Communication Theory
System Design for Bit Interleaved Coded Square
QAM with Iterative Decoding in a Rician Fading
Channel
H ENRIK S CHULZE
University Paderborn, Division Meschede
D-59872 Meschede, Germany. [email protected]
Abstract. One-step iterative decoding for bit-interleaved coded QAM with conventional Gray mapping
can give a significant improvement of performance for fading channels. Furthermore, iterative decoding with
ideally known feed back bits is easier to analyze theoretically than non-iterative decoding. In this paper, we
derive analytical expressions for the pairwise error probabilities for bit-interleaved coded QAM with correct
feed-back bits. They are used to obtain union bounds for the bit error rate. Numerical simulations show that
the performance with only one feedback step comes very close to these ideal theoretical curves. They are
therefore a reasonable guideline for system design to choose the right code rate and modulation level for bit
interleaved coded QAM in a fading channel.
1
I NTRODUCTION
The need for transmitting higher and higher data
rates over band limited fading channels lets system designers venture upon higher level modulation
schemes. As an example, 16-QAM and 64-QAM
are part of the standards for DVB-T [1] and HIPERLAN/2 [2] at least as possible options - in addition to
the the well established and robust standard 4-QAM
(QPSK). Mainly because of pragmatic reasons of easy
and flexible implementation, standard convolutional
codes have been chosen together with conventional
Gray mapping. Both systems use OFDM with symbol
interleaving in frequency direction. For higher level
modulation however, this would not be sufficient, because a deep fade of one QAM symbol would influence several adjacent bits in the coded data stream. To
avoid these error burst, an additional bit interleaver
has to be introduced. This makes DVB-T and HIPERLAN/2 maybe the first two systems that have implemented the general concept of bit interleaved coded
modulation (BICM) [14, 3]. It has been observed and
theoretically founded [3] that, for fading channels, the
general concept of trellis coded modulation (TCM)
with Ungerboeck set partitioning that combines coding and modulation is inferior to the simpler approach
that treats both matters separately. Besides these theoretical reasons there are many practical benefits of
Submission
BICM: There is a high flexibility in the code rate
using only one decoder and punctured convolutional
codes, and it offers the possibility to adjust the coding
to the transmission rate and the channel.
Li and Ritcey [11] have demonstrated for the case
of bit-interleaved coded 8-PSK that iterative decoding (ID) improves the performance. Advantage can
be taken from the knowledge of correctly decoded
bits from preceding decoding steps. However, for
their iterative decoding approach they use a hybrid
set partitioning instead of the conventional Gray mapping. This provides a higher gain with iterative decoding, but it is inferior without it. For this reason,
and because it is implemented in existing transmission sytems, we concentrate ourselves on Gray mapping, even though the gains due to ID may be smaller.
In this paper, we investigate bit-interleaved coded
QAM (BICQAM) with square constellations and the
and the code
trade-off between modulation level
rate
to choose the best combination of both for a
given spectral efficiency. For BICQAM with iterative
decoding, we derive an exact expression for the probability
of an error event of weight that can be
used to calculate union bounds for the bit error rate
(BER). This expression uses the polar representation
of the Gaussian probability integral like described in
[7] to average over the fading and all possible combinations of bits to get an expression for
that leaves
1
H. Schulze
ENC
CH
!
MCU
DEC
ENC
Figure 1: Block diagram of the system model.
only one simple integral that can be easily computed
numerically. Using the same method we can also
calculate the so-called expurgated union bounds (EX
bounds) for
that have been obtained by Caire et
al. [3] by using an inverse Laplace transform method.
These may be used to obtain tight bounds for the BER
without iterative decoding.
A comparison of both types of BER curves gives
an estimate of the possible gain that can be obtained
by iterative decoding. ID improvements turn out to be
significant for low code rates, but small for high code
rates. However, numerical simulations show that the
ID union bounds for the bit error rate is much tighter
than the EX union bound, so that the gain by iterative decoding will be overestimated by this method.
It can be shown that only one additional iteration step
(with hard decision bits) is sufficient to reach practically the ideal ID curves for correctly fed back bits,
so the complexity of iterative decoding is quite low.
This paper is organized as follows: In section 2 we
explain the system model and the notation to be used.
Optimum and sub optimum receivers and their metric
computations are discussed. In section 3 we derive an
expression to calculate bit error rates for BICQAM.
We compare the bit error curves with computer simulations and discuss their relevance for system design.
In section 4 we draw some conclusions.
2
2.1
S YSTEM
MODEL
T HE TRANSMISSION
CHAIN
#
2
$&%('*),+ .-0/21436587:96; 5=<>?@7BAAA*7 $DCFEHG
5% ?
5 %F$IC=E
9 $
% % JLK0M N 7O5 % ?P7 E 7BAA*7 $QCRE
9
/
(1)
determine which ASK-symbol will be transmitted.
