Hindawi Publishing Corporation EURASIP Journal on Applied Signal Processing Volume 2006, Article ID 39026, Pages 1–14 DOI 10.1155/ASP/2006/39026
A Low-Complexity Time-Domain MMSE Channel Estimator for Space-Time/Frequency Block-Coded OFDM Systems
1 Department of Computer Engineering, Kadir Has University, Cibali 34230, Istanbul, Turkey 2 Department of Electrical Engineering, Istanbul University, Avcilar 34850, Istanbul, Turkey 3 Department of Electrical and Electronics Engineering, Bilkent University, Bilkent 06800, Ankara, Turkey
Habib S¸ enol,1 Hakan Ali C¸ ırpan,2 Erdal Panayırcı,3 and Mesut C¸ evik2
Received 1 June 2005; Revised 8 February 2006; Accepted 18 February 2006
Focusing on transmit diversity orthogonal frequency-division multiplexing (OFDM) transmission through frequency-selective channels, this paper pursues a channel estimation approach in time domain for both space-frequency OFDM (SF-OFDM) and space-time OFDM (ST-OFDM) systems based on AR channel modelling. The paper proposes a computationally efficient, pilot- aided linear minimum mean-square-error (MMSE) time-domain channel estimation algorithm for OFDM systems with trans- mitter diversity in unknown wireless fading channels. The proposed approach employs a convenient representation of the channel impulse responses based on the Karhunen-Loeve (KL) orthogonal expansion and finds MMSE estimates of the uncorrelated KL series expansion coefficients. Based on such an expansion, no matrix inversion is required in the proposed MMSE estimator. Sub- sequently, optimal rank reduction is applied to obtain significant taps resulting in a smaller computational load on the proposed estimation algorithm. The performance of the proposed approach is studied through the analytical results and computer sim- ulations. In order to explore the performance, the closed-form expression for the average symbol error rate (SER) probability is derived for the maximum ratio receive combiner (MRRC). We then consider the stochastic Cramer-Rao lower bound(CRLB) and derive the closed-form expression for the random KL coefficients, and consequently exploit the performance of the MMSE channel estimator based on the evaluation of minimum Bayesian MSE. We also analyze the effect of a modelling mismatch on the estimator performance. Simulation results confirm our theoretical analysis and illustrate that the proposed algorithms are capable of tracking fast fading and improving overall performance.
Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.
1. INTRODUCTION
Next generations of broadband wireless communications systems aim to support different types of applications with a high quality of service and high-data rates by employing a variety of techniques capable of achieving the highest possi- ble spectrum efficiency [1]. The fulfilment of the constantly increasing demand for high-data rate and high quality of ser- vice requires the use of much more spectrally efficient and flexible modulation and coding techniques, with greater im- munity against severe frequency-selective fading. The com- bined application of OFDM and transmit antenna diversity appears to be capable of enabling the types of capacities and data rates needed for broadband wireless services [2–8].
OFDM has emerged as an attractive and powerful al- ternative to conventional modulation schemes in the recent past due to its various advantages in lessening the severe ef- fect of frequency-selective fading. The broadband channel undergoes severe multipath fading, the equalizer in a con- ventional single-carrier modulation becomes prohibitively complex to implement. OFDM is therefore chosen over a single-carrier solution due to lower complexity of equalizers [1]. In OFDM, the entire signal bandwidth is divided into a number of narrowbands or orthogonal subcarriers, and sig- nal is transmitted in the narrowbands in parallel. Therefore, it reduces intersymbol interference (ISI), obviates the need for complex equalization, and thus greatly simplifies chan- nel estimation/equalization task. Moreover, its structure also allows efficient hardware implementations using fast Fourier transform (FFT) and polyphase filtering [2]. On the other hand, due to dispersive property of the wireless channel, sub- carriers on those deep fades may be severely attenuated. To robustify the performance against deep fades, diversity tech- niques have to be used. Transmit antenna diversity is an ef- fective technique for combatting fading in mobile in multi- path wireless channels [4, 9]. Among a number of antenna diversity methods, the Alamouti method is very simple to implement [9]. This is an example for space-time block code (STBC) for two transmit antennas, and the simplicity of the receiver is attributed to the orthogonal nature of the code
2 EURASIP Journal on Applied Signal Processing
[10, 11]. The orthogonal structure of these space-time block codes enable the maximum likelihood decoding to be im- plemented in a simple way through decoupling of the signal transmitted from different antennas rather than joint detec- tion resulting in linear processing [9].
The layout of the paper is as follows. In Section 2, a gen- eral model for transmit diversity OFDM systems together with SF and ST coding, AR channel modelling, and unified signal model are presented. In Section 3, an MMSE channel estimation algorithm is developed for the KL expansion co- efficients. Performance of the proposed algorithm is studied based on the evaluation of the modified Cramer-Rao bound of the channel parameters and the SNR and correlation mis- match analysis together with closed-form expression for the average SER probability in Section 4. Some simulation exam- ples are provided in Section 5. Finally, conclusions are drawn in Section 6.
2. SYSTEM MODEL
2.1. Alamouti’s transmit diversity scheme for OFDM systems
In this paper, we consider a transmitter diversity scheme in conjunction with OFDM signaling. Many transmit diversity schemes have been proposed in the literature offering dif- ferent complexity versus performance trade-offs. We choose Alamouti’s transmit diversity scheme due to its simple im- plementation and good performance [9]. The Alamouti’s scheme imposes an orthogonal spatio-temporal structure on the transmitted symbols that guarantees full (i.e., order 2) spatial diversity.
The use of OFDM in transmitter diversity systems mo- tives exploitation of diversity dimensions. Inspired by this fact, a number of coding schemes have been proposed re- cently to achieve maximum diversity gain [6–8]. Among them, ST-OFDM has been proposed recently for delay spread channels. On the other hand, transmitter OFDM also of- fers the possibility of coding in a form of SF-OFDM [6–8]. OFDM maps the frequency-selective channel into a set of flat fading subchannels, whereas space-time/frequency en- coding/decoding facilitates equalization and achieves perfor- mance gains by exploiting the diversity available with trans- mit antennas. Moreover, SF-OFDM and ST-OFDM trans- mitter diversity systems were compared in [6], under the as- sumption that the channel responses are known or can be estimated accurately at the receiver. It was shown that the SF- OFDM system has the same performance as a previously re- ported ST-OFDM scheme in slow fading environments but shows better performance in the more difficult fast fading environments. Also, since, SF-OFDM transmitter diversity scheme performs the decoding within one OFDM block, it only requires half of the decoder memory needed for the ST- OFDM system of the same block size. Similarly, the decoder latency for SF-OFDM is also half that of the ST-OFDM im- plementation.
We consider the Alamouti transmitter diversity coding scheme, employed in an OFDM system utilizing K subcar- riers per antenna transmissions. Note that K is chosen as an even integer. The fading channel between the μth transmit antenna and the receive antenna is assumed to be frequency selective and is described by the discrete-time baseband equivalent impulse response hμ(n) = [hμ,0(n), . . . , hμ,L(n)]T , with L standing for the channel order.
