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行政院國家科學委員會補助專題研究計畫
√ 成 果 報 告
□期中進度報告
（計畫名稱）
計畫類別：
√
個別型計畫 □ 整合型計畫
計畫編號：NSC 96－2221-E－009－053
執行期間： 2007 年 8 月 1 日至 2008 年 10 月 31
日
計畫主持人：Carrson C. Fung
共同主持人：
計畫參與人員：
成果報告類型(依經費核定清單規定繳交)：□精簡報告
√
完整報告
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□涉及專利或其他智慧財產權，□一年□二年後可公開查詢
執行單位：Dept. of Electronics Engineering, National Chiao Tung University
中 華 民 國 2008 年 10 月 31 日
□ 可申請專利 □ 可技術移轉
日期： 年 月 日
計畫名稱：
國科會補助計畫
計畫主持人：
計畫編號： 學門領域：
技術/創作名稱
發明人/創作人
中文：
（100~500 字）
技術說明
英文：
可利用之產業
及
可開發之產品
技術特點
推廣及運用的價值
1.每項研發成果請填寫一式二份，一份隨成果報告送繳本會，一份送 貴單位研
發成果推廣單位（如技術移轉中心）
。
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3.本表若不敷使用，請自行影印使用。
Space-Time Precoder-Equalizer System
Design for Correlated FIR-MIMO Channel
Principal Investigator: Carrson C. Fung
Project Number: NSC-96-2221-E-009-053
Effective Date: August 1, 2007 to October 31, 2008
Abstract
In this project, a SOS-based precoder-assisted blind equalizer for MIMO-OFDM system which can equalize correlated FIR-MIMO channel at the receiver is proposed. The proposed scheme uses a set of orthogonal precoders at the transmitter such that the transmitted signal can be colored temporally thereby satisfying the identifiability condition previously proposed by Hua and Tugnait. The scheme is shown to outperform previously proposed schemes in terms of BER and computational complexity. Simulation results have shown that the BER performance is close to that of a least-squares zero-forcing equalizer with perfect channel knowledge.
I. INTRODUCTION
With the insatiable need to transmit more data at any time and anywhere, next generation wireless com-munication systems, such as IEEE 802.11n and 802.16m, are required to support higher data rate than their predecessors, while the mobile terminals are undergoing faster mobility. It is envisioned that MIMO-OFDM will be used for these systems in order to increase link reliability and capacity. However, large transmission overhead in the form preamble, signal pilot and guard interval severely hamper the performance of such systems. This has made blind channel estimation and equalization techniques for MIMO-OFDM systems an attractive alternative.
Traditionally, blind channel estimation and equalization has been based on higher-order statistics (HOS) [1], [2]. However, much of the research effort has since shifted toward using second-order statistics (SOS) after the seminal work by [3] and [4] since SOS based techniques can also estimate and equalize FIR channels at much lower latency than its HOS counterparts. This has led to the work by [5] which exploited the subspace method to estimate FIR-SIMO channels. Good performance in terms of mean squared error (MSE) can be achieved in high SNR condition. However, its performance degrades at a fast rate in low SNR condition, such
as 0 − 10 dB. An FIR-MIMO extension of the subspace method was proposed in [6], [7] which suffers from
the same problem as its SIMO counterpart in low SNR condition. The subspace channel estimation method requires the channel transfer function matrix,H(z), to be irreducible and column-reduced [13], which limited the application of SOS based methods to a narrow class of communication channels.
Recently, [8] has shown that a weaker condition for the identifiability of H(z) exists, where H(z) can be
identified up to scaling and permutation ambiguity ifH(z) is irreducible and the power spectral density matrix of the channel input signal is a diagonal matrix with distinct diagonal functions. [9] has proposed an algorithm
for estimating H(z) under this weaker condition, but an direct equalization algorithm was never discussed.
[10] has proposed a SOS based blind equalization algorithm which implicitly uses the identifiability conditions stated in [8] for flat fading channels, where the number of transmit antennas, Nt, has to be equal to the number
of receive antennas, Nr. [11] also exploited this condition by designing a novel SOS based channel estimation algorithm to estimate MIMO channels for OFDM based systems. The algorithm uses cyclic power spectral density of the received signal to decouple the MIMO channels into parallel SISO channels for estimation. The technique requires the use of a precoder to inject cyclostationarity into the input bitstream. Although not stated in [11], but the precoder actually colors the signal such that MIMO channel equalization using SOS is possible.
