Channel Prediction Using an Adaptive Kalman Filter

WSA 2015 • March 3-5, 2015, Ilmenau, Germany
Channel Prediction Using an Adaptive Kalman
Filter
Behailu Y. Shikur and Tobias Weber
Institute of Communications Engineering, University of Rostock, 18119 Rostock, Germany
Email: {behailu.shikur, tobias.weber}@uni-rostock.de
Abstract—We present an adaptive Kalman filter based channel
estimation and prediction algorithm for multicarrier systems with
time-varying mobile radio channels. We prove that for a widesense stationary uncorrelated scattering channel in which P plane
waves superpose at the receiver antenna, each current/future
sample of the channel transfer function can be represented
as a linear combination of at least P past channel transfer
function samples. Based on this result, a vector auto-regressive
model is used to model the dynamics of the time-varying
mobile radio channel. The adaptive Kalman filter initially uses
a state transition matrix derived from the Jakes’ model for the
temporal correlation and the one-sided exponential power delay
profile for the spectral correlation between the channel transfer
function samples. Afterwards, the state transition matrix of the
adaptive Kalman filter is periodically updated by computing
the correlation between the estimated channel transfer function
samples. It is shown that the periodical update of the state
transition matrix results in a significant prediction performance
improvement over the one based on the correlation estimates
using the Jakes’ model and the one-sided exponential power delay
profile. Furthermore, the proposed adaptive Kalman filter yields
a satisfactory performance when used as an interpolation filter in
the frequency domain even for cases where the channel transfer
function is sampled at sub-Nyquist rate.
I. I NTRODUCTION
Advanced transmission techniques such as adaptive modulation, bit-loading, precoding techniques, etc, rely on the
availability of reliable channel state information (CSI) at
the transmitter and the receiver. Furthermore, communication
systems which use coherent detection rely on the availability
of accurate CSI at the receiver. The CSI at the receiver side
can be estimated by transmitting a priori known pilot symbols.
However, in time-varying mobile radio channels the channel
changes continually in time which makes the estimated CSI
outdated. This necessitates the need for tracking and predicting
the channel by exploiting the correlations between the channel
transfer function (CTF) samples. In this paper we propose
an adaptive Kalman filter based channel state predictor for
multicarrier systems with time-varying mobile radio channels.
The Kalman filter has been proposed for channel estimation, tracking and prediction for single-input single-output
and multiple-input multiple-output mobile radio channels by
several authors [1], [2], [3]. These authors have exploited the
temporal and spectral correlation between the CTF samples
for channel estimation, tracking and prediction. An autoregressive (AR) model has been used to model the dynamics
of the CTF. The coefficients of the AR model were calculated
using the temporal correlation under the assumption of the
ISBN 978-3-8007-3662-1
1
Jakes’ fading model and the spectral correlation under the
assumption of the one-sided exponential power delay profile.
The Jakes’ model is based on the assumption of infinitely
many propagation paths with scatterers uniformly surrounding
the mobile station. However, in practice this assumption does
not hold true for many mobile radio channels. Several measurements have shown that there are only a handful of propagation paths with significant power [4], [5]. Furthermore, the
Jakes’ model with one-sided exponential power delay profile
only describes the statistics of the whole stochastic process.
However, as the stochastic process for several channels of
practical interest is not ergodic, the Jakes’ model with onesided exponential power delay profile does not reflect the
statistics of a single realization of the stochastic process. Thus
the computed channel correlation using the Jakes’ model with
one-sided exponential power delay profile may not be accurate
enough to be used in practice with high reliability. Towards
this end, we propose to periodically update the state transition
matrix of the Kalman filter by computing the correlation
from the estimated CTF samples. The Jakes’ model with
one-sided exponential power delay profile will only be used
for initializing the Kalman filter. This results in an adaptive
Kalman filter channel estimator and predictor which better
reflects the actual mobile radio channel.
The remainder of this paper is organized as follows. Section
II introduces the system model for channel estimation and
tracking in multicarrier systems with time-varying point-topoint mobile radio channels. The dynamics of the mobile
radio channel are discussed in Section III. In Section IV, the
