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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 46, NO. 11, NOVEMBER 1998
Frequency-Hopped Multiple-Access
Communication with Nonorthogonal
BFSK in Rayleigh Fading Channels
Ho Kyu Choi and Sang Wu Kim
Abstract— A nonorthogonal binary frequency-shift keying is
considered in frequency-hopped multiple-access communication
systems with Reed–Solomon coding. The effect of tone spacing on
the average number of successfully transmitted information bits
per unit time per unit bandwidth (called normalized throughput)
is examined in Rayleigh fading channels. The tradeoff among tone
spacing, code rate, and number of frequency slots in maximizing
the normalized throughput is examined, keeping the total bandwidth fixed. The optimal tone spacing, code rate, and number
b =N0 is
of frequency slots in terms of the number of users and E
also discussed. The throughput gain attained by using the optimal
tone spacing becomes more significant as the number of users is
increased.
I. INTRODUCTION
I
N FREQUENCY-HOPPED multiple-access (FHMA) systems, the total RF bandwidth is divided into subbands
called frequency slots, and there is one carrier frequency
available in each of these slots. The RF signal from a given
transmitter is hopped from slot to slot by changing the carrier
frequency. Frequency hopping (FH) decorrelates the fading
process from hop-to-hop over the span of the interleaver or,
equivalently, reduces the duration of long fades [1]. Therefore, the burst errors caused by slow fading are randomized
by means of the FH, and the application of random errorcorrecting codes becomes effective [2].
In this paper we consider nonorthogonal binary frequencyshift keying (BFSK) with Gaussian pulse shaping in FHMA
communication systems that employ Reed–Solomon (RS)
codes. Gaussian pulse shaping is adopted in FH-type wireless
local area networks standard (IEEE 802.11) [3]. In general, as
the tone spacing decreases, the system is more vulnerable to
background noise, but the required bandwidth for modulation
decreases. Therefore, for a given bandwidth, decreasing the
tone spacing allows more bandwidth for processing gain
(number of frequency slots) and channel coding. Increasing
the number of frequency slots or decreasing the code rate
allows the system to be more robust to multiuser interference
(hit). Therefore, it would be interesting to examine the effect of
tone spacing on the average number of successfully transmitted
Paper approved by C. Robertson, Editor for Spread Spectrum Systems of the
IEEE Communications Society. Manuscript received April 15, 1997; revised
January 22, 1998 and May 1, 1998.
The authors are with the Department of Electrical Engineering, Korea
Advanced Institute of Science and Technology, Yusong–gu, Taejon 305-701,
Korea.
Publisher Item Identifier S 0090-6778(98)08124-0.
information bits per unit time per unit bandwidth (normalized
throughput). We examine the optimum tone spacing and code
rate pair that maximize the normalized throughput in terms of
the number of users and signal-to-noise ratios. We discuss an
adaptive transmission scheme and examine the performance
improvement attained by the adaptation. We also examine
the tradeoff among tone spacing, code rate, and processing
gain in maximizing the normalized throughput, when the total
bandwidth is fixed.
An extensive study of the choice of optimal code rates for
maximizing normalized throughput is discussed in [4]. The
optimal choice of code rate and processing gain for the block
error probability within a bound is examined in [5]. In [4]
and [5], however, the effect of nonorthogonal tone spacing
and the overall bandwidth expansion due to combined coding,
modulation, and spreading are not investigated.
The results given in this paper would be helpful in designing
FHMA communication system. They specify how to choose
design parameters such as tone spacing, number of frequency
slots, and code rate to maximize the normalized throughput
for a given bandwidth.
II. SYSTEM MODEL
In this section we describe the system model. From a
bits enter the Reed–Solomon (RS)
binary data source,
-bit -ary symbols
. The encoder
encoder as
symbols to obtain an
symbol codeword
appends
bits long). One code symbol ( bits long) is transmitted
(
during a hop interval using BFSK modulation symbols. The
received signal is dehopped and demodulated to form -bit
RS codeword symbols. Then the code symbols are decoded
bits of information. We assume that
to obtain the original
the channel is frequency-nonselective Rayleigh fading, and the
received signal strength is effectively constant over a block
of bits (slow fading). This assumption is appropriate when
the transmitter, receiver, and all reflecting surfaces are slowly
moving, relative to the carrier wavelength and symbol rate [6].
