6
Unbiased FIR State Estimation
The arithmetic mean is the most probable value.
Carl F. Gauss [53], p. 244
It is usually taken for granted that the right method for determining the constants is the method of least squares.
Karl Pearson [146], p. 266
6.1 Introduction
Optimal state estimation requires information about noise and initial values, which is not always available. The requirement of initial values is canceled in the optimal unbiased or ML estimators, which still require accurate noise information. Obviously, these estimators can produce large errors if the noise statistics are inaccurate or the noise is far from Gaussian. In another extreme method of state estimation that gives UFIR estimates corresponding to the LS, the observation mean is tracked only under the zero mean noise assumption. Designed to satisfy only the unbiasedness constraint, such an estimator discards all other requirements and in many cases justifies suboptimality by being more robust. The great thing about the UFIR estimator is that, unlike OFIR, OUFIR, and ML FIR, it only needs an optimal horizon of 
points to minimize MSE. It is worth noting that determining 
requires much less effort than noise statistics. Given 
, the UFIR state estimator, which has no other tuning factors, appears to be the most robust in the family of linear state estimators.
In this chapter, we discuss different kinds of UFIR state estimators, mainly filters and smoothers and, to a lesser extent, predictors.
6.2 The a posteriori UFIR Filter
Let us consider an LTV system represented in discrete‐time state space with the following state and observation equations, respectively,
These equations can be extended on 
as
using the definitions of vectors and matrices given for (4.7) and (4.14).
6.2.1 Batch Form
The unbiased filtering problem can be solved if our goal is to satisfy only the unbiasedness condition
and then minimize errors by choosing the optimal horizon of 
points.
For the FIR estimate defined as
and the state model represented with the 
th row vector in (6.3),
the condition (6.5) gives two unbiasedness constraints,
By multiplying both sides of (6.8) from the right‐hand side by the matrix identity 
and discarding the nonzero 
on both sides, we find the fundamental gain 
of the UFIR filter
where 
. Then, referring to the forced gain 
given by (6.9), the a posteriori UFIR filtering estimate can be written as
As can be seen, the gain 
(6.10a) does not contain any information about noise and initial values, which means that the UFIR filter does not have the inherent disadvantages of the KF and OFIR filter.
The error covariance for the a posteriori UFIR filter (6.11) can be written similarly to the OFIR filter (4.31) as
where the error residual matrices are given by
It now follows that (6.12) has the same structure as (4.55) of the a posteriori OUFIR filter with, however, another gain 
in (6.13) and (6.14).
Because 
given by (6.10a) does not require noise covariances, the UFIR filter is not optimal and therefore generally gives larger errors than the OUFIR filter. In practice, however, this flaw can be ignored if a higher robustness is required, as in many applications, especially industrial.
Generalized Noise Power Gain
An important indicator of the effectiveness of FIR filtering is the NPG introduced by Trench in [198]. The NPG is the ratio of the output noise variance 
to the input noise variance 
, which is akin to the noise figure in wireless communications. For white Gaussian noise, the NPG is equal to the sum of the squared coefficients of the FIR filter impulse response 
,
which is the squared norm of 
.
In state space, the homogeneous gain 
represents the coefficients of the FIR filter impulse response. Therefore, the product 
plays the role of a generalized NPG (GNPG) [179]. Referring to (6.10a) and (6.10b), GNPG can be written in the following equivalent forms:
It follows that GNPG is a symmetric square matrix 
, where the main diagonal components represent the NPGs for the system states, and the remaining components the cross NPGs. The main property of 
is that its trace decreases with increasing horizon length, which provides effective noise reduction. On the other hand, an increase in 
causes an increase in bias errors, and therefore 
must be optimally set by choosing an optimal horizon length.
6.2.2 Iterative Algorithm Using Recursions
Recursions for the batch a posteriori UFIR filtering estimate (6.11) can be found by decomposing (6.11) as 
, where
is the homogeneous estimate and 
is the forced estimate.
Using (6.16b), the estimate 
can be transformed to
and, by applying the decomposition
the recursion for the GNPG (6.16b) can be obtained if we transform 
as
This gives the following forward and backward recursive forms:
Using (6.20) and (6.22) and extracting 
from (6.10a), we next transform the product 
as
By combining (6.21) and (6.23), we finally obtain the following recursion for the homogeneous estimate
where 
is the bias correction gain of the UFIR filter.
To find the recursion for the forced estimate (6.18), we first show that
Then, by extracting 
from (6.19), 
can be transformed as
Now, by combining (6.25) and (6.26) and substituting (6.22), the forced estimate (6.18) can be represented with the recursion
which, together with (6.24), gives the recursions for the a priori and a posteriori UFIR filtering estimates, respectively,
The recursive computation of the a posteriori UFIR filtering estimate (6.11) can now be summarized with the following theorem.
Note that the need to compute the initial 
and 
in short batch forms arises from the fact that the inverse in (6.16c) does not exist on shorter horizons.
Modified GNPG
It was shown in [50,195] that the robustness of the UFIR filter can be improved by scaling the GNPG (6.21) with the weight 
as
The weight 
is defined by
where 
, 
is the integer part of 
, and 
is the number of the states. The RMS deviation 
of the estimate is computed using the innovation residual as
where 
means the dimension of the target motion. With this adaptation of the GNPG to the operation conditions, the UFIR filter can be approximately twice as accurate as the KF when tracking maneuvering targets [50].
6.2.3 Recursive Error Covariance
Although the UFIR filter does not require error covariance 
to obtain an estimate, 
may be required to evaluate the quality of the estimate on a given horizon. Since the batch form (6.12) is not always suitable for fast estimation, next we present the recursive forms obtained in [226].
Consider 
given by (6.12) and represent with
where the components are defined as
Using the decomposition 
, we represent matrix 
recursively as
Then, referring to (6.20), (6.22), (6.31), and
and substituting 
with 
, we represent matrices 
and 
recursively with
Similarly, the recursions for 
and 
can be obtained as
By combining (6.31), (6.34), (6.35), and (6.36) with (6.30), we finally arrive at the recursive form for the error covariance 
of the a posteriori UFIR filter
It is worth noting that (6.37) is equivalent to the error covariance of the a posteriori KF (3.77), in which the Kalman gain 
must be replaced with the bias correction gain 
of the UFIR filter. The notable difference is that the Kalman gain minimizes the MSE and is thus optimal, whereas the 
derived from the unbiased constraint (6.17) makes the UFIR estimate truly unbiased, but noisier, since no effort has been made so far to minimize error covariance. The minimum MSE can be reached in the UFIR filter output at an optimal horizon of 
points, which will be discussed next.
6.2.4 Optimal Averaging Horizon
Unlike KF, which has IIR, the UFIR filter operates with data on the horizon 
of 
points, which is sometimes called the FIR filter memory. To minimize MSE, 
should be optimally chosen as 
. Otherwise, if 
, noise reduction will be ineffective, and, for 
, bias errors will prevail as shown in Fig. 6.1.
Figure 6.1 The RMSE produced by the UFIR filter as a function of 
. An optimal balance is achieved when 
, where the UFIR estimate is still less accurate than the KF estimate.
Since the UFIR filter is not selective, there is no exact relationship between filtering order and memory. Optimizing the memory of a UFIR filter in state space requires finding the derivative of the trace 
of the error covariance with respect to 
, which is problematic, especially for LTV systems. In some cases, 
can be found heuristically for real data. If a reference model or the ground truth is available at the test stage, then 
can be found by minimizing 
. For polynomial models, 
was found in [169] analytically through higher order states, which has explicit limitations. If the ground truth is not available, as in many applications, then 
can be measured via the derivative of the trace of the measurement residual covariance, as shown in [153]. Next we will discuss several such cases based on the time‐invariant state‐space model
All methods will be separated into two classes depending on the operation conditions: with and without the ground truth.
Available Ground Truth
When state 
is available through the ground truth measurement, which means that 
is also available, 
can be found by referring to Fig. 6.1 and solving on 
the following optimization problem
There are several approaches to finding 
using (6.40).
- When
in (6.40) reaches a minimum with 
, we can assume that 
and represent the error covariance (6.37) by the discrete Lyapunov equation (A.34)
where 
is required to be stable and matrix 
is symmetric and positive definite. Note that the bias correction gain 
in (6.41) is related to GNPG 
, which decreases as the reciprocal of 
. The solution to (6.41) is given by the infinite sum [83]
whose convergence depends on the model. The trace of (6.42) can be plotted, and then 
can be determined at the point of minimum. For some models, 
can be found analytically using (6.42) [169]. - For
given by (6.37), the optimization problem (6.40) can also be solved numerically with respect to 
by increasing 
until 
reaches a minimum. However, for some models, there may be ambiguities associated with multiple minima. Therefore, to make the batch estimator as fast as possible, 
should be determined by the first minimum [153]. It is worth noting that the solution (6.42) agrees with the fact that when the model is deterministic, 
and 
, and the filter is a perfect match, then we have 
and 
. - From Fig. 6.1 and many other investigations, it can be concluded that the difference between the KF and UFIR estimates vanishes on larger horizons,
[179]. Furthermore, the UFIR filter is low sensitive to 
; that is, this filter produces acceptable errors when 
changes up to 30% around 
[179]. Thus, we can roughly characterize errors in the UFIR estimate at 
in terms of the error covariance of the KF. If the product 
is invertible, then replacing the Kalman gain 
with the bias correction gain 
in the error covariance of the KF as 
and then providing simple transformations give
where
is prior error covariance of the KF. For the given (6.43), the backward recursion (6.22) can be used to compute the GNPG back in time as(6.44)
Because an increase in
always results in a decrease in the trace of 
[179], then it follows that 
can be estimated by calculating the number of steps backward until 
finally becomes singular. Since 
is singular when 
, where 
is the number of the states, the optimal horizon can be measured as 
. This approach can serve when 
or when there are enough data points to find 
. Its advantage is that 
can be found even for LTV systems.
