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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. XX, NO. XX, MONTH 20091

A Fixed-Lag Particle Smoother for Blind SISO Equalization of

Time-Varying Channels

Alberto Guimarães, Boujemaa Ait-El-Fquih and François Desbouvries

Abstract—We introduce a new sequential importance sampling

(SIS) algorithm which propagates in time a Monte Carlo approx-

imation of the posterior fixed-lag smoothing distribution of the

symbols under doubly-selective channels. We perform an exact

evaluation of the optimal importance distribution, at a reduced

computational cost when compared to other optimal solutions

proposed for the same state-space model. The method is applied

as a soft input–soft output (SISO) blind equalizer in a turbo

receiver framework and simulation results are obtained to show

its outstanding BER performance.

Index Terms—Fixed-lag smoothing, particle filter, mixture

Kalman filter, SISO equalization.

I. INTRODUCTION

B

tion recently [1], particularly in the context of blind equaliza-

tion problems. Among other solutions, Mixture Kalman Filter

(MKF) techniques embed Kalman Filter (KF) recursions in a

SIS framework [2]. A smoothing solution for the case of blind

SISO equalization with static channels has been proposed in

[3]. For time-variant channels, a particular fixed-lag smoothing

MKF was studied in [4] and [5]; however, although being

a near-optimal solution, the algorithm was not implemented

because of a computational cost exponential in the smoothing

lag and other parameters1. The aforementioned works then

overcome this constraint by introducing further Monte Carlo

sampling steps, but at the expense of algorithm performance.

By contrast, our method presents a novel solution for the

“optimal” approach, which under typical application scenarios

is remarkably less complex than the alternative proposed so

far. Although this approach is unavoidably constrained by

the exponential growth of complexity, it presents superior

performance than the suboptimal solutions.

We next apply our method to SISO blind equalization of

doubly-selective channels in a turbo equalization setup. Turbo

equalizers [6] have already been proposed for doubly-selective

channels with blind algorithms performing channel estimation

jointly with equalization [7][8]. Computational complexity is

a severe issue for such approach, however the remarkable

tracking performance of our optimal Bayesian estimator, even

in the presence of fast channel variations, enables us to

AYESIAN restoration in Conditionally Gaussian Linear

State-Space Models (CGLSSM) has received much atten-

Manuscript received December 24, 2008; revised July 7, 2009 and October

27, 2009; accepted November 13, 2009. The associate editor coordinating the

review of this letter and approving it for publication was Dr. G. Vitetta.

A. Guimarães is with Instituto Militar de Engenharia, Rio de Janeiro, Brazil

(e-mail: agaspar@ime.eb.br).

B. Ait-El-Fquih and F. Desbouvries are with Institut Telecom, Telecom

SudParis, CITI dpt. and CNRS UMR 5157, 91011, Evry, France (e-mail:

b_aitelfquih@yahoo.fr, Francois.Desbouvries@it-sudparis.eu).

Digital Object Identifier 10.1109/TWC.2009.081694.

1In [5] we refer to the DSIS method with optimal importance point mass

function.

implement a solution in which our SIS-based equalizer is used

only in the very first iterations of the turbo loop, and then

substituted in subsequent iterations by an equalizer with a

reduced computational load. This hybrid solution is validated

by simulation results. The rest of the letter is organized as

follows. In section II we first describe our model. Our fixed-lag

particle smoothing (FLPS) algorithm is developed in section

III. In section IV we show the computational load of our

method and we evaluate through simulations the Bit-Error

Rate performance of a turbo receiver including our algorithm.

II. STATE-SPACE MODEL

Let xT(resp. xH) denote the transpose (resp. Hermitian

transpose) of vector x. We assume that the receiver input yn

is related to the transmitted complex symbols by

yn= xT

nhn+ ωn

(1)

where hn = [h0,n,...,hL−1,n]Trepresents the baseband

channel impulse response of finite length L, and xn =

[xn,xn−1,...,xn−L+1]T

from time n − L + 1 up to time n. The sequence xn are

differentially modulated to resolve phase ambiguity, hence

{xn} is a Markov Chain (MC) and from (1) we get2

p(xn|x1:n−1,y1:n−1) = p(xn|xn−1), and we assume that

p(xn|xn−1) is known for all n. The noise variables {ωn} are

independent circularly symmetric complex Gaussian variables

with zero mean and known variance Λω

of {xn}. For notational convenience we consider in this

letter the case where yn is scalar, but the extension of our

algorithm to Multiple Input Multiple Output (MIMO) systems

is analytically straightforward, although the computational

complexity increases exponentially with the number of input

streams. Finally hnpropagates according to

gathers the transmitted symbols

n, and are independent

hn+1= Fnhn+ vn,

(2)

where {vn} are complex independent variables, independent

of {xn}, of {ωn} and of h0. It is also assumed that vn ∼

N(0,Λv

known.

