# Analysis and Throughput Optimization of Selective Chase Combining for OFDM
Systems

Taniya Shafique, Muhammad Zia and Huy-Dung Han

###### Abstract

In this paper, we present throughput analysis and optimization of bandwidth
efficient selective retransmission method at modulation layer for conventional
Chase Combining (CC) method under orthogonal frequency division multiplexing
(OFDM) signaling. Most of the times, there are fewer errors in a failed packet
and receiver can recover from errors receiving _partial_ copy of original
frame. The proposed selective retransmission method at modulation layer for
OFDM modulation requests retransmission of information corresponding to the
poor quality subcarriers. In this work, we propose cross-layer multiple
selective Chase combining (MSCC) method and Chase combining with selective
retransmission (CCWS) at modulation level. We also present bit-error rate
(BER) and throughput analysis of the proposed MSCC and CCWS methods. In order
to maximize throughput of the proposed methods under OFDM signaling, we
formulate optimization problem with respect to amount of information to be
retransmitted in selective retransmission in the event of packet failure. We
present tight BER upper bounds and tight throughput lower bounds for the
proposed selective Chase combining methods. The simulation results demonstrate
significant throughput gain of the optimized selective retransmission methods
over conventional retransmission methods. The throughput gain of the proposed
selective retransmission at modulation layer are also holds for conventional
for hybrid automatic repeat request (HARQ) methods.

##  I Introduction

The contemporary wireless communication standards such as LTE-advanced [1]
integrate new technologies to meet increasing need of high data rate. The
current and future communication systems employ multiple-input multiple-output
(MIMO) technologies due to their potential to achieve higher data rate and
diversity. In order to assure error-free communication with high throughput,
packet error detection and correction protocols have evolved over time [2].
The automatic repeat reQuest (ARQ) method guarantees error free data transfer
using cyclic redundancy check (CRC) approach. The concept of hybrid ARQ (HARQ)
integrates ARQ and forward error correction (FEC) codes [2, 3] to provide
effective means of enhancing overall throughput of communication systems. In
the event of packet failure, advanced form of HARQ incorporates joint decoding
by combining soft information from multiple transmissions of a failed packet.
Thus, HARQ is one of the most important technologies embedded in the latest
communication systems such as high-speed down link packet access (HSDPA),
universal mobile telecommunications system (UMTS) that pervade 3G and 4G
wireless networks to ensure data reliability.

In type-I HARQ, receiver request retransmission of a failed packet and
discards observation of the failed packet. Type-II HARQ is most commonly used
method and achieves higher throughput. The type-II HARQ is divided into Chase
combining HARQ (CC-HARQ) and incremental redundancy HARQ (IR-HARQ). In CC-
HARQ, receiver preserves observations of the failed packet and request
retransmission of full packet. The rceiever Chase combines [4] observations of
the failed packet and retransmitted packet to decode packet. In the event of
packet failure under IR-HARQ, receiver request retransmission of more parity
information to recover from error. In response to retransmission request,
transmitter sends more parity bits lowering code rate of FEC. The receiver
combines new parity bits with buffer observation for FEC decoding.

Most of the research conducted on HARQ focused on ARQ and FEC [3, 5] without
exploring modulation layer. Throughput of capacity achieving codes FECs such
as LDPC and Turbo codes is optimized for Rayleigh fading channel in [6] with
ARQ and HARQ protocols. Mutual information based performance analysis for HARQ
over Rayleigh fading channel is provided in [7]. Optimal power allocation for
Chase combining based HARQ is optimized in [8, 9, 10, 11, 12]. Without
exploiting channel state information and frequency diversity of the frequency
selective channel, partial retransmission of the original symbol stream of a
failed packet is addressed in [13, 14, 15]. These methods retransmit punctured
packet in predetermined fashion. Furthermore, the complexity of joint
detection for _partial_ retransmission is [14, 13, 16] not tractable. Partial
retransmission of orthogonal space-time block (OSTB) coded [17] OFDM signaling
is proposed in [15]. In [18], for conventional ARQ protocol, full packet
retransmission at modulation layer is employed when channel gain is below a
threshold value without buffering observations of low SNR channel realization.
In a typical failed packet, there are small number of corrupted bits and
retransmission of full packet is not necessary. The receiver can recover from
errors by retransmission of potentially culprit bits. The OFDM signaling
allows to identify poor quality bits corresponding to the subcarriers that
have low signal-to-noise ratio (SNR). Selective retransmission at modulation
layer of OFDM signaling proposed in [19, 20, 21, 22] achieves throughput gain
as compared to conventional HARQ methods. Throughput optimization of selective
retransmission and performance analysis is not addressed in [19].

The motivation of selective retransmission owing to the fact that in the event
of failed packet under OFDM signaling at MAC layer, often receiver can recover
from error by retransmitting _partial_ information corresponding to the poor
quality subcarriers. An OFDM signaling allows retransmission of information
transmitted over poor quality subcarrier at physical layer (PHY) selectively.
After receiving copy of information symbols corresponding to poor quality
subcarriers, receiver jointly decode data in Chase combining fashion. In this
work, we propose low complexity and bandwidth efficient multiple selective
Chase combining (MSCC) and Chase combining with selective retransmission
(CCWS) methods in cross-layer fashion at modulation layer for OFDM signaling.
We also provide bit-error rate (BER) and throughput analysis in terms tight
upper BER bound and lower throughput bound, respectively for the proposed
retransmission schemes. The amount of information to be retransmitted for each
subcarrier in the event of failed packet is function of signal-to-noise ratio
(SNR) of the corresponding subcarrier. Using norm of channel gain for each
subcarriers as channel quality measure, we also optimized threshold $\tau$ on
channel norm for selective retransmission in order to maximize throughput. The
simulation results demonstrate that proposed methods offers substantial
throughput gain over conventional CC method in low SNR regime. Our results
also show that there is marginal gap between analytical bounds and simulation
results (Monte Carlo method ) for BER and throughput of the proposed methods.
The throughput gain of the proposed schemes also hold with FEC.

We organize our manuscript as follows. First, we present system model and
problem formulation (MSCC and CCWS) for selective Chase combining of OFDM
system in Section II-A and Section II-B. In Section III, we present BER
analysis of MSCC for one selective retransmission at modulation layer and CCWS
methods. Throughput analysis for SCC and CCWS are presented in Section IV.
Throughput Optimization is performed in Section V. We discuss our results in
Section VI. Finally, we conclude the purposed work in Section VII.

##  II SYSTEM MODEL and Problem Formulation

###  II-A System Model

We consider OFDM system for selective retransmission with $n_{r}$ receive
antennas. Transmitter transmits information symbol using OFDM signaling over
frequency selective channel of $L$ coefficients. An OFDM signaling converts
frequency selective channel $\mathbf{h}_{i}$, where $i=1,\cdots,n_{r}$, from
transmit antenna to the receive antenna $i$ into $N_{s}$ parallel channel[23].
Thus, a channel gain vector of $\ell$-th subcarrier between transmitter and
receiver pair is $H(\ell)=\big{[}H_{1}^{T}(\ell)\,\,\cdots
H_{n_{r}}^{T}(\ell)\big{]}^{T}$, where elements of channel vector $H(\ell)$
are generated by applying Fourier transformation matrix
$F\in\mathcal{C}^{N_{s}\times N_{s}}$ on frequency selective channel
$\mathbf{h}_{i}$. The elements of vector $H(\ell)$ are independent and
identically distributed (i.i.d.) with distribution $\mathcal{N}_{c}(0,1)$
[23]. Note that for $n_{r}=1,$ SIMO-OFDM system becomes SISO-OFDM system. The
matrix model of the received vector $\mathbf{y}(\ell)$ for the $\ell$-th
subcarrier can be written as

$\displaystyle\mathbf{y}(\ell)=H(\ell)\cdot s(\ell)+\mathbf{w}(\ell),$ (1)

where vector $\mathbf{w}(\ell)\sim\mathcal{N}_{c}(0,N_{0}I)$ is an additive
white Gaussian noise vector. A typical failed packet has few erroneous bits
and bits transmitted over OFDM subcarriers with small channel norm
$\|H(\ell)\|^{2}$ are more susceptible to the impairments. OFDM signaling
allows retransmission of targeted information symbols corresponding to the
poor quality subcarriers instead of unnecessary retransmission of full packet
[19]. The joint detection by combining observation of the first transmission
and subsequent selective retransmission of the poor quality subcarriers
enhances throughput of the transceiver under OFDM signaling. The selective
retransmitted information improves bit-error rate of the poor quality
subcarriers resulting into throughput gain. Note that retransmission of more
information does not increase throughput linearly. The amount of information
to be retransmitted is controlled by threshold $\tau$ on the channel norm
$\|H(\ell)\|^{2}$ of the $\ell$-th subcarriers. The optimization of threshold
$\tau$ in order to maximize throughput $\eta$ of the transceiver under
selective retransmission is the main focus of this work. Throughput of
selective retransmission is function of probability of error, which in fact is
function of $\tau$. We present uncoded BER analysis and throughput analysis in
Section III and IV, respectively, in order to optimize parameter $\tau$ in
Section V. Next, we provides description of proposed MSCC and CCWS methods.

Figure 1:  Selective retransmission model for SIMO-OFDM system at modulation
layer

###  II-B Problem Formulation

Now we present our proposed cross-layer MSCC and CCWS methods for OFDM
signaling. In both MSCC and CCWS methods, medium access control (MAC) layer
initiates retransmission in the event of CRC failure. Based on quality of the
subcarriers, instead of full retransmission under convetional Chase combining,
PHY layer initiates selective retransmission of information symbols
corresponding to the poor quality subcarriers along with NACK signal. We use
norm of subcarrier gain vector $\|{H(\ell)}\|^{2}$ as channel quality measure.
Note that OFDM signaling allows retransmission of information symbols
transmitted over poor quality subcarriers ($\|{H(\ell)}\|^{2}<\tau$ )
selectively avoiding overhead of retransmission of information symbols
corresponding to good quality subcarriers, where $\tau$ is threshold on
channel norm. The receiver feeds back partial channel state information (PCSI)
when each coherent time is elapsed. We assume that due to longer
retransmission delay, each retransmission encounters independent channel.
First, we present MSCC method under OFDM signaling.

