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Deep Learning-Based SC-FDMA Channel
Equalization
Mohamed A. Mohamed1,2,*, Hassan A. Hassan1,2, Mohamed H. Essai1, Hamada Esmaiel2,
Ahmed S. Mubarak2, and Osama A. Omer2
1 Department of Electrical Engineering, Faculty of Engineering, Al-Azhar University, Qena 83513, Egypt
2 Department of Electrical Engineering, Faculty of Engineering, Aswan University, Abulrish 81542, Egypt
Email: mohammed.anbar@azhar.edu.eg (M.A.M.), hassanali2720.el@azhar.edu.eg (H.A.H.),
mhessai@azhar.edu.eg (M.H.E.), h.esmaiel@aswu.edu.eg (H.E.), ahmed.soliman@aswu.edu.eg (A.S.M.),
omer.osama@aswu.edu.eg (O.A.O.)
Abstract—It is very challenging to design an effective
wireless communication system. That’s because of
numerous factors affecting the performance of a typical
wireless communication system, such as nonlinear channel
distortions and impairments. single carrier frequency
division multiple access (SC-FDMA) is a multiple access
scheme that is an important part of the long-term evolution
(LTE) standard for uplink transmission. An advanced
mobile radio system’s multiple access schemes should
indeed meet stringent requirements, such as a low bit error
rate (BER). In this article, we investigate the equalization
problem for nonlinear channel distortions and impairments
using deep neural networks (NN). We introduce a novel
combined deep neural network channel equalization and
symbol detection scheme based on a deep learning (DL)
recurrent feedback (RF) long short-term memory (LSTM)
neural network to achieve blind equalization and decoding
for SC-FDMA systems without knowing the channel state
information (CSI). To train the model efficiently, the
training data is gathered by simulation, with channel effects
and noise treated as a complete black box. CSI and
constellation demapping are learned by a deep neural
network (DNN) model. Then, the frequency-domain
sequences that have been corrupted are implicitly equalized
to get the broadcasted signal back. Our specified SC-FDMA
system, which uses a quadrature phase-shift keying (QPSK)
modulation method and the suggested Deep Learning-based
model channel equalizer, performs better than the existing
equalizers by an average of 1 to 4 dB at moderate signal-to-
noise (SNR) ratios, according to simulation data. A
complexity comparison between the proposed and the
conventional equalizers was conducted in terms of training
time, execution time, and number of operations. On
combined channel equalization and symbol detection, the
suggested system delivers state-of-the-art performance.
I. INTRODUCTION
In recent years, the communication standards and
innovations are driven by the eager desire of customers to
elevate their access to broadband wireless communication
service. Accordingly, mobile networks witness a huge
demand in terms of larger data rates and massive
connected devices. It is anticipated for global mobile data
traffic to margin 230 exabytes (EB) per month, and the
connected devices to pass 90 million by 2026 [1].
Consequently, new wireless transmission techniques
with large data rates and resistance to radio frequency
(RF) impairments have gotten a lot of attention as a
response to this demand. Multicarrier orthogonal
frequency division multiple access (OFDMA) strategies
have become the dominant essence for wireless
broadband applications in recent decades because of their
superior spectral efficiency, which is achieved using a
special set of overlapped orthogonal subcarriers, and their
resistance to channel selectivity [2].
Despite of their multiple advantages, OFDMA has
some downsides, such as a high peak-to-average power
ratio (PAPR), which makes it hard for mobile devices to
save energy [3]. To address this issue, a modified version
of OFDMA, discrete Fourier transform (DFT) pre-coded
OFDM, known as single-carrier FDMA, was investigated
(SC-FDMA). It also has the same effectiveness and
complexities as OFDMA but with a lower PAPR
condition. As time passed, SC-FDMA has proven to be a
great design and has been used for uplink transmission in
the long-term evolution (LTE) standard.
Because of the large amount of inter-symbol
interference (ISI) among the transmitted symbols, the
multipath environment of wireless communication
channels makes it difficult to recover rapidly transmitted
data at the receiver. As a result, overcoming the problem
of inter-symbol interference is a critical task for wireless
communications systems. Powerful equalization
procedures are unavoidable in order to offset the negative
consequences of ISI. There have been a lot of new
equalization algorithms in the last few years that can help
cut down the amount of interference (ISI) in fading
channels. The ISI could indeed span hundreds of symbols
when the data rate is extremely high, and the cost of
designing and making these filters might be too high [4,
5].
Equalizer design problem get more attentions both
academically and in the industrial field since it is crucial
International Journal of Electrical and Electronic Engineering & Telecommunications Vol. 13, No. 1, 2024
67
doi: 10.18178/ijeetc.13.1.67-79
Manuscript received July 2, 2023; revised August 22, 2023; accepted
September 8, 2023; published February 2, 2024.
*Corresponding author
Index Terms—Deep learning, channel equalization, Long
Short-Term Memory (LSTM), Quadrature Phase-Shift
Keying (QPSK), recurrent neural network, single-carrier
frequency-division multiple access
to deal with dynamic and rapidly varying channels.
Linear equalizers often use transversal or lattice linear
filters and adaptation techniques like recursive least
square (RLS), fast RLS, least mean square (LMS),
square-root RLS, gradient RLS, and so on. Linear
equalizers, on the other hand, perform poorly on channels
with profound spectral nulls.
Artificial deep learning neural networks (ADLNN)
algorithms in wireless communications systems have
evolved considerably in recent years, particularly in the
physical layer, owing to their tremendous capability to
learn, recognize, and forecast [6]. In the next subsection,
a brief overview for the literature of neural networks (NN)
in equalizer design problem will be introduced.