We regard it as convenient to interpret a QAM symbol
as a two-dimensional real symbol instead of a onedimensional complex symbol. Let
denote the sequence of ASK symbols. Then
is the
sequence of inphase symbols and
is the
sequence of the quadrature component symbols. Each
ASK-symbol can take the values
7 TS 7 TU 7VA*AA
7 PW 7 OX 7BAA*A
<ZY[- % ]> \_^`7a\_b,^`7BAA*A7a\c3 CRE ;d^ G (2)
of the signal constellation Y . Here we have introduced
a distance unit ^ which is related to the symbol en maps $e% ')H+ bits
ergy. The
on a real
gfa 7 7VA*symbol
AA7 *h mapper
symbol . The Gray
mapping can be written as
h
j
g
a
f
*
h
- 7 7VA*AA*7 i1 % 3 ClE ; dqsmgnpo r ^`A
k8f
(3)
Figure 2 shows a Eut@C QAM configuration with this
mapping.
We denote the energy per (two dimensional)
QAM-Symbol by
and the energy per data bit by
. The relation between both is given by
xv w
(4)
vxwZ%s@')H+ 3 ; vzy
r ^ 7 E ?H^ 7p{ r ^ 7VA*AA
One can easily show that v w %
for 4-QAM, 16-QAM, 64-QAM,... , etc.
The sequence of ASK symbols will be written
as a (row) vector |}% 3 7 7 S 7VA*AA~;€ . We consider
a discrete channel CH given by
 %ƒ‚„|„…‡† A
(5)
We consider
-QAM constellations that are
Cartesian products of two -ASK constellations for
the (inphase) and the
(quadrature) component.
Each ASK symbol is labeled by
bits.
The block diagram for the transmission chain of
our system model is shown in figure 1. The encoder (ENC) with code rate
produces an encoded bit stream . For practical examples, we will
use rate-compatible punctured convolutional (RCPC)
codes like described in [13]. The bit interleaver
"
is a pseudo-random permutation of the time index
together with a pseudo-random serial-parallel (S/P)
conversion. Both are assumed to be statistically independent. This block is given by a random index
map
that chooses for each time index
of the encoded bit a new time position with index
and a labeling position
for the
Gray labeling (
means LSB,
means
MSB) . For each time index , the bits
vxy
ETT
System Design for Bit Interleaved Coded Square QAM with Iterative Decoding in a Rician Fading Channel
% c(0)
c(1)
c(0)
c(1)
1
1
2
2
to be fed into
The optimum metric for
the Viterbi decoder is given by the log-likelihoodratio (LLR)
4
0 1 0 0
3
1 1 0 0
1 0 0 0
0 0 0 0
3 % ?Oœ ; 7
(9)
3 %E œ ;]ž
see [5]. The probability 3 % œ ; that the transmitted bit at label 5 has the value under the condition that has been received may depend on hard
or soft decision values from other bits ‘Ÿ* 7T5– l% ¡ 5
that are known from preceding decoding steps. This
means that the constellation points may have different apriori probabilities 3 ; . Let Y f ‘ and Y be the subset of the the constellation corresponding to
% ? and ` %E , resp., and
Figure 2: 16-QAM with Gray mapping.
V©
E
E
 is the vector of received symbols and † is the white ¢ 3 œ Ž ; % £ ‰ f¥¤B¦L§¨ C ‰ f œ C Ž œ
%Š‰ f`‹ r in the probability density for under the condition (10)
Gaussian noise vector with variance ˆ
that
each real component. The RF signal-to-noise ratio has
been
transmitted
over
a
channel
with
(ideally
(SNR) is given by
known) fading amplitude Ž . The LLR is then given
by
Œ ‰F% vxw
(6)
‰f
™ %s')H+Zšxªs« q­¬]® ¯mgnpo ¢¢ 3 œ Ž ; 3 ; A (11)
The fading is described by the diagonal matrix ‚ .
q­¬]® Mmgnpo 3 œ Ž ; 3 ; ž
«
ª
The diagonal is the vector of (real) fading amplitudes
3Ž 7:Ž 7Ž S 7BAA*A~;€ . The fading amplitudes are normal- In practice, the maxlog- approximation
ized to average power one. We have incorporated per™ ±
°³²µ´ ')H+ 3 ¢ 3 œ Ž ; 3 ;p;
fect phase estimation into the model. We can thereq¬]® ¯¦mgnpo
«
fore work with real quantities. We assume indepenC ²¶q­¬]´® M¦mgnpo ')H+ 3 ¢ 3 œ Ž ; 3 ;p;A (12)
dent Rician fading amplitudes.
«
can be used. If no apriori- information is available,
2.2 R
3 ; %·E ‹ for all . If hard decisionr values for
the other bits are fed back, 3 ; %¸E ‹ for exactly
On the receiver side, the counterpart to the symbol
f <“Y f and one point in <“Y , and
mapper is the metric calculation unit (MCU). For one point
each received symbols , it generates the metric val- 3 ; % ? for all other points. This is just the same
situation like for a usual binary decision between two
ues ‘ , i.e. the soft decision values of the bits points f and . The LLR is then given by
corresponding to labeling positions 5’<“>u?P7 E 7VA*AA7 $=C
r f f … ©
™
by
E]G . The soft metric values ‘ are de-interleaved
%
Ž3 C ; ¨ C Ž r A (13)
the inverse permutation -P36587:96; i1”/ % 3•57:9•; .