adaptive Kalman filter based channel estimator and predictor
is proposed. The performance of the proposed algorithm is
analyzed using Monte Carlo simulations in Section V. Finally,
the conclusions of this paper and future recommendations are
discussed in Section VI.
II. S YSTEM
MODEL
A. Wide-sense stationary uncorrelated scattering (WSSUS)
channel model
We consider a time-varying point-to-point mobile radio
channel. The CTF is modeled by P plane waves which superpose at the receiver antenna. Each of the P propagation paths
is characterized by a random complex weight αp , a random
phase θp , a random delay τp and a random Doppler shift fDp ,
1 ≤ p ≤ P . It is assumed that the complex weights, the
phases, the delays and the Doppler shifts are constant within
© VDE VERLAG GMBH · Berlin · Offenbach, Germany
WSA 2015 • March 3-5, 2015, Ilmenau, Germany
the bandwidth B under consideration. The band limited CTF at
the center frequency f0 with the bandwidth B in the equivalent
low-pass domain can be described as a superposition of P
complex exponential functions
H(f, t) =
P
X
αp · ejθp · e−j2π(f +f0 )τp · ej2πfDp t ,
(1)
p=1
where f is in the range − 21 B, 12 B .
The CTF H(f, t) is assumed to have zero mean due to the
random
complex
phase rotation and is normalized such that
E{H(f, t)|2 = 1. The autocorrelation function of the CTF
is defined as
rf,t (f + ∆f, t + ∆t) = E{H(f + ∆f, t + ∆t)H ∗ (f, t)}. (2)
In this paper we consider WSSUS channels where the secondorder moments are stationary and the scattering at two different
paths are independent.
It is often assumed that the autocorrelation function in (2) is
separable in time-domain and frequency-domain correlations,
i.e.,
rf,t (∆f, ∆t)
=
rf (∆f ) · rt (∆t),
(3)
where rf (∆f ) = E{H(f + ∆f, t)H ∗ (f, t)} and rt (∆t) =
E{H(f, t + ∆t)H ∗ (f, t)}. A commonly used model for the
time-domain correlations is the Jakes’ model where it is
assumed that the paths impinge uniformly from all directions
[6]. Each of these paths has an associated Doppler shift
dependent on the direction of arrival at the receiver. For the
Jakes’ model the time-domain correlation function becomes
rt (∆t) = J0 (2πfD,max ∆t) ,
(4)
where J0 (·) is the zeroth-order Bessel function of the first kind
and fD,max is the maximum Doppler frequency. Concerning
the frequency-domain correlations, a one-sided exponential
power delay profile [7]
Ah (τ )
DYNAMICS
A. Deterministic model
Based on the geometrical channel model, the linear timefrequency dynamics of the CTF have been claimed in [8]. In
this paper we prove that the claim holds true almost surely.
Proposition 1: For the CTF given in (1), each sample of the
CTF H(k+∆k, n+∆n), ∆k ∈ N, ∆n ∈ N can be represented
as a linear combination of at least P previous samples of the
CTF almost surely, i.e.,
H(k+∆k, n+∆n) =
L−1
X M−1
X
ϕ∆k,∆n (l, m)H(k−l, n−m),
l=0 m=0
(7)
where L · M ≥ P and ϕ∆k,∆n (l, m) are the complex filter
coefficients which depend on the scenario but not on the time
and frequency indices k and n.
Proof: Substituting the sampled CTF using (1) in (7) and
making further simplifications results in
P
X
γk,n (p)e−j2πF ·∆kτp ej2πfDp T ·∆n
p=1
=
P
X
p=1
γk,n (p)
L−1
X M−1
X
ϕ∆k,∆n (l, m)e−j2πF ·lτp ej2πfDp T ·m ,
l=0 m=0
where γk,n (p) = αp ejθp e−j2π(F ·k+f0 )τp ej2πfDp T ·n . We can
re-write (8) in vector-matrix form as
is commonly assumed, where h(τ, t) is the time varying
impulse response of the channel and τm is the multipath spread
of the channel. The Fourier transform of the power delay
profile yields the frequency-domain correlation function
1
.
1 + j2π∆f τm
III. C HANNEL
(8)
= E {h(τ, t)h∗ (τ, t)}
1 − ττ
=
e m , τ ≥ 0,
τm
rf (∆f ) =
where H(k, n) is the CTF at frequency kF and time nT ,
s(k, n) is the information symbol carried by the k th subcarrier
at the nth transmit symbol and w(k, n) is the zero mean
2
complex white Gaussian noise with variance σw
.
In the following, for the channel estimation and prediction
process, it is assumed that the information symbol s(k, n) is
either a priori known at the receiver when pilot symbols are
used for channel estimation or fedback from the decoder after
initial channel estimation has been performed using the pilot
symbols, i.e., after the training period.
(5)
T
γT
k,n · ψ ∆k,∆n = γ k,n · Ψ · ϕ∆k,∆n ,
(9)
where γ k,n = (γk,n (1), γk,n (2), . . . , γk,n (P ))T , ψ ∆k,∆n =
(e−j2πF ·∆kτ1 ·ej2πfD1 T ·∆n , . . . , e−j2πF ·∆kτP ·ej2πfDP T ·∆n )T
and ϕ∆k,∆n is the filter coefficient vector. The matrix Ψ is the
Khatri-Rao (row-wise Kronecker) product of the Vandermonde
matrices Ψτ and ΨfD , i.e.,
B. Signal model
Let’s consider a multicarrier transmission scheme. The time
index is n = 1, . . . , N whereas the subcarrier index is
k = 1, . . . , K. The transmit symbol has a duration T and
the subcarrier spacing is F . The received signal z(k, n) from
the k th subcarrier at the nth time instant is
z(k, n) = H(k, n) · s(k, n) + w(k, n),
ISBN 978-3-8007-3662-1
(6)
2
Ψ = Ψτ ⊙ ΨfD



