying (PSK), differential PSK (DPSK) and quadrature ampli-tude modulation (QAM) constellations while Section IV deals with the BEP performance for general linear channel estimates. Section V gives some numerical examples based on our anal-ysis. Finally, Section VI summarizes our main results.
II. SYSTEMMODEL ANDCHANNELESTIMATEALGORITHMS
A. Signal and Channel Models
A baseband OFDM signal during the th symbol interval can be expressed by
(1)
where denotes the number of narrowband channels, , being a symbol duration and represents the data symbol of the th subchannel in the th time interval. The values assumes depend on the signal constellation format. For example, quaternary phase-shift keying (QPSK) is represented by the set while the rectangular
16-QAM is the set . The corresponding
sampled version of is equal to the inverse discrete Fourier transform (IDFT) of , i.e.
(2)
A guard time of s is usually inserted such that the resulting waveform during the th “extended” symbol interval becomes (3), shown at the bottom of the next page, where
. is further modified by a pulse shaping filter, yielding a transmitted baseband waveform that can be expressed as , where is of duration and unit power. The corresponding received baseband signal is then given by
(4)
Fig. 1. A typical pilot symbol distribution in the time-frequency plane of an OFDM system.r and r are the numbers of data symbols between two neighboring pilot symbols in the time and frequency domains, respectively. The parameter values for this distribution arer = 4 and r = 1.
where is the number of paths of the channel, and are the the delay and the complex fading envelope of the th
path and , and being
indepen-dent lowpass white Gaussian noise processes with the same flat spectral density (W/Hz).
is filtered by a matched filter and sampled at sam-ples/second. Removing those samples in guard intervals and performing DFT on , we obtain
(5)
where is a zero-mean
com-plex Gaussian random variable with independent in-phase and quadrature phase components and identical variance
. Furthermore
(6)
represents the corresponding channel effect (response). Equation (5) implies that, if is perfectly known, the maximum-likelihood (ML) receiver would select the signal point in the constellation that is closest to . When the true CR is not known, the receiver needs a CR estimate to make a symbol decision. A practical solu-tion, as mentioned before, is to insert pilot symbols at some predetermined (pilot) locations in the time-frequency plane (see Fig. 1), where the ( )th location denotes the th channel and the th time interval. Usually, a receiver would first estimate the CRs at pilot locations and then, based on this
information, obtain CR estimations at other (data symbols) locations through interpolation.
B. CR Estimates at Pilot Locations
1) The LS Method: One obvious CR estimate at a pilot
lo-cation ( ) is the least-squares (LS) estimate [2]
(7) where is the error term due to the presence of noise
whose conditional variance is
. The corresponding vector representation of (7) is given by
(8) where bold face lowercase letters are vectors of the corre-sponding terms in (7) and is a diagonal matrix with pilot symbols s as its diagonal elements. We shall use bold face uppercase and lowercase letters to denote matrices and vectors throughout our discussion.
2) Linear Minimum Mean-Squared Error (LMMSE) Method: A more elaborate method that is capable of reducing
the effect of is the LMMSE method [2]–[5]. Based on the estimated channel autocorrelation matrix and the noise variance , this method yields the following CR estimation vector at the pilot locations [3]
(9) where represents the Hermitian of .
C. CR Estimates at Data Locations
1) Polynomial Interpolation: After the CRs at pilot
loca-tions are obtained, either by the LS or LMMSE method, the CRs at data locations can then be estimated by various interpolation methods [5]. For example, if we have three pilot CR estimates, , and of a given subchannel, we need to the find the coefficients of the degree-2 polynomial
(10)
such that , and . The
coefficients ( ) can be obtained by solving the Lagrange interpolating polynomial
(11)
2) Model-Based Estimate: The CRs at data locations can
also be estimated by applying regression surface model to LS-estimated CRs of pilots [8], [9]. The model-based approach divides the time-frequency plane into blocks of the same basic
structure and finds a nonlinear two-dimensional surface that best fits the CRs of the pilot locations within a block. The noise effect of LS-estimated pilot CRs is greatly reduced by this solution. Furthermore, this approach does not need channel information such as the channel autocorrelation matrix and the noise level .