‰
f
The de-interleaved metric values are then given by
The distance œ f C œ however, is time varying, be % J–K0M N % T A
(7) cause the values of the other bits change for different 9 . This time-varying distance behaves just like another
fading amplitude. The knowledge of this quanThe decoder DEC will typically be a Viterbi decoder
tity
(if
available) occurs in the metric as a multiplicathat decides for the bit sequence with
tive weighting factor and improves the performance,
j 3 ClE ; y•— %=$ Ž / $Z˜O$ A
(8) just like the knowledge of the channel state informa
tion for fading channels. Soft outputs of the decoder
™ s% ')H+›š
2
x2/δ
1
0 1 1 0
1 1 1 0
1 0 1 0
0 0 1 0
0 1 1 1
1 1 1 1
1 0 1 1
0 0 1 1
0 1 0 1
1 1 0 1
1 0 0 1
0 0 0 1
0
−1
−2
−3
−4
−5
−4
−3
−2
−1
0
x1/δ
1
2
3
4
5
ECEIVER MODELS
The decoded data bits may be used or re-encoded, interleaved and fed back as additional side information
for the next decoding step.
Submission
can be incorporated into the LLR in a straightforward
manner. We will not investigate this further, since
hard feedback already gives very good results (see below).
3
H. Schulze
% Instead of using the optimum LLR metric (11) or
the approximation (12) for
one may use a
soft threshold receiver (TR) metric that is just a directed distance to a threshold. A hard decision receiver compares the received symbols with thresholds
and makes decisions based on the sign. The soft decision threshold receiver (TR) uses the distance to the
threshold as soft decision value. For Gray mapping,
just determines the sign of , so
the MSB
is a reasonable soft decision variable
for the AWGN channel. For a fading channel, we
may use
. For the next less significant bit
, there is a threshold at
if
and at
if
. We
use the metric
for the AWGN
channel and
for the fading
channel. We eventually get the recursive equation for
the metric
*h *h % *h % Ž *h *h % ? %(C º ^ *h
*h % œ œ C º ^
h % Ž œ œ C Ž º ^
%¹…›º ^
%(E
*h » % œ *h » œ C Ž r ^ (14)
»
for ¼½%IE 7 r 7BAA*A7 $·CƒE . This straightforward thresh-
old metric has been used by Morelos-Zaragoza et al.
[12] for the analysis of multilevel coding with block
partitioning in the AWGN channel and, for the special
case of bit-interleaved 4-ASK, by the author [9].
% {
Figure 3 shows the values obtained from 3 different metric expressions for
and SNR=6 dB.
The maxlog approximation is very close to the optimum and becomes practically the same for higher
SNR values. For the LSB, the maxlog and TR curves
are identical. For the MSB, the TR metrics underestimates the reliability for the most reliable values of .
LLR
0
−3
−2
−1
0
yl /δ
1
2
3
4
10
LLR
5
0
SNR=6 dB
Threshold
Maxlog
Optimum
−5
−3
−2
−1
0
yl /δ
1
BIC-
AN
EXPRESSION FOR PAIRWISE ERROR
PROBABILITIES
|À
Ž
À 3 | 1¿
|¾ ;
‚
|
3| Á
1 | ¾ œ ‚ ; % rE V¤ ÂpÏÄÆÅÇÉË ÈÊ { ‰ E f
Ê
j C À ÌÍ
œ A
k Ž œ
(15)
We follow the method described in [7] and use the polar representation of the Gaussian probability integral
(see [8])
JHÐ £
Î
rE ¤BÂ:ÃÄ % }
E Ï f ¤B¦–§
and get
J,Ð × 3 | 1Á| ¾ ; % E Ï f zØ
k
Î
C¨ ÑpÒÓ Õ Ô ©Ö Ô
(16)
E œ ÑpÒCÓ À Ô œ © Ö Ô A
¨ { ‰ f¶Ù
(17)
Here we have used the abbreviation
N OTATION
VARIABLES
−2
2
3
4
Figure 3: Comparison of metric expression for 4-ASK at
SNR=6 dB: LSB (upper) and MSB (lower).
4
PROBABILITIES FOR
Before we can calculate , the probability of an
error event with weight , we need an expression for
the pairwise error probability (PEP)
of the
error event that the sequence of symbols has been
transmitted but the receiver decides for the sequence
of symbols . For a fixed matrix of fading amplitudes , the conditional PEP is given by
3.2
2
−10
−4
3.1
E RROR
QAM
Ø 3Î Ú
(18)
; - % ÝÛ ÜßÞ B¤ ¦–§ 3 C Î Ž ;Bà
Î
©
% E݅‡E݅‡áIá … Î B¤ ¦L§ ¨ C E݅âá áI… Î
for the averaged exponential of the Rician fading amplitude Ž with Rice factor á , see [15].
4
−4
−4
3
AND DEFINITION OF RANDOM
To calculate the error probabilities , recall that
the encoded bit stream will be randomly interleaved
and serial/parallel converted in a random manner
leading to a random bit labeling. This is given by
a random index map
that chooses
for each time index of the encoded bit
a labeling position
. We consider a sequence of encoded bits and ask for the probability
for an error event of length
that the receiver decides for a wrong code word that
differs from in exactly positions. Let and be
the corresponding symbol sequences. We assume that
they also differ in exactly positions. This is justified
by the assumption of an sufficiently large interleaver.