Ψτ = 

(10)
ψτ1
ψτ2
..
.
ψτ21
ψτ22
..
.
...
...
ψτL−1
1
ψτL−1
2
..
.
1 ψτP
ψτ2P
...
ψτL−1
P
1
1
..
.





© VDE VERLAG GMBH · Berlin · Offenbach, Germany
WSA 2015 • March 3-5, 2015, Ilmenau, Germany

ΨfD


= 


1 ψfD1
1 ψfD2
..
..
.
.
1 ψfDP
ψf2D
1
ψf2D
2
..
.
ψf2D
P
. . . ψfM−1
D1
. . . ψfM−1
D2
..
.
. . . ψfM−1
D
P



,


where ψτp = exp(j2πF τp ) and ψfDp = exp(−j2πT fDp ).
Since the filter coefficient vector ϕ∆k,∆n shall be independent of k and n, (7) shall hold for all possible linear
combinations of the CTF samples. Thus for L + ∆k < K and
M + ∆n < N , the CTF samples with the indices k, L + ∆k ≤
k ≤ K and n, M + ∆n ≤ n ≤ N can be expressed as a linear
combinations of the appropriate CTF samples with the indices
k, 1 ≤ k ≤ K − ∆k and n, 1 ≤ n ≤ N − ∆n as per (7). These
(K − ∆k − L + 1) · (N − ∆n − M + 1) linear equations can
be written as
Γ · ψ ∆k,∆n = Γ · Ψ · ϕ∆k,∆n .
(11)
The matrix Γ is defined as:
Γ = Γτ,fD ◦ Γα,θ,τ ,
Γτ,fD = Γτ ⊗ ΓfD

γτL1
 γτL+1
1

Γτ = 
..

