arise in their development. In fact ... the development of non-linear estimators can be by passed by ..... estimates of the symbols s(i) by means of the APPs of the.
Optimal Non-Linear MMSE Channel Estimation for Broadband Wireless TDMA Systems and Necessity of Soft Decisions Stefano Galli Telcordia Technologies Inc., 445 South Street, Morristown, NJ 07960, USA Abstract – At high data rates, TDMA links are degraded by severe inter-symbol interference (ISI). Therefore, it is imperative to gain confidence with optimum channel estimation. This problem is a non-linear estimation problem since the probability density function of the channel sequence conditioned on the observations is non Gaussian. However, the optimal approach to non linear minimum mean square error (NLMMSE) estimation leads to an estimator that is non-recursive. On the other hand, it is also possible to prove that the correct statement of the problem of NL-MMSE channel estimation leads to the necessity of the exploitation of soft-decisions. There are several examples in the past and current literature on the usefulness of soft-decisions in digital communications. The intuitive idea behind the use of soft-decisions is that the use of “estimates” in place of “decisions” is advantageous whenever the reliability of the hard decisions is low. However, the use of soft-decisions in place of hard-decisions is not only a mere intuition but a straightforward consequence of the correct statement of the problem of optimal NL-MMSE estimation. 1. Introduction A first example of the advantages of exploiting softinformation can be found in a seminal work by Taylor [6], where a sub-optimum MMSE estimate of the transmitted symbols was used in the feedback section in place of hard decisions. The usefulness of NL-MMSE estimates for mitigating the effects of erroneous hard decisions has also been recently pointed out, although no method for computing them was suggested [7]. It is important to point out that the use of NL-MMSE estimates in place of hard decisions is not only a mere intuition. In fact, it will be proved here (see Sect. 2) that this is a straightforward consequence of the statement of the problem of optimal, i.e. non-linear, MMSE estimation. However, non-linear estimators have seldom been used in practice because of the intrinsic analytic difficulties that arise in their development. In fact, the usual approach to NLMMSE estimation involves the exploitation of the properties of Martingale Difference (MD) sequences and the application of the so-called “MD Representation Theorem” [12]. Interestingly, this optimal approach leads to estimators that are intrinsically non-recursive and this is certainly a major drawback for their practical utilization (arguments to prove this claim are given in Sect. 3). The above mentioned difficulties that naturally arise in the development of non-linear estimators can be by passed by exploiting the results given in Sect.2 and the recent results which appeared in [15]. In fact, in [15] it has been shown that
the NL-MMSE filtered and fixed-lag smoothed estimates of the transmitted symbols can be efficiently obtained as a linear transformation of the vector of the A Posteriori Probabilities (APPs) of the states of the ISI channel. APPs can be efficiently generated by a Symbol-by-Symbol Maximum A Posteriori (SbS-MAP) receiver. These receivers have been recently “rediscovered” because they are able to generate soft-information in the form of APPs and this new interest has led to the derivation of new algorithms for the computation of the APPs that present a computational complexity of the same order of the Viterbi algorithm (VA), at least for limited values of the decision delay [1]. This appealing feature has recently led to the use of SbS-MAP receivers in several contexts, such as iterative decoding of parallel/serial concatenated coded streams, channel estimation and tracking [2], blind equalization [3], soft-output equalization and decoding [4], interference cancellation [5], and multiuser detection [8]. The results of Sect. 2 together with the results in [15] allow us to design enhanced channel estimators that employ the soft information given by the NL-MMSE estimates of the transmitted symbols in place of the conventional hard decisions. In so doing, the negative effects of erroneous hard decisions can be mitigated. 