For a fixed code word , , as a function of the random map , is a random variable. Let
be
e-c/‡1 •3 57:9•; /
$¸CãE]G
ä 5â<ã>?@¾ 7VA*AA*7 å ‡%æ 3 ä 1 ä ;
ä¾
ä
| |¾
ä |
% ETT
System Design for Bit Interleaved Coded Square QAM with Iterative Decoding in a Rician Fading Channel
one of the bits of the error positions under consideration. Then depends not only on the given value of
, but also on the random variable which labels
the position of the bit between LSB and MSB.
labels a subset
of size
, where the value of
is determined by the value of the random map .
The choice which
will actually be transmitted depends on the (random) value of the
other bits
. These random variables are
also functions of the big random map . For simplicity and without loss of generality we assume that the
error positions correspond to with
and
redefine and as vectors of finite length . We define
as the Cartesian product of all the sets
and write
if
for
. An
erroneous decided symbol is an element of the complement of
inside the signal constellation, i.e.
. Again we
define the Cartesian product and write
In the following, we have to average over all possible
given by all possible random maps
. We will write
for this average. Practically,
for each time index , this means averaging over all
possible labeling positions and all possible values
of the other bits.
be expurgated (still remaining valid) by ignoring all
error events that are not corresponding to the nearest
neighbor
of each
. We write
for this unique nearest neighbor sequence to for a given map .
is thus already
bounded by
À < Y À 3 ç;
À <sY 3
ç;
|F¾ %·‰‰ 3 | 7p
ç;
r‹
|
Y 3­
ç;
å
5
<èY 3
ç;
l÷ %sÛ ï N J Þ, 3 | ¿
1 | ¾ ;Bœï ô kÕößöz ï N J àxA (21)
é
$
ê
C
E
» 7 ¼ % ¡ 5
To illustrate this idea for our C ASK constellations,
assume for example that a bit % ? is an MSB, i.e.
CsE . This means <=> … ^`7VA*AA7 … 3 CsE ;d^ G
5 % $·
9 %QE 7BAA*A7 À
<É> C ^`7BAAA*7 C 3 CQE ;d^ G . The expurgated
À
| |¾
and
union bound would only take into account %éC ^
Yë3
ç;
Y 3
ç; and ignore all possible other wrong symbols À %
| <›Y23­
ç; <ìY 3
ç; 9 %E 7BAA*A7 C bH^]7 C ^]7BAA*A
For both expressions òó and , we have to
calculate PEPs 3 | 1Á| ¾ ; for an error sequence that
À < Y À 3
ç; Y % 3­
ç; YîíuY 3­
ç;7O9ì<¸> E 7BAA*A7 PG À
is uniquely determined by the transmitted sequence
| ¾ < Yë3
ç;
| and then have to average over the transmitted
seÀ
r
quence.
Using
the
fact
that
the
%
C
are
- ofœ the index
œ ‹ ),
| <Y23­
ç; J
i.i.d.
random
variables
(independent
9
Û2ï N
we get for the averaged equation (17)
9
Ð
5
ò aó %ƒÛ JHÐ ï N J > 3 | 1Á| ¾ ; G
(22)
Ð
© ÖÔA
E
òó Ø E ÑpÒ Ó Ô %
Û
‰
Ï f
¨ fÙ
3.3 G
Ð random variable takes different
the (ideal) iterative decoder (ID) the other bits For ID and EX the
» For
values, and Û òaó denotes the respective expecta7 ¼ % ¡ 5 are known thanks to successful prior de- tion
values.
coding steps. This
means
that
for
each
,
there
is
À À
only one unique < Y 3­
ç; and the problem reduces
to the case of antipodal transmission,
but with a timeÀ r
variant amplitude œ C œ ‹ that is known to the re- 3.4 16-QAM
ceiver. This is similar to an additional multiplicative
%
We first illustrate our result for the example
fading amplitude with a statistics given by the statis
G
(16-QAM)
where
.
The
bit
under
<
]
>
_
\
`
^
a
7
_
\
,
b
^
{
tics of . We write |ð
¾ %(ñx‰ 3 | 7p
ç; for this unique
*f
antipodal sequence to | for a given map . The prob- consideration is either the MSB , or the LSB ,
ability of an error event of length is now given by both with the same probability 1/2. Consider an MSB
of value 0. Then %é… bH^ for gfa % ? and %
the average
…
for gf % E . For the ID case ( gfa known), the
J
J
^
À
%ãl òó %ãÛ ï N ÞH 3 | 1Á| ¾ ;Bœï ô kÕõ8öz ï N à (19) erroneous symbols are % C bH^ for gf % ? and
À %éC ^ for gfa %éE , resp., corresponding to %
Without knowledge of the other bits the optimum
receiver must use the metric (11) or (12) that depends bH^ and e% ^ , both with equal probability. For the
on all the points in the constellation. Because it is too EX case, we have À to consider the nearest erroneous
difficult to deal with this metric for calculating error symbol, which is %4C ^ in any case, leading to
r
probabilities, one can use union bound methods that (%
^ and (% ^ , both with equal probability. If
need only pairwise error probabilities. is upper the bit under consideration is the LSB, Á% ^ for
ID and EX and any value of the MSB. It follows that
bounded by the union bound expression given by
¹% ^ with probability
3/4 (ID and EX) and Q% bH^
r ^ (EX) with
üJFýû j
(ID) or %
probability 1/4. We thus
l÷ ù øú %ãÛ ï N
J 3 | 1Á| ¾ ;]ÿ (20) have
ï ô ¬ßþ® Û òaó _Ø © % b _Ø ^ © … E zØ ^ ©
Caire et al. have shown [3] very generally that for
{ ¨
{ ¨ (23)
¨
constellations with Gray mapping, union bounds can
5
ENERAL FORMULAS
AS A FIRST EXAMPLE
Submission
5
H. Schulze
% Hb ^ ÕC % ^
Û
Ø ¨ © % {b Ø ¨ ^ © … {E Ø ¨ ,{ ^ ©
j % z… (31)
{T3 ;d^ A
(24)
k
for the EX case. Here we have used the abbreviation
"! - %ƒ‰ f ÑpÒÓ Ô A
!
