![Fig. 4. Outage probabilities of PRS, ORS, and OSTC as a function of the end-to-end average SNR γ̄ = P/σ2n with K=4. Analytical results of ORS and OSTC are derived from (20) in [12], and that of PRS is from (25).](/figures/fig-4-outage-probabilities-of-prs-ors-and-ostc-as-a-function-3jind0oi.webp)
A Deep Learning-Assisted Cooperative Diversity
Method under Channel Aging
Wei Jiang
Intelligent Networking Research Group
German Research Center for Artificial Intelligence (DFKI)
Kaiserslautern, Germany
https://orcid.org/0000-0002-3719-3710
Hans D. Schotten
Institute for Wireless Communication and Navigation
University of Kaiserslautern
Kaiserslautern, Germany
https://orcid.org/0000-0001-5005-3635
Abstract—Single-relay selection is a simple but efficient scheme
for cooperative diversity among multiple user devices. However,
the wrong selection of the best relay due to aged channel
state information (CSI) remarkably degrades its performance,
overwhelming this cooperative gain. Multi-relay selection is robust
against channel aging but multiple timing offset (MTO) and mul-
tiple carrier frequency offset (MCFO) among spatially-distributed
relays hinder its implementation in practical systems. In this
paper, therefore, we propose a deep learning-based cooperative
diversity method coined predictive relay selection (PRS) that
chooses a single relay with the largest predicted CSI, which can
alleviate the effect of channel aging while avoiding MTO and
MCFO. Performance is evaluated analytically and numerically,
revealing that PRS clearly outperforms the existing schemes with
a negligible complexity burden.
Index Terms—Cooperative diversity, aged CSI, channel aging,
channel prediction, deep learning, LSTM, opportunistic relaying
I. INTRODUCTION
Cooperative diversity [1] is an effective technique to achieve
spatial diversity as same as multi-input multi-output (MIMO),
through the collaboration among multiple single-antenna n-
odes, when there is no possibility of embedding an antenna
array on a mobile terminal. A main difference between MI-
MO and cooperative diversity is the inherent asynchronization
among spatially-distributed relays in the latter. Multiple timing
offset (MTO) [2] and multiple carrier frequency offset (MCFO)
[3] among simultaneously transmitting relays make multi-
relay selection methods, such as distributed beam-forming
[4] and distributed space-time coding (DSTC) [5], hard to
implement for practical systems. In contrast, a single-relay
selection approach called opportunistic relay selection (ORS)
has been extensively recognized as a simple but efficient way to
achieve cooperative diversity [6]. Despite only a single node is
opportunistically selected to retransmit, identical performance
as all-participating strategy using DSTC is expected, while
avoiding the need on multi-relay synchronization.
However, channel state information (CSI) used to select the
best relay may differ from the actual CSI due to feedback delay.
Retransmitting signals on a wrong relay selected in terms of
∗
This work was supported by German Federal Ministry of Education and
Research (BMBF) through TACNET4.0 project (Grant no. KIS15GTI007) and
KICK project (Grant no. 16KIS1105).
aged CSI substantially deteriorates the performance of ORS
[7]–[9]. To remain cooperative diversity under channel aging,
Generalized Selection Combining (GSC) [10] and its enhanced
version called N plus normalized threshold GSC (N+NT-GSC)
[11] have been proposed. But these schemes require at least
N orthogonal channels to retransmit, resulting in around 1/N
spectral efficiency. In [12], one author of this paper proposed
a scheme called opportunistic space-time coding (OSTC) that
alleviates the effect of aged CSI but avoids the decrease of
spectral efficiency. By far, to the best knowledge of the authors,
OSTC can achieve the best result under channel aging, but its
gap to the optimal performance is still large, which motivates
the work in this paper.
