Transmit filter optimization based on partial CSI ... - IEEE Xplore
Transmit Filter Optimization based on Partial CSI Knowledge for Wireless Applications Francesc Rey, Meritxell Lamarca, Gregori V´azquez Department of Signal Theory and Communications, Polytechnic University of Catalonia UPC Campus Nord - M`odul D5, c/Jordi Girona 1-3, 08034 Barcelona (Spain) e-mail:{frey,xell,gregori}@gps.tsc.upc.es
Abstract— This paper deals with closed-loop schemes in OFDM modulation systems. A Bayesian approach is presented to design minimum BER prefiltering matrices in the presence of channel uncertainties. This formulation leads to a robust algorithm that exhibits lower sensitivity to channel estimation errors than classical schemes. The proposed design encompasses the single antenna transmission, the beamforming schemes and the frequency flat fading channels as particular cases.
I. I NTRODUCTION Closed-loop multiple antennas systems are growing in importance because the knowledge of the channel response at the transmitter allows to adapt the modulation and channel coding of the transmitted signal accordingly, obtaining significant performance gains. Nevertheless, the potential of these schemes can not be fully accomplished when neither the transmitter nor the receiver have access to ideal Channel Status Information (CSI). The problem of power allocation for single and multiple antenna transmission in fading channels has been widely studied in the literature assuming ideal CSI. However, only a few references have dealt with the design of algorithms that exhibit low sensitivity to CSI errors (e.g. [1], [2] and [3]) and the design of algorithms that take into account partial knowledge of the CSI is still an open question. This paper deals with closed-loop OFDM MIMO systems. A Bayesian formulation is used to design the robust linear transformation under a minimum BER criterion when the channel estimates at the transmitter are noisy. The proposed solution is based on the SVD decomposition of the ’equivalent channel’. The formulation is quite general, encompassing the single antenna transmission, the beamforming schemes and the frequency flat fading channels as particular cases. Moreover, the proposed algorithm can accept different transmission rates, so it can use the space diversity to improve channel reliability or to increase the transmission rate. The proposed algorithm considers the statistics of channel estimation in the cost function, so it adapts to the channel uncertainty, providing a solution that goes from the configuration for open-loop to the closed-loop with perfect CSI as estimation errors diminish (in a similar way as [3]). This work was partially supported by the European Union through IST-200030116 FITNESS project; the Spanish Government (CICYT) TIC2002-04594-, TIC2001-2356-, TIC2000-1025- C02-01; and CIRIT/Generalitat de Catalunya Grant 2001SGR-00268.
II. P ROBLEM S TATEMENT A. System model This section describes the signal model for an OFDM MIMO communications system over Rayleigh fading channel. The MIMO configuration consists of MT transmit and MR receive antennas, and K subcarriers. The application of OFDM allows to decouple the frequency-selective MIMO channel into K MIMO frequency-flat channels if a certain structure is imposed in the transmitter and receiver. Let xT = [(x1 )T . . . (xK )T ] be the M ·K×1 vector that stacks K blocks xi , where xk contains M symbols to be transmitted in subcarrier k-th and M ≤ min {MT , MR }. The data are assumed to be i.i.d. symbols with zero mean and variance E xxH = σx2 I. The input-output relation, once the cyclic prefix has been removed, can be written in terms of matrices that involve only one subcarrier each as: rk = Hk Fk xk + nk
k = 1...K
(1)
where rk is the MR ×1 vector that contains the symbols received through the different antennas for the k-th subcarrier, Hk is a MR × MT matrix containing the frequency responses of the MIMO channels, Fk is a MT × M matrix that denotes the linear precoding matrix and allocates the power over the MT antennas, and nk is the noise vector after the FFT, which has the same Gaussian statistic as its time-domain = σn2 I. Note counterpart: zero mean and variance E nk nH k that the proposed structure of F has the advantage of providing a scheme where the spatial prefiltering matrix is applied individually to each subcarrier, reducing to the classic OFDM scheme when only one antenna is used at the transmitter. However, it is worth mentioning that it is suboptimum from the diversity point of view, since it does not benefit from the frequency diversity of the channel, a problem that is common to all OFDM systems. In subsequent developments, it will be useful to store the complete channel response in a vector. The MIMO channel response for the k-th subcarrier and for the multicarrier system will be denoted hk and h respectively: T (2) hk = vec {Hk } ; h = hT1 . . . hTK According to this and making use of the identity new notation, vec (ABC) = CT ⊗ A vec (B) [4], the received vector for the k-th subcarrier rk can be rewritten as: T (3) rk = (Fk xk ) ⊗ I hk + nk k = 1 . . . K
0-7803-7802-4/03/$17.00 © 2003 IEEE
2567