!
There
are
such
sequences,
and each occurs with
(25)
S
W . Averaging over all possible
probability W
symbols with and
symbols with has been transmitted so that the SED equals
for the ID case and
Using these expressions, equation (22) can now easily
be evaluated numerically.
For the AWGN channel, a closed-form expression
can be found. For , insert
0
10
(26)
OC
−3
10
9 %E 7BAA*7 œ |ZC | ¾ œ % j œ C À œ % { j A (29)
k
k
À
For the ID case, % œ C œ ‹ r is a random variable
that takes the value % ^ with probability 3/4 and
% bH^ with probability 1/4. We have to calculate
üý
%ãÛ q û rE ¤VÂpÃÄ ÅÇ Ë ÈÊ ‰ E f j ÌÍ ÿ 7 (30)
Ê k
where Û q means averaging over all random variables
. To perform the average, we count all the possible
events. Assume the event that a fixed sequence of
6
Rc=8/12
Rc=8/10
Rc=8/8
Rc=8/24
−5
−6
10
−7
10
−8
10
0
2
4
6
8
10
SNR [dB]
12
14
16
18
20
Figure 4: Bit error rates (union bounds) for 16-QAM for
different code rates in the AWGN channel.
j © ©
E
N
õ
ç
ö
% ¨{
b (28)
k8f ¨
ÌÍ
Er
_
…
3
b
; Ù ‰^ f
Ù ¤BÂ:ÃÄ ÅÇ 0
10
ID
EX
−1
10
16−QAM
Rayleigh
−2
10
−3
10
Rc=8/24
BER
These expressions can also be obtained without
the polar form of the Gaussian probability integral:
Consider an error event where the two code words
differ in positions with time indices
. The
squared Euclidean distance (SED) is then given by
Rc=8/16
−4
10
10
òaó N õçö % ¨ {E © j ¨ © b (27)
k8f
ÌÍ
Er
_
…
; ‰^
3
Ù ¤VÂpÏÄÆÅÇ
Ù f
16−QAM
AWGN
−2
10
into eqs. (23) and (24) and expand the
th powers
of these expressions using binomial coefficients, and
insert into equation (22). Using again the polar form
of the Gaussian probability integral, we finally get the
formulas
and
ID
EX
TR
−1
10
BER
á 1
Î;
Q%
C
3
¤B¦L§
sequences, we get equation (27). For the EX case, the
same method leads to equation (28).
Rc=8/16
−4
10
Rc=8/12
Rc=8/10
−5
10
−6
10
−7
10
−8
10
0
2
4
6
8
10
SNR [dB]
12
14
16
18
20
Figure 5: Bit error rates (union bounds) for 16-QAM for
different code rates in the Rayleigh fading channel.
Using similar arguments, we are able to derive
another expression for
for the soft threshold receiver in the AWGN channel for decoding without additional information about the other bit(s). Consider
an error event where the two code words differ in
positions. For
, let # be the difference of
transmit symbol to the the decision threshold. For
simplicity and without loss of generality, we assume
9 % E 7BAA*A7 ETT
System Design for Bit Interleaved Coded Square QAM with Iterative Decoding in a Rician Fading Channel
0
The probability that this random variable becomes
negative is given by
10
ID
EX
/21
−1
10
16−QAM
Rice (K=6 dB)
−2
10
−3
10
BER
Rc=8/24
Rc=8/10
Rc=8/16
−4
10
Rc=8/12
−5
10
−6
10
−7
10
−8
10
0
2
4
6
8
10
SNR [dB]
12
14
16
18
20
Figure 6: Bit error rates (union bounds) for 16-QAM for
different code rates in the Rician fading channel
dB).
(Rice factor
$&%('
0
10
ID
EX
−1
10
64−QAM
AWGN
−2
10
−3
BER
10
−4
10
Rc=8/24
Rc=8/16
Rc=8/8
Rc=8/12 Rc=8/10
−5
10
r
E
…
3 ?,; % r B¤ Â:ÃÄ ÅÇ 3 ; Ù ‰ ^ f ÌÍ A (35)
3 !
!
!S
sequences
with their reAveraging over all the
W
W
leads to
spective probabilities
(36)
465 N õ îö % ¨ {E © j ¨ © b k8f
r
E
3 _… ; ^ ÌÍ A
r
Ù ¤VÂpÃÄ ÅÇ Ù ‰ f
This formula has been derived in [9]. We note that
the formula in [12] for multilevel coded (MLC) modulation is very similar and has been derived by the
same method. The only difference is that for MLC
and
belong to individually coded
modulation
bit streams and there are different probabilities for the
distances.
