# Spectrum Shaping For Multiple Link Discovery in 6G THz Systems

Farzam Hejazi,Department of Electrical and Computer Engineering University of
Central Florida
Orlando, USA<EMAIL_ADDRESS>
Katarina Vuckovic, Department of Electrical and Computer Engineering
University of Central Florida
Orlando, USA<EMAIL_ADDRESS>
Nazanin Rahnavard, Department of Electrical and Computer Engineering,
University of Central Florida
Orlando, USA<EMAIL_ADDRESS>
###### Abstract

This paper presents a novel antenna configuration to measure directions of
multiple signal sources at the receiver in a THz mobile network via a single
channel measurement. Directional communication is an intrinsic attribute of
THz wireless networks and the knowledge of direction should be harvested
continuously to maintain link quality. Direction discovery can potentially
impose an immense burden on the network that limits its communication capacity
exceedingly. To utterly mitigate direction discovery overhead, we propose a
novel technique called spectrum shaping capable of measuring direction, power,
and relative distance of propagation paths via a single measurement. We
demonstrate that the proposed technique is also able to measure the
transmitter antenna orientation. We evaluate the performance of the proposed
design in several scenarios and show that the introduced technique performs
similar to a large array of antennas while attaining a much simpler hardware
architecture. Results show that the spectrum shaping with only two antennas
placed 0.5 mm, 5 mm, and 1 cm apart performs direction of arrival estimation
similar to a much more complex uniform linear array equipped with 7, 60, and
120 antennas, respectively.

##  I Introduction

6G mobile networks promise to bring a new era of ultra high-speed
communications that surpasses previous generations by several orders of
magnitude in communication capacity [1]. One of the core technologies behind
such a spectacular revolution is massive Multi-Input-Multi-Output (MIMO)
communication at mmWave and THz bands [2]. Mobile networks that work at these
frequency bands are bound to employ highly directional beams [3]. As
directional communication has gained importance in the new generation of
communication systems, direction of arrival (DoA) estimation has obtained
gravity as an enabler of directional communication [4]. To clarify this
necessity, we should note that two devices that exploit directional antennas
cannot communicate unless they ascertain the direction of the other device.
Moreover, this knowledge of direction should be maintained during the
communication period otherwise the link will be disrupted [5].

The process of finding the best beam that maximizes the communication rate is
called _beam selection_ [6]. The problem of beam selection for mmWave and THz
communication has been under extensive research recently. The most common
approach proposes a brute-force search on all beams [7]. Unfortunately brute-
force search compels a huge overhead on the communication system since it
should be harvested swiftly and continuously. As beams become narrower at
higher frequencies, beam selection overhead escalates such that it drastically
restricts the communication rate. Various techniques have been proposed to
ease the burden of beam selection on communication systems. Ali et. al propose
using legacy sub-6 GHz channel information for sparse recovery of mmWave
channel and beam selection [8]. Myers et. al utilize an efficient set of
antenna weight vectors for fast beam alignment through a compressive sensing
(CS) approach [9]. Swift-link [10] is a technique that incorporates randomized
beam training along with a CS algorithm to develop beam selection technique
robust to carrier frequency offset. Although these techniques can ease the
overhead of beam selection, they still require numerous channel measurements.
Many recent studies consider machine learning techniques to address the beam
selection problem. Long et. al cast the beam selection problem as a multi-
class classification problem and employ support vector machine (SVM) to
achieve a statistical classification model that maximizes the sum rate [11].
Myers et. al propose an end-to-end deep learning technique to design a
structured CS matrix based on the underlying channel distribution, leveraging
both sparsity and the particular spatial structure of propagation paths that
appears in a communication channel [12]. Alrabeiah et al. train a deep network
to learn the mapping between sub-6 GHz channel state information (CSI) and
optimal mmWave beam [13]. Recently, several works consider vision, LIDAR,
Radar, and other means of situational awareness to integrate with
communication data for mmWave channel estimation and beam selection [14, 15,
16, 17]. A common limitation among available literature incorporating deep
learning for beam selection is the blindness to unseen data and lack of
adaptability to time-varying dynamic environments.