B. Channel Model This section introduces the model for the channel response and channel estimates used to design and simulate the power allocation algorithm. The propagation channel is modelled as a Rayleigh fading channel with uncorrelated coefficients for all antennas and all taps in the impulse response, and identical power delay profile for all subchannels. Therefore, the channel vector h is modelled as a multivariate Gaussian process with zero mean and covariance: (4) E hhH = P ⊗ I where P is the circulant matrix built as an Hermitian Toeplitz matrix whose first row is the DFT of the power delay profile. The design of the linear precoder in the presence of channel estimation errors requires the definition of a model for the CSI at the transmitter and the receiver. Considering the channel estimation error ε as an additive term, the estimated channel ˆ is written as: h ˆ =h+ε h (5) where ε can be modelled as a Gaussian process independent of the true channel h with covariance matrix: (6) E εεH = E ⊗ I
where E, with the same structure as P, contains the DFT of the variance in the channel estimation error for each tap. The development of the robust algorithms requires the use of the joint statistics of the channel and its estimate. Assuming ˆ and h vectors are jointly Gaussian, the conditional p.d.f. that h fh/hˆ (h) is also a Gaussian, with mean and covariance given by [5]: −1 ˆ mh|hˆ = P (P + E) ⊗ I h (7) −1 Ch|hˆ = P (P + E) E ⊗ I
is ω = 0. On the contrary, in the extreme case where no CSI ˆ eq = 0 and ω = 1 (where the channel impulse is available, h k response is assumed to be normalized to unity). III. M INIMUM BER D ESIGN This section proposes a design for a linear precoder Fk that minimizes the Bit Error Rate (BER) assuming that a Maximum a Posteriori (MAP) detector is implemented at the receiver. The design can be applied to the MIMO OFDM channel, including single antenna transmission and beamforming as particular cases. The cost function to be optimized is described for a general constellation {s1 . . . sN } of size N , even though final equations are shown for the particular case of QPSK modulation. In order to be able to derive a closed form solution, two main assumptions have been made: A1) The receiver is operating at high SNR. √ A2) The function Q ( x) can be approximated by: √ Q x δe−αx (11) where constants α and δ can be chosen to set certain constraints. The Chernoff bound can be regarded as a particular case for α = 1/2 and δ = 1/2. However, √ it is also possible to derive a lower bound for Q ( x) that is very tight in a wide range of values √ around a, following a Taylor expansion of ln (Q ( x)) in the neighborhood of x = a. In this case α and δ are: √ e−|a|/2 α= √ ; δ = Q a eαa √ 2 2πaQ ( a)
For a given channel realization, the BER for the Maximum Likelihood (ML) receiver can be written in terms of the pairwise probability of detecting symbol xkj when the symbol xki was transmitted:
Next section will focus on the conditional covariance for a specific subcarrier hk , that is given by: Chk |hˆk = ωI
(8)
where ω is a scalar that denotes the (k, k)-th element of −1 P (P + E) E and it is independent of the k-th subcarrier, since E and P are circulant matrices. A detailed analysis of (7) shows that the conditional mean can be understood as an equivalent channel that exploits the correlation between subcarriers and the channel uncertainty structure to mitigate the degradation due to CSI errors [5]. For each subcarrier the conditional mean can be expressed as a linear combination of the channel estimates for all subcarriers: ˆ eq = m ˆ = h k h k |h
K
ˆj βk (j) h
(9)
j=1
where βk (j) is the j-th element of vector β k defined as: β Tk = pk (P + E)
−1
(10)
and pk is the k-th row of P. When perfect CSI is available at the transmitter, the equivaˆ k = hk whereas the covariance ˆ eq = h lent channel becomes: h k
(12)
BER =
M
1
N 2M K
K N
B(xki , xkj )P (xki → xkj ) (13)
k=1 i,j=1
where B(xki , xkj ) denotes the number of bits that are different in xki and xkj . The robustness of the algorithm is obtained averaging the BER over channel uncertainty using a Bayesian formulation: M
min ξ =
N
1 2M
K
K N
k=1 i,j=1
B(xki , xkj )Eh/hˆ {P (xki → xkj )} subject to
K
k=1
Tr FH k Fk = P0 (14)
A. Cost Function The pairwise error probability (PEP) in (14) can be written in terms of the Euclidean distance between transmitted codewords xki and xkj as they appear at the receiver, and is upper bounded by:
1 2 P (xki → xkj ) ≤ Q (15) |Hk Fk eijk | 2σn2
2568
where eijk = (xki − xkj ). Accordingly, under assumption A2), the PEP can be approximated by: −αk
P (xki → xkj ) δk e
1 |Hk Fk eijk |2 2σn2
(16)
This equation can be rewritten in a more compact form in order to simplify the notation in subsequent equations: P (xki
H → xkj ) δk e−h Mijk h
(17)
where Mijk is the QMT MR × QMT MR matrix defined 0 ... 0 ... .. . . .. . . . H (Fk eijk ) (Fk eijk ) ⊗ I ... Mijk = γk 0 . . . . . .. .. .. . 0 ... 0 ...
as:
0 .. . 0 .. . 0 (18) 0 is a square all zero matrix of size MT MR , I is the identity matrix of size MR and γk = αk 2σ12 . n Using the results in (7) to expand the Gaussian p.d.f. fh/hˆ , the averaged PEP becomes: Eh/hˆ {P (xki → xkj )} = ˆ eq )H C−1 (h−h ˆ eq ) H δk −(h−h ˆ h/h e−h Mijk h e dh K π |Ch/hˆ | h∈C
(19) ˆ eq is defined in (9). Previous integral can be easily where h solved rewriting its integrand as: H (20) e− (h − µ) β (h − µ) − η dh h∈C
where:
β = Mijk + C−1 ˆ h/h −1 ˆ eq h µ = Mijk Ch/hˆ + I −1 ˆ eq ˆ eqH Mijk C ˆ + I η=h Mijk h h/h
(21)
The solution to the integral in (20) can be found by comparing its integrand with a complex Gaussian p.d.f., whose integral equals to one. Accordingly, (19) becomes: Eh/hˆ {P (xki → xkj )} =
ˆ eqH (Mijk C
δk e−h
)
ˆ +I h/h
−1
ˆ eq Mijk h
|Ch/hˆ Mijk + I|
(22) And the function to minimize, subject to a power constraint, becomes: ξ=
1 N 2M K
M
K N
B(xki , xkj ) δk
−
2
1 + γk ω |Fk eijk | k=1 i,j=1
e
2
Fk eijk | |Hˆ eq k 2 1+γk ω |Fk eijk |
γk
(23) where (18) and vec (ABC) = CT ⊗ A vec (B) has been applied and ω is obtained from (8).