####  II-B1 Multiple selective Chase combining

Motivation of selective Chase combining (SCC) stems from the fact that under
OFDM signaling, in the event of CRC failure, receiver can recover from errors
by selectively retransmitting information symbols corresponding to the poor
quality subcarriers without changing modulation. In SCC, instead of requesting
full retransmission for conventional Chase combining, PHY layer requests
retransmission of information symbols corresponding to the poor quality
subcarriers and retains observation of the failed data in a buffer. In
response to NACK signal, transmitter retransmits requested information
symbols. The receiver combines observation of the first transmission and
selective retransmission of the poor quality subcarriers for joint decoding in
Chase combining fashion and discards buffered observations.This is defined as
one round of SCC. Note that one round of conventional CC consists of first
transmission followed by full retransmission in the event of CRC failure. The
receiver decodes information from observations of first transmission and
selective retransmission jointly. Let $\mathcal{M}$ be the maximum allowed
number of rounds of CC at MAC layer. At the end of each retransmission round
of CC, receiver discards buffered observations and initiates new CC round (if
CRC check fails). In selective retransmission, the amount of information to be
retransmitted is controlled by threshold $\tau$ on the norm of subcarrier gain
$\|{H(\ell)}\|^{2}$. The threshold $\tau$ can be optimized to maximized
throughput for given average SNR. In SCC, one round of selective
retransmission consists of one round of CC. In [19] and [22], SCC method is
presented for SISO-OFDM and MIMO-OFDM under OSTB coded signals, respectively.
In this work, we optimize threshold $\tau$ on channel quality to maximize
throughput for SCC and present throughput analysis. We also propose MSCC
method that achieves significant throughput gain over SCC method. We present
provide tight upper BER performance bound and throughput analysis of proposed
MSCC method.

One retransmission round of MSCC method in PHY consists of at most $\Omega$
many conventional CC rounds at MAC layer. In one round of MSCC method, a
receiver continues to request retransmission of information symbols over
$\ell$th subcarrier of a failed packet until sum of norm-square of the
subcarrier gains from all iterations of selective retransmission of MSCC
method is larger than threshold $\tau$ or allowed number of retransmissions is
reached. If CRC check is satisfied, selective retransmission and CC rounds are
terminated followed by ACK signal to the transmitter. Let $J$ be the count of
retransmission rounds of CC at MAC layer. For the $I$-th iteration of
selective retransmission at modulation layer of the $J$-th retransmission
round at MAC layer, if needed at modulation layer, receiver request
retransmission of poor quality subcarrier which have
$\sum_{i=1}^{I}\|H_{i}(\ell)\|^{2}<\tau$ for $I=1,\ldots,\Omega$, where
$\Omega$ is the maximum number of retransmissions considered for joint
detection in $J$-th transmission round at MAC layer. At the end of each
conventional CC round at MAC layer, when MSCC method is enabled, receiver
jointly detects information bits by combining observations of all previous $I$
selective retransmissions iterations. After $I=\Omega$ iterations of selective
retransmission, which constitute one round of proposed MSCC method, receiver
discards buffered observation and initiates new selective retransmission
round.

Note that $I=1$ represents first transmission of a packet and
$\mathcal{T}_{I}$ is set of indices of poor quality subcarriers of the $I$-th
selective retransmission. The set of indices of poor quality subcarriers of
$I+1$-th selective retransmission $\mathcal{T}_{I+1}$ is subset of set of poor
quality indices $\mathcal{T}_{I}$ of the $I$-th selective retransmission. That
is, $\mathcal{T}_{I+1}\subseteq\mathcal{T}_{I}$. The receiver stops
retransmission of information symbols corresponding to the subcarriers in the
iterations of selective retransmission at modulation layer for which
$\sum_{i=1}^{I}\|H_{i}(\ell)\|^{2}\geq\tau$. Let $\beta_{I}$ be the
cardinality $|\mathcal{T}_{I}|$ of set $\mathcal{T}_{I}$, which represents
number of poor subcarriers for the $I$-th iteration of selective
retransmission of the $J$-th round at MAC layer. The flow graph of MSCC
selective retransmissions method is presented in Algorithm 1.

Algorithm 1 MSCC protocol

1:$J=1$ corresponds to the first transmission of the $k$-th packet

2:Set $I=1$

3:Detection and CRC check of first iteration of MSCC at PHY layer and $J$-th
round at MAC layer of $k$-th packet

4:if CRC satisfies then $k=k+1$, discard observations and go to 1

5:if $I=\Omega$ then go to 9

6:Selective retransmission for the $\beta_{I}$ subcarriers which have
$\displaystyle\sum_{i=1}^{I}\|H_{i}(\ell)\|^{2}<\tau$ and preservation of
observations

7:Joint detection from $\displaystyle N_{s}+\sum_{i=1}^{I}\beta_{i}$
observations

8:Set $I=I+1$, $J=J+1$ and go to 4

9:if $J={\mathcal{M}}$ then, discard observations, declare packet loss,
$k=k+1$ and go to 1

10:Discard observations and go to 2

We present tight upper bound on uncoded BER and lower bound on throughput of
MSCC method under OFDM signaling over Rayleigh fading channel in Section III
and Section IV-A, respectively. We also optimized threshold on channel quality
that controls amount of information need to be transmitted that maximizes
throughput in Section V.

####  II-B2 Chase combining with selective retransmission

Now we discuss proposed CCWS protocol for OFDM signaling. In proposed CCWS
method, MAC layer, similar to conventional CC method, initiates full
retransmission in the event of packet failure. The proposed CCWS method is
different from convention CC method in the sense that CCWS method employs a
selective retransmission corresponding to the poor quality subcarriers at PHY
level independent of MAC layer for the first transmission and full
retransmission. That is, first transmission of each packet and full
retransmission of a failed packet both undergo a selective retransmission of
the information symbols corresponding to the poor quality subcarriers at PHY
layer. The channel quality ($\|H(\ell)\|^{2}$) of subcarriers for the first
transmission of a packet under CCWS method is compared with threshold $\tau$
prior to decoding at PHY layer. The receiver request selective retransmission
request to the transmitter and preserves observations of the first
transmission of a packet. Then, receiver decodes information by fusing
observations of the first transmission and selective retransmission jointly.
If CRC check is not satisfied, receiver buffers observation and requests full
retransmission of failed packet by sending NACK signal. Similar to the first
transmission, receiver initiates selective retransmission of the information
symbols corresponding to poor quality subcarriers of full retransmission for
which $\|H_{c}(\ell)\|^{2}<\tau$. Note that $H(\ell)$ and $H_{c}(\ell)$ are
the gains of $\ell$-th subcarrier for the first transmission and full
retransmission, respectively. One round of CCWS consists of a full
transmission, a full retransmission and two selective retransmissions. After
response of the NACK signal from transmitter, the receiver decodes information
by combining observation of the one round of CCWS. If CRC check is satisfied,
receiver send ACK signal to the transmitter. Otherwise, receiver clears
buffered observations and initiates next round of CCWS. When maximum number of
rounds of CCWS are elapsed, packet failure is declared to RLC layer. In order
to maximize throughput of CCWS, we optimizes threshold $\tau$ on channel
quality.

Let $\beta_{1}$ be the number of subcarriers with channel gains
$\|H(\ell)\|^{2}<\tau$ for the first transmission of CCWS method. The
modulation layer initiates selective retransmission of symbols corresponding
to the $\beta_{1}$ subcarriers prior to joint detection. Due to selective
retransmission and joint detection at modulation layer, BER performance
$P_{e_{1}}$ of the first transmission of CCWS method at MAC layer is same as
that of MSCC method with $\Omega=1$.

Note that channel vector of the poor quality subcarriers of OFDM signaling of
CCWS method of the first transmission as a result of selective retransmission
is stack of two channel realizations similar to SCC. The decoded
$N_{s}\mbox{log}_{2}M$ bits are processed for CRC check. In the event of CRC
failure for the first transmission of a packet, receivers preserves
observations of the first transmission ($\beta_{1}+N_{s}$ symbols) and
initiates full retransmission request of packet. The modulation layer
initiates selective retransmission of $\beta_{2}$ poor quality subcarrier for
which $\|H_{c}(\ell)\|^{2}<\tau$ similar to the first transmission, where
$E\left[\beta_{1}\right]=E\left[\beta_{2}\right]=P(\|H_{1}(\ell)\|^{2}<\tau)=P(\|H_{c}(\ell)\|^{2}<\tau)=m$.
The joint detection as a result of full retransmission at MAC layer processes
$\beta_{1}+\beta_{2}+2N_{s}$ observations to decode $N_{s}\mbox{log}_{2}M$
bits and achieves BER performance $P_{e_{2}}$. Let $\mathcal{M}$ be the
maximum number of allowed rounds of transmissions of a packet and $J$ be the
round counter for the transmission of the $k$-th packet of CCWS method. The
flow graph of CCWS protocol for the $k$-th packet is presented in Algorithm 2.
We present tight upper bound on uncoded BER and lower bound on throughput of
CCWS method under OFDM signaling over Rayleigh fading channel in Section III-B
and Section IV-B, respectively. We also optimized threshold on channel quality
that controls amount of information need to be transmitted that maximizes
throughput in Section V.

Algorithm 2 CCWS protocol

1:$J=1$ corresponds to the first transmission of the $k$-th packet

2:Selective retransmission of $\beta_{1}$ subcarriers and preservation of
$N_{s}$ observations

3:Joint decoding from $N_{s}+\beta_{1}$ observations and CRC check

4:if CRC satisfies then $k=k+1$, discard observations and go to 1

5:NACK signal for full retransmission of packet at MAC layer and preservation
$\beta_{1}+N_{s}$ observations.

6:Selective retransmission of $\beta_{2}$ poor quality subcarriers
corresponding to full retransmission and preservation of $N_{s}$ observations

7:Joint detection from $\beta_{1}+\beta_{2}+2N_{s}$ observations

8:if CRC satisfies then $k=k+1$, discard observations and go to 1

9:if $J>\mathcal{M}$ then declare packet loss, discard observation, $k=k+1$
and go to 1

10:$J=J+1$, discard observation and go to 2

Performance analysis of SCC and CCWS is presented next.

##  III Performance Analysis

In this section, we present tight upper bounds on BER of joint detection for
MSCC and CCWS methods at PHY layer. We use these tight upper bound on BER to
optimize throughput of the proposed selective retransmission methods in
Section IV. We first derive upper bound on BER of MSCC method discussed in
Section II-B1 for $\Omega=1$ (SCC).