In the most recent breakthrough, Ghasempour et. al. introduce the idea of THz
rainbow for single-shot link discovery [18]. In this work, authors name the
process of finding the DoA of signal from the transmitter (TX) via only a
single measurement, single-shot link discovery. The authors employ leaky wave
antennas at both TX and receiver (RX). When excited by a broadband source
(i.e. a pico second pulse with a flat spectrum between [0.1THz,1THz]), the TX
antenna propagates a different and unique frequency within the spectrum at
each angle, thus forming a THz rainbow. The RX employs THz time-domain
spectroscopy (THz-TDS) to measure the spectrum of the received signal and
consequently estimates its DoA. THz-TDS is a technique to measure the THz
electric field that can be used to measure the received spectrum in the range
[0.1THz,10THz] and currently is implementable on chip [19]. Other than DoA,
the THz rainbow technique is able to measure the angle of departure (AoD) of
the signal from the TX, yet in a limited range. Although measuring DoA and AoD
in a single shot is a groundbreaking achievement, the THz rainbow technique
suffers from following inefficiencies:

  * •

very limited DoA and AoD observability range ($[10^{o},80^{o}]$ for DoA, and
$DoA+[-40^{o},20^{o}]$ for AoD),

  * •

incapablity in estimating multiple DoAs, thus inapplicable in harsh multipath
environments,

  * •

limited applicability to only dry indoor environments

  * •

spectral inefficiency as a result of using only a very tiny range of available
spectrum for DoA estimation.

In our work, we propose a new design to address the inefficiencies of THz
rainbow, which we refer to as _Spectrum Shaping (SSH)_. We adopt a well-known
array design with antenna spacing that typically makes use of the time
difference of arrival (TDoA) of signals between two antennas to estimate DoA
in LoS communication [20, 21]. However, since we reduce the antenna spacing to
less than a centimeter, measuring TDoA requires several Tera-samples per
seconds which is not accessible. On the other hand, instead of measuring TDoA,
by carefully adding delay lines in the design we try to shape the received
spectrum in such a way that enables us to estimate multiple concurrent DoAs
leveraging the spectrum of the spectrum of the received signal. Moreover, we
devise a novel antenna placement at the receiver and the TX to estimate AoD
and DoA using only a single measurement. To the best of our authors knowledge,
this is the first time that such an antenna spacing design is introduced for
AoD and DoA estimation in a single-shot for THz band applicable in the
presence of strong multi-path. In spectrum shaping, we incorporate the same
broadband signal source at the TX and TDS-THz at RX as the THz rainbow. The
main characteristics of our novel design can be encapsulated as

  * •

capability of measuring DoA and AoD in a wide range ($[0^{o},180^{o}]$),

  * •

capability of measuring multiple incoming signals with multiple DoAs in a
single shot,

  * •

performing similar to a large array while benefiting from a much simpler
architecture,

  * •

utilizing the whole available spectrum for DoA estimaton,

  * •

no requirement for synchronization between RX and TX,

  * •

applicability in humid indoor and outdoor environments, and

  * •

resiliency to attenuation due to atmospheric gases and presence of intense
water vapor.

The rest of the paper is organised as follows. In Section II, we present the
system model and our novel RX design. We demonstrate the capability of the
proposed design in measuring multiple DoAs in a single shot. We propose a new
TX design and show the capability of the system in measuring DoA and AoD in a
wide angular range in Section III. We derive the Cramer Rao Lower Band of
error for DoA and AoD estimation in Section IV. In Section V, we demonstrate
the performance of spectrum shaping by several simulations and show that it
performs similar to a large antenna array. Finally, we conclude the paper in
Section VI.

