IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 48, NO. 11, NOVEMBER 2000
1865
Adaptive Orthogonal Signaling with Diversity in a
Frequency-Nonselective Rayleigh Fading Channel
Sang Wu Kim, Senior Member, IEEE
M
Abstract—Power and rate adaptations with diversity are considered for -ary orthogonal signals in a frequency-nonselective
Rayleigh fading channel. We derived the minimum required
Eb =N0 for a given probability of error, the maximum average data
rate, and the outage probability, when maximal-ratio combining
is performed with an imperfect channel estimation. We analyzed
the effects of channel estimation errors on the required Eb =N0
and outage probability, as a function of diversity order and adaptation methods. The relative benefits of rate and power control
techniques are further presented in terms of channel estimation
error, diversity order, and outage probability. The minimum
b =N0 with Reed–Solomon codes is also presented for
required E
comparison purposes.
Index Terms—Adaptive orthogonal signaling, diversity, power
and rate control, Rayleigh fading, Reed–Solomon code.
I. INTRODUCTION
I
T IS well known that when communicating over a white
-ary orthogonal signaling can
Gaussian noise channel,
achieve capacity because the error probability can be made ar[1]. For a Rayleigh
bitrarily small, provided that
fading channel, however, an infinite signal-to-noise ratio (SNR)
is needed for an arbitrarily small error probability [2].
Several techniques have been developed for reliable transmission of data in fading channels with finite power. Coding and interleaving techniques are common methods to improve the performance in fading channels [3]. In this paper, we will consider
a Reed–Solomon (RS) code, because RS codes can achieve the
greastest possible minimum distance for a given rate, i.e., satisfy
the Singleton bound with equality [4]. Therefore, the guaranteed number of correctable errors is maximized. RS codes are
currently used in many commercial and military systems. Diversity is a powerful communication technique that reduces the
depth of the fades and/or the fade duration by supplying the receiver with multiple replicas of the transmitted signal that have
passed over independently fading channels [3]. Diversity can
often be combined with other fading mitigation techniques. The
diversity combining requires the availability of channel state information (CSI) at the receiver.
When the transmitter is provided with the CSI, adaptive modulation techniques, power control, and rate control can be used
Paper approved by K. B. Letaief, the Editor for Wireless Systems of the IEEE
Communications Society. Manuscript received March 20, 1999; revised October
25, 1999 and May 1, 2000. This work was supported in part by the Korea Science
and Engineering Foundation (KOSEF).
The author is with the Department of Electrical Engineering, Korea Advanced Institute of Science and Technology, Taejon 305-701, Korea (e–mail:
swkim@san.kaist.ac.kr).
Publisher Item Identifier S 0090-6778(00)09882-2.
to combat the fading [5]–[8]. Power control is to maintain a con[or bit-error rate (BER)] by inverting the
stant received
channel fading. This technique ensures a constant bit rate, but
requires a large amount of the transmitted power to compensate
for deep channel fades. A better approach is to invert the channel
fading only above a fixed-cutoff fade depth , and otherwise declare an outage. In real systems, transmitters are usually subject
to a maximum transmit power limit. We will see later that is
related to this maximum transmit power limit. This type of power
control is discussed in [8] and [9]. Rate control maintains a con(or BER) by adapting the data rate relative
stant received
to the channel fading: transmitting at high rates under favorable
channel conditions and responding to channel degradation by reducing data rates, while the transmit power is held fixed.
In most mobile radio communication systems, the transmit
required to achieve a
power is limited. Therefore, the
given probability of error should be minimized, and specified
when designing radio communication systems. In this paper,
for a given probwe investigate the minimum required
ability of error with orthogonal signals, the maximum average
data rate, and the outage probability when both power control
and rate control techniques are combined with maximal-ratio
combining (MRC) diversity. MRC is the optimal diversity combining method in the sense of maximizing the combiner output
SNR, and therefore provides the maximum performance improvement relative to all other combinig methods [10].
-ary orthogonal signaling is a power-efficient modulation
method that is useful for low power and low data rate applications. Power efficiency of this modulation method increases
increases at the expense of increased
as the signal alphabet
bandwidth. In this paper, we first consider the case of an infiand present the lower limits on the required
for
nite
also serves as
all values of . The asymptotic minimum
a lower bound for all other types of modulation, because -ary
orthogonal signaling can achieve capacity. In Section V, we consider the case of finite , and then discuss how to evaluate the
for a given probability of error with finite .
required
We investigate the effects of channel estimation errors on
, and also outage probability
the required minimum
as a function of diversity order and adaptation methods.
