Filtering of Discrete-Time State-Space models with the p

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Recent Researches in Telecommunications, Informatics, Electronics and Signal Processing
Filtering of Discrete-Time State-Space models with the p-Shift
Kalman-like Unbiased FIR Algorithm
OSCAR IBARRA-MANZANO
Guanajuato University
Department of Electronics, DICIS
Ctra. Salamanca-Valle, 3.5+1.8km, Palo Blanco, Salamanca
MEXICO
[email protected]
Abstract: In this paper, we show a simple way to derive the p-shift finite impulse response (FIR) unbiased estimator (UE) recently proposed by Shmaliy for time-invariant discrete-time state-space models. We also examine
its iterative Kalman-like form. We conclude that the Kalman-like algorithm can serve efficiently as an optimal
estimator with large averaging horizons. It has better engineering features than the Kalman one, being independent
on noise and initial conditions. Both algorithms produce similar errors, although the proposed one overperforms
the Kalman filter if the noise covariance matrices are filled incorrectly. The full horizon Kalman-like and Kalman
algorithms produce equal errors only within some range of averaging horizons. With smaller horizons, the Kalman
filter is more accurate and, with larger ones, the proposed solution provides better denoising. Simulation results
are obtained for the 3-state space polynomial model and quadratic noiseless signal measured with noise.
Key–Words: Kalman-like filtering, FIR estimator, State-space
1
Introduction
where xn and x̃n|n are the state vector and its estimate1 and E denotes an average of the succeeding expression. The condition (1) means that the average
of the estimate x̃n|n must be equal to that of its origin xn in order for the bias to be removed from the
estimate. We find (1) imbedded in many linear estimators, as shown in [4]. Most recently, the unbiasedness has been employed by Zhou and Wang in [11] to
design FIR-median hybrid filters with polynomial fitting, Shmaliy et. al in [9, 12–14] for unbiased FIR filtering, prediction and smoothing of state space models, Kim and Lee in [15] to design a FIR filter for
state estimation, and Ahn and Kim in [16] for fixedlag maximum likelihood FIR smoothing of state space
models.
Any FIR filter obeying (1) can be said to be optimal in the minimum bias sense with the following recognized engineering features: its gain does not depend
on noise and initial conditions [2, 4]; for polynomial
signals, its gain can be represented with unique finitedegree polynomials, existing on an averaging horizon
of N points [9]; and it becomes virtually optimal in
the minimum mean square error (MSE) sense when
N À 1 [2] or the mean square initial state dominates
the noise components in order of magnitudes [17], because the estimate variance reduces as a reciprocal
Unbiased estimators play an important role in solving problems in tracking [1], timekeeping and clock
synchronization [2], positioning, etc. Unbiasedness is
strongly desired for median estimators, channel estimation in wireless systems, channel identification and
equalization, estimation of systems with unknown inputs and periodic time-varying structures, image denoising, etc. Even the IEEE Standard 1139-1999 [3]
for frequency and time metrology states that “an efficient and unbiased estimator is preferred”.
Signals and models are often estimated using the
finite impulse response (FIR) filters [4–9] owing to
important advantages of the latter against the infinite impulse response (IIR) ones. FIR filters are
known to have an imbedded bounded input/bounded
output (BIBO) stability and better robustness against
the model uncertainties [10]. It also follows from [4]
that the optimal FIR filter is more robust against the
Kalman one. Summarizing, Jazwinski stated in [10]
that the limited memory (or FIR) filter appears to be
the only device for preventing divergence in the presence of unbounded perturbation in the system.
In state space FIR modeling and estimation
of discrete-time signals, unbiasedness is commonly
achieved by satisfying the unbiasedness condition
E{x̃n|n } = E{xn } ,
ISBN: 978-1-61804-005-3
1
Here and in the following x̃k|v means the estimate at k via
measurement from the past to v.
(1)
71
of N [7, 8]. A payment for this is a higher order of
FIR filters against the IIR ones that causes computational burden. Therefore, methods of fast convolution
computation are often required in order to reduce the
computation time. That can be provided either employing the circular convolution theorem in the discrete Fourier transform (DFT) domain or designing
fast recursive or iterative computation forms in discrete time.
Below, we first show a simple way to derive
the p-shift FIR unbiased estimator (UE) proposed by
Shmaliy in [18] for discrete-time state-space models
and then examine its iterative Kalman-like form in
comparison to the standard Kalman one.
positive step p. Finally, smoothing can be organized
at a past point n + p with a negative lag p.
Inherently, all FIR estimates are N -dependent.