.

ΓfD


= 





Γα,θ,τ = 

γτL2
γτL+1
2
..
.
...
...
(12)
γτLP
γτL+1
P
..
.
. . . γτK−∆k
γτK−∆k
γτK−∆k
P
2
1
γfMD
γfMD
...
γfMD
1
2
P
γfM+1
γfM+1
. . . γfM+1
D1
D2
DP
..
..
..
.
.
.
N −∆n
N −∆n
N −∆n
γfD
γfD
. . . γfD
1
2
P

ϕ̂∆k,∆n = Ψ† · ψ ∆k,∆n ,




B. Stochastic model





γα2 ,θ2 ,τ2
γα2 ,θ2 ,τ2
..
.
...
...
γαP ,θP ,τP
γαP ,θP ,τP
..
.
γα1 ,θ1 ,τ1
γα2 ,θ2 ,τ2
...
γαP ,θP ,τP
The deterministic channel dynamics discussed in Section
III-A can be extended to a stochastic channel dynamics
by introducing noise. Consequently, the stochastic channel
dynamics of the CTF can be modeled by a VAR model of
order M with L subcarriers exploited such that M · L ≥ P .
The M th order VAR model of the CTF is



,

where γτp = exp(−j2πF τp ), γfDp = exp(j2πT fDp ) and
γαp ,θp ,τp = αp ejθp e−j2πf0 τp . The signs ◦ and ⊗ denote the
Hadamard product and the column-wise Kronecker product of
two matrices, respectively.
If there is a filter coefficient vector ϕ∆k,∆n which satisfies
ψ ∆k,∆n = Ψ · ϕ∆k,∆n ,
(13)
then (11) is also satisfied. Thus (13) is a sufficient condition for
(11). If the matrix Γ has full rank, then (13) is a necessary and
sufficient condition for (11). Thus (13) is a necessary condition
if (7) shall hold for all k and n, i.e., if the filter coefficient
vector ϕ∆k,∆n shall be independent of k and n.
In the following we will show that the matrix Γ has full rank
and hence (13) is a necessary condition for (7) to hold. For
(K −∆k−L+1)·(N −∆n−M +1) > P the matrices Γτ and
ΓfD , which are submatrices of Vandermonde matrices, have a
full rank of P as the delays and Doppler shifts are assumed
to be generated from a continuous distribution [9]. The matrix
ISBN 978-3-8007-3662-1
(14)
where Ψ† represents the the Moore-Penrose pseudoinverse of
the matrix Ψ.

