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Improving Link Reliability Complexity Trade-off by
Exploiting Reliable Feedback Signaling
Mohamed A. M. Hassanien and Pavel Loskot
School of Engineering
Swansea University, Swansea, United Kingdom, SA2 8PP
{443387,p.loskot}@swan.ac.uk
Abstract— In this paper, two retransmission schemes that rely
on reliable reverse link for error-free feedback message delivery
are proposed. Unlike the conventional automatic repeat request
schemes, the proposed schemes do not require the use of cyclic-
redundancy-check bits for error detection. The first scheme is
based on choosing one or more predefined packet segments for
retransmission. In the second scheme, random permutations are
exploited to locate the bits for retransmission in the predefined
window within the packet. The complexity-performance trade-
offs of the proposed schemes are investigated by computer
simulations. For the same forward transmission rate, the bit
error rate performance of the second scheme is found to be
superior to the first scheme, and depending on the particular
implementation, the second scheme may require less feedback
bits.
I. INTRODUCTION
The reliability of information transmission over unreliable
communication links can be achieved by exploiting the trans-
mission diversity. The transmission diversity is often obtained,
for example, by forward error correction (FEC) coding, auto-
matic repeat request (ARQ) retransmissions or by multiple
antennas located at the transmitter and at the receiver. In
general, the diversity signaling requires that the degrees-of-
freedom of the communication link are used jointly which
may significantly increase the computational as well as im-
plementation complexity of the associated signal processing,
particularly at the receiver.
It is well-known that the feedback link does not improve
the information theoretic capacity of memoryless channels,
however, it can greatly simplify the encoding and decoding
complexity [1]. A good example is the ARQ schemes that
are often used to reduce the implementation complexity [2].
On the other hand, the feedback does increase the channel
capacity of channels with memory [3]. The go-back-N ARQ
retransmission scheme was optimized in [4]. The hybrid FEC
and ARQ scheme for reliable delivery of control messages
is designed in [5]. The permutations of the retransmitted
packets to improve the performance of the ARQ schemes
were proposed in [6] assuming a bit-to-symbol mapping for
multilevel linear modulations, and in [7] assuming a symbol-
to-subcarrier mapping for orthogonal frequency division mul-
tiplexing (OFDM). Furthermore, MacKay [8, Ch. 50] com-
ments: “The best solution to the communication problem is
to combine a simple, pseudo-random code with a message-
passing decoder.”
In this paper, we improve the bit error rate (BER) perfor-
mance of a transmission link by designing two new retrans-
mission schemes that exploits the reliable feedback signaling.
These schemes do not use the forward error correction (FEC)
coding nor they use the cyclic redundancy check (CRC) bits
for error detection unlike the most frequently used conven-
tional type II hybrid ARQ schemes [9]. On the other hand,
the ARQ schemes typically require one bit of feedback per
retransmission while the proposed schemes benefit from using
more than one bit of feedback. Both proposed schemes assume
that there exists a reliable reverse link that can deliver error-
free feedback messages from the receiver to the transmitter.
In the first retransmission scheme, we propose how to choose
one of the predefined packet segments for the retransmission.
The second retransmission scheme uses random permutations
at the receiver to decide on particular bits that should be re-
transmitted. The second scheme resembles the random coding
approach described in [8] except that, in our scheme, the
random selection is done at the receiver rather than at the
transmitter.
The rest of this paper is organized as follows. System
model is presented in Section II including definitions of the
transmission rates. The segmentation based retransmission
scheme and the random permutations based retransmission
scheme are proposed in Section III. Numerical examples to
illustrate the performance of the proposed schemes are studied
in Section IV. Conclusions are given in Section V.
II. SYSTEM MODEL
Consider a point-to-point transmission between a source and
a destination that consists of the forward and reverse links as
shown in Fig. 1. It is assumed that the communication channel
corresponding to the forward link has much larger information
theoretic capacity (equivalently, the achievable transmission
rate) than the communication channel corresponding to the
reverse link. More importantly, the transmission rate in the
reverse link is assumed to be sufficiently small, so that the
feedback signaling from the destination to the source over the
reverse link can be considered to be free of the transmission
errors.