2. Necessity of the NL-MMSE Estimates of the Transmitted Symbols Let us consider the problem of channel estimation in a time varying environment. Referring to a TDMA-based digital link impaired by time varying multipath phenomena, thermal noise and co-channel interference, the baud-rate sampled sequence received at the output of the equivalent low-pass randomly time varying ISI channel can be modeled as: y (i ) =
L −1
∑ g (i; m)s(i − m) + v(i) ≡ G T (i) x(i) + v(i),
m =0
(1)
where the transmitted data sequence {s(k)∈A≡{s1,...,sM}⊂C1} is comprised of M-ary generally complex independent identically distributed symbols taken from an assigned modulation constellation A; {g(i;m)∈C1, 0≤m≤L-1, i≥1 } is the TS-sampled time-variant impulse-response of the overall link including the transmitting filter, the multipath-faded radio channel and the receiving filter;
G T (i ) ≡ [g (i;0)...g (i; L − 1)] ∈ C L is the resulting L-variate complex impulse-response vector of the channel at the i-th
epoch; x(i ) ≡ [s (i ) ... s (i − L + 1)]T is the state-transition sequence of the ISI channel; and {v(i)∈C1} is a complex zero-mean white Gaussian random sequence which accounts for the combined effects of the receiver thermal-noise and cochannel interference [9]. On the basis of the innovations approach proposed by Kailath in [10], it is possible to express the MMSE estimate of the channel sequence G(i) as a linear combination of the innovations process of the observations y(i):
{
} ∑ c η (k ) , i
Gˆ (i / i ) ≡ E G (i ) | y1i =
(2)
k
k =1
where the innovations η(1), η(2), …, η (i) are orthogonal to each other and the coefficients ck are to be determined so as to minimize the mean square value of the estimation error ε (i ) = G (i ) − Gˆ (i / i ) . The innovations of the observations can be easily expressed by resorting to the Doob-Meyer representation [11] of the random variable y(i). This decomposes y(i) into the sum of two terms, a predictable term and a non-predictable term, the former being the MMSE one step prediction of the observations and the latter being the innovation of the observations. In so doing, we obtain: y (i ) = yˆ (i / i − 1) + η (i ) = E y (i ) | y1i −1 + η (i ) . (3) Eq.(3) can be rewritten as the following:
{
{
} ∑ c η (k ) + c η (i) =
Gˆ (i / i ) ≡ E G (i ) | y1i = =
}
i −1
k
i
k =1
.
i −1
(4)
∑ c kη (k ) + ci [y(i) − yˆ (i / i − 1)] k =1
By definition, the summation on the right-hand side of eq.(4) represents the estimate of the vector G at the (i-1)-th epoch given the observations up to step (i-1), i.e. Gˆ (i − 1 / i − 1) ≡ E G (i − 1) | y i−1 , so that we can finally write(1):
{
1
}
Gˆ (i / i ) = Gˆ (i − 1 / i − 1) + ci [ y (i ) − yˆ (i / i − 1)] .
(5)
As far as the one step prediction of the observations yˆ (i / i − 1) is concerned, we can write the following: yˆ (i / i − 1) ≡ E y (i ) | y i −1 = Gˆ (i / i − 1) T E x(i ) | y i −1 , (6)
{
1
}
{
{
}
1
}
It is now clear that the term E x (i ) | y1i −1 , i.e. the MMSE one step prediction of the state sequence of the ISI channel, appears in the problem of optimum MMSE estimation of the channel impulse response G(i). Usually, it is (1)
If {G(i)} were a Gauss-Markov process, the estimate
Gˆ (i − 1 / i − 1) would be replaced by the one-step MMSE prediction Gˆ (i / i − 1) .