γα1 ,θ1 ,τ1
γα1 ,θ1 ,τ1
..
.
Γτ,fD which is the column-wise Kronecker product of Γτ and
ΓfD has a rank P . The simple proof is that the matrix Γτ,fD
is the Kronecker product of Γτ and ΓfD , which has rank P 2 ,
with P 2 − P columns removed which results in a matrix with
rank P . The (K − ∆k − L + 1)·(N − ∆n− M + 1)× P matrix
Γα,θ,τ has a rank of 1. The matrix Γ, which is the Hadamard
product of the matrices Γτ,fD and Γα,θ,τ , has a rank equal
to the product of the rank of the matrices Γτ,fD and Γα,θ,τ .
Thus the matrix Γ has a full rank of P .
According to the Rouché-Capelli theorem, (13) has at least
one solution if the rank of the augmented matrix is equal to
the rank of the coefficient matrix. The Khatri-Rao product of
the Vandermonde matrices Ψτ ∈ CP ×L and ΨfD ∈ CP ×M
whose 2P complex exponential parameters τp and fDp are
drawn from a continuous distribution, with respect to the
Lebesgue measure in C2P , has almost surely full rank [10].
Thus the matrix Ψ has almost surely full rank. Furthermore,
for L · M ≥ P the augmented matrix Ψ|ϕ∆k,∆n has a rank
equal to the rank of the coefficient matrix Ψ. Thus (13) has at
least one solution if L · M ≥ P and a unique filter coefficient
vector ϕ∆k,∆n if L · M = P almost surely.
The optimal filter coefficient vector ϕ̂∆k,∆n , in the least
squares sense, can be determined from (13) as
3
h(k, n) +
M
X
A(m) · h(k, n − m) = u(k, n),
(15)
m=1
where A(m) are the VAR model coefficient matrices,
h(k, n) = (H(k, n), H(k − 1, n), . . . , H(k − L + 1, n))T and
u(k, n) ∼ CN (0, Ruu ) is a vector white Gaussian process.
The VAR model coefficient matrices A(m) and the VAR
model noise covariance matrix Ruu can be determined from
the Yule-Walker equation using the channel autocorrelation
function [11].
IV. K ALMAN
FILTER BASED CHANNEL ESTIMATOR AND
PREDICTOR
We can develop a state space model from the VAR
model in (15) and the signal model in (6) as follows.
The state vector and the measurement vector are defined as
x(k, n) = (hT (k, n), hT (k, n − 1), . . . , hT (k, n − M + 1))T
and z(k, n) = (z(k, n), z(k − 1, n), . . . , z(k − L + 1, n))T ,
respectively. Using (6) and (15) the state equation and the
© VDE VERLAG GMBH · Berlin · Offenbach, Germany
WSA 2015 • March 3-5, 2015, Ilmenau, Germany
measurement equation of the state space model can be defined
as
Φ · x(k, n − 1) + B · u(k, n)
C(k, n) · x(k, n) + w(k, n),
x(k, n) =
z(k, n) =
(16)
(17)
where



Φ=

−A(1) · · ·
IL
···
..
.
0L
···

−A(M − 1) −A(M )

0L
0L


..
..

.
.
IL
0L
(18)
B = (Ruu , 0L , . . . , 0L )
C(k, n) = (diag (s(k, n), . . . , s(k − L + 1, n)) , 0L , · · · , 0L )
w(k, n) = (w(k, n), w(k − 1, n), . . . , w(k − L + 1, n))T .
IL and 0L are the identity and the zero matrices of size L,
respectively.
The Kalman filter is an optimal sequential linear minimum
mean square-error estimator of a signal corrupted by a noise.
We will use the Kalman filter to estimate, track and predict
the CTF. The Kalman filter tracks the estimate of the state
vector x̂(n|n) and the correlation matrix of the estimation error
M(n|n) based on the measurements z(k, 1), . . . , z(k, n). The
subcarrier index k has been dropped from x̂(n|n) and M(n|n)
for brevity.
The Kalman filter starts with an initial estimate x̂(0|0) =
0P ×1 and an initial correlation matrix of the estimation error
M(0|0) = IP . The Kalman filter computes the following
equations sequentially for each n.
• Prediction stage:
M(n|n − 1) = Φ · M(n − 1|n − 1) · ΦH + B · BH
x̂(n|n − 1) = Φ · x̂(n − 1|n − 1)
•
Update stage:
M(n|n − 1) · CH (k, n)
2I
C(k, n) · M(n|n − 1) · CH (k, n) + σw
L
z̃(k, n) = z(k, n) − C(k, n) · Φ · x̂(n|n − 1)
K(k, n) =
x̂(n|n) = Φ · x̂(n|n − 1) + K(k, n) · z̃(k, n)
M(n|n) = (IM·L − K(n) · C(k, n)) · M(n|n − 1)
K(k, n) is the Kalman gain and z̃(k, n) is the innovation
vector.
The estimated and predicted CTF samples at time instance
n for the subcarriers k, k − 1, . . . , k − L + 1, i.e., ĥ(k, n)
and ȟ(k, n), are obtained from x̂(k, n|n) and x̂(k, n|n − 1),
respectively.
The state transition matrix Φ and the VAR process noise
covariance matrix Ruu can be determined from the YuleWalker equation using (4) and (5) for computing the channel
autocorrelation function. However, as mentioned in the Section
I, the Jakes’ model with one-sided exponential power delay
profile is a reasonable model only for rich scattering environments where the transmitted radio signal is assumed to be
ISBN 978-3-8007-3662-1
4
scattered by many objects before arriving at the receiver. This
assumption is not valid in many propagation environments.
Thus channel estimation and prediction algorithms based on
the Jakes’ model with one-sided exponential power delay
profile might not always deliver a satisfactory performance.
In this paper we propose to periodically update the correlation
matrix of the channel in order to get a better estimate of the
state transition matrix and the VAR process noise covariance
matrix. Doing so, we obtain an adaptive Kalman filter. The
Jakes’ model with one-sided exponential power delay profile
is only used for getting initial estimates of the CTF samples
Ĥ(k, n) for n ≤ M which are then used to compute the
estimates of the correlation matrix of the channel. Thus the
state transition matrix and the VAR process noise covariance
matrix are thus time-varying, i.e., Φ(n) and Ruu (n).
The time-varying state transition matrix Φ(n) is computed
by solving the Yule-Walker equations as follows. The M th
order VAR model equation in (15) can be re-written as
 h(M + 1) 
 −A(1)   u(M + 1) 