Closed-form expressions are provided for the required
of both power and rate control techniques that are subject to a
maximum transmit power limit. The relative benefits of rate and
power control techniques are also presented in terms of channel
estimation error, diversity order, and outage probability.
We model the channel as a frequency-nonselective Rayleighfading with white Gaussian background noise. We assume that
the channel gain is effectively constant over a block of symbols
0090–6778/00$10.00 © 2000 IEEE
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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 48, NO. 11, NOVEMBER 2000
(slow fading). This model is appropriate when the transmitter,
receiver, and all reflecting surfaces are slowly moving relative
to the carrier wavelength and symbol rate.
where
is a constant. It follows from (2) and (7) that
for error-free communications
the average transmit power
is given by
II. TRUNCATED POWER CONTROL
(8)
It is well known that the limiting symbol-error probability
for -ary orthogonal signals in the limit of large
in a faded,
additive white Gaussian noise (AWGN) channel is [1]
(1)
(9)
where is the channel power gain associated with the fading,
is the transmit power,
is the data rate in bits/s, and
is the one-sided noise spectral density. In this section, we
relative to
with a fixed
subject
consider adapting
constraints, where
is the maximum
to
transmit power.
It follows from (1) that error-free communication requires
(2)
should be controlled acThat is, the transmit power
cording to (2) in order to compensate for fading when
. If
, then
an outage is declared, i.e., data transmission is suspended
with no service. Thus, the data transmission suffers an outage
of
probability1
(3)
and the outage probability
is given by
(10)
(11)
Thus, it follows from (6), (9), and (11) that error-free communications require
(12)
or
. The conventional power conwhere
.
trol technique corresponds to a special case of
The average duration of an outage is proportional to the outage
probability
(4)
(13)
is the level-crossing rate at . In Rayleigh fading
where
channels, the level-crossing rate is given by [10]
The left-hand side of (12) may be interpreted as the average
received energy per bit to noise spectral density ratio (denoted
). If we require a constant bit rate (i.e.,
or
), then it follows from (12) and (13) that
(5)
is the maximum Doppler frequency. The average
where
outage duration will impact systems where mobile users have a
,
delay constraint. Since no data is transmitted when
is
the average long-term data rate
(14)
or
(6)
paths with power gains
If we assume that there are
, and the gains are independent and identically
, then the pdf
of an MRC
distributed with mean
combiner output is given by [10]
(7)
1One may consider the average fade duration of SNR going below a threshold
in defining the outage probability. The notion of average duration outage will
lead to a different performance measure.
(15)
decreases on the
Equation (14) shows that the required
when the diversity of order is applied with
order of
the conventional power control.
III. TRUNCATED RATE CONTROL
When the transmit power is fixed, the data rate
can be
adapted to the channel fading, such that a desired probability of
error is maintained at the receiver. In this section, we consider
KIM: ADAPTIVE ORTHOGONAL SIGNALING WITH DIVERSITY IN A RAYLEIGH FADING CHANNEL
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a truncated rate control technique that suspends transmission
. Otherwise, it controls the data rate
with a
when
such that a desired probability of
fixed transmission power
error is maintained. It follows from (1) that the data rate
should be controlled by
Then, it follows from (10) and (21) that the outage probability
is given by
(16)
for error-free communications
The minimum required
with the truncated power control can be obtained from (8) and
(21)
for error-free communications. Therefore, the average data rate
is given by
(23)
(17)
(18)
(24)
is the average transmission power.
where
for error-free communicaConsequently, the required
tions is
with the truncated rate control can be
Also, the required
obtained from (17) and (21)
(19)
(25)
Note that if we require zero-outage
from (19) that
, then it follows
(20)
A comparison of (14) and (20) indicates that power control reby a factor of
, when
quires an increase in
and
, relative to the rate control. However, both
techniques yield the same performance in the limit of a large .