##  II System model

As illustrated in Fig. 1, we consider a link where RX is equipped with an
antenna pair. The gap between the two antennas is denoted by $D$. The
transmitter (TX) is excited by a broadband source generating a single pulse of
broadband emission whose spectral coverage is broad enough to cover the entire
relevant band (0.1 to 1 THz). The first ($RX_{1}$) and the second ($RX_{2}$)
antennas receive the signal emitted by the TX. After passing through a delay
line with length $D$, the received signal at $RX_{2}$ is superimposed with the
signal received at $RX_{1}$ and fed into a THz-TDS receiver.

Figure 1: Block diagram of the spetrum shaping for detecting incoming DoAs.

###  II-A Line of Sight (LoS) DoA Estimation

In this section, we assume there is only one LoS path between the TX and the
RX. We denote the DoA of the signal to the RX by $\theta_{i}$. Given the far
field assumption, the signal emitted by the TX is received at the two antennas
of the RX with a time shift $\delta t_{i}$ given by

$\delta t_{i}=-\frac{D}{c}\cos(\theta_{i})\,,$ (1)

where $c$ is the speed of light. The broadband RX then measures the spectrum
of the superimposition of the signals received at $RX_{1}$ and $RX_{2}$ (with
a delay equal $\frac{D}{c}$) using THz-TDS technique. Subsequently, the output
spectrum of the broadband receiver can be expressed as

$\displaystyle\boldsymbol{E}_{r}(f,\theta_{i})=|\mathcal{F}\\{r(t)+r(t-\frac{D}{c}-\delta
t_{i})\\}|^{2}=|R(f)+R(f)e^{-j2\pi f(\delta t_{i}+\frac{D}{c})}|^{2}$ (2)
$\displaystyle=|2R(f)|^{2}|cos(\pi f\frac{D}{c}(1-\cos(\theta_{i})))|^{2}$
$\displaystyle=|2R(f)|^{2}|cos(\pi
f\frac{2D}{c}\sin^{2}(\frac{\theta_{i}}{2}))|^{2}\,,$

where $\boldsymbol{E}_{r}(f,\theta_{i})$ is the spectrum of the
superimposition of the two received signals, $\mathcal{F\\{.\\}}$ is the
Fourier transform operator, $r(t)$ is the received signal at the $RX_{1}$, and
$R(f)$ is the Fourier transform of the $r(t)$. We assume only one LoS path
between the RX and the TX; hence, $r(t)=a(t)*s(t)$, where $a(t)$ models THz
channel (which is typically a frequency selective channel at the interested
band), $s(t)$ is the transmitted signal, and $*$ is the convolution operator.
Considering $s(t)$ has a flat spectrum over the relevant band, (2) can be
expressed as

$\boldsymbol{E}_{r}(f,\theta_{i})=C|a(f)|^{2}+C|a(f)|^{2}\cos(2\pi
f\frac{2D}{c}\sin^{2}(\frac{\theta_{i}}{2}))\,.$ (3)

where $C$ is a constant and $a(f)$ is the channel frequency response. Applying
Fourier transform over $\boldsymbol{E}_{r}(f,\theta_{i})$, defining
$\mathbf{a}(\zeta)=\mathcal{F}\\{|a(f)|^{2}\\}$ we have

$\mathcal{F}\left\\{\boldsymbol{E}_{r}(f,\theta_{i})\right\\}=C\left(\mathbf{a}(\zeta)+\frac{1}{2}\mathbf{a}\left(\zeta-\frac{2D}{c}\sin^{2}(\frac{\theta_{i}}{2})\right)+\frac{1}{2}\mathbf{a}\left(\zeta+\frac{2D}{c}\sin^{2}(\frac{\theta_{i}}{2})\right)\right)\,.$
(4)