3) General Linear CR Estimate: Although estimating the
CRs at data locations may involve a nonlinear interpolating polynomial, e.g., (10), the derived data CR estimate, e.g., (11), is still a linear combinations of LS- or LMMSE-estimated pilot CRs. Since each LMMSE-estimated pilot CR is also a linear function of the LS-estimated pilot CRs [see (9)], the resulting data CR estimate is a linear function of the LS-estimated pilot CRs. This conclusion is also valid for the model-based CR estimate [8], [9] and other modified algorithms based on the LS or LMMSE method. In summary, for a general class of linear channel estimates, we can express the CR estimate as
(12)
where is the noise component in and
can be viewed as the channel estimate in the absence of noise. The weighting vector is a function of both the pilot CR esti-mation algorithm and the interpolation method.
III. BEP ANALYSIS
Throughout our analysis, we shall assume that the true CR, , is a bandpass wide sense stationary zero-mean complex Gaussian process so that at a given location, ( ), is Rayleigh distributed and is uniformly distributed within . We begin with BEP analysis for systems with a channel estimate that can also be modeled as a bandpass stationary zero-mean complex Gaussian process. Dropping the time and frequency indices for convenience, we have
(13) (14) where is the true CR and is the estimated CR. The equation , as can be seen from (12), decomposes into the noise-absent estimate and the noise component . The ad-ditive white Gaussian noise (AWGN) is complex Gaussian distributed with zero mean and variance . Be-cause is a function of noise samples at pilot locations while is the noise component at a data symbol location, is inde-pendent of . On the other hand, is a linear combination of LS-estimated CRs, which, as can be seen from (7), are complex Gaussian distributed, it is also complex Gaussian distributed. We, therefore, conclude that , , and are all Gaussian
distributed, and are Rayleigh distributed and and are uniformly distributed in .
Invoking the general Rayleigh fading process assumption, , are independent and identically distributed (i.i.d.) processes with
(15) we can show that , are also i.i.d. processes (see Appendix A) with
(16) (17)
Re (18)
Im (19)
The correlation coefficients and average SNR per bit are de-fined, respectively, as
(20)
(21) where denotes the number of bits represented by one symbol.
For example, for QPSK, for 16-QAM.
Note that (18) and (19) imply
(22) and for a reasonably good estimate, i.e., , (22) then leads
to and ( ). Furthermore, the
mean-squared estimation error (MSEE) is
(23) which is independent of ( ). However, as we will see shortly that the BEP performance of systems with imperfect channel estimate does indeed depend on ( ). Hence, the MSEE, as a performance measure, is not necessarily proper in general. An exception can be found in single-carrier narrowband channels based on Jakes or similar models in which we have
; see (66).
The joint probability density function (pdf) of ( ) [11] is given by (24), shown at the bottom of the page. Trans-forming the rectangular coordinate ( ) into the
polar coordinate ( ) and making the changes of
variables, , we have in (25).
(25)
The parameters , , and in (25) are functions of the channel statistic and the CR estimation method used. More detailed expressions for them will be given in Section IV-A.
Expressing an equalized symbol as
(26)
we notice that is a zero-mean complex Gaussian random variable whose variance is the same as that of . It can be shown that is also independent of even though the phase-shift and are correlated. Using (13) and (14), we can rewrite (26) as
(27)
A. Binary Phase Shift Keying (BPSK) and QPSK Systems
Coherent BPSK and QPSK receivers base their symbol de-cisions solely on the phase location of the associated matched filter output. Equation (27) indicates that such decisions are in-dependent of . Their respective BEPs conditioned on a fixed phase error and fading envelope are given in [15, 10.14a]
(28) for BPSK and
(29)
for QPSK, where . We can express
the (unconditional) BEPs in a Rayleigh fading channel as
(30) Substituting (25), (28), and (29) into (30) and using the integral formula (B.1) of Appendix B, we obtain
(31)
for an OFDM-BPSK system and
(32)
for an OFDM-QPSK system. In the above two equations, (31) and (32)
(33)
is the average SNR per bit for both BPSK and QPSK systems, as can be seen from (21).