B. Power Allocation Design This section derives a suboptimal solution for the minimization of (23). As the direct optimization of (23) leads to very intricate equations, the minimization has been obtained assuming the following structure for Fk : Fk = Vk Φk T
(24)
where Vk contains the right singular vectors of the equivalent channel matrix, Φk is a diagonal matrix and T is a M × M unitary matrix. This configuration has been proved to be optimum for minimum BER in a zero forcing receiver with perfect CSI [6], and can also be shown to be optimum for minimum BER in a MAP receiver when multiple transmit antennas are used for beamforming [7], but it has not been proved to be optimum when multiple symbols are transmitted at the same time. However, there are several reasons to support its choice. It keeps the same structure as other papers published previously. It leads to a simple suboptimum closed form design, as will be shown next. Finally, the introduction of matrix T alleviates the main drawback of the use of matrices Vk and Φk : the loss of space diversity caused by the decomposition of the MIMO channel into a set of parallel multiplicative subchannels. In the context of frequency flat Rayleigh fading channels, it is known that the receiver could benefit from the diversity of the fading channel provided the transmitter used a linear transformation that spread the symbols in time, obtaining significant performance gains [8]. In this paper T has been set to the DFT matrix. This choice is supported by [8, eq.17], where it is shown that the DFT or the Walsh-Hadamard matrices could provide the desired fading diversity with minimum complexity. Besides, the same condition was required for optimum T design in the zero forcing receiver [6, Lemma 1]. Once the structure of Fk has been forced to be (24), the design of Fk reduces to design matrix Φk . Unfortunately a closed form solution for the cost function resulting from (23)(24) can not be found. In this paper the BER optimization is achieved by means of a gradient algorithm [9]. Alternatively, a closed form solution is also derived for high SNR replacing the true BER by an upper bound as shown next. Although this solution is suboptimum for M > 1 it can be shown that it reduces to the optimum solution when M = 1 (beamforming) [7]. As a result of the√application of Jensen’s inequality to the convex function Q ( x), it can be shown that the following inequality holds for high SNR: M N 2 |Λ Φ Te | k k ijk 1 ≤ B(xki , xkj )Q N 2M 2σn2 i,j=1 (25) M N 2 |Λ Φ e | k k ijk 1 B(xki , xkj )Q N 2M 2σn2 i,j=1
where equality applies if Λk Φk is proportional to the identity matrix (zero forcing solution). Therefore, if the true BER is replaced by the upper bound, an expression is obtained that does not depend on T, allowing to get a simple closed form expression for Φk .
2569
Expanding the cost function in (23) according to the structure of Fk , the new cost function, including the average power constraint (14) becomes: ξ=
1
N 2M K
M
K N
B(xki , xkj ) δk
−
2 e 1 + γk ω |Φk eijk | k=1 i,j=1 K M 2 −µ |φk (l)| − P0
2 eq γk Λ Φk eijk k 2
|
|
1+γk ω Φk eijk
| |
which is the same one found in [1] for uncorrelated antennas. Indeed, as it will be seen in the simulations, the power allocation tends to the open loop solution.
k=1 l=1
(26) where µ is the Lagrange multiplier. As the cost function depends on |φk (l)|2 we will choose φk (l) as a real positive variable. The minimization problem follows deriving the cost function with respect to φk (l): M
N B(xki , xkj ) δk γk |eijk (l)|2 φk (l) ∂ξ 1 = − 2M 2 N K i,j=1 ∂φk (l) 2 1 + γk ω |Φk eijk | 2 eq γk |Λ Φk eijk | k 2 − 2 γk ω |Λeq k Φk eijk | e 1+γk ω|Φk eijk | ω + λeq (l)2 − 1+γ ω|Φ − µφk (l) 2 e | k
When the channel uncertainty is very high, in the extreme ˆ eq = 0 and ω = 1, the robust BER cost function case of H k converges to: δk Eh/hˆ {P (xki → xkj )} = (30) |Ch/hˆ Mijk + I|
k ijk
(27) where eijk (l) denotes the l-th element in vector eijk and ˆ eq . λeq (l) is the l-th singular value of channel matrix H k Under the high SNR assumption A1), the error probability is dominated by the minimum distance between any pairs of symbols (xki ,xkj ). Therefore, the summation in (27) can be approximated considering the terms that only differ in one symbol, i.e. xki − xkj = dim , where im is one of the columns of the identity matrix and d is the minimum distance between any two constellation symbols. Using this approximation in equation (27), and setting the derivative to zero, the√following equation is obtained for QPSK constellation (d = 2σx ):
γ˜k λeq (l)2 φk (l)2 − 2 δk γ˜k e 1 + γ˜k ωφk (l) λeq (l)2 −K ω + =µ 2 1 + γ˜k ωφk (l)2 (1 + γ˜k ωφk (l)2 ) (28) where γ˜k = 2σx2 γk and K is a non-relevant constant. A closed form solution for this identity can not be derived. However under the assumption that channel uncertainty is low, the following approximations can be used after applying the logarithmic function at both sides of the equality: ln (1 + x)
x and 1/ (1 + x) 1, obtaining a closed form solution for φk (l)2 : + µ − ln (δk γ˜k ) − ln ω + λeq (l)2 2 φk (l) = − (29) γ˜k (2ω + λeq (l)2 )
IV. S IMULATION R ESULTS In order to illustrate the performance of the proposed algorithm, simulations were carried out for different scenarios. The simulation parameters were selected according to HIPERLAN/2 standard. The bit stream to be transmitted was mapped into a QPSK constellation, multiplexed in MT antennas and modulated in OFDM symbols including pilot tones and empty carriers (see HIPERLAN/2 standard). The simulated Rayleigh MIMO channel obeys an exponential power delay profile with 50ns of delay spread, typical values for an office indoor scenario. Channel uncertainty modelled channel estimation errors due to noise and to channel prediction error in timevarying channels. In both cases the variance in the channel estimation error for each tap σ2 , was assumed to be constant for all taps of the channel impulse response. Channel uncertainty at the transmitter was assumed to be independent of the SNR’s as would be the case of channel tracker based on linear prediction, whereas channel estimation error at the receiver was assumed to be proportional to the noise variance σ2 = χσn2 , as would be the case in a linear channel estimator. All simulations were carried out using the MAP detector at the receiver. A. Performance vs EbNo Figure 1 compares the performance in terms of BER of the algorithms proposed in this paper with the MMSE design in [5] for the 3x3 antenna configuration with M = 3. In order to show the losses due to the mismatch between the channel response and the transmitter pre-filtering matrix, it was assumed that the receiver had perfect CSI knowledge, whereas the channel estimate at the transmitter had constant variance equal to σ2 = 0.02. Note that, the iterative algorithm outperforms the suboptimum closed form solution. This different performance is due to the application of the upper bound in (25) and the approximation of the BER in (13) considering only the contribution of minimum distance terms. Both designs have better performance than the MMSE and the Non-Robust power allocation polices.
where µ is determined forcing the power constraint (14). C. Asymptotic performance In this section the robust BER algorithm is analyzed for the extreme cases where the uncertainty is very high or very low. When channel knowledge at the transmitter is perfect, ˆ eq = H and ω = 0, and only one symbol is i.e. when H k transmitted per subcarrier (i.e. M = 1), the solution for the robust cost function in (29) coincides with that one of [10].