We can thus recognize the fixed horizon estimation
with T = τ (N − 1), N = const, and m = n − N + 1;
variable horizon estimation with T = τ (N − 1), N =
var, and m = n − N + 1; and full horizon estimation
via all the data available with T = τ (N − 1) = τ n,
N = n + 1, and m = 0. If the condition (1) is satisfied, then all these estimates will exist in the unbiased
sense.
Being essentially an averaging structure, the unbiased FIR filter relies on large N in order to reduce
noise substantially in the output. That causes the main
problem associated with such kind of filters: their
practical implementation has typically large computational complexity featured to batch discrete convolution forms, when N À 1. To circumvent this problem, fast methods of discrete convolution computations need to be used. In discrete time, this means
finding fast recursive forms for the batch unbiased FIR
estimator.
2 Signal Model
Let us consider a general discrete real-time-invariant
linear model [19] represented in state space with the
state and observation equations, respectively,
xn = Axn−1 + Bwn ,
(2)
yn = Cxn + Dvn ,
(3)
3
where the K × 1 state vector and M × 1 observation
vector are given by, respectively,
xn = [x1n x2n . . . xKn ]T ,
(4)
yn = [y1n y2n . . . yM n ]T .
(5)
It has been shown in [4] that the unbiasedness constraint for linear FIR filters is equal to the deadbeat
constraint associated with noiseless both system and
measurement. That means that the unbiased FIR estimator can be derived, if we remove noise from (1) and
(2) and represent the remaining deterministic equations on a horizon of N points, similarly to [9], with
recursively computed forward-in-time solutions [19]
as follows, respectively,
The K × K transition matrix A projects the nearest
past state xn−1 to the present state xn . The measurement matrix C has M × K dimensions. Most generally, B and D have K × K and M × M dimensions,
respectively. The K × 1 input noise vector and M × 1
measurement noise vector, respectively,
wn = [w1n w2n . . . wKn ]T ,
(6)
vn = [v1n v2n . . . vM n ]T ,
(7)
Xn,m = An−m xm ,
(8)
Yn,m = Cn−m xm ,
(9)
where the KN × 1 state vector Xn,m and the M N ×
1 observation vector Yn,m are specified on the time
interval from m to n by, respectively,
£
¤T
Xn,m = xTn xTn−1 . . . xTm
,
(10)
have zero-mean components, E{wn } = 0 and
E{vn } = 0. It is implied that wn and vn are mutually uncorrelated, E{wi vjT } = 0, having arbitrary
covariances Rw = E{wi wjT } and Qv = E{vi vjT },
respectively, for all i and j. The vectors wn and vn
are thus supposed to be not obligatorily Gaussian and
delta-correlated.
If we now apply a linear convolution operator to
measurement yn on a horizon of N points, we can
expect solving three basic estimation problems. Filtering can be provided if the estimate is related to the
current point n. Prediction can be obtained if the estimate is forwarded to the future point n + p with a
ISBN: 978-1-61804-005-3
p-Shift Unbiased FIR Estimator
£
¤
T
T T
Yn,m = ynT yn−1
. . . ym
.
(11)
Here, the KN × K transition matrix An−m and the
M N ×K observation matrix Cn−m are time-invariant
and dependent on the averaging interval length N −
1 = n − m, respectively,
£
¤T
,
(12)
Ai = (Ai )T (Ai−1 )T . . . AT I
Ci =
72
£
(CAi )T
(CAi−1 )T
...
(CA)T
¤T
CT
.
(13)
Let us now assign some K × M N gain matrix
H(p) and claim that the estimate x̃n+p|n of xn is
x̃n+p|n = H(p)Yn,m
= H(p)Cn−m xm .
Table 1: Iterative p-Shift Kalman-like FIR UE Algorithm
(14)
(15)
Stage
Because Yn,m in (14) can be said to be an input and
x̃n+p|n an output of the estimator, the gain matrix
H(p) must realize the convolution principle.
The estimate x̃n+p|n can now be found in the unbiased sense and assigned as x̄n+p|n . To enable finding the relevant unbiased gain H̄(p), first rewrite the
unbiasedness condition (1) for the deterministic state
vector as
E{x̄n+p|n } = E{xn+p } = xn+p .