The previously considered methods make it possible to obtain 
by minimizing 
. However, if the ground truth is unavailable and the noise covariances are not well known, then 
measured in this ways may be incorrect. If this is the case, then another approach based on the measurement residual rather than the error covariance may yield better results.
Unavailable Ground Truth
In cases when the ground truth is not available, 
can be estimated using the available measurement residual covariance 
[153]. To justify the approach, we represent 
as
where the two last terms strongly depend on 
. Now, three limiting cases can be considered:
- When
, the UFIR filter degree is equal to the number of the data points, noise reduction is not provided, and the error 
with the sign reversed becomes almost equal to the measurement noise 
. Replacing 
with 
makes (6.45) close to zero
- At the optimal point
, the last two terms in (6.45) vanish by the orthogonality condition, the measurement noise prevails over the estimation error, 
, and the residual becomes
- When
, the estimation error is mainly associated with increasing bias errors. Accordingly, the first term in (6.45) becomes dominant, and 
approaches it,
The transition between the 
values can be learned if we take into account (6.46), (6.47), and (6.48). Indeed, when 
, the bias errors are practically insignificant. They grow with increasing 
and approach the standard deviation in the estimate at 
when the filter becomes slightly inconsistent with the system due to process noise.
Since 
decreases monotonously with increasing 
due to noise reduction, 
is also a monotonic function when 
. It grows monotonously from 
given by (6.46) to 
given by (6.47) and passes through 
with a minimum rate when 
is minimum. It matters that the rate of 
around 
cannot be zero due to increasing bias errors. And this is irrespective of the model, as the UFIR filter reduces white noise variance as the reciprocal of 
, and the effect of bias is still small when 
.
Another behavior of 
can be observed for 
, when the bias errors grow not equally in different models. Although 
is always convex on 
in stable filters, bias errors affecting its ascending right side can grow either monotonously in polynomial models or eventually oscillating in harmonic models. The latter means that 
can reach its value (6.48) with oscillations, even having a constant average rate.
It follows from the previous that the derivative of 
with respect to 
can pass through multiple minima and that the first minimum gives the required value of 
. The approach developed in [153] suggests that 
can be determined in the absence of the ground truth by solving the following minimization problem
where minimization should be ensured by increasing 
, starting from 
, until the first minimum is reached. To avoid ambiguity when solving the problem (6.49), the number of points should be large enough. Moreover, 
may require smoothing before applying the derivative.
Let us now look at the 
properties in more detail and find a stronger justification for (6.49). Since the difference between the a priori and a posteriori UFIR estimates does not affect the dependence of 
on 
, we can replace 
with 
, 
with 
, and 
with 
. We also observe that, since 
, 
, 
, and 
can be replaced by the UFIR estimate (6.29), the last two terms in (6.45) can be represented as
This transforms (6.45) to
Again we see the same picture. By 
, the bias correction gain in the UFIR filter is close to unity, 
, and (6.50) gives a value close to zero, as in (6.46). When 
, 
and 
are small enough and (6.50) transforms into (6.47). Finally, for 
, 
becomes large and 
small, which leads to (6.48).
Although we have already discussed many details, it is still necessary to prove that the slope of 
is always positive up to and around 
and that this function is convex on 
. To show this, represent the derivative of 
as 
, assuming a unit time‐step for simplicity. For (6.50) at the point 
we have 
and, therefore, 
can be written as
Since the GNPG 
decreases with increasing 
, then we have 
, and it follows that 
and so is the derivative of 
,
Thus, 
passes through 
with minimum positive slope. Moreover, since bias errors force 
to approach 
as 
increases, function 
is reminiscent of 
. Finally, further minimizing 
with 
yields 
, which can be called the key property of 
.
6.3 Backward a posteriori UFIR Filter
The backward UFIR filter can be derived similarly to the forward UFIR filter if we refer to (5.23) and (5.26) and write the extended model as
for which the definitions of all vectors and matrices can be found after (5.23) and (5.26). Batch and recursive forms for this filter can be obtained as shown next.
6.3.1 Batch Form
The batch backward UFIR estimate 
can be defined as
where 
is given by (6.53). The homogenous gain 
and the forced gain 
can be determined by satisfying the unbiasedness condition 
applied to (6.54) and the model
which is the last 
th row vector in (6.52). This gives two unbiasedness constraints
and the first constraint (6.56) yields the fundamental gain
For 
obtained by (6.58), the forced gain is defined by (6.57), and we notice that the same result appears when the noise is neglected in the backward OFIR estimate (5.28). The backward a posteriori UFIR filtering estimate is thus given in a batch form by (6.54) with the gains (6.58) and (6.57).
The error covariance 
for the backward UFIR filter can be written as
where the error residual matrices are specified by
Note that, as in the forward UFIR filter, matrices (6.60) and (6.61) provide an optimal balance between bias and random errors in the backward UFIR filtering estimate 
if we optimally set the averaging horizon of 
points, as shown in Fig. 6.1 and Fig. 6.2.
6.3.2 Recursions and Iterative Algorithm
The batch estimate (6.54) can be computed iteratively on 
if we find recursions for 
and 
.
By introducing the backward GNPG
we write the homogeneous estimate as
and represent recursively the inverse of the GNPG (6.62) by
This gives two recursive forms
Referring to 
taken from (6.63a) and (6.63b), we next represent the product 
as
By combining (6.65), (6.66), and (6.67), a recursion for 
can now be written as
where 
is the bias correction gain of the UFIR filter.
To derive a recursion for the forced gain 
, we use the decompositions
follow the derivation of the recursive forms for the backward OFIR filter, provide routine transformations, and arrive at
A simple combination of (6.68) and (6.69) finally gives
where the prior estimate is specified by
The backward iterative a posteriori UFIR filtering algorithm can now be generalized with the pseudocode listed as Algorithm 12. The algorithm starts with the initial 
and 
computed at 
in short batch forms (6.62) and (6.54). The iterative computation is performed back in time on the horizon of 
points so that the filter gives estimates from zero to 
.
It is worth noting that the typical differences between KFs and UFIR filters illustrated in Fig. 6.4 are recognized as fundamental [179]. Another thing to mention is that forward and backward filters act in opposite directions, so their responses appear antisymmetric.
6.3.3 Recursive Error Covariance
The recursive form for the error covariance (6.59) of the backward a posteriori UFIR filter can be found similarly to the forward UFIR filter. To this end, we first represent (6.59) using (6.60) and (6.61) as
We then find recursions for each of the matrix products in the previous relationship, combine the recursive forms obtained, and finally come up with
It can now be shown that there is no significant difference between the error covariances of the forward and backward UFIR filters. Since these filters process the same data, but in opposite directions, it follows that the errors on the given horizon 
are statistically equal.
6.4 The 
‐lag UFIR Smoother
In postprocessing and when filtering stationary and quasi stationary signals, smoothing may be the best choice because it provides better noise reduction. Various types of smoothers can be designed using the UFIR approach, although many of them appear to be equivalent as opposed to OFIR smoothing. Here we will first derive the 
‐lag FFFM UFIR smoother and then show that all other UFIR smoothing structures of this type are equivalent due to their ability to ignore noise.
Let us look again at the 
‐lag FFFM FIR smoothing strategy illustrated in Fig. 5.2. The corresponding UFIR smoother can be designed to satisfy the unbiasedness condition
where the 
‐lag estimate can be defined as
and the state model represented by the 
th row vector in (6.3) as
where 
is the 
th row vector in (4.9) and so is 
in 
.
6.4.1 Batch and Recursive Forms
Condition (6.73) applied to (6.74) and (6.75) gives two unbiasedness constraints,
and after simple manipulations the first one in (6.76) gives a fundamental UFIR smoother gain 
,
Referring to 
, we next transform (6.78) to
where 
is the homogeneous gain (6.10a) of the UFIR filter, and express the 
‐lag homogeneous UFIR smoothing estimate in terms of the filtering estimate 
as
that does not require recursion; that is, the recursively computed filtering estimate 
is projected into 
in one step by the matrix 
.
For the forced estimate, the recursive form
appears if we first represent the last row vector 
of the matrix 
given by (4.9) as
where matrix 
becomes matrix 
if we replace 
by 
. Also 
can be written as
and then the subsequent modification of (6.81) gives
Next, using the decomposition 
, we transform the forced estimate to the form
where the 
‐varying product correction term 
is computed recursively as
By combining (6.80) and (6.82), we finally arrive at the recursion
using which, the recursive 
‐lag FFFM a posteriori UFIR smoothing algorithm can be designed as follows. Reorganize (6.83) as
set 
, assign 
, and compute for 
until this recursion gives 
. Given 
and 
, the smoothing estimate is obtained by (6.84). It is worth noting that in the particular case of an autonomous system, 
, the 
‐lag a posteriori UFIR smoothing estimate is computed using a simple projection (6.80).