B. Performance vs CSI quality at the transmitter To test the performance of the minimum BER design when channel uncertainty increases, Fig. 2 shows the minimum EbNo required to achieve a BER ≤ 10−3 for different MIMO configurations (1x1, 2x2,3x3). The required EbNo is plotted as a function of the channel uncertainty degree at the transmitter ρ defined as: H ˆ ˆ h (31) ρ = E εH ε /E h
2570
MIMO 3x3 − UncTx 0.02 (per tap) ; UncRx Perfect CSI
0
10
BER Robust Iterative Solution BER Non−Robust BER Robust Closed Form Solution MMSE Non−Robust MMSE Robust
35
30
−1
Target EbNo (dB) − BER=10−3
Raw BER
10
−2
10
−3
25 Open Loop − 1x1
Non Robust Algorithm. MIMO 1x1 Robust Algorithm. MIMO 1x1 Non Robust Algorithm. MIMO 2x2 Robust Algorithm. MIMO 2x2 Non Robust Algorithm. MIMO 3x3 Robust Algorithm. MIMO 3x3
5
0
−4
10
−5
0
10
5
15
20
3x3
15
10
10
2x2
20
0.1
0.2
0.3
0.4
0.5 0.6 Uncertainty ρ
0.7
0.8
0.9
1
Eb/No (dB)
Fig. 1. Raw BER comparison between different power allocation strategies. 2 = 0.02 Receiver MT = 3, MR = 3. Transmitter uncertainty: σn uncertainty: Perfect CSI
where ρ = 0 denotes perfect CSI, whereas ρ = 1 means no channel knowledge. The receiver uncertainty was fixed to σ2 = σn2 /16. As a reference the point ρ = 0 was simulated assuming perfect CSI at both transmitter and receiver. The robustness of the proposed algorithm is evidenced since ’BER Robust’ solution always outperforms ’BER Non-Robust’ one. Note that the ’BER Robust’ design tends to the openloop solution (i.e. equally power allocation for all subcarriers and antennas) when CSI quality at the transmitter degrades (ρ → 1). A second conclusion from Fig. 2 is drawn when comparing the performance for different MIMO scenarios. Note that when the number of antennas increases, the relative gain between the robust and non-robust solutions is substantially enlarged. Accordingly, the robust solution becomes an interesting scheme not only when the CSI quality is highly degraded, but also when ρ is close to zero. V. C ONCLUSIONS This paper has presented a Bayesian approach to the design of a linear pre-coding scheme that is robust to channel estimation errors. The linear transformation has been designed following a minimum BER criterion subject to an average power constraint. The minimum BER design has been formulated √ based on a generalized exponential bound of the function Q( x) that includes the Chernoff bound as a particular case. An iterative algorithm and a suboptimum closed-form solution have been proposed, that is based on the SVD of the so called equivalent channel, and it has been proved to converge to results published previously in the literature in the extreme cases of very high or very low uncertainty in the CSI. Besides, it has been shown that the design can be regarded as a reconfigurable algorithm that adapts the transmitted signal to the available channel knowledge, providing a solution that converges to the open-loop design for the case of no channel
Fig. 2. Minimum EbNo that achieves Raw BER ≤ 10−3 . Transmitter 2 /16. uncertainty defined by ρ. Receiver uncertainty σ2 = σn
knowledge and to the closed-loop design with perfect CSI for the case of no uncertainty. R EFERENCES [1] H.Sampath and A.Paulraj, “Linear Precoding for Space-Time Coded Systems With Known Fading Correlations,” IEEE Communications Letters, vol. 6, no. 6, pp. 239–241, Jun. 2002. [2] M.Iarossi O.Gasparini, E.Marinis, “Evaluation of MIMO spatial multiplexing for Wireless LAN with channel models from experimental data,” in Proc. of IST Mobile & Wireless Telecommunications Summit, Thessaloniki (Grece), Jun. 2002. [3] G. Jongren, M. Skoglund, and B. Ottersten, “Combining Beamforming and Orthogonal Space-Time Block Coding,” IEEE Trans. on Information Theory, vol. 48, no. 3, pp. 611–627, Mar. 2002. [4] J.R. Magnus and H. Neudecker, Matrix Differential Calculus with Applications in Statistics and Econometrics,, John Wiley, 1999. [5] F. Rey M. Lamarca and G. V´azquez, “Optimal Power Allocation with Partial Channel Knowledge for MIMO Multicarrier Systems,” in Proceedings of VTC-fall’02, Vancouver (Canada), Sep. 2002. [6] Y.Ding, T.Davidson, J.Zang, Z Luo, and K.Wong, “Minimum BER Block Precoders for Zero-Forcing Equalization,” in Proc. of ICASSP’02, Orlando (USA), May 2002, pp. 2261–2264. [7] FITNESS IST-2000-30116 project, “Performance analysis of reconfigurable MTMR transceivers for WLAN,” Deliverable D3.2.1 available at http://www.ist-fitness.org, Nov. 2002. [8] M.Lamarca and G.V´azquez, “Transform Modulations for Mobile Communications,” in Proc. of PIMRC’97, Helsinki (Finland), Sep. 1997, pp. 462–466. [9] O.L. Frost, “An Algorithm for Linearly Constrained Adaptive Array Processing,” Proc. of IEEE, vol. 60, no. 8, pp. 926–935, Aug. 1972. [10] E. N. Onggosanusi, A. M. Sayeed, and B. D. Van Veen, “Efficient Signaling Schemes for Wideband Space-Time Wireless Channels Using Channel Side Information,” Submitted to IEEE Trans. on Communications, Jan. 2001.
