Iterative bi directional kalman DFE equalizer for the high data rate HF waveforms in the HF channel
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Iterative Bi-directional Kalman-DFE Equalizer for the
High Data Rate HF Waveforms in the HF Channel
Mahmoud A. Elgenedy
Comm. System Design
Varkon Semiconductors
Cairo, Egypt
Essam Sourour
Dept. of Electrical Engineering
Faculty of Engineering, Alexandria University
Alexandria, Egypt
Magdy Fikri
Dept. of Electrical Engineering
Faculty of Engineering, Cairo University
Cairo, Egypt
Abstract— In this paper, we present the Kalman decision
feedback equalizer (Kalman-DFE) as a solution for the high data
rate HF waveforms. We show that the basic performance for the
Kalman-DFE is far from the standard requirements of the HF
and the performance breaks down for the higher constellation.
As a result, we add two major enhancements for the Kalman-
DFE: the bi-directional and iterative structures. The bi-
directional structure successfully prevents the performance
break down at the higher constellation and enhances the overall
performance. The iterative structure then enhances the
performance significantly (about 6dB gain at 1e-5 BER for
64QAM after three iterations). We show that the proposed
equalizer in the final form achieves significant performance
enhancements over the normal one. Moreover, the proposed
iterative Kalman-DFE is much simpler than turbo equalizer,
especially for the fractional-spaced model, for two reasons: first,
the direct adaptation structure used in Kalman filter is much
simpler than the indirect adaptation structure usually used with
the turbo structure (it does not require matrix inversion).
Secondly, the iterative structure (exchanges hard information) is
simpler than turbo (exchanges soft information) as the later
requires a lot of changes in both equalizer and decoder to
support soft information. We perform the simulations on the
MIL-STD-188-110 (Appendix C) waveforms, transmitted over an
ITU-R poor channel (commonly used channel to test HF modem).
Keywords— HF modems, Kalman equalizer, DFE, Iterative
equalization.
I. INTRODUCTION
Transmission in the high frequency band (HF, 3 to 30MHz)
has a lot of difficulties and challenges due to severe channel
fading, especially at high latitudes, which impedes the
mitigation (equalization) process. The task of equalizer
becomes more difficult when high data rates (high
constellation orders) are assumed.
Various types of Kalman-DFE in HF channel for a medium
data rate are proposed by many researches [1-4].
Eleftheriou and Falconer in [1] propose both LMS (the least
mean square) and FRLS (the fast recursive least squares)
adaptation algorithms with periodic restart. They indicate that
FRLS adaptation yields a superior performance to LMS in
rapid fading conditions. A more numerically stable algorithm
for fixed point implementation is the square root Kalman,
presented by HSU in [4].
Bi-directional DFE processing has been previously
proposed in [5-7]. The time-reversal equalization mode
improves decision feedback equalizer (DFE) with a small
number of forward filter taps to perform equally well for both
minimum-phase and maximum-phase channel characteristics.
It converts the non minimum-phase to the minimum-phase
condition by the time-reversal operation.
The mean-squared error (MSE) between the input and
output of the DFE decision device is used as the criterion to
choose between the results of forward and reverse processing
in [6] and [7]. The MSE criterion is applied to an entire frame
of data. In contrast to the preceding global MSE criterion, a
local MAP decision between two candidate sequences is used
in [5], since each decision is based only on a window around
the bit of interest. In this paper, we use the global MSE as
selection criteria.
Most of equalizer errors result from the bad reliability of
hard decisions of equalizer output, which affect the adaptation
process. The main idea behind iterative or turbo structures is
to exchange the information (hard or soft information)
between the decoder and the equalizer iteratively (through an
interleaver). A returned data from decoder are much more
reliable than equalizer output hard decisions and are improved
with iterations. The main difference between iterative and
turbo (in this paper) is the type of data exchanged. If hard
information is used, we call this iterative, but the famous
expression (turbo) is used with soft information.