n), h0 ∼ N(0,Λh

0), and that Fn, Λv

nand Λh

0are

III. AN FLPS ALGORITHM

In this section we focus on the computation of the prob-

ability mass function (pmf) p(xn|y1:n+M), where M > 0

is some fixed delay. We resort to an SIS approximation

(see e.g. [9][10] and references therein). So the a posteriori

joint pmf of symbols x1:n−1 at time n − 1 is approxi-

mated by p(x1:n−1|y1:n+M−1) ≈

?N

i=1λi

n−1δ(x1:n−1 −

2Depending on the context, let ui:j denote either {ui,··· ,uj} or the

vector with components uk,i ≤ k ≤ j.

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2 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. XX, NO. XX, MONTH 2009

xi

an importance distribution q(x1:n−1|y1:n+M−1), and the

importance weight λi

tory xi

?N

can be computed recursively as

1:n−1), in which the samples xi

1:n−1are generated from

n−1 associated to the i-th trajec-

1:n−1 is given by λi

i=1λi

q(xn|x1:n−1,y1:n+M)q(x1:n−1|y1:n+M−1), then the weights

n−1

∝

p(xi

q(xi

1:n−1|y1:n+M−1)

1:n−1|y1:n+M−1),

n−1 = 1. If we assume that q(x1:n|y1:n+M) =

λi

p(xi

n∝

n|xi

1:n−1,y1:n+M)p(yn+M|xi

q(xin|xi

1:n−1|y1:n+M−1)

q(xi

?

where xi

1:n

=

drawn from the conditional importance distribution (CID)

q(xn|xi

proximates p(x1:n|y1:n+M), and therefore?N

well known to suffer from weights degeneracy. Classical

rescues consist in resampling from?N

CID q(xn|xi

good choice is to sample particles from the “optimal" CID

(see [10]), i.e. the distribution which minimizes the variance

of the importance weights, conditionally on the observations

and past samples. In our case, the optimal distribution reads

1:n−1,y1:n+M−1)

1:n−1,y1:n+M)

×

p(xi

1:n−1|y1:n+M−1)

???

∝λi

n−1

,

(3)

[xi

1:n−1,xi

n] and each particle xi

?N

n is

1:n−1,y1:n+M). Finally

i=1λi

nδ(x1:n− xi

1:n) ap-

nδ(xn−

i=1λi

xi

n) approximates p(xn|y1:n+M). Now, SIS algorithms are

i=1λi

nδ(xn−xi

n) (either

systematically or according to some strategy) and in choosing

1:n−1,y1:n+M) carefully. To that respect, one

qopt(xn|xi

and under that choice (3) becomes

1:n−1,y1:n+M) = p(xn|xi

1:n−1,y1:n+M),

(4)

λi

n∝ p(yn+M|xi

?

nin (5).

1:n−1,y1:n+M−1)

???

˜λi

n

λi

n−1.

(5)

From now on, we thus focus on the computation of (4) and

of factor˜λi

A. Computing the Optimal CID

Let us address (4). For each n and i, we should sample a

new particle xi

naccording to

p(xn|xi

1:n−1,y1:n+M) =

p(xn|xi

?

p(xn|xi

p(yn:n+M|xn,xi

as

p(yn:n+M|xn,xi

?

with

θi

xn:n+M

(˜ x1:n+M,y1:n−1), i.e., we set ˜ xk = xi

and ˜ xk

= xk if n

˜ xn

= [˜ xn, ˜ xn−1,..., ˜ xn−L+1]T.

n−1)p(yn:n+M|xn,xi

xnp(xn|xi

n−1)

1:n−1,y1:n−1),

1:n−1,y1:n−1)

n−1)p(yn:n+M|xn,xi

known, so

1:n−1,y1:n−1);

remains

whichcan

(6)

isitto

be

compute

written

1:n−1,y1:n−1) =

xn+1

..