B. DPSK System (Without Channel Equalization)
The need of channel equalization can be avoided by using DPSK modulation if the difference of channel-induced phase rotation remains relatively small during two or more symbol intervals. The performance of a narrowband DPSK or an OFDM-DPSK system can also be analyzed by the same method presented in the above subsection. Neglecting the subscript denoting the channel number and leaving only the time index, we have
(34) where the phase of contains the data information of a DPSK signal. The parameter now represents the phase differ-ence between and the pseudo CR estimate
. Comparing (34) with (26) and assuming a Jakes Rayleigh channel [14], we obtain, from the definitions given by (15)–(21), (35) (36) (37) (38) (39)
Fig. 2. Constellation of Gray-coded rectangular 16-QAM and the corresponding I-Q bit mapping.
where denotes the complex conjugate of . Substi-tuting and into (31) and (32) we have, for differential
BPSK ( ),
(40) which is the same as (13-34) of [13] and, for differential QPSK
( )
(41) Note that for slow fading (zero or very small Doppler) channels,
, (40) then becomes
(42) while (41) is reduced to
(43) which coincides with (50b) of [16].
C. QAM Systems
This subsection gives the BEP performance analysis of the rectangular 16-QAM constellation with gray-bit mapping shown in Fig. 2. The performance of other QAM constellations, like rectangular 64-QAM or 128-QAM, can be derived in a similar manner. Unlike PSK or DPSK signals, the performance
of QAM systems does depend on the the distribution of .
A 16-QAM symbol can be written as
, where , . From (27), we have
the equalized data symbol
(44) where . If the transmitted symbol is, say,
, then the corresponding bit pattern is 0001 with and components being 00 and 01, respectively. The conditional BEPs for the component channels are thus, given by
(45)
(46) The above equations, (44)–(46) lead to
where . We can obtain the conditional BEPs of other transmitted bit patterns
in a similar manner and the overall conditional BEP is
(48) where each term on the right hand side is either of the form
(49) or
(50)
with ,
. As in (30), the BEP is given by
(51) For the rectangular 16-QAM, the average SNR per bit defined by (21) becomes
(52) Substituting (25), (48), and (52) into (51) and using (B.1) in Appendix B, we obtain (53) shown at the bottom of the page,
where the associated coefficients (sgn )
are listed in in Table I.
The BEP expression of the 16-QAM system in flat Rayleigh fading derived in [10] involves a few finite-range integrals. Using the above equation with , , we can obtain a closed-form expression instead.
IV. PERFORMANCE OFGENERALLINEARCHANNELESTIMATES As discussed in Section II-B, a linear pilot-assisted CR esti-mate can be written as
(54)
TABLE I
COEFFICIENTSASSOCIATEDWITHVARIOUSTERMS IN(53)
where is the vector contains the true CRs at pilot locations and is the weighting vector that depends on the CR estimation method.
A. Procedure for Computing BEP
The general procedure for evaluating the BEP of an OFDM receiver with a linear channel estimate algorithms can be de-scribed as follows.
1) Derive the weighting vector from the CR estimate used. 2) Evaluate the following parameters:
(55) (56) (57) Re (58) sgn (53)
Im
(59)
where and .
3) Substitute , , and into a proper BEP formula. To illustrate how the weighting vector is obtained, we con-sider the OFDM-16QAM system. The receiver first estimates the pilot CRs by the LMMSE method (9) then compute the CR of a data location by using a polynomial interpolation, i.e.,
(60)
where is the vector of the interpolating coefficients. In-voking the Lagrangian interpolation formula and equating (60) and (11), we obtain
(61)
Therefore, the final weighting vector for the CR estimate is
(62)
B. Channel Correlation Functions
To evaluate the parameters, , , , , we need to know channel correlation functions like and . We consider two classes of Rayleigh channels based on Jakes model [14]. The first class has a maximum Doppler frequency and an exponentially distributed delay profile with mean delay and channel correlation
(63)
where is the Bessel function of the first kind of order zero. The second class assumes an -tap impulse response
(64)
where and are independent stationary
com-plex zero-mean Gaussian processes with unit variance and is the relative delay of the th cluster. The discrete version of
this model is equivalent to a special case of the frequency-do-main model of (6). The resulting channel correlation function is given by
(65) For a single-carrier narrowband channel, both (63) and (65) become
(66)
C. Optimal Linear CR Estimate
As has been shown before, the problem of finding the optimal linear CR estimate is equivalent to finding a weighting vector that minimizes the BEP performance. Appendix C shows that, for BPSK signals, the optimal weighting vector is given by (67) where can be any nonzero real number. It is straight-forward to show that the same conclusion holds for QPSK signals. For QAM signals, the corresponding has the same form as (67) but is a constant that depends on the system and channel parameters. Numerical experiment has revealed that is in the vicinity of 1 and if
.