$\mathbf{a}(\zeta)$ is the frequency response of the THz channel frequency
response. Fig. 2 and 3 depict $\mathbf{a}(\zeta)$ for a dry (zero water vapor
density) and a humid (water vapor density equals $10\frac{g}{m^{3}}$)
environments, respectively. Although, channels are frequency selective,
$\mathbf{a}(\zeta)$ shows a very strong and distinctive global pick at zero
and at other frequencies is roughly zero. In Section V, we will show by
simulation that THz channel can potentially result in a reduction of angular
resolution of the technique in humid environments which can be compensated by
increasing $D$. Since $\sin^{2}(\frac{\theta_{i}}{2})$ is positive,
$\sin^{2}(\frac{\theta_{i}}{2})$ can be simply estimated via finding the only
positive element of the spectrum of $\boldsymbol{E}_{r}(f,\theta_{i})$. Then
if $\theta_{i}\in[0,\pi]$, $\theta_{i}$ can be uniquely estimated. Moreover,
since we exclusively make use of the amplitude of the Fourier transform of the
received signal (and not its phase), we eliminate the requirement of
synchronization between the TX and the RX [18]. It should be mentioned that,
without the delay line after $RX_{2}$, the $\boldsymbol{E}_{r}(f,\theta_{i})$
turns out to be

$\boldsymbol{E}_{r}(f,\theta_{i})=C|a(f)|^{2}(1+\cos(2\pi
f\frac{D}{c}\cos(\theta_{i})))\,.$ (5)

In this case,
$\boldsymbol{E}_{r}(f,\theta_{i})=\boldsymbol{E}_{r}(f,-\theta_{i})=\boldsymbol{E}_{r}(f,\pi-\theta_{i})$.
Therefore, DoA is uniquely observable _only_ if
$\theta_{i}\in[0,\frac{\pi}{2}]$. Adding the delay line, we _double_ the DoA
observability range to $[0,{\pi}]$. Regarding (2), we observe that the
proposed design ends up in multiplication of $|\cos(\pi
f\frac{2D}{c}\sin^{2}(\frac{\theta_{i}}{2}))|^{2}$ to the received spectrum,
which provide us with enough information to estimate DoA in the whole range
from $0^{o}$ to $180^{o}$. Hence, we call the multiplied term _spectrum
shaper_ and the proposed technique _spectrum shaping_.

Figure 2: Specific attenuation due to atmospheric gases (above figure) and
frequency response of the frequency response of the channel in a dry
environment (typically an indoor environment). Although THz channel is
naturally a frequency selective channel, the rate of change of the channel
based on frequency is very slow, thus the frequency response of the frequency
response of it contains a sole distinctive peak at zero. Figure 3:  Specific
attenuation due to atmospheric gases (above figure) and frequency response of
the frequency response of the channel in a humid environment (path length is
1000m). The rate of change of the channel versus frequency is more than a dry
environment, however, it is relatively slow such that frequency response of
it’s frequency response contains a sole distinctive peak at zero.

###  II-B Multiple Non-LoS (NLoS) DoA Estimation

In this section, we assume that there are multiple paths between the TX and RX
(or similarly there are multiple TXs in the environment). We intend to detect
powers and angles of all incoming paths to the RX. Suppose there are $m$ paths
between the TX and the RX and $a_{k}(t),\theta_{k},T_{k};k\in\\{1,\dots,m\\}$
denote the THz channel through path $k$, DoA of path $k$, and the time of
flight (ToF) of the signal through path $k$, respectively (Fig. 1). The
received signal at the RX antenna pair can be expressed as

Figure 4:  spectrum shaping can be employed for detecting multiple incoming
DoAs. There are 3 paths between the TX and The RX.
$\theta_{1},\theta_{2},\theta_{3}$ and $a_{1},a_{2},a_{3}$ are DoAs the and
path loss of the three paths, respectively.