At each time index n, the input serial information sym- bols with symbol duration Ts are converted into a data vec- tor X(n) = [X(n, 0), . . . , X(n, K − 1)]T by means of a serial- to-parallel converter. Its block duration is KTs. Moreover, X(n, k) denote the kth forward polyphase component of the serial data symbols, that is, X(n, k) = X(nK + k) for k = 0, 1, 2, . . . , K − 1 and n = 0, 1, 2, . . . , N − 1. Polyphase com- ponent X(n, k) can also be viewed as the data symbol to be transmitted on the kth tone during the block instant n. The transmitter diversity encoder arranges X(n) into two vectors X1(n) and X2(n) according to an appropriate coding scheme described in [6, 9]. The coded vector X1(n) is modulated by an IFFT into an OFDM sequence. Then cyclic prefix is added to the OFDM symbol sequence, and the resulting sig- nal is transmitted through the first transmit antenna. Sim- ilarly, X2(n) is modulated by IFFT, cyclically extended, and transmitted from the second transmit antenna.
At the receiver side, the antenna receives a noisy super- position of the transmissions through the fading channels. We assume ideal carrier synchronization, timing, and perfect symbol-rate sampling, and the cyclic prefix is removed at the receiver end. Channel estimation for transmit diversity OFDM sys- tems has attracted much attention with pioneering works by Li et al. [4] and Li [5]. A robust channel estimator for OFDM systems with transmitter diversity has been first de- veloped with the temporal estimation by using the correla- tion of the channel parameters at different frequencies [4]. Its simplified approaches have been then presented by iden- tifying significant taps [5]. Among many other techniques, pilot-aided MMSE estimation was also applied in the con- text of space-time block coding (STBC) either in the time do- main for the estimation of channel impulse response (CIR) [12, 13] or in the frequency domain for the estimation of transfer function (TF) [14]. However channel estimation in the time domain turns out to be more efficient since the number of unknown parameters is greatly decreased com- pared to that in the frequency domain. Focusing on transmit diversity OFDM transmissions through frequency-selective fading channels, this paper pursues a time-domain MMSE channel estimation approach for both SF-OFDM and ST- OFDM systems. We derive a low complexity MMSE channel estimation algorithm for both transmiter diversity OFDM systems based on AR channel modelling. In the development of the MMSE channel estimation algorithm, the channel taps are assumed to be random processes. Moreover, orthogonal series representation based on the KL expansion of a random process is applied which makes the expansion coefficient ran- dom variables uncorrelated [15, 16]. Thus, the algorithm es- timates the uncorrelated complex expansion coefficients us- ing the MMSE criterion.
Tx − 1
Pilot insertion & IFFT & add cyclic prefix
X(n, 0) −X ∗(n, 1) . . . X(n, K − 2) −X ∗(n, K − 1)
Habib S¸ enol et al. 3
X(n)
Tx − 2
Serial to parallel
Space- frequency encoding
X(n, 1) X ∗(n, 0) .. .
Pilot insertion & IFFT & add cyclic prefix
X(n, K − 1) X ∗(n, K − 2)
Figure 1: Space-frequency coding on two adjacent FFT frequency bins.
(cid:2)
vectors of X(n) as
(cid:3) T , (cid:3) T , (2)
X(n, 0), X(n, 2), . . . , X(n, K − 4), X(n, K − 2) (cid:2) X(n, 1), X(n, 3), . . . , X(n, K − 3), X(n, K − 1) Xe(n) = Xo(n) =
The generation of coded vectors X1(n) and X2(n) from the information symbols leads to corresponding transmit diversity OFDM scheme. In our system, the generation of X1(n) and X2(n) is performed via the space-frequency cod- ing and space-time coding, respectively, which were first sug- gested in [9] and later generalized in [7, 8].
(cid:5)
then the space-frequency block code transmission matrix may be represented by Space-frequency coding
Space −→ (cid:4) (3) Frequency ↓ . Xe(n) Xo(n) o (n) X∗ −X∗ e (n)
If the received signal sequence is parsed in even and odd blocks of K/2 tones, Ye(n) = [Y (n, 0), Y (n, 2), . . . , Y (n, K − 2)]T and Yo(n) = [Y (n, 1), Y (n, 3), . . . , Y (n, K − 1)]T , the re- ceived signal can be expressed in vector form as
o (n)H1,o(n) + X†
(4) Ye(n) = Xe(n)H1,e(n) + Xo(n)H2,e(n) + We(n), Yo(n) = −X† e (n)H2,o(n) + Wo(n),
(cid:2)
We first consider a strategy which basically consists of coding across OFDM tones and is therefore called space-frequency coding [6–8]. Resorting to coding across tones, the set of generally correlated OFDM subchannels is first divided into groups of subchannels. This subchannel grouping with ap- propriate system parameters does preserve diversity gain while simplifying not only the code construction but decod- ing algorithm significantly as well [6]. A block diagram of a two-branch space-frequency OFDM transmitter diversity system is shown in Figure 1. Resorting subchannel grouping, X(n) is coded into two vectors X1(n) and X2(n) by the space- frequency encoder as
X1(n) =
(1) X2(n) =
where Xe(n) and Xo(n) are K/2 × K/2 diagonal matri- ces whose elements are Xe(n) and Xo(n), respectively, and (·)† denotes conjugate transpose. Let Hμ,e(n) = [Hμ(n, 0), Hμ(n, 2), . . . , Hμ(n, K − 2)]T and Hμ,o(n) = [Hμ(n, 1), Hμ(n, 3), . . . , Hμ(n, K − 1)]T be K/2 length vectors denoting the even and odd component vectors of the channel attenu- ations between the μth transmitter and the receiver. Finally, We(n) and Wo(n) are zero-mean, i.i.d. Gaussian vectors with covariance matrix σ 2IK/2. X(n, 0), −X ∗(n, 1), . . . , X(n, K − 2), (cid:3) T , − X ∗(n, K − 1) (cid:2) X(n, 1), X ∗(n, 0), . . . , X(n, K − 1), (cid:3) T , X ∗(n, K − 2)
Space-time coding
(cid:5)
In contrast to SF-OFDM coding, ST encoder maps every two consecutive symbol blocks X(n) and X(n+1) to the following 2K × 2 matrix: where (·)∗ stands for complex conjugation. In space- frequency Alamouti scheme, X1(n) and X2(n) are transmit- ted through the first and second antenna elements, respec- tively, during the OFDM block instant n. Space −→ (cid:4) (5) X(n) Time ↓ . X(n + 1) −X∗(n + 1) X∗(n) The operations of the space-frequency block encoder can best be described in terms of even and odd polyphase component vectors. If we denote even and odd component
Tx − 1
X(n, 0) X(n, 1) . . .
X(n, K − 1)
−X ∗(n + 1, 0) −X ∗(n + 1, 1) . . . −X ∗(n + 1, K − 1)
Pilot insertion & IFFT & add cyclic prefix
Tx − 2
4 EURASIP Journal on Applied Signal Processing
X(n)
Serial to parallel
Space- time encoding
X(n + 1, 0) X(n + 1, 1) .. .
X(n + 1, K − 1)
X ∗(n, 0) X ∗(n, 1) .. . X ∗(n, K − 1)
Pilot insertion & IFFT & add cyclic prefix
Figure 2: Space-time coding on two adjacent OFDM blocks.