For simplicity, the forward link is modeled as an additive
white Gaussian noise (AWGN) channel. The source wants
to transmit a packet of 𝑁information bits using binary
phase shift keying (BPSK) modulation. In this case, the
978-1-4244-6317-6/10/$26.00 © 2010 IEEE ISWCS 2010775
link
link
forward
reverse
destinationsource
Fig. 1. A point-to-point communication system.
reliabilities (i.e., the log-likelihood ratios) of the received
bits are proportional to the absolute value of the received
BPSK symbols [10]. The signal-to-noise ratio (SNR) per bit is
denoted as 𝛾b=𝐸b/𝑁0where 𝐸bis the average energy of the
transmitted BPSK symbols normalized by the forward rate 𝑅f,
and 𝑁0=2𝜎2
wis the one-sided power spectral density of the
AWGN with the variance 𝜎2
w. Assuming the BPSK modulated
symbols ‘+1’ and ‘-1’, the variance, 𝜎2
w=(2𝑅f𝛾b)−1.More
importantly, note that the reverse rate 𝑅ris not considered in
normalization of the SNR since the source does not expend
any energy on transmitting the feedback bits.
After receiving the last of the 𝑁BPSK symbols, the
destination uses the received reliabilities to decide which of the
𝑊binary symbols, 0≤𝑊≤𝑁, should be retransmitted. The
retransmission request is a feedback message of 𝐶≥1bits
sent to the source via the reverse (feedback) link. The values
of 𝑊and 𝐶are assumed to be predetermined and remain
constant during all retransmissions. Consequently, after 𝐷
retransmissions, 𝐷=1,2,⋅⋅⋅, the total number of additional
bits sent over the forward link is 𝑃=𝐷𝑊, so that the
total number of bits sent from the source to the destination
is 𝑃+𝑁. The total number of bits sent from the destination
to the source over the reverse link is 𝑄=𝐷𝐶. For example,
the conventional stop-and-wait ARQ system is described by
the parameters 𝑊=𝑁and 𝐶=1[2].
The transmission rates for the communication system in
Fig. 1 can be defined as follows. For the reverse link, the
reverse rate is defined as,
𝑅r=𝑁
𝑁+𝑄=𝑁
𝑁+𝐷𝐶 .
Note that lim𝑁→∞ 𝑅r=𝑅r∣𝐷=0 =1. Similarly, for the
forward link, the forward rate is defined as,
𝑅f=𝑁
𝑁+𝑃=𝑁
𝑁+𝐷𝑊
and again, lim𝑁→∞ 𝑅f=𝑅f∣𝐷=0 =1. Since 𝑊≤𝑁,
always, 𝑅f≥(1 + 𝐷)−1. Hence, given 𝑃and 𝑄,therates𝑅f
and 𝑅rare increasing with the number of information bits 𝑁.
Also, given 𝑊and 𝐶,therates𝑅fand 𝑅rare decreasing with
the number of retransmissions 𝐷. Equivalently, given the fixed
rates 𝑅fand 𝑅r, the product 𝐷𝑊 and the product 𝐷𝐶 should
be constant, i.e., in this case, the number of retransmitted bits
𝑊can be expressed as,
𝑊=𝑁
𝐷(1
𝑅f
−1).(1)
Finally, the total (overall) rate 𝑅tcan be defined as,
𝑅t=𝑁
𝑁+𝑃+𝑄=𝑁
𝑁+𝐷(𝑊+𝐶).