h(M + 2)
..
.
h(n)


 = H(n) · 
−A(2)
..
.
−A(M )
 
+
u(M + 2)
..
.
u(n)


(19)
where



H(n) = 

h(M )
h(M − 1) · · ·
h(M + 1)
h(M )
···
..
..
.
.
h(n − 1) h(n − 2) · · ·
H
H
h(1)
h(2)
..
.
h(n − M )





H
h(n) = (h (K, n), h (K − 1, n), . . . , h (L, n))T .
The Yule-Walker equations are generated from (19) using
the initial estimates of the CTF samples Ĥ(k, n) instead of
H(k, n). The resulting Yule-Walker equations are then solved
using the correlations computed from the time-average, i.e.,
1, . . . , n. The time-varying VAR process noise covariance
matrix Ruu (n) can be calculated in a similar fashion.
It must be noted that the Kalman filter gives more weight
initially to the measurement data than the predicted state
vector. This reduces the impact of error propagation due to
the possibly erroneous estimate of the state transition matrix
using the Jakes’ model with one-sided exponential power delay
profile.
V. S IMULATION
RESULTS
A. Simulation setup
In this section we analyze the performance of the proposed
channel estimation and prediction algorithm using Monte
Carlo simulation with 104 independent trials. The subcarrier
spacing is F = 2 kHz and the transmit symbol duration is
T = 500 µsec. The transmitted signals have a center frequency
of 2.4 GHz. For each run of the Monte Carlo simulations the
path delays τp and Doppler shifts fDp were generated using
the one-sided exponential power delay profile and the Jakes’
model discussed in Section II-A using a multipath spread
τm = 20 µsec and fD,max = 222.22 Hz, respectively. The
© VDE VERLAG GMBH · Berlin · Offenbach, Germany
WSA 2015 • March 3-5, 2015, Ilmenau, Germany
0
B. Channel prediction
Fig. 1 shows the NMSE performance of the proposed
channel prediction algorithms, with respect to the PSNR, for
different dimensions of the filter coefficient, i.e., L · M , for a
scenario with P = 20 propagation paths. It can be seen that
the performance of the prediction algorithms is bounded by
the reference Kalman filter. Also shown is the performance
of the stochastic model predictor which is a Kalman filter
based solely on the Jakes’ model with one-sided exponential
power delay profile. The performance of the stochastic model
predictor is rather poor regardless of the dimension of the
filter. The proposed adaptive Kalman filter shows a superior
performance over the stochastic model Kalman filter.
It can be seen that even though the minimal required
dimension of the filter is P = 20, higher values of L and M ,
which yield filter dimensions L · M > P , yield a significant
improvement in performance for both the reference and the
adaptive Kalman filter. This is due to the gain from noise
suppression from exploitation of more measurements for each
predicted CTF sample as the filter dimension is increased. The
case where the filter order is less than the minimum filter order,
i.e., L = 4, M = 4 yields an inferior performance.
ISBN 978-3-8007-3662-1
5
−10
−15
−20
−25
−10
−5
0
5
PSNR ρ/dB
10
15
20
Fig. 1. Impact of the dimension of the filter on the performance of the
reference Kalman filter (RKF) and the adaptive Kalman filter (AKF), P = 20
The performance metric is the normalized mean square error
(NMSE) of the predicted CTF samples Ȟ(k, n)
n
2 o
E Ȟ(k, n) − H(k, n)
n
o
,
2
E |H(k, n)|
0
RKF, P = 4
AKF, P = 4
RKF, P = 10
AKF, P = 10
RKF, P = 20
AKF, P = 20
stochastic predictor
−5
NMSE /dB
where in the simulations the expectations are computed using
time and frequency averages. In the following the NMSE of
the proposed algorithms, averaged for the last 10 snapshots,