[12] has extended the SOS algorithm in [10] such that any FIR-MIMO channelH(z) can be equalized up to
a scaling, phase, and block delay ambiguity given that the identifiability conditions in [8] are satisfied. This was accomplished by designing the blind FIR equalizer within the space-time precoder-equalizer system where redundancy is injected into the transmitted bitstream to make FIR-MIMO channel equalization possible using an FIR equalizer. In [12], the independently distributed input signal streams were colored using a set of low complexity filters to satisfy the power spectral density condition stated in [8] such that the algorithm in [10] can be extended to be applicable to ISI channels. However, the precoder that was proposed was not optimally designed. As shown in the sequel, this not only impacts the equalization performance, but also increases the
computational complexity at the receiver.
In this project, we proposed a new set of precoder to perform direct channel equalization for MIMO-OFDM systems such that improved BER performance and lower receiver complexity can be achieved compared to that of [12], even when the spatial correlation exists. In Section II, we will give a description of the system model, followed by a review of the equalization algorithm in Section III-A. We will then propose a novel precoder design in Section III-B. Simulation results are given in Section IV followed by the conclusion in Section ??.
Notation: Upper (lower) bold face letters indicate matrices (column vectors). SuperscriptHdenotes Hermitian,
T denotes transposition. E[·] stands for expectation. diag(x) denotes a diagonal matrix with x on its main
diagonal;IN denotes an N× N identity matrix; 0M×N denotes an M× N all zero matrix.
II. SYSTEMMODEL
We consider a MIMO-OFDM system with Nttransmit antennas and Nrreceive antennas. Let s(i)m, denotes
the complex-valued data symbol transmitted on the mth tone in the thOFDM symbol from the ithantenna
for i= 1, 2, . . . , Nt. Also, let K = M + v denote the overall OFDM symbol length, where M is the size of
the FFT and v is the length of the cyclic prefix. Then the transmitted signal ui[n] can be written as [11]
ui[n] = g[n − K] M−1 m=0 s(i)m,ej2πMm(n−K)
where g[n] is a rectangular function rect[0,K−1][n] with rect[T1,T2][n] = ⎧ ⎨ ⎩ 1, n = T1, T1+ 1, . . . , T2 0, otherwise. Then the received signal at the kth receive antenna can be written as
xk[n] = Nt i=1 hk,i[]ui[n − ] + ηk[n] (1)
where ηk[n], for k = 1, 2, . . . , Nr, is the stationary additive white channel noise at the kthreceive antenna and
hk,i[] is the discrete-time impulse response of the channel. Defining
x[n]=[ x1[n] x2[n] · · · xNr[n] ]T,
u[n]=[ u1[n] u2[n] · · · uNt[n] ]T,
η[n]=[ η1[n] η2[n] · · · ηNr[n] ]T
as the receive signal vector, transmit signal vector and the channel noise vector, respectively, then (1) can be written as
x[n] =
Hu[n − ] + η[n] (2)
where[H]k,i = hk,i[] is the Nr× Nt channel matrix of order q, that is,
H(z) =
q
=0
Hz−
is the channel transfer function matrix.
H ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ H0 H1 · · · Hq 0 · · · · 0 0 H0 H1 · · · Hq 0 · · · 0 .. . ... ... ... ... ... ... ... 0 · · · · 0 H0 H1 · · · Hq ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , and ˇx[n] xT[n] xT[n − 1] · · · xT[n − L + 1]T, ˇu[n] uT[n] uT[n − 1] · · · uT[n − L − q + 1]T. ˇη[n] ηT[n] ηT[n − 1] · · · ηT[n − L + 1]T.
as the received signal vector, transmitted signal vector and noise vector, respectively, then (2) can be written as ˇx[n] = Hˇu[n] + ˇη[n].
