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Charalambous, Charalambos D.; Kourtellaris, Christos K.; Charalambous, Themistoklis
A General Coding Scheme for Signaling Gaussian Processes over Gaussian Decision Models
Published in:
Proceedings of the IEEE 19th International Workshop on Signal Processing Advances in Wireless
Communications, SPAWC 2018
DOI:
10.1109/SPAWC.2018.8445982
Published: 24/08/2018
Document Version
Peer reviewed version
Please cite the original version:
Charalambous, C. D., Kourtellaris, C. K., & Charalambous, T. (2018). A General Coding Scheme for Signaling
Gaussian Processes over Gaussian Decision Models. In Proceedings of the IEEE 19th International Workshop
on Signal Processing Advances in Wireless Communications, SPAWC 2018 (Vol. 2018-June). [8445982] (IEEE
International Workshop on Signal Processing Advances in Wireless Communications). IEEE.
https://doi.org/10.1109/SPAWC.2018.8445982
A General Coding Scheme for Signaling Gaussian Processes over
Gaussian Decision Models
Charalambos D. Charalambous, Christos K. Kourtellaris and Themistoklis Charalambous
Abstract— In this paper, we transform the n−finite transmis-
sion feedback information (FTFI) capacity of unstable Gaussian
decision models with memory on past outputs, subject to an
average cost constraint of quadratic form derived in [1], into
controllers-encoders-decoders that control the output process,
encode a Gaussian process, reconstruct the Gaussian process via
a mean-square error (MSE) decoder, and achieve the n−FTFI
capacity. For a Gaussian RV message X ∼ N (0, σ
2
X
) it is
shown that the MSE decays according to E|X −
b
X
n|n
|
2
=
exp{−2C
0,n
(κ)}σ
2
X
, κ ∈ (κ
min
, ∞), where C
0,n
(κ) is the
n−FTFI capacity, and κ
min
is the threshold on the power to
ensure convergence.
Index Terms— Coding, n−finite transmission feedback infor-
mation, Gaussian decision models, capacity.
I. INTRODUCTION
It has been recently shown [2] that randomized strategies
in decision systems are operational, in the sense that not only
they stabilize the system but they also encode information,
which can be decoded at the output of the control system
with arbitrary small probability of decoding error. In other
words, the control system is used to communicate informa-
tion, with an operational meaning as defined by Shannon’s
capacity of communication channels.
Our main objective of this paper is to synthesize controller-
encoder strategies for Gaussian recursive models (G-RMs),
with input process {A
t
: t = 0, 1, . . . , n} that simultaneously
control the output process {Y
t
: t = 0, 1, . . . , n}, encode an
information process {X
t
: t = 0, 1, . . . , n}, and synthesize
a decoder at the output {Y
t
: t = 0, 1, . . . , n}, such that the
process {X
t
: t = 0, 1, . . . , n} is transmitted reliably to the
decoder.
If the G-RM is a control system model, then the system,
as depicted in Figure 1, gives rise to several application.
One application is that of tracking the dynamics of {X
t
:
t = 0, 1, . . . , n} at the output of the decoder. If {X
t
:
t = 0, 1, . . . , n} is generated by a discrete deterministic
recursion, then we show that it is possible for the decoder
to track {X
t
: t = 0, 1, . . . , n} with arbitrary small MSE.
This is contrary to standard tracking design systems in
which the noise of the G-RM, i.e., {V
t
: t = 0, 1, . . . , n}
imposes limitations on the minimum tracking error. Another
application is that of signaling digital messages available
to the controller, such as, values associated with actuating
devices, for failure detection and monitoring applications.
C. D. Charalambous and C. K. Kourtellaris are with the Department
of Electrical Engineering, University of Cyprus, Nicosia, Cyprus. E-mails:
{chadcha,kourtellaris.christos}@ucy.ac.cy.
T. Charalambous is with the Department of Electrical Engineering and
Automation, School of Electrical Engineering, Aalto University, Finland.
E-mail: themistoklis.charalambous@aalto.fi
II. PROBLEM STATEMENT AND RELATED WORK
A. Notation
h·, ·i denotes inner product of elements of linear spaces,
S
q×q
+
denotes the set of symmetric positive semi-definite q×q
matrices and S
q×q
++
the subset of positive definite matrices,
with real entries.