In summary, the optimal linear CR estimate has the form of an LMMSE estimate that combines LS-estimated CRs at pilot locations by using the optimal weighting vector given by (67). The complexity of this optimal estimate is very high since we need to estimate for each location.
D. Perfect Channel Estimation
When the perfect channel estimation is available, we have
, , , and the resulting BEP
performances can serve as the lower bound for the performance of various OFDM systems in Rayleigh fading channels.
1) PSK Constellation: For this ideal case, both (31) and (32)
reduce to the same BEP expression
(68)
which coincides with the well-known result given by (14-3-7) of [11].
2) QAM Constellation: As for the rectangular 16-QAM
system, setting , , and in (53) yields
Fig. 3. BEP performance of differential BPSK and QPSK systems as functions of and f T .
Another way to obtain this lower bound is by noticing that, as a result of , (26) becomes
(70) and the corresponding BEP becomes
(71) where is the BEP of the rectangular 16-QAM system in AWGN [15]
(72)
Substituting (72) into (71) gives (69).
3) Asymptotic Performance: With perfect CR information,
the asymptotic BEP behaviors of various constellations con-verge to known results. For both BPSK and QPSK systems, (68)
indicates that for , while for DPSK,
, according to (42) and (43). This is consistent with (14-3-13) of [11]. In the case of the rectangular 16-QAM
system, (69) reveals that when is large
enough.
V. NUMERICALEXAMPLES
This section presents some application examples of our BEP analysis. We first examine cases (Figs. 3–4) in which the BEP performance depends explicitly on some system or channel pa-rameters, say or . Then, we consider the other cases when we must resort to the procedure given in Section IV-A in order to calculate the BEP performance.
Using (40) and (41), Fig. 3 shows the BEP performance of DPSK signals in Rayleigh fading, parameterized by , the normalized Doppler shift. Letting in (40) and (41),
Fig. 4. BEP performance of an OFDM-16QAM system for various values of
; = 0 is assumed.
we find that the corresponding BEPs approach a constant that depends on . The constants correspond to different are the error floors in the figure. The behavior of the error floor level as a function of is predictable as is monotonic decreasing in the neighborhood of . The fact that the dif-ference of the error floor levels of differential BPSK and QPSK systems is an increasing function of can also be predicted by (40) and (41).
The BEP behavior of an OFDM-16QAM system as a function
of and is plotted in Fig. 4, assuming and .
When , represents the correlation between the true CR and estimated CR, higher not only results in smaller mean squared estimation error [see (23)] but smaller BEP as well. In a time-varying environment, smaller or may be caused by higher fading rate, less frequent channel estimation (i.e., larger estimation period) and/or smaller pilot density, which, as ex-pected, leads to higher BEP floor.
Now consider an OFDM-16QAM system whose receiver uses the LMMSE method to estimate the pilot CRs and then obtains the CRs of the data locations by a polynomial interpo-lation. We consider the time-varying multipath Rayleigh-fading channel based on (6). The parameter values used are ,
and . All s are independent
stationary complex zero-mean Gaussian processes with unit variance while the relative path strengths and delays are , 0.7305, 0.3175, 0.1137, and , 0.1, 0.5, 1 s, respectively. Assuming this receiver knows the correlation matrix perfectly but not , we depict the BEP performance in Fig. 5 where denotes the estimated and five pilot symbols are uniformly inserted in each block with four data symbols between two neighboring pilot symbols. These curves indicate that overestimating causes negligible degradation at smaller s while underestimation leads to much greater degradation when the true is much larger. It is, therefore, preferred to assume a larger than the designed operating . Fig. 6 illustrates the impact of the frequency domain CR correlation on OFDM-QPSK system explicitly. Let be the pilot symbol vector at
Fig. 5. BEP performance of OFDM-16QAM systems using the LMMSE method and polynomial interpolation; = the estimated and
f T = 2:76 1 10 .
Fig. 6. BEP performance of the OFDM-QPSK system for different normalized mean channel delay =T which affects the channel correlation function via (73).
the th symbol interval. The correlation values for various components of can be obtained from (63) by setting
(73)
The receiver first estimates the pilot CRs by the LMMSE method, then interpolates the CRs at data locations
( ) by a polynomial. Fig. 6 plots the BEP
performance for different s, assuming . Smaller normalized mean channel delay yields larger CR corre-lation and smaller channel variation in frequency domain, thus reduce the channel estimation error and improve the system performance.