###  III-A BER analysis of SCC

One round of MSCC consists of $\Omega$ round of conventional CC method. In
case of CRC failure of the first transmission, receiver buffers observations
corresponding to the first transmission and requests retransmission of the
poor quality subcarrier as shown in Figure 1. Let $h(t)$ and $h_{s}(t)$ be the
channel impulse responses of the first transmission and selective
retransmission, respectively shown in Figure 1. We assume that $h(t)$ and
$h_{s}(t)$ are independent due to longer retransmission delay. Note that norm
of subcarrier gain and norm of subcarriers channel vector are channel quality
measures for SISO-OFDM and SIMO-OFDM systems, respectively. In a typical
failed packet, there are fewer error bits and retransmission of potentially
culprit bits corresponding to the poor quality sub-carries can help receiver
to recover from error by performing joint detection. In SCC, PHY layer is
aware of retransmission due to cross-layer design and performs joint detection
for poor quality subcarriers combining observations from the first
transmission and subsequent selective retransmission.

Each element of the complex channel vector $H(\ell)$ of the $\ell$-th
subcarrier follows Gaussian distribution with mean zero and unit variance
obtained by applying $N_{s}$ point DFT matrix on frequency selective channel
$h(t)$ of the first transmission of a CC round . The estimate of information
symbol over the $\ell$-th subcarrier after equalization is

$\displaystyle\hat{s}(\ell)=s(\ell)+\frac{H^{*}(\ell)\mathbf{w}(\ell)}{\|H(\ell)\|^{2}}=s(\ell)+\mathbf{u}(\ell),$
(2)

where $\mathbf{u}(\ell)$ is the effective noise with distribution
$\mathcal{N}(0,\|H(\ell)\|^{-2}N_{0})$. Note that channel $H(\ell)$ and
additive noise $\mathbf{w}(\ell)$ are scalars and vectors for SISO-OFDM and
SIMO-OFDM systems, respectively. Selective retransmission method at modulation
layer retransmits _partial copy_ of symbols by targeting symbols corresponding
to the poor quality subcarriers for retransmission. Thus, information symbols
corresponding to the subcarriers with channel norm-square $\|H(\ell)\|^{2}$
less than threshold $\tau$ are selected for retransmission. That is,
$(\|H(\ell)\|^{2}\leq\tau)$. The symbols transmitted over subcarriers for
which $\|H(\ell)\|^{2}>\tau$ are omitted from retransmission due to the fact
that they are less susceptible to impairments. In this work, we search for
threshold $\tau$, which optimizes the amount of information to be
retransmitted in order to maximize throughput of an OFDM system.

We denote the outcomes $\|H(\ell)\|^{2}>\tau$ and $\|H(\ell)\|^{2}\leq\tau$ by
the events $\xi$ and $\xi^{c}$, respectively. The probabilities of events
$\xi$ and $\xi^{c}$ are

$\displaystyle P(\xi^{c})$
$\displaystyle=P(\|H(\ell)\|^{2}\leq\tau)\;\mbox{and}\;P(\xi)=P(\|H(\ell)\|^{2}>\tau),$

respectively, where random variable $\chi_{1}=\|H(\ell)\|^{2}$ has Chi-square
distribution of degree $2n_{r}$ [24] and
$P(\xi^{c})=1-P(\xi)=P(\chi_{1}\leq\tau)$ . Now for Rayleigh fading channel,

$\displaystyle P(\xi^{c})$
$\displaystyle=1-\exp\left(-\frac{\tau}{2\sigma^{2}}\right)\sum_{k=0}^{n_{r}-1}\frac{1}{k!}\,\left(\frac{\tau}{2\sigma^{2}}\right)^{k},$
(3) $\displaystyle P(\xi)$
$\displaystyle=P(\chi_{1}>\tau)=\exp\left(-\frac{\tau}{2\sigma^{2}}\right)\sum_{k=0}^{n_{r}-1}\frac{1}{k!}\,\left(\frac{\tau}{2\sigma^{2}}\right)^{k},$
(4)

where $\sigma^{2}=\frac{1}{2}$ is variance of real and imaginary components of
complex channel coefficient of a subcarrier. When event $\xi^{c}$ occurs,
receiver performs joint detection by combing observation of the first
transmission and subsequent retransmission. The combined channel response
$\mathcal{H}(\ell)=\big{[}H^{T}(\ell)\,\,H_{s}^{T}(\ell)\big{]}^{T}$ is
constructed by stacking channel of the first transmission and selective
retransmission. If there are $\beta_{1}$ many subcarriers with
$\|H(\ell)\|^{2}\leq\tau$, then there will be joint detection for $\beta_{1}$
subcarrier for SCC method ($\beta_{1}=N_{s}\implies\mbox{full
retransmission}$). Estimate of information symbol as a result of joint
detection is

$\displaystyle\hat{s}(\ell)=$ $\displaystyle
s(\ell)+\underbrace{\|\mathcal{H}(\ell)\|^{-2}\mathcal{H}^{H}(\ell)\tilde{\mathbf{w}}(\ell)}_{\tilde{\mathbf{u}}(\ell)},$
(5)

where
$\tilde{\mathbf{u}}(\ell)\thicksim\mathcal{N}_{C}(0,\|\mathcal{H}(\ell)\|^{-2}N_{0})$
and $\tilde{\mathbf{w}}(\ell)\thicksim\mathcal{N}_{C}(0,\,N_{0}I_{n_{r}})$.
Note that the random variable $\|\mathcal{H}(\ell)\|^{2}$ also has Chi-square
distribution of degree $4n_{r}$. Also that
$\|\mathcal{H}(\ell)\|^{2}=\chi_{1}+\chi_{2}$, where Chi-square random
variables $\chi_{1}$ and $\chi_{2}=\|H_{s}(\ell)\|^{2}$ are i.i.d. of degree
$2n_{r}$ each. The bit-error probability of joint detection for selective
retransmission over Rayleigh fading channel is

$\displaystyle P_{e_{s}}$
$\displaystyle=E_{H}\Big{[}P(\xi^{c})\,\,\,P_{e|\xi^{c}}+P(\xi)\,\,\,P_{e|\xi}\Big{]},$
(6)

where $P_{e|\xi}$ and $P_{e|\xi^{c}}$ are the conditional bit-error
probabilities of single detection and joint detection, respectively. Now we
evaluate threshold $\tau$ to achieve target BER. Let $P_{e\tau}$ be the bit-
error rate and $\tau$ be the corresponding channel norm of a subcarrier. Then,
for constellation set of cardinality $M$, relationship between $P_{e\tau}$ and
$\tau$ is

$\displaystyle P_{e\tau}=c\cdot
Q\left(\sqrt{g\frac{\tau}{N_{0}}}\right)\Longrightarrow\tau=\frac{N_{0}}{g}\left(Q^{-1}\left(\frac{P_{e\tau}}{c}\right)\right)^{2},$
(7)

where $c$ and $g$ are modulation constants of constellation set [23]. Since
$Q(.)$ is monotonically decreasing function SNR, in order to achieve BER
smaller than $P_{b\tau}$, $\|H(\ell)\|^{2}>\tau$. That is, $P_{e|\xi}\leq
P_{e\tau}$. Therefore, threshold on the gain of a subcarrier is

$\displaystyle\|H(\ell)\|^{2}$
$\displaystyle\geq\tau=\frac{N_{0}}{g}\left(Q^{-1}\left(\frac{P_{b\tau}}{c}\right)\right)^{2}.$
(8)

We use this threshold for selective retransmission at modulation layer to
ensure $P_{e}\leq P_{e\tau}$. In Section V, we search for threshold $\tau$
that maximized throughput . The upper bound on probability of error for joint
detection of selective Chase combining (SCC) is

$\displaystyle P_{e_{s}}$ $\displaystyle=P(\xi^{c})\cdot c\cdot
E_{H|\xi^{c}}\left[Q\left(\sqrt{g\frac{\chi_{1}}{N_{0}}}\right)\right]+P(\xi)\cdot
c\cdot
E_{\mathcal{H}|\xi}\left[Q\left(\sqrt{g\frac{\chi_{1}+\chi_{2}}{N_{0}}}\right)\right],$
(9)

where $E_{H|\xi^{c}}$ and $E_{\mathcal{H}|\xi}$ are conditional expectations.
The subscript $s$ in (9) represent selective Chase combining method. The
conditional probability density function $f_{\chi_{1}|\xi^{c}}(x_{1})$ of
$f_{\chi_{1}}(x_{1})$ when $\chi_{1}>\tau$ is
$f_{\chi_{1}|\xi^{c}}(x_{1})=\dfrac{f_{\chi_{1}}(x_{1})}{P(\xi^{c})},\mbox{
where }\chi_{1}>\tau$. In order to solve first term of (9) [25] , we have

$\displaystyle\begin{split}\displaystyle
E_{H|\xi^{c}}\left[Q\left(\sqrt{g\frac{\chi_{1}}{N_{0}}}\right)\right]=&\int_{\tau}^{\infty}Q\left(\sqrt{g\frac{x_{1}}{N_{0}}}\right)\dfrac{f_{\chi_{1}}(x_{1})}{P(\xi^{c})}dx_{1}.\end{split}$
(10)

Similarly,

$\displaystyle\begin{split}\displaystyle
E_{H|\xi}\left[Q\left(\sqrt{g\frac{\chi_{1}+\chi_{2}}{N_{0}}}\right)\right]=&\int_{x_{1}=0}^{\tau}\int_{x_{2}=0}^{\infty}Q\left(\sqrt{g\frac{x_{1}+x_{2}}{N_{0}}}\right)\dfrac{f_{\chi_{1}}(x_{1})f_{\chi_{2}}(x_{2})}{P(\xi)}dx_{2}dx_{1}.\end{split}$
(11)

The upper bound on $P_{es}$ in (9) using approximation of Q-function [26],
(10) and (11) can be written as [22, 25]