III. PROPOSED RETRANSMISSION SCHEMES
Recall that neither FEC coding nor CRC bits are used
in the proposed retransmission schemes. Hence, the number
of retransmissions 𝐷as well as the retransmission window
size 𝑊are typically fixed for the given forward transmission
rate 𝑅f. Consequently, the main performance criterion for
designing such retransmission schemes while avoiding the use
of the CRC bits for error detection is the average BER. On
the other hand, when the number of retransmissions 𝐷is not
fixed, but can vary for each received packet of 𝑁bits, one
has to consider, in general, also the average throughput; cf.
conventional ARQ schemes. In our schemes, even though it is
possible to terminate the retransmissions before the maximum
number of retransmissions is reached, for example, when the
reliabilities of all 𝑁information bits are above a specified
threshold, this scenario is not considered in this paper. Note
also that an early termination of the retransmissions without
using the CRC bits requires, in general, knowledge of the SNR
at the destination.
A. Segmentation Based Retransmissions
In this scheme, the source packet of 𝑁information bits is
divided into 𝑆non-overlapping segments. We can show that,
for the channel model considered on the forward link, the
optimum segments are of the same length. Hence, we assume
that the number of information bits can be written as 𝑁=2
𝐿
where 𝐿≥1is an integer. Then, the 𝑆=2
𝑖segments, 𝑖=
0,1,2,⋅⋅⋅ ,𝐿, are of equal length 𝑊=𝑁/𝑆 =2
𝐿−𝑖bits.
The feedback message to request a retransmission of exactly
one segment consists of 𝐶=log
2𝑆=𝑖bits.
More importantly, many criteria can be devised to select
which segments should be scheduled for retransmission. Since
it is straightforward to show that the average BER of bits in the
received packet is dominated by the bits having the smallest
reliabilities, we consider the following heuristic algorithm.
The least reliable bit is first identified within each segment.
The segments are then ordered according to the reliabilities
of their least reliable bits. The segments containing the least
reliable bits are requested for retransmission. The retransmitted
segments are combined using the maximum ratio combining
(MRC) which is particularly simple for our case of the AWGN
channel model with the time-invariant noise power spectral
density 𝑁0. The bits in the MRC combined segments are
scaled in order to allow comparison of the reliabilities of bits
across all segments. In this paper, we assume that exactly
one segment containing the bit with the smallest reliability
among all 𝑁received bits is requested for retransmission.
After the MRC and scaling of the reliabilities corresponding to
the retransmitted segment, another segment with the smallest
bit reliability is selected for the next retransmission. This
procedure is repeated 𝐷times, 𝐷=1,2,⋅⋅⋅, in total.
776
B. Random Permutations Based Retransmissions
The underlying assumption of this scheme is that the source
and the destination can generate an identical pseudo-random
sequence. The pseudo-random sequence is used to generate
random permutations of 𝑁bits in the packet. For example,
the source and the destination share the initial random seed.
Since the source and the destination are assumed to be symbol-
synchronized in time, we can assume that a given finite number
of random permutations is generated every symbol period. In
this case, the feedback message is the permutation number
within one symbol period. This retransmission scheme then
operates as follows.
As soon as the last of 𝑁information bits is transmitted
by the source, the destination starts to generate 2𝐶random
permutations of the received 𝑁bits during each consecutive
symbol period. The purpose of random permutations is to
place the bits with the minimum reliabilities into a predefined
window of length 𝑊bits; for example, it is convenient to
choose the first 𝑊bit positions to be such a window. Many
stopping criteria can be devised to stop the generation of the
random permutations, and to request the retransmission of
the bits corresponding to the permutation that placed most of
the low reliability bits into the the predefined window. More
importantly, we note that evaluation of the stopping condition
to determine whether sufficiently many bits of small reliabil-
ities are located in the predetermined window is non-trivial
and represents a significant proportion of the implementation
(and also simulation) complexity. One possible approach is
to order the received bits according to their reliabilities, and
then test the cardinality of the intersection of the integer sets
of bit indices. Once the desired permutation is found (i.e.,
the stopping condition is satisfied), the feedback message is
the permutation number modulo the number of permutations
tested during one symbol period, i.e., modulo 2𝐶. The choice
of the value of 𝐶is a design trade-off. On one hand, the larger
the value of 𝐶the more permutations are tested during one
symbol period and the smaller the delay before the feedback
message is sent. On the other hand, increasing the value of
𝐶reduces the reverse rate 𝑅rdue to feedback signaling.