{
}
the approximation E x(i ) | y1i −1 ≅ x (i ) , i.e. correct hard decisions, that gives rise to conventional RLS-like decisiondriven channel estimators. However, this method turns an intrinsically non-Gaussian problem into a Gaussian one and, therefore, the conventional RLS-like decision-driven channel estimators represent the suboptimal linear MMSE solution to the problem of the estimation of the channel vector G(i). The non-Gaussian nature of the problem of estimating the sequence s(i) on the basis of the available observation y(i) in (1) arises because the probability density function of s(i) conditioned to the available observations y1i is nonGaussian. Similarly, the direct estimation of the channel G given y1i is a non-Gaussian problem as well. The only way to turn the channel estimation problem into a Gaussian problem is to assume correct decisions ~ s (i ) and condition G to both i ~ y and s (i ) . 1
On the basis of the previous considerations, the optimal estimator of the channel state vector is truly nonlinear so that we can write: sˆ NL− MMSE (i / i − 1) ˆ s NL− MMSE (i − 1 / i − 1) E x (i ) | y1i −1 = , (7) M sˆ NL −MMSE (i − L + 1 / i − 1)
{
}
where we have exploited the property that the MMSE estimate of a random vector is the vector constituted by the MMSE estimates of the single elements. As eqs.(6), (7) clearly show, the MMSE estimate of x(i) requires the MMSE smoothed estimates of the transmitted symbols and the use of the hard decisions is only an approximation. Remark. It is important to point out that estimator in (5) and (7) is not the optimum, i.e. truly non-linear, MMSE estimator of the channel impulse response G(i). In fact, the non-linear nature of the estimator Gˆ (i / i ) would be ensured by the observation-dependency of the filtering gain ci in (5). The exact NL-MMSE estimate of the channel vector G(i) should be approached by exploiting a particular property of the innovations of the observations, the Martingale Difference property. This approach is the only optimal approach to NLMMSE estimation and will be addressed in the following Section. 3. Optimal Nonlinear MMSE Channel Estimation In order to address properly the problem of optimal NLMMSE estimation of {G(i)}, let us define the following two sequences ζ (i ) ≡ Gˆ (i / i ) − Gˆ (i / i − 1) = E G (i ) | y i − E G (i ) | y i −1 ,
{
} { η (i ) ≡ y (i ) − yˆ (i / i − 1) = y (i ) − E {y (i ) | y }, 1
i −1 1
1
}
(8) as the estimation and observation innovations sequences, respectively. It is possible to show that the innovations
sequences ζ(i) and {η(i)} have the property of being Martingale Difference (MD) sequences. Roughly speaking, the MD property is something intermediate between statistical independence and uncorrelation, so that independence implies the MD property that in turn implies uncorrelation. If a suboptimal linear MMSE solution for the estimation of G(i) is desired, it is sufficient to consider the sequences in (8) as uncorrelated. However, if the optimal NLMMSE solution is desired, the additional MD structure embedded in the innovations sequences cannot be neglected and has to be taken into account. This may be done by exploiting the so called MD Representation Theorem [12]. MD Representation Theorem. The estimation innovations sequence is an MD sequence with respect to the σ-algebra built on { y1i } and, therefore, may be represented as a transformation of the observations innovation sequence as in the following: ζ (i ) = CG (i / i )η (i ) (9) where CG(i) is an adapted sequence with respect to the σi algebra built on { y1 } and can be computed as follows: CG (i / i ) =
{ {
E ζ (i )η * (i ) | y1i E η (i )η * (i ) | y1i
} }
(10)
The theorem states that the gain sequence is an adapted sequence, thus the optimal NL-MMSE estimator of the sequence {G(i)} does not appear to be recursive, since recursivity is only ensured by the property of predictability. However, it has been shown that in some special cases the gain sequence is predictable with respect to the σ-algebra y1i −1 }.
built on { the following:
C G (i / i ) =
In those cases it is possible to rewrite (10) as
{ E {η (i )η
E ζ (i )η * (i ) | y1i −1
}≡ C }
(11) G (i / i − 1) y1i −1 The predictable form of the theorem has been proven only in the following three cases: 1. Continuous-time case for signals observed in Gaussian white noise [13]; 2. Continuous-time case for signals observed through point processes [14]; 3. Discrete-time case for signals observed through binary point processes [14]. Unfortunately, the predictability of the gain sequence has been proven to be false in the important case of discrete-time observations in white Gaussian noise, i.e. the most recurrent case in communications problems. So, in principle, the optimal NL-MMSE estimator of any sequence observed in white Gaussian noise is intrinsically non-recursive. If a recursive solution is desired, it is necessary to impose recursivity as a system constraint by using the following approximation: * (i ) |