AR channel model in SF-OFDM
μ,o(n),
The even and odd component vectors of the channels Hμ,e(n) and Hμ,o(n) between the μth transmitter and the receiver can be modelled as a first-order AR process. An AR process can be represented as The columns are transmitted in successive time intervals with the upper and lower blocks in a given column sent simul- taneously through the first and second transmit antennas, respectively, as shown in Figure 2. If we focus on each re- ceived block separately, each pair of two consecutive received blocks Y(n) = [Y (n, 0), . . . , Y (n, K − 1)]T and Y(n + 1) = [Y (n + 1, 0), . . . , Y (n + 1, K − 1)]T are given by (7) Hμ,o(n) = αHμ,e(n) + η
Y(n) = X(n)H1(n) + X(n + 1)H2(n) + W(n), (6) Y(n + 1) = −X†(n + 1)H1(n + 1)
μ,o(n).
+ X†(n)H2(n + 1) + W(n + 1), where α can be obtained from the normalized exponential discrete channel correlation for different subcarriers in SF- OFDM case. Moreover, using (7), simple manipulations lead μ,o (n) = (1 − |α|2)IK/2 of zero- to the covariance matrix Cη mean Gaussian AR process noise η
AR channel model in ST-OFDM
Similarly, the channel frequency response Hμ(n) between the μth transmitter and the receiver antenna at the nth time slot varies accordingly:
μ(n + 1),
where X(n) and X(n + 1) are K × K diagonal matrices whose elements are X(n) and X(n + 1), respectively. Hμ(n) is the channel frequency response between the μth transmit- ter and the receiver antenna at the nth time slot which is ob- tained from channel impulse response hμ(n). Finally, W(n) and W(n + 1) are zero-mean, i.i.d. Gaussian vectors with covariance matrix σ 2IK per dimension. (8) Hμ(n + 1) = αHμ(n) + η Having specified the received signal models (4) and (6), we proceed to explore channel models.
μ(n + 1) as Cη
2.2. AR models considerations
where α is related to Doppler frequency fd and symbol dura- tion Ts via α = Jo(2π fdTs) in ST-OFDM. Using (8), we ob- tain the covariance matrix of zero-mean Gaussian AR process noise η μ(n+1) = (1 − |α|2)IK .
2.3. Unifying SF-OFDM and ST-OFDM signal models
(cid:4)
(cid:5)
(cid:4)
(cid:5) (cid:4)
(cid:5)
(cid:4)
(cid:5)
=
The transmitter diversity OFDM schemes considered here can be unified into one general model for channel estima- tion. Considering signal models (4) and (6) with correspond- ing AR models (7) and (8), we unify SF-OFDM and ST- OFDM in the following equivalent model:
+ (9) . Y1 Y2 H1 H2 W1 W2 X1 X2 X† −X† 1 2 Channel estimation in transmit diversity systems results in ill-posed problem since for every incoming signal, extra un- knowns appear. However, imposing structure on channel variations render estimation problem tractable. Fortunately many wireless channels exhibit structured variations hence fit into some evolution model. Among different models, the AR model is adopted herein for channel dynamics. Since only the first few correlation terms are important to finitely parametrize structured variations of a wireless channel in the design of a channel estimator, low-order AR models can cap- ture most of the channel tap dynamics and lead to effective estimation techniques. Thus this paper associates channel ef- fect in SF/ST-OFDM systems with a first-order AR process.
5 Habib S¸ enol et al.
(cid:4)
(cid:5)
(cid:5)
(cid:4)
(cid:4)
(cid:5)
(cid:4)
(cid:5)
=
corresponding to pilot symbols as follows:
(cid:7)
(cid:6)
=
(cid:11)
(cid:8)
(cid:11)
(cid:11)
(cid:8)
(cid:8)
(cid:8)
(cid:7) ,
(cid:4)
(cid:4)
(cid:5)
=
(cid:7)
=
For convenience, we list the corresponding vectors and ma- trices for SF-OFDM as (cid:6) + , (cid:11) Y1,p Y2,p (cid:9)(cid:10) Yp H1,p H2,p (cid:9)(cid:10) Hp W1,p W2,p (cid:9)(cid:10) Wp X1,p X2,p X† −X† 2,p 1,p (cid:9)(cid:10) Xp (12) ,
(cid:6)
(cid:7)
(cid:6)
=
(cid:3)
(cid:7) ,
(cid:2) 1/α
o (n)η
e (n)η
1,o(n) + X†
2,o(n)
pWp.
p
Ye(n) Yo(n)/α Xe(n) Xo(n) −X† o (n) X† e (n) (cid:7) (cid:6) , where (·)p is introduced to represent the vectors correspond- ing to pilot locations. Y1 Y2 (cid:5) X1 X2 X† −X† 1 2 (cid:6) H1 H2 H1,e(n) H2,e(n) We(n) W1 W2 Wo(n) − X† (10) X† (13) XpHp + X† For a class of QPSK-modulated pilot symbols, the new observation model can be formed by premultiplying both sides of (12) by X† p: pYp = X†
pYp and (cid:13)Wp =
p
(cid:6)
(cid:7)
(cid:6)
pWp, (13) can be rewritten as
(cid:7) ,
(cid:4)
where W1 ∼ N (0, σ 2IK/2), W2 ∼ N (0, σ 2 + 2(1 − |α|2)/ |α|2IK/2). Similarly for ST-OFDM, Xp = 2I2Kp , and letting (cid:12)Yp = X† Since X† X†
= (cid:4)
(cid:5)
(cid:12)Yp = 2Hp + (cid:13)Wp
=
(cid:6)
(cid:7)
(cid:6)
(cid:4)
(cid:5)
(cid:4)
(cid:5)
(cid:4)
(cid:5)
=
Y1 Y2 (cid:5) (14) Y(n) Y(n + 1)/α X(n) , namely, X1 X2 X† −X† 1 2
= 2
(cid:6)
(cid:7)
(cid:12)Y1,p (cid:12)Y2,p
(cid:13)W1,p (cid:13)W2,p
X(n + 1) −X†(n + 1) X†(n) (cid:7) , (15) + , H1 H2 H1(n) H2(n) H1,p H2,p
(cid:5)
(cid:3)
=
(cid:2) 1/α
1(n+1)+X†(n)η
2(n+1)
(cid:12)Y1,p = X† (cid:12)Y2,p = X† (cid:13)W1,p = X† (cid:13)W2,p = X†
1,pY1,p − X2,pY2,p, 2,pY1,p + X1,pY2,p, 1,pW1,p − X2,pW2,p, 2,pW1,p + X1,pW2,p,
where W1 W2 (cid:4) W(n) . W(n+1)−X†(n+1)η (11) (16)
Note that W1 ∼ N (0, σ 2IK ) and W2 ∼ N (0, σ 2 + 2(1 − |α|2)/ |α|2IK ).
and note that (cid:13)W1,p ∼ N (0, σ 2IKp ) and (cid:13)W2,p ∼ N (0, σ 2IKp ) where σ 2 = (σ 2(1 + |α|2) + 2(1 − |α|2))/|α|2. By writing each row of (16) separately, we obtain the following obser- vation equation set to estimate the channels H1,p and H2,p:
(cid:12)Yμ,p = 2Hμ,p + (cid:13)Wμ,p
(17) μ = 1, 2.
Relying on the unifying model (9), we will develop a channel estimation algorithm according to the MMSE crite- rion and then explore the performance of the estimator. An MMSE approach adapted herein explicitly models the chan- nel parameters by the KL series representation since KL ex- pansion allows one to tackle the estimation of correlated pa- rameters as a parameter estimation problem of the uncorre- lated coefficients.