A. Related Work
Modulation classification problem has been tackled in
multiple research works. In [7–9], an effective
modulation classifier has been developed using
convolutional neural networks (CNN). The authors of
[10–12] demonstrated higher performance and faster
convergence for NN-based decoders for BCH and polar
code channel decoding. Additionally, in [13], signal
detection in multiple-input-multiple-output (MIMO)
orthogonal frequency-division multiplexing (OFDM)
systems was presented to attain highly developed
accuracy with much less complexity while giving
robustness under intricate channel interference.
Additionally, [14–16] provide examples of their
opportunities for enhancing wireless communication
systems.
Artificial DNNs have recently attracted attention in the
domain of channel equalization due to their abilities to
accomplish the mapping between input and output
domains in a way that is not linear and because
equalization and symbol detection might be considered as,
classification tasks [17–19].
In the equalization process that uses an adaptive
technique based on a NN to recover the target signal, the
NN minimizes the difference exist between the equalizer
outcome and the delayed signal to recover the channel’s
nonlinear properties from the data that was received.
Patra et al. demonstrated in [20–22] that for signals with
either pulse amplitude modulation (PAM) or quadrature
amplitude modulation (QAM), NN-based nonlinear
equalizers outperformed conventional linear equalizers in
terms of bit error rate (BER).
For a long time, NNs have been employed for channel
equalization [18, 23–25], therefore it appears reasonable
to use DNN as an end-to-end approach for optimizing
channel equalization and decoding simultaneously.
Numerous machine learning techniques were used to
tackle the nonlinear equalization problem in the absence
of accurate channel state information (CSI). Among these
techniques are DNN [26, 27], Gaussian processes for
classification (GPC) [28], convolutional neural network
(CNN) [19], and support vector machine (SVM) [29, 30],
which allow the receiver to achieve adaptive equalization.
The DNN equalizer described in [26] requires a
significant number of parameters and is only useful for
short codes. In [28, 31, 32] these equalizers determine the
channel filter coefficients of ISI using multiple training
sequences. Additionally, these methods require prior
knowledge of channel filter coefficient distributions and
the variance of AWGN. In actuality, such assumptions
may be false. As a result, without a priori information, it
is difficult to generate correct CSI analysis results. As
shown in [31], after the decoder, the SVM equalizer
performs poorly.
Given the aforementioned positive developments, DL
(deep learning) has a lot of power and is very promising
for dealing with more difficult situations and meeting the
strict needs of 5G and beyond communication systems.
B. Motivations and Contributions
Deep learning has lately gained wide popularity in the
areas of natural language processing, text translation, and
computer vision [33, 34]. The application of deep
learning in wireless communications has also experienced
significant growth [35].
The basic feedforward NN only creates the weighted
connections between the cascading layers, but in the
recurrent neural networks (RNNs), the adjacent neurons
in the same layer are also connected to each other. RNNs
could be trained to learn sequential or time-varying
patterns since they have memory or feedback connections.
In the last few years, RNNs have been prevented from
being a mainstream network model because of the
challenges in training and computing complexity. Due to
the advancement of deep learning theory, RNNs are
growing quickly right now.
The main feature of RNNs is that they have a hidden
layer with memory capabilities, which provides them
with a structural advantage when processing time-series
data. Therefore, using an RNN as a channel equalizer can
help the CS equalizer learn faster and work better. Deep
learning techniques are much better for channel
equalization than other methods because they can
automatically find problem features without needing a lot
of information in advance.
Standard RNNs have the drawback of being unable to
use information about future input [36]. In tasks like text
translation, the content that comes after the current text
often has a big effect on how it is understood, and it is
evident that the standard RNN is unable to account for
this. The bidirectional RNN (B-RNN) was suggested in
[36, 37] as a remedy for this shortcoming. In order to
accomplish this, the B-RNN uses two distinct hidden
layers that analyses the data in both directions before
feeding the results into a single output layer [33].
Standard RNNs also have the issue that they do not have
much of a chance of picking up informational features
over long distances. This is because as the time step
increases during the training phase, they might run into
gradient elimination and gradient explosion. A long
short-term memory (LSTM) architecture was suggested
in [38] as a solution to this issue. One could consider the
LSTM network to be an improved form of the RNN’s
simple hidden unit [33, 39]. Long-distance dependencies
can be handled well by the LSTM [40]. A traditional
LSTM has three gates (forget, input, and output), a
memory cell, a block input, and an output activation
International Journal of Electrical and Electronic Engineering & Telecommunications Vol. 13, No. 1, 2024
68
function (AF). Each of the gates and the block’s input are
linked to the block’s output [39].
In this work, we model the channel equalization
problem in SC-FDMA systems as a DL task, and propose
a new idea for combined channel equalization and signal
detection (CE-SD) based on DL recurrent feedback
LSTM-NN. This idea takes features from the SC-FDMA
system’s received messages and labels them based on the
constellation map used at the transmitter. Compared to
conventional non-neural network approaches, the neural
network approach is more flexible since it can be used for
various channel circumstances and does not need to
worry about channel details. In the proposed scheme,
channel equalization and signal detection are treated as a
complete black box, and the box functions are
continuously approached by a DNN model. The DNN
model has the ability to perform equalization and symbol
decoding simultaneously, even in the absence of channel
state information (CSI). In terms of bit error rate (BER),
simulation results showed that our suggested scheme
outperforms other commonly used signal equalization
approaches. This successful example highlights the utility
of DL in SC-FDMA systems.
Fig. 1. The proposed SC-FDMA scheme.
The most significant contributions are listed below.
1) We embed the DL strategy into the SC-FDMA system
to equalize the channel effects and detect symbols to
exploit the capabilities of the DNN at recognizing and
representing things, the training process can make the
CS equalization at data subcarriers work better.
2) We evaluate the performance of the suggested
framework for channel equalization under various
conditions. The accuracy of the channel equalization is
explicitly assessed using a simulation of the bit error
rate (BER). Additionally, simulations and comparisons
have demonstrated that the suggested framework is
effective and trustworthy across a variety of channel
circumstances.