6.4.2 Error Covariance
In batch form, the 
‐varying error covariance 
of the FFFM UFIR smoother is determined by (6.12), although with the renewed matrices,
where the error residual matrices are given by
To find a recursive form for (6.85), we write it in the form
where 
is the error covariance (6.37) of the UFIR filter, and represent the 
‐varying amendment 
as
It can be seen that 
naturally becomes zero at 
due to 
. Furthermore, the structure of the matrix 
suggests that 
and also 
.
By considering several cases of 
for 
,
and reasoning deductively, we obtain the following recursion for 
,
where the matrix 
is still in batch form as
Similarly, we represent 
in special cases as
and then replace the sum in the parentheses with
which gives the following recursion for 
,
By combining (6.90) and (6.93) with (6.89), we finally obtain the recursive form for the error covariance as
which should be computed starting with 
for the initial value
using the matrix 
given by (6.91) and 
by (6.92). Noticing that the recursive form for the batch matrix 
given by (6.92) is not available in this procedure, we postpone to “Problems” the alternative derivation of recursive forms for (6.85).
Time‐Invariant Case
For LTI systems, the matrix 
, given in batch form as (6.92), can easily be represented recursively as
and the 
‐lag FFFM UFIR smoothing algorithm can be modified accordingly. Computing the initial 
by (6.95) and knowing the matrix 
, one can update the estimates for 
as
It should be noted that the computation of the error covariance 
using (6.99) does not have to be necessarily included in the iterative cycle and can be performed only once after 
reaches the required lag‐value.
6.4.3 Equivalence of UFIR Smoothers
Other types of 
‐lag UFIR smoothers can be obtained if we follow the FFBM, BFFM, and BFBM strategies discussed in Chapter . To show the equivalence of these smoothers, we will first look at the FFBM UFIR smoother and then draw an important conclusion.
The 
‐lag FFBM UFIR smoother can be obtained similarly to the FFFM UFIR smoother in several steps. Referring to the backward state‐space model in (5.23) and (5.26), we define the state at 
) as
and the forward UFIR smoothing estimate as
From (6.100) at the point 
, we can also obtain
where 
is the last row vector in (5.41) and so is 
in 
.
The unbiasedness condition 
applied to (6.100) and (6.101) gives
Taking into account that 
and replacing 
extracted from (6.102) with 
, the previous relationship can be split into two unbiasedness constraints
What now follows is that the first constraint (6.103) is exactly the constraint (6.76) for the FFFM UFIR smoother. We then observe that for 
we have 
and thus 
can be transformed as
Finally, we transform the second constraint (6.104) to
and conclude that this is constraint (6.77) for FFFM UFIR smoothing.
Now the following important conclusion can be drawn. Because the FFBM and FFFM UFIR smoothers obey the same unbiasedness constraints in (6.76) and (6.77), then it follows that these smoothers are equivalent. And this is not surprising, since all UFIR structures obeying only the unbiasedness constraint ignore noise. By virtue of that, the forward and backward models become identical at 
, and thus the FFFM and FMBM UFIR smoothers are equivalent. In addition, since the forward and backward UFIR filters are equivalent for the same reason, then it follows that the BFFM and BFBM UFIR smoothers are also equivalent, and an important finding follows.
Equivalence of UFIR smoothers: Satisfied only the unbiasedness condition,
‐lag FFFM, FFBM, BFFM, and BFBM UFIR smoothers are equivalent.
In view of the previous definition, the 
‐lag FFFM UFIR smoother can be used universally as a UFIR smoother in both batch and iterative forms. Other UFIR smoothing structures such as FFBM, BFFM, and BFBM have rather theoretical meaning.
6.5 State Estimation Using Polynomial Models
There is a wide class of systems and processes whose states change slowly over time and, thus, can be represented by degree polynomials. Examples can be found in target tracking, networking, and biomedical applications. Signal envelopes in narrowband wireless communication channels, remote wireless control, and remote sensing are also slowly changing.
The theory of UFIR state estimation developed for discrete time‐invariant polynomial models [172] implies that the process can be represented on 
using the following state‐space equations
where 
and 
are zero mean noise vectors. The 
power of the system matrix 
has a specific structure
which means that, by 
, each row in 
is represented with the descending 
th degree, 
, Taylor or Maclaurin series, where 
is the number of the states.
6.5.1 Problems Solved with UFIR Structures
The UFIR approach applied to the model in (6.106) and (6.107) to satisfy the unbiasedness condition gives a unique 
th degree polynomial impulse response function 
for each of the states separately, where 
is a discrete time shift relative to the current point 
. Function 
has many useful properties, which make the UFIR estimate near optimal. Depending on 
, the following types of 
can be recognized. When 
, function 
is used to obtain UFIR filtering. When 
, function 
serves to obtain 
‐lag UFIR smoothing filtering. Finally, when 
, function 
is used to obtain 
‐step UFIR predictive filtering.
Accordingly, the following problems can be solved by applying UFIR structures to a polynomial signal 
measured as 
in the presence of zero mean additive noise:
- Filtering provides an estimate at
based on data taken from 
,
- Smoothing filtering provides a
‐lag, 
, smoothing estimate at 
based on data taken from 
,
- Predictive filtering provides a
‐step, 
, predictive estimate at 
based on data taken from 
,
Note that one‐step predictive UFIR filtering,
, was originally developed for polynomial models in [71]. In state space, it is known as RH FIR filtering [106] used in state‐feedback control and MPC. - Smoothing provides a
‐lag smoothing estimate at 
, 
, based on data taken from 
,
- Prediction provides a
‐step, 
, prediction at 
based on data taken from 
,
In the previous definitions of the state estimation problems, the functions 
and 
are not equal for 
, but can be transformed into each other. More detail can be found in [181].
6.5.2 The 
‐shift UFIR Filter
Looking at the details of the UFIR strategy, we notice that the approach that ignores zero mean noise allows us to solve universally the filtering problem (6.109), the smoothing filtering problem (6.110), and the predictive filtering problem (6.111) by obtaining a 
‐shift estimate [176]. The UFIR approach also assumes that a shift to the past can be achieved at point 
using data taken from 
with a positive smoother lag 
, a shift to the future at point 
using data taken from 
with a positive prediction step 
, and that 
means filtering.
Thus, the 
‐shift UFIR filtering estimate can be defined as
where the components of the gain 
are diagonal matrices specified by the matrix
whose components, in turn, are the values of the function 
. The unbiasedness condition applied to (6.114) gives the unbiasedness constraint
where 
.
For the 
th system state, the 
‐shift UFIR filtering estimate is defined as
and the constraint (6.115) is transformed to
where 
means the 
th row in 
and the remaining 
th rows are given by
It is worth noting that the linear matrix equation 6.117 can be solved analytically for the 
th degree polynomial impulse response 
. This gives the 
‐varying function
where 
, 
, and the coefficient 
is defined by
where the determinant 
of the 
‐varying Hankel matrix 
is specified via the Vandermonde matrix 
as
and 
is the minor of 
. The 
th component 
, 
, of matrix (6.122) is the power series
where 
is the Bernoulli polynomial.
The coefficients of several low‐degree polynomials 
, which are most widely used in practice, are given in Table 6.1. Using this table, or (6.121) for higher‐degree systems, one can obtain an analytic form for 
as a function of 
, where 
is for UFIR filtering, 
for UFIR smoothing filtering, and 
for UFIR predictive filtering.
Table 6.1 Coefficients 
of Low‐Degree Functions 
.
 | Coefficients |
|---|---|
| Uniform: |  |
| Ramp: |  |
 | |
| Quadratic: |  |
 | |
 | |
 |
Properties of 
‐shift UFIR Filters
The most important properties of the impulse response function 
are listed in Table 6.2, which also summarizes some of the critical findings [181]. If we set 
, we can use this table to examine the properties of the UFIR filter. We can also characterize the UFIR filter in terms of system theory. Indeed, since the transfer function 
of the 
th degree UFIR filter is equal to unity at zero frequency, 
, then it follows that this filter is essentially a low‐pass (LP) filter. Moreover, if we analyze other types of FIR structures in a similar way, we can come to the following general conclusion.
Table 6.2 Main Properties of 
.
| Property | |
|---|---|
| Region of existence: |  |
 ‐transform at  : |  (UFIR filter is an LP filter) |
| Unit area: |  |
| Energy (NPG): |  |
| Value at zero: |  |
| Zero moments: |  ,  |
 ,  | |
| Orthogonality: |  , |
 , | |
 | |
| Unbiasedness: |  |
 |
State estimator in the transform domain: All optimal, optimal unbiased, and unbiased state estimators are essentially LP structures.
Since the sum of the values of 
is equal to unity, it follows that the UFIR filter is strictly stable. More specifically, it is a BIBO stable filter due to the FIR. The sum of the squared values of 
represents the filter NPG, which is equal to the energy of the function 
. The important thing is that NPG is equal to the function 
at zero, which, in turn, is equal to the zero‐degree coefficient 
. This means that the denoising properties of the UFIR filter can be fully explored using the value 
. It is also worth noting that the family of functions 
and 
, 
, establish an orthogonal basis, and thus high‐degree impulse responses can be computed through low‐degree impulse responses using a recurrence relation.