$\displaystyle\begin{split}P_{e_{s}}\leq&\frac{c}{12}\int_{\tau}^{\infty}\exp(-g\frac{x_{1}}{2N_{0}})f_{\chi_{1}}(x_{1})dx_{1}+\frac{c}{4}\int_{\tau}^{\infty}\exp(-g\frac{4x_{1}}{3.2N_{0}})f_{\chi_{1}}(x_{1})dx_{1}\\\
&+\frac{c}{12}\int_{0}^{\tau}\exp(-g\frac{x_{1}}{2N_{0}})f_{\chi_{1}}(x_{1})dx_{1}.\int_{0}^{\infty}\exp(-g\frac{x_{2}}{2N_{0}})f_{\chi_{2}}(x_{2})dx_{2}\\\
&+\frac{c}{4}\int_{0}^{\tau}\exp(-g\frac{4x_{1}}{3.2N_{0}})f_{\chi_{1}}(x_{1})dx_{1}.\int_{0}^{\infty}\exp(-g\frac{4x_{2}}{3.2N_{0}})f_{\chi_{2}}(x_{2})dx_{2}\end{split}$
(12)

Simplifying (12), we have upper bound on probability of error of the joint
detection as follows

$\displaystyle\begin{split}P_{e_{s}}\leq&\frac{c}{12}\left(\frac{\rho}{\sigma}\right)^{2n_{r}}\exp\left(-\frac{\tau}{2\rho^{2}}\right)\sum_{k=0}^{n_{r}-1}\frac{1}{k!}\left(\frac{\tau}{2\rho^{2}}\right)^{k}+\frac{c}{4}\left(\frac{\rho_{1}}{\sigma}\right)^{2n_{r}}\exp\left(-\frac{\tau}{2\rho_{1}^{2}}\right)\sum_{k=0}^{n_{r}-1}\frac{1}{k!}\left(\frac{\tau}{2\rho_{1}^{2}}\right)^{k}\\\
&+\frac{c}{12}\left(\frac{\rho}{\sigma}\right)^{4n_{r}}\left(1-\exp\left(-\frac{\tau}{2\rho^{2}}\right)\sum_{k=0}^{n_{r}-1}\frac{1}{k!}\left(\frac{\tau}{2\rho^{2}}\right)^{k}\right)\\\
&+\frac{c}{4}\left(\frac{\rho_{1}}{\sigma}\right)^{4n_{r}}\left(1-\exp\left(-\frac{\tau}{2\rho_{1}^{2}}\right)\sum_{k=0}^{n_{r}-1}\frac{1}{k!}\left(\frac{\tau}{2\rho_{1}^{2}}\right)^{k}\right),\end{split}$
(13)

where $\rho=\sqrt{\frac{\sigma^{2}N_{o}}{g\sigma^{2}+N_{o}}}$,
$\rho_{1}=\sqrt{\frac{\sigma^{2}N_{o}}{g_{1}\sigma^{2}+N_{o}}}$ and
$g_{1}=\frac{4g}{3}$. For 4-QAM constellation $g=2$ and
$c=\frac{2}{\text{log}_{2}M}$ [24]. Note that in (13), $P_{e_{j}}$ is function
of $\tau$, which controls amount of information to be retransmitted in
selective retransmission.The probability of error of the first transmission
with $n_{r}$ receiver antenna can be derived from (13) by setting parameter
$\tau\rightarrow 0$, which means that probability of selecting a subcarrier
for retransmission is almost zero. For $\tau\rightarrow 0$, (13) becomes

$\displaystyle\begin{split}P_{e}\leq&\frac{c}{12}\left(\frac{\rho}{\sigma}\right)^{2n_{r}}+\frac{c}{4}\left(\frac{\rho_{1}}{\sigma}\right)^{2n_{r}},\end{split}$
(14)

which is probability of error of the first transmission with $n_{r}$ receive
antennas. Similarly, for full retransmission, $\tau\rightarrow\infty$ and BER
upper bound can be achieved by setting $\tau=\infty$ in (13). Thus,

$\displaystyle\begin{split}P_{e_{f}}\leq&\frac{c}{12}\left(\frac{\rho}{\sigma}\right)^{4n_{r}}+\frac{c}{4}\left(\frac{\rho_{1}}{\sigma}\right)^{4n_{r}}.\end{split}$
(15)

In low SNR regime, receiver needs more information during retransmission to
recover from errors. Therefore, threshold $\tau$ on channel norm has large
value in low SNR regime. Throughput $\eta$ is function of probability of error
of first transmission $P_{e}$ and probability of error of joint detection
$P_{e_{s}}$, which is function of SNR and threshold $\tau$. We use (13) and
(14) to search of optimal value of $\tau$ to maximize throughput of SCC method
in Section IV . Now we provide BER analysis of CCWS method.

###  III-B BER analysis of CCWS

The probability of bit-error $P_{e_{1}}$ of the first transmission of CCWS is
in fact $P_{e{}_{s}}$ for SCC given in (13). There are two outcomes from the
first transmission related to the $\ell$-th OFDM subcarrier denoted by $\xi$
and $\xi^{c}$ with probabilities $P(\xi)=P(\|H(\ell)\|^{2}>\tau)$ and
$P(\xi^{c})=P(\|H(\ell)\|^{2}\leq\tau)=1-P(\xi)$ similar to SCC. It is
important to note that BER $P_{e_{1}}$of the first transmission of CCWS is
upper bounded by (13). That is, $P_{e_{1}}=P_{e_{s}}$ .

Now we evaluate upper bound on BER for joint detection when CRC failure occurs
for CCWS method. Let $H_{c}(\ell)$ be the channel of the $\ell$-th subcarrier
during full retransmission of packet of Chase combining protocol. Similar to
SCC method, modulation layer initiates retransmission of the poor subcarriers
$\|H_{c}(\ell)\|^{2}<\tau$ for the full retransmission. We denote the event
$\|H_{c}(\ell)\|^{2}\leq\tau$ by $\xi_{c}^{c}$. The subscript $c$ represents
retransmission of Chase combining and superscript with event $\xi_{c}$
represent complement of $\xi_{c}$. The channel realizations during first
transmission and conventional Chase combining are $H(\ell)$ and $H_{c}(\ell)$,
respectively, are i.i.d. and
$P(\xi)=P(\|H(\ell)\|^{2}>\tau)=P(\xi_{c})=P(\|H_{c}(\ell)\|^{2}>\tau)$. In
response to full retransmission of failed packet, receiver combines
$\beta_{s}+N_{s}$ observations of the first transmission and $\beta_{c}+N_{s}$
of full retransmission for joint detection. Note that $\beta_{c}$ is the
number of poor subcarriers retransmitted at modulation layer due to full
retransmission at data link layer. Probability of error of joint detection is
as follows:

$\displaystyle\begin{split}P_{e_{2}}=&E_{H}\big{[}P(\xi_{1})P_{e|\xi_{1}}+P(\xi_{2})P_{e|\xi_{2}}+P(\xi_{3})P_{e|\xi_{3}}+P(\xi_{4})P_{e|\xi_{4}}\big{]},\end{split}$
(16)

where events $\xi_{1},\;\xi_{2},\,\xi_{3}$ and $\xi_{4}$ defined as follows:

  1. 1.

Event $\xi_{1}$ occurs when $\|H(\ell)\|^{2}>\tau$ and
$\|H_{c}(\ell)\|^{2}>\tau$. The resulting joint channel for joint detection of
CCWS into $\mathcal{H}_{1}(\ell)=[H^{T}(\ell)\;\;H_{c}^{T}(\ell)]^{T}$, where
$H(\ell)$ and $H_{c}(\ell)$ are i.i.d. channel realization with Gaussian
distribution of zero mean and unit variance. The channel norm
$\|\mathcal{H}_{1}(\ell)\|^{2}=\|H(\ell)\|^{2}+\|H_{c}(\ell)\|^{2}$ has Chi-
square distribution.

  2. 2.

Event $\xi_{2}$ occurs when $\|H(\ell)\|^{2}\leq\tau\mbox{ and
}\|H_{c}(\ell)\|^{2}>\tau$. The resulting joint channel response for joint
detection of CCWS into
$\mathcal{H}_{2}(\ell)=[H^{T}(\ell)\;\;H_{s}^{T}(\ell)\;\;H_{c}^{T}(\ell)]^{T}$.

  3. 3.

Event $\xi_{3}$ occurs when $\|H(\ell)\|^{2}>\tau\mbox{ and
}\|H_{c}(\ell)\|^{2}\leq\tau$. The resulting joint channel response for joint
detection of CCWS into
$\mathcal{H}_{3}(\ell)=[H^{T}(\ell)\;\;H_{c}^{T}(\ell)\;\;H_{sc}^{T}(\ell)]^{T}$,
where $H_{sc}(\ell)$ is channel gain of the $\ell$-th subcarrier selected for
retransmission during full retransmission of failed packet at data link layer.

  4. 4.

Event $\xi_{4}$ occurs when $\|H(\ell)\|^{2}\leq\tau$ and
$\|H_{c}(\ell)\|^{2}\leq\tau$. The resulting joint channel for joint detection
of CCWS into
$\mathcal{H}_{4}(\ell)=[H^{T}(\ell)\;\;H_{s}^{T}(\ell)\;\;H_{c}^{T}(\ell)\;\;H_{sc}^{T}(\ell)]^{T}$,
where $H_{s}(\ell)$ and $H_{sc}(\ell)$ are the channels corresponding to the
selective channels from the first transmission and full retransmission of CCWS
method. Note that random variables $\|H_{s}(\ell)\|^{2}$ and
$\|H_{sc}(\ell)\|^{2}$ are i.i.ds. with Chi-square distribution of degree
$2n_{r}$ each.