III. SOS BASEDPRECODER-ASSISTEDBLINDEQUALIZERDESIGN
A. Methodology of Equalization
To satisfy the blind identification conditions in [8], the transmitted signal vector ˇu(n) are assumed to be spatially uncorrelated but temporally correlated with distinct power. Without loss of generality,ˇu(n) can assume to have unit variance and zero mean. Define the correlation matrix of ˇu[n] as Rˇuˇu(τ) Eˇu[n]ˇuH[n + τ],
thenRˇuˇu(0) = INtL. The autocorrelation matrix of ˇu(n) can be expressed as
Rˇuˇu(τ) = Eˇu[n]ˇuH[n + τ]= = ⎧ ⎨ ⎩ INt(L+q), for τ = 0, diagρ1(τ), . . . , ρNt(L+q)(τ), for τ = 0,
where ρ1(τ) = · · · = ρNt(L+q)(τ) = 0. We further assume that ˇη[n] is white Gaussian distributed and is
mutually uncorrelated with ˇu[n]. Then the autocorrelation matrix of the channel output ˇx(n) can be written as Rˇxˇx(τ) = ⎧ ⎨ ⎩ HRˇuˇu(0)HH+ ση ˇ2ˇηINrL, for τ = 0, HRˇuˇu(τ)HH, for τ = 0, (3) where σ2 ˇ
ηˇη is the variance of the noise signal ˇη[n]. Defining ˇv[n] = Hˇu[n] as the channel output vector and
Rˇvˇv(τ) Eˇv[n]ˇv[n + τ]H.
Since Rˇuˇu(0) = INt(L+q), therefore
Rˇvˇv(0) = HHH.
LetW be a whitening matrix that whitens ˇv[n] such that
EWˇv[n]ˇvH[n]WH= INrL,
where
W = Σ−12
[ ]
1x n
(0)
xxR
1 2=
− H x xW
Σ Q
x
( )
τ
j zzR
JD
U estimation
[ ]
2x n
z
y
Fig. 1. Block diagram of equalization process with 2 receive antennas where JD denotes joint diagonalization.
withΣˇvbeing the square root inverse of the eigenvalue matrix ofRˇvˇv(0), and Qˇvbeing the eigenvector matrix
ofRˇvˇv(0). Then we can obtain
WRˇvˇv(0)WH=EWˇv[n]ˇvH[n]WH
=WHHHWH
=INrL (4)
According to (4), the effective channelU = WH is a unitary matrix. Applying W to the received signal vector
ˇx[n], we can obtain
ˇz[n]=Wˇx[n]
=W [Hˇu[n] + ˇη[n]]
=Uˇu[n] + Wˇη[n] (5)
From (5), we see thatU can be equalized by
U−1ˇz[n] = UHˇz[n] = ˇu[n] + UHWˇη[n] (6)
From (6), the problem of equalization becomes finding the unitary equalization matrix of U. Defining the
correlation matrices for ˇz[n] and ˇη[n] as Eˇz[n]ˇzH[n + τ] and Eˇη[n]ˇηH[n + τ], respectively. From (3),
Eˇη[n]ˇηH[n + τ]= 0, for τ = 0. Thus, the correlation matrix of ˇz[n] can be written as
Rˇzˇz(τ) = URˇuˇu(τ)UH, for τ = 0. (7)
Thus, the equalizer U can be obtained by diagonalizing Rˇzˇz(τ). According to [12], we can find U that
equalizes frequency-selective channels if the source signal has different spectral energy. In addition, the chance of eigenvalue degeneracy can also be reduced by performing a joint diagonalization on a set ofRˇzˇz(τ) with
various τ = 0, i.e.
UHR
ˇzˇz(τp)U = diagρ1(τp), ρ2(τp), . . . , ρNt(L+q)(τp),
Ant 1 IFFT P/S S/P IFFT P/S S/P Ant Nt ( ) v v M v M × − ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ 0 I I ( ) v v M v M × − ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ 0 I I CP insertion CP insertion Coloring precoders , i m α β,in ISI channel H Blind Equalizer %1( )n η %2( )n η % ( ) r N n η S/P S/P FFT P/S FFT P/S CP deletion [0MxvIM] CP deletion [0MxvIM] Decoloring decoders , 1 i m α , 1 i n β
Fig. 2. MIMO-OFDM precoder-assisted blind equalizer system withNt/Nr Tx/Rx antennas.
where τ0, τ1, . . . , τP are non-zero time lags. The overall joint diagonalization equalization process is illustrated
in Figure 1.