Recently, a technique referred to as channel prediction [13],
[14], which can improve the timeliness of CSI by forecasting
future CSI in advance, attracts the attention of researchers. In
this paper, leveraging its capability on time-series prediction, a
deep recurrent neural network with Long Short-Term Memory
(LSTM) [15] is employed to build a channel predictor. Upon
this, we propose a novel cooperative diversity method coined
predictive relay selection (PRS). Its key idea is to choose a
single relay (in order to avoid MTO and MCFO in multi-
relay selection) with the largest predicted CSI, earning a
prediction horizon to counteract induced delay. A closed-form
expression of outage probability for PRS is derived and then
verified by simulations. Performance evaluation reveals that
it clearly outperforms the existing schemes, without bring
complexity burden. The rest of this paper is organized as
follows: Section II introduces the system model. Section III
and IV present the proposed scheme and analyze its outage
probability, respectively. Numerical results are given in Section
V. Finally, Section VI concludes this paper.
II. SYSTEM MODEL
A. Model of Cooperative Networks
Consider a two-hop decode-and-forward (DF) cooperative
network where a source s communicates with a destination d
with the help of K relays, neglecting the direct link due to line-
of-sight blockage. The received signal in link A→B is modeled
as y
B
=h
A,B
x
A
+z
B
, where x
A
∈ C is the transmitted symbol
from Node A with average power P
A
=E[|x
A
|
2
] (E denotes the
expectation operator), h
A,B
represents channel coefficient that
is a zero-mean circularly-symmetric complex Gaussian random
variable with variance σ
2
h
, i.e., h∼CN (0, σ
2
h
), under Rayleigh
flat-fading channels, and z
B
stands for additive white Gaussian
noise with zero-mean and variance σ
2
n
, i.e., z∼CN (0, σ
2
n
).
The instantaneous signal-to-noise ratio (SNR) of link A→B
is denoted by γ
A,B
=|h
A,B
|
2
P
A
/σ
2
n
and the average SNR
¯γ
A,B
=E[γ
A,B
]=σ
2
h
P
A
/σ
2
n
. Node A can be the source A=s
or k-th relay A=k, k∈{1, ..., K}, corresponding to B=k or
B=d, respectively.
Because of severe signal attenuation, the relays with a single
antenna should operate in half-duplex transmission mode to
avoid harmful self-interference between the circuits of trans-
mitter and receiver. Without loss of generality, time-division
multiplexing is applied for analysis hereinafter and therefore
the signal transmission is organized into two phases. In the
first phase, as shown in Fig.1, the source (e.g., the drone in
the figure) transmits a signal and those of relays which can
correctly decode this signal form a decoding subset (marked
by DS) of source-relay link
DS ,
k
1
2
log
2
(1 + γ
s,k
) > R
, (1)
where R is the end-to-end target rate for the two-hop cooper-
ative network. Note that the required rate for either link raises
to 2R due to the half-duplex mode.
The best relay
˙
k in the conventional ORS is opportunistically
selected from DS in terms of
˙
k = arg max
k∈DS
ˆγ
k,d
, where
ˆγ
k,d
is the SNR of relay-destination link at the instant of
relay selection, which is an outdated version of γ
k,d
at the
time of actual signal transmission. In comparison, the proposed
PRS scheme replaces the aged CSI with the predicted CSI
ˇ
h, and determines
˙
k in terms of
˙
k = arg max
k∈DS
ˇγ
k,d
,
where ˇγ
k,d
=|
ˇ
h
k,d
|
2
P
k
/σ
2
n
. In our notation, h is actual CSI,
ˆ
h denotes aged CSI, and
ˇ
h means predicted CSI. In addition
to the best relay, OSTC needs to select another relay with
the second strongest SNR, i.e.,
¨
k = arg max
k∈DS−{
˙
k}
ˆγ
k,d
.
In the first phase, the source broadcasts a pair of symbols
(x
1
, x
2
) to all relays on two consecutive symbol durations.
The regenerated signals are encoded by means of the Alamouti
scheme, a unique space-time code achieving both full-rate and
1
...
1 2
: [x]
: [x , x ]
ORS/PRS
OSTC
Broadcasting
Relaying
2
3
1
...
2
3
K
K
RelayingBroadcasting
source transmits
relay(s) retransmits
.