The maximum value of 𝐶is also practically limited by the
real-time computation processing capability available at the
destination.
The source counts the number of elapsed symbol periods
until the feedback message arrives, and thus, it can deduce
the total number of permutations tested at the destination.
Knowledge of the permutation number is used to select 𝑊bits
from the original 𝑁bits to be retransmitted to the destination.
At the destination, the retransmitted bits are combined using
the MRC, and again, the reliabilities are scaled as explained
in the previous subsection.
IV. NUMERICAL EXAMPLES
The proposed retransmission schemes are investigated here
for a given fixed rate 𝑅fand a given fixed number of
retransmissions 𝐷and number of segments 𝑆. The length of
TAB LE I
TRANSMISSION RATES FOR SEGMENTATION BASED RETRANSMISSIONS
reverse rates 𝑅r
𝑁=64 𝑁= 128 𝑁= 1024
𝐷=1 0.955 0.977 0.997
𝐷=8 0.571 0.727 0.955
𝐷=16 0.364 0.534 0.901
𝐷= 128 0.048 0.09 0.445
total rates 𝑅t
𝑁=64 𝑁= 128 𝑁= 1024
𝐷=1 0.854 0.871 0.887
𝐷=8 0.534 0.667 0.979
𝐷=16 0.334 0.5 0.81
𝐷= 128 0.043 0.084 0.421
the retransmission window 𝑊is computed using eq. (1). As a
reference, the proposed retransmission schemes are compared
with the stop-and-wait ARQ scheme having the same fixed
number of retransmissions 𝐷. Such ARQ scheme corresponds
to a binary repetition code (𝐾, 1,𝑑
min)where 𝐾=𝐷is the
block length, and the minimum Hamming distance 𝑑min =𝐷
[2]. More importantly, recall that the SNR 𝛾bis normalized
by the forward rate 𝑅f, so that the repetition code (2,1,2)
corresponding to exactly one retransmission of the whole
packet of 𝑁information bits has the coding gain 0dB over an
uncoded BPSK. Consequently, the coding gain of the proposed
retransmission schemes as a SNR reduction to achieve the
same target BER as an uncoded BPSK is also equal to the
SNR reduction to achieve the same target BER as a binary
repetition code (2,1,2). If the SNR was not normalized by
the forward rate 𝑅f, then the coding gain of the proposed
schemes would be 10 log10(1/𝑅f)dB larger, i.e., at least 3
dB larger for the rates 𝑅f>1/2.
Consider first the segmentation based retransmission scheme
assuming 𝑆=8segments, i.e., one feedback message contains
𝐶=log
28=3bits. In this case, the forward rate, 𝑅f=
1
1+1/𝑆 =8/9=0.89 is independent of the packet length
𝑁. Table. I summarizes the transmission rates 𝑅rand 𝑅tfor
various values of 𝑁and 𝐷. Note that these transmission rates
decrease quickly with the number of retransmissions which
fundamentally limits the maximum usable value of 𝐷.
Given the values of 𝑁and 𝐷, Fig. 2 and Fig. 3 show the
average BER versus SNR 𝛾b. The BER values are averaged
over all 𝑁information bits in the packet assuming that
asymptotically infinitely many packets have been sent from
the source to the destination. We observe that the achievable
coding gain of the segmentation based retransmission scheme
decreases with the increasing packet length 𝑁although the
reduction of the coding gain with increasing value of 𝑁is
less severe for larger number of retransmissions 𝐷. Also, as
expected, the BER performance improves with the number
of retransmissions 𝐷. For example, the coding gain of 2.7
dB is obtained for the target BER 10−4, the packet length of
𝑁=64bits and 𝐷=8 retransmissions corresponding to the
transmission rates 𝑅f=0.889,𝑅r=0.571 and 𝑅t=0.534.