978-1-4673-2821-0/13/$31.00 ©2013 IEEE
In the case of the iterative structure, we use the same
equalizers and decoders as conventional receivers. The main
difference as explained previously is to use the data returned
from decoder instead of hard decisions of equalizer output.
Returned data from decoder should first pass on the interleaver
then the encoder and finally to the modulator.
The iterative structure enhances the estimated error signal
used for equalizer adaptation. In case of Decision feedback
equalizers, the use of good quality decoder output data instead
of equalizer output decisions enhances the ISI cancellation.
The structure of turbo equalizer is quite similar to the
iterative structure except exchanging soft values (LLR) instead
of hard decisions. Exchanging soft values requires some
modifications for both equalizer and decoder, as both of them
should accept soft information at its input and provide soft
information as output (SISO modules). Also, the modulator
block is replaced by another block which accepts input soft
values and provides both mean and variance for encoded
symbols. Finally, turbo equalizer structure can be viewed as
serially concatenated SISO modules.
Linear MMSE with soft ISI cancellation in a turbo
equalization structure (assuming symbol-spaced model) is
proposed by Roald Otnes in [8] for the HF medium data rate
standard, and he extends his results for high data rates in [9].
In this paper, we first investigate the performance of basic
Kalman-DFE with detailed tuning for all equalizer parameters.
The bi-directional structure is then tested and successfully
prevents the performance break down at the higher
constellation. Finally, the iterative structure achieves a
significant performance enhancement (about 6 dB gain at 1e-5
BER for the 64QAM using three iterations).
The rest of the paper is organized as follows. The next
section presents the system model. In Section III, a study for
the normal Kalman-DFE performance is introduced. Section IV
presents a trial of testing the Kalman-DFE in the backward
direction. In Section V, we present the bi-directional structure.
Finally, Section VI shows the final form of the iterative bi-
directional Kalman-DFE structure and the final BER
performance results.
II. SYSTEM
MODEL
A. Transmitter
The transmitter structure for high data rate serial tone HF is
defined in (Annex B of STANAG 4539 and Appendix C of
MIL-STD-188-110B). The transmitter starts with the blocking
operation which is responsible of dividing the input raw bits
into blocks of size dependent on the rate and the interleaver
length. Each code block shall be interleaved within a single
interleaver block of the same size. The full-tail-biting and
puncturing techniques shall be used with a rate of 1/2
convolutional code to produce a rate of 3/4 block code that is
the same length as the interleaver. The interleaver output is
then scrambled and input to modulation block. The
modulation used (QPSK, 8PSK, 16QAM, 32QAM and
64QAM) shall depend upon the data rate (3200, 4800, 6400,
8000 and 9600 bps). An 8PSK training symbols (initial
preamble, mini-probes and reinserted preamble) are then
inserted within the data frames (256 symbols per frame)
through the framing process. Finally, the output frames are
filtered using SRRC filter (roll-off 0.25 – filter length 12
symbol time) to get the transmitted data samples .
B. Channel Model
In an ionospheric HF communication system, the
transmitter and receiver are not moving (or moving slowly
relative to the wavelength), while the radio waves are being
reflected by a large number of randomly moving ions. This
suggests that the Doppler shift has a Gaussian distribution, as
was verified experimentally by Watterson in [10].
A Gaussian Doppler spectrum can be written as
(1)
where
is the mean value and
is the variance of the
Doppler shift. For a Gaussian fading spectrum, the Doppler
spread
is by convention defined as twice the standard
deviation of the Doppler shift (
).
Using this definition and assuming that
is zero, (1)
becomes,
(2)
C. Receiver Model
The receiver operations start with the SRRC filter matched
to the TX (could be removed if fractional equalizer is used)
followed by the equalization process then the reverse
operations for the transmitter blocks. The decoder is a Viterbi
decoder supports tail biting initialization. The receiver assumes
perfect time and frequency synchronization.