B. Precoder Design
If all the source data streams are uncorrelated, the required temporal correlation property can be easily achieved by shaping the power spectral density of each data stream [8]. [12] proposed to use a set of low complexity filters to color the source signal stream such that FIR blind equalization is possible at the receiver to equalize FIR-MIMO channels. However, the proposed filters were chosen arbitrarily without regards on its effects on BER performance. Moreover, no investigation was carried out about how the precoder can be used to reduce computational complexity at the receiver while sustaining equalization performance. In this paper, a new set of precoders are proposed that will allow us to select a subset of{Rˇzˇz(τp)} such that it not only
reduces the computational complexity at the receiver, but it also does not impact the equalization performance compare to the case when the full set of autocorrelation matrices are used. As seen in Figure 2, the precoders
are applied in the frequency domain (prior to IFFT) to all Nt transmit antennas of a MIMO-OFDM system.
The set of coloring precoders are denoted as{P0(z), P1(z), . . . , PNt−1(z)}, where
Pi(z) = diag (αi,0, αi,1, . . . , αi,M−1) for i = 1, 2, . . . , Nt.
αi,m is the multiplier coefficient of the mth path of IFFT for the ith transmit antenna as illustrated in Figure
2. A scaling matrix is then applied in the time domain (after the IFFT), which is given as Si(z) = diag (βi,0, βi,1, . . . , βi,M−1) , for i = 1, 2, . . . , Nt,
where βi,n is a scaling factor for satisfying the condition of distinct power. At the receiver, the inverse
manipulation of the transmitters is used to decolor the colored signal. The proposed real-valued multiplier αi,m is formed with two parts. The first part generates the orthogonality among different precoders, and the second part introduces temporal correlation to the transmitted signal. Since the performance of the joint diagonalization algorithm is based on spectral overlap of the source signals [10], this led to the use of orthogonal precoders.
αi,m can be expressed as αi,m = Oi(m) 1 −P −1 p=0 Ci,τpcos 2πmτp M (9)
where Oi(m) is a function having only two possible values +1 and −1. Oi(m) can be designed to generate
orthogonality among different precoders by being assigned different shape for different precoders. Ci,τp
deter-mines the magnitude of corresponding cosine term. Distinct values of Ci,τp must be used for various values
of n and p in order to satisfy the distinct power conditions in [8]. The number of cosine term can be decided arbitrarily by choosing P . Furthermore, different τp is used for different cosine terms with τp= 1, 2, . . .. The reason for using cosine is because we can completely control how many autocorrelation matrices in{Rˇzˇz(τp)}
we need in (8) for the joint diagonalization. This can be seen by considering the inverse Fourier transform of cos2πmτp M : F−1 cos 2πmτp M = M−1 m=0 cos 2πmτp M ej2πmnM = 1 2 M−1 m=0 ej2πm(n+τp)M + M−1 m=0 ej2πm(n−τp)M = 12(δ[n + τp] + δ[n − τp]) (10)
Using (10), the time domain signal after IFFT can be written as ˇu(n) ∗ {1 −Ci,τ0
2 (δ[n + τ0] + δ[n − τ0])}
= ˇu(n) −Ci,τ0
2 (ˇu[n + τ0] + ˇu[n − τ0])
(11)
where ∗ denotes convolution and P = 1. From (11), it is easy to see that temporal correlation of delay τ0
can be generated. Therefore, only Rˇzˇz(τ0) will have to be used in the joint diagonalization process at the
receiver. In fact, using the rest of the Rˇzˇz(τp), ∀p = 0 will not improve the equalization performance. This
will be shown in the simulation results in the next section when we compare equalization performance of our proposed algorithm using different τp. Besides varying the parameter τp, the parameter P can also be used to improve performance of the equalizer. This can be achieved by increasing the value of P such that more temporal correlation is added to the transmitted bitstream. However, as will be seen in Section IV, P cannot be increased indefinitely because the precoder will introduce too much amplitude variation into the bitstream
which degrades the BER performance, even though a better estimation ofU can be obtained.
