IEEE COMMUNICATIONS LETTERS, VOL. 3, NO. 1, JANUARY 1999
1
Logical AND Diversity Combining in
FH/SSMA Packet Radio Networks
Jik Dong Kim, Student Member, IEEE, and Sang Wu Kim, Member, IEEE
Abstract— We propose the logical AND diversity combining
in slotted frequency-hopped spread spectrum multiple access
(FH/SSMA) packet radio networks. The proposed diversity
scheme employs the symbol-by-symbol logical AND operation
between the currently received packet and the previously
combined packet. Our results show that the proposed lowcomplexity diversity combining scheme provides a significant
performance improvement over existing diversity combining
schemes.
Index Terms—Diversity combining, frequency-hop spread spectrum, multiple access.
I. INTRODUCTION
P
ACKET combining utilizes the additional diversity available in the previous (or failed) packets to achieve the
minimum number of transmissions for correct decoding. There
are two different approaches in combining packets: symbolby-symbol combining (termed as diversity combining) and
codeword-by-codeword combining (termed as code combining) [1], [2]. In this letter we consider the symbol-by-symbol
combining.
There are a number of papers that study the symbol-bysymbol combining. The combining scheme in [3] assumes that
the receiver knows the presence of multiple access interference
(MAI), and chooses a symbol that has not been hit by MAI. In
[4], soft decision demodulator outputs are added to previously
summed outputs at each combining step, and the symbol
corresponding to the largest is decided as the transmitted
symbol. Li [5] analyzed the performance of ratio-threshold
diversity combining (RTDC) in which Viterbi’s ratio-threshold
test is performed for each symbol, and the quality bit is utilized
in combining. Harvey [6] considered the averaged diversity
combining (ADC) which generates the averaged soft decision
value based on all received copies, and then considered the
weighted diversity combining (WDC) whose weights depend
on the channel state. However, these combining schemes
require a high computational complexity and large memory.
The logical AND diversity combining proposed in this letter
is based on the observation that the desired symbol is not
changed in all retransmitted packets, whereas the interfering
symbols change. Therefore, the desired symbol can be detected
from the symbol-by-symbol logical AND operation between the
newly received packet and the previously combined packet.
Manuscript received May 12, 1998. The associate edtor coordinating the
review of this letter and approving it for publication was Dr. B. R. Vojcic.
The authors are with the Department of Electrical Engineering, Korea
Advanced Institute of Science and Technology, Taejon 305-701, Korea (email: swkim@san.kaist.ac.kr).
Publisher Item Identifier S 1089-7798(99)01261-2.
The desired symbol produces logic “1,” whereas the interfering
symbols produce logic “0” with high probability.
II. SYSTEM DESCRIPTION
We consider a slotted frequency-hopped spread spectrum
multiple access (FH/SSMA) network, where there are
transmitter-receiver pairs communicating over
frequency
Reed–Solomon
slots. Each packet is encoded by an
, and each -ary code symbol is
(RS) code over
-ary orthogonal FSK signals.
composed of
We assume that the frequency hopping is synchronous, and
one code symbol is transmitted per hop. The receiver block
energy detector
diagram is shown in Fig. 1. Each of the
outputs is compared with a threshold, , and logic “1” is
if the detector output
is larger than
generated
, and logic “0” is generated
,
the threshold
otherwise.
The proposed combining scheme is illustrated for the case
and
in Fig. 2. Let
, and
of
denote the th code symbol in the previously combined packet,
the newly received packet, and the newly combined packet,
is the logical AND of
and
.
respectively, where
and
are hit,
As illustrated in Ex. 1, even when both
logic “1” is generated for the desired symbol and logic “0” for
interfering symbols. Thus, the desired symbol can be correctly
detected. When more than one logic “1’s” are generated after
combining, as illustrated in Ex. 2, we declare an erasure.
If we assume that each symbol is transmitted equally likely,
, then the probability of correct
i.e., with probability
detection increases with larger , because the probability that
the interfering symbols are identical (producing logic “1”)
. Also, as the number of combined
decreases with larger
packets increases, the probability that logic “1” is generated by
the interfering symbols decreases, which leads to an additional
performance improvement.