III. F
ORWARD KALMAN FILTER
First, we introduce the normal operation for the Kalman-
DFE which is the case of adapting in the forward direction.
Fig. 1 shows the equalizer structure, where [n] is the
equalizer input (received data),
and
are the
equalizer filter coefficients with lengths
forward and
in
feedback, is the equalizer soft output, is the hard
decision output symbol, and is the corresponding error
signal. The following equations describe the Kalman
algorithm as derived in [11],
Define the vector with the total length of ,
, contains the data received following to current symbol
and past decisions with lengths
respectively:
Also define a vector of equalizer forward and backward
filter coefficients;
The equalizer soft output is
.
(3)
Now, define the Kalman gain vector as follows
(4)
where represents a weighting factor and
is the inverse of the correlation matrix D
N
[n] of the received
signal R
N
[n] (measurement),
(5)
In the current notation, we define
as the inverse of
the received signal (measurement) correlation matrix as
derived in [11]; however, the more popular definition for the
matrix is the correlation matrix of the estimated signal.
The
matrix can be calculated recursively as follows
(6)
Finally, update the filter coefficients
(7)
or
(8)
The initial value for the
matrix is
, where
is a small positive constant whose value is small for high SNR
and large for low SNR and is the identity matrix. Also, the
initial value for filter taps is all zeros except the position of the
output which may start with one. Note that previous equations
and equalizer structure consider the case of the symbol-spaced
model.
]N[
a
+nr
][ nr
ba ,
a
N−
a
1N
a
+−
a
0
a
1
b
2
b
b
N
b
∑
∑
][
ˆ
nx
][
~
nx
]N[
~
b
−nx
Fig.1. Kalman-DFE equalizer.
Start considering the case of reference-mode, where the
perfect data are fed back instead of the hard decisions. It is an
important starting point for testing the adaptive equalizer for
two reasons. First, examine the equalizer performance
limitation (get the best performance). Second, find an easy
way to set optimum values for equalizer parameters.
In the following, we tune the equalizer parameters in the
reference-mode and show the final simulation results.
A. Number of forward and backward coefficients
As indicated in [1], number of equalizer taps is the major
source of performance limitation for the directly adapted
equalizers. Choosing the optimum number of taps for both
forward and backward filters is a major problem in the finite
length equalizers [13], [14] and there is no closed form for
these lengths. In a classic design of the adaptive equalizer, the
filter lengths are usually fixed to some compromise values.
Other approaches use exhaustive search algorithms which
increase the complexity with a great amount. Suboptimum
search algorithms are introduced by [13]. In our simulation,
we choose to fix the forward and backward number of taps to
a suboptimum value, getting it from the simulation.
The main parameter that controls the number of filter taps is
the channel delay spread. Simulation results show that the
forward number of taps
should not be less than and
the performance is improved with increasing that number but
with limits. However, it is sufficient to put the backward
taps equals to (famous choice as indicated in [13] and [8]).
B. Tap of decision (decision delay)
In finite length adaptive equalizers, decision delay choice is
also an interesting problem, especially for short filter lengths
[13-16], and to get an optimum choice, we should study it
together with filter lengths [13, 14, 16]. Also, this is not the
scope of this paper and then we use a fixed suboptimum
choice for the decision delay. For the case of forward direction
adaptation model we found that the last tap is the correct
choice in most cases (it could be seen as collecting symbol
power from the successive symbols in the forward filter and
subtracting the ISI from the preceding symbols in the feedback
filter).
C. Periodic restart
We choose to periodically restart the filter taps and all
related variables each training sequence. Our simulation
shows a slight change in the performance (loss) if we continue
without restarting, as the mini-probe length (31 symbols) is
sufficient in all cases to achieve initial convergence.