3. MMSE ESTIMATION Since our goal is to develop channel estimation in time do- main, (17) can be expressed in terms of hμ by using Hμ,p (cid:2) Fhμ in (17). Thus we can conclude that the observation mod- els for the estimation of channel impulse responses hμ are
(cid:12)Yμ,p = 2Fhμ + (cid:13)Wμ,p,
(18) μ = 1, 2,
Pilots-symbols-assisted techniques can provide information about an undersampled version of the channel that may be easier to identify. In this paper, we therefore address the prob- lem of estimating channel parameters by exploiting the dis- tributed training symbols. where F is a Kp × L FFT matrix generated based on pilot in- dices and Kp is the number of pilot symbols per one OFDM block.
3.1. MMSE estimation of the multipath channels
Since both SF and ST block-coded OFDM systems have sym- metric structure in frequency and time, respectively, the pi- lot symbols should be uniformly placed in pairs. Specifically, we also assume that even number of symbols are placed be- tween pilot pairs for SF-OFDM systems. Based on these pi- lot structures, (9) is modified to represent the signal model Since (18) offers a Bayesian linear model representa- tion, one can obtain a closed-form expression for the MMSE estimation of channel vectors h1 and h2. We should first make the assumptions that impulse responses h1 and h2 are i.i.d. zero-mean complex Gaussian vectors with covari- ance Ch, and h1 and h2 are independent from (cid:13)W1,p ∼ N (0, σ 2IKp ) and (cid:13)W2,p ∼ N (0, σ 2IKp ) and employ PSK pi- lot symbolassumption to obtain MMSE estimates of h1 and
6 EURASIP Journal on Applied Signal Processing
(cid:15)
(cid:16)−1
h2 [17]:
(cid:14)hμ =
2F†F + (19) μ = 1, 2. F† (cid:12)Yμ,p, C−1 h σ 2 2 MMSE estimator of g requires 4L2 +4LKp +2L real multi- plications. From the results presented in [18], ML estimator of gμ which requires 4L2 + 4LKp real multiplications can be obtained as
(cid:14)gμ = 1 2Kp
(25) μ = 1, 2. Ψ†F† (cid:12)Yμ,p,
(cid:17)
(cid:18)−1
Under the assumption that uniformly spaced pilot symbols are inserted with pilot spacing interval Δ and K = Δ × Kp, correspondingly, F†F reduces to F†F = KpIL. Then according to (19), and F†F = KpIL, we arrive at the expression
(cid:14)hμ =
(20) μ = 1, 2. 2KpIL + F† (cid:12)Yμ,p, C−1 h σ 2 2
}L−1 Since the trailing L − r variances {λgl l=r are small com- }r−1 pared to the leading r variances {λgl l=0 , the trailing L − r variances are set to zero to produce the approximation. How- ever, typically the pattern of eigenvalues for Λ splits the eigenvectors into dominant and subdominant sets. Then the choice of r is more or less obvious. The optimal truncated KL (rank-r) estimator of (23) now becomes
It is clear that the complexity of the MMSE estimator in (20) is reduced by the application of KL expansion. However, the complexity of the (cid:14)gμ can be further reduced by exploiting the optimal truncation property of the KL expansion [15]. A truncated expansion gμr can be formed by selecting r or- thonormal basis vectors from all basis vectors that satisfy ChΨ = ΨΛ. Thus, a rank-r approximation to Λr is defined as Λr = diag{λ0, λ1, . . . , λr−1, 0, . . . , 0}.
As it can be seen from (20) MMSE estimation of h1 and h2 for SF-OFDM and ST-OFDM systems still requires the inver- sion of C−1 h . Therefore it suffers from a high computational complexity. However, it is possible to reduce complexity of the MMSE algorithm by expanding multipath channel as a linear combination of orthogonal basis vectors. The orthog- onality of the basis vectors makes the channel representa- tion efficient and mathematically convenient. KL transform which amounts to a generalization of the DFT for random processes can be employed here. This transformation is re- lated to diagonalization of the channel correlation matrix by the unitary eigenvector transformation,
= ΓrΨ†F† (cid:12)Yμ,p,
(cid:14)gμr
1, . . . , ψ
0, ψ
(26) (21)
}.
(cid:18)−1
where
(cid:17) 2KpΛr + (cid:19)
= diag
Ch = ΨΛΨ†, L−1], ψ where Ψ = [ψ l’s are the orthonormal basis vectors, and gμ = [gμ,0, gμ,1, . . . , gμ,L−1]T is zero-mean Gaus- sian vector with diagonal covariance matrix Λ = E{gμg† μ Γr = Λr IL σ 2 2 2λ1 (27)
2λ0 4Kpλ0 + σ 2 , 2λr−1 4Kpλ1 + σ 2 , . . . , (cid:20) 4Kpλr−1 + σ 2 , 0, . . . , 0 . Thus the vectors h1 and h2 can be expressed as a lin- ear combination of the orthonormal basis vectors, that is, as hμ = Ψgμ where μ is the multipath channel index. As a result, the channel estimation problem in this application is equiva- lent to estimating the i.i.d. complex Gaussian vectors g1 and g2 which represent KL expansion coefficients for multipath channels h1 and h2.
Thus, the truncated MMSE estimator of gμ (26) requires 4Lr + 4LKp + 2r real multiplications.
(cid:12)Yμ,p = 2FΨgμ + (cid:13)Wμ,p,
3.2. MMSE estimation of KL coefficients Substituting hμ = Ψgμ in unified observation model (18), we can rewrite it as 3.3. Estimation of Hμ,o(n) and Hμ(n + 1) (22) μ = 1, 2,
(cid:18)−1
(cid:4)
(cid:5)
(cid:4)
(cid:5) (cid:4)
(cid:5)
(cid:4)
(cid:5)
For the Bayesian MMSE estimation of the channel param- eters Hμ,o(n) and Hμ(n + 1) for SF-OFDM and ST-OFDM, respectively, the unified signal model in (9) can be rewritten by exploiting AR representation in (7) and (8) as
= 1 α
= ΓΨ†F† (cid:12)Yμ,p,
which is also recognized as a Bayesian linear model, and re- call that gμ ∼ N (0, Λ). As a result, the MMSE estimator of KL coefficients gμ is (cid:17) 2KpΛ + (cid:14)gμ = Λ Ψ†F† (cid:12)Yμ,p IL σ 2 2 (23) + (28) . Y1 Y2 H1+ H2+ W1+ W2+ X1 X2 X† −X† 1 2 μ = 1, 2,
(cid:18)−1
(cid:4)
(cid:5)
(cid:6)
(cid:7)
where The corresponding vectors for SF-OFDM can be listed as
=
(cid:17) 2KpΛ + (cid:19)
(cid:20)
(cid:5)
(cid:4)
= diag
(cid:7)
(cid:6)
1,o(n) − Xo(n)η
2,o(n)]
=
Γ = Λ IL , H1,o(n) H2,o(n) 2λ1 σ 2 2 2λ0 4Kpλ0 + σ 2 , 4Kpλ1 + σ 2 , . . . , 2λL−1 4KpλL−1 + σ 2 H1+ H2+ We(n) − 1/α[Xe(n)η . (24) W1+ W2+ 1/αWo(n) (29) and λ0, λ1, . . . , λL−1 are the singular values of Λ.