?

xn+M

p(yn:n+M|θi

xn:n+M)

M

?

k=1

p(xn+k|xn+k−1)

(7)

def

=(xi

1:n−1,xn:n+M,y1:n−1)

def

=

kif k ≤ n − 1

n + M. Let also

Onecan

≤

k

≤

show that

p(yn:n+M|θi

a Gaussian density. Let us thus set p(yn:n+M|θi

N(yn:n+M;μyi

foracircular complex

function with parameters (μ,Σ). In practice, parameters

μyi

M+1

and Σyi

M+1

can be computed recursively, and

ithas proven advantageous (as far as computational

cost is concerned)tocompute

recursively as well. This makes the difference between

the calculus routine developed here and those considered

in [4] and [5]. Let us set p(yn:n+k−1,hn+k|θi

N((yn:n+k−1,hn+k);μi

?

quadratic form in p(yn:n+k−1|θi

(yn:n+k−1− μyi

denote a block-diagonal matrix and Ik−1the (k−1)×(k−1)

identity matrix. Finally our algorithm (see the Appendix) is

as follows:

1) Recursive computation of μi

• Compute parameters of p(hn|xi

N(hn;?hi

μi

Fn

?˜ xT

• Recursion (k − 1) → k, for all k = 2,··· ,M.

Compute

xn:n+M) in (7) is the value at point yn:n+M of

xn:n+M) =

;μ,Σ) stands

probability

M+1,Σyi

M+1), where N(

Gaussian

·

density

p(yn:n+M|θi

xn:n+M)

xn:n+M) =

Σyi

k

Σyhi

Σhyi

k

denote

k,Σi

k) with Σi

?

k=

?

us

k

Σhi

k

?

and

μi

k

=

μyi

k

μhi

k

,andlet the

xn:n+M) by QF(k)

k). Let diag( )

=

k)H(Σyi

k)−1(yn:n+k−1− μyi

Mand Σi

M.

1:n−1, y1:n−1) =

n|n−1,Λhi

?

n|n−1) by the KF;

?

Λhi

n|n−1

Fn

• Compute

1=

˜ xT

n

?hi

n|n−1

?˜ xT

Σi

1=

n

Fn

?

n

?H

+

?Λω

n

0

0Λv

n

?

;

(8)

μi

Σi

k= Ai

k= Ai

diag(0k−1,Λw

k−1μi

k−1Σi

k−1

k−1(Ai

n+k−1,Λv

k−1)H+

n+k−1)

⎤

(9)

with

Ai

k−1=

⎡

⎣

Ik−1

0

0

0

˜ xT

Fn+k−1

p(yn:n+M|θi

n+k−1

⎦.

xn:n+M)

2) Recursive

N(yn:n+M;μyi

• Compute (Σyi

QF(1)= (yn− μyi

• Recursion (k − 1) → k, for k = 2,··· ,M + 1.

Compute

computation

M+1,Σyi

1)−1= ([1

of

=

M+1).

0]Σi

1[1 0]T)−1and

1);

1)T(Σyi

1)−1(yn− μyi

(Σyi

k)−1

=

?

γ−1× detΣyi

QF(k−1)+ Y2γ,

k−1)−1b[bT(Σyi

−αbD−1and γ

=

Σyhi

n+k−1; and Y

k−1)T(Σyi

α

βT

β

γ

?

,

(10)

detΣyi

k

=

=

k−1,

(11)

(12)

QF(k)

where α

D]−1bT(Σyi

bT(Σyi

xT

xT

=

k−1)−1; β

k−1)−1b]−1, with b

n+k−1(Σhi

n+k−1μhi

(Σyi

k−1)−1− (Σyi

=

k−1)−1b −

=[D −

k−1xn+k−1 and D

=

k−1)xn+k−1 + Λw

k−1) − (yn:n+k−1− μyi

=(yn+k −

k−1)−1b.

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B. Updating the Importance Weights

We now address the computation of˜λi

easily that

p(yn:n+M|xi

p(yn:n+M−1|xi

The numerator of (13) is equal to the denominator of (6)

which has been computed before. The denominator of (13)

is the value of a Gaussian density N( · ;μyi

point yn:n+M−1. Its determinant detΣyi

QF(M)have been computed at step (k = M) of (11) and (12) .

So the computation of˜λi

of the CID in (6).

nin (5). One can see

˜λi

n=

1:n−1,y1:n−1)

1:n−1,y1:n−1).