The fact that all moments of the function 
are equal to zero means nothing more and nothing less than the UFIR filter is strictly unbiased by design. The unbiasedness of FIR filters can also be checked by equating the area of the transfer function and the area of the squared magnitude frequency response. The same test for unbiasedness in the discrete Fourier transform (DSP) domain is ensured by equating the sum of the DSP values and the sum of the squared magnitude values. At the end of this chapter, when we will consider practical implementations, we will take a closer look at the properties of UFIR filters in the transform domain.
6.5.3 Filtering of Polynomial Models
The UFIR filter can be viewed as a special case of the 
‐shift UFIR filter when 
. This transforms the impulse response function (6.120) to
where the coefficient 
is defined by (6.121) as
The constant (zero‐degree, 
) FIR function 
is used for simple averaging. The ramp (first‐degree, 
) FIR function
is applicable for linear signals. The quadratic (second‐degree, 
) FIR function
is applicable for quadratically changing signals, etc.
There is also a recurrence relation [131]
that can be used when 
and 
to compute 
of any degree in terms of the lower‐degree functions. Several low‐degree functions 
computed using (6.128) are shown in Fig. 6.5.
Figure 6.5 Low‐degree polynomial FIR functions 
.
The NPG of a UFIR filter, which is defined as 
, suggests that the best noise reduction associated with the lowest NPG is obtained by simple averaging (case 
in Fig. 6.5). An increase in the filter degree leads to an increase in the filter output noise, and the following statements can be made.
Noise reduction with polynomial filters: 1) A zero‐degree filter (simple averaging) is optimal in the sense of minimum produced noise; 2) An increase in the filter degree or, which is the same, the number of states leads to an increase in random errors.
Therefore, because of the better noise reduction, the low‐degree UFIR state estimators are most widely used. In Fig. 6.5, we see an increase in 
caused by an increase in the degree 
at 
.
6.5.4 Discrete Shmaliy Moments
In [131], the class of the 
th‐degree polynomial FIR functions 
, defined by (6.124) and having the previously listed fundamental properties, was tested using the orthogonality condition
where 
is the Kronecker symbol and 
. It was found that the set of functions 
for 
and 
is orthogonal on 
with the square of the weighted norm 
of 
given by
where 
and 
for 
is the Pochhammer symbol. The non‐negative weight 
in (6.129) is the ramp probability density function
An applied significance of this property for signal analysis is that, due to the orthogonality, higher‐degree functions 
can be computed in terms of lower‐degree functions using a recurrence relation (6.128).
Since the functions 
have the properties of discrete orthogonal polynomials (DOP), they were named in [61,162] discrete Shmaliy moments (DSM) and investigated in detail. It should be noted that, due to the embedded unbiasedness, DSM belong to the one‐parameter family of DOP, while the classical Meixner and Krawtchouk polynomials belong to the two‐parameter family, and the most general Hahn polynomials belong to the three‐parameter family of DOP [131]. This makes the DSM more suitable for unbiased analysis and reconstruction if signals than the classical DOP and Tchebyshev polynomials. Note that DSM are also generalized by Hanh's polynomials along with the classical DOP.
6.5.5 Smoothing Filtering and Smoothing
Both the 
‐lag UFIR smoothing filtering problem (6.110) and the UFIR smoothing problem (6.112) can be solved universally using (6.120) and (6.121) if we assign 
, 
.
Smoothing Filtering
A feature of UFIR smoothing filtering is that the function 
exists on the horizon 
, and otherwise it is equal to zero, while the lag 
is limited to 
. Zero‐degree UFIR smoothing filtering is still provided by simple averaging. First‐degree UFIR smoothing filtering can be obtained using the 
‐varying ramp response function
where 
is the UFIR filter ramp impulse response (6.126). What can be observed is that better noise reduction is accompanied in (6.132) by loss of stability. Indeed, when 
approaches unity, the second term in (6.132) grows indefinitely, which also follows from NPG of the first‐degree given by
Approximation (6.134) valid for large 
and 
clearly shows that increasing the horizon length 
leads to a decrease in NPG, which improves noise reduction. Likewise, increasing the lag 
results in better noise reduction.
The features discussed earlier are inherent to UFIR smoothing filters of any degree, although with the important specifics illustrated in Fig. 6.6. This figure shows that all smoothing filters of degree 
provide better noise reduction as 
increases, starting from zero. However, NPG can have multiple minima, and therefore the optimal lag 
does not necessarily correspond to the middle of the averaging horizon 
. For odd degrees, 
can be found exactly in the middle of 
, while for even degrees at other points. For more information see [174].
Figure 6.6 The 
‐varying NPG of a UFIR smoothing filter for several low‐degrees.
Smoothing
The smoothing problem defined by (6.112) and discussed in state space in Chapter can be solved if we introduce a gain 
as
which exists on 
with the same major properties as 
. Indeed, the ramp UFIR smoother can be designed using the FIR function
The NPG for this smoother is defined by
As expected, NPG (6.138) is exactly the same as (6.133) of the UFIR smoothing filter, since noise reduction is provided by both structures with equal efficiency. Note that similar conclusions can be drawn for other UFIR smoothing structures applied to polynomial models.
6.5.6 Generalized Savitzky‐Golay Filter
A special case of the UFIR smoothing filter was originally shown in [158] and is now called the Savitzky‐Golay (SG) filter. The convolution‐based smoothed estimate appears at the output of the SG filter with a lag 
in the middle of the averaging horizon as
where the set of 
convolution coefficients 
is determined by the linear LS method to fit with typically low‐degree polynomial processes. Since the coefficients 
can be taken directly from the FIR function 
, the SG filter is a special case of (6.110) with the following restrictions:
- The horizon length
must be odd; otherwise, a fractional number appears in the sum limits. - The fixed‐lag is set as
, while some applications require different lags and the optimal lag may not be equal to this value.
It then follows that the UFIR smoothing filter (6.110), developed for arbitrary 
and lags 
, generalizes the SG filter (6.139) in the particular case of odd 
and 
. Also note that the lag in (6.110) can be optimized for even‐degree polynomials as shown in [174].
6.5.7 Predictive Filtering and Prediction
The predictive filtering problem (6.111) can be solved directly using the 
‐shift FIR function (6.120) if we set 
. Like filtering and smoothing, zero‐degree UFIR predictive filtering is provided by simple averaging. The first‐degree predictive filter can be designed using a 
‐varying ramp function
which makes a difference with smoothing filtering. The NPG of the predictive filter is determined by
and we note an important feature: increasing the prediction step 
leads to an increase in NPG, and denoising becomes less efficient. Looking at the region around 
in Fig. 6.5 and referring to (6.138) and (6.142), we come to the obvious conclusion that prediction is less precise than filtering and filtering is less precise than smoothing.
The prediction problem (6.113) can be solved similarly to the smoothing problem by introducing the gain
which exists on 
with the same main properties as 
. The ramp UFIR predictor can be designed using the FIR function
and its efficiency can be estimated using the NPG (6.141). Note that similar conclusions can be drawn for other UFIR predictors corresponding to polynomial models.
6.6 UFIR State Estimation Under Colored Noise
Like KF, the UFIR filter can also be generalized for Gauss‐Markov colored noise, if we take into account that unbiased averaging ignores zero mean noise. Accordingly, if we convert a model with colored noise to another with white noise and then ignore the white noise sources, the UFIR filter can be used directly. In Chapter, we generalized KF for CMN and CPN. In what follows, we will look at the appropriate modifications to the UFIR filter and focus on what makes it better in the first place: the ability to filter out more realistic nonwhite noise. Although such modifications require tuning factors, and therefore the filter may be more vulnerable and less robust, the main effect is usually positive.
6.6.1 Colored Measurement Noise
We consider the following state‐space model with Gauss‐Markov CMN, which was used in Chapter to design the GKF,
where 
and 
have the covariances 
and 
. The coloredness factor 
is chosen such that the Gauss‐Markov noise 
is always stationary, as required. Using measurement differencing as 
, we write a new observation as
where 
is the new observation matrix, the auxiliary matrices are defined as 
and 
, and the noise
is zero mean white Gaussian with the properties 
, 
, and 
, where 
.
It can be seen that the modified state‐space model in (6.145) and (6.148) contains white and time‐correlated noise sources 
and 
. Unlike KF, the UFIR filter does not require any information about noise, except for the zero mean assumption. Therefore, both 
and 
can be ignored, and thus the UFIR filter is unique for both correlated and de‐correlated 
and 
. However, the UFIR filter cannot ignore CMN 
, which is biased on a finite horizon 
.
The pseudocode of the a posteriori UFIR filtering algorithm for CMN is listed as Algorithm 13. To initialize iterations avoiding singularities, the algorithm requires a short measurement vector 
and an auxiliary block matrix
It can be seen that for 
this algorithm becomes the standard UFIR filtering algorithm. More details about the UFIR filter developed for CMN can be found in [183].
The error covariance for the UFIR filter is given by [179]
where the matrices 
, 
, and 
are defined earlier and the reader should remember that the GNPG matrix 
is symmetric.
Typical RMSEs produced by the KF and UFIR algorithms versus 
are shown in Fig. 6.8 [183], where we recognize several basic features. It can be seen that the KF and UFIR filter modified for CMN give fewer errors than the original ones. It should also be noted that GKF performs better when 
, and that the general UFIR filter is more accurate when 
. This means that a more robust UFIR filter may be a better choice when measurement noise is heavily colored.