CG (i / i ) ≅ CG (i / i − 1) ≡
{ }. E {η (i )η (i ) | y }
E ζ (i )η * (i ) | y1i −1
(12)
i −1 1
*
The approximation in (12) ensures the possibility of updating the estimate in (9) recursively. At this point it is possible to state that the NL-MMSE solution in (9) and (12) is the optimal solution among the set of NL-MMSE recursive estimators. 4. Applications to Channel Estimation and Tracking Given the random transmitted data sequence {s(k)∈A≡{s1,...,sM}⊂C1}, constituted by M-ary generally complex independent identically distributed equiprobable symbols, it has been recently shown [15] that it is possible to generate NL-MMSE filtered and fixed-lag smoothed estimates of the symbols s(i) by means of the APPs of the states of the ISI channel x(i ) ≡ [s (i ) ... s (i − L + 1) ] . In particular, it is possible to show that: sˆNL−MMSE(i / i ) = Ξ π (i / i) (13) T
where (L is the memory of the channel and N≡ML): L x (i )∈ A ≡ {ξ 1 , ..., ξ N } sˆNL− MMSE(i / i ) ˆ (i −1/ i ) s , sˆNL− MMSE(i / i) ≡ E x(i) | y1i = NL− MMSE M sˆNL− MMSE(i − L + 1/ i )
{
( ( (
}
) ) , )
P x (i ) = ξ 1 | y i 1 i π (i / i ) = P x (i ) = ξ 2 | y1 M i P x (i ) = ξ N | y1
Lσ Lσ M O M Lσ
σ 1(1) σ 1( 2) (1) ( 2) σ σ2 Ξ= 2 σ L(1) σ L( 2)
M
(N) 1 (N) 2
(N) L
,
are the ISI channel state vector, the NL-MMSE estimate of the ISI channel state vector, the APPs of the state sequence of the ISI channel, and a mapping matrix, respectively. The columns of the LxN mapping matrix Ξ are comprised of the
vectors {ξi} (i.e. Ξ ≡ [ξ 1 , ..., ξ N ] ) so that σ i ∈ A (1≤ i ≤L, 1≤ j ≤N) represents the i-th component of the j-th determination ξj of the channel-state x(i). The proposed approach allows us to obtain the filtered i NL-MMSE estimate of s(i), i.e. E s (i ) | y1 , and the fixed-lag ( j)
{
}
{
}
i smoothed NL-MMSE estimates of s(i), i.e. E s (i − d ) | y1 for 1 ≤ d ≤ L − 1 . The latter estimates are more reliable than the former ones and their use is optimal for the case of systems with memory.
5. Example of Application The availability of NL-MMSE filtered and fixed-lag smoothed estimates of the transmitted symbols may be useful in many applications, e.g. in the case of adaptive equalization for fading channels. In fact, during deep faded periods, the hard-decisions usually employed to update and track the channel estimate are not reliable. The use of NL-MMSE estimates of the transmitted symbols, in place of the corresponding hard-decisions, would make the receiver more robust with respect to deep fades. In principle, any channel estimator that employs hard decisions for channel estimation and tracking can be modified in order to employ the NLMMSE estimates of the transmitted symbols in place of the hard decisions. As an example, the NL-MMSE estimates of the transmitted symbols given by (13) can be directly employed in the recursive estimator (5). Among the several NL-MMSE based adaptive receivers that can be foreseen, the one that gave rise to the best performance results was the receiver equipped with an RLSlike channel estimator fed with the NL-MMSE estimates of the transmitted symbols and an MLS equalizer (see Figure 1). The performance of this particular adaptive receiver has been evaluated via computer simulations and compared to other well-known receivers. The performance index is expressed in terms of Bit Error Probability (BEP) versus Signal-to-Noise Ratio (SNR). In all simulations the TDMA-slot assumed for the transmitted sequence consists of frames of Lf symbols of which Lp are known at the receiver, so that the resulting net throughput is ρ ≡ 1 - Lp /Lf . No form of channel coding has been considered. Two typical scenarios have been considered in the simulations. Channel A. The first channel considered in the simulations of Figs.2, 3 is the radio channel explicitly recommended by the GSM standard for test purposes (see Fig.8.25.d, [9]). This link is constituted by six equal-powered, TS-spaced, Uncorrelated Scattering (US) taps affected by Rayleighdistributed multipath phenomena with time correlation modeled by the usual zero-order Bessel function Jo(⋅). The adopted modulation is BPSK and the values of the product Doppler bandwidth-signaling period BDTS are 10-4 and 5⋅10-4 (in the GSM environment, this corresponds to a mobile speed of 30 and 150 Km/h, respectively). Channel B. The third channel considered (simulation in Fig.4) is the HF “Moderate” CCIR channel [16]. This channel consists of two equal-powered paths spaced 1 ms apart. The time correlation is modeled by a Gaussian-shaped fading