Habib S¸ enol et al. 7
(cid:6)
(cid:6)
(cid:7)
=
(cid:22)
(cid:21)
(cid:22) ,
Moreover for ST-OFDM, (cid:7) , the modified FIM can be obtained by a straightforward mod- ification of J(gμ) FIM as (cid:22) H1(n + 1) H2(n + 1) (cid:2) J (35) JM gμ
(cid:21) gμ
(cid:21) gμ
(cid:7)
(cid:6)
(cid:6)
1(n+1)−X(n+1)η
2(n+1)]
=
(cid:7) .
+ JP H1+ H2+ W(n)−(1/α)[X(n)η (1/α)W(n+1) W1+ W2+ (30)
(cid:22)
=
where JP(gμ) represents the a priori information. Under the assumption that gμ and (cid:13)Wμ,p are independent of each other and (cid:13)Wμ,p is a zero mean, from [19] and (31) the conditional PDF is given by
(cid:21) (cid:12)Yμ,p | gμ
(cid:23) (cid:23)
(cid:22)†
× exp
(cid:12)Yμ,p = 2 α
×
(cid:21) (cid:12)Yμ,p − 2FΨgμ
Note that W1+ ∼ N (0, (σ 2 + 2(1 − |α|2)/|α|2)I) and W2+ ∼ N (0, σ 2/|α|2I). According to the unified model in (28), cor- responding pilot model in (12), and Hμ+ = FΨgμ+, the ob- servation model becomes p πKp (31) μ = 1, 2, (36) FΨgμ+ + (cid:13)Wμ+,p, C−1 (cid:13)Wμ,p 1 (cid:23) (cid:23)C(cid:13)Wμ,p (cid:21) (cid:24) (cid:12)Yμ,p − 2FΨgμ − (cid:22)(cid:25) where
(cid:13)W1+,p = X† (cid:13)W2+,p = X†
1,pW1+,p − X2,pW2+,p, 2,pW1+,p + X1,pW2+,p,
(cid:22)
(cid:22)†
from which the derivatives follow as (32)
(cid:12)Yμ,p − 2FΨgμ
(cid:21) = 2 (cid:22)
(cid:21) (cid:12)Yμ,p | gμ ∂gT μ ∂2 ln p
∂ ln p FΨ, C−1 (cid:13)Wμ,p and note that (cid:13)Wμ+,p ∼ N (0, σ 2I). Thus, the estimation of the KL coefficient vector gμ+ is (37)
= −4Ψ†F†C−1 (cid:13)Wμ,p
(cid:14)gμ+ = (cid:12)ΓΨ†F† (cid:12)Yμ,p,
(cid:21) (cid:12)Yμ,p | gμ μ ∂gT μ
FΨ, (33) μ = 1, 2, ∂g∗
(cid:17)
(cid:18)−1
(cid:12)Γ = Λ
where
(cid:19)
(cid:22)
(cid:7)
(cid:6)
= diag
(cid:21) (cid:12)Yμ,p | gμ μ ∂gT μ (cid:7)
(cid:6)
−
= −E
=
σ 2IL where the superscript (·)∗ indicates the conjugation opera- = σ 2IKp , Ψ†Ψ = IL, and F†F = KpIL, and tion. Using C(cid:13)Wμ,p taking the expected value yields the following simple form: 2 α∗ KpΛ + 2α∗λ1 (34) ∂2 ln p J(gμ) = −E 4Kpλ1 + |α|2σ 2 , . . . , (cid:20) ∂g∗ . α 2 2α∗λ0 4Kpλ0 + |α|2σ 2 , 2α∗λL−1 4KpλL−1 + |α|2σ 2 (38) 4Kp σ 2 IL
4Kp σ 2 IL.
(cid:24)
(cid:26)
Note that, choosing α = 1 results in Hμ,o = Hμ,e and Hμ(n + 1) = Hμ(n), respectively, which significantly simpli- fies the channel estimation task in transmit diversity OFDM systems. Second term in (35) is easily obtained as follows. Consider the prior PDF of gμ(n) as
− g† μ
(39) . p(gμ) = 1 Λ−1gμ The performance analysis issues elaborated in the next section only consider the Bayesian MMSE estimator of gμ for Hμ,e(n) and Hμ(n). However extensions for gμ+ are straight- forward. πL|Λ| exp
(cid:22)
The respective derivatives are found as 4. PERFORMANCE ANALYSIS
(cid:21) gμ
= −g† μ (cid:22)
Λ−1,
= −Λ−1.
(40)
∂ ln p ∂gT μ ∂2 ln p ∂g∗
(cid:21) gμ μ ∂gT μ
In this section, we turn our attention to analytical per- formance results. We first exploit the performance of the MMSE channel estimator based on the evaluation of mod- ified Cramer-Rao lower bound, Bayesian MSE together with mismatch analysis. We then derive the closed-form expres- sion for the average SER probability of MRRC.
(cid:7)
(cid:6)
(cid:2)
Upon taking the negative expectations, second term in (35) becomes 4.1. Cramer-Rao lower bound for random KL coefficients JP(gμ) = −E
= −E = Λ−1.
(41) ∂2 ln p(gμ) ∂g∗ μ ∂gT μ (cid:3) − Λ−1
In this paper, the estimation of unknown random parameters gμ is considered via MMSE approach; the modified Fisher in- formation matrix(FIM) therefore needs to be taken into ac- count in the derivation of stochastic CRLB [19]. Fortunately,
8 EURASIP Journal on Applied Signal Processing
(cid:22)
(cid:22)
(cid:22)
= J
4.3. Mismatch analysis
=
(cid:17)
(cid:18)
Substituting (38) and (41) in (35) produces for the modified FIM the following: (cid:21) gμ JM
(42) Λ−1
(cid:21) (cid:21) gμ + JP gμ 4Kp σ 2 IL + Λ−1 2KpIL +
σ 2 2
= 2 σ 2 = 2 σ 2
Γ−1. In mobile wireless communications, the channel statistics depend on the particular environment, for example, in- door or outdoor, urban or suburban, and change with time. Hence, it is important to analyze the performance degrada- tion due to a mismatch of the estimator with respect to the channel statistics as well as the SNR, and to study the choice of the channel correlation and SNR for this estimator so that it is robust to variations in the channel statistics. As a perfor- mance measure, we use Bayesian MSE (45).
(cid:22)
(cid:22)
Inverting the matrix JM(gμ) yields
(cid:21) (cid:14)gμ
CRLB
(cid:21) gμ
= J−1 M = σ 2 2
(43) Γ.