B. Main Problem
We consider a multiple-input multiple output (MIMO)
time-verying unstable G-RM given by
Y
i
= C
i−1
Y
i−1
+ D
i
A
i
+ V
i
, Y
−1
= s, i = 0, . . . , n, (1)
subject to an average cost of quadratic form described by
1
n + 1
E
s
n
n
X
i=0
γ
i
(A
i
, Y
i−1
)
o
≤ κ, (2)
A
n
4
= {A
0
, A
1
, . . . , A
n
} is the input process,
Y
n
4
= {Y
0
, A
1
, . . . , Y
n
} is the output process,
V
n
4
= {V
0
, . . . , V
n
} is an independent zero mean Gaussian
noise process, denoted by V
i
∼ N (0, K
V
i
), K
V
i
0, i =
0, . . . , n,
γ
i
(a
i
, y
i−1
) , ha
i
, R
i
a
i
i + hy
i−1
, Q
i−1
y
i−1
i,
(C
i−1
, D
i
) ∈ R
p×p
× R
p×q
, (Q
i−1
, R
i
) ∈ S
p×p
+
× S
q×q
++
.
The G-RM may correspond to a control system or a com-
munication channel, as shown in Figure 1. The distribution of
the G-RM is P
Y
i
|A
i
,Y
i−1
,S
= Q
i
(dy
i
|y
i−1
, a
i
), i = 1, . . . , n,
and for i = 0, the distribution is Q
0
(dy
0
|s, a
0
).
By [1], the characterization of the n−finite transmission
feedback information (FTFI) capacity is
C
0,n
(κ) = J
A
n
→Y
n
|s
(π
∗
, κ)
4
= sup
◦
P
[0,n]
(κ)
E
π
s
n
n
X
i=0
log
Q
i
(·|Y
i−1
, A
i
)
P
π
(·|Y
i−1
)
(Y
i
)
o
(3)
P
π
(dy
i
|y
i−1
) =
Z
A
i
Q
i
(dy
i
|y
i−1
, a
i
) ⊗ π
i
(da
i
|y
i−1
), (4)
for i = 1, . . . , n, and
P
π
(dy
0
|s) =
Z
A
0
Q
0
(dy
0
|s, a
0
) ⊗ π
0
(da
0
|s),
for i = 0. The set of randomized strategies
◦
P
[0,n]
(κ) are
conditionally Gaussian and Markov defined by
◦
P
[0,n]
(κ)
4
=
n
π
i
(da
i
|y
i−1
), i = 0, . . . , n :
1
n + 1
E
π
s
n
X
i=0
γ
i
(A
i
, Y
i−1
)
≤ κ
o
⊂ P
[0,n]
(κ). (5)
Source or
Tracking Signal
Encoder or
Control Object
Deterministic Part
(Stabilization)
Random Part
(Innovation)
Channel or
Controlled Object
Decoder or
Estimator
I(A
n
→Y
n
)
Unit Delay
P
X
i
|X
i−1
P
A
i
|A
i−1
,Y
i−1
P
Y
i
|Y
i−1
,A
i
P
b
X
i
|
b
X
i−1
,Y
i
Control
Process
Controlled
Process
X
i
A
i
Y
i
b
X
i
Fig. 1. Depicts Shannon’s communication block diagram and its analogy to stochastic control systems.
The corresponding joint process {A
0
, Y
0
, . . . , A
n
, Y
n
} and
output process {Y
0
, . . . , Y
n
}, for fixed S = s, are Gaussian.
Further, under the standard detectability and stabilizability
conditions [1, Theorem 4.1] the feedback capacity is
C(κ)
4
= lim
n−→∞
1
n + 1
J
A
n
→Y
n
|s
(π
∗
, κ), κ ∈ [κ
min
, ∞).. (6)
The main objective is to transform the distribution
π
i
(da
i
|y
i−1
), i = 0, . . . , n that achieves the n−FTFI capac-
ity and capacity C
0,n
(κ), into a controller-encoder and to
construct a decoder, such that
(1) the controller-encoder operates at C
o,n
(κ),
(2) the decoder is MSE optimal,
for Gaussian, Markov process, described by the recursion
X
i+1
= F
i
X
i
+ G
i
W
i
, X
0
= x ∈ X
i
4
= R
q
(7)
where W
i
∼ N
0, K
W
i
, i = 0, . . . , n − 1 are W
i
=
R
k
−valued zero mean Gaussian, independent of V
i
, i =
0, 1, . . . , n, and X
0
∼ N
0, K
X
0
.