Second and third terms in (16) are equivalent due to the fact that
$P(\xi_{2})=P(\xi_{3})$ and random variables $\|\mathcal{H}_{2}(\ell)\|$ and
$\|\mathcal{H}_{3}(\ell)\|$ are i.i.ds. Therefore,
$E_{H}\Big{[}P(\xi_{2})P_{e|\xi_{2}}+P(\xi_{3})P_{e|\xi_{3}}\Big{]}=2E_{H}\Big{[}P(\xi_{2})P_{e|\xi_{2}}\Big{]}$.
Note that all channel realizations of the $\ell$-th subcarrier of an OFDM
system are i.i.d. with Gaussian distribution of mean zero and unit variance.
In order to achieve joint upper bound on BER for joint detection of CCWS
method, we rewrite (16) as follows:

$\displaystyle
P_{e_{2}}=cE_{H}\Bigg{[}P(\xi_{1})Q\left(\sqrt{\frac{g\|\;\mathcal{H}_{1}(\ell)\|^{2}}{N_{0}}}\right)+2P(\xi_{2})\cdot$
$\displaystyle
Q\left(\negmedspace\negmedspace\sqrt{\frac{g\|\;\mathcal{H}_{2}(\ell)\|^{2}}{N_{0}}}\negmedspace\right)\negmedspace\negmedspace+\negmedspace
P(\xi_{4})Q\left(\negmedspace\negmedspace\sqrt{\frac{g\|\;\mathcal{H}_{4}(\ell)\|^{2}}{N_{0}}}\right)\negmedspace\negmedspace\Bigg{]}.$
(17)

Note that $\|\mathcal{H}_{1}(\ell)\|^{2}=\chi+\chi_{c}$ in the first term of
(17) is sum if two i.i.d. chi-square random variables , where $\chi>\tau$ and
$\chi_{c}>\tau$. Using approximation of Q-function in [26] and, following (10)
and (11), we have

$\displaystyle\begin{split}E_{H}\left[P(\xi_{1})P_{e}|\xi_{1}\right]=&cE_{H}\left[P(\xi_{1})Q\left(\sqrt{\frac{g\|\mathcal{H}_{1}(\ell)\|^{2}}{N_{0}}}\right)\right]\\\
\leq&\frac{c}{12}\int_{\tau}^{\infty}\exp(-g\frac{x}{2N_{0}})f_{X}(x)dx.\int_{\tau}^{\infty}\exp(-g\frac{x_{c}}{2N_{0}})f_{X_{c}}(x_{c})dx_{c}+\\\
&\frac{c}{4}\int_{\tau}^{\infty}\exp(-g\frac{4x}{3.2N_{0}})f_{X}(x).dx\int_{\tau}^{\infty}\exp(-g\frac{4x_{c}}{3.2N_{0}})f_{X_{c}}(x_{c})dx_{c}\\\
=&\frac{c}{12}\left(\frac{\rho}{\sigma}\right)^{4n_{r}}\left(\exp\left(-\frac{\tau}{2\rho^{2}}\right)\sum_{k=0}^{n_{r}-1}\frac{1}{k!}\left(\frac{\tau}{2\rho^{2}}\right)^{k}\right)^{2}+\\\
&\frac{c}{4}\left(\frac{\rho_{1}}{\sigma}\right)^{4n_{r}}\left(\exp\left(-\frac{\tau}{2\rho_{1}^{2}}\right)\sum_{k=0}^{n_{r}-1}\frac{1}{k!}\left(\frac{\tau}{2\rho_{1}^{2}}\right)^{k}\right)^{2}.\end{split}$
(18)

Similarly,

$\displaystyle\begin{split}2E_{H}&\left[P(\xi_{2})P_{e}|\xi_{2}\right]=2cE_{H}\left[P(\xi_{2})Q\left(\sqrt{\frac{g\|\mathcal{H}_{2}(\ell)\|^{2}}{N_{0}}}\right)\right]\\\
\leq&\frac{c}{6}\int_{0}^{\tau}\exp(-g\frac{x}{2N_{0}})f_{X}(x)dx.\int_{0}^{\infty}\exp(-g\frac{x_{s}}{2N_{0}})f_{X_{s}}(x)dx_{s}\int_{\tau}^{\infty}\exp(-g\frac{x_{c}}{2N_{0}})f_{X_{c}}(x_{c})dx_{c}+\\\
&\frac{c}{2}\int_{0}^{\tau}\exp(-g\frac{4x}{3.2N_{0}})f_{X}(x)dx.\int_{0}^{\infty}\exp(-g\frac{4x_{s}}{3.2N_{0}})f_{X_{s}}(x)dx_{s}\int_{\tau}^{\infty}\exp(-g\frac{4x_{c}}{3.2N_{0}})f_{X_{c}}(x_{c})dx_{c},\end{split}$
(19)

where $\chi<\tau$, $\chi_{s}\in\mathcal{R}$ and $\chi_{c}>\tau$. Simplifying
(19), we have

$\displaystyle\begin{split}2E_{H}&\left[P(\xi_{2})P_{e}|\xi_{2}\right]\leq\frac{c}{6}\left(\frac{\rho}{\sigma}\right)^{6n_{r}}\left(1-\exp\left(-\frac{\tau}{2\rho^{2}}\right)\sum_{k=0}^{n_{r}-1}\frac{1}{k!}\left(\frac{\tau}{2\rho^{2}}\right)^{k}\right)\left(\exp\left(-\frac{\tau}{2\rho^{2}}\right)\sum_{k=0}^{n_{r}-1}\frac{1}{k!}\left(\frac{\tau}{2\rho^{2}}\right)^{k}\right)+\\\
&\frac{c}{2}\left(\frac{\rho_{1}}{\sigma}\right)^{6n_{r}}\left(1-\exp\left(-\frac{\tau}{2\rho_{1}^{2}}\right)\sum_{k=0}^{n_{r}-1}\frac{1}{k!}\left(\frac{\tau}{2\rho_{1}^{2}}\right)^{k}\right)\left(\exp\left(-\frac{\tau}{2\rho_{1}^{2}}\right)\sum_{k=0}^{n_{r}-1}\frac{1}{k!}\left(\frac{\tau}{2\rho_{1}^{2}}\right)^{k}\right).\end{split}$
(20)

Also, it can be shown that

$\displaystyle\begin{split}E_{H}&\left[P(\xi_{4})P_{e}|\xi_{4}\right]=cE_{H}\left[P(\xi_{4})Q\left(\sqrt{\frac{g\|\mathcal{H}_{4}(\ell)\|^{2}}{N_{0}}}\right)\right]\\\
\leq&\frac{c}{12}\Big{(}\frac{\rho}{\sigma_{h}}\Big{)}^{8n_{r}}\Bigg{(}1-\exp\Big{(}-\frac{\tau}{2\rho^{2}}\Big{)}\sum_{k=0}^{n_{r}-1}\frac{1}{k!}\Big{(}\frac{\tau}{2\rho^{2}}\Big{)}^{k}\Bigg{)}^{2}+\\\
&\frac{c}{4}\Big{(}\frac{\rho_{1}}{\sigma_{h}}\Big{)}^{8n_{r}}\Bigg{(}1-\exp\Big{(}-\frac{\tau}{2\rho_{1}^{2}}\Big{)}\sum_{k=0}^{n_{r}-1}\frac{1}{k!}\Big{(}\frac{\tau}{2\rho_{1}^{2}}\Big{)}^{k}\Bigg{)}^{2},\end{split}$
(21)

where $\chi<\tau$, $\chi_{s}\in\mathcal{R}$, $\chi_{c}<\tau$ and
$\chi_{sc}\in\mathcal{R}$. Now using (18), (20) and (21) in (17), we have

$\displaystyle\begin{split}P_{e_{2}}&\leq\frac{c}{12}\Big{(}\frac{\rho}{\sigma}\Big{)}^{4n_{r}}\Bigg{(}\exp\Big{(}-\frac{\tau}{2\rho^{2}}\Big{)}\sum_{k=0}^{n_{r}-1}\frac{1}{k!}\Big{(}\frac{\tau}{2\rho^{2}}\Big{)}^{k}\Bigg{)}^{2}+\frac{c}{4}\Big{(}\frac{\rho_{1}}{\sigma}\Big{)}^{4n_{r}}\Bigg{(}\exp\Big{(}-\frac{\tau}{2\rho_{1}^{2}}\Big{)}\sum_{k=0}^{n_{r}-1}\frac{1}{k!}\Big{(}\frac{\tau}{2\rho_{1}^{2}}\Big{)}^{k}\Bigg{)}^{2}+\\\
&\frac{c}{6}\Big{(}\frac{\rho}{\sigma}\Big{)}^{6n_{r}}\Bigg{(}\exp\Big{(}-\frac{\tau}{2\rho^{2}}\Big{)}\sum_{k=0}^{n_{r}-1}\frac{1}{k!}\Big{(}\frac{\tau}{2\rho^{2}}\Big{)}^{k}\Bigg{)}\Bigg{(}1-\exp\Big{(}-\frac{\tau}{2\rho^{2}}\Big{)}\sum_{k=0}^{n_{r}-1}\frac{1}{k!}\Big{(}\frac{\tau}{2\rho^{2}}\Big{)}^{k}\Bigg{)}+\\\
&\frac{c}{2}\Big{(}\frac{\rho_{1}}{\sigma}\Big{)}^{6n_{r}}\Bigg{(}\exp\Big{(}-\frac{\tau}{2\rho_{1}^{2}}\Big{)}\sum_{k=0}^{n_{r}-1}\frac{1}{k!}\Big{(}\frac{\tau}{2\rho_{1}^{2}}\Big{)}^{k}\Bigg{)}\Bigg{(}1-\exp\Big{(}-\frac{\tau}{2\rho_{1}^{2}}\Big{)}\sum_{k=0}^{n_{r}-1}\frac{1}{k!}\Big{(}\frac{\tau}{2\rho_{1}^{2}}\Big{)}^{k}\Bigg{)}+\\\
&\frac{c}{12}\Big{(}\frac{\rho}{\sigma_{h}}\Big{)}^{8n_{r}}\Bigg{(}1-\exp\Big{(}-\frac{\tau}{2\rho^{2}}\Big{)}\sum_{k=0}^{n_{r}-1}\frac{1}{k!}\Big{(}\frac{\tau}{2\rho^{2}}\Big{)}^{k}\Bigg{)}^{2}\\\
&+\frac{c}{4}\Big{(}\frac{\rho_{1}}{\sigma_{h}}\Big{)}^{8n_{r}}\Bigg{(}1-\exp\Big{(}-\frac{\tau}{2\rho_{1}^{2}}\Big{)}\sum_{k=0}^{n_{r}-1}\frac{1}{k!}\Big{(}\frac{\tau}{2\rho_{1}^{2}}\Big{)}^{k}\Bigg{)}^{2}.\end{split}$
(22)

In next section, we present throughput analysis and optimization with respect
parameter $\tau$ for SCC and CCWS.