2571
Abstract— This paper deals with closed-loop schemes in OFDM modulation systems. A Bayesian approach is presented to design minimum BER prefiltering matrices in the presence of channel uncertainties. This formulation leads to a robust algorithm that exhibits lower sensitivity to channel estimation errors than classical schemes. The proposed design encompasses the single antenna transmission, the beamforming schemes and the frequency flat fading channels as particular cases.
I. I NTRODUCTION Closed-loop multiple antennas systems are growing in importance because the knowledge of the channel response at the transmitter allows to adapt the modulation and channel coding of the transmitted signal accordingly, obtaining significant performance gains. Nevertheless, the potential of these schemes can not be fully accomplished when neither the transmitter nor the receiver have access to ideal Channel Status Information (CSI). The problem of power allocation for single and multiple antenna transmission in fading channels has been widely studied in the literature assuming ideal CSI. However, only a few references have dealt with the design of algorithms that exhibit low sensitivity to CSI errors (e.g. [1], [2] and [3]) and the design of algorithms that take into account partial knowledge of the CSI is still an open question. This paper deals with closed-loop OFDM MIMO systems. A Bayesian formulation is used to design the robust linear transformation under a minimum BER criterion when the channel estimates at the transmitter are noisy. The proposed solution is based on the SVD decomposition of the ’equivalent channel’. The formulation is quite general, encompassing the single antenna transmission, the beamforming schemes and the frequency flat fading channels as particular cases. Moreover, the proposed algorithm can accept different transmission rates, so it can use the space diversity to improve channel reliability or to increase the transmission rate. The proposed algorithm considers the statistics of channel estimation in the cost function, so it adapts to the channel uncertainty, providing a solution that goes from the configuration for open-loop to the closed-loop with perfect CSI as estimation errors diminish (in a similar way as [3]). This work was partially supported by the European Union through IST-200030116 FITNESS project; the Spanish Government (CICYT) TIC2002-04594-, TIC2001-2356-, TIC2000-1025- C02-01; and CIRIT/Generalitat de Catalunya Grant 2001SGR-00268.
II. P ROBLEM S TATEMENT A. System model This section describes the signal model for an OFDM MIMO communications system over Rayleigh fading channel. The MIMO configuration consists of MT transmit and MR receive antennas, and K subcarriers. The application of OFDM allows to decouple the frequency-selective MIMO channel into K MIMO frequency-flat channels if a certain structure is imposed in the transmitter and receiver. Let xT = [(x1 )T . . . (xK )T ] be the M ·K×1 vector that stacks K blocks xi , where xk contains M symbols to be transmitted in subcarrier k-th and M ≤ min {MT , MR }. The data are assumed to be i.i.d. symbols with zero mean and variance E xxH = σx2 I. The input-output relation, once the cyclic prefix has been removed, can be written in terms of matrices that involve only one subcarrier each as: rk = Hk Fk xk + nk
k = 1...K
(1)
where rk is the MR ×1 vector that contains the symbols received through the different antennas for the k-th subcarrier, Hk is a MR × MT matrix containing the frequency responses of the MIMO channels, Fk is a MT × M matrix that denotes the linear precoding matrix and allocates the power over the MT antennas, and nk is the noise vector after the FFT, which has the same Gaussian statistic as its time-domain = σn2 I. Note counterpart: zero mean and variance E nk nH k that the proposed structure of F has the advantage of providing a scheme where the spatial prefiltering matrix is applied individually to each subcarrier, reducing to the classic OFDM scheme when only one antenna is used at the transmitter. However, it is worth mentioning that it is suboptimum from the diversity point of view, since it does not benefit from the frequency diversity of the channel, a problem that is common to all OFDM systems. In subsequent developments, it will be useful to store the complete channel response in a vector. The MIMO channel response for the k-th subcarrier and for the multicarrier system will be denoted hk and h respectively: T (2) hk = vec {Hk } ; h = hT1 . . . hTK According to this and making use of the identity new notation, vec (ABC) = CT ⊗ A vec (B) [4], the received vector for the k-th subcarrier rk can be rewritten as: T (3) rk = (Fk xk ) ⊗ I hk + nk k = 1 . . . K
0-7803-7802-4/03/$17.00 © 2003 IEEE
2567
B. Channel Model This section introduces the model for the channel response and channel estimates used to design and simulate the power allocation algorithm. The propagation channel is modelled as a Rayleigh fading channel with uncorrelated coefficients for all antennas and all taps in the impulse response, and identical power delay profile for all subchannels. Therefore, the channel vector h is modelled as a multivariate Gaussian process with zero mean and covariance: (4) E hhH = P ⊗ I where P is the circulant matrix built as an Hermitian Toeplitz matrix whose first row is the DFT of the power delay profile. The design of the linear precoder in the presence of channel estimation errors requires the definition of a model for the CSI at the transmitter and the receiver. Considering the channel estimation error ε as an additive term, the estimated channel ˆ is written as: h ˆ =h+ε h (5) where ε can be modelled as a Gaussian process independent of the true channel h with covariance matrix: (6) E εεH = E ⊗ I