3) We provide a dataset that will help the research
community to evaluate against linear equalizers (LEs),
such as the minimum main square error (MMSE) and
zero forcing (ZF) and optimization algorithms as well.
The following sections will organize the remainder of the
paper: The second section, which follows this one, is
devoted to describing the system. The third and fourth
sections introduce the Deep Learning model and the
offline training of the suggested model, respectively. The
simulation results are then shown. Finally, to conclude
the study.
II. SYSTEM MODEL
The SC-FDMA system is demonstrated in Fig. 1, as in
[41]. The overall system subcarriers are M. From among
those Nu users, each N subcarriers are designated to only
one user, where M= Nu×N. All of this is accomplished
following the N points fast Fourier transform (FFT). A
length Lcp of cyclic prefix (CP), longer than or equal to
the length of the channel’s transfer function Lch, will be
added following the M points inverse fast Fourier
transform (IFFT). The time domain (TD) transmitted
signal that corresponds to the kth user without the Lcp in
vector form is
where sk is the kth user’s
(N×1) symbol vector, Tk is an M×N sub-carrier mapping
matrix, and FN and
are the FFT and IFFT matrices,
respectively, with dimensions N×N and M×M. Assume
that Hk is the transfer function of the channel between the
kth user and the base station, with maximum delay spread
Lch less than the Lcp, to entirely remove the ISI.
The procedure will be reversed at the receiving end.
The CP is removed first, then the FFT transforms the SC-
FDMA symbols into FD by M points FFT accompanied
by sub-carrier demapping to extract the FD signal
received for the kth user.
The TD signal received relating to all Nu users for the
tth SC-FDMA symbol presented by
(1)
where
is the (M×M) channel circular convolution
matrix where the first column contains the impulse
response of the channel between the lth user and the BS,
and nt is an (M×1 ) AWGN vector with variance
. After
M FFT, the transform of the received signal is expressed as
(2)
It will be easier to follow if we remove the symbol index
from Eq. (2), as shown below:
(3)
where
The FD signal received for the kth user is calculated
International Journal of Electrical and Electronic Engineering & Telecommunications Vol. 13, No. 1, 2024
69
after demapping as:
(4)
Because of the mapping matrix’s orthogonality property,
The FD signal received for the kth user could well
resemble
(5)
where
is a N×N diagonal sub-matrix of
the M×M
matrix.
The signal received can be expressed using the bth
frequency bin as follows:
(6)
Then perform frequency domain equalization by any
traditional linear techniques (ZF or MMSE) like in [41] to
counteract the ISI’s effects. Where the frequency domain
channel equalization is carried out on a sub-carrier basis.
The FD-LE is conducted through multiplying W, to the
signal received. The signal equalized in FD domain is
given by
(7)
where
and
represent the diagonal matrix
used for channel equalization. The bth entry along the
diagonal is given by
for the ZF criterion and
for the MMSE criterion [42].
Then perform FD equalization by any traditional
technique like in [41] to counteract the ISI’s effects. Then
demodulate and detect the kth user original transmitted
symbols after N points IFFT TD transformation. However,
in the suggested method, a DNN is used instead of
conventional channel equalization techniques, which
results in an end-to-end scheme that can get back the
original information from the information that was sent.
III. DEEP LEARNING MODEL
Due to deep learning’s powerful capabilities, it has
been successfully applied in a wide variety of
applications, including natural language processing [43],
computer vision [44], speech recognition [33], and others.
Here, we will discuss the most fundamental theories and
concepts underlying deep learning and how they apply to
our model. For a detailed explanation of deep learning
and machine learning, please see [45].
LSTM NNs are discussed in this section for combined
channel equalization and symbol detection. Offline
training with simulated data is used to train the proposed
DL-LSTM-based channel equalizer.
The LSTM network is a form of RNN that is smart
enough to learn long-term correlations between time step
sequences [38]. Numerous LSTM-based systems have
been developed to address issues such as speech
recognition [46], handwriting recognition [47], and online
translation with tools like Facebook translation systems
[48], and Google neural machine translation [49].
Input, output, and forget gates, as well as a memory
cell, comprise the LSTM-NN structure. The LSTM-NN
properly stores the long-term memory via the forget and
input gates. The LSTM cell’s primary structure is
depicted in Fig. 2 in [38]. The forget gate allows the
LSTM-NN to eliminate the unwanted data from the last
process by using the present used input xt and the cell
output ht. On the basis of the preceding cell output ht-1
and the present cell’s input xt, the input gate determines
the data that will be utilized in conjunction with the
preceding LSTM cell state ct-1 to generate a new state of
the cell ct. LSTM may decide which data is discarded and
which is maintained by using the forget and input gates.
The output gate determines the present cell output ht
by utilizing the preceding cell’s output ht-1 at the present
state of the cell ct and input xt.
Fig. 2. LSTM neural network architecture.
The mathematical formulation for the LSTM-NN
configuration is given by (8) to (13) as in [50].
(8)
(9)
(10)
(11)
(12)
(13)
where is the input vector at time t with
entries. are the hidden layer states
vectors of the LSTM at time t and t1, respectively. wi,
wf, are the forward or the input
trainable weight matrices. are
the recurrent trainable weight matrices.
are the biases vectors. i, o, and f represent the input,
output, and forget gates, respectively.
represent the cell candidate, the gate activation function
(sigmoid function), and the state activation function (tanh
function), respectively. denote the Hadamard Product
(Elementwise Multiplication). LSTM-NN evaluates only
the prior sequence’s impact on the present sequence,
neglecting the subsequent information.
The DL LSTM-NN was constructed using an array of
the following five layers as shown in Fig. 3, to perform
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combined channel equalization and symbol detection: A
layer for the sequence input (with a size equal to the
amount of features in the input data, which is 128), an
LSTM layer (with 128 hidden units), and eventually, 256
classes are accomplished by the use of a 256 fully
connected layer, accompanied by a SoftMax layer and a
classification layer.