Given:
m = n − N + 1, s = m + K − 1
PK−1 = (CTK−1 CK−1 )−1
FK−1 = AK−1 PK−1 (AK−1 )T
x̄s+p|s
(16)
= AK−1+p PK−1 CK−1 T Ys,m
Update:
Fv = AFv−1 AT − AFv−1
×(I + ΞFv−1 )−1 ΞFv−1 AT
x̄m+v+p|m+v = Ax̄m+v+p−1|m+v−1
+Ap Fv CT (ym+v − CA1−p
×x̄m+v+p−1|m+v−1 )
Instruction:
Use x̄m+v+p|m+v as the output
when v = N − 1
3.1.1
Full Horizon Algorithm
It follows that the algorithm (Table 1) can be used
straightforwardly when N is fixed or variable. In a
specific case of the model distinct over all observation time, estimation must be provided employing all
the data available. That will guarantee best denoising. The relevant full horizon Kalman-like unbiased
FIR algorithm attains hence a simpler form (Table
2). This algorithm suggests that the computation must
start with n = K, because CTK−1 CK−1 is singular
otherwise. Its splendid property is that only two parameters are required, K and p.
(19)
and relevant p-shift unbiased FIR estimate
x̄n+p|n = An−m+p (CTn−m Cn−m )−1 CTn−m Yn,m .
(20)
Although the batch convolution form (20) can be
used whenever the unbiased estimate is needed via
measurement from m to n, the computational problem
arises instantly when N is large. That often causes
computation time delays such that x̄n+p|n cannot be
used in real time. Below we shall show that the problem can efficiently be overcome by representing (20)
in a recursive Kalman-like form.
4
Simulations
Below, we apply the Kalman-like algorithm to the
3-state space polynomial model and quadratic signal
measured with noise in a comparison to the standard
Kalman one.
3.1 Kalman-like Form of the FIR UE
Following [18], the FIR UE can be represented in an
iterative form as shown in Table 1. One can observe
that the iterative computation starts at m + K with
v = K and finishes at n when v = N − 1. The true
estimate is taken at each v = N − 1 and the procedure
repeated recursively. It is important that the algorithm
(Table 1) does not involve noise and initial conditions.
ISBN: 978-1-61804-005-3
Ξ = AT CT CA
Set:
In order to solve (15) for H̄(p) fitting (16), the
state vector xn+p needs to be specified via the initial
state xm . That can be done if we write the first component of (8) at n as xn = An−m xm . Because n can
be arbitrary, we can further induce a time shift p and
go to
xn+p = An−m+p xm .
(17)
Now substitute (17) instead of x̃n+p|n in (15), remove the initial state vector xm from both sides, and
go to the p-variant unbiasedness (or deadbeat) constraint
An−m+p = H̄(p)Cn−m .
(18)
Further
multiplying
(18)
from
the
right-hand side with the identity matrix
(CTn−m Cn−m )−1 CTn−m Cn−m gives us the p-shift
unbiased gain matrix
H̄(p) = An−m+p (CTn−m Cn−m )−1 CTn−m
K, p, v = K, ..., N − 1
4.1
Filtering of the 3-state space model
In the first experiment, we consider the 3-state space
polynomial model
xn = Axn−1 + wn ,
73
(21)
Table 2: Full Horizon Kalman-like FIR UE Algorithm
Given:
Set:
First state × 103
Stage
K, p, and n > K
Ξ=
Kalman-like
Kalman
1
AT CT CA
x1n
PK−1 = (CTK−1 CK−1 )−1
FK−1 =
0
1
AK−1 PK−1 (AK−1 )T
2
n × 103
(a)
x̄K−1+p|K−1
150
= AK−1+p PK−1 CK−1 T YK−1
100
Kalman
Kalman-like
Error
Update:
2
Fn = AFn−1 AT − AFn−1
50
2
0
×(I + ΞFn−1 )−1 ΞFn−1 AT
1
? 50
x̄n+p|n = Ax̄n−1+p|n−1 + Ap Fn CT
×(yn − CA1−p x̄n−1+p|n−1 )
150
n × 103
(b)
Kalman
Kalman-like
Error
100
50
2
yn = Cxn + vn ,
(22)
1
? 50
in which xn = [x1n x2n x3n ]T , C = [ 1 0 0 ], and

2 
1 τ τ2
A= 0 1 τ .
0 0 0
Figure 1: Kalman and Kalman-like estimates of x1n
under the temporary measurement uncertainty: (a)
measurement and estimates, (b) estimate errors, and
(c) errors in the Kalman filter estimate affected by incorrect noise description.
The process was generated recursively at 3000
discrete points by setting τ = 1 s, x10 = 100,
x20 = 10−2 /s, and x30 = 10−5 /s2 . All noise
sources were assumed to be independent and uncorrelated white sequences with the standard deviations
of σx1 = 1, σx2 = 5 × 10−3 /s, σx3 = 3 × 10−5 /s2 ,
and σv = 100. To investigate effect of a temporary
measurement uncertainty upon the estimates, we voluntary added a bias of 200 to the measurement noise
vn in the time interval from 1200 to 1250 points.