where E, with the same structure as P, contains the DFT of the variance in the channel estimation error for each tap. The development of the robust algorithms requires the use of the joint statistics of the channel and its estimate. Assuming ˆ and h vectors are jointly Gaussian, the conditional p.d.f. that h fh/hˆ (h) is also a Gaussian, with mean and covariance given by [5]: −1 ˆ mh|hˆ = P (P + E) ⊗ I h (7) −1 Ch|hˆ = P (P + E) E ⊗ I
is ω = 0. On the contrary, in the extreme case where no CSI ˆ eq = 0 and ω = 1 (where the channel impulse is available, h k response is assumed to be normalized to unity). III. M INIMUM BER D ESIGN This section proposes a design for a linear precoder Fk that minimizes the Bit Error Rate (BER) assuming that a Maximum a Posteriori (MAP) detector is implemented at the receiver. The design can be applied to the MIMO OFDM channel, including single antenna transmission and beamforming as particular cases. The cost function to be optimized is described for a general constellation {s1 . . . sN } of size N , even though final equations are shown for the particular case of QPSK modulation. In order to be able to derive a closed form solution, two main assumptions have been made: A1) The receiver is operating at high SNR. √ A2) The function Q ( x) can be approximated by: √ Q x δe−αx (11) where constants α and δ can be chosen to set certain constraints. The Chernoff bound can be regarded as a particular case for α = 1/2 and δ = 1/2. However, √ it is also possible to derive a lower bound for Q ( x) that is very tight in a wide range of values √ around a, following a Taylor expansion of ln (Q ( x)) in the neighborhood of x = a. In this case α and δ are: √ e−|a|/2 α= √ ; δ = Q a eαa √ 2 2πaQ ( a)
For a given channel realization, the BER for the Maximum Likelihood (ML) receiver can be written in terms of the pairwise probability of detecting symbol xkj when the symbol xki was transmitted:
Next section will focus on the conditional covariance for a specific subcarrier hk , that is given by: Chk |hˆk = ωI
(8)
where ω is a scalar that denotes the (k, k)-th element of −1 P (P + E) E and it is independent of the k-th subcarrier, since E and P are circulant matrices. A detailed analysis of (7) shows that the conditional mean can be understood as an equivalent channel that exploits the correlation between subcarriers and the channel uncertainty structure to mitigate the degradation due to CSI errors [5]. For each subcarrier the conditional mean can be expressed as a linear combination of the channel estimates for all subcarriers: ˆ eq = m ˆ = h k h k |h
K
ˆj βk (j) h
(9)
j=1
where βk (j) is the j-th element of vector β k defined as: β Tk = pk (P + E)
−1
(10)
and pk is the k-th row of P. When perfect CSI is available at the transmitter, the equivaˆ k = hk whereas the covariance ˆ eq = h lent channel becomes: h k
(12)
BER =
M
1
N 2M K
K N
B(xki , xkj )P (xki → xkj ) (13)
k=1 i,j=1
where B(xki , xkj ) denotes the number of bits that are different in xki and xkj . The robustness of the algorithm is obtained averaging the BER over channel uncertainty using a Bayesian formulation: M
min ξ =
N
1 2M
K
K N
k=1 i,j=1
B(xki , xkj )Eh/hˆ {P (xki → xkj )} subject to
K
k=1
Tr FH k Fk = P0 (14)
A. Cost Function The pairwise error probability (PEP) in (14) can be written in terms of the Euclidean distance between transmitted codewords xki and xkj as they appear at the receiver, and is upper bounded by:
1 2 P (xki → xkj ) ≤ Q (15) |Hk Fk eijk | 2σn2
2568
where eijk = (xki − xkj ). Accordingly, under assumption A2), the PEP can be approximated by: −αk
P (xki → xkj ) δk e
1 |Hk Fk eijk |2 2σn2
(16)
This equation can be rewritten in a more compact form in order to simplify the notation in subsequent equations: P (xki
H → xkj ) δk e−h Mijk h
(17)
where Mijk is the QMT MR × QMT MR matrix defined 0 ... 0 ... .. . . .. . . . H (Fk eijk ) (Fk eijk ) ⊗ I ... Mijk = γk 0 . . . . . .. .. .. . 0 ... 0 ...
as:
0 .. . 0 .. . 0 (18) 0 is a square all zero matrix of size MT MR , I is the identity matrix of size MR and γk = αk 2σ12 . n Using the results in (7) to expand the Gaussian p.d.f. fh/hˆ , the averaged PEP becomes: Eh/hˆ {P (xki → xkj )} = ˆ eq )H C−1 (h−h ˆ eq ) H δk −(h−h ˆ h/h e−h Mijk h e dh K π |Ch/hˆ | h∈C
(19) ˆ eq is defined in (9). Previous integral can be easily where h solved rewriting its integrand as: H (20) e− (h − µ) β (h − µ) − η dh h∈C
where:
β = Mijk + C−1 ˆ h/h −1 ˆ eq h µ = Mijk Ch/hˆ + I −1 ˆ eq ˆ eqH Mijk C ˆ + I η=h Mijk h h/h
(21)
The solution to the integral in (20) can be found by comparing its integrand with a complex Gaussian p.d.f., whose integral equals to one. Accordingly, (19) becomes: Eh/hˆ {P (xki → xkj )} =
ˆ eqH (Mijk C
δk e−h
)
ˆ +I h/h
−1
ˆ eq Mijk h
|Ch/hˆ Mijk + I|
(22) And the function to minimize, subject to a power constraint, becomes: ξ=
1 N 2M K
M
K N
B(xki , xkj ) δk
−
2
1 + γk ω |Fk eijk | k=1 i,j=1
e
2
Fk eijk | |Hˆ eq k 2 1+γk ω |Fk eijk |
γk
(23) where (18) and vec (ABC) = CT ⊗ A vec (B) has been applied and ω is obtained from (8).