The BEP performance of the model-based estimate proposed in [9], with the same delay profile as the example in Fig. 5, can also be derived from our analysis. We present some typical per-formance in Fig. 7, where , are defined in Fig. 1 and
Fig. 7. BEP performance of the OFDM-16QAM system that uses the model-based channel estimate. The parameters (N ; N ; r ; r ) are defined in Fig. 1 and Section V,f T = 2:76 1 10 .
are the numbers of pilot symbols in the time and frequency do-mains per time-frequency block. The 4-tuple ( ) thus completely specify the pilot density and the channel es-timation period. As expected, increasing pilot density does in-deed enhance the system BEP performance. Simulation results are also given to validate the accuracy of our analysis.
VI. CONCLUSION
In this work, we have presented a unified approach for ana-lyzing the BEP performance of various OFDM systems using an arbitrary linear pilot-assisted CR estimate. Our analysis can be applied to estimate the BEP behavior of various signal con-stellation like QAM, PSK (BPSK, QPSK), and DPSK as long as the fading process can be modeled as a bandpass stationary zero-mean complex Gaussian process. Some of the previous known results become special cases of our general analysis, i.e., (40), (42), (43), and (68). The correctness of our analysis is ver-ified by the fact that the system performance predicted by our analysis and computer simulation yields almost the same result. These closed-form BEP formulae enable us to derive the op-timal linear CR estimate and to easily predict the influence of both the channel statistics and the CR estimation method on the system performance.
Finally, we notice that fast channel fading may cause inter-channel interference (ICI) amongst subinter-channels. By modeling the ICI as an equivalent Gaussian noise, , we can also obtain an estimation of the BEP performance by adding an equivalent variance term, , the variance of the ICI, to the corresponding BEP expression, if the variance can be evaluated.
APPENDIX A
CROSS-CORRELATIONPROPERTIES OFESTIMATEDCRS From (12), we see that the CR estimate is a linear combination of the true CRs at pilot locations plus an independent complex Gaussian noise i.e.
where is the vector of the true CRs at pilot locations,
is the weighted vector. In this appendix, for simplicity, we use single letter subscripts to indicate the locations of pilot symbols. The real and imaginary parts of are denoted by and , respectively. The real and imaginary parts of
in (A.1) can be decomposed as and
. The stationarity assumption of the true CR process implies [12]
where we denote the ( )th entry of a matrix by .
Therefore, we have and
, which implies
(A.2) and
Hence
(A.3) Equations (A.2) and (A.3) lead to
(A.4) Because and are independent, we have
(A.5) (A.6) (A.7) (A.8)
APPENDIX B
DERIVATION OF ANINTEGRALIDENTITY We want to establish the identity
(B.1)
where is defined by (25) and . The
change of variable, , , 2, on the first
line of (B.1) yields
(B.2)
The changes of variables, ,
and definition , give
(B.3)
If we define and ,
(B.4) We now further simplify the two integrals, and .
sgn
sgn
(B.5)
Using the identity , we obtain
sgn
sgn
(B.6)
where the sign function sgn is defined by
sgn (B.7)
Substituting (B.5) and (B.6) into (B.4), we then obtain the de-sired equation.
APPENDIX C
OPTIMALLINEARCR ESTIMATE
Consider an arbitrary linear pilot-assisted CR estimate of the form (12). To find the optimal CR estimate is equivalent to find the weighting vector that minimizes the BEP performance. We shall only derive the optimal for the BPSK system in this appendix; those for QPSK and QAM signal constellations can be obtained by analogous procedures as we can see that the BEP expressions for BPSK, QPSK, and QAM systems, (31), (32), and (53), all have similar forms.
Rewriting (31) as
(C.1)
taking derivative with respect to and setting it to zero, we obtain (C.2) Substituting Re Im into (C.2), we have (C.3) which implies (C.4) where is a scalar to be determined. Substituting (C.4) into
(C.3), we have , or , hence,
can be any nonzero real number. The same conclusion holds for OFDM-QPSK signals. As for OFDM-QAM signals, replacing (C.1) with (53) and following a similar approach, we find that the solution of (C.2) is still of the form of (C.4), but now
depends on other parameters like Doppler frequency, channel delay spread and . Numerical investigation indicates that is usually in the vicinity of one.