For such processes with known initial conditions
and noise variances, the standard Kalman algorithm
can be applied straightforwardly to produce optimal
estimates as described in [2] (see Appendix B). The
Kalman-like unbiased algorithm (Table 1) can also be
applied straightforwardly, if we let K = 3, specify
C2 , by (13), as

1 2τ
C2 =  1 τ
1 0
2τ 2
τ2
2
4.1.1
Filtering of the first state
Figure 1a sketches filtering estimates, p = 0, of
the first state x1n provided with two algorithms and
N = 700. One can observe that the estimates are
consistent, except for the initial transient region that
is removed in Fig. 1b and Fig. 1c. Both filters produce similar errors that can be seen in Fig. 1b. It can
also be seen that the Kalman filter exhibits a bit larger
peak-excursion in the uncertainty region in Fig. 1b.
A situation changes if we recall that the state noise is
not always observable in applications and increase the
standard deviation in the first and second states by the
factor of 2 and 4, respectively. Under such conditions,
the Kalman filter is no longer optimal and one can expect for errors in its estimate. In fact, the Kalman filter demonstrates larger excursions in Fig. 1c under
the temporary uncertainty and larger noise beyond this
region, although its estimate remains unbiased. We
meet a similar picture reported in [8] as related to the
current clock state filtering via the Global Positioning
System (GPS)-based measurement of the clock time
errors in the presence of GPS time uncertainties.

,
0
and set N corresponding to the minimum difference
with the Kalman estimate in the absence of uncertainty.
ISBN: 978-1-61804-005-3
n × 103
(c)
74
Kalman
Kalman
2
1.5
x3n
Third state × 103/s2
Second state × 103/s
1.5
1
x2n
Kalman-like
1
0.5
0.5
Kalman-like
2
1
0
? 0.5
Kalman
n × 103
(a)
n × 103
(a)
Kalman-like
2
1
0
?1
?2
0.5
1
2
? 0.5
1.5
n × 103
(b)
Kalman
1
5
4
3
2
1
0
?1
?2
?3
Error × 103/s
Kalman-like
0.5
2
1
? 0.5
Kalman-like
Kalman
2
1
n × 103
(b)
Kalman
Error × 103/s2
Error × 103/s
1
2
1
Error × 103/s2
0
Kalman-like
2
1
n × 103
(c)
n × 103
(c)
Figure 3: Kalman and Kalman-like Estimates of x3n
(c) errors in the Kalman filter estimate caused by incorrect noise description in the second and third states.
Figure 2: Kalman and Kalman-like Estimates of x2n
(c) errors in the Kalman filter estimate caused by incorrect noise description in the second and third states.
4.1.2
Quadratic Process × 102
Kalman-Like (Full-Horizon)
Filtering of the second and third states
In line with the first state, the same tendencies can be
traced in the estimates of the second and third states
sketched in Fig. 2 and Fig. 3. In fact, both filters produce similar errors in Fig. 2b and Fig. 3b, except for
the region close to n = 2 × 103 , in which the Kalmanlike filter exhibits excursions, owing to transients in
the response to the uncertainty. It is also seen that the
Kalman-like filter strongly overperforms the Kalman
one in Fig. 2c and Fig. 3c when we increase the noise
variances of the second and third states in the covariance matrix.
4.2
Kalman
1
0
1.5
0.5
?5
x1n
yn
?10
n × 103
(a)
240
Error
Kalman
80
1.5
0.5
1
? 80
Denoising of a quadratic signal
Kalman-Like (Full-Horizon)
Equal
Errors
In the second typical experiment, we generated a
quadratic noiseless signal x1n observed as yn in the
presence of measurement white noise sequence vn
having the standard deviation of σv = 200 as shown
in Fig. 4a. The process was generated with the initial states x10 = 100, x20 = −5 × 10−2 /s, and
x30 = −5×10−4 /s2 . Because the Kalman filter needs
ISBN: 978-1-61804-005-3
5
? 240
n × 103
(b)
Figure 4: Denoising of a quadratic signal with the
Kalman and full horizon Kalman-like algorithms: (a)
measurement and estimates and (b) estimate errors.
75
the system covariance matrix, we voluntary accepted
σx1 = 2, σx2 = 5 × 10−2 /s, and σx3 = 10−5 /s2 .