![Schematic illustration of typical RMSEs produced by KF and UFIR filter for a two-state model versus the coloredness factor Ψ [183].](https://imgdetail.ebookreading.net/2023/10/9781119863076/9781119863076__9781119863076__files__images__c06f008.png)
Figure 6.8 Typical RMSEs produced by KF and UFIR filter for a two‐state model versus the coloredness factor 
[183].
6.6.2 Colored Process Noise
Unlike the CMN, which always needs to be filtered out, the CPN, or at least its slow spectral components, can be tracked to avoid losing information about the process behavior. Let us show how to deal with CMN based on the following the state‐space model
where matrices 
, 
, and 
are nonsingular, 
, and 
is the Gauss‐Markov CPN. Noise vectors 
and 
are mutually uncorrelated with the covariances 
and 
. The coloredness factor matrix 
is chosen such that the CPN 
is always stationary.
Using state differencing (3.175a), a new state equation can be written as
where 
is white Gaussian, 
, 
, and 
is defined by solving for initial 
the NARE 
, where 
.
Using (3.188), we write the modified observation equation as
where 
can be substituted with the available past estimate 
.
The pseudocode of the UFIR algorithm developed for CPN is listed as Algorithm 14. To initialize iterations, Algorithm 14 employs a short data vector 
and an auxiliary matrix
Note that, by setting 
and 
, Algorithm 14 becomes the standard iterative UFIR filtering algorithm.
The error covariance of the UFIR filter modified for CPN can be found if we notice that 
, where 
. Since this estimate is subject to the constraint 
[182], the error 
for 
can be transformed to
and the corresponding error covariance 
found to be
This finally gives
Typical RMSEs produced by the modified and original filters for a two‐state polynomial model with CPN are shown in Fig. 6.9 as functions of the scalar coloredness factor 
[182]. The filtering effect here is reminiscent of the effect shown in Fig. 6.6 for CMN, and we notice that the accuracy of the original filters has been improved. However, a significant improvement in performance is recognized only when the coloredness factor is relatively large, 
. Otherwise, the discrepancies between the filter outputs are not significant.
![Schematic illustration of typical RMSEs produced by the two-state UFIR filter, KF, and modified KF and UFIR filter in the presence of CPN as functions of the coloredness factor θ [182].](https://imgdetail.ebookreading.net/2023/10/9781119863076/9781119863076__9781119863076__files__images__c06f009.png)
Figure 6.9 Typical RMSEs produced by the two‐state UFIR filter, KF, and modified KF and UFIR filter in the presence of CPN as functions of the coloredness factor 
[182].
Considering the previous modifications of the UFIR filter, we finally conclude that the filtering effect in the presence of CMN and/or CPN is noticeable only with strong coloration.
6.7 Extended UFIR Filtering
Representation of physical processes and approximation of systems using linear models does not always fit with practical needs. Looking at the nonlinear model in (3.226) and (3.227) and analyzing the Taylor series approach that results in the extended KF algorithms, we conclude that UFIR filtering can also be adapted to nonlinear behaviors [178], as will be shown next.
Given a nonlinear state‐space model
where 
and 
are mutually uncorrelated, zero mean, and not obligatorily Gaussian additive noise vectors. In Chapter, when derived EKF, it was shown that (6.156) and (6.157) can be approximated using the second‐order Taylor series expansion as
where 
is the modified observation vector and 
and 
represent the components resulting from the linearization,
in which 
, 
, and 
and 
are Cartesian basis vectors with the 
th and 
th components unity, and all others are zeros. The nonlinear functions are represented by
where 
and 
are Jacobian matrices and 
and 
are Hessian matrices.
Based on (6.158) and (6.159), we can now develop the first‐ and second‐order extended UFIR filtering algorithms. Similarly to the EKF‐1 and EKF‐2 algorithms, we will refer to the extended UFIR algorithms as EFIR‐1 and EFIR‐2.
6.7.1 First‐Order Extended UFIR Filter
Let us look at the model in (6.158) and (6.159) again and assume that the mutually uncorrelated 
and 
are zero mean and white Gaussian. The EFIR‐1 (first‐order) filtering algorithm can be designed using the following recursions [178]
The pseudocode of the EFIR‐1 filtering algorithm is listed as Algorithm 15. As in the EKF‐1 algorithm, here the prior estimate 
is obtained using a nonlinear projection 
, and then 
is projected onto the observation as 
. Also, the system and observation matrices 
and 
are Jacobian. The error covariance for this filter can be computed using (6.37) and the Jacobian matrices 
and 
.
The EFIR‐1 filtering algorithm developed in this way turns out to be simple and in most cases truly efficient. However, it should be noted that the initial state 
, computed linearly in line 5 of Algorithm 15, may be too rough when the nonlinearity is strong. If so, then 
can be computed using multiple projections as 
, 
, ..., 
. Otherwise, an auxiliary EKF‐1 algorithm can be used to obtain 
.
6.7.2 Second‐Order Extended UFIR Filter
The derivation of the second‐order EFIR‐2 filter is more complex, and its application may not be very useful because it loses the ability to ignore noise covariances. To come up with the EFIR‐2 algorithm, we will mainly follow the results obtained in [178].
We first define the prior estimate by
for which, by the cyclic property of the trace operator, the expectation of 
can be found as
We can show that for 
given by (6.164), the expectation of the prior error 
becomes identically zero,
Averaging the nonlinear function 
gives
where the expectation of 
can be found as
that allows us to obtain the error covariance as will be shown next.
Prior Error Covariance
Using (6.158) and (6.164) and taking into account that, for zero mean Gaussian noise, the vector 
and other similar vectors are equal to zero, the a priori error covariance 
can be transformed as
where matrix 
is specialized via its 
th component
Posterior Error Covariance
Reasoning similarly, the a posteriori error covariance 
can be transformed using extended nonlinearities if we first represent it as
take into account that 
and 
, provide the averaging, and obtain
Due to the symmetry of the matrices 
, 
, and 
, the following expectations can be transformed as
Then, taking into account that the expectations of other products are matrices with zero components, we finally write the covariance 
in the form
where, the 
th component of matrix 
is defined by
and the 
th component of matrix 
is
It has to be remarked now that when developing the second‐order extended algorithms, the authors use two forms of 
. The complete form in (6.170) can be found in [12,157]. On the contrary, in [72,185] only first‐order components are preserved.
The pseudocode of the EFIR‐2 filtering algorithm is listed as Algorithm 16. For the given 
, 
, and 
, the set of auxiliary matrices is computed and updated at each 
. Then all matrices and vectors are updated iteratively, and the last updated 
and 
go to the output when 
. Although the second‐order EFIR‐2 algorithm has been designed to improve accuracy, empirical studies show that its usefulness is questionable due to the following drawbacks:
- Unlike the EFIR‐1 Algorithm 15, the EFIR‐2 Algorithm 16 requires
and 
and is thus less robust. It thus has more in common with the EKF rather than with the EFIR‐1 filter. - As noted in [178], nothing definite can be said about the accuracy of the EFIR‐2 algorithm compared to the EFIR‐1 algorithm. The same conclusion was made earlier in [185] regarding the EKF‐2 algorithm.
Overall, it can be said that the EFIR‐2 and EKF‐2 filtering algorithms do not demonstrate essential advantages over the EFIR‐1 and EKF‐1 algorithms. Moreover, these algorithms are more computationally complex and slower in operation.
6.8 Robustness of the UFIR Filter
Practice dictates that having a good estimator is not enough if the operating conditions are not satisfied from various perspectives, especially in unknown environments. In such a case, the required estimator must also be robust. In Chapter, we introduced the fundamentals of robustness, which is seen as the ability of an estimator not to respond to undesirable factors such as errors in noise statistics, model errors, and temporary uncertainties. We are now interested in investigating the trade‐off in robustness between the UFIR estimator and some other available state estimators. To obtain reliable results, we will examine along the KF, which is optimal and not robust, and the robust game theory recursive 
filter (3.307)–(3.310), which was developed in [185]. We will use the following state‐space model
where the noise covariances 
and 
in the noise vectors 
and 
are not necessarily known, as required by KF. To apply the 
filter, 
and 
will be thought of as norm‐bounded with symmetric and positive definite matrices 
and 
.
In our scenario, there is not enough information about 
and 
to optimally tune KF, and the maximum values of the matrices 
and 
are also not known to make the 
filter robust. Therefore, we organize a scaling test for robustness by representing any uncertain component 
in (6.173) and (6.174) as 
, where the increment 
is due to a positive‐valued scaling factor 
such that 
means undisturbed 
. This test can be applied with the following substitutions: 
, 
, 
, 
, 
, and 
, where the set of scaling factors 
can either change matrices or not when 
. For 
, the product matrix 
becomes
We can learn about the robustness of recursive estimators by examining the sensitivity of the bias correction gain to interfering factors and its immunity to these factors. Taken from the UFIR Algorithm 11, the alternate KF (3.92)–(3.96) [184], and the 
filter (3.307)–(3.310) [185], the bias corrections gains can be written as, respectively,
where 
, 
, and matrix 
is constrained by the positive definite matrix 
. Note that the tuning factor 
is required by the game theory 
filter in order to outperform KF.
The influence of 
on (6.176)–(6.178) can be learned if we use the following rule: if the error factor causes a decrease in 
, then the bias errors will grow; otherwise, random errors will dominate. The increments 
caused by the error factors in the bias correction gains can now be represented for each of the filters as
and we notice that lower 
means higher robustness. There is one more important preliminary remark to be made. While more tuning factors make an estimator potentially more accurate and precise, it also makes an estimator less robust due to possible tuning errors (Fig. 1.3). In this sense, the two‐parameter 
UFIR filter appears to be more robust, but less accurate, while the four‐parameter 
KF and five‐parameter 
game theory 
filter are less robust but potentially more accurate.