It should be mentioned that controllers-encoders have con-
tradictory goals. The controller aims at stabilization, while
the encoder aims at communicating new information. A low-
cost control strategy would want the state process to be kept
near the origin, with as little randomness as feasible injected
by the coding part, while a communication strategy requires
informative deviations.
C. Related Work
The Shannon coding capacity For the memoryless addi-
tive Gaussian noise (AGN) channel
Y
i
= A
i
+ V
i
, i = 0, . . . , n,
1
n + 1
E
n
n
X
i=0
|A
i
|
2
o
≤ κ (8)
with or without feedback, the capacity is given by [3]
C
Sh
(κ)
4
=
1
2
log
1 +
κ
K
V
. (9)
The input process that achieves it is independent and
identically distributed (IID) Gaussian A
∗
i
∼ P
∗
A
(da) ∼
N(0, κ), i = 0, 1, . . ..
Elias Coding Scheme of a Gaussian Message. Elias [4]
introduced a coding scheme to communicate a Gaussian RV
X ∼ N(0, σ
2
X
) reliably over the memoryless AGN channel
(8), that achieves C
Sh
(κ), given by
A
i
=
v
u
u
t
κ
E
X−E
n
X
Y
i−1
o
2
X−E
n
X
Y
i−1
o
. (10)
b
X
i−1
4
= E
X
Y
i−1
is linear in Y
i−1
, i = 0, . . . , n, and it
is computed recursively, using the Kalman-filter.
MSE Decoder of Gaussian Messages. When the Elias
coding scheme is applied with a MSE decoder, the error is
Σ
n
4
= E|X −
b
X
n
|
2
=
σ
2
X
1 +
κ
K
V
n+1
, n = 0, 1, . . . . (11)
Maximum Likelihood (ML) Decoder of Digital Mes-
sages. Schalkwijk and Kailath [5] showed that, when the
Elias coding scheme is applied to a set of equiprobable
messages
0, 1, . . . , M
n
4
= exp{(n + 1)R} : n = 0, 1, . . .
,
then the probability of ML decoding error at time n, decays
doubly exponentially.
Butman Conjecture. For an AGN channel with stable and
stationary noise V
n
4
= {V
0
, . . . , V
n
} (with limited memory),
Butman [6] “conjectured” that the Elias coding scheme of
transmitting the error achieves capacity. Cover and Pombra
[7] derived the characterization of feedback capacity for
the AGN channel (8), when the noise V
n
is nonstationary,
nonergodic, with distribution P
V
n
. Kim [8] revisited the
limited memory, stationary ergodic version of the Cover
and Pombra [7] AGN channel, and applied frequency do-
main methods to conclude that Butman’s conjecture is true.
Variations of the Elias and Schalkwijk-Kailath schemes for
network communication over memoryless AGN channels are
extensive and given in [9]–[12].
III. THE n−FTFI CAPACITY AND
CONTROLLER-ENCODER-DECODER
By [1] the n−FTFI capacity of the G-RM is achieved by
the input process
A
i
=e
i
(Y
i−1
, Z
i
)=U
i
+Z
i
= Γ
i
Y
i−1
+Z
i
(12)
(i) U
i
4
= Γ
i
Y
i−1
, i = 0, . . . , n is the control process,
(ii) Z
i
independent of (A
i−1
, Y
i−1
), i = 0, . . . , n,
(iii) Z
i
independent of V
i
, for i = 0, . . . , n,
(iv) {Z
i
∼ N(0, K
Z
i
) : i = 0, . . . , n} an independent
Gaussian process.
The corresponding output process is
Y
i
=
C
i−1
+ D
i
Γ
i
Y
i−1
+ D
i
Z
i
+ V
i
, Y
−1
= s. (13)
Furthermore, the optimal control and innovations parts of the
strategy are found in [1, Section IV]. We include them below
for completeness.
