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ISlT 1998,
Cambridge
MA
USA.
August 16 August 21
Turbo Codes For Time Varying AWGN Channels With Application
To
FH/SSMA
Hesham
El
Gam al Evaggelos Geranioti s
Dept. of Electrical Engineering Dept. of Electrical
Engineering
Inst itute for Systems Research Inst itute for Systems
Research
University of Maryland University
of
Maryland
College Park, MD 20742
College Park, MD 20742
helgamalOeng.umd.edu evagelosOeng.umd.edu
A b s t r a c t
In t h i s paper
we
invest igate the ap-
plication
of parallel concatenated
convolutional
codes
i.e., Turbo codes)
to
A d d i ti v e W h i t e Gaussian Noise
channels
with
t ime-varying variance. Appli cations of
this
model
include multiple-access interference and
part ia l -band
noise
j amming
in
frequency hopped net-
works.
Three schemes for
evaluat ing
the
likelihood
ra t io
o the
channel
output are compared.
These
schemes provide tradeoffs between simplicity
of
im-
plementat ion,
BER
performance,
and
robustness to
inaccuracies in the measurements.
I.
CHANNELODEL
As in [l], he frequency hopping channel is modeled by a dis-
crete time-varying AWGN channel. The channel adds zero
mean white Gaussian noise with time varying variance U : ) .
We assume perfect symbol synchronization and perfect power
control. Withou t loss
of
generality, we also assume that the
transmitted symbols
dt E
1,
-1 .
Hence, th e log-likelihood
ratio at time
t
assuming prior knowledge of th e channel vari-
ance) is
2Yt
Lt
=
From
l ) ,
we can see the importan ce of the accurate estimation
of th e channel variance in t he Tu rbo decoder
for
two reasons:
1-In
the Turbo decoding algorithm, the log-likelihood ratio
is
updated after each decoding step by the extrinsic
information.
Hence, we need to have an accurate estimate of the channel
variance, even if it does not change with time, t o calculate
the
log-likelihood ratio.
2
The weighting function in the log-likelihood ratio should
be inversely proportional to the noise variance, otherwise
the
highly corrupted symbols will lead to long burs ty error
blocks
at the decoder output.
11. EST~MATIONCHEMES
In this section, we briefly describe three log-likelihood
ratio
estimation schemes. The
first
scheme atte mpt s to take advan-
tage of th e iterative structur e of th e Turbo decoder in
order
to improve the channel variance estimate after each decod-
ing step. In order to use this technique, the variance of
the
channel must be constant over more than one symbol, and
the receiver must have a priori knowIedge of which symbols
were corrupted by noise vector with const ant variance. The
extrinsic information supplied by th e previous decoding
step
is
used
as
a priori probability
in
the variance estimator.
Let
pot
be the probability th at (dt
=
- l ) , and
{yl,yz,
..... }
are the received symbols which are known to be corrupted by
a constant variance noise vector.
A
suboptimum
estimate of
0-7803-5000-6/O8/ 10.001998 EEE.
the noise variance to be used in calculating the
log-likelihood
ratio at time
is
given by
1
n 1
e 8 =
[POt(Yt
1 1 2 1 Ot)(Yt q2] c
l < t < n , t k
2)
where c = 2 1
~ 0 6 ) ~
is a constant added to unbias the
estimator; and
pot =
[eq+%,t)
] -
3)
in which
LeZt is
the extrinsic information provided by the
previous decoding step.
When the receiver does not know which received symbols
are corrupted by a constant variance noise, or when the
noise
variance changes sufficiently fast such tha t each symbol is
af-
fected by
a
noise sample with different variance, we need to
find a robust estimator
for
the log-likelihood ratio such that
the highly corrupted symbols do not affect their surrounding
symbols. The generalized maximum likelihood ratio test is
used
for
obtaining this robust estimator
where TI and ~ 7 re the variance values which maximizep yl1)
and p yl0) respectively. These values are given by:
6 1 = Y t 11;U0 = lyt 11
5)
substituting these values back in 4)we get
lYt
11
Lt
= l o g -
IYt
11
The t hird scheme is th e simplest and is only used for com-
parison purposes. The log-likelihood ratio used is equal to
the
channel output yt.
Upper bounds for the BER achieved by the three schemes
were obtained following [Z]. The reader
is
referred to [3] for
the detailed numerical results obtained through both
analysis
and simulation.
REFERENCES
[I] E. Geraniotis, Multiple Access Capability of Requency
Hopped Spread Spectrum Revisited, IEEE
Pans.
on
Com-
munications, pp.
1066-1077,
ul
1990.
[2] D.
Divsalar, S. Dolinar, and
F.
Pollara, Transfer function
bounds on the performance of turbo codes, Telecom. nd
ata
Acquisit ion Progress R eport
42-122
Jet Propulsion Laboratory,
August 1995.
[3]
H.
El Gamal, and
E.
Geraniotis,
Turbo
Codes
with
Chan-
nel Estimation and Dynamic Power Allocation
for
Anti-Jam
SFHISSMA,
submitted
to MIL OM
98,
October 1998.
457
http://helgamaloeng.umd.edu/http://evagelosoeng.umd.edu/http://evagelosoeng.umd.edu/http://helgamaloeng.umd.edu/