$\displaystyle r_{1}(t)=\sum_{k=1}^{m}a_{k}(t)*s(t-T_{k})$ (6) $\displaystyle
r_{2}(t)=$
$\displaystyle\sum_{k=1}^{m}a_{k}(t)*s(t-T_{k}+\frac{D}{c}\cos(\theta_{k}))\,,$
(7)

where $r_{1}(t)$ and $r_{2}(t)$ are the received signals at $RX_{1}$ and
$RX_{2}$, respectively. Passing $r_{2}(t)$ through the delay line in Fig. 1,
the spectrum of the superimposition of the two signal turns out to be

$\displaystyle\boldsymbol{E}_{r}(f,\theta_{1},\dots,\theta_{m})=|\mathcal{F}\\{r_{1}(t)+r_{2}(t-\frac{D}{c})\\}|^{2}$
$\displaystyle=\left|\sum_{k=1}^{m}a_{k}(f)S(f)e^{-j2\pi
fT_{k}}+\sum_{k=1}^{m}a_{k}(f)S(f)e^{-j2\pi fT_{k}}e^{-j2\pi
f\frac{D}{c}(1-\cos(\theta_{k}))}\right|^{2}$
$\displaystyle=|2S(f)|^{2}\left|\sum_{k=1}^{m}a_{k}(f)e^{-j2\pi
f(T_{K}+\frac{D}{c}\sin^{2}(\frac{\theta_{k}}{2}))}\cos{(\pi
f\frac{2D}{c}\sin^{2}(\frac{\theta_{k}}{2}))}\right|^{2}$
$\displaystyle=|2S(f)|^{2}(\sum_{k=1}^{m}a^{2}_{k}(f)\cos^{2}(\pi
f\frac{2D}{c}\sin^{2}(\frac{\theta_{k}}{2}))+$
$\displaystyle\sum_{k=1}^{m}\sum_{p=k+1}^{m}2a_{k}(f)a_{p}(f)\cos{(\pi
f\frac{2D}{c}\sin^{2}(\frac{\theta_{k}}{2}))}\cos{(\pi
f\frac{2D}{c}\sin^{2}(\frac{\theta_{p}}{2}))}\cos{(2\pi
f(T^{\prime}_{k}-T^{\prime}_{p}))})\,.$ (8)

where
$T^{\prime}_{k}=T_{k}+\frac{D}{c}\sin^{2}(\frac{\theta_{k}}{2}),k\in{1,\dots,m}$.
Given $s(t)$ has a flat frequency response over the relevant band,
$\boldsymbol{E}_{r}(f,\theta_{1},\dots,\theta_{m})$ is proportional to

$\displaystyle\boldsymbol{E}_{r}(f,\theta_{1},\dots,\theta_{m})\propto$
$\displaystyle(\sum_{k=1}^{m}\frac{a^{2}_{k}(f)}{2}+\sum_{k=1}^{m}\frac{a^{2}_{k}(f)}{2}\cos(2\pi
f\frac{2D}{c}\sin^{2}(\frac{\theta_{k}}{2}))+$
$\displaystyle\sum_{k=1}^{m}\sum_{p=k+1}^{m}a_{k}(f)a_{p}(f)\cos{(\pi
f\frac{2D}{c}\sin^{2}(\frac{\theta_{k}}{2}))}\cos{(\pi
f\frac{2D}{c}\sin^{2}(\frac{\theta_{p}}{2}))}\cos(2\pi
f(T^{\prime}_{k}-T^{\prime}_{p})))\,.$ (9)

Considering (9), $\boldsymbol{E}_{r}(f,\theta_{1},\dots,\theta_{m})$ is a
summation of multiple cosine functions modulated with different
frequencies111The third term in (9) consists of multiplications of three
cosine terms. From basic trigonometry we know that any arbitrary multiples of
cosines equals sum of cosines. , hence, the spectrum of
$\boldsymbol{E}_{r}(f,\theta_{1},\dots,\theta_{m})$ exhibit multiple spikes
corresponding to each cosine function.

$\boldsymbol{Lemma\>1.}$ Assuming
$|T^{\prime}_{k}-T^{\prime}_{p}|\gg\frac{2D}{c};p\neq
k;p,k\in\\{1,\dots,m\\},$ the third term in (9) can be filtered out by
applying a low-pass filter with cut-off frequency $\frac{2D}{c}$ on
$\boldsymbol{E}_{r}(f,\theta_{1},\dots,\theta_{m})$.