REFERENCES
[1] J. A. C. Binghama, “Muticarrier modulation for data transmission: An idea whose time has come,” IEEE Commun. Mag., pp. 5–14, May 1990. [2] J.-J. van de Beek, O. Edfors, M. Sandell, and S. K. Wilson, “On channel estimation in OFDM systems,” in Proc. 45th IEEE Vehicular Technology
Conf., Chicago, IL, July 1995, pp. 815–819.
[3] O. Edfors, M. Sandell, and J.-J. van de Beek et al., “OFDM channel estimation by sigular value decomposition,” IEEE Trans. Commun., vol. 46, pp. 931–939, July 1998.
[4] Y.(G.) Li and L. J. Cimini et al., “Robust channel estimation for OFDM systems with rapid diversity fading channels,” IEEE Trans. Commun., vol. 46, pp. 902–915, July 1998.
[5] M. Hsieh and C. Wei, “Channel estimation for OFDM systems based on comb-type pilot arrangement in frequency selective fading channels,”
IEEE Trans. Consumer Electron., vol. COM-46, pp. 931–939, July 1998.
[6] Y. Zhao and A. Huang, “A novel channel estimation method for OFDM mobile communication systems based on pilot signals and transform domain processing,” in Proc. IEEE 47th Vehicular Technology Conf., Phoenix, AZ, May 1997, pp. 2089–2093.
[7] V. Mignone and A. Morello, “CD3-OFDM: A novel demodulation scheme for fixed AHD mobile receivers,” IEEE Trans. Commun., vol. 44, pp. 1144–1151, Sept. 1996.
[8] M.-X. Chang and Y. T. Su, “2-D regression channel estimation for equal-izing OFDM signals,” in Proc IEEE 5th Vehicular Technology Conf., Tokyo, May 2000, pp. 240–244.
[9] , “Model-based channel estimation for equalizing OFDM signals,”
IEEE Trans. Commun., vol. 50, pp. 540–544, Apr. 2002.
[10] X. Tang, M. Alouini, and A. K. Goldsmith, “Effect of channel estimation error on M-QAM BER performance in Rayleigh fading,” IEEE Trans.
Commun., vol. 47, pp. 1856–1864, Dec. 1999.
[11] J. G. Proakis, Digital Communications, 3rd ed. Englewood Cliffs, NJ: Prentice-Hall, 1995.
[12] C. W. Therrien, Discrete Random Signals and Statistical Signal
Pro-cessing. Englewood Cliffs, NJ: Prentice-Hall, 1992.
[13] W. C. Y. Lee, Mobile Communications Engineering, 2nd ed. New York: McGraw-Hill, 1998.
[14] W. C. Jakes, Microwave Mobile Communications. New York: Wiley, 1974.
[15] M. K. Simon, S. M. Hinedi, and W. C. Lindsey, Digital Communication
Techniques. Englewood Cliffs, NJ: Prentice-Hall, 1995.
[16] L. E. Miller, “BER expressions for differentially detected=4 DQPSK,”
IEEE Trans. Commun., vol. 46, pp. 71–78, Jan. 1998.
Ming-Xian Chang received the B.S. degree in
elec-trical engineering from National Taiwan University, Taipei, Taiwan, in 1995, and the M.S. and Ph.D. de-grees in communication engineering from National Chiao Tung University, Hsinchu, Taiwan, in 1997 and 2002, respectively.
He is currently an Assistant Professor in the Department of Electrical Engineering, National Cheng Kung University, Tainan, Taiwan. His current research interests include multicarrier communica-tion systems, blind estimacommunica-tion, and coding theory.
Yu T. Su (S’81–M’83) received the B.S. degree
from Tatung Institute of Technology, Taipei, Taiwan, in 1974, and the M.S. and Ph.D. degrees from the University of Southern California, Los Angeles, in 1983, all in electrical engineering.
From 1983 to 1989, he was with LinCom Corpo-ration, Los Angeles, where he was involved in the design of various measurement and digital satellite communication systems. Since September 1989, he has been with National Chiao-Tung University, Hsinchu, Taiwan, where he is presently Head of the Department of Communication Engineering. He is also affiliated with the Microelectronics and Information Systems Research Center of the same uni-versity and served as a Deputy Director from 1997 to 2000. His main research interests include communication theory and statistical signal processing.