##  IV Throughput Analysis

Now we present throughput analysis of the proposed SCC and CCWS methods. In
throughput analysis, we consider infinite many rounds of retransmission. One
round of retransmission consists of first transmission followed by a
retransmission. The receiver preserves observation corresponding to the first
transmission for joint detection. We consider joint detection for
retransmission by combining observation from the first transmission and
subsequent _partial or full_ retransmission. In practice, transceiver pair
continues retransmission rounds until error free packet is received or maximum
number of retransmission rounds are reached. For throughput analysis, we
follow conventional definition of throughput $\eta$, which is ratio of error-
free information bits received $k$ to the total number of bits transmitted $n$
( $\eta=\frac{k}{n}$). Let $P_{e}$ and $P_{e_{s}}$ be the bit-error
probabilities of the first transmission and joint detection followed by
retransmission, respectively. Assuming that each bit in the frame is
independent, probability of receiving an error-free packet of length $L_{f}$
with probability of bit-error $P_{e}$ is $p_{c}=(1-P_{e})^{L_{f}}$. Then
probability of receiving a bad packet is $p_{\epsilon}=1-p_{c}$. Similarly,
for joint detection, $p_{c_{s}=(1-P_{e_{s}}})^{L_{f}}$ and
$p_{\epsilon_{s}}=1-p_{c_{s}}$. Next, we present throughput analysis of SCC
method.

###  IV-A Throughput analysis of SCC

Now we provide throughput analysis of SCC method. In response to the NACK
signal in the event of packet failure, receiver initiates selective
retransmission of poor quality subcarriers instead of full retransmission. Let
$m$ be the fraction of subcarriers selected for retransmission. That is, at a
given SNR, fraction $m=P(\|H(\ell)\|^{2}\leq\tau)$. In SCC, each
retransmission round consists of first transmission followed by _partial_
retransmission resulting in to transmission of $k(1+m)$ bits in each round.
Let $p_{c}$ and $p_{c_{s}}$ be the probabilities of receiving correct frame
from the first transmission and joint detection of SCC method, respectively.
The probabilities of receiving bad frame from the first transmission and joint
detection of SCC are $p_{\epsilon}$ and $p_{\epsilon_{s}}$, respectively, for
SCC. The probabilities of frame error $p_{\epsilon}$ and $p_{\epsilon_{s}}$
are functions of BER given in (13) and (14), respectively. Thus, expected
number of information bits $n$ to be transmitted to receive $k$ error-free
bits are [27]

$\displaystyle\begin{split}n=&kp_{c}+k(1+m)p_{\epsilon}p_{c_{s}}+k(2+m)p_{\epsilon}p_{\epsilon_{s}}p_{c}\negthinspace\negthinspace+k(2+2m)p_{\epsilon}^{2}\\\
&p_{\epsilon_{s}}p_{c_{s}}\negthinspace\negthinspace+k(3+2m)p_{\epsilon}^{2}p_{\epsilon_{s}}^{2}p_{c}+k(3+3m)p_{\epsilon}^{3}p_{\epsilon_{s}}^{2}p_{c_{s}}\negthinspace\negthinspace+k(4+3m)\\\
&p_{\epsilon}^{3}p_{\epsilon_{s}}^{3}p_{c}+k(4+4m)p_{\epsilon}^{4}p_{\epsilon_{s}}^{3}p_{c_{s}}+k(5+4m)p_{\epsilon}^{4}p_{\epsilon_{s}}^{4}p_{c}+\cdots\end{split}$
(23)

Now rearranging (23), we have

$\displaystyle\begin{split}n=&kp_{c}\Big{(}1+(2+m)p_{\epsilon}p_{\epsilon_{s}}+(3+2m)p_{\epsilon}^{2}p_{\epsilon_{s}}^{2}+(4+3m)\\\
&p_{\epsilon}^{3}p_{\epsilon_{s}}^{3}\negthinspace\negthinspace\negthinspace\negthinspace+(5+4m)p_{\epsilon}^{4}p_{\epsilon_{s}}^{4}+\cdots\Big{)}\negthinspace\negthinspace+kp_{\epsilon}p_{c_{s}}\Big{(}(1+m)+(2+2m)\\\
&p_{\epsilon}p_{\epsilon_{s}}+(3+3m)p_{\epsilon}^{2}p_{\epsilon_{s}}^{2}+(4+4m)p_{\epsilon}^{3}p_{\epsilon_{s}}^{3}+\cdots\Big{)}\\\
=&kp_{c}\Big{(}1+(2+m)\alpha+(3+2m)\alpha^{2}+(4+3m)\alpha^{3}+\ldots\Big{)}\\\
&+kp_{\epsilon}p_{c_{s}}\Big{(}(1+m)+(2+2m)\alpha+(3+3m)\alpha^{2}+\\\
&(4+4m)\alpha^{3}+(5+5m)\alpha^{4}+\cdots\Big{)}\\\
=&k.p_{c}(1+m\alpha)(1+2\alpha+3\alpha^{2}+4\alpha^{3}+\cdots)\\\
&+kp_{\epsilon}p_{c_{s}}(1+m)\Big{(}1+2\alpha+3\alpha^{2}+4\alpha^{3}+\cdots\Big{)}\end{split}$
(24)

where $\alpha=p_{\epsilon}p_{\epsilon_{s}}$. Note that
$1+2\alpha+3\alpha^{2}+4\alpha^{3}+\cdots=\frac{1}{(1-\alpha)^{2}}$ [28].
Therefore,

$n=\frac{kp_{c}(1+m\alpha)+p_{\epsilon}p_{c_{s}}(1+m)}{(1-\alpha)^{2}}.$ (25)

Thus, throughput of SCC methods is

$\displaystyle\eta_{s}=$
$\displaystyle\dfrac{k}{n}=\dfrac{(1-\alpha)^{2}}{p_{c}(1+m\alpha)+p_{\epsilon}p_{c_{s}}(1+m)}.$
(26)

It is clear from (13), (14) and (26) that throughput $\eta_{s}=f(\tau,SNR)$.
Next, we present throughput analysis of CCWS method.

###  IV-B Throughput analysis of CCWS

In CCWS method, modulation layer executes selective retransmission for first
transmission of a packet and full retransmission in the event of CRC failure.
Thus, for each of the first transmission and subsequent full retransmission of
OFDM symbol initiated by Chase combining protocol at data-link layer, $k(1+m)$
information bits are transmitted, where additional $km$ bits are the results
of selective retransmission for the first transmission and full
retransmission. Similar to SCC method, $P_{e_{1}}$ and $P_{e_{2}}$ be the
probability of error for the first transmission and joint detection
corresponding to the retransmission. Since CCWS method employs selective Chase
combining during first transmission, the bit-error probability of the first
transmission is given in (13). The bit-error rate $P_{e_{2}}$ for joint
detection CCWS is given in (22), respectively. We denote probabilities of
receiving error-free frame of length $L_{f}$ for the first transmission and
selective retransmission by $p_{c_{1}}$ and $p_{c_{2}}$, respectively. The
corresponding probabilities of receiving incorrect frame for first
transmission and joint detection in the event of packet failure are
$p_{\epsilon_{1}}=1-p_{c_{1}}$ and $p_{\epsilon_{2}}=1-p_{c_{2}}$,
respectively. The expected number of information bits transmitted to deliver
error-free $k$ information bits for CCWS method are [27]

$\displaystyle\begin{split}n=&k.(1+m)p_{c_{1}}+2k(1+m)p_{\epsilon_{1}}.p_{c_{2}}+k3(1+m)p_{\epsilon_{1}}p_{\epsilon_{2}}p_{c_{1}}+\\\
&4k(1+m)p_{\epsilon_{1}}^{2}p_{\epsilon_{2}}p_{c_{2}}+k.5(1+m)p_{\epsilon_{1}}^{2}p_{\epsilon_{2}}^{2}p_{c_{1}}+6k(1+m)\\\
&p_{\epsilon_{1}}^{3}p_{\epsilon_{2}}^{2}p_{c_{2}}+k.7(1+m)p_{\epsilon_{1}}^{3}p_{\epsilon_{2}}^{3}p_{c_{1}}+\ldots.\\\
=&k.(1+m)p_{c_{1}}\Big{(}1+3p_{\epsilon_{1}}p_{\epsilon_{2}}+5p_{\epsilon_{1}}^{2}p_{\epsilon_{2}}^{2}+\ldots\Big{)}+\\\
&2k(1+m)p_{\epsilon_{1}}p_{c_{2}}\Big{(}\\!1+2p_{\epsilon_{1}}p_{\epsilon_{2}}\\!+\\!\\!3p_{\epsilon_{1}}^{2}p_{\epsilon_{2}}^{2}\\!+\ldots\Big{)}.\end{split}$
(27)

Let $\alpha=p_{\epsilon_{1}}p_{\epsilon_{2}}$, then

$\displaystyle\begin{split}n=&k(1+m)p_{c_{1}}\Big{(}1+3\alpha+5\alpha^{2}+7\alpha^{3}+\ldots\Big{)}+\\\
&2k(1+m)p_{\epsilon_{1}}.p_{c_{2}}\Big{(}1+2\alpha+3\alpha^{2}+4\alpha^{3}+\ldots\Big{)}.\end{split}$
(28)

After simplification of (28), we have [28]

$n=k(1+m)p_{c_{1}}\frac{1+\alpha}{(1-\alpha)^{2}}+2k(1+m)p_{\epsilon}p_{c_{2}}\frac{1}{(1-\alpha)^{2}}.$
(29)

Throughput of CCWS method is

$\displaystyle\eta_{c}$
$\displaystyle=\frac{k}{n}=\frac{(1-\alpha)^{2}}{\Big{(}p_{c_{1}}(1+\alpha)+2(1-p_{c_{1}})p_{c_{2}}\Big{)}\quad(1+m)}.$
(30)

Note that similar to the throughput $\eta_{s}$ for SCC method, throughput
$\eta_{c}$ of CCWS is also function of parameter $\tau$ that controls the
information to be transmitted during selective retransmission. The parameter
$\tau$ can be optimize to maximize throughput under OFDM signaling. Next, we
discuss search for optimal $\tau$ for SCC and CCWS to enhance throughput of
OFDM transceiver.