.
.
2
3
K
Decoding
Subset
: [ ]ORS/PRS x
:
*
1 2
*
2 1
x -x
OSTC
x x
é ù
ê ú
ê ú
ë û
Fig. 1. Schematic diagram of DF cooperative diversity with different relaying
strategies: ORS, PRS, and OSTC.
full-diversity, at the pair of selected relays. In the second
phase, a relay transmits (x
1
, −x
∗
2
) while another sends (x
2
, x
∗
1
)
simultaneously at the same frequency.
B. Model of Aged CSI
From a practical point of view, the CSI
ˆ
h used to select
relay(s) may remarkably differ from the actual CSI h at the
instant of using the selected relay(s) to forward regenerated
signals, leading to performance deterioration. To quantify such
CSI inaccuracy, the correlation coefficient between h and
ˆ
h is
introduced, i.e.,
ρ
o
=
E[h
ˆ
h]
q
E[|h|
2
]E[|
ˆ
h|
2
]
. (2)
According to [16], we have
ˆ
h = σ
ˆ
h
ρ
o
σ
h
h + ε
p
1 − ρ
2
o
, (3)
where ε∼CN (0, 1) and σ
2
ˆ
h
is the variance of
ˆ
h. Under the
assumption of a Jakes’ model, the correlation coefficient takes
the value ρ
o
= J
0
(2πf
d
τ), where f
d
is the maximal Doppler
frequency, τ stands for the delay between the outdated and
actual CSI, and J
0
(·) denotes the zeroth order Bessel function
of the first kind.
C. Model of Predicted CSI
To train a deep learning (DL) predictor, the applied objective
is to generate predicted CSI
ˇ
h that approximates to the actual
CSI (zero-mean complex Gaussian random variable) as close
as possible. Hence, we can assume that
ˇ
h also follows zero-
mean complex Gaussian distribution, i.e.,
ˇ
h∼CN (0, σ
2
ˇ
h
). The
relationship between
ˇ
h and h can be modeled as
ˇ
h = h + e, (4)
where e is the prediction error that is zero-mean complex
Gaussian variable with variance σ
2
e
. Like (2), the correlation
coefficient between
ˇ
h and h can be obtained. Replacing
ˆ
h with
ˇ
h and substituting (4) into (2), yields
ρ
p
=
E[h
ˇ
h]
q
E[|h|
2
]E[|
ˇ
h|
2
]
=
σ
h
σ
ˇ
h
=
1
p
1 + σ
2
e
. (5)
In the field of machine learning (ML), normalized mean
squared error (NMSE) is an usual metric applied to measure the
accuracy of data fitting, which can be easily acquired during
both the training and predicting phase. In our case of channel
prediction, the NMSE is
NMSE =
E[|h −
ˇ
h|
2
]
E[|h|
2
]
, (6)
and it can be straightforward derived that the NMSE is re-
lated to e by NMSE = σ
2
e
/σ
2
h
. The model-less ML tech-
niques make traditional statistics-based performance analysis
intractable, but the availability of NMSE provides another
method for performance evaluation.
The actual CSI h and its predicted version
ˇ
h follow joint
complex Gaussian distribution. Then, the instantaneous SNR
of relay-destination link γ
k,d
conditioned on ˇγ
k,d
follows non-
central Chi-square distribution with two degrees of freedom.
Substituting (5) into Eq. (12) of [17], the probability density
function (PDF) in terms of σ
2
e
is obtained, that is
f
γ
k,d
|ˇγ
k,d
(γ|ˇγ)
=
(1 + σ
2
e
)e
−
ˇγ+γ(1+σ
2
e
)
σ
2
e
¯γ
k,d
σ
2
e
¯γ
k,d
I
0
2
p
(1 + σ
2
e
)γˇγ
¯γ
k,d
σ
2
e
!
, (7)
where ¯γ
k,d
means the average SNR of relay-destination link,
and I
0
(·) denotes the zeroth order modified Bessel function
of the first kind.