Fig. 3. DL LSTM-NN framework for the proposed joint channel equalizer and symbol detector.
In the current work each user is assigned four
subcarriers from the SC-FDMA frame, and each
subcarrier can be one of four QPSK constellation points.
The number of labels in the training equals
, where Ms
is the constellation (modulation) order and N is the
subcarriers that are assigned to only one user. Therefore,
the number of labels in the training equals 44=256, so the
number of classes is 256. Consequently, the size of the
fully connected layer in the LSTM-NN should be 256 to
match the number of classes. If we use higher-order
modulations or give each user more subcarriers, the
number of labels will also go up. This will make the
system harder to use, take longer to train, and lead to an
impractical system. As a result, we recommend using
QPSK.
As the proposed DL LSTM-based channel equalizer
and symbol detector is created, the weights and biases of
the recommended equalizer must be adjusted (tuned)
before deployment, using the appropriate optimization
algorithm. The optimal parameters (weights and biases)
are learned on a training set with predefined outputs. To
figure out the best parameters, a loss function is used to
figure out how far the network output is from the desired
output, and then three different DLNN optimization
methods are used to change the parameters. A number of
different optimization methods are used to get the best
possible channel equalization and symbol detection for
the SC-FDMA wireless communication system. Some of
them are stochastic gradient descent with momentum
(SGDm), root mean square propagation (RMSProp), and
adaptive moment estimation (Adam).
A. Optimization Algorithms
The method of back-propagation (BP) is frequently
utilized to train NNs. The training method repeats two
phases: propagation and weight update. The error
propagates from the output layer backward to the rest of
the nodes. Using these errors, we can determine the
gradient of the loss function relative to the network
weights. In order to minimize the loss function, the
optimization method takes the gradient and uses it to
make adjustments to the weights [51].
Learning processes are greatly aided by optimization
algorithms. By adjusting the model’s weights and biases
to minimize the loss function, the learning process seeks
to find a model that will yield improved results. Learning
deep neural networks is analogous to solving an
optimization problem, with the goal of achieving global
optimization via a stable training trajectory and rapid
convergence with gradient descent algorithms [52].
It is generally usual to use a gradient descent
optimization algorithm to seek the minimum loss function.
The gradient descent method changes the weights and
biases incrementally in small steps in the direction of the
negative gradient of the loss function.
(14)
(15)
where t is the iteration number, l and α represents the
layer number and learning rate, respectively. The learning
rate, α ∈ [0,1], determines how much the weights is
updated each time. When this parameter’s value is too
large, the output does not converge to the solution but
instead floats around it. To the contrary, if it is set too low,
the calculation will take too long to arrive at a solution.
In the typical gradient descent approach, the gradient
of the loss function is evaluated using the complete
training set at once and updates the parameters (w, and b)
after scanning the whole training set, this method is
referred to as “batch gradient descent”.
In the case of a convex problem, batch gradient
descent is ensured to converge to the global minimum,
while in the case of a non-convex problem, it is
confirmed to converge to a local minimum. However, in
deep learning-related tasks, the training set typically
contains thousands or even trillions of samples, making it
impractical to calculate the gradient via a quick scan.
This makes even a single update to the parameters too
time-consuming. It is also challenging to feed all the data
into the model at once due to the computation memory’s
capacity limitations. As a result, the batch gradient
descent method for solving the optimization problem is
rarely used in deep learning models.
Stochastic gradient descent (SGD) is a technique that
can be used to calculate the gradient and update the
parameters for each training sample rather than using the
entire set of training data. However, due to the large
variation between the training samples, updating
parameters often results in significant fluctuations in the
objective function. However, while a low learning rate
allows SGD to converge to a great point, it also slows
down training. The frequent data transfers between GPU
memory and local memory also reduce efficiency when
we are employing GPUs to carry out the computation.
Iterations utilize a different subset of the data, referred
to as a “mini batch”. The benefits of both batch gradient
descent and stochastic gradient descent are combined in
mini-batch gradient descent, which also updates the
parameters after obtaining the gradient of a small group
of samples. Unfortunately, good convergence is not
guaranteed by mini-batch gradient descent, and tuning the
learning rate also requires some expertise. So, to further
enhance the convergence, some researchers have added
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additional helpful tricks and techniques. All mini batches
of the entire training dataset are processed by the training
algorithm during one epoch.
Algorithms using stochastic gradient descent permit
oscillations along the path taken by the steepest fall to
reach the ideal. One mechanism for stabilizing this
oscillation is the use of momentum [53]. The weights and
biases of a neural network can be adjusted using
stochastic gradient descent with momentum (SGDm) in
the following manner:
(16)
(17)
where the contribution of the prior gradient step to the
current iteration is given by ϵ.
All of SGDm’s parameters are learned at the same rate.
By implementing learning rates that vary according to
parameters and can automatically adapt to the loss function
being utilized, network training can be made better. Root
mean square propagation (RMSProp) is one such approach.
To do this, it calculates the squares of the gradients of the
parameters at each element using the formula:
(18)
(19)
where µ
2 is the rate of decay of the moving average.
Overall, the rate of decay seems to be 0.9, 0.99, or 0.999.
The associated squared gradient averaging lengths are
equivalent to 1/(1µ
2), particularly, 10, 100, or 1000
parameter updates, respectively.
With the help of a moving average, the RMSProp
method normalizes the updates of the weights and bias
parameters, as shown below:
(20)
(21)
This allows RMSProp to increase learning rates for
parameters with small gradients while decreasing learning
rates for parameters with large gradients. A minor constant
is added to prevent a division by zero. where an
element-by-element division is used.
