# Benchmarking Machine Learning Techniques for THz Channel Estimation Problems

Mounir Bensalem and Admela Jukan  Technische Universität Braunschweig, Germany
{mounir.bensalem<EMAIL_ADDRESS>
###### Abstract

Terahertz communication is one of the most promising wireless communication
technologies for 6G generation and beyond. For THz systems to be practically
adopted, channel estimation is one of the key issues. We consider the problem
of channel modeling and estimation with deterministic channel propagation and
the related physical characteristics of THz bands, and benchmark various
machine learning algorithms to estimate THz channel, including neural networks
(NN), logistic regression (LR), and projected gradient ascent (PGA). Numerical
results show that PGA algorithm yields the most promising performance at SNR=0
dB with NMSE of -12.8 dB.

##  I Introduction

THz communication systems are positioned as the main driver in the evolution
towards next generation wireless communications, also referred to as 6G [1].
In THz systems, operating in a frequency range 300 GHz - 10 THz and enabling
unprecedented wireless transmission rates, several major physical limitations
make the channel modeling and estimation fundamentally hard [2]. Recent work
focused on developing systems and algorithms for estimating mmWave channels in
MIMO systems [3, 4]. MIMO channel estimation is challenging due to the
quantization of the linear combination of transmitted signals from multiple
input antennas [5]. Using higher frequencies, and larger bandwidths needs high
precision analog-to-digital-converters (ADCs), which motivates the utilization
of low resolution ADCs for MIMO systems [3], and more practically 1-bit ADCs
[6].

Today, significant efforts are underway to accurately model and estimate the
channel with machine learning (ML). The trend of applying machine learning
algorithms in order to estimate mmWave and furthermore THZ channels is rather
notable in fact, also considering low resolution ADCs [7, 3, 8, 6]. [8]
proposed a deep convolutional neural network (DCNN)-based spherical-wave CE
algorithms for THz systems, the THz channel is simulated using a channel model
developed for mmWave systems and adapting the frequency parameters to 0.6 THz.
We refer to a some ML algorithms studied in [6] for MIMO systems, such as
projected gradient ascent (PGA) and Franke-Wolfe techniques, to benchmark
their performance in a THz scenario alongside with basic neural network
estimator, due to their good performance for a carrier frequency of 28 GHz.
Overall, ML-based channel estimation problem in THz communication systems is
still a rather open research direction, with experimental setups and
analytical models still few and far between.

In this paper, we benchmark different machine learning algorithms used for THz
channel estimation. Our reference model is THz-MIMO channel with 1-bit ADCs,
as proposed in [9] and [10] due to the limited quantization level of ultra-
high speed low power ADCs. We model the THz channel using a deterministic
channel propagation model from the literature [11], considering the existence
molecular absorption loss and showing that data rates around 1 Tbps can be
obtained with a short distance of around 1 meter. Our model advances the basic
model of THz band proposed in [12], which uses an updated molecular High
Resolution Transmission (HITRAN) 2012 database. We adopt an indoor scenario
with reflecting walls in order to consider the line-of-sight (LoS) and several
non-line-of-sight (NLoS) reflected rays. We apply the analytical model to
obtain the channel state information. Using a pilot generation methods,
Discrete Fourier Transform (DFT) and Zadoff-Chu (ZC) based sequence training,
we generate a training dataset of pilots input and signal response at the
receiver. Based on this reference model, we apply different machine learning
algorithms for channel estimation, including neural networks (NN), logistic
regression (LR), and PGA We chose these specific algorithms as they proved
good performance for similar problems, with complexity similar to the
complexity of gradient descent learning Numerical results, including analysis
and simulations, show that PGA algorithm yields the most promising performance
at SNR=0 dB with NMSE of -12.8 dB.

The rest of the paper is organize as follows: Section II describes the channel
and system model. Section III describes the studied ML algorithms for Channel
Estimation. Section IV shows the simulation results. We conclude the paper in
Section V.