6.8.1 Errors in Noise Covariances and Weighted Matrices
The additional errors produced by estimators applied to stochastic models and caused by 
and 
occur due to the practical inability to collect accurate estimates of noise covariances and norm‐bounded weights, especially for LTV systems.
UFIR Filter
The UFIR filter does not require any information about zero mean noise. Thus, this filter is insensitive (robust) to variations in 
and 
, unlike the KF and game theory 
filter.
Kalman Filter
For 
and 
, errors in the KF can be learned in stationary mode if we allow 
. Accordingly, the error covariance can be represented using the DARE
The solution to (6.179) does not exist in simple form. But if we accept 
, which is true for low measurement noise, 
, then (6.179) can be replaced by the discrete Lyapunov equation [184] (A.34)
which is solvable (A.35). Although approximation (6.180) can be rough when 
is not small, it helps to compare the KF and 
filter for robustness, as will be shown next.
When 
, we can assume that KF is tracking state with high precision and let 
that gives 
. This gives 
and 
, and we conclude that 
. Note that this relationship is consistent with the equality in (6.183), which was obtained for the isolated cases of 
and 
.
Referring to lemma 6.1, the bias correction gain (6.177) can now be approximated for 
by the inequality
which does not hold true for 
. It is worth noting that 
and 
act in opposite directions, and therefore the worst case will be when 
and 
or vice versa.
Game Theory 
Filter
Reasoning similarly for equal weighting errors obtained with 
, the bias correction gain (6.178) for the recursive game theory 
filter can be approximated by the inequality [180]
It can be seen that 
makes no difference with the KF gain (6.184) for Gaussian models implying 
and 
. Otherwise, we can expect KF to be less accurate and the 
filter to perform better, provided that both 
and 
are properly maximized. Indeed, if we assume that 
and 
, then we can try to find some small 
such that the effect of 
and 
is compensated. However, if 
is not set properly, the 
filter can cause even more errors than the KF. Moreover, even if 
is set properly, any deviation of 
from unity will additionally chance the gain through the product of 
. More information and examples on this topic can be found in [180].
What follows behind the approximations (6.184) and (6.185) is that, if properly tuned with 
, the 
filter will be more accurate than the KF in the presence of errors in noise covariances. In practice, however, it can be difficult to find and set the correct 
, especially for LTV systems that imply a new 
at each point in time. If so, then the 
filter may give more errors and even diverge. Thus, the relationship between the filters in terms of robustness to errors in noise statistics should be practically as follows:
where 
holds true for properly maximized 
and 
and properly set 
. Otherwise, the KF may perform better. In turn, the UFIR filter that is 
‐invariant may be a better choice, especially with large errors in noise covariances.
6.8.2 Model Errors
Permanent estimation errors arise from incorrect modeling and persistent model uncertainties, known as mismodeling, when the model is not fully consistent with the process over all time [69]. If no improvement is expected in the model, then the best choice may be to use robust estimators, in which case testing becomes an important subject of confirmation of the estimator accuracy and stability.
UFIR Filter
In UFIR filtering, errors caused by 
are taken into account only on the horizon 
. Accordingly, the bias correction gain 
can be expressed as
and approximated using the following lemma.
The gain 
can finally be approximated with
It is seen that the factor 
affects 
directly and as a reciprocal. However, the same cannot be said about the factor 
raised to the power 
. Indeed, for 
, the gain 
can significantly grow. Otherwise, when 
, it decreases. Thus, we conclude that the effect of errors in the system matrix on the bias correction gain of the UFIR filter is more complex than the effect of errors in the observation matrix.
Kalman Filter
When examining Kalman filter errors associated with mismodeling, 
and 
, it should be kept in mind that such errors are usually observed along with errors in noise covariances. If we ignore errors in the description of noise, the picture will be incomplete due to multiplicative effects. Therefore, we assume that 
and 
, and analyze the errors similarly to the UFIR filter. To do this, we start with the discrete Lyapunov equation 6.180, the solution of which can be approximated using the following lemmas.
Referring to lemma 6.3 and lemma 6.4, the bias correction gain 
of the KF, which is given by (6.177), can be approximated as
and we notice that inequality (6.196) is inapplicable to some models.
Analyzing (6.195), it can be concluded that the mismodeling errors, 
, and errors in noise covariance, 
, cause multiplicative effects in the Kalman gain. In the best case, these effects can compensate for each another, although not completely. In the worst case, which is of practical interest, errors can grow essentially. Again, we see that 
and 
act in opposite directions, as in the case of 
.
Game Theory 
Filter
The game theory 
filter obtained in [185] does not take into account errors in the system and observation matrices. Therefore, it is interesting to know its robustness to such errors in comparison with the UFIR filter and KF. Referring to (6.195) and (6.196), the bias correction gain 
of the 
filter, which is given by (6.178), can be approximated for 
and 
by
These inequalities more clearly demonstrate the key advantage of the 
filter: a properly set 
can dramatically improve the estimation accuracy. Indeed, if the tuning factor 
constrained by 
is chosen such that the relationships in parentheses become zero, then it follows that the effects of 
are reduced to zero, and errors in the observation matrix will remain the main source. Furthermore, by adjusting 
, the effect of 
can be fully compensated.
Unfortunately, the previous remedy is effective only if the correct 
is available at any given time. Otherwise, the 
filter may produce more errors than the KF, and can even diverge. For more information, the reader can read [180], where a complete analysis for robustness is presented with typical examples.
The general conclusion that can be drawn from the analysis of the impact of mismodeling errors is that, under 
and 
, the KF performance can degrade dramatically and that the 
can improve it if we set a proper tuning factor 
at each time instant. Since this is hardly possible for many practical applications, the UFIR filter is still preferred, and the relationship between the filters in terms of robustness to model errors will generally be the same (6.186), where 
holds true for properly maximized errors in 
and 
if 
and 
are properly maximized and 
is set properly. Otherwise, the 
filter may give large errors and go to divergence, while the KF may perform better.
6.8.3 Temporary Uncertainties
Certainly persistent model errors can seriously degrade the estimator performance, causing uncontrollable bias errors. But their effect can be mitigated by improving the model, at least theoretically. What cannot be done efficiently is the elimination of errors caused by temporary uncertainties such as jumps in phase, frequency, and velocity. Such errors are not easy to deal with due to the unpredictability and short duration of the impacts. The problem is complicated by the fact that temporary uncertainties exist in different forms, and it is hard to imagine that they have a universal model.
One way to investigate the associated estimation errors is to assume that the model has no past errors up to the time index 
and that the uncertainty caused by 
occurs at 
. The estimator that is less sensitive to such an impact caused by 
at 
can be said to be the most robust.
UFIR Filter
Suppose the model is affected by 
over all time and that 
is known at 
. At the next step, at 
, the system experiences an unpredictable impact caused by 
and 
. Provided that 
is also known at 
, the bias correction gain 
of the UFIR filter (6.176) can be transformed to
As can be seen, 
and 
act in the same way in (6.199) as in the case of model errors represented by (6.192). However, the effect in (6.199) turns out to be stronger because of the square of 
, in contrast to (6.192). This means that temporary data errors have a stronger effect on the UFIR filter than permanent model errors. The conclusion just made should not be unexpected, given the fact that rapid changes cause transients, the values of which usually exceed steady‐state errors.
Kalman Filter
The Kalman gain can be approximated similarly to the bias correction gain of the UFIR filter. Substituting the prior estimate 
into (6.177), where the matrices are scaled by the error factors 
and 
taking into account that 
, we obtain the approximation of the Kalman gain in the form
Analyzing (6.199) and (6.200), we draw attention to an important feature. Namely, the effect of the first 
is partially compensated by the second 
in the parentheses, which is not observed in the UFIR filter. Because of this, the KF responds to temporary uncertainty with lower values than the UFIR filter. In other words, the UFIR filter is more sensitive to temporary uncertainties. On the other hand, all transients in the UFIR filter are limited by 
points, while the KF filter generates long transients due to IIR. Both these effects, which are always seen in practice, represent one of the fundamental differences between the UFIR filter and the KF [179].
Transients in the UFIR filter and KF: Responding to the stepwise temporary uncertainty, the UFIR filter produces
points transient with a larger overshoot, and the KF produces a longer transient with a shorter overshoot.
Examples of transients in the UFIR filter and KF are given in Fig. 4.2 and Fig. 6.3, and we notice that the difference between the overshoots is usually 
.
Game Theory 
Filter
Providing similar transformations as for (6.200), we approximate the bias correction gain of the 
filter (6.178) as
and notice an important difference with (6.200). In (6.201), there is an additional term containing 
, which can be chosen such that the effect of 
is fully compensated. This is considered to be the main advantage of the game theory 
filter over KF. The drawback is that the tuning factor 
is an unknown analytical function of the set 
, which contains uncertain components. Therefore, if 
cannot be set properly at every point of time, the 
filter performance may significantly degrade, and the filter may even demonstrate the divergence [180]. Analyzing (6.201), it can also be noted that the problem is complicated by the higher sensitivity of 
to 
.