(a) The optimal control part of the strategy {U
∗
i
: i =
0, . . . , n}, is given by
U
∗
i
= Γ
∗
i
Y
∗
i−1
, i = 0, . . . , n, (14)
Γ
∗
i
= −
D
T
i
P (i+1)D
i
+R
i
−1
D
T
i
P (i+1)C
i−1
, (15)
where Γ
∗
n
= 0,
P (i) : i = 0, . . . , n
is a solution of the
Riccati difference matrix equation
P (i) = C
T
i−1
P (i + 1)C
i−1
+ Q
i−1
− C
T
i−1
P (i + 1)D
i
D
T
i
P (i + 1)D
i
+ R
i
−1
C
T
i−1
P (i + 1)D
i
T
, P (n) = Q
n−1
(16)
(b) The optimal innovations part of the strategy
K
∗
Z
i
0 :
i = 0, . . . , n
is the solution of the following problem.
J
A
n
→Y
n
|s
(π
∗
, κ) = C
0,n
(κ
∗
0
, . . . , κ
∗
n
)≡
n
X
i=0
C
i
(κ
∗
i
) (17)
4
= sup
K
Z
i
0,i=0,...,n:
P
n
i=0
κ
i
(K
Z
i
)=κ(n+1)
n
X
i=0
C
i
(κ
i
) (18)
where
C
i
(κ
i
)
4
=
1
2
log
|D
i
K
Z
i
D
T
i
+ K
V
i
|
|K
V
i
|
, i = 0, . . . , n (19)
κ
i
≡ κ
i
(K
Z
i
) (20)
4
=
tr
R
n
K
Z
n
, i = n
tr
P (i + 1)
D
i
K
Z
i
D
T
i
+ K
V
i
+R
i
K
Z
i
, i = 1, . . . , n − 1
tr
P (1)
D
0
K
Z
0
D
T
0
+ K
V
0
+ R
0
K
Z
0
+hs, P (0)si, i = 0.
Example 3.1: Let p = q = 1. Then the solution of the
Riccati difference matrix equation (16) does not depend on
the covariance K
Z
i
, i = 0, . . . , n, and hence this simplifies
the computation of the optimal K
∗
Z
i
in the optimization
problem (18). By the Kuhn-Tucker conditions we obtain
K
∗
Z
n
=
n
1
2λR
n
−
K
V
n
D
2
n
o
+
,
x
+
4
= max
0, x
, (21)
K
∗
Z
i
=
n
1
2λ
P (i + 1)D
2
i
+ R
i
−
K
V
i
D
2
i
o
+
(22)
for i = n − 1, n − 2, . . . , 0, where λ = λ
n
(κ) ≥ 0 is chosen
to satisfy the average constraint with equality given by
n−1
X
i=0
nn
1
2λ
−
P (i + 1)D
2
i
+ R
i
K
V
i
D
2
i
o
+
+P (i+1)K
V
i
o
+
n
1
2λ
−
R
n
K
V
n
D
2
n
o
+
+ s
2
P (0) = κ(n + 1). (23)
Upon substituting into (19) then we obtain
C
0,n
(κ)
4
= J
A
n
→Y
n
|s
(π
∗
, κ) =
1
2
n
X
i=0
log
|D
2
i
K
∗
Z
i
+ K
V
i
|
|K
V
i
|
(24)
Clearly, in general, for each i, C
i
(κ
∗
i
) > 0 provided κ
∗
i
∈
(κ
min,i
, ∞) and these critical values depend on whether the
coefficients of the G-RM (13) lie outside or inside the unit
circle, i.e., |C
i−1
| ≥ 1 or |C
i−1
| < 1, for i = 0, . . . , n
(see [1]). Expressions (21), (22) are known as water-filling
in information theory. Clearly, the following hold.
If κ
4
= κ
min
∈ [0, ∞) is such that (25)
λ
n
(κ
min
) >
D
2
n
2K
V
n
R
n
, (26)
λ
i
(κ
min
) >
D
2
i
2K
V
i
P (i + 1)D
2
i
+ R
i
, i = 0, . . . , n − 1,
(27)
then C
0,n
(κ) = 0, and (28)
κ
min
4
= κ
0,n
(K
∗
Z
)
K
Z
∗=0
=
n−1
X
i=0
P (i + 1)K
V
i
+ s
2
P (0).
Hence, κ
min
is the minimum power above which a non-
negative information rate occurs, i.e., C
0,n
(κ) > 0, ∀κ ∈
(κ
min
, ∞). κ
min
is the solution of the Linear-Quadratic
Gaussian stochastic optimal control problem of minimizing
the average power, when A
i
= g
i
(Y
i−1
, s), i = 0, . . . , n,
i.e., non-random; see the numerical example in Fig 2.