##  II Channel and System Model

Notation: H is a complex matrix and H denotes a real matrix. $\textbf{H}^{T}$
and $\textbf{H}^{\ast}$ represent the transpose and conjugate transpose of H,
respectively. Re(H), Im(H) are the real and imaginary part of H.
$\|\textbf{H}\|$ is the Frobenius norm of H. $\mathbbm{1}_{[.]}$ is the
indicator function.

###  II-A THz Propagation Model

We consider a single transmitter (Tx) and a single receiver (Rx) THz
transmission system, using a point-to-point line-of-sight communication
scenario with omnidirectional antennas [12], separated by a distance $d$. The
main difference between Thz bands and other frequency bands appears from the
molecular absorption loss, which varies with the signal frequency, the
transmission distance and the concentration and the mixture of molecules
coming across the path.
We consider the medium loss due to the vibration changes of molecules, which
is denoted as the absorption loss $L_{abs}(f,d)$. This loss accurs due the
transformation of a part of the wave energy into a kinetic energy, and it
depends on the operating frequency the distance between the sender and
receiver, and the composition of the meduim. The equation is obtained from
[12] as follows:

$L_{abs}(f,d)=e^{k(f)\cdot d}=k(f)\cdot d\cdot 10\cdot log(e)\;\;(dB)$ (1)

Where f is the operating frequency, and $k(f)$ is is the overall absorption
coefficient of the medium available from HITRAN database [13], which includes
HITRAN line-transition parameters and cross-sections, that can be used to
model signal transmission.

Similar to [12], we assume that the spreading loss, which represents the
attenuation due to the expansion of a wave that propagates through the medium,
is given by:

$L_{spread}(f,d)=\left(\frac{4\pi fd}{c}\right)^{2}=20\cdot
log\left(\frac{4\pi fd}{c}\right)\;\;(dB)$ (2)

Where c is the speed of light in the vacuum. The received signal power
spectral densit (psd) in THz band is given by:

$\begin{split}P_{Rx}&=\frac{P_{Tx}}{L_{abs}\cdot L_{spread}}\\\ &=P_{Tx}\cdot
C\cdot f^{-2}\cdot
d^{-2}e^{-k(f)d},\;\;\;C=\frac{c^{2}}{16\pi^{2}}\end{split}$ (3)

Where $P_{Tx}$ represents the transmitted signal psd.
Molecular absorption introduces noise to the transmitted signal, which needs
to be considered to evaluate the signal to noise ratio (SNR). The molecular
noise psd is given by [12]:

$P_{M}=\frac{P_{Tx}(f)}{L_{spread}(f,d)}(1-e^{-k(f)\cdot d})$ (4)

Additionally, we consider the Johnson-Nyquist (JN) noise generated by thermal
agitation of electrons in conductors. The psd of JN noise is given by[2]:

$P_{JN}=\frac{hf}{(\exp(\frac{hf}{k_{B}T})-1)}$ (5)

Where $k_{B}$ denotes the Boltzmann constant and $T$ denotes the temperature
in Kelvin.
In order to simplify our analysis, we approximate eq. (5) using series
expansion, then we keep the first order term as follows:

$\begin{split}P_{JN}&=\frac{hf}{(\exp(\frac{hf}{k_{B}T})-1)}\\\
&=\frac{hf}{\frac{hf}{k_{B}T}+\left(\frac{hf}{k_{B}T}\right)^{2}\cdot\frac{1}{2!}+\left(\frac{hf}{k_{B}T}\right)^{3}\cdot\frac{1}{3!}+..}\\\
&\approx k_{B}T,\;\;\;\text{1${}^{\text{st}}$ order approx.}\end{split}$ (6)

Using eq. (4) and (6), the total noise psd is given by:

$P_{N}=k_{B}T+P_{Tx}\cdot C\cdot f^{-2}\cdot d^{-2}(1-e^{-k(f)d})$ (7)

Therefore, the signal-to-noise ratio (SNR) at the receiver is represented as :

$\begin{split}SNR&=\frac{P_{Rx}}{P_{N}}\end{split}$ (8)

###  II-B System Model

We consider a narrowband THz-MIMO system with $M_{t}$ antennas at the
transmitter, $M_{r}$ antennas at the receiver. Our studied system is depicted
in Fig. 1. We assume that the number of antennas at both transmitter and
receiver are equal. The arrival paths are considered to include two main
components of a THz channel: the LoS and several NLoS reflected rays caused by
reflection. We assume that the scattering and diffraction can be neglected due
to their negligible effect on propagation in the THz band.