(13)

M, Σyi

M) at

Mand quadratic form

nfollows directly from the evaluation

IV. PERFORMANCE RESULTS

A. Computational Load

We first compare our FLPS algorithm with that of the

algorithm in [5] with optimal importance pmf (DSIS Op-

timal). The main difference relies in the computation of

(7) : in [5] one writes p(yn:n+M|x1:n+M,y1:n−1)

?M

The number of floating operations (flops) required by our

method to evaluate (4) at time n, for a given trajectory i,

is approximately

?M2− M

XMM

22

On the other hand, the number of flops required by the method

of [5] is approximately

?3

In Fig. 1 N1and N2are plotted for M = 3, X = 2 or 4, and

L = 1 to 5. We see above that the computational complexity

of both proposals increases exponentially with M, but in all

scenarios N1< N2and the difference significantly increases

as L increases3.

=

k=0p(yn+k|x1:n+M,y1:n+k−1), so if X denotes symbol

alphabet size one needs to implement (M + 1)XMKF.

N1=3

2ML3+

?M + 3

2

+ M

?

?

L2+

L2

?M

+ 1

L +5

6M2+ 2M +53

12

?

.

N2= XM× (M + 1)

2L3+ 3L2+7

2L + 3

?

.

(14)

B. BER Performance

The following simulation setup is considered to evaluate

the performance of our algorithm used as a SISO equalizer

embedded in a turbo equalization receiver. A set of 80 random

data bits is encoded using a 1/2-rate (1 + D2,1 + D + D2)

convolutional encoder. Next the coded bits {cn} are inter-

leaved, mapped to ±1 symbols (BPSK), and transmitted over

a channel with dynamics given by (2) with Fn = κ1/2I2,

Λv

to 0.999 or 0.992 corresponding respectively to the slow and

fast-fading scenarios4, and Λω

n= 0.5(1 − κ)I2and h0∼ N(0,0.5I). Parameter κ is set

n= N0/2. At the receiver the

3If hnis constant the algorithm presented in [3] requires less computations

than ours when M > 3 and M ≈ L. Nevertheless, the solution proposed

therein cannot be extended to scenarios involving time-variant channels.

4Definingthe normalized fading

0.98?∞

κ = 0.992 yields fd≈ 10−2.

rate

fd

from

?fd

0

S(f)df

≈

0

S(f)df, where S(f) is the power spectrum density of the channel

coefficients, κ = 0.999 corresponds to a fading rate of fd≈ 10−3, and

1 1.52 2.53 3.54 4.55

0

1

2

3

4

5

6

7

8x 10

4

Channel Length (L)

N1, N2

M=3

N1, PSK−2

N2, PSK−2

N1, PSK−4

N2, PSK−4

Fig. 1.

proposal and the method of [4],[5].

Comparison in terms of number of floating operations between our

BCJR algorithm is used as the SISO decoder. Our algorithm is

implemented with M = 3 and N = 30 samples. We resample

from p(xn|y1:n+M) ≈

effective sample size Neff≈ (?N

where hi

encoding is employed to combat phase ambiguity and the soft

information outputted from the FLPS equalizer is computed

as in [8]: P(cn= Cj|y1:n+M) =?N

2 correspond to the first and fourth iterations, averaged over

1000 channel realizations under a slow fading scenario. We

observe a noticeable performance gain (2 dB for BER <

5 × 10−3) over the iterations of our FLPS-based receiver,

which shows its efficiency as a SISO processing block. Even

though a thorough investigation of robustness is out of the

scope of this note, in Fig. 2 we considered the effect of

channel order overestimation by setting Fn= 0.9991/2I3and

Λv

this mismatched model causes a 2 dB performance loss at

BER < 10−2, but the degradation for the fourth iteration is

only 1 dB.

We next show the performance of our FLPS turbo re-

ceiver when the equalizer has some information about h0,

i.e. this algorithm is initialized with hi

(i = 1,...,N). In this case differential encoding is not used so

{xn} are independent. Consequently, p(xk|xk−1) = p(xk) in

eqs. (6) and (7) and P(cn= Cj|y1:n+M) =?N

presented in [11], which performs an iterative channel estima-

tion from second-order statistics of each transmitted symbol,

fed back from the decoder in the previous iteration. This

turbo scheme with soft input channel estimator (SICE) is

implemented with a BCJR algorithm as the SISO equalizer,

initialized by a preamble with 40 pilot symbols.

The BER performances of the turbo receivers with the

SICE and FLPS based equalizers are displayed in Figs. 3 and

4. The figures also include the performance achieved by a

turbo scheme after the fourth iteration using a clairvoyant

?N

i=1λi

nδ(xn− xi

i=1(λi

n) whenever the

n)2)−1< N/3.