can be represented by
The BFSK signal
(1)
is the bit duration,
is
where is the signal magnitude,
is the pulse shape. In this paper,
the tone frequency, and
0090–6778/98$10.00  1998 IEEE
CHOI AND KIM: FHMA COMMUNICATION
1479
probability of a particular code symbol being hit is
, where
we consider the Gaussian pulse shape represented by
elsewhere
where reflects the rate at which pulse decay occurs.
between
and
The correlation coefficient
defined as
(2)
synchronous hopping
(9)
asynchronous hopping
is
(3)
is the tone spacing. If a sufficient
where
amount of the tails of the Gaussian pulse is included in the
, then the limits on the integrals in (3) and the
interval
may, for approximate computational purposes, be
tails of
. Then
is evaluated as [7]
extended to
is the probability of another transmitter hopping to the same
frequency slot [8]. In this paper, we assume the use of
asynchronous frequency hopping.
We will consider the case where the receiver knows the
presence of a hit (perfect side information) [2] and erases the
symbol that is hit. We will also look at the case where such side
information is not available and the signal detection is made by
hard decision. If perfect side information is available, the code
, given
simultaneous
symbol erasure probability,
transmissions, is given by
(10)
III. PERFORMANCE ANALYSIS
(4)
is the 3-dB bandwidth of the spectrum
where
of the infinitely extended Gaussian pulse, and it is very close to
that of the finite duration pulse [7]. If we define the bandwidth
as the difference between the upper 3-dB frequency of
the upper band and the lower 3-dB frequency of the lower
band, then
If we require
then we get
bandwidth
The conditional code symbol error probability,
,
simultaneous transmissions and the
given that there are
received signal-to-noise ratio (SNR) is , is represented as
(11)
(5)
is the conditional bit error probability given that
where
there is no hit, and the received SNR is . For nonorthogonal
is given by [[9], p. 293]
BFSK signals,
(i.e., 2% at the tail end),
. In this paper, we will assume
. Then the correlation coefficient
and the
are given by
(12)
(6)
where
is the modified Bessel function of order , and
and
(13)
(7)
is the tone spacing factor. When the
where
rectangular pulse shaping is used, BFSK signals are non. The total bandwidth
coherently orthogonal if
is
, because there are frequency slots and
. If the source
each frequency slot has a bandwidth of
(bits/s), then
generates information at a rate of
(8)
.
because
transmitter–receiver pairs commuWe assume there are
nicating over frequency slots. Each pair uses a frequency
hopping pattern that randomly hops among all frequency
slots with equal probability, independent of previous hop
frequencies (i.i.d. hopping). Whenever two or more signals
from different transmitters are transmitted simultaneously in
the same frequency slot, we say a “hit” occurs. Then the
(14)
Then the code symbol error probability,
simultaneous transmission, is given by
, given
(15)
where
(16)
is the probability density function for the received SNR , and
is the average received SNR. When an RS code of rate is
, where
is the average
used, is equivalent to
is the one-sided noise
received information bit energy and
spectral density of the background noise. The Gauss–Laguerre
numerical integration [[10], p. 923] may be used in evaluating
the integration in (15).
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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 46, NO. 11, NOVEMBER 1998
Fig. 1. Normalized throughput W versus the number of users K for various
(k; ) pairs; (32; k ) RS code, Rs = 10 kb/s, BT = 2:56 MHz, q = 64;
Eb =N0 = 25 dB.
Fig. 2. Optimal tone spacing o and optimal code rate ro , versus the number
users K ; (32; k ) RS code, Rs = 10 kb/s, BT = 2:56 MHz, q = 64;
Eb =N0 = 25 dB.
As the
RS code can correct any set of errors and
erasures, provided
[11], the probability of not
decoding correctly is given by
(17)
Therefore, the normalized throughput in bits/(s Hz) is given by
(18)
IV. NUMERICAL RESULTS AND DISCUSSIONS
A. Optimum
Pair
versus
Fig. 1 is a plot of the normalized throughput
for various
pairs when the
the number of users
and the number of frequency slots are
total bandwidth
pair that
fixed. We find that there exists an optimum
pairs are
maximizes the normalized throughput. Here,
chosen such that the total bandwidths are the same. Fig. 2
is a plot of the optimal tone spacing factor and code rate,
and , that maximize the normalized throughput. We
and
decrease as the number of users
find that
b =N0 ; (32; k) RS code,
Fig. 3. Maximum throughput Wmax versus E
K = 20; Rs = 10 kb/s, BT = 2:56 MHz, q = 64.