Fig. 4 compares the coding gains 𝐺(in dB) versus the
number of segments 𝑆for several values of the number of
777
4 5 6 7 8 9 10
10−5
10−4
10−3
10−2
uncoded BPSK
N=64
N=128
N=1024
𝛾b[dB]
BER(𝛾b)
Fig. 2. The BER of the segmentation based retransmission scheme versus
SNR 𝛾𝑏for 𝑆=8segments and assuming 𝐷=1 retransmission.
4 5 6 7 8 9 10
10−5
10−4
10−3
10−2
uncoded BPSK
N=64
N=128
N=1024
𝛾b[dB]
BER(𝛾b)
Fig. 3. The BER of the segmentation based retransmission scheme versus
SNR 𝛾𝑏for 𝑆=8segments and assuming 𝐷=8 retransmissions.
retransmissions 𝐷assuming the target BER 10−3and the
packet length 𝑁=64bits. We observe that the coding gain
increases with the number of segments 𝑆for any number of
retransmissions 𝐷. The coding gain improves with 𝑆since
the BER performance benefits from more bits being reliably
transmitted as feedback messages from the destination to the
source. Note also that the maximum value of 𝐷that can be
used is given as 𝐷=𝑁/𝑆, for any given value of 𝑆. Similarly,
the coding gain increases with the number of retransmissions
𝐷for any number of segments 𝑆. However, recall that the
reverse and total rates rapidly decrease with the increasing
value of 𝐷; see Table. I.
Consider now the second proposed retransmission scheme
that is based on random permutations. Importantly, the BER
performance of this retransmission scheme does not depend
on the packet length 𝑁since only the bits having small
reliabilities are retransmitted. This is not the case for the first
scheme where some of the retransmitted bits may have large
−0.5
0
0.5
1
1.5
2
2.5
212223242526
D=1
D=2
D=4
D=8
𝑆
𝐺[dB]
Fig. 4. The coding gain 𝐺[dB] of the segmentation based retransmission
scheme versus the number of segments 𝑆for 𝑁=64bits and the target
BER 10−3.
0 1 2 3 4 5 6 7 8 9
10−4
10−3
10−2
10−1
uncoded BPSK
Rf = 0.99
Rf = 0.9
Rf = 0.8
Rf = 0.7
𝛾b[dB]
BER(𝛾b)
Fig. 5. The BER of the random permutations based retransmission scheme
versus SNR 𝛾bfor 𝐷=1retransmission.
reliabilities, and thus, its BER performance depends on 𝑁.
The average BER versus the SNR 𝛾bfor several values of
the forward rate 𝑅fis shown in Fig. 5. We observe that there
exists an optimum value of the rate 𝑅ffor which the coding
gain is maximized. This behavior is confirmed by Fig. 6 which
shows the average BER as a function of the forward rate 𝑅ffor
several values of 𝐷and the SNR 𝛾b=3dB. We observe that
the optimum rate 𝑅fminimizing the BER is dependent on 𝐷.
Note that, in general, larger values of 𝛾bcan be translated to
smaller number of retransmitted bits 𝑃effectively increasing
the forward rate 𝑅f. Finally, Fig. 7 shows the BER versus
SNR for the forward rate 𝑅f=0.9and for several values of
the number of retransmissions 𝐷. We observe that the coding
gain differences between different values of 𝐷are decreasing
with the increasing SNR 𝛾b.
V. C ONCLUSIONS
Two novel retransmission schemes were proposed and their
BER performance analyzed by computer simulations. Both
778
0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
10−3
10−2
10−1
D=1
D=2
D=3
D=4
𝑅f[-]
BER(𝑅f)
Fig. 6. The BER of the random permutations based retransmission scheme
versus the rate 𝑅ffor the SNR 𝛾b=3dB.