will be investigated.
As a performance benchmark the performance of a reference
Kalman filter with the state transition matrix calculated using
(14) for each run of the Monte Carlo simulation is considered
whenever applicable.
RKF, M = 5, L = 20
AKF, M = 5, L = 20
RKF, M = 5, L = 10
AKF, M = 5, L = 10
RKF, M = 5, L = 4
AKF, M = 5, L = 4
AKF, M = 4, L = 4
stochastic predictor
−5
NMSE /dB
real and imaginary parts of the paths’ complex weight are
generated from a uniform distribution U(0, √12 ) whereas the
phase θp is generated from a uniform distribution U(0, 2π).
The CTF samples are generated using (1).
The time spacing 5T and the frequency spacing 12F
between the pilots/transmit symbols which are exploited for
channel estimation and prediction are chosen such that the
CTF is sampled almost at the Nyquist rate, i.e, 5T · fD,max ≈
1/2 and 12F · τm ≈ 1/2. The number of the exploited
subcarriers for channel estimation and prediction is 40 and
100 snapshots of these pilot/transmit symbols in time domain
are considered. In all the simulations the impact of possible incorrect decoding of the transmitted signals s(k, n) is excluded
by assuming perfect decision feedback. The performance of
the proposed prediction algorithm is analyzed under different
pseudo signal-to-noise-ratios (PSNRs)
o
n
E |s(k, n)|2
.
ρ=
2
σw
−10
−15
−20
−25
−10
−5
0
5
PSNR ρ/dB
10
15
20
Fig. 2. Impact of the number of paths on the performance of the reference
Kalman filter (RKF) and the adaptive Kalman filter (AKF), L = 20, M = 5
Fig. 2 shows the influence of the number of paths P on
the performance of the proposed prediction algorithms. The
numbers of paths are 4, 10 and 20 and for each case the
values L = 20 and M = 5 are chosen to get satisfactory
performance. Apart from the stochastic model based predictor,
which yields a poor performance for all cases, the proposed
prediction algorithms show a decrease in performance as the
number of paths is increased. This is due to the fact that for
the case where the number of paths is less, with the dimension
of the filter unchanged, effectively more measurements than
the required minimum are exploited than the case with more
number of paths. This results in an improved performance
from noise suppression.
For the sake of fairness, we have assumed perfect knowledge of the maximum Doppler frequency and the multipath
spread which is crucial when initially estimating the state
© VDE VERLAG GMBH · Berlin · Offenbach, Germany
WSA 2015 • March 3-5, 2015, Ilmenau, Germany
0
0
−5
−5
NMSE / dB
NMSE / dB
stochastic predictor
−10
adaptive Kalman filter
−15
−10
AKF, interpolated
−15
AKF, un-interpolated
reference Kalman filter
−20
−25
−10
−5
0
5
PSNR ρ /dB
10
−20
15
20
−25
RKF, un-interpolated
−10
−5
0
RKF, interpolated
5
PSNR ρ /dB
10
15
20
Fig. 3. Impact of over-sampled CTF samples on the performance of the
proposed algorithms, P = 20, L = 20, M = 5
Fig. 4. The NMSE prediction interpolation performance of the proposed
algorithms for P = 20, L = 20, M = 5
transition matrix. For the case otherwise, the performance
improvement by the adaptive Kalman filter over the stochastic
model based predictor would be even higher. However, the
adaptive Kalman filter would only be slightly affected.
Fig. 4 shows the performance of the proposed algorithms
as an interpolation filter when the number of paths P = 20.
Measurements are available only for every other CTF sample
in the frequency domain. The adaptive Kalman filter state
transition matrix is initially computed from the adaptive
Kalman filter filtered CTF samples from 50 time snap shots,