III. PERFORMANCE ANALYSIS
In this section, we derive the packet error probability in
the presence of MAI only. The effect of thermal noise and
fading will be examined by computer simulations. In the
presence of MAI only, the combined symbol can be either
RS code can correct up to
erased or correct. Since the
erasures [7], the packet containing more than
symbol erasures is declared unsuccessful, and a retransmission
, where
is
is requested. If we let
the number of packets simultaneously transmitted in the th
1089–7798/99$10.00  1999 IEEE
2
Fig. 1.
IEEE COMMUNICATIONS LETTERS, VOL. 3, NO. 1, JANUARY 1999
Receiver block diagram.
In (2),
is the conditional probability that a modulation
symbol is erased after combining, given a hit pattern , and
is given by
(4)
and
is the number of logic
where
(see Fig. 1) for the th
“1’s” in
denotes the maximum of , i.e.,
transmitted packet, and
. The conditional probability
that a
tone pattern occurs given a hit pattern , is given by [8]
(5)
is the conditional probability that a modulation
In (4),
symbol is erased after combining given a tone pattern , which
is given by
Fig. 2.
Proposed combining algorithm:
M
= 8;
N
= 1.
transmission,
, then the average packet error
, after combining packets is given by
probability,
(1)
is the conditional probability that a code
where
symbol is erased after combining packets, given , which
can be represented by
(6)
represents the number of logic “1’s” in
(see Fig. 1) after receiving the
th packet
, and
and
are the minimum and maximum of
, respectively.
is the conditional probability
logic “1’s” given
and , which is
that there are
given by [9, p. 60]
where
(7)
(2)
and
is the number of inwhere
is the
terfering users in the th transmission, and
conditional probability that a hit pattern occurs given ,
represented by
(3)
IV. NUMERICAL RESULTS
In obtaining numerical results, we consider the case where
the number of transmitted packets is fixed in all transmissions
), and that the packet length
(i.e.,
is equal to the code symbol size (i.e., extended RS code).
Since ADC in [6] is identical to the scheme in [4] when applied
to FH/SSMA systems, we consider WDC in [6].
KIM AND KIM: FH/SSMA PACKET RADIO NETWORKS
Fig. 3. Packet error probability PE versus the number of users, no background noise, no fading: q = 50; M = 8, (64, 48) RS code.
3
Fig. 5. Packet error probability PE versus E b =N0 in a
frequency-nonselective Rayleigh fading channel with background noise:
K = 50, q = 50, L = 3, M = 8, (64, 32) RS code.
, used in deciding
or 1 (see Fig. 1) is optimized
such that the packet error probability is
for each
minimized. Also, the threshold for Viterbi’s ratio-threshold
test in scheme [5] is optimized. We find that the proposed
low-complexity combining scheme out performs much more
complex alternatives discussed in [3]–[6].
V. CONCLUSIONS
Fig. 4. Packet error probability PE versus the number of users, no background noise, no fading: q = 50; L = 3, (64, 48) RS code.
We proposed the logical AND diversity combining scheme in
slotted FH/SSMA packet radio networks. The proposed lowcomplexity scheme is simple to realize, and gives a (more)
significant performance improvement over the existing much
more complex schemes as the number of combined packets
or the modulation symbol size
increases.
REFERENCES
Fig. 3 is a plot of packet error probability
versus the
number of users for several values of in the presence of MAI
only. Computer simulation is also performed for the proposed
logical AND combining scheme to confirm the accuracy of
the analytical results derived in Section III. We find that the
proposed scheme yields a much lower packet error probability
than existing schemes in [3]–[5]. As discussed in Section
II, the performance improvement is more significant as
increases.
versus the
Fig. 4 is a plot of packet error probability
in the presence of
number of users for several values of
increases, the packet error probability deMAI only. As
creases, as expected. This indicates that the proposed scheme
is more effective with larger .
versus
Fig. 5 is a plot of packet error probability
in a frequency-nonselective Rayleigh fading channel with
background noise. The packet error probability is evaluated
by computer simulations. We assume that fading magnitudes
for different users are independent, and the fading magnitude
is constant during a modulation symbol period. The threshold,
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