CHOI AND KIM: FHMA COMMUNICATION
1481
Fig. 5. Optimal (; r; q ) triple versus the number of users
b =N0 = 25 dB.
code, Rs = 10 kb/s, BT = 2:56 MHz, E
K ; (32; k) RS
o and optimal code rate ro , versus Eb =N0 ;
K = 20; Rs = 10 kb/s, BT = 2:56 MHz, q = 64.
Fig. 4. Optimal tone spacing
(32; k ) RS code,
increases, when the number of frequency slots and the total
are fixed. An explanation for this is that, as
bandwidth
increases, the decoding error is dominantly caused by the
multiuser interference (hit) rather than by background noise.
Because tone spacing has nothing to do with the suppression
of multiuser interference, it is desirable to decrease the tone
spacing and use the remaining bandwidth for channel coding
to correct more erasures, as the number of users is increased.
versus
Fig. 3 is a plot of the normalized throughput
for various
pairs. We find that there exists
pair that maximizes
for each
.
an optimum
This suggests an adaptive combined modulation and coding
pair being adaptively changed relative to
scheme, i.e.,
. This has the potential of achieving the upper envelope
pairs. Fig. 4 is a plot of
and
of a family of
versus
. We find that when the background noise is
), it is more desirable to reduce
dominant (i.e., low
the error occurrence by increasing the tone spacing than to
correct errors occurred by channel coding. But when the multi), the erasure
user interference is dominant (i.e., high
correction capability has to be enhanced (by decreasing the
code rate at the cost of reducing the tone spacing) to mitigate
the multiuser interference.
B. Optimum
Triple
, and versus the number
Fig. 5 is a plot of the optimal
of users , when the total bandwidth is fixed. We find that the
and the optimum code
optimum number of frequency slots
increase, while the optimum tone spacing factor
rate
b =N0 ; (32; k) RS code,
Fig. 6. Optimal (; r; q ) triple versus E
kb/s, BT = 2:56 MHz, K = 20.
Rs
= 10
decreases, as the number of users is increased. This indicates
that it is more valuable to prevent hits (erasures) by increasing
number of frequency slots than to correct erasures occurred or
to decrease the error rate by increasing the tone spacing, as
the number of users is increased.
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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 46, NO. 11, NOVEMBER 1998
b =N0 for several number of users K ;
Fig. 7. Maximum throughput versus E
Rs = 10 kb/s, BT = 2:56 MHz.
(32; k ) RS code,
b =N0 for
Fig. 9. Normalized throughput W with hard decision versus E
various (k; ) pairs; synchronous hopping, (32; k ) RS code, K = 20;
Rs = 10 kb/s, BT = 2:56 MHz, q = 64.
ground noise is dominant (i.e., noise-limited), the tone spacing
has to be increased to mitigate the background noise, but
when the multiuser interference is dominant (i.e., interference
limited), the number of frequency slots has to be increased to
mitigate the multiuser interference. The optimal code rate is
.
almost insensitive to
attained
Fig. 7 is a plot of the maximum throughput
and that by using
by using the optimal triple
versus
. We find that the throughput
case is more significant with larger
gain over the
at high
.
C. No Side Information: Hard Decision
Fig. 8. Normalized throughput W with hard decision versus the number of
users K for various (k; ) pairs; synchronous hopping, (32; k ) RS code,
Rs = 10 kb/s, BT = 2:56 MHz, q = 64, Eb =N0 = 25 dB.
Fig. 6 is a plot of the optimal
, and versus
,
when the total bandwidth is fixed. We find that when the back-
So far, we considered the case where perfect side information regarding the presence of hit is available at the
receiver. In this subsection, we consider the case where
such side information is not available and hard decision
demodulation is made at the receiver. Since the exact analysis is quite complicated, we present a computer simulation result. Fig. 8 is a plot of the normalized throughput
. We find that the adjustversus the number of users
ment of tone spacing still presents the same advantage in
throughput as in the perfect side information case. Fig. 9 is
. We find
a plot of the normalized throughput versus
pair for each
,
that there exists an optimum
and the adaptive combined coding and modulation may also
be applicable.