##  V Throughput Optimization

In this section, we optimize throughput of the proposed selective
retransmission methods at modulation layer. The amount of information that a
receiver request to the transmitter in the event of a packet failure has
direct impact on the throughput of the transceiver. Most of the time,
especially in high SNR regime, receiver can recover from bit errors by
receiving little more information and employing joint detection. In selective
retransmission at modulation level, threshold $\tau$ on channel norm of a
subcarrier is measure of quality of a channel. By choosing proper threshold
$\tau$, receiver can request minimum information needed to recover from errors
for a failed packet. The threshold $\tau$ is function of SNR and modulations
such as 4-QAM and 16-QAM. It is clear from (26) and (30) that throughput of
SCC and CCWS methods, respectively, are function of frame-error rate (FER).
Furthermore, FER is not a linear or quadratic function of SNR and parameter
$\tau$. Now we write unconstrained optimization problem for throughput $\eta$
with respect to parameter $\tau$ as follows:

$\mathbf{\tau_{0}}=\underset{\tau}{\text{arg max}}\quad\eta=f(\tau,SNR).$ (31)

Since throughput $\eta$ is non-convex function in parameter $\tau$, optimal
$\tau$ that maximizes throughput $\eta$ for each SNR can be computed off-line
using exhaustive search. Thus, a table of optimal threshold $\tau$ to maximize
throughput for SNR operating points can be generated using throughput
expression in (26) and (30) for SCC and CCWS methods, respectively. Note that
each retransmission method has different throughput function
$\eta=f(\tau,\,SNR)$. For example, analytical throughput function for SCC and
CCWS is given in (26) and (30), respectively. In Section VI, we maximize
$\eta=f(\tau,\,SNR)$ with respect to parameter $\tau$. Note that parameter
$\tau$ appears in probability of frame error $p_{\epsilon}$, which is function
of probability of bit-error presented in Section (III). Although
$\eta=f(\tau,\,SNR)$ is not linear or quadratic function of $\tau$ at a given
SNR, Figure 2 shows that for a given SNR, $\tau_{o}$ that maximizes throughput
$\eta$ is unique for SCC method. The optimal $\tau_{o}$ can be computed off-
line from throughput lower bound for SCC and CCWS using (26) and (30),
respectively. Based on channel condition, amount of information to be
transmitted can be controlled using vector $\tau_{o}$. Figure 3 shows optimal
threshold $\tau_{o}$ for SNR points that maximizes throughput CCWS method. In
low SNR regime, throughput $\eta$ is more sensitive to threshold $\tau$ as
compared to high SNR regime. This is due to the fact that in high SNR regime,
very few errors occur during first transmission resulting in to fewer
retransmissions. We use threshold vector $\tau_{o}$ in next section to compute
throughput.

Figure 2: Optimal $\tau_{o}$ for different SNR operating point that maximizes
analytical throughput of SCC method in (26). Figure 3: Optimal $\tau_{o}$ for
different SNR operating point that maximizes analytical throughput of CCWS
method in (30).

##  VI Simulation

Figure 4: BER performance comparison of analytical bit-error rate in (13) and
Monte Carlo simulation for SCC method of SISO-OFDM system. In analytical and
simulation results, parameter $\tau=0$, $\tau_{f}$ and $\tau_{o}$ are used to
target BER without selective retransmission(single), full retransmission and
optimize throughput, respectively, for SCC method.

Now we present efficiency for aforementioned SCC, CCWS and MSCC methods for
optimized threshold $\tau_{o}$ on the throughput of OFDM systems. In
simulation setup, we consider 4-QAM constellation for OFDM signaling under
Rayleigh fading frequency selective channel. We use OFDM signaling with
$N_{s}=512$ subcarriers for 10-tap frequency selective channel. Each complex
OFDM channel realization has Gaussian distribution with zero-mean and unit
variance ($\sigma^{2}_{h}=1$). We assume block fading channel in quasi-static
fashion such that channel remains highly correlated during transmission of one
OFDM symbol. First, we present comparison of analytical BER upper bound and
BER from Monte Carlo simulation denoted by $BER_{a}$ and $BER_{m}$,
respectively. We also provide throughput results of SCC and CCWS methods in
comparison with conventional Chase combining method. We denote analytical and
simulation throughput by $\eta_{a}$ and $\eta_{m}$, respectively. In order to
maximize throughput, threshold $\tau$ on channel norm of OFDM subcarriers is
optimized for each SNR point of SCC and CCWS protocols. We analytically
compute threshold vector $\tau_{o}$ off-line to maximize throughput of SCC and
CCWS methods from the analytical throughput using (26) and (30), respectively.
We also demonstrate that throughput gain our proposed CCWS and MSCC methods
hold under CC-HARQ method at MAC layer. We consider half-rate LDPC code (648,
324) to evaluate efficacy of CCWS and MSCC methods. We denote CCWS and MSCC
methods with FEC enabled by CCWS-HARQ and MSCC-HARQ in simulation results.

First, we compare BER performance for single (first) transmission and
subsequent retransmission in the event of failed packet. In particular, we
compare BER of full retransmission (Chase combining) and SCC in Figure 4. For
analytical and Monte Carlo runs of SCC, we use optimal threshold vector
$\tau_{o}$ that maximizes throughput. In Figure 4, we present BER comparison
of upper bound and BER from Monte Carlo simulation of SCC under SISO-OFDM
system using optimal threshold vector $\tau_{o}$. Note that BER performance of
SCC with threshold $\tau_{o}$ and $\tau_{f}$ is similar, where threshold
$\tau_{f}$ targets BER of SCC equals full retransmission. Figure 5 compares
threshold vector $\tau_{o}$ that optimizes throughput of SCC and CCWS at
different SNR points and threshold vector $\tau_{f}$ to target BER of SCC and
CCWS equals the corresponding full retransmission. Difference between vectors
$\tau_{o}$ and $\tau_{f}$ is marginal at moderate and high SNR. At high SNR,
BER of SCC with threshold $\tau_{o}$ is marginally higher than that of BER for
threshold $\tau_{f}$ (BER of full retransmission). It is clear from Figure 4
that retransmission of partial information corresponding to poor quality
subcarriers achieves BER of full retransmission. The threshold $\tau$ on
channel norm of subcarriers controls amount of partial information to be
retransmitted.

Figure 5: Threshold comparison $\tau_{f}$ and $\tau_{o}$ corresponding to
target BER equals full retransmission and optimal throughput for both SCC and
CCWS protocols, respectively.

Figure 6 compares BER performance of CCWS method with conventional CC method
for different values of threshold vector $\tau$ on channel norm of
subcarriers. First transmission of packet under CCWS method, which employs
selective retransmission at modulation level, achieves BER of full
retransmission with threshold vector $\tau_{o}$ as shown in Figure 6.
Threshold vector $\tau_{o}$ for CCWS method optimizes throughput of CCWS. The
comparison of $\tau_{o}$ for SCC and CCWS is presented in Figure 5. Figure 6
also reveals that using threshold $\tau_{o}$ that maximizes throughput of CCWS
method, BER of joint detection under CCWS method is similar to the BER of four
receive antennas. Note that threshold $\tau_{\infty}$ implies retransmissions
of all subcarriers and $\tau_{\infty}$ for CCWS in fact is equivalent to four
transmissions of a packets or four receive antennas. In Figure 6, we also
compare analytical BER ($\text{BER}_{a}$) of joint detection given in (22)
with Monte Carlo BER ($\text{BER}_{m}$) of CCWS for threshold vector
$\tau_{o}$. It is clear from Figure 4 and Figure 6 that (13) and (22) are
tight BER upper bounds for joint detections of SCC and CCWS methods,
respectively. In CCWS method, threshold $\tau_{o}$ is computed using (31) to
maximize throughput $\eta$. Similar to SCC method, we compute threshold
$\tau_{f}$ that targets BER of CCWS equals to the BER of joint detection of
four transmissions. For CCWS method, $\mbox{BER}_{m}$ for threshold vector
$\tau_{f}$ is lower than $\mbox{BER}_{m}$ for threshold vector $\tau_{o}$.

Figure 6: BER performance comparison of analytical bit-error rate in (22) and
Monte Carlo simulation for CCWS method of SISO-OFDM system. In analytical and
simulation results, parameter $\tau=0,\tau_{f}$ and $\tau_{o}$ are used to
target BER corresponding to full retransmission and optimize throughput for
both first transmission and retransmission, respectively, for CCWS method.

Figure 7:  Simulation and Analytical Throughput comparison for SCC, CCWS and
CC using (26) and (30) for $\tau_{o}$.

.

Throughput is the key performance metric of a communication system and
throughput optimization of selective retransmission based on channel quality
for OFDM system is the main focus of this work. For throughput computation, we
use standard definition of throughput of a communication system [2] as

$\displaystyle\eta=\frac{\mbox{\,Error-free bits received}}{\mbox{Total bits
transmitted}}=\dfrac{k}{n}.$ (32)

Now we provide throughput results to demonstrate the efficacy of selective
retransmission on spectral efficiency for SCC and CCWS methods at modulation
layer. We use throughput gain of optimized $\tau_{o}$ for SCC and CCWS methods
over conventional Chase combining method as a measure of spectral efficiency.
Figure 7 provides throughput comparison of conventional Chase combining, SCC
and CCWS methods. Monte Carlo throughput (simulation) $\eta_{om}$ for SCC and
CCWS is computed using their respective optimal thresholds $\tau_{o}$. The
throughput performance of both SCC and CCWS is much higher than conventional
Chase combining method. Throughput in Figure 7 also reveals that CCWS
outperforms SCC in low SNR regime. This is due to the fact that at low SNR,
joint detection using more retransmission improves throughput of the
communication systems. Figure 7 also compares analytical throughput
$\eta_{oa}$ and simulation throughput $\eta_{om}$ of SCC and CCWS methods.
Analytical throughput $\eta_{oa}$ of SCC and CCWS is computed using (26) and
(30), respectively. The throughput expressions for SCC and CCWS in (26) and
(30), respectively, provide tight lower bounds on throughput as shown in
Figure 7.

Now we present impact of thresholds $\tau_{o}$ and $\tau_{f}$ on the
throughput of SCC and CCWS methods. Figure 8 compares simulation results of
throughput $\eta_{om}$ and $\eta_{fm}$ corresponding to the optimal threshold
$\tau_{o}$ that maximizes throughput and threshold $\tau_{f}$ that achieves
BER of full retransmission of the proposed methods (SCC and CCWS). The
simulation results suggest that throughput of SCC method for $\eta_{om}$ and
$\eta_{fm}$ are similar. Throughput for $\eta_{om}$ are marginally better than
for $\eta_{fm}$. However, throughput $\eta_{om}$ of CCWS with optimal
threshold vector $\tau_{o}$ is higher than throughput $\eta_{fm}$ of CCWS with
threshold vector $\tau_{f}$ that achieves BER of full retransmission.