Doppler spectrum (Watterson model [16]) with Doppler spread BD is equal to 0.5 Hz. The adopted modulation is QPSK at 1,200 baud (i.e., 2,400 bps), thus the value of the product Doppler bandwidth-signaling period BDTS is 4.17⋅104 . In order to take into account the lowpass interpolating effects operated by the receiver filter, the length L of the equivalent discrete-time channel impulse response assumed at the receiver has been set equal to four (L=4). The performance of several receivers has been evaluated by means of Monte Carlo. The simulated receivers are here listed according to the labels used in the plots: RLS-MLSE: The conventional adaptive receiver equipped with a RLS channel estimator fed with tentative d-delayed hard decisions and an MLS equalizer that delivers final Ddelayed hard decisions. The values chosen for d and D are optimal, i.e. yield the lowest BEP. I-MLSE: Ideal MLSE receiver operating at a decision delay D and perfect channel state information. DFE (a,b): The conventional LMS-DFE adaptive receiver where a and b indicate the sizes of the feedforward and feedback sections, respectively. The values of a and b, as well as the value of the step-size parameter of the LMS algorithm for the updating of the taps of the DFE, have been chosen by testing several values and selecting those resulting in the lowest BEP. I-DFE (a,b): Same as for DFE (a,b) but for the ideal case of correct symbols fed-back. Soft-MAP: The adaptive receiver proposed in [2] that employs the APP-fed optimum channel estimator and a MAP equalizer that that delivers final D-delayed hard decisions. Proposed: The receiver in Fig. 1 equipped with an RLS-like channel estimator fed with the NL-MMSE estimates of the transmitted symbols and an MLS equalizer. Final hard decisions are delivered after having received the whole slot and no intermediate tentative hard decisions are generated. As the simulation results clearly show, the proposed adaptive receiver outperforms the conventional hard decision based receivers and gains ranging between 8 and 18 dB are achieved at a BEP of 10-4. When the channel time-variations are small (see Figure 2), the Proposed receiver exhibits nearly optimum performances and no apparent BEP-floor, whereas the BEPfloor of the conventional RLS-MLSE receiver is around 10-4. Modest gains around 1 dB are also achieved over the SoftMAP receiver. When the channel variability increases (see Figure 3), all the considered receivers suffer some performance degradation but the Porposed receiver still outperforms the other ones and exhibits no BEP-floor. Moreover, the gain over the Soft-MAP receiver increases to 4 dB at a BEP of 105 .
Finally, in the case of the HF link (see Figure 4) the Proposed receiver yields to gains of 3 dB and 7 dB at a BEP of 10-3 over the Soft-MAP and RLS-MLSE receivers, respectively. As the simulations have shown, the Proposed receiver always exhibits a moderate to substantial gain over the receiver proposed in [2] that employs the optimum NLMMSE channel estimator fed by the APPs of the states of the ISI channel. Although the channel estimator proposed in [2] is superior to the one proposed here, the exploitation of a much higher decision delay turns out to be the key feature that allows the proposed receiver to yield better performances. In particular, the performance gap is bigger when the product Doppler bandwidth-signaling period BDTS is higher. These results tend to indicate that when the channel variations are higher the exploitation of larger decision delays is more effective. 6. Conclusions A new and efficient method for generating NL-MMSE filtered and fixed-lag smoothed estimates of the transmitted symbols with a SbS-MAP receiver has been recently proposed [15]. This method makes the use of SbS-MAP receivers very appealing because they can now generate three kinds of information: hard-statistics based information (the hard decisions), soft-statistics based information (the APPs) and an intermediate case represented by the NL-MMSE estimates of the transmitted symbols. Several applications can be foreseen in a wide variety of detection and estimation problems such as in multiuser detection, iterative decoding and in all those situations where hard decisions are heavily employed despite the low reliability of the detection process. 7. References [1] Y. Li, B. Vucetic, Y. Sato, “Optimum Soft-Output Detection for Channels with Intersymbol Interference”, IEEE Trans. on Inform. Theory, vol.41, no.3, pp.704713, May 1995. [2] E. Baccarelli, R. Cusani, S. Galli, “A Novel Adaptive Equalizer with Enhanced Channel-Tracking Capability for TDMA-Based Mobile Radio Communications”, IEEE JSAC, vol.16, no. 9, Dec. 1998.