IV. CHANNEL ESTIMATION ERROR
The effects of channel estimation error on the performance
of MRC systems have been first addressed by Bello and Nelin
[11], where the channel estimation error was shown to be
Gaussian. The pdf of the output of a maximal-ratio combiner
with Gaussian distributed weighting (estimation) errors is given
by [12]
(21)
where
(perfect estimation) is substituted in (21),
Note that if
(24), and (25), then we get (7), (12), and (19), respectively. Also,
, then it follows from (25)
if we require zero-outage
is
that the required
(26)
increases with
Equation (26) indicates that the required
for the rate control, when the channel estimation is not perfect
. This is due to the effect that the average combiner
. The maximum average data
output decreases with for
for both a truncated power control and truncated rate
rate
control can be obtained by substituting (24) and (25) in
(27)
that the truncated rate conrespectively. The power gain
trol provides over the truncated power control can be further
case, yields
obtained by dividing (24) by (25), which, for
(22)
is the correlation between the actual channel
where
gains and their estimates. It follows from (21) that
, which is a strictly decreasing function of
for
. This indicates that the average combiner output
decreases with when the chennel estimation is not perfect
.
(28)
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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 48, NO. 11, NOVEMBER 2000
TABLE I
(p ; M ) AT p = 10
V. FINITE SIGNAL ALPHABET
In previous sections, we have considered the asymptotic case
is finite, the bit-error probability
of an infinite . When
for orthogonal modulations with noncoherent detection is given
by [3]
(29)
is the channel power gain. In order to maintain
, we require
for some
, which can be found by a numerical search.
constant
at
Table I presents the specific values of
for several values of
. Therefore, the minimum required
for
with truncated power control can be
in (12) and (24) by
.
obtained by replacing
for
Similarly, we may obtain the minimum required
with truncated rate control by replacing
in (19)
. Therefore, the observations made with
and (25) by
can also apply to systems with finite .
an infinite
where
=N needed for error-free communications versus outage
Fig. 1. Minimum E
probability P : perfect channel estimation ( = 1).
or
VI. RS CODES
It is well known that an
RS code can correct any
symbol errors provided that
[4]. As
,
the relative error frequencies in a codeword will approach the
probability of symbol error computed asymptotically. Hence,
error-free communications can be achieved in the limit if
(30)
where
is the code rate and
(31)
(35)
Even better performance is obtained from RS codes if some
side information is provided to the decoder so that unreliable
symbols can be erased prior to decoding. A ratio-threshold test
(RTT)-type demodulation is a simple method for generating information on the quality of the channel [13]. RTT declares an
erasure whenever the ratio of the largest to the second largest
of the energy detector outputs does not exceed a fixed threshold
for
greater than one. It is shown in [14] that the required
error-free communications is given by
dB
is the asymptotic symbol-error probability when hard-decision
(HD) demodulation is employed [2]. It follows from (30) and
(31) that error-free communications can be achieved if
(32)
(36)
or
(37)
when the optimum ratio threshold and the optimum code rate
are used.
or
VII. NUMERICAL RESULTS AND DISCUSSION
(33)
The optimum code rate that minimizes the right-hand side of
(32) is 0.46, and for this optimum value of rate
6.8 dB
(34)
Fig. 1 is a plot of the minimum
needed for error-free
communications, as a function of the outage probability . We
than
find that the truncated rate control requires a lower
the truncated power control. We also find that the transmission
power can be traded with the outage probability. That is, traffic,
which can tolerate a longer outage duration (or a larger delay),
KIM: ADAPTIVE ORTHOGONAL SIGNALING WITH DIVERSITY IN A RAYLEIGH FADING CHANNEL
Fig. 2.
Power gain G versus outage probability P .
requires a smaller
. This can be utilized in minimizing
transmission power at the mobile unit in multimedia communications, where a population of users simultaneously transmits
a variety of traffic types having different delay requirements.
can be reduced by increasing the diverThe required
for the truncated power control, except at high
sity order
outage probabilities, where the optimal that minimizes the
depends on . For the truncated rate control,
required
is almost independent of except at high
the required
increases as
outage probabilities, where the required
increases. We also plotted the minimum required
with
RS codes for comparison purposes in Fig. 1.
that the truncated rate
Fig. 2 is a plot of the power gain
control provides over the truncated power control versus the
for several values of correlation coeffioutage probability
cient and diversity order . Note that the power gain increases
as decreases. This indicates that the truncated rate control
is more effective when the channel estimation is less accurate.
decrease. This is
Also, the power gain increases as and/or
because the truncated power control uses a lot of the transmission power in compensating for deep fades at low outage probabilities, and the fade depth or fade duration increases as decreases. On the other hand, the truncated rate control uses a constant transmission power and compensates for deep fades by reducing rate, rather than using a large power. Thus, the truncated
rate control saves power during bad channels, and uses power
more effectively for better channels. However, the truncated rate
control experiences a time delay when the channel gain is low.