B. Power Allocation Design This section derives a suboptimal solution for the minimization of (23). As the direct optimization of (23) leads to very intricate equations, the minimization has been obtained assuming the following structure for Fk : Fk = Vk Φk T
(24)
where Vk contains the right singular vectors of the equivalent channel matrix, Φk is a diagonal matrix and T is a M × M unitary matrix. This configuration has been proved to be optimum for minimum BER in a zero forcing receiver with perfect CSI [6], and can also be shown to be optimum for minimum BER in a MAP receiver when multiple transmit antennas are used for beamforming [7], but it has not been proved to be optimum when multiple symbols are transmitted at the same time. However, there are several reasons to support its choice. It keeps the same structure as other papers published previously. It leads to a simple suboptimum closed form design, as will be shown next. Finally, the introduction of matrix T alleviates the main drawback of the use of matrices Vk and Φk : the loss of space diversity caused by the decomposition of the MIMO channel into a set of parallel multiplicative subchannels. In the context of frequency flat Rayleigh fading channels, it is known that the receiver could benefit from the diversity of the fading channel provided the transmitter used a linear transformation that spread the symbols in time, obtaining significant performance gains [8]. In this paper T has been set to the DFT matrix. This choice is supported by [8, eq.17], where it is shown that the DFT or the Walsh-Hadamard matrices could provide the desired fading diversity with minimum complexity. Besides, the same condition was required for optimum T design in the zero forcing receiver [6, Lemma 1]. Once the structure of Fk has been forced to be (24), the design of Fk reduces to design matrix Φk . Unfortunately a closed form solution for the cost function resulting from (23)(24) can not be found. In this paper the BER optimization is achieved by means of a gradient algorithm [9]. Alternatively, a closed form solution is also derived for high SNR replacing the true BER by an upper bound as shown next. Although this solution is suboptimum for M > 1 it can be shown that it reduces to the optimum solution when M = 1 (beamforming) [7]. As a result of the√application of Jensen’s inequality to the convex function Q ( x), it can be shown that the following inequality holds for high SNR: M N 2 |Λ Φ Te | k k ijk 1 ≤ B(xki , xkj )Q N 2M 2σn2 i,j=1 (25) M N 2 |Λ Φ e | k k ijk 1 B(xki , xkj )Q N 2M 2σn2 i,j=1
where equality applies if Λk Φk is proportional to the identity matrix (zero forcing solution). Therefore, if the true BER is replaced by the upper bound, an expression is obtained that does not depend on T, allowing to get a simple closed form expression for Φk .
2569
Expanding the cost function in (23) according to the structure of Fk , the new cost function, including the average power constraint (14) becomes: ξ=
1
N 2M K
M
K N
B(xki , xkj ) δk
−
2 e 1 + γk ω |Φk eijk | k=1 i,j=1 K M 2 −µ |φk (l)| − P0
2 eq γk Λ Φk eijk k 2
|
|
1+γk ω Φk eijk
| |
which is the same one found in [1] for uncorrelated antennas. Indeed, as it will be seen in the simulations, the power allocation tends to the open loop solution.
k=1 l=1
(26) where µ is the Lagrange multiplier. As the cost function depends on |φk (l)|2 we will choose φk (l) as a real positive variable. The minimization problem follows deriving the cost function with respect to φk (l): M
N B(xki , xkj ) δk γk |eijk (l)|2 φk (l) ∂ξ 1 = − 2M 2 N K i,j=1 ∂φk (l) 2 1 + γk ω |Φk eijk | 2 eq γk |Λ Φk eijk | k 2 − 2 γk ω |Λeq k Φk eijk | e 1+γk ω|Φk eijk | ω + λeq (l)2 − 1+γ ω|Φ − µφk (l) 2 e | k
When the channel uncertainty is very high, in the extreme ˆ eq = 0 and ω = 1, the robust BER cost function case of H k converges to: δk Eh/hˆ {P (xki → xkj )} = (30) |Ch/hˆ Mijk + I|
k ijk
(27) where eijk (l) denotes the l-th element in vector eijk and ˆ eq . λeq (l) is the l-th singular value of channel matrix H k Under the high SNR assumption A1), the error probability is dominated by the minimum distance between any pairs of symbols (xki ,xkj ). Therefore, the summation in (27) can be approximated considering the terms that only differ in one symbol, i.e. xki − xkj = dim , where im is one of the columns of the identity matrix and d is the minimum distance between any two constellation symbols. Using this approximation in equation (27), and setting the derivative to zero, the√following equation is obtained for QPSK constellation (d = 2σx ):
γ˜k λeq (l)2 φk (l)2 − 2 δk γ˜k e 1 + γ˜k ωφk (l) λeq (l)2 −K ω + =µ 2 1 + γ˜k ωφk (l)2 (1 + γ˜k ωφk (l)2 ) (28) where γ˜k = 2σx2 γk and K is a non-relevant constant. A closed form solution for this identity can not be derived. However under the assumption that channel uncertainty is low, the following approximations can be used after applying the logarithmic function at both sides of the equality: ln (1 + x)
x and 1/ (1 + x) 1, obtaining a closed form solution for φk (l)2 : + µ − ln (δk γ˜k ) − ln ω + λeq (l)2 2 φk (l) = − (29) γ˜k (2ω + λeq (l)2 )
IV. S IMULATION R ESULTS In order to illustrate the performance of the proposed algorithm, simulations were carried out for different scenarios. The simulation parameters were selected according to HIPERLAN/2 standard. The bit stream to be transmitted was mapped into a QPSK constellation, multiplexed in MT antennas and modulated in OFDM symbols including pilot tones and empty carriers (see HIPERLAN/2 standard). The simulated Rayleigh MIMO channel obeys an exponential power delay profile with 50ns of delay spread, typical values for an office indoor scenario. Channel uncertainty modelled channel estimation errors due to noise and to channel prediction error in timevarying channels. In both cases the variance in the channel estimation error for each tap σ2 , was assumed to be constant for all taps of the channel impulse response. Channel uncertainty at the transmitter was assumed to be independent of the SNR’s as would be the case of channel tracker based on linear prediction, whereas channel estimation error at the receiver was assumed to be proportional to the noise variance σ2 = χσn2 , as would be the case in a linear channel estimator. All simulations were carried out using the MAP detector at the receiver. A. Performance vs EbNo Figure 1 compares the performance in terms of BER of the algorithms proposed in this paper with the MMSE design in [5] for the 3x3 antenna configuration with M = 3. In order to show the losses due to the mismatch between the channel response and the transmitter pre-filtering matrix, it was assumed that the receiver had perfect CSI knowledge, whereas the channel estimate at the transmitter had constant variance equal to σ2 = 0.02. Note that, the iterative algorithm outperforms the suboptimum closed form solution. This different performance is due to the application of the upper bound in (25) and the approximation of the BER in (13) considering only the contribution of minimum distance terms. Both designs have better performance than the MMSE and the Non-Robust power allocation polices.
where µ is determined forcing the power constraint (14). C. Asymptotic performance In this section the robust BER algorithm is analyzed for the extreme cases where the uncertainty is very high or very low. When channel knowledge at the transmitter is perfect, ˆ eq = H and ω = 0, and only one symbol is i.e. when H k transmitted per subcarrier (i.e. M = 1), the solution for the robust cost function in (29) coincides with that one of [10].