L−1(cid:29)
In practice, the true channel correlations and SNR are not known. If the MMSE channel estimator is designed to match the correlation of a multipath channel impulse re- sponse Ch and SNR, but the true channel parameters (cid:12)hμ have the correlation C(cid:12)h and the true (cid:31)SNR, then average Bayesian MSE for the designed channel estimator is extended from [21] as follows 4.2. Bayesian MSE (i) SNR mismatch:
i=0
(cid:27)
(cid:22)(cid:28)−1
=
(cid:21) 2FΨ
(47) From the performance of the MMSE estimator for the Bayesian linear model theorem [17], the error covariance matrix is ob- tained as BMSE( gμ) = 1 L λi(cid:12)σ 2 4Kpλi + σ 4/(cid:12)σ 2 (4Kpλi + σ 2)2 ,
(cid:17)
(cid:18)−1
C(cid:2)μ Λ−1 + (2FΨ)†C−1 (cid:13)Wμ,p where
= σ 2 2 = σ 2 2
Λ−1 (44) 2KpIL + , + σ 2 2 (48) Γ. + . 2(1 − |α|2) |α|2 2(1 − |α|2) |α|2 σ 2 = 1 + |α|2 |α|2 SNR (cid:12)σ 2 = 1 + |α|2 |α|2(cid:31)SNR
(cid:22)
L−1(cid:29)
(cid:12)λiσ 2 + 4Kpλi
(cid:21) (cid:12)λi + λi − 2βi
(ii) Correlation mismatch:
i=0
Comparing (43) with (44), the error covariance matrix of the MMSE estimator coincides with the stochastic CRLB of the random vector estimator. Thus, (cid:14)gμ achieves the stochastic CRLB. , (49) BMSE((cid:14)gμ) = 1 L σ 2 + 4Kpλi
(cid:22)
(cid:22)
We now formalize the Bayesian MSE of the full-rank es- timator which is actually an extension of previous evaluation methodology presented in [20, 21]:
(cid:21) (cid:14)gμ
where (cid:12)λi is the ith diagonal element of (cid:12)Λ = Ψ†C(cid:12)h Ψ, and βi is ith diagonal element of the real part of the crosscorrelation matrix between (cid:12)gμ and gμ. BMSE
(cid:21) C(cid:2)μ tr (cid:17)
(cid:18)
L−1(cid:29)
= 1 L = 1 L
= 1 L
i=0
2 ]T and cast (9) in a matrix/vector
(cid:5)
(cid:4)
(45) 4.4. Theoretical SER for SF/ST-OFDM systems Γ , tr σ 2 2 σ 2λi σ 2 + 4Kpλi
(cid:5)
(cid:4)
(cid:5)
(cid:4)
(cid:5)
=
(cid:11)
(cid:8)
(cid:11)
(cid:8)
(cid:8)
(cid:11)
(cid:8)
−H † 1 (cid:9)(cid:10) H
r−1(cid:29)
L−1(cid:29)
Let us define Y = [Y1 Y∗ form: (cid:4) where, substituting σ 2 = 1/SNR in σ 2, σ 2 = 1 + |α|2/ |α|2 SNR +2(1 − |α|2)/|α|2, and tr denotes trace operator on matrices. + H1 H2 H † 2 (50) , (cid:11) X1 X2 (cid:9)(cid:10) X Y1 Y∗ 2 (cid:9)(cid:10) Y W1 W∗ 2 (cid:9)(cid:10) W Following the results presented in [20, 21], BMSE((cid:14)gμ) given in (45) can also be computed for the truncated (low- rank) case as follows:
i=0
i=r
(cid:4)
(cid:5)
(cid:4)
(cid:5)
=
+ (46) λi. BMSE((cid:14)gμr ) = 1 L 1 L σ 2λi σ 2 + 4Kpλi where Hμ = diag(Hμ). By premultiplying (50) by H † the signal model for maximal ratio receive combiner (MRRC) can be obtained as
(cid:6) H1 (cid:6)2 + (cid:6) H2 (cid:6)2 0 (cid:4)
(cid:5)
(cid:5)
(cid:4)
(cid:30)
×
0 (cid:6) H1 (cid:6)2 + (cid:6) H2 (cid:6)2 ˘Y1 ˘Y2 (51) + , X1 X2 ˘W1 ˘W2, Notice that the second term in (46) is the sum of the powers in the KL transform coefficients not used in the truncated estimator. Thus, truncated BMSE((cid:14)gμr ) can be lower bounded L−1 by (1/L) i=r λi which will cause an irreducible error floor in the SER results.
Habib S¸ enol et al. 9
where
#
# ∞
π/2
If we now apply ρ1 = ζ cos(α) and ρ2 = ζ sin(α) transfor- mations, we arrive at the following SER expression for ST- OFDM and SF-OFDM systems:
0
0
(cid:28)
(cid:27)"
(cid:28)(cid:28)
(cid:27)
1 Y1 + H2Y∗ 2 , 2 Y1 − H1Y∗ 2 , 1 W1 + H2W∗ 2 , 2 W1 − H1W∗ 2 .
×
− Q2
# ∞
(cid:27)"
(cid:27)"
(cid:28)(cid:28)
=
(cid:27) 2Q
− Q2
(cid:23) (cid:23)2 +
−
(cid:21) γ2
2γ1
0 = 3 4
1 +ρ2
(52) Pr(e) = 2ζ 3 sin(2α)e−ζ 2 (cid:27)" ˘Y1 = H † ˘Y2 = H † ˘W1 = H † ˘W2 = H † 2Q dα dζ ζ 2 SNR ζ 2 SNR (cid:28) Thus, at the output of MRRC the signal for kth subchan- nel is ζ 2 SNR dζ 2ζ 3e−ζ 2 (cid:17) ˘Yμ(k) = + arctan (53) (cid:22) ζ 2 SNR (cid:18) (cid:22) 2γ3 − γ2 γ3 1 2 1 π (58)
(cid:23) (cid:23)2(cid:22) Xμ(k) + ˘Wμ(k). (cid:21) ˘Wμ(k) | ρ1, ρ2, θ1, θ2 ∼ 2)σ 2, and the faded signal energy 2)2Es. Thus, the symbol error probabil-
1 + ρ2
(cid:21)(cid:23) (cid:23) (cid:23)H1(k) (cid:23)H2(k) Assuming that Hμ(k) = ρμe− jθμ, N (0, ˘σ 2), where ˘σ 2 = (ρ2 at MRRC ˘Es = (ρ2 ity of QPSK for given ρ1, ρ2, θ1, θ2 is
(cid:22)
(cid:15)!
(cid:15)!
(cid:16)
2 γ3,
= 2Q
− Q2
or by neglecting the Q2(·) term in (58) we get simplified form as Pr (59) Pr(e) = 1 − γ3
(cid:21) e | ρ1, ρ2, θ1, θ2 (cid:16) ˘Es ˘σ 2
(cid:15)!
(cid:16)
(cid:15)!
(cid:16)
= 2Q
1 + ρ2 2)
1 + ρ2 2)
(cid:27)"
− Q2 (cid:28)
(cid:27)"
where ˘Es ˘σ 2 (54) , γ1 = (ρ2 (ρ2 1 2π(SNR +1) ! Es σ 2 Es σ 2
= 2Q
− Q2
(cid:28) .
1 + ρ2
2) SNR
1 + ρ2
2) SNR
(60) , (ρ2 (ρ2
(cid:22)
. SNR γ2 = SNR +2 γ3 = SNR +3 SNR
(cid:21)
=
(cid:22) dθ2dθ1
##
−π π
(cid:21)
(cid:22)
(cid:22)
=
Pr 5. SIMULATIONS Bearing in mind that Pr(e|ρ1, ρ2, θ1, θ2) does not depend on θ1 and θ2, note that (cid:21) e | ρ1, ρ2 ## π Pr e, θ1, θ2 | ρ1, ρ2
(cid:21) θ1
(cid:21) θ2
(cid:22) dθ2dθ1
π
(cid:22)
(cid:22)
= Pr
(55) Pr p p e | ρ1, ρ2, θ1, θ2 ##
(cid:21) θ1
(cid:21) θ2
(cid:22) dθ2dθ1
−π
= Pr
(cid:22) .