Let $\textbf{H}\in\mathbb{C}^{M_{t}\times M_{r}},$ denotes the narrowband
channel matrix of THz channel impulse response. The channel matrix is
expressed as follows [11]:

$\begin{split}\textbf{H}=&\alpha_{L}G_{t}G_{r}a_{r}(\theta^{r},\psi^{r})a_{t}(\theta^{t},\psi^{t})\\\
&+\sum_{i=1}^{N_{clu}}\sum_{l=1}^{L_{ray}^{i}}\alpha_{i,l}^{NL}G_{t}G_{r}a_{r}(\theta_{i,l}^{r},\psi_{i,l}^{r})a_{t}(\theta_{i,l}^{t},\psi_{i,l}^{t})P_{r}(f,\tau_{i,l})\end{split}$
(9)

Where $\alpha_{L}$ is the complex gains of the LOS ray component, given by
$|\alpha_{L}|^{2}=L_{spread}(f,d)L_{abs}(f,d)$. $\alpha_{i,l}^{NL}$ is the
complex gains of the NLOS ray component, given by
$|\alpha_{i,l}^{NL}|^{2}=\mathcal{F}_{i,l}(f)L_{spread}(f,d_{1}+d_{2})L_{abs}(f,d_{1}+d_{2})$,
with $\mathcal{F}_{i,l}$ is the Fresnel reflection coefficient, $d_{1}$ is the
distance between the transmitter and the reflector, and $d_{2}$ is the
distance between the reflector and the receiver [14]. $P_{r}(\tau_{i,l})$ is
the pulse-shaping function at time delay $\tau_{i,l}$. $G_{t}$ and $G_{r}$ are
the associated transmit and receive antenna gains, respectively, $a_{t}(.)$
and $a_{r}(.)$ are the antenna array response vectors at the transmitter and
receiver, $\theta^{t}$ , $\psi^{t}$ , $\theta^{r}$ and $\psi^{r}$ are the
azimuth AoD, elevation AoD, azimuth AoA, and elevation AoA of the LOS ray
component, $\theta_{i,l}^{t}$ , $\psi_{i,l}^{t}$ , $\theta_{i,l}^{r}$ and
$\psi_{i,l}^{r}$ are the corresponding parameters for the $l^{t}h$ NLOS ray in
the $i^{t}h$ cluster.

At the receiver, after obtaining the signal the FFT is computed. Let
$\textbf{r}\in\mathbb{C}^{M_{r}\times 1}$ denote the unquantized received
signal, and given as follows:

$\textbf{r}=\textbf{H}\textbf{x}+\textbf{z},$ (10)

where $\textbf{z}\in\mathbb{C}^{M_{r}\times 1}$ is the additive white gaussian
noise (AWGN) matrix, whose samples are independent and identically distributed
(i.i.d.) $\textbf{z}\sim\mathcal{CN}(0,N_{0})$, and $N_{0}$ is the noise
power.

Figure 1: THz-MIMO with $M_{t}$ transmit antennas, $M_{r}$ receiver antennas.
Two 1-bit ADCs are used to equalize each received signal.