with full measurement data, rather than the from the Jakes’
model with one-sided exponential power delay profile as the
previous cases. It can be seen from the figure that a satisfactory
interpolation performance can be obtained by the adaptive and
the reference Kalman filters. The interpolated CTF samples
have a slightly higher NMSE than the CTF samples whose
measurement data is available.
C. Impact of over-sampling
Channel prediction and estimation algorithms based on
stochastic models commonly consider CTF samples which are
sampled at several multiples of the Nyquist rate so that the
CTF samples have strong correlation, making the prediction
problem relatively easier [1], [2], [3]. Fig. 3 shows the
performance of the proposed algorithms for the case where
the CTF is sampled at twice the Nyquist rate in time and
frequency. Unlike the previous cases where the performance
of the stochastic model based predictor was unsatisfactory,
we see a considerable improvement in performance owing to
the strong correlation between the CTF samples. However,
the proposed adaptive Kalman filter still shows a superior
performance over the stochastic model based predictor.
E. Convergence of the Kalman filter
From the simulation results it has been observed that, in general, both the reference Kalman filter and the adaptive Kalman
filter converge after a handful multiple of M iterations.
VI. C ONCLUSIONS
D. Channel interpolation
The proposed adaptive Kalman filter can also be used as
an interpolation filter in the frequency domain. In this case
the length of the measurement vector z(k, n) is less than the
number of entries in the state vector x(k, n) corresponding
to the current time instant n. Thus fewer measurements are
available than the number of CTF samples to be estimated.
The estimation task is even more difficult as the available
measurements are for CTF samples which are sampled less
than the Nyquist rate. The interpolation is achieved by the
cumulative effect of extracting the available information from
the measurement data about the CTF sample to be interpolated
and recursively incorporating this information with the next
measurement data to predict the CTF sample to be interpolated.
ISBN 978-3-8007-3662-1
6
In this paper we have presented an algorithm for estimating,
tracking and predicting a time-varying mobile radio channel
using an adaptive Kalman filter. A vector auto-regressive
model has been used to model the linear dynamics of the channel. The state transition matrix of the adaptive Kalman filter
is periodically updated by computing the correlation matrix
of the channel from the estimated CTF samples. Simulation
results have shown the superior performance of the proposed
adaptive Kalman filter over the Kalman filter based on the
Jakes’ model with one-sided exponential power delay profile.
Future works may include adapting the proposed adaptive
Kalman filter for uplink-downlink transformation of the CTF
in frequency-division-duplex (FDD) systems. Furthermore, an
extension of the proposed algorithm for multiple-input and
multiple-output mobile radio systems could be investigated.
© VDE VERLAG GMBH · Berlin · Offenbach, Germany
WSA 2015 • March 3-5, 2015, Ilmenau, Germany
ACKNOWLEDGMENT
The authors are indebted to the German Research Foundation for sponsoring this work under the research grant No.
WE2825/10-1.
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ISBN 978-3-8007-3662-1
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© VDE VERLAG GMBH · Berlin · Offenbach, Germany