III. P
REDICTIVE RELAY SELECTION
This section introduces the principles of deep learning with
LSTM and the corresponding channel predictor, analyzes its
computational complexity, and then depicts the protocol design
to implement predictive relay selection.
A. DL-based Channel Predictor
Unlike feed-forward neural networks, recurrent neural net-
works (RNNs) can memorize historical information in its
internal state, exhibiting great power in time-series prediction.
But back-propagated error signals in RNN tend to infinity
(gradient exploding), resulting in oscillating weights, or apt
to zero (gradient vanishing) that implies a prohibitively-long
training time. To this end, Long Short-Term Memory were
proposed by Hochreiter and Schmidhuber in their pioneer work
of [15], where special units called memory cells and mul-
tiplicative gates that control information flow are introduced
into the RNN structure. Each LSTM memory cell contains
three gates: an input gate protecting the memory contents
from perturbation by irrelevant interference, a forget gate to
filter out useless memory, and an output gate that controls the
extent to which the memory information applied to generate an
output activation. Despite of its short history, LSTM has been
successfully applied to popular commercial products such as
Apple Siri and Google Translate.
The upper part of Fig.2 shows a deep LSTM network
consisting of an input layer, multiple hidden layers, and an
output layer. At time t, the instantaneous CSI h[t] is acquired
at the receiver through estimating a pilot symbol. Because
the relay selection relies on the value of SNR, only real-
valued amplitude |h[t]| is enough, rather than complex-valued
h[t], which in turn can simplify the implementation of neural
network by using real-valued weights. Feeding |h[t]| into the
input feed-forward layer to get an intermediate activation d
(1)
t
,
further activating the memory cells in the first hidden layer.
Along with the recurrent unit from the previous time step, d
(2)
t
is generated and then forwarded to the second hidden layer.
This recursive process continues until the output layer gets the
predicted CSI |
ˇ
h[t+1]|. As illustrated in the lower part of Fig.2,
a memory block has two internal states: the short-term state
and the long-term state. At the l
th
hidden layer, the short-term
Hadamard
Addition
tanh
Sigmoid
t
g
t
i
t
o
t
c
t
s
t
f
1t-
s
1t-
c
( 1)l
t
+
d
[t+1]h
[t]h
Channel
Estimator
DL Predictor
( )l
t
d
Fig. 2. Block diagram of a DL-based predictor and an LSTM memory block.
state s
(l)
t−1
getting at time step t−1, together with the input
vector d
(l)
t
, activates four different fully connected (FC) layers
to generate the gate vectors:
f
(l)
t
= δ
g
W
(l)
f
d
(l)
t
+ U
(l)
f
s
(l)
t−1
+ b
(l)
f
, (8)
i
(l)
t
= δ
g
W
(l)
i
d
(l)
t
+ U
(l)
i
s
(l)
t−1
+ b
(l)
i
, (9)
o
(l)
t
= δ
g
W
(l)
o
d
(l)
t
+ U
(l)
o
s
(l)
t−1
+ b
(l)
o
, (10)
where W and U represent weight matrices for the FC layers, b
denotes bias vector, the subscripts f , i, and o associate with the
forget, input, and output gate, respectively, and δ
g
represents
the Sigmoid activation function δ
g
(x) =
1
1+e
−x
. Besides, there
is an intermediate element
g
(l)
t
= δ
h
W
(l)
g
d
(l)
t
+ U
(l)
g
s
(l)
t−1
+ b
(l)
g
, (11)
where δ
h
is the hyperbolic tangent (tanh) function δ
h
(x) =
e
2x
−1
e
2x
+1
. Traversing the block, the previous long-term state c
(l)
t−1
first discards some outdated memories at the forget gate, on-
boards new information selected by i
(l)
t
, and then transforms
into c
(l)
t
= f
(l)
t
⊗ c
(l)
t−1
+ i
(l)
t
⊗ g
(l)
t
, where ⊗ denotes the
Hadamard product (element-wise multiplication) for matrices.