We first illustrate the performanceof the blind turbo receiver

0|−1∼ N(0,0.5I) (i = 1,...,N). Differential

i=1λi

nδ(xi

n×xi

n−1−Xj),

where Xj is the BPSK mapping of bit Cj. The plots in Fig.

n= 10−3× diag(0.43,0.43,0.13). For the first iteration

0|−1∼ N(h0,0.1I)

i=1λi

nδ(xi

n−

Xj). We compare this proposal against the turbo equalizer

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2 2.53 3.54 4.55 5.56 6.57

10

−4

10

−3

10

−2

10

−1

BER

Eb/N0 (dB)

fd=0.001

FLPS (1st it.)

FLPS (4st it.)

FLPS (1th it.) mismat.

FLPS (4st it.) mismat.

Fig. 2. BER performance of blind FLPS-based turbo receiver for fd= 10−3

(κ = 0.999).

1 1.52 2.53 3.54 4.55 5.56

10

−4

10

−3

10

−2

10

−1

BER

Eb/N0 (dB)

fd=0.001

FLPS (1st it.)

SICE (1st it.)

FLPS (4th it.)

SICE (4th it.)

Clairv. (4th it.)

Fig. 3.

non-blind context) with the SICE-based and clairvoyant BCJR-based turbo

receivers for fd= 10−3(κ = 0.999).

BER performance of FLPS-based turbo receiver compared (in a

BCJR equalizer. In Fig. 3 there is only a 1.5–2 dB gap

between the FLPS receiver performance and the upper limit

shown by the receiver with the clairvoyant BCJR equalizer.

Our method clearly outperforms the SICE based receiver,

mainly when SNR increases. In Fig. 4 we observe that the

performance of the FLPS receiver is far superior under such

scenario after four iterations. Accordingly, the BER plot has

a noticeably decreasing slope, whereas the performance of

the SICE receiver does not improve significantly when the

observations samples become less noisy, even after some turbo

iterations.

Finally, we have run a turbo processing experiment (again

under partial knowledge of h0) where we use the FLPS SISO

equalizer for the initial iterations, and for the subsequent

iterations a BCJR-based equalizer is used based on the channel

estimateˆhn =

produced by the FLPS algorithm in the previous iterations.

Results are shown in Figs. 5 and 6, respectively for the

slow and fast fading channels, under the same transmission

conditions. For the slow fading case we used the FLPS

?

iλi

nˆhi

n|n−1and on the soft information

45678910 1112

10

−4

10

−3

10

−2

10

−1

BER

Eb/N0 (dB)

fd=0.01

FLPS (1st it.)

SICE (1st it.)

FLPS (4th it.)

SICE (4th it.)

Clairv. (4th it.)

Fig. 4.

non-blind context) with the SICE-based and clairvoyant BCJR-based turbo

receivers for fd= 10−2(κ = 0.992).

BER performance of FLPS-based turbo receiver compared (in a

1 1.52 2.53 3.54 4.55 5.56

10

−4

10

−3

10

−2

10

−1

BER

Eb/N0 (dB)

fd=0.001

FLPS (4th it.)

Hybrid (4th it.)

SICE (4th it.)

Fig. 5. BER performance of turbo receiver with the hybrid scheme (FLPS and

BCJR) compared with the turbo receivers with the FLPS and SICE equalizers

for fd= 10−3(κ = 0.999).

equalizer only in the first iteration, and in the fast fading case

until the second loop. Both figures show that the performance

degradation is indeed small. So this hybrid solution still

outperforms the SICE-based receiver, with a computational

time which is comparable to that proposal.

V. CONCLUSIONS

In this paper we propose a novel SIS-based Bayesian

restoration method for CGLSSM using a FLPS methodology.

We sample particles from the optimal importance function

exactly evaluated by using an algorithm with reduced com-

putational cost when compared to existing alternatives. Sim-

ulations show the performance of our technique as a SISO

blind equalizer embedded in a turbo equalization receiver.

Due to the optimality of the equalizer here developed, our

turbo receiver shows outstanding performance when compared

(in a non-blind context) to another turbo scheme designed

for doubly-selective channels, specially under a fast fading

scenario. Simulations also demonstrate the robustness of the

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456789 101112

10

−3

10

−2

10

−1

BER

Eb/N0 (dB)

fd=0.01

FLPS (4th it.)

Hybrid (4th it.)