0 2 4 6 8 10
0
0.5
1
1.5
2
Fig. 2. Example in which the rate of a scalar system is shown for different
values of power level, κ, for n = 100 and n = 1000. Note that C
i
is
chosen such that |C
i
| > 1.
A. Optimal Controllers-Encoders
Now, we use the calculation of the n−FTFI capacity to
show that linear controller-encoder strategies in (x
i
, y
i−1
),
denoted by {µ
L
i
(x
i
, y
i−1
) : i = 0, . . . , n}, achieve the
n−FTFI, and that among such linear controller-encoders,
then the conditional mean decoder minimizes the MSE.
Theorem 3.2: In the class of linear controller-encoders
that encode the Gaussian Markov process X
n
defined by (7)
and operate at the n−FTFI capacity, the optimal controller-
encoder exists, the conditional mean decoder minimizes the
MSE, and these are given below.
Let
n
Γ
∗
i
, K
∗
Z
i
: i = 0, . . . , n
o
be the optimal strat-
egy corresponding to (15) and (18), and joint process
(A
∗
i
, Y
∗
i
, Z
∗
i
) : i = 0, . . . , n
.
Define the filter estimates
1
and conditional covariances
b
X
i|i−1
4
= E
s
n
X
i
Y
∗,i−1
o
,
b
X
i|i
4
= E
s
n
X
i
Y
∗,i
o
,
Σ
i|i−1
4
= E
s
n
X
i
−
b
X
i|i−1
X
i
−
b
X
i|i−1
T
Y
∗,i−1
o
,
Σ
i|i
4
= E
s
n
X
i
−
b
X
i|i
X
i
−
b
X
i|i
T
Y
∗,i
o
, i = 0, . . . , n.
(a) Controller-Encoder. The mutual information between X
n
and Y
n
for fixed S = s, denoted by I
µ
L,∗
(X
n
; Y
n
|s) of the
controller-encoder strategy
2
A
∗
i
=µ
L,∗
i
(X
i
, Y
∗,i−1
)=Γ
∗
i
Y
∗
i−1
+Θ
∗
i
n
X
i
−
b
X
i|i−1
o
, (29)
Θ
∗
i
=K
∗,
1
2
Z
i
Σ
−
1
2
i|i−1
, Θ
∗
i
0, i = 0, . . . , n, (30)
Y
∗
i
=
C
i−1
+D
i
Γ
∗
i
Y
∗
i−1
+D
i
Θ
∗
i
n
X
i
−
b
X
i|i−1
o
+V
i
, (31)
operates at the n−FTFI capacity of the G-RM, i.e.,
I
µ
L,∗
(X
n
; Y
n
|s) = J
A
n
→Y
n
|s
(π
∗
, κ). (32)
Moreover, the following hold.
(b) Filter Estimates. Define the innovations process by
ν
∗
i
4
=
Y
∗
i
− E
s
Y
∗
i
Y
∗,i−1
: i = 0, . . . , n
. Then the optimal
filter estimates satisfy the following recursions.
b
X
i+1|i
= F
i
b
X
i|i−1
+Ψ
i|i−1
n
Y
∗
i
−
C
i−1
+D
i
Γ
∗
i
Y
∗
i−1
o
,
= F
i
b
X
i|i−1
+Ψ
i|i−1
ν
∗
i
,
b
X
0|−1
= Given, (33)
b
X
i+1|i+1
= F
i
b
X
i|i
+ Ψ
i|i−1
ν
∗
i
, (34)
Σ
i+|i
= F
i
Σ
i|i−1
F
T
i
+G
i
K
W
i
G
T
i
− F
i
Σ
i|i−1
D
i
Θ
∗
i
T
h
D
i
K
∗
Z
i
D
T
i
+ K
V
i
i
−1
D
i
Θ
∗
i
Σ
i|i−1
F
T
i
, (35)
Σ
0|−1
= E
s
n
X
0
−
b
X
0|−1
X
0
−
b
X
0|−1
T
o
(36)
Σ
i|i
= Σ
i|i−1
− Ψ
i|i−1
D
i
Θ
∗
i
Σ
i|i−1
(37)
Σ
i+1|i
=F
i
Σ
i|i
F
T
i
+ G
i
K
W
i
G
T
i
(38)
1
Without loss of generality we may assume σ{Y
∗
−1
} = σ{Ω, ∅}, and
omit the dependence of E
s
on e.
