Complex Information Filter and Complex Kalman Filter Comparison:
Selection of the Faster Filter
ATHANASIOS POLYZOS1, CHRISTOS TSINOS1, MARIA ADAM2, NICHOLAS ASSIMAKIS1
1Department of Digital Industry Technologies,
National and Kapodistrian University of Athens,
34400 Psachna Evias,
GREECE
2Department of Computer Science and Biomedical Informatics,
University of Thessaly,
2-4 Papasiopoulou Str., 35131, Lamia,
GREECE
Abstract: - Complex Kalman filters are used in complex signal processing. A comparison between the complex
Kalman filter and the complex Information filter is presented in the general case of discrete-time widely linear
models. The complex Kalman filter and the complex Information filter compute iteratively the same
estimations. The computational burdens of these complex filters are determined and a method is derived to
decide which filter is the faster one, taking into account only the model dimensions.
Key-Words: - Linear estimation, Widely linear model, Complex Kalman filter, Complex Information filter,
Complex signals, Computational burden.
Received: April 19, 2024. Revised: August 24, 2024. Accepted: September 15, 2024. Published: November 13, 2024.
1 Introduction
Kalman filter [1], [2] and Information filter [2], [3]
are known estimation algorithms that have been
successfully used in many linear estimation
problems: applications of Kalman filter are referred
in [2], [4], [5], [6] and applications of Information
filter are referred in [3], [7], [8], [9], [10], [11].
Applications of complex Kalman filter include
tracking, oceanography, array processing,
communications, biomedicine [6], [12], tracking for
Global Navigation Satellite System (GNSS) meta-
signals [13], power system frequency [14], [15],
unbalanced grids [16], proper and improper signals
applications [17], two-dimensional local navigation
system using complex Kalman and particle filters
[18].
The basic statistical properties of a complex
signal are (a) the covariance matrix that concerns
the total power of the complex signal and (b) the
pseudo-covariance matrix (complementary
covariance) that concerns the correlations between
the real and imaginary parts of the complex signal,
[12]. In linear estimation, the complex Kalman filter
is derived assuming (a) the traditional state space
model which takes into account the covariance
matrix only; the derived conventional complex
Kalman filter (CCKF) takes into account the
covariance matrix only and hence it is applicable to
proper (circular) signals (b) the augmented model or
widely linear model, which takes into account the
covariance as well as the pseudo-covariance
matrices; the derived augmented complex Kalman
filter (ACKF) is applicable to improper (non-
circular) complex signals that are correlated with
their complex conjugates. It is worth noting that the
use of the pseudo-covariance matrix in ACKF can
improve the performance of CCKF, [19].
Motivated by minimizing the computational
time, the paper derives the augmented complex
Information filter (ACIF) from the equations of the
augmented complex Kalman filter. A comparison
between the complex Kalman filter and the complex
Information filter is presented in this paper. The
origin of this idea is a comparison of the
corresponding filters in linear estimation, where real
signals are involved, [3]. The key contributions of
this paper are as follows: a) the calculation burdens
of the augmented complex Kalman and complex
Information filters are derived, b) a method is
derived to determine the faster complex filter.
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2 Augmented Model
Let a complex variable . The complex conjugate is
denoted as . Let a complex matrix . The
transpose matrix is denoted as and the conjugate
transpose matrix as . The augmented matrix is
and
.
The augmented matrix inversion is
with
The following widely linear model [19] is used
in complex Kalman filters:
Here, is the state complex vector,
is the measurement complex vector,
and are the transition complex
matrices, and are is the output
complex matrices, is the state noise
complex vector and is the measurement
noise complex vector at (discrete) time.
Furthermore, the state noise is a Gaussian
process with zero mean and known
dimensional covariance and pseudo-
covariance . The measurement noise is a
Gaussian process with zero mean and known
dimensional known covariance and pseudo-
covariance .
The initial state is Gaussian with known
mean , covariance and pseudo-covariance .
Consider the augmented state vector
and the augmented
measurement vector
.
Using the complex augmented state and
measurement vectors, the augmented model (or
widely linear model) becomes:
(1)
(2)
Here, the augmented noise vectors are
and
and the augmented
matrices are
and
.
Furthermore assume that:
- the augmented state noise process is non-
circular Gaussian with zero mean and covariance
matrix
,
- the augmented measurement noise process
is non-circular Gaussian with zero mean and
covariance matrix
,
- the augmented initial state is non-circular
Gaussian with mean
and covariance
matrix
.
It is worth to note that a) and are
Hermitian matrices (M is a Hermitian matrix when
), as covariance matrices and b) and
are symmetric matrices (N is a symmetric
matrix when ), as pseudo-covariance
matrices, [9].
3 Augmented Complex Kalman Filter
Let denote a) the state prediction as with
covariance and pseudo-covariance
; then, the augmented state prediction is
with covariance matrix
, b) the state estimation
as with covariance and pseudo-
covariance
; then the augmented state
estimation is with covariance matrix
. Let denote the
augmented Kalman filter gain as
.