D. Adaptation rate
Adaptation rate depends mainly on the channel variation
speed (fading rate) and the transmission data rate. The
adaptation rate should be increased (w decreased) with fading
rate increase. Fig. 2 shows the final results after optimizing the
equalizer parameters in reference-mode for all supported rates.
Fig.2. Forward Kalman-DFE, reference-mode, 72 frame interleaver size,
=11,
=5, Doppler=1 Hz, delay spread=2 msec, adaptation rate = [0.99-
0.98-0.97-0.96-0.93] for [QPSK-8PSK-16QAM-32QAM-64QAM].
Now, let us test the more realistic scenario by replacing the
perfect data in the feedback by the hard decision one.
Simulation results show that Kalman-DFE adapting in the
forward direction can not cope with channel parameters (Poor
channel) for all supported rates. Fig. 3 shows the BER curves
for all rates after reoptimizing the equalizer parameters (some
changes are needed in the realistic scenario like the adaptation
rate). We note that only QPSK and 8PSK can work properly.
IV. B
ACKWARD KALMAN FILTER
In the current standard specification, there is a similarity for
training sequences around the data frame and hence we can
adapt the Kalman filter in the reverse direction. Fig. 4 describes
the behavior of the backward equalizer (Fig. 4b) compared to
the forward one (Fig. 4a) assuming 3-path symbol spaced
channel and filter lengths are
=3 and
=2. The most
important parameter that we have to change in the backward
case is the tap of decision (decision delay).
Fig.3. Forward Kalman-DFE performance, decision-directed mode, all
data rates, 72 frame interleaver size,
=11,
=5, Doppler = 1 Hz, delay
spread = 2 msec. adaptation rate = [0.99-0.97-0.93-0.86-0.72] for [QPSK-
8PSK-16QAM-32QAM-64QAM].
]0[
0
xh
]1[
0
xh
+
]0[
1
xh
]2[
0
xh
+
]1[
1
xh
+
]0[
2
xh
]3[
0
xh
+
]2[
1
xh
]1[
2
xh
+
][
0
nxh
]1[
1
−nxh
]2[
2
−nxh
+
+
]1[
0
+nxh
][
1
nxh
+
]1[
2
−nxh
+
]2[
0
+nxh
+
]1[
1
+nxh
][
2
nxh
+
2−
a
1−
a
0
a
1
b
2
b
][nr
]1[ +nr
]2[ +nr
]1[
~
−nx
]2[
~
−nx
Fig.4a. Forward Kalman-DFE (adaptation behavior), delay of decision is
the last tap, 3-paths symbol-spaced channel,
=3,
=2.
]0[
0
xh
]1[
0
xh
+
]0[
1
xh
]2[
0
xh
+
]1[
1
xh
+
]0[
2
xh
]3[
0
xh
+
]2[
1
xh
]1[
2
xh
+
][
0
nxh
]1[
1
−nxh
]2[
2
−nxh
+
+
]1[
0
+nxh
][
1
nxh
+
]1[
2
−nxh
+
]2[
0
+nxh
+
]1[
1
+nxh
][
2
nxh
+
2−
a
1−
a
0
a
][nr
]1[ +nr
]2[ +nr
1
b
2
b
]1[
~
+nx
]2[
~
+nx
Fig.4b. Backward Kalman-DFE (adaptation behavior), 3-path symbol-
spaced channel,
=3,
=2.
As shown in Fig. 4b, the tap of decision should be
transferred to the first tap (it could be seen as collecting
symbol power from the successive symbols in the forward
filter and subtracting the ISI of these successive symbols in
the feedback filter). Fig. 5 shows the performance for the
backward Kalman-DFE assuming reference-mode. We notice
that the performance is almost the same as forward adaptation
but they are independent as will be shown.