Figure 8: Throughput comparison of Optimal SCC and CCWS using Target BER of
Full retransmission $\tau=0$ Vs optimal SCC form Figure 5.

.

Figure 9: BER comparison of SCC and MSCC for $\tau_{o}$ and $\tau_{\infty}$.

.

Figure 10: Throughput comparison of SCC, CCWS and MSCC.

.

Figure 11: BER comparison of ARQ, SCC, CCWS, CC Vs HARQ, SCC-HARQ, CCWS-HARQ
and CC-HARQ at $\tau=0$, $\tau=\tau_{o}$ and $\tau=\infty$ respectively using
half rate LDPC code.

.

BER of MSCC for retransmission count $\Omega=1,2\mbox{ and }3$ for optimal
threshold $\tau_{0}$ on channel norm is presented in Figure 9. The threshold
values of vector $\tau_{0}$ that maximize throughput $\eta$ at corresponding
SNR point are obtained from exhaustive search using (31). Figure 9 reveals
that lower BER does not result into higher throughput. The BER performance
corresponding to $\tau_{\infty}$ is lower than $\tau_{0}$. However,
$\eta_{o}>\eta_{\infty}$. Figure 10 presents throughput gain of MSCC method
over SCC (MSCC with single retransmission) and CCWS method. It is clear from
Figure 10 that for two retransmissions ($\Omega=2$), MSCC achieves significant
throughput gain as compared to SCC. Throughput of MSCC with joint detection of
three selective retransmissions is higher than CCWS as well. As shown in
Figure 10, there is throughput improvement by jointly detecting MSCC for three
and four selective retransmissions. Throughput gain of MSCC method is
significant in low and moderate SNR regime.

Figure 12: Throughput comparison of ARQ, SCC, CCWS and CC Vs HARQ, SCC-HARQ,
CCWS-HARQ and CC-HARQ at $\tau=0$, $\tau=\tau_{o}$ and $\tau=\tau_{f}$,
respectively using half rate LDPC code.

.

Figure 11 presents simulated BER performance of conventional retransmission
methods and proposed selective retransmission methods with half-rate LDPC
(648,324) code. We denote hybrid SCC and hybrid CCWS by SCC-HARQ and CCWS-
HARQ, respectively in figure 11. Note that SCC-HARQ with threshold $\tau=0$ is
type-I HARQ, which is just single transmission without combining. The
threshold $\tau=\infty$ represents full retransmission and conventional CC-
HARQ. In figure 11, $BER_{om}$ of SCC-HARQ represents BER of SCC-HARQ with
optimal threshold $\tau_{o}$ on channel norm that maximizes throughput of the
SISO-OFDM system. It is clear from figure 11 that in high SNR regime,
$BER_{om}$ for SCC-HARQ converges to $BER_{m}$ of single transmission. The
joint detection of CCWS-HARQ method with $\tau=\infty$ is equivalent to having
four receive antennas. BER of CCWS-HARQ with optimal threshold $\tau_{o}$ is
very close to that of with $\tau=\infty$.

Figure 12 provides Monte Carlo simulation results of throughput for ARQ, SCC,
CCWS, HARQ, CC-HARQ, SCC-HARQ and CCWS-HARQ . As it is clear from figure 12
that in low SNR regime, throughput of conventional and proposed methods with
FEC is higher than conventional and proposed methods without FEC. In high SNR
regime, normalized throughput of uncoded methods is higher than coded schemes.
Figure 12 also reveals that throughput of CCWS has higher throughput as
compared to SCC method. Also, proposed methods have higher throughput than
conventional coded and uncoded CC methods. Thus, simulation results
demonstrate efficacy of the proposed methods with and without channel coding.

##  VII Conclusion

We presented performance analysis and throughput optimization of the proposed
selective retransmission methods at modulation layer for OFDM system. We
provide tight BER upper bound and lower throughput bound for SCC and CCWS
methods. We optimize threshold $\tau$ to maximize throughput of the proposed
selective Chase combining methods. We also generalize SCC method to multiple
retransmission at modulation layer in response to packet failure. The
simulation results show that there is marginal gap between throughput from
Monte Carlo runs and that of from throughput analysis. The simulation results
also demonstrate significant throughput gain of optimized selective
retransmission methods over conventional Chase combining methods with and
without channel coding.

## References

  * [1] “Long Term Evolution of the 3GPP Radio Technology,” _http:/www.3gpp.org_.
  * [2] S. Lin and P. S. Yu, “A Hybrid ARQ Scheme with Parity Retransmission for Error Control of Satellite Channels,” _IEEE Trans. Commun._ , vol. 30, no. 7, pp. 1701–1719, Jul. 1982.
  * [3] S. Y. Park and D. Love, “Hybrid ARQ Protocol for Multi-Antenna Multicasting Using a Common Feedback Channel,” _IEEE Trans. Commun._ , vol. 59, no. 6, pp. 1530–1542, Jun. 2011.
  * [4] D. Chase, “Code combining - A maximum-likelihood decoding approach for combining an arbitrary number of noisy packets,” _IEEE Tran. Commun._ , vol. 33, pp. 385–393, May 1985.
  * [5] C. Shen and M. P. Fitz, “Hybrid ARQ in Multiple-Antenna Slow Fading Channels: Performance Limits and Optimal Linear Dispersion Code Design,” _IEEE Trans. Inf. Theory_ , vol. 57, no. 9, pp. 5863–5883, Sep. 2011.
  * [6] P. Wu and N. Jindal, “Coding versus ARQ in Fading Channels: How Reliable Should the PHY Be?” _IEEE Trans. Commun._ , vol. 59, no. 12, pp. 3363–3374, December 2011.
  * [7] ——, “Performance of Hybrid-ARQ in Block-Fading Channels: A Fixed Outage Probability Analysis,” _IEEE Trans. on Comm._ , vol. 58, no. 4, pp. 1129–1141, April 2010.
  * [8] T. C. E. Larsson, “Optimal power allocation for hybrid arq with chase combining in i.i.d. rayleigh fading channels,” _IEEE Trans. on Commun._ , vol. 61, no. 5, pp. 1835 – 1846, May 2013.
  * [9] ——, “Outage-optimal power allocation for hybrid arq with incremental redundancy,” _IEEE Trans. Wireless Commun._ , vol. 10, no. 7, pp. 2069–2074, 2011.
  * [10] ——, “Adaptive power allocation for harq with chase combining in correlated rayleigh fading channels,” _IEEE Commun. Lett._ , vol. 3, no. 2, pp. 169 – 172, 2014.
  * [11] Dechene, J. Dan, and A. Shami, “Energy-aware resource allocation strategies for lte uplink with synchronous harq constraints,” _IEEE Trans. Mobile Computing_ , vol. 13, no. 2, pp. 422–433, 2014.
  * [12] B. Makki, G. Amat, and T. Eriksson, “Green Communication via Power-Optimized HARQ Protocols,” _IEEE Trans. Veh. Technol._ , vol. 63, no. 1, pp. 161–177, Jan 2014.
  * [13] M. Zia and Z. Ding, “Joint ARQ Receiver Design for Bandwidth Efficient MIMO Systems,” in _Proc. IEEE Global Telecommunication Conference (GLOBECOM’08)_ , New Orleans, LO, Dec. 2008, pp. 1–5.
  * [14] X. Liang, C. M. Zhao, and Z. Ding, “Piggyback Retransmissions over Wireless MIMO Channels: Shared Hybrid-ARQ (SHARQ) for Bandwidth Efficiency,” _IEEE Trans. Wireless Commun._ , vol. 12, no. 8, pp. 3770–3782, Aug. 2013.
  * [15] M. Zia and Z. Ding, “Bandwidth Efficient Variable Rate HARQ Under Orthogonal Space-Time Block Codes,” _IEEE Trans. Signal Process._ , vol. 62, no. 3, pp. 3360–3390, Jul. 2014.
  * [16] R. Jermey and Z. Ding, “Channel Estimation and Equalization Techniques in Downsampled ARQ Systems,” _IEEE Trans. Signal Process._ , vol. 55, no. 5, pp. 2251–2262, May. 2007.
  * [17] S. Alamouti, “A Simple Transmit Diversity Technique for Wireless Communications,” _IEEE J. Sel. Areas Commun._ , vol. 16, no. 8, pp. 1451–1458, Oct. 1998.
  * [18] M.-W. Wu and P.-Y. Kam, “ARQ with Channel Gain Monitoring,” _IEEE Trans. Commun._ , vol. 60, no. 11, pp. 3342–3352, Nov. 2012.
  * [19] M. Zia, H. Mahmood, A. Ahmed, and N. Saqib, “Selective HARQ Transceiver Design for OFDM System,” _IEEE Commun. Lett._ , vol. 17, no. 12, pp. 2229 – 2232, Dec. 2013.
  * [20] T. Kiani, M. Zia, N. Abbas, and H. Mahmood, “Bandwidth-efficient selective retransmission for mimo-ofdm,” _ETRI_ , Feb. 2015.
  * [21] A. Ahmed, M. Zia, N. Abbas, and H. Mahmood, “Partial automatic repeat request transceiver for bandwidth and power efficient multiple-input multiple-output orthogonal frequency division multiplexing systems,” _IET Communications_ , Feb. 2015.
  * [22] M. Zia, “Selective Chase Combining for OSTB Coded MIMO-OFDM System,” _International Journal Of Electronics and Communication_ , 2015.
  * [23] E. G. Larsson and P. Stoica, _Space-Time Block Coding for Wireless Communications_. Cambridge, 2003.
  * [24] J. Proakis and M. Salehi, _Digital Communications_. McGraw-Hill, 2007.
  * [25] H. Stark and J. W. Woods, _Probability and Random Processes with Applications to Signal Processing_. Prentice Hall, 2001.
  * [26] M. Chiani, D. Dardari, and M. K. Simon, “New Exponential Bounds and Approximations for the Computation of Error Probability in Fading Channels,” _IEEE Trans. Wireless Commun._ , vol. 2, no. 4, pp. 840–845, July 2003.
  * [27] S. Lin, D. J. Costello, and M. J. Miller, “Automatic Repeat Request Error Control Schemes,” _IEEE Commun. Mag._ , vol. 22, no. 12, pp. 5–14, Dec. 1984.
  * [28] T. Bromwich, _An Introduction to the Theory of Infinite Series_. Macmillan and Company, limited, 1908.