−π (cid:21) e | ρ1, ρ2, θ1, θ2 (cid:21) e | ρ1, ρ2, θ1, θ2
p p
##
## ∞
π
(cid:22)
We then substitute (55) in the following equation:
0
(cid:22) dθ2dθ1dρ2dρ1
(cid:21) ρ1, ρ2, θ1, θ2 (cid:21) e | ρ1, ρ2, θ1, θ2
##
## ∞
(cid:22)
=
Pr(e) = p
0
−π × Pr
(56) p In this section, we investigate the performance of the pilot-aided MMSE channel estimation algorithm proposed for both SF-OFDM and ST-OFDM systems. The diversity scheme with two transmit and one receive antenna is consid- ered. Channel impulse responses hμ are generated according to Ch = (1/K 2)F †CHF where CH is the covariance matrix of the doubly-selective fading channel model. In this model, Hμ(k)’s are with an exponentially decaying power-delay pro- file θ(τμ) = C exp(−τμ/τrms) and delays τμ that are uniformly and independently distributed over the length of the cyclic prefix. C is a normalizing constant. Note that the normal- ized discrete channel correlations for different subcarriers and blocks of this channel model were presented in [3] as follows:
## ∞
(cid:2)
(cid:22)(cid:3)
=
c f (k, k(cid:8))
−π × Pr (cid:21) π ρ1, ρ2, θ1, θ2 (cid:22) (cid:21) e | ρ1, ρ2 dθ2dθ1dρ2dρ1 (cid:21) (cid:22) e | ρ1, ρ2
(cid:22) dρ2dρ1.
(cid:21) ρ1, ρ2
(cid:22)(cid:22)(cid:21)
(cid:21)
=
(cid:22) ,
0
(cid:22) /K
(cid:2) − L (cid:21) −L/τrms (cid:21) 2π(n − n(cid:8)) fdTs
## ∞
(cid:22)
(cid:22)
(cid:21)
Pr p 1 − exp (cid:21) 1 − exp k − k(cid:8) τrms 1/τrms + 2π j(k − k(cid:8))/K 1/τrms + 2π j (cid:22) , ct(n, n(cid:8)) = Jo (61) Since channels H1 and H2 are independent, ρ1 and ρ2 are also independent, p(ρ1, ρ2) = p(ρ1)p(ρ2). Thus (56) takes the fol- lowing form:
(cid:21) ρ1
(cid:22) dρ2dρ1
0 ## ∞
=
0
Pr(e) = p p e | ρ1, ρ2 where Jo is the zeroth-order Bessel function of the first kind and fd is the Doppler frequency. Pr (cid:22)
(cid:28)
(cid:21) ρ2 (cid:21) 1+ρ2 ρ2 4ρ1ρ2e− 2 (cid:27)"(cid:21) (cid:27)
(cid:22)
×
(57)
(cid:28)(cid:28)
1 + ρ2 ρ2 2 (cid:27)"(cid:21)
− Q2
1 + ρ2 ρ2 2
2Q SNR (cid:22) SNR dρ2dρ1. The scenario for SF-OFDM simulation study consists of a wireless QPSK OFDM system. The system has a 2.344 MHz bandwidth (for the pulse roll-off factor a = 0.2) and is di- vided into 512 tones with a total period of 136 microseconds, of which 5.12 microseconds constitute the cyclix prefix (L = 20). The uncoded data rate is 7.813 Mbits/s. We assume that
10−2
10−2
) E S M
) E S M
(
(
r o r r e
r o r r e
10−3
10−3
10−4
10−4
e r a u q s - n a e m e g a r e v A
e r a u q s n a e m e g a r e v A
0
5
10
15
20
25
0
5
10
15
20
25
Average SNR (dB)
Average SNR (dB)
Theoretical stochastic CRLB for τrms = 5 MMSE simulation for τrms = 5 Theoretical CRLB MLE simulation for τrms = 5 MMSE simulation for τrms = 9 MMSE simulation for τrms = 9
Theoretical stochastic CRLB MMSE simulation for fd = 0 Hz Theoretical CRLB MLE simulation for fd = 0 Hz MMSE simulation for fd = 100 Hz MLE simulation for fd = 100 Hz
EURASIP Journal on Applied Signal Processing 10
Figure 3: Performance of the proposed MMSE and MLE together with BMSE and CRLB for ST-OFDM.
Figure 4: Performance of the proposed MMSE and MLE together with BMSE and CRLB for ST-OFDM.
10−1
10−2
) R E S (
the rms width is τrms = 5 samples (1.28 microseconds) for the power-delay profile. Keeping the transmission efficiency 3.333 bits/s/Hz fixed, we also simulate ST-OFDM system.
e t a r
10−3
r o r r e
l
10−4
o b m y S
10−5
0
5
10
15
20
25
1 , gT
Average SNR (dB)
5.1. Mean-square-error performance of the channel estimation
The proposed MMSE channel estimators of (23) are imple- mented for both SF-OFDM and ST-OFDM, and compared in terms of average Bayesian MSE for a wide range of signal-to- noise ratio (SNR) levels. Average Bayesian mean-square er- ror(BMSE) is defined as the norm of the difference between 2 ]T and (cid:14)g, representing the true and the the vectors g = [gT estimated values of channel parameters, respectively. Namely,
(cid:6)g − (cid:14)g(cid:6)2.
Theoretical SER MMSE simulation for τrms = 5 MLE simulation for τrms = 5 MMSE simulation for τrms = 9 MLE simulation for τrms = 9
(62) MSE = 1 2L
5.2. MMSE approach
Figure 5: Symbol error rate results for SF-OFDM.
the fact that the orthogonality condition between the subcar- riers at pilot locations is satisfied. In other words, the curves labeled as simulation results for MMSE estimator and ML es- timator fit to the theoretical curve at high SNRs. It also shows that the MMSE-estimated channel SER results are better than ML-estimated channel SER especially at low SNR.
SNR design mismatch
We use a pilot symbol for every ten (Δ = 10) symbols. The MSE at each SNR point is averaged over 10 000 realizations. We compare the experimental MSE performance and its the- oretical Bayesian MSE of the proposed full-rank MMSE es- timator with maximum likelihood (ML) estimator and its corresponding Cramer-Rao lower bound (CRLB) for SF and ST-OFDM systems. Figures 3 and 4 confirm that MMSE esti- mator performs better than ML estimator at low SNR. How- ever, the two approaches have comparable performance at high SNRs. To observe the performance, we also present the MMSE and ML estimated channel SER results together with theoretical SER in Figures 5 and 6. Due to the fact that spaces between the pilot symbols are not chosen as a factor of the number of subcarriers, an error floor is observed in Figures 3, 4, 5, and 6. In the case of choosing the pilot space as a factor of number of subcarriers, the error floor vanishes because of In order to evaluate the performance of the proposed full- rank MMSE estimator to mismatch only in SNR design, the estimator is tested when SNRs of 10 and 30 dB are used in the design. The MSE curves for a design SNR of 10, 30 dB are
10−2
10−1
) E S M
(
10−2
) R E S (
r o r r e
10−3
e t a r
10−3
r o r r e
l
10−4
10−4
o b m y S
e r a u q s - n a e m e g a r e v A
10−5
0
5
10
15
20
25
0
5
10
15
20
25
Average SNR (dB)
Average SNR (dB)
Theoretical stochastic CRLB: SNR design = 30 dB Simulated: SNR design = 30 dB Theoretical stochastic CRLB: SNR design = 10 dB Simulated: SNR design = 10 dB
Theoretical SER MMSE simulation for fd = 0 Hz MLE simulation for fd = 0 Hz MMSE simulation for fd = 100 Hz MLE simulation for fd = 100 Hz
Habib S¸ enol et al. 11
Figure 6: Symbol error rate results for ST-OFDM.