0123456789
10−4
10−3
10−2
10−1
uncoded BPSK
D=1
D=2
D=3
D=4
𝛾b[dB]
BER(𝛾b)
Fig. 7. The BER of the random permutations based retransmission scheme
versus SNR 𝛾bfor 𝑅f=0.9.
schemes does not use the FEC coding nor they rely on the
CRC bits to detect the transmission errors. The main idea of
the proposed retransmission schemes is to exploit the reliable
reverse link to deliver, error-free, the feedback messages from
the source to the destination. This has a benign effect on
the implementation complexity since only bits within the
received packet having small reliabilities are retransmitted; this
condition is much better satisfied for the random permutations
based retransmission scheme. The proposed schemes also
benefit from the low complexity diversity combining of the
retransmitted bits.
In general, the selective retransmissions of the predefined
packet segments can provide larger coding gains than the
conventional stop-and-wait ARQ having the same number of
retransmissions. The random permutations based retransmis-
sion scheme can further improve these coding gains. One
drawback of the random permutations based retransmission
scheme is a possibly large delay when only a finite number
of bit permutations can be tested over the duration of one
symbol period. On the other hand, the coding gain in the
segmentation based retransmission scheme increases with the
number of segments 𝑆and the number of retransmissions 𝐷,
however, at the expense of the growing number of feedback
bits 𝑄=𝐷𝐶. The largest coding gain for a given value
of 𝑆can be obtained by considering shorter packets and by
increasing the number of retransmissions 𝐷. The schemes
with shorter packets are particularly suitable for delay limited
applications. Our results indicate that, for short packets and
larger values of SNR, using the segments of length 𝑁/8can
provide valuable coding gains with very small implementation
complexity. These coding gains can be increased further by
exploiting multiple feedback messages while keeping the
number of bits sent over the forward link constant. The random
permutations based retransmission scheme is well-suited for
applications requiring larger coding gains for a given fixed
forward rate.
Future work will consider the design of retransmission
schemes with the variable number of retransmissions 𝐷, and
the design of low complexity signaling schemes for reliable
transmission over links where the reverse link is used for low
rate reliable transmission of messages from the source to the
destination as indicated by the dashed arrow in Fig. 1.
REFERENCES
[1] T. M. Cover and J. A. Thomas, Elements of information theory, 2nd ed.
John Wiley & Sons, 1991.
[2] S. Lin and D. J. Costello, Error Control Coding: Fundamentals and
Applications. Prentice-Hall, 1983.
[3] A. Goldsmith and P. Varaiya, “Capacity of fading channels with channel
side information,” IEEE Trans. Inform. Theory, vol. 43, no. 6, pp. 1986–
1992, Nov. 1997.
[4] S. Fujii, T. Kaneko, and Y. Hayashida, “Effectiveness of copy-
transmission in Go-Back-N ARQ system with selective repeat in intra-
block and with limited retransmissions,” in Proc. Int. Conf. Comm. Syst.
Networks, 2010, pp. 1–6.
[5] I. Khazali and A. Agarwal, “Control messages delivery protocol,” in
Proc. Bien. Symp. Commun., 2010, pp. 251–254.
[6] C. Wengerter, A. G. E. von Elbwart, E. Seidel, G. Velev, and M. Schmitt,
“Advanced hybrid ARQ technique employing a signal constellation
rearrangement,” in Proc. VTC, vol. 4, 2006, pp. 2002–2006.
[7] J. Jung, H. Park, and J. Lim, “A novel hybrid ARQ scheme based on
shift column permutation bit interleaving for OFDM systems,” in Proc.
VTC, 2006, pp. 1–5.
[8] D. J. C. MacKay, Information Theory, Inference, and Learning Algo-
rithms. Cambridge Univ. Press, 2003.
[9] K. C. Beh, A. Doufexi, and S. M. D. Armour, “Performance evaluation
of hybrid ARQ schemes of 3GPP LTE OFDMA system,” in Proc.
PIMRC, 2007, pp. 1–5.
[10] S. Benedetto and E. Biglieri, Principles of Digital Transmission with
Wireless Applications, 2nd ed. Kluwer Academic, 1999.
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