All that follows from the previous analysis of estimation errors caused by temporary uncertainties is that the UFIR filter can still be considered the most robust solution. The game theory 
filter can improve the performance of KF by properly setting the tuning factor 
. However, since the exact 
is not available at every point of time, this filter can become unstable and diverge.
6.9 Implementation of Polynomial UFIR Filters
Hardware implementation of digital filters is usually not required in state space. In contrast, scalar input‐to‐output structures have found much more applications. Researchers often prefer to use UFIR polynomial structures, whenever suboptimal filtering is required. Examples can be found in tracking, control, positioning, timekeeping, etc. If such a UFIR filter matches the model, then the group delay reaches a minimum. Otherwise, the group delay grows at a lower rate than in IIR filters. This makes UFIR structures very useful in engineering practice of suboptimal filtering. Next we will present and analyze the polynomial UFIR structures in the 
domain [29] and discrete Fourier transform (DFT) domain [30].
6.9.1 Filter Structures in 
‐Domain
The transfer function 
of the 
th degree polynomial UFIR filter is specialized by the 
‐transform applied to the FIR function 
given by (6.120) with 
; that is,
where 
, 
is an angular frequency, 
is the sampling time, 
is given by (6.121), and conventionally we will also use 
in 
as the imaginary sign 
. The following properties of 
can be listed in addition to the inherent 
‐periodicity, symmetry of 
, and antisymmetry of 
.
- Transfer function at
. By 
, the transfer function becomes
(6.203)for all
, which means that the UFIR filter is an LP filter. - Impulse response at
. By the inverse 
‐transform, the value of 
at 
is defined as
Because
is always positive‐valued, 
, and for all 
, the counterclockwise circular integration in (6.204a) always produces a positive imaginary value, which gives(6.205)
- Transfer function at
. By 
, the transfer function becomes
(6.206)where
is the Euler polynomial. For low‐degree impulse responses, the Euler polynomials become 
, 
, 
, and 
. - Energy. By Parseval's theorem, the following relations hold
(6.207a)
(6.207b)
and it follows that the energy (or squared norm) of the FIR function in the
domain is equal to the value of 
at 
. - Unbiasedness in the
domain. The following theorem establishes an unbiasedness condition that is fundamental to digital FIR filters in the 
domain.
- Noise power gain. The squared norm of
, which represents the NPG 
of the UFIR filter (Table 6.2), is also the energy of 
. By Parseval's theorem, NPG takes the following equivalent forms
(6.209a)
(6.209b)
(6.209c)
It is worth noting that using to the properties listed, it becomes possible to design and optimize polynomial FIR filter structures in the 
domain with maximum efficiency for real‐time operation.
Transfer Function in the 
‐Domain
Although the inner sum in (6.202b) does not have a closed form, the properties of the FIR filter in the 
‐domain allow one to represent 
as
By assigning the following subtransfer functions
the generalized block diagram of the 
th degree UFIR filter can be shown as in Fig. 6.10. The low‐degree coefficients 
, 
, and 
are listed for this structure in Table 6.3 [29].
Figure 6.10 Generalized block diagram of the 
th degree UFIR filter.
Table 6.3 Coefficients 
, 
, and 
of Low‐Degree UFIR Filters.
 | ||||
|---|---|---|---|---|
| 0 | 1 | 2 | 3 | |
 |  |  |  |  |
 | 0 | − | − | − |
 | 0 | 0 |  |  |
 | 0 | 0 | 0 | − |
 | − |  | − |  |
 | 0 | − |  | − |
 | 0 | 0 | − |  |
 | 0 | 0 | 0 | − |
 | −1 | −2 | −3 | −4 |
 | 0 | 1 | 3 | 6 |
 | 0 | 0 | −1 | −4 |
 | 0 | 0 | 0 | 1 |
The magnitude response functions 
of low‐degree polynomial UFIR filters are shown in Fig. 6.12. A distinctive feature of LP filters of this type is a negative slop of 20 dB per decade in the Bode plot (Fig. 6.12b) in the transient region. Another feature is that increasing the degree of the polynomial filter expands the bandwidth. It can also be seen that the transfer function has multiple intensive side lobes that are related to the shape of the impulse response (Fig. 6.5). The later property is discouraged in the design of standard LP filters. However, it is precisely this shape of 
that guarantees the filter unbiasedness.
Figure 6.12a assures that bias is eliminated at the filter output by shifting and lifting the side lobes of a zero‐degree uniform FIR filter, which provides simple averaging. Accordingly, the 
‐degree filter passes the spectral content close to zero without change, magnifies the components falling into the first lobe, and attenuates the power of the higher‐frequency components with 10 dB per decade (Fig. 6.12b).
The phase response functions 
of low‐degree polynomial UFIR filters are shown in Fig. 6.13a, and the group delay functions obtained by 
are shown in Fig. 6.13b. The phase response function of this filter is linear on average (Fig. 6.13a). However, the phase response changes following variations in the magnitude response, which, in turn, leads to changes in the group delay around a small constant value (Fig. 6.13b). From the standpoint of the design of basic LP filters, the presence of periodic variations in the phase response is definitely a disadvantage. But it is also a key condition that cannot be violated without making the filter output biased.
Figure 6.13 Phase response functions of the low‐degree UFIR filters for 
: (a) phase response 
and (b) group delay 
.
6.9.2 Transfer Function in the DFT Domain
The DFT of the 
th degree UFIR filter impulse response 
is obtained as
where 
. In addition to the inherent 
‐periodicity, the symmetry of 
, and the antisymmetry of 
, the following properties are important for the implementation of UFIR filters in the DFT domain.
- DFT value at
. By 
, the function 
becomes unity, 
, and, by the properties listed in Table 6.2, the DFT function (6.214) is transformed to
(6.215)for all
, which fits with LP filtering. - Impulse response at
. For 
and 
, the inverse DFT (IDFT) applied to 
gives
(6.216)
Since
holds for all 
, the sum of the DFT coefficients is real and positive. It then follows that(6.217)
- DFT value at
. At the point of symmetry 
for even 
, the function 
becomes 
and the following relation holds
(6.218)where
is the Euler polynomial. For low degrees, the Euler polynomials take the following form: 
, 
, 
, and 
. - Noise power gain. As a measure of noise reduction at the UFIR filter output, NPG
for white Gaussian noise is determined by the energy of 
. The NPG has the following equivalent forms in the DFT domain:
(6.219a)
(6.219b)
(6.219c)
- Unbiasedness condition in the DFT domain. The following theorem establishes an unbiasedness condition that is fundamental to UFIR filters in the DFT domain.
Proof. To prove (6.220), use the following fundamental condition for optimal filtering: the order of the optimal and/or unbiased filter must be the same as that of the system. Then represent
by (6.124),
where the coefficient
is given by (6.125). Recall that function (6.124) has the following main properties, given in Table 6.2: the sum of its coefficients is equal to unity, and the moments zero; that is,(6.221)
(6.222)
Use Parseval's theorem and obtain the following relationships,
(6.223a)
(6.223b)
(6.223c)
Now observe that the IDFT applied to
at 
gives 
that finally leads to (6.220) and completes the proof. - Estimation error boundary. The concept of NPG
can be used to specify the bound 
for the estimation error in the three‐sigma sense as
(6.224)where
is the standard deviation of the measurement white noise 
.
Note that the previously discussed properties of polynomial UFIR filters in the DFT domain are preserved for any polynomial degree, but in practice only low‐degree digital UFIR filters, 
, are usually found. The reason for using low‐degree filters is explained earlier: increasing the degree leads to larger random errors at the filter output.
Transfer Function in the DFT Domain
Using the properties listed, the transfer function 
of the 
th degree UFIR filter can be represented in the DFT domain with the sum
in which the inner sum has no closed‐form solution for an arbitrary 
. However, solutions can be found for low‐degree polynomials as shown in [30]. Next we will illustrate such a solution for a first‐degree UFIR filter and postpone to “Problems” the search for solutions associated with higher degrees.
The magnitude response 
, Bode plot, and phase response 
functions of the 
th degree, 
, polynomial UFIR filter are shown in Fig. 6.14. For the 1st degree, the functions are computed by (6.226b) and for the next two low degrees the transfer functions can be found in [30]. In addition to the inherent properties of periodicity with a repetition period of 
points, symmetry of 
, and antisymmetry of 
, the transfer function 
has several other features that are of practical importance.
- The magnitude response
is a monotonically decreasing function of 
, which changes from 
to 
(Fig. 6.14a), and has a transition slope 
on the Bode plot (Fig. 6.14b). So we have further proof that the UFIR filter is essentially an LP filter. - The phase response
is an antisymmetric function existing from 
to 
on 
with a positive slope, except for the transient region where an increase in the filter degree makes it more complex (Fig. 6.14c). This function is close to linear for low‐degree filters, and it is strictly linear for the first‐degree filter.
Figure 6.14 DFT of the low‐degree polynomial UFIR filters: (a) magnitude response 
for 
, (b) Bode plot of 
for 
, and (c) phase response 
for 
.
Finally, we come to an important practical conclusion. Since the transfer function of the polynomial UFIR filter is monotonic and does not contain the periodic variations observed in the 
‐domain (Fig. 6.13), it follows that the filter can be easily implemented with all of the advantages of suboptimal unbiased filtering.