Momentum terms have been added to the parameter
updates in Adam, similar to RMSProp. When it comes to
deep learning, Adam is one of the most popular
optimization algorithms used. The term “Adam” comes
from the term “adaptive moment estimation”, which
describes how Adam calculates first and second moment
estimates of gradients to determine individual adaptive
learning rates for various parameters [54]. An element-
wise moving average of both the parameter gradients and
their squared values is retained by the algorithm. The
moving average of the parameter gradients can indeed be
described as follows:
(22)
(23)
where is rate of decay. Adam updates the network
parameters utilizing moving averages as follows:
(24)
(25)
If the gradients over multiple iterations are consistent,
a moving average of the gradient can be used to gain
momentum with parameter updates in a particular
direction. If the gradients are noisy, the moving average
of the gradient will be smaller, leading to smaller
parameter updates.
IV. OFFLINE TRAINING OF THE SUGGESTED DL MODEL
While DLNNs are the state-of-the-art approach for
wireless communication systems, they have a huge
amount of computational complexity and a lengthy
training period. GPUs have become the most effective
training equipment for DLNNs.
Due to the lengthy training period required for the
proposed DL model and the large number of parameters
that must be tuned during training, like weights and
biases, training must be conducted offline. The trained
model is utilized to extract the transmitted data during
online implementation.
Because machine learning-based models aren’t built by
experts, they learn from data instead of being made by
them. For the bulk of machine learning tasks, obtaining a
huge amount of labeled data for training is a difficult
challenge. Alternatively, training data for channel
equalization issues can be easily gotten by simply
conducting a simulation. Obtaining the training data is
straightforward once the channel parameters and model
are known.
Offline training of the neural networks is carried out
using simulated data. When you run a simulation, you
start with a random message s and send the SC-FDMA
frames to the receiving end through a simulated channel
model. Each frame has one SC-FDMA symbol in it. To
retrieve the received SC-FDMA signal, SC-FDMA
frames with varying channel defects are used. After
undergoing the distortion of the channel and removing
the CP, the incoming signals y are gathered as training
samples. As shown in Fig. 1, the network’s input data are
the signals that are received y, and the actual information
messages s. These signals act as the supervision labels.
By minimizing the loss function and updating the
weights and biases, the optimization algorithms train the
model. The loss function, in its simplest form, is the
difference between the network’s output and the original
messages, which can be expressed in a variety of ways.
The neural network toolbox in MATLAB gives the user
the option of selecting a loss function from a list of
possible options (i.e., MAE, crossentropyex, and MSE).
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The loss function that we used in our experiments is the
crossentropyex, and it can be expressed as:
(26)
where c is the class number, N is the sample number,
is the ith transmitted data sample for the jth class and
is the DLLSTM model response for sample i class j.
Because the activation function (AF) is a sigmoid
function, each output element has been compressed into
the range [0, 1], which can be understood as the
likelihood that the output bit is 1.
All classical equalizers are heavily reliant on tractable
channel models that are believed to be stationary, linear,
and Gaussian in nature. Practical wireless communication
systems, on the other hand, contain additional defects and
unknown environmental influences that precise channel
models are incapable of adequately addressing. As a
result, researchers have devised numerous channel
models capable of accurately describing practical channel
statistics. In this way, it is possible to get reliable and
relevant training datasets by modelling with these channel
models.
Pilot symbols can be used to figure out the channel
models in the wireless communication systems. Then, the
data broadcasted can be found by using the channel that
was figured out. The vehicular A channel model is being
used in this work to figure out how a real wireless
channel works and how this can hurt the performance of
the suggested model and the whole SC-FDMA wireless
communication system.
Finally, the model can recover data automatically,
without the need for explicit channel estimation and
symbol detection processes. These processes are
accomplished together. Fig. 4 shows how to train offline
to get a learned model based on LSTM-NN.
Fig. 4. Offline training of the DLLSTM-NN.
V. SIMULATION RESULTS
Several experiments were carried out to demonstrate
the efficiency of the proposed DLNN-based channel
equalizer and symbol detector technique for the SC-
FDMA wireless communication system. The proposed
DLNN-based equalizer was trained and compared to the
conventional Zero-Forcing (ZF) and Minimum Mean
Square Error (MMSE) equalizers in terms of bit error
rates (BERs) at different signal-to-noise ratios (SNRs)
using the collected data sets. The training dataset is
gathered for four subcarriers. The transmitter sends the
SC-FDMA packets to the receiver, each containing one
SC-FDMA data symbol. The SC-FDMA system and
channel specifications are listed in Table I. The employed
DL LSTM-NN architecture parameters and training
settings are summarized in Table II.
TABLE I: SC-FDMA SYSTEM SPECIFICATIONS
Parameter
Value
No. of Subcarrier = M-IFFT
64
Subcarriers allocated to each
user = N-IFFT
4
Subcarrier spacing
15KHz
Cyclic prefix length
20
Modulation Format
QPSK
Channel model
Vehicular A
Channel estimation
Perfect
Equalization
ZF, MMSE, and proposed DL Model
TABLE II: DL MODEL ARCHITECTURE
Parameter
Value
Sequence input t size
128
LSTM layer size
128
Fully connected layer size (No. of Classes)
256
Loss function
Crossentropyex
Mini-batch size
1000
Numbers of Epochs
3
Optimization approaches
Adam, RMSProp,
and SGdm
Gate Activation Function (GAF)
Sigmoid
State Activation Function (SAF)
Tanh
Training Options
Initial learning rate
0.05
Learning rate drop factor
0.45
Also, in the present simulations, several learning
optimizers namely: The SGdm, RMSProp, and Adam
will be used to train the proposed equalizer to investigate
how well it performs under these optimization techniques
[55]. Also, the learning rate has been chosen after an
enormous number of trials to ensure convergence for all
learning algorithms, and it is fixed for all channel
conditions.
In the case of deep fading channels, it is well known that
the linear equalization may amplify the noise at the spectral
null, which has a negative impact on the performance of
the SC-FDMA system. So, it is clear from Fig. 5, that the
proposed equalizer using the Adam learning algorithm and
crossentropyex loss functions outperforms both the ZF and
the MMSE equalizers at SNRs ranging from 7 to 20 dB.