######  Definition 1.

An ADC is a device that applies a binary operation on a complex space by
mapping its elements into binary numbers. We define the following mappings:

  * •

1-bit ADC mapping:

$\begin{split}\text{sign}:\mathbb{R}&\longrightarrow\mathbb{Z}\\\
x&\longmapsto\begin{cases}1&\text{if $x\geq 0$ }\\\
-1&\text{otherwise}\end{cases},\end{split}$ (11)

  * •

Vector ADC mapping:

$\begin{split}\text{Sign}:\mathbb{R}^{n}&\longrightarrow\mathbb{Z}^{n}\\\
(x_{1},...,x_{n})&\longmapsto(\text{sign}(x_{1}),...,\text{sign}(x_{n})),\end{split}$
(12)

  * •

Matrix ADC mapping:

$\begin{split}\text{Sign}:\mathbb{R}^{n\times
m}&\longrightarrow\mathbb{Z}^{n\times m}\\\
\\{x_{i,j}\\}_{\begin{subarray}{c}1\leq i\leq n\\\ 1\leq j\leq
m\end{subarray}}&\longmapsto\\{\text{Sign}(x_{i,j})\\}_{\begin{subarray}{c}1\leq
i\leq n\\\ 1\leq j\leq m\end{subarray}},\end{split}$ (13)

  * •

Element-wise ADC quantization operator for channel response:

$\begin{split}\widehat{g}:\mathbb{C}^{n\times
m}&\longrightarrow\mathbb{Z}[j]^{n\times m}\\\
a+jb&\longmapsto\text{Sign}(a)+j\>\text{Sign}(b)\end{split}$ (14)

We equalize the analog received signal to obtain a binary output, defined
using the previous mappings as follows:

$\textbf{y}=\widehat{\text{ g}}(\textbf{r})=\text{
Sign}(\operatorname{Re}(\textbf{r}))+j\text{
Sign}(\operatorname{Im}(\textbf{r}))$ (15)

##  III ML algorithms for Channel Estimation

The aquisition of accurate CSI is crucial for THz communication systems.
Channel estimation in THz systems is a challenging task due to the tiny
wavelength of THz signals, the hardware limitations eg. high speed ADCs, and
difficulty of estimating large amount of parameters with small size
measurements. Moreover, channel estimation needs a minimum SNR value to ensure
the reliability of measurments, while the bandwidth in THz systems is in the
order of gigahertz, which presents a high thermal noise. As a results,
traditional channel estimation techniques, which are based on approximative
channel models may not be sufficiant to provide an accurate channel parameters
estimation. In this section we define the problem of channel estimation, and
present different solutions and various ML based techniques, which learn
channel parameters over any type of channel without prior assumptions.

###  III-A Problem Definition

The goal of CE problem is the estimation of channel response parameters H from
the received noisy symbols $\textbf{y}_{n},\forall n\in\mathcal{N}$, given
perfect knowledge of the sent pilots $\textbf{x}_{n},\forall n\in\mathcal{N}$,
where $\mathcal{N}$ defines the set of indexes of the training blocks,
$N=|\mathcal{N}|$ is the training block size assumed to be equal to $M_{t}$.
We convert the complex definition of parameters into real definition, as
follows:

$y_{n}=[\operatorname{Re}(\textbf{y}_{n}),\operatorname{Im}(\textbf{y}_{n})]$
(16)

$H=[\operatorname{Re}(\textbf{H}),\operatorname{Im}(\textbf{H})]$ (17)

$x_{n}=\begin{bmatrix}\operatorname{Re}(\textbf{x}_{n})&\operatorname{Im}(\textbf{x}_{n})\\\
-\operatorname{Im}(\textbf{x}_{n})&\operatorname{Re}(\textbf{x}_{n})\end{bmatrix}$
(18)

$z=[\operatorname{Re}(\textbf{z}),\operatorname{Im}(\textbf{z})]$ (19)

Let $y_{n}^{e}$ be the expected equalized binary output of received signal. In
order to study a variety of ML algorithms, we define more than one loss
function for CE problem, considering models from the literature and proposed
variations as follows:

  * •

The non-linear least squares minimization: it aims at minimizing the error
between the expected output and the calculated one.

$\begin{split}[\widehat{H},\widehat{z}]=\arg{\min_{H,z}}&\|y_{n}^{e}-y_{n}\|_{2},\\\
&y_{n}=f(Hx_{n}+z),\forall n\in\mathcal{N}\end{split}$ (20)

    * –

with linear activation function [15]: In this case, we consider $f(.)$ to be
the identity function, which is equivalent in neural network terminology to a
perceptron with linear activation function.