[3] E. Baccarelli, S. Galli, “Blind Deconvolution and Data-
Detection Exploiting Second-Order “Soft” Statistics”, submitted to the IEEE Trans. on Commun. [4] Y. Chen, K. B. Letaief, J. C.-I. Chuang, “Soft-Output Equalization and TCM for Wireless Personal Communications Systems”, IEEE J. Select. Areas Commun., vol.15, no.9, Dec. 1998. [5] X. Wang, H.V. Poor, “Iterative (Turbo) Soft Interference Cancellation and Decoding for Coded CDMA”, IEEE Trans. on Comm., vol.47, no.7, July 1999. [6] D.P. Taylor, “The Estimate Feedback Equalizer: A Suboptimum Nonlinear Receiver”, IEEE Trans. on Comm., vol. 21, pp.979-990. Sept. 1973. [7] F. Tarköy, “MMSE-Optimal Feedback and its Applications”, Proceedings of the IEEE Intern. Symp. on Infor. Theory, ISIT’95, Whistler, Canada, 17-22 Sep. 1995. [8] S. Verdú, Multiuser Detection, Cambridge, U.K., Cambridge Univ. Press, 1998. [9] R. Steele, Mobile Radio Communications, Pentech Press, London 1992. [10] T. Kailath, “The Innovations Approach to Detection and Estimation Theory”, Proc. IEEE, vol. 58, pp. 680-695, 1970. [11] G. Kallianpur, Stochastic Filtering Theory, Springer, New York, 1980. [12] A. Segall, “Stochastic Processes in Estimation Theory”, IEEE Trans. on Inform. Theory, vol.22, no.3, pp.275286, May 1976. [13] M. Fujisaki, F. Kallianpur, H. Kunita, “Stochastic Differential Equations for the Nonlinear Filtering Problem”, Osaka J. Math., vol.9, pp.19-40, 1972. [14] P.A. Meyer, “Martingales and Stochastic Integrals”, Lecture Notes in Mathematics, vol.284, New York, Springer, 1972. [15] S. Galli, “Non Linear MMSE Estimation and SbS-MAP Receivers”, IEEE International Symp. on Inform. Theory, ISIT’00, Sorrento, Italy, 25-30 June, 2000. [16] “HF ionospheric channel simulators”, CCIR, 13th Plen. Ass., 3, Rep.549, 66-75, 1974.
NL-MMSE based channel estimator A Known training sequence
Zero-delayed G(i/i) channel estimates
S
B NL-MMSE sequence
^SNL-MMSE(i/i)
Ξ ~ a(i-D)
π(i/i)
r(i) APPs Computer
APP sequence
Observations
Equalizer Final hard-decisions
Figure 1 – Adaptive equalizer considered in the simulations (label proposed)
Channel A @ B D T S =10 -4 (L p =12, L s =60, ρ =0.8) 1.E-01
Bit Error Probability
1.E-02 1.E-03 1.E-04 1.E-05 1.E-06
RLS-MLSE (d=5,D=30) Soft-MAP (D=5) Proposed I-MLSE (D=30)
1.E-07 1.E-08 0
5
10
15
20
25
30
Eb/No (dB) Figure 2 - Performance comparison on the six-tap test channel (Channel A) for the case of a BPSK modulation. The TDMA-slot is 60 symbols long, the preamble is 12 symbols long and BDTS=10-4.
Channel A @ B D T S =5x10 -4 (L p =8, L s =40, ρ =0.8) 1.E-01
RLS-MLSE (d=5,D=30) Soft-MAP (D=5) Proposed I-MLSE (D=30)
Bit Error Probability
1.E-02 1.E-03 1.E-04 1.E-05 1.E-06 1.E-07 1.E-08 5
10
15
20
25
30
35
40
Eb/No (dB) Figure 3 - The same as in Fig.4 but for the case of a TDMA-slot 40 symbols long, a preamble 8 symbols long and BDTS=5⋅10-4.
Channel B @ B D T S =4.17x10 (L p =15, L s =50, ρ =0.7)
-4
Bit Error Probability
1.E-01
1.E-02
1.E-03 DFE (5,3) I-DFE (5,3) RLS-MLSE (d=3,D=15) Soft-MAP (D=3) Proposed
1.E-04 5
10
15
20
25
30
Eb/No (dB) Figure 4 - Performance comparison on the “Moderate” HF link (Channel B) for the case of a QPSK modulation. The TDMA-slot is 50 symbols long, the preamble is 15 symbols long and BDTS=4.17⋅10-4.