Further, c
(l)
t
goes through the tanh function and then is filtered
by o
(l)
t
to update the short-term memory, which serves also as
the output activation, i.e., s
(l)
t
= d
(l+1)
t
= o
(l)
t
⊗ δ
h
c
(l)
t
.
B. Computational Complexity
The computational complexity brought by deep learning is
a general concern. Here, let’s assess the predictor’s complexity
through calculating the number of complex multiplications.
The applied deep recurrent network can be quantified as
follows: an input layer with n
i
neurons, an output layer with
n
o
neurons, and L hidden layers, which has n
l
c
LSTM cells at
layer l= 1, . . . , L. According to [14], the number of parameters
including both weights and biases can be computed by:
N
DL
= 4(n
i
× n
1
c
+ n
1
c
× n
1
c
+ n
1
c
)
+
L
X
l=2
4
#
n
l−1
c
× n
l
c
+ n
l
c
× n
l
c
+ n
l
c
+ n
L
c
× n
o
+ n
o
. (12)
Under the typical stochastic gradient descent training, each
parameter requires O(1) at each time step. Consequently, the
complexity per time step in the training phase is measured by
O(N
DL
). During the predicting phase, each weight requires
one complex-valued multiplication, amounting to the complex-
ity of O(N
DL
) per prediction.
C. Predictive Relay Selection
The implementation of cooperative relaying schemes can be
mainly divided into two categories: distributed [6] and central-
ized. The former relies on a timer at each relay, and applies
a contention period (CP) to determine the best relay. The
latter has a centralized controller, e.g., the destination, which
collects global CSI, makes the selection decision, and informs
the selected relays to retransmit. The information exchange
between the controller and the relays not only requires extra
signaling, but also brings the feedback delay that exacerbates
the aged CSI problem. By introducing channel prediction, the
CSI got at the current frame is applied to generate predicted
CSI that will be used at the next frame, such a prediction
horizon provides a new degree of freedom to design a relaying
protocol. Here, we depict a distributed implementation for
predictive relay selection, as follows:
1) At frame t, as shown in Fig.3, the source broadcasts a
packet containing a pilot called Ready-To-Send (RTS)
and data payload. The channel gain h
s,k
[t] is acquired at
relay k by estimating RTS and is used for detecting the
data symbols. Those relays which correctly decode the
source signal comprise DS and will participate in the
relay selection process.
2) Clear-To-Send (CTS) is sent from the destination, and
relay k estimates h
d,k
[t] from the received pilot y
cts
[t],
and then knows h
k,d
[t] due to channel reciprocity. It
feeds h
k,d
[t] into its embedded channel predictor to
generate
ˇ
h
k,d
[t + 1], and buffers it in the memory for
its usage at the upcoming frame t + 1.
3) On the other hand, relay k fetches
ˇ
h
k,d
[t] from the buffer
that is stored at frame t − 1. This operation starts once
the arrival of CTS, parallel with step 2.
4) Then, a timer with a duration inversely proportional to
ˇ
h
k,d
[t] is started at relay k.
5) The timer on the relay with the largest channel gain
expires first, which sends a short packet to announce.
6) Once receive the best relay’s packet of its presence, other
relays terminate their timers and keep silent. The selected
relay retransmits the regenerative signal until the end of
this frame.
It is possible that the number of relays in D S is zero or the
duration of timer is too long due to a very small channel gain.
To deal with these anomalies, a maximal duration is required to
set for CP. If this duration expires, the relay selection process
is interrupted regardless of the presence of the best relay.
Frame t
CTS
RTS
source CP
CSI-E CSI-P CSI-B
relay
Frame t+1
CTS
RTS
source CP relay
[ 1]h t +
[ 1
]
h t
[ 1[ 1
[ 1
[ 1
[ ]
cts
y t
Fig. 3. Frame structure of PRS. CSI-E: CSI Estimation, CSI-P: CSI Prediction,
CSI-B: CSI Buffering, CP: Contention Period.