2
For any square matrix M with real entries M
1
2
is its square root.
where the filter gains are difined by
Ψ
i|i−1
4
= F
i
Ψ
i|i−1
,
Ψ
i|i−1
4
= Σ
i|i−1
D
i
Θ
∗
i
T
h
D
i
K
∗
Z
i
D
T
i
+ K
V
i
i
−1
. (39)
(c) Conditional Mean Decoder. For the controller-encoder
µ
L,∗
i
(X
i
, Y
∗,i−1
, s) = bµ
L,∗
i
(X
i
, Y
∗
i−1
,
b
X
i|i−1
, s) ≡
bµ
L,∗
i
(X
i
, Y
∗
i−1
,
b
X
i−1|i−1
, s), then the conditional mean
decoder
b
X
i|i
=
b
d
∗
i
(Y
∗,i
, µ
L,∗
(·), s), i = 0, . . . , n is optimal,
in the sense of minimizing the MSE.
Proof: (a)-(c) are easily verified from [13].
Next, we give an illustrative example of Theorem 3.2 to
demonstrate the properties of the optimal controller-encoder,
and its relation to the MSE, which is a generalization of the
material discussed in Section II-C.
Example 3.3: Consider Theorem 3.2 with p = q = 1. By
solving Riccati equation (36) we obtain
Σ
i+1|i
= F
2
i
D
2
i
K
∗
Z
i
+ K
V
i
K
V
i
−1
Σ
i|i−1
+ G
2
i
K
W
i
, Σ
0|−1
= F
2
i
e
−2C
i
(κ
∗
i
)
Σ
i|i−1
+ G
2
i
K
W
i
, i = 0, . . . , n (40)
Σ
i|i
= F
2
i−1
e
−2C
i
(κ
∗
i
)
Σ
i−1|i−1
+ e
−2C
i
(κ
∗
i
)
G
2
i−1
K
W
i−1
.
(41)
Σ
0|0
= e
−2C
0
(κ
∗
0
)
Σ
0|−1
(42)
Clearly, by the above solutions there is a direct rela-
tion between the sequence of the MSEs at each time
{Σ
i|i−1
, Σ
i|i−1
}, the information rates at each time instant
{C
i
(κ
∗
i
) : i = 0, 1, . . . , n}, and the parameters of the
information process {(F
i
, G
i
, K
W
i
) : i = 0, . . . , n − 1}.
Case 1. Information Process X
i+1
= F
i
X
i
, X
0
∼
N(0, σ
2
X
0
), i = 0, . . . , n, that is, we set G
i
= 0. Then
Σ
n|n
=|F
0
|
2
|F
1
|
2
. . . |F
n−1
|
2
e
−2
P
n
j=0
C
j
(κ
∗
j
)
Σ
0|−1
,
n = 0, 1, . . . , Σ
0|0
= e
−2C
0
(κ
∗
0
)
Σ
0|−1
,
Then Σ
n|n
, n = 0, 1, . . . converge monotonically to zero, i.e.,
If
n
X
i=0
C
i
(κ
∗
i
) >
X
i∈{0,...,n−1}:|F
i
|>1
log |F
i
|, ∀n = 0, . . .
then lim
n−→∞
Σ
n|n
= 0. (43)
Conditions (43) states that the larger the F
i
’s, i.e., more
unstable, then the larger total power κ is needed to ensure
the MSE of the decoder converges to zero.
Case 2. RV X ∼ N (0, σ
2
X
). The MSE is obtained from
Case 1, by setting F
i
= 1, i = 0, . . . , n − 1, giving
Σ
n|n
= e
−2
P
n
i=0
C
i
(κ
i
)
Σ
0|−1
, n = 0, 1, . . . . (44)
This is the analog of Elias MSE decoding error (11); it is
identical to (11), if the G-RM is memoryless, i.e., if C
i
=
0, i = 0, . . . , n−1 and Q
i
= 0, i = 0, . . . , n−1, R
i
= 1, i =
0, . . . , n.
Numerical example. In Fig. 4 we observe that as we
increase the total power κ the estimation error Σ
n|n
given
iteratively in (41) is reduced and eventually it converges
exponentially to zero. However, below a certain value, then
the constraint set is not feasible, i.e., it is empty.