The proposed equalizer can achieve BER=at 13 dB
compared to 16 dB by MMSE and 20 dB by ZF,
respectively. Additionally, for higher SNR (i.e., >14 dB)
the DL model was successfully able to detect the signal
perfectly such that the BER is zero for these SNR values,
which reflects the capabilities of our DL model.
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73
Fig. 5. BER curves of the proposed DL LSTM-based equalizer and the
traditional linear equalizers using the Adam learning algorithm and the
crossentropyex loss function.
Fig. 6. BER curves of the proposed DL LSTM-based equalizer and the
traditional linear equalizers using the RMSProp learning algorithm and
the crossentropyex loss function.
Fig. 7. BER curves of the proposed DL LSTM-based equalizer and the
traditional linear equalizers using the SGdm learning algorithm and
crossentropyex loss functions.
On the other hand, the MMSE outperforms the
proposed model when the SNRs drop by less than 7 dB,
and both linear equalizers outperform the proposed model
when the SNRs drop by less than 3 dB.
Moreover, it is clear from Fig. 6 that the proposed
equalizer using the RMSProp learning algorithm and the
crossentropyex loss functions outperforms both the ZF
and the MMSE equalizers at SNRs ranging from 7.5 to 20
dB. The proposed equalizer can achieve BER=103 at 10
dB compared to 11 dB by MMSE and 14 dB by ZF,
respectively. Additionally, for higher SNR (i.e., >12 dB)
the DL model was successfully able to detect the signal
perfectly such that the BER is zero for these SNR values,
which reflects the capabilities of our DL model. However,
when the SNRs decrease by less than 7.5 dB, the MMSE
outperform the suggested model, and when the SNRs
drop by less than 4 dB, both linear equalizers beat the
suggested model.
Furthermore, it is obvious from Fig. 7 that the
proposed equalizer using the SGdm learning algorithm
and the crossentropyex loss function have approximately
comparable performance to the MMSE equalizer at SNRs
ranging from 8.5 dB to 14.25 dB. After SNR = 18 dB, the
proposed equalizer produces zero BER, while the MMSE
and ZF produce a specific value of error, which reflects
the capabilities of our DL model to detect the signal
perfectly. On the contrary, the MMSE outperforms the
suggested model when the SNRs decrease by less than
8.5 dB, and both linear equalizers outperform the
suggested model when the SNRs drop by less than 4.25
dB.
Also, it is obvious from Fig. 5, Fig. 6, and Fig. 7 that
the MMSE equalizer outperforms the ZF equalizer at all
SNR examination ranges because it uses the channel
second-order statistics in the equalization process.
It is obvious from Fig. 8 that all models have
approximately comparable performance at SNR from 0 to
6 dB. After SNR = 6 dB, the SGDm model begins to have
the worst performance, while both the Adam and
RMSProp models were almost the same until SNR=12
dB. After SNR=12 dB, the RMSProp model outperforms
both the Adam and SGDm models. Finally, we can say
that the SGDm models have the worst performance in all
SNR ranges.
Optimization techniques are critical for the
improvement of deep learning systems. DNN training can
be viewed as an optimization issue, with the objective of
achieving a global optimum via a trustworthy training
trajectory and rapid convergence via gradient descent
techniques [55]. The goal of the DL method is to develop
a model that produces more accurate and faster outcomes
by modifying the biases and weights to minimize the loss
function. Selecting the best optimizer for a certain
scientific issue is a difficult task. By selecting an
inadequate optimizer, the network may remain in the
local minima (stay in the same place) during training,
resulting in little progress in the learning process. As a
result, the inquiry is required to look at how different
optimizers perform based on the model and dataset used
to make the best DL model.
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74
This section compares the performance of three
optimization algorithms: Adam, RMSProp, and SGDm,
using an experimental approach.
Fig. 8. Performance comparison of the proposed DL Equalizers using
different optimization algorithms.
Fig. 9. Loss function comparison of the DL equalizers using different
optimization algorithms.
Fig. 10. Accuracy curves comparison of the DL equalizers using
different optimization algorithms.
It is beneficial to monitor the training processes of the
DL equalizers during the training process. By plotting
loss and accuracy measures during the training process,
we can monitor how the training is proceeding. Fig. 9,
and Fig. 10 show that the sgdm optimization approach
achieves the highest loss (worst performance) when
compared to the Adam and rmsprop optimization
strategies and takes a long time to converge to 100%
accuracy as compared to both Adam and rmsprop, which
is supported by Fig. 8, which shows that the trained DL
equalizer using the sgdm algorithm has the greatest BER
values. Furthermore, the loss and accuracy curves of both
the Adam and rmsprop optimization techniques
emphasize the obtained results in Fig. 8.
A. Computational Complexity Comparison
The computational complexity of the proposed LSTM-
based channel equalization and symbol detection deep
learning models in the SC-FDMA is provided empirically
in terms of the training time which is performed offline.
Then these models’ computational complexity is
compared to the conventional models during the online
deployment in terms of the execution time and number of
operations (complex multiplications) of each model.
1) Training time
Training time can be defined as the amount of time
expended to get the best NN parameters (e.g., weights
and biases) that will minimize the error using a training
dataset. Because it involves continually evaluating the
loss function with multiple parameter values, the training
procedure is computationally complex.
Table III lists the consumed training time for LSTM-
based channel equalization and symbol detection deep
learning models. The used computer is equipped with
Windows 10 operating system and an Intel(R) Core(TM)
i5-2450M CPU @ 2.50GHz, and 8 GB of RAM.