The power gain translates into a capacity gain in code-division
multiple-access (CDMA) communication systems, because the
interference to other users decreases due to the transmission
power reduction.
for error-free commuFig. 3 is a plot of the minimum
nications versus the correlation coefficient square for several
decays almost
values of . We find that the minimum
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=N needed for error-free communications versus
Fig. 3. Minimum E
correlation coefficient square : = 0:1.
Fig. 4.
0:2.
Outage probability P versus correlation coefficient square : =
exponentially (linearly in [decibels] scale) with for the truncated power control. Also, for the truncated power control, the
decreases as the diversity order increases,
minimum
and the diversity effect is most pronounced when the channel estimation is perfect. As the channel estimation becomes worse,
even
the diversity effect diminishes, and the minimum
increases with when the accuracy of channel estimation falls
in Fig. 3). However, for the trunbelow a threshold (
increases as increases
cated rate control, the required
for all , as implied by (26).
versus the correFig. 4 is a plot of the outage probability
lation coefficient square for several values of . We find that
the estimation error results in an increase in the outage probability, and the outage probability is more sensitive to estimation
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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 48, NO. 11, NOVEMBER 2000
errors for a larger . When the accuracy of the channel estimain Fig. 4), the outage
tion falls below a threshold (
probability even increases with . This is due to the effect of
inefficient diversity combining when estimation (weighting) errors are large, leading to a higher probability of exhibiting a low
combiner output for a larger .
It should be noted that both power control and rate control
require the availability of CSI at the transmitter, and the diversity combining requires the availability of CSI at the receiver,
whereas the RS code with an HD does not require the CSI. In
the case where CSI is available at either the transmitter or the
receiver, the performance of RS codes can be improved. Also,
RS codes require a sufficiently large interleaving depth in order
to make the statistical behavior of each coded symbol independent, and to achieve its full potential error correction capability.
) and truncated rate control
Truncated power control (for
provide a variable bit rate, whereas RS codes provide a constant
bit rate. So, these schemes apply in different situations.
VIII. CONCLUSION
We considered truncated power control and truncated rate
control with diversity in transmitting -ary orthogonal signals
over a frequency-nonselective Rayleigh fading channel. We defor a given probability of
rived the minimum required
error, the maximum average data rate, and the outage probability, when the MRC is performed with an imperfect channel
estimation.
We found that the transmission power can be traded with the
outage probability. That is, traffic which can tolerate a higher
. The required
outage probability requires a smaller
can also be reduced by increasing the diversity order
for the truncated power control. But, for the truncated rate
control, it is independent of , when the outage probability is
low. If the outage probability is high enough, then the optimum
that minimizes the required
depends on the outage
probability for the truncated power control. Whereas for the
increases as
truncated rate control the required
increases.
We also found that the truncated rate control requires a lower
than the truncated power control, and the power savings
by using the truncated rate control is more significant as the
channel estimation becomes worse, and the diversity order
and/or outage probability decreases. The channel estimation
error results in increasing the outage probability along with the
. The diversity effect in reducing the outage
required
for the truncated power control
probability and required
is most pronounced when the channel estimation is perfect.
As the channel estimation becomes worse, the diversity effect
diminishes, and when the accuracy of the channel estimation
falls below a threshold, the outage probability and the required
even increase as the diversity order increases.
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M
Sang Wu Kim (M’88–SM’99) received the Ph.D. degree in electrical engineering from the University of
Michigan, Ann Arbor, in 1987.
Since 1987, he has been with the Korea Advanced
Institute of Science and Technology (KAIST),
Taejon, where he is currently a Professor of Electrical Engineering. His research interests include
spread-spectrum communications, adaptive 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 currently serving as an Associate Editor for the IEEE
COMMUNICATIONS LETTERS, and an Editor for Journal of Communications
and Networks. He received the Best Paper Award at the IEEE International
Symposium on Spread Spectrum Techniques and Applications (IEEE ISSTA
2000) held in New Jersey in September 2000.