IV. OUTAGE PROBABILITY ANALYSIS
In information theory, outage is defined as the event that
instantaneous channel capacity falls below a target rate R,
where reliable communication cannot be realized whatev-
er coding used. The metric to measure the probability of
outage is referred to as outage probability that is defined
by P (R)=P {log
2
(1 + γ) < R}, where P is the notation of
mathematical probability. Let DS
L
denotes the set of all de-
coding subsets having L relays, and DS
p
L
denotes p
th
element
of DS
L
, namely, DS
L
={DS
p
L
|p=1, ..., |DS
L
|}, where | · |
represents the cardinality of a set. Then, the outage probability
of PRS can be calculated by
P
prs
(R) =
K
X
L=0
|DS
L
|
X
p=1
P(R|DS
p
L
)P(DS
p
L
), (13)
where P(DS
p
L
) is the occurrence probability of DS
p
L
, and
P(R|DS
p
L
) is the outage probability conditioned on DS
p
L
. Sup-
pose that all source-relay links are independent and identically-
distributed (i.i.d.) Rayleigh channels, the values of P(DS
p
L
) are
the same for any p∈{1, ..., |DS
L
|}, and as well P(R|DS
p
L
)
if all relay-destination channels are i.i.d. Then, (13) can be
simplified to
P
prs
(R) =
K
X
L=0
P (R||DS| = L) P (|D S| = L) , (14)
where P(|DS|=L) denotes the probability that the number
of relays in decoding subset is L. In Rayleigh channels, the
instantaneous SNR of each source-relay channel is exponen-
tially distributed, i.e., γ
s,k
∼EXP
1
¯γ
s,k
, whose Cumulative
Distribution Function (CDF) can be expressed by
F
γ
s,k
(x) = 1 − e
−x/¯γ
s,k
, x > 0. (15)
According to (1), the probability that a relay falls into DS
equals to 1−F
γ
s,k
(γ
o
), where γ
o
=2
2R
−1 is the threshold
SNR corresponding to the target rate R. The probability of
successfully decoding L out of K relays follows Binomial
distribution, we have
P(|DS| = L) =
K
L
e
−
γ
o
¯γ
s,k
L
1 − e
−
γ
o
¯γ
s,k
K−L
(16)
Thus, the second term in (14) is determined. Let’s turn to the
first term P (R||DS| = L), which is derived, conditioned on
the number of L, as follows:
a) L=0: In the case that no relay can decode the source’s
signal, the relaying will definitely fail, i.e.,
P(R||DS| = 0) = 1 (17)
b) L=1: Only a unique relay successfully decodes the
signal, it becomes
˙
k directly and a process of relay selection
is skipped. Similar to (15), the CDF of SNR over this relay-
destination link is given by F
γ
˙
k,d
(x)=1−e
−x/¯γ
k,d
. The outage
probability conditioned on L=1 is equal to
P(R|L = 1) = F
γ
˙
k,d
(γ
o
) =
1 − e
−
γ
o
¯γ
k,d
. (18)
c) L>1: In this case, a relay is opportunistically selected
from the decoding set according to the predicted CSI in relay-
destination links. For the sake of mathematical tractability, we
further rewrite ˇγ
k,d
, k∈DS
L
as ˇγ
l
, l∈{1, ..., L}. Defining A
˙
k
as the event that:
A
˙
k
:=
( ˇγ
1
, ..., ˇγ
L
)
˙
k = arg max
l=1,...,L
ˇγ
l
, (19)
which means that A
˙
k
is a set of L elements (ˇγ
1
, ..., ˇγ
L
)
where ˇγ
˙
k
is the largest. However, ˇγ
˙
k
is only for selection, the
post-processing SNR for performance evaluation should be the
actual SNR γ
˙
k
, whose CDF can be calculated by
F
γ
˙
k
(y) =
L
X
˙
k=1
P(γ
˙
k
6 y|A
˙
k
)P
#
A