Comparing eqs. (27), (28), and (36) it follows
from
)
)
(37)
−6
10
_…
3 … r ; ‹ _… b
that
−7
10
−8
10
gf
10
12
14
16
18
20
SNR [dB]
22
24
26
28
30
Figure 7: Bit error rates (union bounds) for 64-QAM for
different code rates in the AWGN channel.
# *)
òaó N õîö ÷ 465 N õîö ÷ N õçö A
(38)
We note that the reason why the ID receiver is superior compared to TR receiver is because it makes use
of the known value of as a weighting factor.
We have used these three formulas for
in the
AWGN channel to obtain union bounds of the type
? for all 9 . # is a random variable that takes
j
the value # % ^ with probability 3/4 and the value
(39)
y ÷ 87:9<;=; uå
# % b,^ with probability 1/4. Let + be the difference
k
of the receive symbol to the decision threshold, i.e.
+ ,
% r # .… - , where - is real AWGN with variance for the bit error rate y of RCPC coded transmis‰ f ‹ . Given a fixed transmit vector, an error occurs sion with the rate 1/3 memory and 6 mother code
if the random variable/
3 E bHbP7 E?>–E 7 E { ;A@ =B and the error coefficients tabulated by Hagenauer [13]. Figure 4 shows the BER
j (32) curves for the three bounds and code rates %
% +
‹ r O{ 7 ‹ EVt 7 ‹ E r 7 ‹ E ? , and for uncoded transmisk
sion. We observe that the three bounds lie very close
becomes negative.
Assume the event that a fixed setogether at relevant BER values for code rate 8/16 and
quence of/ symbols with # % b,^ and C
symbols
higher.
with # % ^ has been transmitted. For such a fixed seFigure 5 shows the BER curves in a Rayleigh fadquence, is a Gaussian random variable with mean
value
0 % bH^ … 3 lC ;d^ % 3 … r ;d^
and variance
ˆ 0 %ƒ ‰ r f A
Submission
(33)
(34)
ing channel for the ID and EX bounds and the same
code rates as above. We note that for this channel, the
gap between both curves becomes larger, indicating
that iterative decoding may give some noticeable gain
in performance. Figure 6 shows the same curves for a
DC .
Rician channel with Rice factor
áé%ãt2
7
H. Schulze
3.5
S YSTEM
-QAM
DESIGN FOR GENERAL
0
10
ID
EX
CONSTELLATIONS
−1
10
5 ·% % ?
rh ^
% ^
5 % E rh rh % b,^ 5 % r r h S
¹% ^ % bH^ % ^
% ^
r h … r h … AA*A … r …R E¥%
ð
%
^
r h CFE
r h Cƒ E
% b,^
rh C
ð% ^
% ^
E
/ = ?
r
r
r
ã
…
E
…
C
E ;p^ G r h C„<.E >L3 ;p^`7Vr 3 h b,;p^`7BAAA*7u3
E ‹ 3$ Ù ;
% {O7 Eut 7 t {P7 r ]t
¹% ^`7 r ^`7bH^`7VA*AA % {P7 Eut 7 t {P7 r Ht
0
10
64−QAM
Rice (K=6 dB)
−2
10
−3
BER
10
−4
10
Rc=8/16
Rc=8/24
Rc=8/12
Rc=8/10
−5
10
−6
10
−7
10
−8
10
10
12
14
16
18
20
SNR [dB]
22
24
26
28
30
Figure 9: Bit error rates (union bounds) for 64-QAM for
different code rates in the Rician fading channel
dB).
(Rice factor
$,%('
0
10
No Iteration (sim.)
One Iteration (sim.)
Ideal (sim.)
ID Bound
EX Bound
−1
10
64−QAM, Rc=1/2
−2
10
Rayleigh
BER
% ?
5
5% ?
5 %s$½C E
‹r % rh $ÉCðE
r‹
E‹$
E ‹ 3$ Ù r h ;
We now calculate the probabilities for the possible
values of that may occur. Let the relevant bit
have a fixed value, say
. Depending of the labeling position between
(LSB) and
(MSB), which is a random variable, it defines a certain subset of
configuration points. It
depends on the random values of the
other
bits which of these
point is the actually transmitted one. Consequently, each labeling position will
be chosen which probability
and each of point in
the subset then with probability
. Each
of these events corresponds to a certain value of ,
but some of them are equal. We now count their occurance.
, all the
We first consider the ID case. For
points in the subset correspond to . For
,
points in the subset correspond to and the other
points in the subset correspond
to . For
,
points in the subset
correspond to ,
, E , and F> ,
respectively, and so on. Summing up all events, we
find that occurs
times. The value occurs
times, the values E and E> occurs
times, and so on. We find that for any integer IH G ,
the values for occur
times. To calculate probabilities,
we have to multiply by
.
Table 1 shows the probabilities for the possible
. For the EX case,
values of for
the same probabilities occur, but takes the values
.
.
−3
10
−4
10
−5
10
12
13
14
15
SNR [dB]
16
17
18
Figure 10: Comparison of the theoretical bounds with simulation results for 64-QAM and
and
the Rayleigh fading channel.