    * –

with tanh activation function: In this case, $f(x)=tanh(x)$ is an activation
function defined on $\mathbb{R}\rightarrow[-1,1]$, which gives an output close
to the $Sign$ function when the input is absolutely high, while being
differentiable. The differentiability of the function is important for any
neural network algorithm in order to calculate the derivative.

  * •

The log-likelihood minimization [16, 6]: is designed to estimate a transformed
varaible $\textbf{X}=X^{Re}+jX^{Im}\in\mathbb{C}^{M_{t}\times M_{r}}$
corresponding to the matrix Hx. The noise component is removed from the
optimization because it is considered to be constant.

$\begin{split}[\widehat{X^{Re}}]=\arg{\min_{X^{Re}}}&\sum_{i=1}^{M_{t}}\sum_{j=1}^{M_{r}}\mathbbm{1}_{[y_{i,j}^{Re}=1]}log(\phi(X_{i,j}^{Re}/\sigma))\\\
&+\mathbbm{1}_{[y_{i,j}^{Re}=-1]}log(1-\phi(X_{i,j}^{Re}/\sigma))\\\
\end{split}$ (21)

where $\phi(.)$ is the cumulative distribution function of the standard normal
random variable, and $\sigma$ represents its standard deviation.

###  III-B Learning based Approach

Before introducing the proposed benchmark solutions, it is essential to
reformulate the minimization problem in eq. (20) and eq. (21) as a binary
classification problem. First, we create a training dataset of $N$ data points
{$(x_{n},y_{n}),\forall n\in\mathcal{N}$}, where each data point represents
one pilot. The minimization problem in eq. (20) can be seen as a minimization
of the $L2$-norm loss function. It is well known that this loss function can
provide stable solutions unlike $L1$-norm, but it suffers from overfitting
problem. To prevent this the coefficients from fitting perfectly to the pilot
training data, we introduce a regularization term to the loss function called
$L2$-regularization.

$\begin{split}[\widehat{H},\widehat{z}]=\arg{\min_{H,z}}&[\mathcal{J}(H,z)],\\\
\mathcal{J}(H,z)=&\frac{1}{N}\sum_{n=1}^{N}\|y_{n}^{e}-\text{tanh}(Hx_{n}+z)\|_{2}\\\
&+\lambda\|H\|_{2}\end{split}$ (22)

where $\lambda$ denotes a regularization coefficient.

####  III-B1 Processing Units

The basic processing unit is called artificial neuron and it computes additive
and multiplicative operations over input variables, before applying an
activation function. Fig. 2 illustrates both the architecture of a general
artificial neuron and the instantiated version for CE.
Let $W=(w_{1},w_{2},...,w_{n})$ be the weight vector of the artificial neuron,
$x=(x_{1},x_{2},...,x_{n})$ is an input variables, and a bias value $b$. The
process of a neuron can be represented as follows:

$y=f(\sum_{i=1}^{n}w_{i}x_{i}+b)$ (23)

where y denotes the output, and $f(.):\mathbb{R}\longrightarrow\mathbb{R}$
represents an activation function.

Commonly, $f(.)$ is a non-linear function, such as sigmoid, hyperbolic, and
rectified linear function. In CE problem, the activation function of a neuron
corresponds to the $sign(.)$ function applied by ADC device, which fulfills
the characteristics of an activation function.

Figure 2: Neuron architecture.

####  III-B2 Network Components

Generally, neurons are grouped into layers to form a multilayered network
architecture, and based on the stacking configuration of processing units, we
obtain different types of neural networks, such as fully-connected,
convolutional, and recurrent networks. In our case, the CE problem is seen as
a basic neural network architecture, which has one hidden layer with $N$
neurons. Fig 3 shows the neural network based channel estimation NN-CE
architecture, where the vectors $H_{n}=[h_{n,1},...,h_{nN}],\forall
n\in\mathcal{N}$ represent the channel response vector for the receive antenna
$n\in\mathcal{N}$. The vector $z=[z_{1},...,z_{N}]$ denotes the bias
parameters of NN and it can be explained as the channel noise associated to
each receive antenna $n\in\mathcal{N}$.