The augmented (widely linear) complex Kalman
filter (ACKF) computes the augmented state
prediction and estimation and the corresponding
covariances, using the augmented Kalman filter
gain, which is derived by minimizing the cost
function based on the MSE criterion, and is
summarized as follows, [6]:
Augmented Complex Kalman Filter (ACKF)
initial conditions
iterations
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Here, is the identity matrix.
In the special case of time-invariant models,
where the augmented model parameters
are constant matrices,
the Time Invariant Augmented Complex Kalman
Filter is derived.
4 Augmented Complex Information
Filter
The idea [2], [3] is the use of the Information
matrix, which is the inverse of the covariance matrix
and the information state vector which is connected
to the estimation vector through the information
matrix. Let define:
Then we can derive the augmented complex
Information filter equations strictly from the
augmented complex Kalman filter equations and
using the matrix inversion lemma. In fact:
- concerning the Kalman filter gain, we get
- concerning the estimation, we get
and
- concerning the prediction, we get
Thus, the augmented complex Information filter
and is summarized as follows:
Augmented Complex Information Filter (ACIF)
initial conditions
iterations
In the special case of time-invariant models,
where the augmented model parameters
are constant matrices,
the Time Invariant Augmented Complex
Information Filter is derived. Note that in this case
the matrices are
calculated off-line.
5 Computational Requirements
It is shown that ACKF and ACIF are algebraically
equivalent filters. Then it becomes clear that the two
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filters calculate the same state estimations and state
predictions. Moreover, it is obvious that the two
filters are iterative algorithms. As a result, the
comparison of the filters’ computational time, is
equivalent to the comparison of their per iteration
calculation burden (CB); the calculation burden of
the off-line calculations is not taken into account.
Table 1. ACKF per iteration calculation burden
Augmented Complex Kalman Filter (ACKF)
Matrix Operation
Calculation Burden
Note that in the time-invariant case, the
calculation burden remains the same as in the time-
varying case.
Table 2. ACIF per iteration calculation burden
Augmented Complex Information Kalman Filter (ACIF)
Matrix Operation
Calculation Burden
It is evident that the calculation burdens of the
two complex filters involve complex matrix
operations, the calculation burdens of which are
given in the Appendix.
The per iteration calculation burdens of the
augmented complex Kalman filter and the complex
Information filter are analytically calculated in
Table 1 and Table 2, respectively.
Note that in the time-invariant case, the matrices
are calculated off-line.
The per iteration calculation burdens of the
augmented complex Kalman filter and the complex
Information filter are and summarized in Table 3.
Table 3. ACKF and ACIF calculation burdens
Model
Filter
Per Iteration
Calculation Burden
time
varying
Kalman
Information
time
invariant
Kalman
Information
6 Selection of the Faster Filter
In this section, a method is derived to select the
faster complex filter. From Table 3, it is obvious
that the calculation burdens of the two complex
filters depend on the state vector dimension n and
the measurement vector dimension m. Hence, the
selection of the faster complex filter depends on the
relation between the known dimensions and .
In the general time-varying models case, we get:
Figure 1 presents the areas where the complex
Information filter or the complex Kalman filter is
faster; subject to the model dimensions. The
obtained Rule of Thumb for time-varying models
has as follows:
The ACIF is faster than ACKF, when
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Fig. 1: The faster filter – time-varying model
In the special time-invariant models case, we get:
Figure 2 presents the areas where the complex
Information filter or the complex Kalman filter is
faster; subject to the model dimensions. The
obtained Rule of Thumb for time-invariant models
has as follows:
The ACIF is faster than ACKF, when
Fig. 2: The faster filter – time-invariant model
It becomes obvious that the knowledge of the
model dimensions is sufficient in order to determine
which filter is faster.
7 Conclusions
The augmented complex Kalman filter and the
complex Information filter are equivalent with
respect to a) the derivation of the state estimations
and predictions and the corresponding error
covariances, b) their stability, since the stability of
the Kalman filter is classically ensured by the
controllability and the observability of linear time-
varying models.
In this paper, a comparison study between the
(discrete time) augmented complex Kalman filter
and complex Information filter was obtained. The
computational requirements of both these complex
filters were derived. It was established that the
computational burdens of the filters are functions of
the state dimension and the measurement
dimension . A method was derived and described
to select, before the implementation of the filters,
the faster complex filter. The basic result is:
- in the general case of time-varying models,
the complex Information filter is faster than the
complex Kalman filter when ,
- in the special case of time-invariant models,
the complex Information filter is faster than
the complex Kalman filter when
The impact of this result on Kalman filtering
combined with AI techniques to determine the
fastest filter, can be to reduce processing time in
complex signal processing applications.
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APPENDIX
Calculation Burdens of Complex Matrix
Operations
In the following, are real numbers and
are complex numbers.