For such a distinct model over all time, the full
horizon Kalman-like filter (Table 2) was run along
with the Kalman filter. Figure 4a shows both estimates and Fig. 4b gives us the relevant errors. Under the above-specified conditions, the Kalman filter
tracks the deterministic signal with errors associated
with the accepted components in the covariance matrix. Just on the contrary, the Kalman-like filter inherently starts with large noise when N is small.
5
IEEE Trans. Ultrason. Ferroel. Freq. Contr.,
Vol. 54, No. 11, 2007, pp. 2403–2406.
[8] Y. S. Shmaliy, Optimal gains of FIR estimators for a class of discrete-time and state-space
models, IEEE Signal Process. L., Vol. 15, 2008,
pp. 517–520.
[9] Y. S. Shmaliy, Unbiased FIR filtering of
discrete-time polynomial state-space models,
IEEE Trans. Signal Process., Vol. 57, No. 4,
2009, pp. 1241–1249.
[10] A. H. Jazwinski, Stochastic Processes and Filtering Theory, Academic Press, 1970.
Conclusions
[11] X. Zhou and X. Wang. FIR-median hybrid filters with polynomial fitting, Digital Signal Process., Vol. 39, No. 2, 2004, pp. 112–124.
In this paper, we proposed a short way to derive the
discrete-time p-shift Kalman-like FIR UE originally
presented in [18]. Based upon several examples, we
showed that this algorithm can serve efficiently as an
optimal estimator if N occurs to be large. In turn, the
Kalman algorithm requires noise description that often cannot be provided correctly and its estimate thus
cannot always be optimal.
[12] Y. S. Shmaliy, L. Morales-Mendoza L, FIR
smoothing of discrete-time polynomial signals
in state space, IEEE Trans. Signal Process.,
Vol. 58. No. 5, 2010, pp. 2544–2555.
[13] Y. S. Shmaliy, An unbiased p-step predictive
FIR filter for a class of noise-free discrete-time
models with independently observed states,
Signal Image Video Process., Vol. 3, No. 2,
2009, pp. 127–135.
References:
[1] E. Brookner, Tracking and Kalman Filtering
Made Easy, John Wiley & Sons, 1998.
[14] L. Arceo-Miquel, Y. S. Shmaliy, O. IbarraManzano, Optimal synchronization of local
clocks by GPS 1PPS signals using predictive
FIR filters, IEEE Trans. Instrum. Measur., Vol.
58, No. 6, 2009, pp. 1833–1840.
[2] Y. S. Shmaliy, An unbiased FIR filter for TIE
model of a local clock in applications to GPSbased timekeeping, IEEE Trans. Ultrason. Ferroel. Freq. Contr., Vol. 53, No. 6, 2006, pp.
862–870.
[15] P. S. Kim, M. E. Lee, A new FIR filter for state
estimation and its application, J. Comput. Sci.
Techn., Vol. 22, No. 5, 2007, pp. 779-784.
[3] Definitions of physical quantities for fundamental frequency and time metrology – random
instabilities. IEEE Standard 1139-1999.
[16] C. K. Ahn, P. S. Kim, Fixed-lag maximum likelyhood FIR smoother for state-space models,
IEICE Electonics Express, Vol. 5, No. 1, 2008,
pp. 11–16.
[4] W. H. Kwon, P. S. Kim, S. H. Han, A receding
horizon unbiased FIR filter for discrete-time
state space models, Automatica, Vol. 38, No.
3, 2002, pp. 545–551.
[17] Y. S. Shmaliy, O. Ibarra-Manzano, Optimal
FIR filtering of the clock time errors, Metrologia, Vol. 45, No. 5, 2008, pp. 571–576.
[5] Y. S. Shmaliy, A simple optimally unbiased
MA filter for timekeeping, IEEE Trans. Ultrason. Ferroel. Freq. Contr., Vol. 49, No. 6, 2002,
pp. 789–797.
[18] Y. S. Shmaliy, Linear optimal FIR estimation
of discrete time-invariant state-space models,
IEEE Trans. Signal Process., Vol. 58, No. 6,
2010, pp. 3086–3096.
[6] Z. Quan, S. Han, W. H. Kwon, A robust FIR
filter for linear discrete-time state-space signal
models with uncertainties, IEEE Signal Process. L., Vol. 14, No. 8, 2007, pp. 553–556.
[19] H. Stark, J. W. Woods, Probability, random
processes, and estimation theory for engineers,
2nd ed., Prentice Hall, 1994.
[7] Y. S. Shmaliy, On real-time optimal FIR estimation of linear TIE models of local clocks,
ISBN: 978-1-61804-005-3
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Filtering of Discrete-Time State-Space models with the p

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