CHOI AND KIM: FHMA COMMUNICATION
1483
V. CONCLUSIONS
We have examined the effects of tone spacing of BFSK
signals on the normalized throughput of an FHMA system
with Reed–Solomon coding in Rayleigh fading channels. We
first considered the case where the number of frequency slots
and the total RF bandwidth are fixed. It is found that the tone
spacing and the code rate have to be reduced as the number
is increased. This is because the decoding error is
of users
caused dominantly by the multiuser interference rather than by
background noise, at large , and tone spacing has nothing to
do with mitigating multiuser interference. However, when the
background noise is the dominant cause of error (i.e., noiselimited), it is more desirable to reduce the error occurrence
by increasing tone spacing than to correct errors occurred by
channel coding. Then, we examined the tradeoff among tone
, code rate
, and number of frequency slots
spacing
under the constraint of fixed total bandwidth. It is found that
the optimum decreases and the optimum and increases, as
the number of users is increased. This indicates that it is more
valuable to prevent hits (erasures) by increasing the number of
frequency slots than to correct erasures occurred or to decrease
the error rate by increasing the tone spacing, as the number of
users is increased. In a noise-limited system, the tone spacing
has to be increased to mitigate the background noise at the
cost of reducing the number of frequency slots. The optimum
when
and are
code rate is almost insensitive to
optimized together.
[6]
[7]
[8]
[9]
[10]
[11]
networks,” IEEE Trans. Commun., vol. COM-41, pp. 307–311, Feb.
1993.
T. S. Rappaport, Wireless Communication. Englewood Cliffs, NJ:
Prentice-Hall, 1995.
G. L. Turin, “Error probabilities for binary symmetric ideal reception
through nonselective slow fading and noise,” Proc. IRE, vol. 46, pp.
1603–1619, Sept. 1958.
E. A. Geraniotis and M. B. Pursley, “Error probabilities for slow
frequency-hopped spread-spectrum multiple-access communications
over fading channels,” IEEE Trans. Commun., vol. COM-30, pp.
996–1009, May 1982.
J. G. Proakis, Digital Communications, 2nd ed. New York: McGrawHill, 1989.
M. Abramowitz and I. A. Stegun, National Bureau of Standards, Applied
Mathematics Series 55: Handbook of Mathematical Functions with
Formulas, Graphs, and Mathematical Tables, U.S. Dept. of Commerce,
1983.
R. E. Blahut, Theory and Practice of Error Control Codes. Reading,
MA: Addison-Wesley, 1983.
Ho Kyu Choi (S’98) received the B.S. degree in
electronical engineering from KyungPook National
University, Taegu, Korea, in 1991, and the M.S.
degree in electrical engineering from the Korea Advanced Institute of Science and Technology, Taejon,
in 1993. He is currently working toward the Ph.D.
degree in electrical engineering at the Korea Advanced Institute of Science and Technology, Taejon.
Since March 1998, he is also working at the
Information and Telecommunication R&D Center
of Samsung Electronics Company Ltd., Sangun. His
research interests include spread spectrum techniques in wireless mobile radio
communication systems, and error correction coding.
REFERENCES
[1] M. Mizuno, “Randomization effect of errors by means of frequencyhopping techniques in a fading channel,” IEEE Trans. Commun., vol.
COM-30, pp. 1052–1056, May 1982.
[2] M. B. Pursley, “Frequency-hop transmission for satellite packet switching and terrestrial packet radio networks,” IEEE Trans. Inform. Theory,
vol. IT-32, pp. 652–667, Sept. 1986.
[3] V. Hayes et al., “Article on IEEE 802.11 standardization status,”
Electron. Eng. Times, Feb. 1994.
[4] S. W. Kim and W. E. Stark, “Optimum rate Reed–Solomon codes
for frequency-hopped spread-spectrum multiple-access communication
systems,” IEEE Trans. Commun., vol. COM-37, pp. 138–144, Feb. 1989.
[5] K. Cheun and W. E. Stark, “Optimal selection of Reed–Solomon code
rate and the number of frequency slots in asynchronous FHSS-MA
Sang Wu Kim (M’87) received the Ph.D. degree
in electrical engineering from the University of
Michigan, Ann Arbor, in 1987.
Since then, he has been with the Korea Advanced
Institute of Science and Technology where he is currently an Associate Professor of Electrical Engineering. His research interests include spread-spectrum
communications, wireless communications, and error correction coding. During 1996–1997, he was
a Visiting Associate Professor at the California
Institute of Technology, Pasadena.
Dr. Kim is a member of Tau Beta Pi.