TABLE III: COMPARISON OF OFFLINE TRAINING TIME
LSTM-based CE-SD
Adam (Min: Sec)
LSTM-based CE-SD
RMSprop (Min: Sec)
LSTM-based CE-SD
SGDm (Min: Sec)
14:06
14:29
15:25
From Table III, the LSTM-based CE-SD trained with
Adam optimizer consumes the lowest training time,
followed by LSTM-based CE-SD trained with RMSprop
optimizer, while the highest training time is consumed by
LSTM-based CE-SD trained with SGDm optimizer, at
the same training options. The LSTM-based CE-SD
trained with SGDm optimizer training time indicates its
high computational complexity in comparison to its peers.
The conventional models have no computational
complexity in the offline phase since they do not need
training.
2) Execution time
Here we will compare the execution time that is
necessary for our proposed model and that is necessary
for the conventional ways that use ZF or MMSE
equalizers, based on MATLAB MathWorks execution
time measurement functions.
At this point, the proposed model and the traditional
methods will be compared based on how long it takes for
each to run on the same computer settings as in offline
training.
From Table IV, the proposed LSTM-based CE-SD
consumes the highest execution time, while the
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75
conventional ways that use conventional equalizers such as
ZF or MMSE consume lower execution time. The
execution time of the ZF equalizer is the lowest one as well.
All of this is because of the number of operations
(complex multiplications and complex additions) in the
proposed LSTM-based CE-SD function in the input size,
hidden units’ size, and the fully connected layer size.
While the number of operations in the ways that use
conventional equalizers (ZF or MMSE) is linearly
dependent on the input size only. Therefore, the
computational complexity of the proposed LSTM-based
CE-SD will be the highest. Also, the number of operations
in the conventional way that use the ZF equalizer is
lower than that used with the MMSE equalizer.
TABLE IV: COMPARISON OF EXECUTION TIMES
LSTM-based CE-SD
With MMSE Equalizer
With ZF Equalizer
0.172989 s
0.011331 s
0.009307 s
3) Number of operations:
The computational complexity of our proposed model
can be computed in terms of the number of operations
(multiplications).
The goal of this high-level metric is to solely take into
account multipliers, neglecting additions, because the
latter can be implemented in hardware or software for a
lower cost, whereas multipliers are typically the slowest
component in the system and take up the most chip space
[56, 57].
According to the mathematical formulations and the
structure of our proposed deep learning model, mainly
depending on the LSTM-NN in Fig. 3, we can drive the
computational complexity. As in [58], the total number of
operations in a standard LSTM network can be calculated
as follows:
(27)
where is the number of hidden (memory cells) units in
the LSTM cell. , is the dimensional or the number of
features in the input vector at time t.
Then we have to compute the number of operations of
the fully connected layer to compute the overall
computational complexity of the proposed model.
Also as in [58], the size of the fully connected layer is
equal to the number of the classes we have to classify, let
us symbolize it with. The input to the fully connected
layer equal to the number of hidden states from the
LSTM layer. Hence, the number of operations of the
fully connected layer equal:
Number of Operations FC = (28)
By ignoring the SoftMax layer computational complexity
for simplicity, the overall complexity of the proposed
model will be
LSTM
FC
Number of Operations Number of Operations
Number of Operations
4 4 4
h i h k h
n n n n n
(29)
As we can see, the computational complexity depend
on the input size, the hidden state size, and the number of
classes we have to classify.
Now we will find the computational complexity of
both ZF and MMSE equalizers. Since the inverse of
N×N diagonal matrices in (7) require, a complexity of
[59]. And the FFT and IFFT require, a complexity
of
and
, respectively, the
overall complexity will be [60]:
LE 2
2
Overall complexity / 2log
/ 2log
N M M
NN
(30)
Therefore, the overall computational complexity of the
proposed and conventional models as a function of the
number of operations (mainly the complex
multiplications) can be summarized in Table V.
The number of addition operations in the denominator
of MMSE is more than that in ZF, so the execution time
for MMSE will be more than ZF, which confirms our
findings in Table IV.
TABLE V: COMPLEXITY COMPARISON OF THE PROPOSED AND
CONVERSATIONAL MODELS
LSTM-based CE-SD
With MMSE equalizer
With ZF equalizer
Fig. 11. BER curves of the proposed DL LSTM-based equalizer and the
traditional linear equalizers under ITU Indoor channel model.
Fig. 12. BER curves of the proposed DL LSTM-based equalizer and the
traditional linear equalizers under ITU Pedestrian channel model.
International Journal of Electrical and Electronic Engineering & Telecommunications Vol. 13, No. 1, 2024
76
B. Generalization Ability and the Robustness
In these experiments, several practical channel models
have been adopted. These channel models have been
established based on lots of measurements (such as the
indoor and pedestrian models) released by ITU [61, 62].
Fig. 5, Fig. 6, Fig. 7, Fig. 11, and Fig. 12 illustrate the
BER performance versus the SNR of the proposed
scheme and linear MMSE and ZF equalizers under three
different ITU channel models. It can be noted that the
proposed equalizer provides stable performance where it
outperforms the ZF equalizer in all studied channel
models. Also, it strongly competes with the MMSE
equalizer and beats it in most of the studied channel
models. The obtained results emphasize the
generalization ability and robustness of the proposed
equalizer as it has been tested using datasets (corrupted
by the 3 different ITU channel models) that it has not
used in the training process before.
VI. CONCLUSION
In conclusion, an online DL-LSTM-CE-SD-based SC-
FDMA system is proposed. The suggested equalizer is first
trained offline, then used in the communication system to
keep track of the channel statistics. Finally, the channel is
equalized, and the transmitted symbol is recovered from
the transmitted data stream. The suggested equalizer’s
performance is studied and compared to other standard
equalizer approaches, such as ZF and MMSE. The
proposed equalizer beats both the ZF and MMSE
equalizers in terms of BER, exhibits significant
enhancements, and is adaptable for various channel
conditions. In addition, a comparison of three different
optimization methods for DL was done to study how the
proposed equalizer performs at each. A complexity
comparison between the proposed and conventional
equalizers was investigated. Since recovering the
transmitted signal and delivering the information to the
receiving end is crucial in most cases, the focus of such
applications is on good performance rather than
mathematical complexity. Even though the computational
complexity of the proposed DL model is higher than that of
traditional models, the fast growth of technology in
designing and manufacturing high-speed GPUs gives
priority to the proposed model. Due to the exceptional
learning and generalization characteristics of the proposed
DL LSTM-CE-SD model, the suggested equalizer seems
promising for channel equalization in SC-FDMA
communication systems, especially in worse channel
circumstances.