IV. SIMULATIONRESULTS
A MIMO-OFDM system is simulated to evaluate the performance of the proposed scheme. In all simulations,
Nt= 2, M = 64, and v = 16. The channels are randomly generated. Two channels were chosen to show the
coefficients H(z) = 2 6 6 6 6 6 4 0.3487 0.7220 0.5121 0.5970 −0.3651 0.6136 0.6202 0.4880 3 7 7 7 7 7 5 + 2 6 6 6 6 6 4 −0.4650 0.6189 0.7682 −0.3980 −0.9129 0.3835 0.2481 0.7807 3 7 7 7 7 7 5 z−1 + 2 6 6 6 6 6 4 −0.8137 −0.3094 −0.3841 0.6965 0.1826 0.6903 0.7442 −0.3904 3 7 7 7 7 7 5z −2. (12)
The second one is a 7-tap channel with 5 received antennas, i.e. Nr= 5, with coefficients
H(z) = 2 6 6 6 6 6 6 6 4 0.5671 −0.1796 −0.2803 0.2466 −0.4485 0.2949 −0.4709 0.2253 0.1899 0.4222 3 7 7 7 7 7 7 7 5 + 2 6 6 6 6 6 6 6 4 0.4962 0.2694 −0.4484 0.3288 0.1495 −0.2949 −0.4036 −0.1931 0.2279 0.3518 3 7 7 7 7 7 7 7 5 z−1 + 2 6 6 6 6 6 6 6 4 −0.1418 0.2694 −0.5045 −0.4110 0.4485 −0.5160 0.3363 0.4507 0.6078 −0.2814 3 7 7 7 7 7 7 7 5 z−2+ 2 6 6 6 6 6 6 6 4 −0.3544 −0.3592 0.3363 −0.5754 0.2242 −0.1474 −0.4709 0.1931 −0.3799 0.5629 3 7 7 7 7 7 7 7 5 z−3 + 2 6 6 6 6 6 6 6 4 0.3544 0.4490 0.3363 0.1644 −0.2242 −0.5160 0.1345 −0.5151 −0.5318 −0.2814 3 7 7 7 7 7 7 7 5 z−4+ 2 6 6 6 6 6 6 6 4 0.2836 −0.4490 −0.1962 −0.4932 −0.5232 0.3686 0.2018 −0.4507 0.3039 −0.4222 3 7 7 7 7 7 7 7 5 z−5 + 2 6 6 6 6 6 6 6 4 −0.2836 0.5388 0.4484 −0.2466 −0.4485 −0.3686 0.4709 0.4507 0.1519 0.2111 3 7 7 7 7 7 7 7 5 z−6. (13)
The input data stream to the IFFT at the transmitter is uniformly distributed QPSK signal with zero mean and unit variance. Except for the precoders in Figure 8, the precoder used in the simulations is of the form
αi,m = Oi(m)[1 − Ci,1cos(2πm/64) − Ci,2cos(4πm/64)], for i = 1 and 2. In other words, P = 2. Oi(m)
is chosen to make the inner product of the precoder spectrum between two transmit antennas equal to 0. This,
however, can easily be generalized to any number of transmit antennas. Ci,1 and Ci,2 are set to range from
0.025 to 0.25, and varies for different transmit antennas and different time indicies in order to distinctly color the signal in time. For the simulation results below, for n= 0, C1,1 = 0.025, C1,2 = 0.25, C2,1 = 0.1, and C2,2 = 0.175. For n = 1, C1,1= 0.25, C1,2 = 0.1, C2,1 = 0.175, and C2,2 = 0.025. For n = 2, C1,1 = 0.1,
C1,2 = 0.175, C2,1 = 0.025, and C2,2 = 0.25. For n = 3, C1,1 = 0.175, C1,2 = 0.025, C2,1 = 0.25, and
C2,2= 0.1. The sequence will then repeat for subsequent time index. This is done in order to satisfy the distinct
power condition. Since the distinct power condition has already been satisfied by varying Ci,τp, therefore, βi,n can be assigned to have a value of 1 for all i and n.
0 5 10 15 20 25 10−5 10−4 10−3 10−2 10−1 100 SNR [dB] BER BER vs. SNR, Nt=2, Nr=4, q=2
LS equalizer with perfect CSI JD equalizer with prefilter JD equalizer without prefilter
Fig. 3. Comparison of BER performance between system without coloring and with coloring for 3-tap channel.