B. Performance vs CSI quality at the transmitter To test the performance of the minimum BER design when channel uncertainty increases, Fig. 2 shows the minimum EbNo required to achieve a BER ≤ 10−3 for different MIMO configurations (1x1, 2x2,3x3). The required EbNo is plotted as a function of the channel uncertainty degree at the transmitter ρ defined as: H ˆ ˆ h (31) ρ = E εH ε /E h
2570
MIMO 3x3 − UncTx 0.02 (per tap) ; UncRx Perfect CSI
0
10
BER Robust Iterative Solution BER Non−Robust BER Robust Closed Form Solution MMSE Non−Robust MMSE Robust
35
30
−1
Target EbNo (dB) − BER=10−3
Raw BER
10
−2
10
−3
25 Open Loop − 1x1
Non Robust Algorithm. MIMO 1x1 Robust Algorithm. MIMO 1x1 Non Robust Algorithm. MIMO 2x2 Robust Algorithm. MIMO 2x2 Non Robust Algorithm. MIMO 3x3 Robust Algorithm. MIMO 3x3
5
0
−4
10
−5
0
10
5
15
20
3x3
15
10
10
2x2
20
0.1
0.2
0.3
0.4
0.5 0.6 Uncertainty ρ
0.7
0.8
0.9
1
Eb/No (dB)
Fig. 1. Raw BER comparison between different power allocation strategies. 2 = 0.02 Receiver MT = 3, MR = 3. Transmitter uncertainty: σn uncertainty: Perfect CSI
where ρ = 0 denotes perfect CSI, whereas ρ = 1 means no channel knowledge. The receiver uncertainty was fixed to σ2 = σn2 /16. As a reference the point ρ = 0 was simulated assuming perfect CSI at both transmitter and receiver. The robustness of the proposed algorithm is evidenced since ’BER Robust’ solution always outperforms ’BER Non-Robust’ one. Note that the ’BER Robust’ design tends to the openloop solution (i.e. equally power allocation for all subcarriers and antennas) when CSI quality at the transmitter degrades (ρ → 1). A second conclusion from Fig. 2 is drawn when comparing the performance for different MIMO scenarios. Note that when the number of antennas increases, the relative gain between the robust and non-robust solutions is substantially enlarged. Accordingly, the robust solution becomes an interesting scheme not only when the CSI quality is highly degraded, but also when ρ is close to zero. V. C ONCLUSIONS This paper has presented a Bayesian approach to the design of a linear pre-coding scheme that is robust to channel estimation errors. The linear transformation has been designed following a minimum BER criterion subject to an average power constraint. The minimum BER design has been formulated √ based on a generalized exponential bound of the function Q( x) that includes the Chernoff bound as a particular case. An iterative algorithm and a suboptimum closed-form solution have been proposed, that is based on the SVD of the so called equivalent channel, and it has been proved to converge to results published previously in the literature in the extreme cases of very high or very low uncertainty in the CSI. Besides, it has been shown that the design can be regarded as a reconfigurable algorithm that adapts the transmitted signal to the available channel knowledge, providing a solution that converges to the open-loop design for the case of no channel
Fig. 2. Minimum EbNo that achieves Raw BER ≤ 10−3 . Transmitter 2 /16. uncertainty defined by ρ. Receiver uncertainty σ2 = σn
knowledge and to the closed-loop design with perfect CSI for the case of no uncertainty. R EFERENCES [1] H.Sampath and A.Paulraj, “Linear Precoding for Space-Time Coded Systems With Known Fading Correlations,” IEEE Communications Letters, vol. 6, no. 6, pp. 239–241, Jun. 2002. [2] M.Iarossi O.Gasparini, E.Marinis, “Evaluation of MIMO spatial multiplexing for Wireless LAN with channel models from experimental data,” in Proc. of IST Mobile & Wireless Telecommunications Summit, Thessaloniki (Grece), Jun. 2002. [3] G. Jongren, M. Skoglund, and B. Ottersten, “Combining Beamforming and Orthogonal Space-Time Block Coding,” IEEE Trans. on Information Theory, vol. 48, no. 3, pp. 611–627, Mar. 2002. [4] J.R. Magnus and H. Neudecker, Matrix Differential Calculus with Applications in Statistics and Econometrics,, John Wiley, 1999. [5] F. Rey M. Lamarca and G. V´azquez, “Optimal Power Allocation with Partial Channel Knowledge for MIMO Multicarrier Systems,” in Proceedings of VTC-fall’02, Vancouver (Canada), Sep. 2002. [6] Y.Ding, T.Davidson, J.Zang, Z Luo, and K.Wong, “Minimum BER Block Precoders for Zero-Forcing Equalization,” in Proc. of ICASSP’02, Orlando (USA), May 2002, pp. 2261–2264. [7] FITNESS IST-2000-30116 project, “Performance analysis of reconfigurable MTMR transceivers for WLAN,” Deliverable D3.2.1 available at http://www.ist-fitness.org, Nov. 2002. [8] M.Lamarca and G.V´azquez, “Transform Modulations for Mobile Communications,” in Proc. of PIMRC’97, Helsinki (Finland), Sep. 1997, pp. 462–466. [9] O.L. Frost, “An Algorithm for Linearly Constrained Adaptive Array Processing,” Proc. of IEEE, vol. 60, no. 8, pp. 926–935, Aug. 1972. [10] E. N. Onggosanusi, A. M. Sayeed, and B. D. Van Veen, “Efficient Signaling Schemes for Wideband Space-Time Wireless Channels Using Channel Side Information,” Submitted to IEEE Trans. on Communications, Jan. 2001.
2571