6.10 Summary
By abandoning the requirement for initial values and ignoring zero mean noise, the UFIR state estimator appears to be the most robust among other linear estimators such as OFIR, OUFIR, and ML state estimators. Moreover, its iterative algorithm is more robust than the recursive KF algorithm. The only tuning factor required for the UFIR filter is the optimal horizon, which can be determined much more easily than noise statistics. Furthermore, at given averaging horizons, the UFIR state estimator becomes blind, which is very much appreciated in practice. In general, it is only within a narrow range of error factors caused by various types of uncertainties that optimal and ML estimators are superior to the UFIR estimator. For the most part, this explains the better performance of the UFIR state estimator in many real‐world applications.
Like other FIR filters, the UFIR filter operates on the averaging horizon 
of 
points, from 
to 
. Its discrete convolution‐based batch resembles the LS estimator. But, the latter is not a state estimator. The main performance characteristic of a scalar input‐to‐output UFIR filter is NPG. If the UFIR filter is designed to work in state space, then its noise reduction properties are characterized by GNPG.
The recursive forms used to iteratively compute the batch UFIR estimate are not Kalman recursions. An important feature is that UFIR filter recursions serve any zero mean noise, while Kalman recursions are optimal only for white Gaussian noise. It is worth noting that the error covariance of the UFIR and Kalman filters are represented by the same Riccati equation. The difference lies in the different bias correction gains. The optimal bias correction gain of the KF is called the Kalman gain 
, while the bias correction gain of the UFIR filter is computed in terms of GNPG as 
.
Since the UFIR filter does not involve noise covariances to the algorithm, it minimizes the MSE on the optimal horizon of 
points (Fig. 6.1). The significance of 
is that random errors increase if the horizon length 
is chosen such that 
. Otherwise, bias errors grow if 
. Using (6.49), the optimal horizon 
can be estimated through the measurement residual without using ground truth. In response to stepwise temporary uncertainties, the UFIR filter generates finite time transients at 
points with larger overshoots, while the KF generates longer transients with lower overshoots. This property represents the main difference between the transients in both filters.
It turns out that, since zero mean noise is ignored by UFIR state estimators, the FFFM, FFBM, BFFM, and BFBM 
‐lag UFIR smoothing algorithms are equivalent. It should also be kept in mind that all unbiased, optimal, and optimal unbiased state estimators are essentially LP filters. Moreover, the zero‐degree (simple average) UFIR filter is the best in the sense of the minimum produces noise. This is because an increase in the filter degree or the number of the states leads to an increase in random errors at the estimator output. It also follows that the set of degree polynomial impulse responses of the UFIR filter establishes a class of discrete orthogonal polynomials that are suitable for signal analysis and restoration due to the built‐in unbiasedness.
Although the UFIR filter ignores zero mean noise, it is not suitable for colored noise that is biased on short horizons. If the colored noise is well‐approximated by the Gauss‐Markov process, then the general UFIR filter becomes universal for time‐correlated and uncorrelated driving white noise sources. Like GKF, a general UFIR filter can be designed to work with CMN and CPN. It can also be applied to state‐space models with smooth nonlinearities using the first‐ or second‐order Taylor series expansions. However, it turns out that the second‐order EFIR‐2 filter has no practical advantage over the first‐order EFIR‐1 filter.
The implementation of polynomial UFIR filters can be most efficiently obtained in the DFT domain due to the following critical property: the DFT transfer function is smooth and does not exhibit periodic variations inherent to the 
‐domain.
6.11 Problems
- Explain the difference between polynomial fitting and filtering (smoothing) of polynomial models.
- NPG for scalar signals is given by (6.15). Give the interpretation of GNPG defined by (6.16a) for state vectors. Why does the NPG decrease with an increase in the averaging horizon?
- Solved problem: Recursive forms for UFIR filter. Consider the batch UFIR filtering estimate
with the fundamental gain given by (6.10b). Another way to obtain recursive forms for this filter (Algorithm 11) is the following [179]. Represent the inverse of the GNPG 
as
Since
holds for 
, write 
as
substitute to the previous relation, and arrive at the recursion (6.21) for
.Similarly, represent the product
as
combine with
, obtain 
, substitute 
, and come up with
From (6.21) find
, substitute into the previous relation, and arrive at the recursion (6.29) for the UFIR estimate. - The error covariance of the UFIR filter is given by (6.37) as
Compare this relation to the error covariance of the KF and explain the difference between the bias correction gain
of the UFIR filter and the Kalman gain 
. Why can the Kalman gain not be less than the optimal value? Why does the bias correction gain 
of the UFIR decrease with increasing the horizon length? - To minimize MSE, the averaging horizon for the UFIR filter should be optimal
(Fig. 6.1) by solving the following minimization problem
Why does the OFIR filter not need an optimal horizon? Why does the error in the OFIR filter decrease with increasing the horizon length, while the UFIR filter minimizes the MSE by only
? Give simple and intuitive explanations. - Measurement data transmitted over a wireless communication channel with latency are represented at the receiver with the following state‐space equations
where
, 
, is a deterministic weight, which can be either unity or zero. The time‐stamped data indicate a value of 
for which the weight is unity and the other weights are zero. Transform the model to another without delay and develop a UFIR filter. - A useful property of the measurement residual
of the UFIR filter is that the derivative of its trace with respect to the horizon length reaches a minimum when 
approaches 
(Fig. 6.2). This makes it possible to determine 
by (6.49) or by solving the minimization problem
Give an explanation to this fact and discuss the problems that may arise in the practical implementations of this method.
- An open and challenging problem is to find an analytical relationship between
and noise covariances 
and 
similarly to (6.51) [153]. Find a solution to this problem for a periodical process observed as
where
and 
, 
, and 
are state variables. Write the state equation with additive white Gaussian noise. - The GNPG of the backward UFIR filter is defined by (6.62) as
Represent GNPG as
and specify 
. - The error covariance
of the backward UFIR filter is given by
Reasoning similarly as for the forward UFIR filter, provide the derivation of (6.72). Analyze this relationship and find out what will happen to
if the noise covariance 
is set equal to zero. - The error covariance of the batch
‐lag FFFM UFIR smoother is given by (6.85). Recursive computation of (6.85) is provided by (6.94), where the matrix 
still has a batch form (6.92) given by
Find a recursive form for (6.92) or, otherwise, develop a different approach for recursive computation of (6.85).
- Using deductive reasoning, the recursive form for
was found as (6.90) and for 
as (6.93), respectively,
Show other possible ways to obtain recursions for (6.90) and (6.93).
- Referring to the optimal state estimation problem and considering the transform of the UFIR filter gain
, find an explanation for the fact that the optimal, optimal unbiased, ML, and unbiased FIR filters are LP filters. - It follows from (6.129) that polynomial FIR functions
, 
, given by (6.124), establish a new class discrete orthogonal polynomials related by the recurrence relation (6.128). Reasoning in a similar way, examine for orthogonality the class of more general 
‐shift FIR functions 
, 
, 
, given by (6.120), and find the recurrence relation, if any. - Find an explanation for the fact that increasing the number of states makes the state estimator less effective in terms of noise reduction. In this sense, explain why simple averaging is best in terms of the minimum produced noise.
- It follows from Fig. 6.5 that prediction errors grow with increasing the
‐step. Which method is more efficient in predicting future states: stochastic prediction requiring future noise values? Deterministic prediction ignoring future noise? Or linear deterministic prediction? - General UFIR and Kalman filters serve for correlated and de‐correlated Gauss‐Markov process noise
and measurement noise 
, where 
and 
. Can we modify the UFIR filter and KF for non‐Gaussian colored noise? If not, explain why. - The periodic system is represented in state space by the equations
where
is a constant angular measure, and 
and 
are zero mean noise sources. Using the derivation of the UFIR filter for polynomial models, derive the scalar FIR functions for the first state 
and the second state 
to satisfy the unbiasedness condition 
. - The bias correction gain of the KF is approximated by (6.184). Referring to the derivation of (6.184), give a proof of the corresponding approximation (6.185) for the
filter. - Lemma 6.3 states that the prior error covariance
of the KF is approximated by
where
and 
are given constants. Using the proof of lemma 6.1, give a corresponding proof of the lemma 6.3. - Using the derivation procedure given in [180], provide a proof of approximation (6.194) of the prior error covariance
of the KF stated by lemma 6.4. - A linear digital system operates normally without uncertainties from the past to the time index
inclusive. Then it experiences uncertainties in the coefficients 
and 
at the time index 
. Check that the approximate gains of the UFIR filter (6.199), KF (6.200), and 
filter (6.201) are correct. - For the
th degree polynomial scalar FIR function 
existing on 
and the transfer function 
, the unbiasedness constraint in the 
‐domain is established by theorem 6.2 as
Obtain the corresponding constraint for the general UFIR filter matrix gain case
. - Given a digital UFIR filter with the
th degree transfer function 
corresponding to a polynomial FIR function 
existing on 
, theorem 6.3 establishes the unique unbiasedness constrain for this filter in the DFT domain as
Obtain the corresponding constraint for the UFIR filter matrix gain
. - The transfer function
of the 
th degree polynomial UFIR filter is given in the 
‐domain by (6.210), which for arbitrary 
has no closed‐form solution. Using the properties of the UFIR filters in the 
‐domain and Table 6.3, find the closed forms of this function for the second and third degrees and represent the filter with block diagrams similar to that shown in Fig. 6.11. - The transfer function
of the 
th degree polynomial UFIR filter is given by (6.225) and has no closed‐form solution. Find solutions in closed-form for special cases of the second and third degrees.