Figure 3: NN-CE architecture.

####  III-B3 Training

After modeling the NN architecture, we need to define a loss function, in
order evaluate and improve its results. The main goal of the training is to
minimize the error loss function, while adjusting the weights and bias
parameters, which is eventually the same goal of CE problem. The loss function
was discribed in III-A.

Some optimization algorithm like Stochastic Gradient Descent (SGD) can be used
to train the NN-CE, while minimizing the loss function. The optimization
process updates gradually the weights representing the channel response
parameters, and the bias representing the noise, in order to find the optimal
solution:

$\begin{split}h_{i,j}=&h_{i,j}-\alpha\frac{\partial\mathcal{J}(H,z)}{\partial
h_{i,j}}\\\ z_{i}=&z_{i}-\alpha\frac{\partial\mathcal{J}(H,z)}{\partial
z_{i}}\end{split}$ (24)

where $\alpha$ is a parameter called the learning rate, which determines how
much the model learns in each step. The backpropagation algorithm is used to
calculate both the weights and bias partial derivatives of the cost function.

The training steps of NN-CE can be described in two main steps: (i) the feed-
forward pass, where the information cross all the network layers (one layer in
our case) from the input layer to the output layer which contains the
classification decision, i.e. the channel response, and (ii) the
backpropagation pass, where the error generated by the NN-CE is calculated and
propagated through the layers in the opposite direction, from the output layer
to the input layer.

###  III-C Benchmark Algorithms

Variety of ML algorithms can be tested for the CE problem, where the concept
of iterative learning is preserved, while modifying some details in the way we
update the CSI at each step. In the previous subsection, we described how NN
is modeled and used for channel estimation. Similar to it we can use Logistic
Regression (LR) while changing the loss function. In our paper, we consider
two other learning algorithms: Projected Gradient Ascent (PGA) and Franke-
Wolfe techniques, developed in [6] for low-rank mmWave MIMO channel, in order
to check its performance on THz MIMO channel, which is extremely sensitive to
weather condition, distance, and suffers from molecular attenuation. For PGA
algorithm, the learning process is based on gradient ascent, a similar
learning algorithm to gradient descent with a positive addition of the
gradient at each update. PGA includes a projection step at each iteration,
assuming that the matrix H has lower rank $r$ than N, $r\ll N$, where
$N=M_{t}=M_{r}$. Singular Value Decomposition (SVD) and simplex projection are
used to find the closest low rank matrix to the updated estimation of H. For
Franke-Wolfe algorithm, an additional step is included to the basic gradient
ascent learning, where we update the channel matrix differently: instead of
adding a weighted gradient matrix, we compute the top singular vector of the
gradient, then we soustract it from the gradient and update the channel matrix

###  III-D Pilot Generation Methods

In order to simulate the transmission process in our THz MIMO scenario, it is
required to define a pilot generation method, regarding its role in TX-RX
synchronization in wireless communication systems, and its importance in
learning the CSI. Similar to [6] we adopt two methods from the literature: DFT
and ZC based sequence training. DFT and ZC sequences can be pre-determined
without resorting to the knowledge about the channel, which makes both methods
suitable for our scenario.

##  IV Simulation Results

In the simulation, we consider a THz-MIMO system with $Mt=16$, $M_{r}=16$, and
a frequency $f=0.3$ THz. We simulate the THz channel using the proposed model
in section II. The deterministic channel model proposed in [11] works in the
range of frequencies between 0.1 and 1 THz, which explains our choice of the
frequency $f=0.3$ THz in the simulation. We consider a dry air environment,
where we adopt an updated molecular HighResolution Transmission (HITRAN) 2012
database. We consider the following gas composition of the air including their
mixing ratios: $N_{2}$: 0.78, $O_{2}$: 0.21, $CO_{2}$: $365\cdot 10^{-6}$,
$O_{3}$: $10\cdot 10^{-6}$, $CH_{4}$: $1.7\cdot 10^{-6}$, $H_{2}$: $500\cdot
10^{-9}$, $N_{2}O$: $320\cdot 10^{-9}$, $H_{2}O$: 0.0096. We set the operating
temperature equal to the standard temperature 296 K, and the pressure to 1
atm. We set the distance between the transmitter and the receiver $d=1$ meter,
where this value is set to be similar to the distance used in some experiments
[17]. The azimuth AoA and AoD are generated from the uniform distribution in
the range of $[-\frac{\pi}{2},\frac{\pi}{2}]$. The CSI is generated using eq.
(9), and the channel response at the receiver is measured as described in
section II. The developed THz simulator will be documented and open source,
along with the generated dataset and evaluated algorithms.