V. B
I-DIRECTIONAL KALMAN FILTER
An efficient structure for the equalizer can be achieved by
combining the two previous structures (forward and
backward) [5-7]. In this case, the equalizer is running in the
two directions (for each data frame) then chooses the best
result by comparing the mean square error between the two
directions (MSE is measured between soft equalizer output
and corresponding hard decisions). The new structure shown
in Fig. 6 introduces a real diversity in the results as we found
that convergence may be achieved in a direction while the
other is failed.
The reference-mode simulation results for the bi-directional
equalizer for all data rates are shown in Fig. 7 and they show a
significant improvement over one directional (forward or
backward) equalizer.
Fig.5. Backward Kalman-DFE, reference-mode, 72 frame interleaver
size,
=11,
=5, Doppler = 1 Hz, delay spread = 2 msec.
Fig.6. Bi-directional Kalman-DFE structure.
Fig.7. Bi-directional Kalman-DFE, reference-mode, 72 frame interleaver
size,
=11,
=5, Doppler = 1 Hz, delay spread = 2 msec.
Fig. 8 shows the simulation results for decision-directed bi-
directional equalizer. Although we have a noticeable
enhancement in the performance compared to one directional
structure, unfortunately, the performance is still far from the
standard expectations.
VI. I
TERATIVE STRUCTURE
Previous tests and results show that using the hard decision
data in the feedback (real decision-directed mode) degrades
the performance (from the optimum reference-mode) with a
big margin. Iterative equalization through the decoder aims to
enhance these hard decisions by feeding back the decoder
enhanced results.
There is no doubt that iterative structure will increase the
complexity and latency of the system as each iteration should
wait for the entire block of data (interleaver block) to be
deinterleaved and decoded then interleave and encode the
decoder output again to input to the equalizer. System model
for one direction is shown in Fig. 9.
Fig. 10 shows the BER for the 64QAM constellation with
the equalizer iteration increase. Simulation results show a
great enhancement, but we notice that zeroth iteration always
limits the performance, i.e., if the BER of the zeroth iteration
is less than 10
-3
, a great enhancement can be achieved with
iterations; otherwise, no improvement can be achieved.
The BER curves for all data rates for the iterative structure
of the bi-directional Kalman-DFE are shown in Fig. 11.
Fig.8. Bi-directional Kalman-DFE, decision-directed, 72 frame
interleaver size,
=11,
=5, Doppler=1 Hz, delay spread=2 msec.
Fig.9. Iterative Kalman-DFE structure (one direction).
Fig.10. Iterative Kalman-DFE with iterations, decision-directed, 64QAM,
72 frame interleaver size,
=11,
=5, adaptation rate=0.91 for first iteration
and 0.93 for other iterations, Doppler=1Hz, delay spread=2 msec.
Fig.11. Iterative Bi-directional Kalman-DFE, decision-directed, 72 frame
interleaver size,
=11,
=5, Doppler = 1 Hz, delay spread = 2 msec.
VII. CONCLUSIONS
In this paper, we treated the ITU-R Poor channel for the HF
high data rates using directly adapted fractional spaced
Kalman-DFE with two major enhancements in the structure.
The bi-directional structure successfully prevents the
performance break down at the higher constellation and
enhances the overall performance for all rates. In the bi-
directional structure, a real diversity is achieved when
adapting the equalizer in the reverse direction. The iterative
structure then enhances the performance significantly (about
6dB gain at 1e-5 BER for 64QAM using three iterations). The
iterative structure enhances the reliability for the feedback
data by replacing the hard decisions with the decoder feedback
data.
We can notice that the proposed solution is much simpler
than turbo equalizers as it uses direct adaptation structure
(eliminates the use of matrix inversion) and also uses the less
complicated iterative technique to enhance the reliability of
the feedback data.
Finally, both structures (bi-directional and iterative) can be
used also with indirect adaptive equalizers (adaptation through
channel estimation) and should improve its performance.
A
CKNOWLEDGMENT
Some parts of this research were a part of a project in
MTSE Company. We are greatly thankful for the CEO Essam
El Aasar and the department director Ehab Samy.
R
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