Figure 3 compares the BER performance of the proposed precoder with the joint diagonalization based equalizer with a similar system that uses the joint diagonalization based equalizer but without any precoder. A least-squares (LS) zero-forcing equalizer with perfect channel state information (CSI) is used as benchmark. For the precoder-equalizer system, the equalizer uses two correlation matrices,Rˇzˇz(τ1) and Rˇzˇz(τ2), for joint
diagonalization. The 3-tap channel in (12) was used. Since the input signal s(i)m, is independently distributed, the result reaffirms the idea that if the diagonal entries of the input signal power spectral density matrix are not distinct (no precoder is used), then it is not possible to identify and equalize FIR-MIMO channels using SOS of the received signal.
Figure 4 shows the BER results of the proposed precoder-equalizer system as the number of OFDM symbols varies for different SNR values. Similar to previous simulations, Rˇzˇz(τ1) and Rˇzˇz(τ2) are used for joint
diagonalization. As the figure shows, the BER of the proposed precoder-equalizer system approaches that of the LS equalizer when the number of symbols increases. This is because as more symbols are used, more accurate estimation of the correlation matrix can be obtained. Furthermore, at SNR = 18 dB, the proposed
algorithm is able to equalize the channel using only 350 symbols with a BER of about 10−3. Compare to
higher-order statistics techniques such as [1], when the number of symbols needed for equalization is in the order of103, only a small amount of latency is incurred in the proposed technique in order to equalize
FIR-MIMO channels.
Figures 5 and 6 compare the BER performance of the proposed equalizer system with the precoder-equalizer scheme in [12]. Results using a LS precoder-equalizer and an identical system that uses no equalization are also shown as benchmarks. As seen in the figures, the performance gap between the LS equalizer and the proposed one remains virtually unchanged as the channel spectrum changes. This shows that the performance of the proposed scheme is insensitive to various channel responses. This can be explained by observing the equation of the precoder in (9). Since the precoder is composed of cosine functions, the spectrum of the precoder will fluctuate periodically in the frequency domain. Since the amplitude of the cosine, Ci,τp, is set to a small value, even if the minimum value of the cosine term coincides with the spectral null of the channel, this will not greatly impact the BER. Compared with the precoders in [12], the proposed precoders perform better by at least
100 200 300 400 500 600 700 800 900 1000 10−5 10−4 10−3 10−2 10−1 100
Number of OFDM symbols
BER
BER vs. Number of OFDM symbols, N=2, M=4, q=2
LS equalizer with perfect CSI at SNR = 18dB JD equalizer with proposed prefilter at SNR = 18dB JD equalizer with proposed prefilter at SNR = 14dB JD equalizer with proposed prefilter at SNR = 10dB
Fig. 4. BER vs. different number of received OFDM symbols for 3-tap channel at SNR =10, 14, 18 dB.
0 5 10 15 20 25 10−5 10−4 10−3 10−2 10−1 100 SNR [dB] BER BER vs. SNR, Nt=2, Nr=4, q=2
LS equalizer with perfect CSI JD equalizer with proposed prefilter JD equalizer with prefilter in [12] No equalization
Fig. 5. Comparison of BER vs. SNR with different algorithms for 3-tap channel.
1 dB in all of the simulated channel conditions. Furthermore, using the proposed precoder, the performance of the 3-tap channel outperforms that of the 7-tap because there are more coefficients in the 7-tap channel which need to be identified for equalization.
Figure 7 shows the BER result when different lags are chosen for joint diagonalization at the receiver. As explained in Section III, if τ1 = 1 and τ2 = 2, then Rˇzˇz(1) and Rˇzˇz(2) become the most important
correlation matrices for joint diagonalization. This is reaffirmed by the simulation results in Figure 7 when the best BER performance is attained when onlyRˇzˇz(1) and Rˇzˇz(2) are used for equalization. When other
correlation matrices are used, the BER curves saturate to a noise floor. Since the choice for τp is chosen at the transmitter and it determines exactly which, as well as how many, correlation matrices should be used at the receiver for equalization, the transmitter has complete control on the computational complexity of the equalizer. Table I compares the computational complexity and latency of the proposed algorithm with those of [11] and [12]. As indicated in the table, the latency for the proposed algorithm can be controlled by the proposed precoder while the latency for [11] and [12] are fixed. In addition, the computational complexity of the proposed
0 5 10 15 20 25 30 35 40 45 50 10−5 10−4 10−3 10−2 10−1 100 SNR [dB] BER BER vs. SNR, N=2, M=5, q=6
LS equalizer with perfect CSI JD equalizer with proposed prefilter JD equalizer with prefilter in [12] No equalization
Fig. 6. Comparison of BER vs. SNR with different algorithms for 7-tap channel.