JKL%&M8NPO
ID
EX
−1
10
á % t2
æ
_}% ‹ r {P7 ‹ EVt 7 ‹ E r 7 ‹ E ?
64−QAM
Rayleigh
−2
10
−3
BER
10
Rc=8/24
−4
10
Rc=8/16
Rc=8/10
Rc=8/12
−5
10
−6
10
−7
10
−8
10
10
12
14
16
18
20
SNR [dB]
22
24
26
28
30
Figure 8: Bit error rates (union bounds) for 64-QAM for
different code rates in the Rayleigh fading channel.
Figures 7,8,9 show the ID and EX curves for 64
QAM and the AWGN, the Rayleigh fading channel,
8
and the Rician fading channel with factor
QC
for code rates
. The
gap between the ID and EX curves becomes larger
than for 16-QAM. This can be understood from the
fact that in a larger constellation, more useful information can be gained from the successful decoding of
the other bits.
The question arises if the ID curves for iteration
with ideal knowledge of the other bits reflect a real
situation where there can be bit errors that may influence further iteration steps. One must also ask how
many iterations are necessary. We have carried out
numerical simulations for several code rates and several values of
. Figure 10 shows as an example
a simulation for 64-QAM and code rate
in comparison with the theoretical ID and EX curves.
We have simulated the first decoding step without iteration (stars), and then one additional iterative de-
_}%ÉE ‹ r
ETT
System Design for Bit Interleaved Coded Square QAM with Iterative Decoding in a Rician Fading Channel
¹%
%% EV{ t
% %ãr t ]{ t
^
bH^
^
^
Table 1: Probabilities for
1
3/4
7/12
15/32
1/4
3/12
7/32
1/12
3/32
>
R
1/12
3/32
1
^
(ID).
1/32
E,E ^ E b,^ E? ^
1/32
1/32
1/32
1
10
10
bits per symbol
4−QAM
16−QAM
64−QAM
256−QAM
bits per symbol
4−QAM
16−QAM
64−QAM
256−QAM
3.2 bits per symbol
1.6 bits per symbol
1.33 bits per symbol
0
10
0
0
5
10
15
SNR [dB]
20
25
30
MTSVUXW
Figure 11: SNR needed for BER=
for different spectral efficiencies (bits per symbol) in the AWGN
channel.
coding step using only the decoded hard decision values for the information from the other bits (circles).
A third curve shows the iterative decoding with ideal
knowledge of the other bits (squares). The first curve
is tightly bounded by the theoretical EX curve, but
there is still an observable gap of about 0.3 dB at
. The third curve is extremely tightly
bounded by the theoretical ID curve at relevant BERs
). The ef(much less than 0.1 dB below
fective gain due to iterative decoding is about 0.5 dB
at
, which seems to be worth enough to
carry out this one iteration. Our most important simulation result is that the second curve with one hard
decision iteration lies between these curves at relevant BERs. We conclude that indeed the theoretical
ID curves may serve as a guidance for practical system design for the lowest complexity iterative decoding with only one hard decision step. For our simulations, we have used the maxlog-approximation (12)
of the optimum metric. We also simulated the soft
threshold receiver (TR) metric which leads to a very
slight degradation in the performance.
We have evaluated the ID bounds for code rates
8/24, 8/23, ... , 8/9, and for uncoded transmission
to get a diagram of the spectral efficiency (in bits per
QAM-symbol) as a function of the SNR that is needed
for a certain BER. Figure 11 shows this diagram for
the AWGN channel and a required
and
y %±E ?@ W
åy›% E ?@ W
y %ŠE ?@ S
y % E ?@ U
Submission
10
0
5
10
15
SNR [dB]
20
25
30
MTSVUZY
Figure 12: SNR needed for BER=
for different spectral efficiencies (bits per symbol) in the Rayleigh
fading channel.
% {P7 EVt 7 t {O7 r ]t . We note that the lower level
modulation scheme performs better than the higher
level scheme at almost all spectral efficiencies. At 2
bits per symbol, there is a gain of approx. 2.5 dB for
the rate 1/2 coded 16-QAM compared to the uncoded
4-QAM. But this gain is quite poor compared to even
the simplest TCM schemes. This is not surprising,
since BICM does not maximize the squared Euclidean
distance which is needed to get an optimized transmission scheme for the AWGN channel.
For the Rayleigh fading channels, things become
different. Figure 12 shows this diagram for the
Rayleigh channel and a required
and
. We observe that 16-QAM performs always better than 4-QAM at the spectral efficiencies under consideration. For 1.33 bits per symbol, the lowest spectral efficiency for 16-QAM that
[ ),
can be achieved with our code family (
there is still a gain of approx. 1.6 dB compared to
4-QAM with
, . Even without iterative decoding, using the EX bound, we see from figure 5 that
16-QAM is still at least 1 dB better than 4-QAM. At
1.6 bits per symbol, i.e. 16-QAM with
\
and 4-QAM with
] , the gain of 3.7 dB is
even more significant. We conclude that it is always
favorable to use 16-QAM instead of 4-QAM for spectral efficiencies above 1 bit per symbol. As a rule of
thumb we can state that one should avoid weak codes
% {P7 EVt 7 t {O7 r Ht
_% ‹ E r
_Æ% ‹ E ?
y%³E ?– W
% ‹r{
_’% ‹ r ?