We train the proposed algorithms: Logistic Regression, NN, PGA, and Frank
Wolf, with 10 channel matrices generated with a fixed seed. The number of
epochs is set to 100, which ensures the convergences based on our simulations.
The stopping criterion is set the same for all learning algorithms as
$\epsilon=10^{-10}$. The learning rate is intiialized with 0.01, and to
$1/N_{p}$ , where $N_{p}$ is the number of pilot transmissions for PGA
algorithm. The learning rate is decreasing by 0.7 for LR and NN, when the loss
function decreases, and by 0.5 with other algorithms. The previous settings
are empirically chosen. The pilots used for channel measurements are generated
using ZC-based and the DFT-based training matrices. The performance of channel
state information estimation algorithms are compared using the normalized mean
squared error (NMSE) over all channel realizations, which is defined as the
average of the Frobenius norm of $\|H-\beta\widehat{H}\|^{2}/\|H\|^{2}$, where
$\beta$ is given by
$\beta=\|\widehat{H}^{\ast}H\|_{1}/\|\widehat{H}^{\ast}\widehat{H}\|_{1}$ [6].

Figure 4 shows the NMSE of the channel estimation algorithms with different
number of pilots, where the SNR is set to 0 dB, and the frequency to 0.3 THz.
Each algorithm is tested using two different pilot training algorithms ZC and
DFT based training. We remark that for all algorithms the error decreases when
we increase the number of pilot transmissions. The ZC-based training
outperform DFT-based training for all simulations. PGA and Frank Wolf
algorithms has almost the same performance with different number of pilot
transmissions number, compared to LR and NN, which can be explained by the
fact that those algorithms uses the knowledge about the low rank
characteristics of the channel matrices, in order to reduce the dimentionality
of the estimated channel information. Figure 5 shows that the PGA and Frank
Wolf algorithms give better results than LR and NN, for different values of
SNR. We can remark also that using DFT pilot generation is giving better
results with NN than ZC method which is not the case with other algorithms. NN
is using a basic network structure that can be improved in future work,
regarding the potential of NN-based algorithms to learn complex structure. In
THz transmission systems, it is challenging to achieve high SNR values, where
$\text{SNR}=0$ is considered as a good reachable value. Both PGA and Frank
Wolf algorithms give an NMSE value around -12 dB, which is a promising result.

Figure 4: Performance of several learning algorithms with different pilot
tranmissions number with $SNR=0$ and frequency $f=0.3\;THz$. Figure 5:
Performance of several learning algorithms with different $SNR$ values, with
240 pilot transmissions and frequency $f=0.3\;THz$.

##  V Conclusion

In this paper, we have modeled the channel estimation problem in THz-MIMO
systems, considering a deterministic channel propagation model from the
literature, and one-bit ADCs at the receiver. We have developed several
machine learning algorithms to estimate the CSI, and evaluated two different
pilot training techniques: DFT and ZC based training. Our results showed a
comparison between few ML algorithms that are using gradient descent as a
learning optimization. PGA algorithm has given the best performance at SNR=0
dB with NMSE of -12.8 dB. In a future work, we will compare the current
algorithms to more ML and DL algorithms and other approaches from the
literature, and consider studying the complexity and the hardware
requirements.

## Acknowledgment

This work was partially supported by the DFG Project Nr. JU2757/12-1,
”Meteracom: Metrology for parallel THz communication channels.”

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