Figure 8: Effects of SNR mismatch on MSE for ST-OFDM ( fd = 100 Hz).
10−2
10−2
) E S M
(
) E S M
(
r o r r e
10−3
r o r r e
10−3
10−4
10−4
e r a u q s - n a e m e g a r e v A
e r a u q s - n a e m e g a r e v A
0
5
10
15
20
25
0
5
10
15
20
25
Average SNR (dB)
Average SNR (dB)
Theoretical stochastic CRLB: SNR design = 30 dB Simulated: SNR design = 30 dB Theoretical stochastic CRLB: SNR design = 10 dB Simulated: SNR design = 10 dB
Theoretical: true correlation for MMSE estimator Theoretical: true correlation for ML estimator Simulated: true correlation for MMSE estimator Simulated: correlation mismatch for MMSE estimator
Figure 7: Effects of SNR mismatch on MSE for SF-OFDM.
Figure 9: Effects of correlation mismatch on MSE for SF-OFDM.
(cid:22)
(cid:21) 2π jLk/K
evaluated for an exponentially decaying power-delay profile. The uniform channel correlation between the attenuations can be obtained by letting τrms → ∞ in (61), resulting in
(63) . c f (k) = 1 − exp 2π jk/K shown in Figures 7 and 8. The performance of the MMSE es- timator for high-SNR (30 dB) design is better than low-SNR (10 dB) design across a range of SNR values (0–28 dB). This result confirms that channel estimation error is concealed in noise for low-SNR whereas it tends to dominate for high- SNR. Thus, the system performance degrades especially for low-SNR design.
Correlation mismatch
To analyze full-rank MMSE estimator’s performance further, we need to study sensitivity of the estimator to design errors, that is, correlation mismatch. We therefore designed the es- timator for a uniform channel correlation which gives the worst MSE performance among all channels [20, 22] and Figures 9 and 10 demonstrate the estimator’s sensitivity to the channel statistics in terms of average MSE performance measure. As can be seen from Figures 9 and 10 only small performance loss is observed for low SNRs when the estima- tor is designed for mismatched channel statistics. This justi- fies the result that a design for worst correlation is robust to mismatch.
10−2
10−2
) E S M
) E S M
(
(
r o r r e
r o r r e
10−3
10−3
10−4
10−4
e r a u q s - n a e m e g a r e v A
e r a u q s - n a e m e g a r e v A
0
5
10
15
20
25
5
15
10
30
25
35
40
Average SNR (dB)
20 Number of KL expansion coefficients
Theoretical: true correlation for MMSE estimator Theoretical: true correlation for ML estimator Simulated: true correlation for MMSE estimator Simulated: correlation mismatch for MMSE estimator
Theoretical stochastic CRLB: SNR = 10 dB Simulated: SNR = 10 dB Theoretical stochastic CRLB: SNR = 20 dB Simulated: SNR = 20 dB Theoretical stochastic CRLB: SNR = 30 dB Simulated: SNR = 30 dB
12 EURASIP Journal on Applied Signal Processing
Figure 10: Effects of correlation mismatch on MSE for ST-OFDM ( fd = 100 Hz).
Figure 12: MSE as a function of KL expansion coefficients for ST- OFDM ( fd = 100 Hz).
10−2
) E S M
(
r o r r e
10−3
10−4
e r a u q s - n a e m e g a r e v A
5
15
10
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40
20 30 Number of KL expansion coefficients
Theoretical stochastic MSE: SNR = 30 dB Simulated: SNR = 30 dB Simulated: SNR = 20 dB Simulated: SNR = 10 dB
6. CONCLUSION
Figure 11: MSE as a function of KL expansion coefficients for SF- OFDM.
We consider the design of low-complexity MMSE channel es- timators for SF/ST-OFDM systems in unknown wireless dis- persive fading channels. We first derive the MMSE estimator based on the stochastic orthogonal expansion representation of the channel via KL transform. Based on such represen- tation, we show that no matrix inversion is needed in the MMSE algorithm. Therefore, the computational cost for im- plementing the proposed MMSE estimator is low and com- putation is numerically stable. Moreover, the performance of our proposed method was first studied through the deriva- tion of stochastic CRLB for Bayesian approach. Since the actual channel statistics and SNR may vary within OFDM block, we have also analyzed the effect of modelling mis- match on the estimator performance and shown both analyt- ically and through simulations that the performance degra- dation due to such mismatch is negligible for low-SNR val- ues. Obvious directions for future work include developing sequential MMSE, Kalman filtering, and sequential Monte- Carlo-based approaches to track channel variations.
ACKNOWLEDGMENTS
Performance of the truncated estimator
This research has been conducted within the NEWCOM Network of Excellence in Wireless Communications funded through the EC 6th Framework Programme. The present work was also supported in part by the Research Fund of Istanbul University Project numbers BYP-938/02032006, and UDP-599/28072005, UDP-582/12072005 and by The Scientific and Technological Research Council of Turkey (T ¨UB˙ITAK) under Grant 104E166.
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The truncated estimator performance is also studied as a function of the number of KL coefficients. Figures 11 and 12 are plotted for L = 40, τrms = 5 samples and L = 40, fd = 100 Hz, respectively. Figures 11 and 12 present the MSE re- sult of the truncated MMSE estimator for SNR = 10, 20, and 30 dB. If only a few expansion coefficients are employed to reduce the complexity of the proposed estimator, then the MSE between channel parameters becomes large. However, if the number of parameters in the expansion is increased, the irreducible error floor still occurs.
Habib S¸ enol et al. 13
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Hakan Ali C¸ ırpan received the B.S. de- gree in 1989 from Uludag University, Bursa, Turkey, the M.S. degree in 1992 from the University of Istanbul, Istanbul, Turkey, and the Ph.D. degree in 1997 from the Stevens Institute of Technology, Hoboken, NJ, USA, all in electrical engineering. From 1995 to 1997, he was a Research Assistant with the Stevens Institute of Technology, working on signal processing algorithms for wireless communication systems. In 1997, he joined the faculty of the De- partment of Electrical-Electronics Engineering at the University of Istanbul. His general research interests cover wireless communica- tions, statistical signal and array processing, system identification and estimation theory. His current research activities are focused on signal processing and communication concepts with specific attention to channel estimation and equalization algorithms for space-time coding and multicarrier(OFDM) systems. He received the Peskin Award from Stevens Institute of Technology as well as the Professor Nazim Terzioglu Award from the Research Fund of the University of Istanbul. He is a Member of IEEE and Member of Sigma Xi.
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