The calculation burdens (CB) of real and
complex scalar operations are summarized in Table
4 and Table 5, respectively.
Table 4. Real scalar operations
code
real
scalar
operation
real
scalar
adds
real
scalar
mults
real
scalar
divs
CB
R1
1
0
0
1
R2
0
1
0
1
R3
0
0
1
1
Table 5. Complex scalar operations
code
complex
scalar
operation
real
scalar
adds
real
scalar
mults
real
scalar
divs
CB
C1
2
0
0
2
C2
2
4
0
6
C3
1
0
0
1
C4
1
0
0
1
C5
0
2
0
2
C6
1
2
0
3
C7
1
2
0
3
C8
0
0
2
2
C9
3
6
2
11
In the following, are complex vectors;
C, C1, C2 are general complex matrices; H, H1, H2
are complex Hermitian matrices; S, S1, S2 are
complex symmetric matrices; I is the identity matrix
of dimension n; are augmented complex
vectors of the form ; are augmented
complex matrices of the form
;
are special augmented complex matrix
of the form
, is the identity matrix of
dimension 2n.
The calculation burdens (CB) of complex
matrices addition and complex augmented matrices
addition operations are summarized in Table 6 and
Table 7, respectively.
Table 6. Complex matrices addition
code
Complex
Matrices
Addition
oper
CB
total CB
A1
C1
A2
C1
A3
R1
A4
C1
A5
C1
A6
C1
A7
C1
C3
A8
C1
C3
A9
R1
C1
A10
R1
C1
A11
C1
A12
C1
A13
C1
Table 7. Complex augmented matrices addition
code
Complex
Augmented
Matrices
Addition
oper
A14
A1
A15
A2
A16
A3
A17
A9
2
A11
A18
A10
A12
The calculation burdens (CB) of complex
matrices multiplication and complex augmented
matrices multiplication operations are summarized
in Table 8 and Table 9, respectively.
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Table 8. Complex matrices multiplication
code
Complex
Matrices
Multiplication
oper
CB
total CB
M1
C1
C2
M2
C1
C2
M3
C1
C2
M4
C1
C2
M5
C1
n
C2
C5
M6
C1
C2
M7
C1
C2
M8
C1
C2
M9
C1
C2
M10
C1
C2
M11
C1
C2
M12
C1
C2
C5
M13
C1
C2
C5
M14
C1
C2
C5
M15
C1
C2
C5
M16
R1
C1
C2
C6
M17
R1
C1
C2
C6
M18
R1
C1
C2
C6
M19
C1
C2
C5
M20
C1
C2
M21
C1
C2
C5
M22
C1
C2
C5
M23
C1
C2
M24
C1
C2
M25
C1
C2
M26
C1
C2
Table 9. Complex augmented matrices
multiplication
code
Complex
Augmented
Matrices
Multiplication
oper
(times)
M27
M1(2)
A1(1)
M28
M3(2)
A1(1)
M29
M4(2)
A2(1)
M30
M5(1)
M2(1)
A1(1)
M31
M6(4)
A6(2)
M32
M8(4)
A6(2)
M33
M12(2)
M23(2)
A5(2)
M34
M13(2)
M24(2)
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A4(2)
M35
M15(2)
M25(2)
A6(2)
M36
M6(4)
A7(1)
A13(1)
M37
M11(4)
A10(1)
A12(1)
M38
M8(4)
A9(1)
A11(1)
M39
M14(2)
M26(2)
A4(2)
M40
M15(2)
M25(2)
A7(1)
A13(1)
The following recursive algorithm is used for the
complex Hermitian matrix inversion:
Dimensions:
The calculation burden (CB) of the recursive
algorithm for the complex Hermitian matrix
inversion is summarized in Table 10.
Table 10. Complex Hermitian matrix inversion
recursive algorithm
Matrix operation
oper
times
total CB
C1
8
C2
Real
C3
C6
real
R1
Real
R3
C5
Hermitian
C2
C6
Hermitian
C5
Hermitian
R1
Hermitian
C1
Here
Then
with
So
Thus
Hence, the calculation burden (CB) of the
recursive algorithm for the complex Hermitian
matrix inversion is:
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The calculation burdens (CB) of complex
Hermitian matrices inversion operations are
summarized in Table 11.
Table 11. Complex Hermitian matrices inversion
code
Complex
Hermitian
Matrix
Inversion
method
CB
I1
recursive
algorithm
I2
recursive
algorithm
I3
computation
for I1 with 2n
I4
computation
for I2 with 2m
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
Athanasios Polyzos: Methodology, Investigation
Writing - original draft preparation, Writing -
review and editing and Visualization, Validation.
Christos Tsinos: Formal analysis, Investigation.
Maria Adam: Formal analysis, Investigation.
Nicholas Assimakis: Conceptualization, Software.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare
that are relevant to the content of this article.
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