9
H. Schulze
% ‹E?
^ . It is better to use a higher level modlike
ulation scheme instead. For example, at 3.2 bits per
symbol, 64-QAM with
_ ` performs 2 dB
better than 16-QAM with
a . Even 64-QAM
with
b performs more than 0.5 dB better
than 16-QAM with
at a slightly better
c
spectral efficiency. We note that 256-QAM has no
advantage for less than 5 bits per symbol.
_.% ‹ E
Ý% ‹ E ?
_% ‹ EVt
½% ‹ E,E
Comparing Figs. 11 and 12 with Figs. 4-7 of
[3], we observe that our numerically evaluated values are quite close (less than 1 dB) to the cutoff
rate, which therefore seems to be helpful as a ruleof-thumb guideline for system design. There is still
a gap of approx. 3 dB between the capacity curve as
the theoretical limit and and our values.
R EFERENCES
[1] ETS 300 744. Digital broadcasting systems for television, sound, and data services; Framing structure,
channel coding and modulation for digital terrestrial television. European Telecommunication Standard, 1997.
[2] ETSI TS 101 475 Broadband Radio Access Networks
(BRAN); HIPERLAN Type 2 Physical Layer. European Telecommunication Standard, 2000.
[3] G. Caire, G. Taricco, E. Biglieri. Bit-Interleaved
Coded Modulation. IEEE Transactions on Information Theory, Vol. IT-44(3), pages 927-946, May 1998.
[4] J. Hagenauer. Fehlerkorrektur und Diversity in
Fading-Kanälen. AEÜ Vol. 36, pages 337-344, 1982.
[5] J. Hagenauer. Source-Controlled Channel Decoding.
IEEE Trans. on Communications, Vol. COM-43(9),
pages 2449-2459, Sep. 1995.
[6] A. Proakis. Digital Communications,
McGraw-Hill, New York, 1995.
4
C ONCLUSIONS
Iterative decoding for bit-interleaved coded QAM
with conventional Gray mapping provides a signifiand less).
cant gain at low code rates (for
Numerical simulations show that only one iteration
with hard decision feedback bits is then necessary to
obtain practically the full gain of ideal error-free feedback bits. Therefore, because of the low complexity, it is recommended to make use of iterative decoding. For theoretical investigations, exact expressions for the pairwise error probabilities for ideal iterative decoding can be obtained instead of only (expurgated) union bounds for non-iterative decoding. It
is therefore reasonable to base the system design on
the theoretical iterative decoding performance curves:
For low code rates the one iteration gives a significant gain and it should be carried out. For high code
rates the gain is small, so the performance curve reflect also the situation without iterative decoding. The
theoretical curves for iterative decoding can be compared with the expurgated union bounds to estimate
the iteration gain.
_ ° E ‹ r
have produced performance curves for
We
C QAM and RCPC codes for several values of and , resp., to show which SNR is needed for these
combinations and to give some hints for practical system design. The theoretical curves are very easy to
produce and may serve as a tool for system designers
that helps to avoid time-consuming exhaustive channel simulations.
Manuscript received on September 13, 2000
10
3rd ed.,
[7] M.K. Simon, D. Divsalar. Some New Twists to Problems Involving the Gaussian Probability Integral.
IEEE Trans. on Communications, Vol. COM-46(2),
pages 200-210, Feb. 1998
[8] M.K. Simon, Mohamed-Slim Alouini, Digital Communications over Fading Channels. Wiley, New York,
2000.
[9] M. Ruf, H. Schulze. Erhöhung der Datenkapazität
bei DAB durch hierarchische Modulation. ITGFachtagung Codierung, Aachen (Germany), ITG
Fachbericht 146, pages 249-257, March 1998.
[10] U. Wachsmann, R.F.H. Fischer, J.B. Huber. Multilevel Codes: Theoretical Concepts and Practical Design Rules. IEEE Transactions on Information Theory, Vol. IT-45, pages 1361-1391, July 1999
[11] X. Li, J.A. Ritcey. Bit-Interleaved Coded Modulation with Iterative Decoding. Communications Letters, Vol. COMML-1, pages 169-171, Nov. 1999.
[12] R.H. Morelos-Zaragoza, M.P.C. Fossorier, S. Lin, H.
Imai. Multilevel Coded Modulation for Unequal Error
Protection and Multistage Decoding-Part I: Symmetric Constellations IEEE Transactions on Communications, COM-48(2), pages 204-213, Feb. 2000.
[13] J. Hagenauer. Rate Compatible Punctured Convolutional Codes (RCPC Codes) and their Applications.
IEEE Transactions on Communications, Vol. COM36, pages 389-400, 1988 .
[14] E. Zehavi. 8-PSK codes for a Rayleigh channel. IEEE
Transactions on Communications, COM-40(5), pages
873-884, May 1992.
[15] E. Biglieri, D. Divsalar, P.J. McLane, M.K. Simon.
Introduction to Trellis-Coded Modulation with Applications, Macmillan, New York, 1991.
[16] S. Benedetto, E. Biglieri. Priciples of Digital Transmission, Kluwer Academic Press, New York 1999.
ETT