Here are some suggestions for future research:
Evaluating the performance of the proposed
equalizer utilizing multiple optimization techniques,
including Adagrad, AdaMax, and Nadam.
Evaluating the efficacy of the proposed equalizer
with different states and gate activation functions.
Developing and employing loss functions that are
more robust than the cross-entropy function to
enhance the efficacy of the proposed model.
CONFLICT OF INTEREST
The authors declare no conflict of interest.
AUTHOR CONTRIBUTIONS
All of the authors of this research paper took part in
planning, carrying out, and analyzing the study. They
have all read and approved the final version.
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distributed under the Creative Commons Attribution License (CC BY-
NC-ND 4.0), which permits use, distribution and reproduction in any
medium, provided that the article is properly cited, the use is non-
commercial and no modifications or adaptations are made.
Mohamed A. Mohamed Omar was born in
Sohag, Egypt in 1987. He received his B.Sc.
with an honour degree in electronics and
communication engineering from Al-Azhar
University, Egypt, in 2010. He received his
M.Sc. degree in telecommunications
engineering from King Fahd University of
Petroleum and Minerals (KFUPM), Dhahran,
KSA, in 2018. From 2012 to 2015 and from
2018 to 2020, he was a teaching assistant at
Al-Azhar University. Since 2020, he has
been an assistant lecturer at Al-Azhar University. Currently, he is a
Ph.D. student at Aswan University. His research interests include
Wireless Communications, Communication & Signal Processing, and
machine learning.
Hassan A. Hassan was born in Luxor,
Egypt in 1984. He received a B.S. degree in
electrical engineering from Al-Azhar
University, Egypt, in 2007, and an M.S.
degree in electrical engineering from Al-
Azhar University, Egypt, in 2017. From
2012 to 2017 he was a teaching assistant
with Al-Azhar University. Since 2017, he
has been an assistant lecturer with Al-Azhar
University. Currently, he is a Ph.D. student
at Aswan University. His research interests
include wireless communication, channel estimation of signals, and
machine learning.
Mohamed H. Essai Ali was born in El
Balyana town, Sohag, Egypt in 1978. He
received the B.S. degree in electrical
engineering from the Al-Azhar University,
Egypt, in 2001, M.S. degree in electrical
engineering from Assuit University, Egypt,
in 2007, Egypt and the Ph.D. degree in
mechanical engineering from Novosibirsk
State Technical University, Novosibirsk,
Russia, in 2012. From 2001 to 2008, he was
a demonstrator and lecturer assistant with Al-Azhar University. From
2009 to 2012 he was a PhD student with Novosibirsk State Technical
University. From 2012 to 2018 he was Assistant Professor with Al-
Azhar University. From 1st April 2014 to 25 Dec. 2014, he was a Guest
Researcher, with Novosibirsk State Technical University. Since 2018,
he has been an Associate Professor with the Electrical Engineering
Department, Al-Azhar University, Faculty of Engineering. He is the
author of five textbooks, more than 50 articles. His research interests
include theory and applications of robust statistics, wireless
communication, channel estimation of signals in terms of a priori
uncertainty for the problems of telecommunications, optical wireless
communication, artificial intelligence-based signal processing
applications, and FPGA based applications.
Hamada Esmaiel received the B.Sc. degree
in electrical engineering from South Valley
University, Egypt, in 2005, and the MSc in
wireless communications from South
Valley University, Egypt in 2010, and the
Ph.D. degree in communication engineering
from University of Tasmania, Australia in
2015. In 2011 he was a researcher assistant
in wireless communication Lab. Wonkwang
University, Iksan, South Korea. Since 2015,
he has been an assistant professor at Aswan University, Egypt. He is the
general co-chair of IEEE IEEE ISWC’ 18. He is a technical committee
member in many international conferences and a reviewer in many
international conferences, journals and transactions. His current
research interests are 5G networks, Li-Fi technology, millimeter wave
transmissions, underwater communication and MIMO systems. He is an
IEEE member.
Osama A. Omer (Member, IEEE) received
the B.Sc. and M.Sc. degrees from South
Valley University, in 2000 and 2004,
respectively, and the Ph.D. degree from the
Tokyo University of Agriculture and
Technology, in 2009. He has spent six
months as a Postdoctoral Researcher with the
Medical Engineering Department, Luebeck
University, Germany. He has also spent three
months as a Postdoctoral Researcher at
Kyushu University, Japan. The last but not least, he has spent six
months as a Research and Development Scientist Engineer at the
NOKIA Research and Development Center, Tokyo/Japan, in 2008. He
is currently a Full Professor at the Faculty of Engineering, Aswan
University. His research interests include wireless communications,
deep learning, and image/signal processing.
Ahmed S. Mubarak received his B.Sc.
degree in electrical engineering from South
Valley University, Egypt, Egypt, in 2001,
and his M.Sc. degree in electrical
engineering from Assiut University, Egypt,
in 2012, and his Ph.D. degree in electrical
engineering from Aswan University, Egypt,
in 2019. Currently, he is an assistant
professor in Aswan Faculty of Engineering,
Aswan, Egypt. His research interests are in
the broad areas of wireless communications
including 5G & 6G networks, millimeter wave/THz transmissions and
MIMO systems, communication theory, signal processing, and deep
learning.
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