SICE (4th it.)

Fig. 6. BER performance of turbo receiver with the hybrid scheme (FLPS and

BCJR) compared with the turbo receivers with the FLPS and SICE equalizers

for fd= 10−2(κ = 0.992).

equalizer and the validity of a hybrid proposal, in which our

algorithm initializes a BCJR-based turbo equalizer.

APPENDIX - COMPUTING p(yn:n+M|θi

xn:n+M) IN (7).

From

{(hn+k,yn+k−1)}M

(1)(2), conditionally on

θi

xn:n+M,

k=0is a (vector) MC. So

p(yn:n+M|θi

?

?

p(hn+M,yn+M−1|hn+M−1,yn:n+M−2,θi

p(yn+M|hn+M,yn:n+M−1,θi

xn:n+M) =

p(hn:n+M,yn:n+M|θi

xn:n+M)dhn:n+M=

p(hn|θi

xn:n+M)p(hn+1,yn|hn,θi

xn:n+M)···×

xn:n+M)×

xn:n+M)dhn:n+M.

So p(yn:n+M|θi

starting

hn, ···, hn+M. Other simplifications result from (1)

(2):

p(hn|θi

p(yn+k−1,hn+k|yn:n+k−2,hn+k−1,θi

as p(yn+k−1|hn+k−1, ˜ xn+k−1)p(hn+k| hn+k−1). Let us

summarize the computation of p(yn:n+M|θi

following recursion :

xn:n+M) can be computed recursively,

p(hn|θi

=

p(hn|xi

xn:n+M)

from

xn:n+M)

andintegratingw.r.t.

xn:n+M)

1:n−1,y1:n−1); and

factorizes

xn:n+M) in the

1) Initialization.

Compute p(hn|θi

next p(yn,hn+1|θi

xn:n+M) = p(hn|xi

xn:n+M) from p(hn|θi

?

p(yn|hn, ˜ xn)p(hn+1|hn)

?

1:n−1,y1:n−1), and

xn:n+M) as

p(yn,hn+1|θi

xn:n+M) =

p(hn|θi

xn:n+M)×

???

p(yn,hn+1|hn,θi

xn:n+M)

dhn

(A.1)

2) Recursion (k − 1) → k, for all k = 2,··· ,M. Com-

pute p(yn:n+k−1,hn+k|θi

xn:n+M) from p(yn:n+k−2,

hn+k−1|θi

p(yn:n+k−1,hn+k|θi

?

p(yn+k−1|hn+k−1, ˜ xn+k−1)p(hn+k|hn+k−1)

?

dhn+k−1;

xn:n+M) as

xn:n+M) =

p(yn:n+k−2,hn+k−1|θi

xn:n+M)×

???

p(yn+k−1,hn+k|yn:n+k−2,hn+k−1,θi

xn:n+M)

×

(A.2)

3) Step M → (M+1). Compute p(yn:n+M|θi

p(yn:n+M−1,hn+M|θi

p(yn:n+M|θi

?

p(yn+M|hn+M, ˜ xn+M)

?

In practice, the Gaussian assumption in (1) (2) im-

plies that all densities in (A.1), (A.2) and (A.3) are

Gaussian as well. Let p(hn|xi

Λhi

p(yn:n+M|θi

p(hn+k|hn+k−1)

p(yn+k|hn+k,θi

ters?hi

reduces to that of μyi

whence (8) and (9). Equations (10)-(12) are easily verified.

xn:n+M) from

xn:n+M) as

xn:n+M) =

p(yn:n+M−1,hn+M|θi

xn:n+M)×

???

p(yn+M|yn:n+M−1,hn+M,θi

xn:n+M)

dhn+M.

(A.3)

1:n−1,y1:n−1)∼ N(?hi

M+1,Σyi

∼ N(Fn+k−1hn+k−1 , Λv

xn:n+M)

∼ N(˜ xT

n|n−1and Λhi

the computation of p(yn:n+M|θi

M+1and Σyi

n|n−1,

k) and

n|n−1), p(yn:n+k−1,hn+k|θi

xn:n+M) ∼ N(μyi

xn:n+M) ∼ N(μi

M+1). From (1) (2)

k,Σi

(2)

n+k−1) and

n+k). Parame-

(1)

n+khn+k, Λw

n|n−1are computed via the KF, and finally

xn:n+M) from p(hn| θi

M+1from?hi

REFERENCES

xn:n+M)

n|n−1,

n|n−1and Λhi

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