0 5 10 15 20 25 10−5 10−4 10−3 10−2 10−1 100 SNR [dB] BER BER vs. SNR, Nt=2, Nr=4, q=2
LS equalizer with perfect CSI Proposed prefilter with τ range=1 and 2 Proposed prefilter with τ range=3 and 4 Proposed prefilter with τ range=5 and 6 Proposed prefilter with τ range=7 and 8 Proposed prefilter with τ range=9 and 10
Fig. 7. BER vs. SNR with different lag for 3-tap channel.
algorithm is a lot lower than that of [11] and [12] because N M.
As discussed earlier, the parameter P cannot be increased indefinitely in order to enhance equalization performance at the expense of increase computational complexity. Likewise, P also cannot be made too small since it will adversely affect the equalization performance. Figure 8 shows the result of the proposed algorithm
when P is allowed to vary from1 to 5. As seen from the figure, the BER is smallest when P = 2. Therefore,
P cannot be made arbitrary small in order to minimize computational complexity at the receiver, but it also
cannot be made arbitrarily big since it will adversely impact the BER performance since this will induce too much amplitude variation into the transmitted bitstream.
V. CONCLUSION
A SOS-based precoder-assisted blind equalizer system has been proposed to equalize correlated FIR-MIMO channels for MIMO-OFDM system. Simulation results have shown that using the proposed precoder cannot only outperform the precoder proposed in [12] in terms of BER, but also allows the transmitter to dictate which, and how many, correlation matrices are to be used for equalization at the receiver. This decreases the amount
TABLE I
PERFORMANCE COMPARISON
Performance Indices Proposed Precoder-Blind System Periodic Precoder System [11] Using Precoder in [12]
SNR at BER =10−4for3-tap channel 19.5 dB 19 dB 21 dB
Number of symbols for SOS at SNR=18 dB 350 500 350
Latency from correlation matrix Controlled by precoders 3M + 2v + q − 3 10 ∼ 20
Computational Complexity O(N3)∗ O(M3) O(N5)
*N denotes the size of the correlation matrices
0 5 10 15 20 25 10−5 10−4 10−3 10−2 10−1 100 SNR [dB] BER BER vs. SNR, Nt=2, Nr=4, q=2
LS equalizer with perfect CSI Proposed prefilter, P=1 Proposed prefilter, P=2 Proposed prefilter, P=3 Proposed prefilter, P=4 Proposed prefilter, P=5
Fig. 8. BER vs. SNR for differentP for 3-tap channel.
of computational complexity at the receiver compared to the scheme in [12] since the number of correlation matrices needed for joint diagonalization can be predetermined before transmission.
In the future, a semi-blind equalization scheme can be extended from this work. High spectrum efficiency and channel capacity can be achieved by blind techniques. However, the BER loss compared with non-blind algorithms is an obstacle to practical application. To mitigate the BER, a short training sequence can be introduced to constitute a semi-blind system, which has better BER performance than blind systems while its channel capacity is still higher than that of non-blind systems. Moreover, SBTC or orthogonal STBC can be taken into account since these coding algorithms are often used to increase transmit diversity for a reliable link of MIMO systems. It is also possible to combine coloring precoders with precoders of multi-user MIMO systems.
VI. ACHIEVEMENTS
1) K.-L. Kuo, T.-W. Chen, C.C. Fung and C.-Y. Lee, “Second-Order Statistics Based Prefilter-Blind Equal-ization for MIMO-OFDM,” Proc. of the Asian Pacific Conference in Communications, Oct. 2008. 2) K.-L. Kuo, T.-W. Chen, C.C. Fung and C.-Y. Lee, “Second-Order Statistics Based Precoder-Assisted Blind
Equalization for MIMO-OFDM,” in preparation.
Thesis, National Chiao Tung University, Jul. 2008.
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