˙
k
, (20)
where P(A
˙
k
) denotes the occurrence probability of A
˙
k
, e-
qualing to
1
L
since each relay has the same chance to be
selected under i.i.d channel assumption. P(γ
˙
k
6 y|A
˙
k
) notates
the probability that the actual SNR is below a threshold y
conditioned on A
˙
k
, which can be computed by:
P(γ
˙
k
6 y|A
˙
k
) =
Z
y
0
Z
∞
0
f
γ
˙
k
|ˇγ
˙
k
(γ|ˇγ)f
ˇγ
˙
k
|A
˙
k
(ˇγ)dγdˇγ, (21)
where f
γ
˙
k
|ˇγ
˙
k
(γ|ˇγ) stands for the PDF of γ
˙
k
conditioned on its
predicted version ˇγ
˙
k
, which is already given in (7). f
ˇγ
˙
k
|A
˙
k
(ˇγ)
denotes the PDF of the largest predicted SNR conditioned
on A
˙
k
, analogue to multi-user selection with a max-SNR
scheduler [18], we can write it as:
f
ˇγ
˙
k
|A
˙
k
(ˇγ) =
Le
−
ˇγ
¯γ
k,d
¯γ
k,d
1 − e
−
ˇγ
¯γ
k,d
L−1
(22)
Substituting (7), (21), and (22) into (20), yields
F
γ
˙
k
(y) = L
L−1
X
l=0
L − 1
l
(−1)
l
l + 1
1 − e
−
y(l+1)(1+σ
2
e
)
¯γ
k,d
(1+σ
2
e
(1+l))
!
.
(23)
Thus, the conditional outage probability at L > 1 is
P(R||DS| = L) = F
γ
˙
k
(γ
o
). (24)
If setting L=1 in (24), we can get a result equaling to (18),
thus, (24) can be extended to cover the case of L =1.
Now, the closed-form expression for the first term in (14) is
available. Substituting (16), (17), and (24) into (14), the overall
outage probability of PRS in the presence of aged CSI can be
computed as
P
prs
(γ
o
)=
1 − e
−
γ
o
¯γ
s,k
K
+
K
X
L=1
L
L−1
X
l=0
L − 1
l
(−1)
l
l + 1
1 − e
−
γ
o
(l+1)(1+σ
2
e
)
¯γ
k,d
(1+σ
2
e
(1+l))
!
·
K
L
e
−
γ
o
¯γ
s,k
L
1 − e
−
γ
o
¯γ
s,k
K−L
. (25)
V. N
UMERICAL RESULTS
In this section, we make use of Monte-Carlo simulations
to validate the correctness of analytical analyses and evaluate
performance. Given i.i.d. Rayleigh channels with a normalized
gain σ
2
h
= 1, outage probabilities of PRS, ORS, and OSTC
in the presence of aged CSI are provided. The maximal
Doppler frequency is set to f
d
=100Hz, emulating fast fading
environment, and an end-to-end target rate of R=1bps/Hz is
applied for outage calculations. Training data sets are built
by sampling a series of 7500 consecutive channel response
{h[t] |t=1, 2, . . . , 7500}, with and without considering the
impact of noise in channel estimation. The cooperative net-
work has K=4 DF relays and equal power allocation among
nodes is used. Assuming the end-to-end power is P , the
source transmits with P
s
=0.5P , resulting in an average SNR
¯γ
s,k
=0.5P/σ
2
n
for source-relay channels, while ¯γ
k,d
=0.5P/σ
2
n
for relay-destination channels. Detailed simulation parameters
are summarized in Table I.
As illustrated in Fig.4, the markers indicating the numerical
results fall into their corresponding curves that are the analyti-
cal results, corroborating our theoretical analyses in this paper.
As the benchmark, the curve of ORS when the knowledge of
TABLE I
S
IMULATION CONFIGURATION
Parameters Values
Frame length 2ms
Max. Doppler shift f
d
= 100Hz
Channel model Rayleigh (Jakes’s model)
Training length 7500 samples
Deep learning L=2 LSTM netwok
Hidden neurons 20/10
Training algorithm Adam optimizer
Batch size 256
Cost function MSE
Prediction length 2ms
Actuation function tanh