Proof. The frequency of all the elements in the first and the second terms of
(9) is less than $\frac{2D}{c}$, while all the elements in the third term are
modulated by $\cos(2\pi f(T^{\prime}_{k}-T^{\prime}_{p}))$ harmonics. Thus, if
the condition in Lemma 1 holds, the third term can be removed using a low-pass
filter with cut-off frequency $\frac{2D}{c}$. $\square$

The condition in Lemma 1 expresses that the third term of (9) is removable by
filtering if the difference between ToF of all paths to the RX are far larger
than $\frac{2D}{c}$; or equivalently the difference between paths’ length of
all paths to the RX are far larger than $2D$. In Section V, we will discuss
that $D$ is typically in the sub centimeter range (i.e. 1 mm to 1 cm). Thus
the condition of Lemma 1 will hold in most practical use cases. After passing
$\boldsymbol{E}_{r}(f,\theta_{1},\dots,\theta_{m})$ through a low-pass filter
with cut-off frequency $\frac{2D}{c}$, referring to (3), the output
$\boldsymbol{\hat{E}}_{r}(f,\theta_{1},\dots,\theta_{m})$ can be expressed as

$\displaystyle\boldsymbol{\hat{E}}_{r}(f,\theta_{1},\dots,\theta_{m})=C\left(\sum_{k=1}^{m}{a^{2}_{k}(f)}+\sum_{k=1}^{m}{a^{2}_{k}(f)}\cos(2\pi
f\frac{2D}{c}\sin^{2}(\frac{\theta_{k}}{2}))\right)=\sum_{k=1}^{m}\boldsymbol{E}^{(k)}_{r}(f,\theta_{k})\,,$
(10)

where $\boldsymbol{E}^{(k)}_{r}(f,\theta_{k})$ is the spectrum of the received
signal if only one LoS path arrived at the antenna pair from path $k$.
According to (10), the output spectrum of THz-TDS after lowpass filtering will
result in a linear weighted summation of
$\boldsymbol{E}^{(k)}_{r}(f,\theta_{k})$’s. In other words, the proposed
design is a linear system facing multiple incoming signals. Since
$\boldsymbol{\hat{E}}_{r}(f,\theta_{1},\dots,\theta_{m})$ composes of linear
summation of harmonic signals, by using harmonic decomposition techniques such
as Fourier transform, MUSIC [22], or Pisarenco harmonic decomposition [23] we
can separate and detect different harmonics of
$\boldsymbol{\hat{E}}_{r}(f,\theta_{1},\dots,\theta_{m})$ and consequently we
can determine the angles and the powers of the incoming paths.

##  III AoD Estimation

THz rainbow [18] introduces a technique to measure AoD of signal from the TX.
THz rainbow takes advantage of the derivative of the maximum and the minimum
received frequency with respect to the RX rotation to estimate AoD. Here, we
introduce a scheme that can be used for AoD estimation in a single shot
without any reliance of RX rotation leveraging a TX antenna configuration
similar to the RX as illustrated in Fig. 5.

Figure 5: Block diagram of spectrum shaping for estimating AoD and DoA
together. The TX antenna configuration is almost identical to the RX unless
the delay line before $TX_{2}$ is 3 times longer than the delay line after
$RX_{2}$.

Assuming there is only LoS path between the TX and the RX, we propose using
the same antenna placement of the RX at the TX. There is a delay line between
the source and $TX_{2}$ with length $3D$. Thus, $TX_{2}$ propagates the pulse
generated by the source with a delay equal to $\frac{3D}{c}$ with respect to
the $TX_{1}$ emission. The two pulses will be received at the $RX_{1}$ and
$RX_{2}$ with a delay due to the AoD and the delay between the transmission of
the two pulses equals to

$\delta_{d}=-\frac{D}{c}\cos(\theta_{d})+\frac{3D}{c}\,,$ (11)

where $\delta_{d}$ denotes the difference time of arrival of the two
transmitted pulses at $RX_{1}$, and $\theta_{d}$ denotes AoD. Denoting the
emitted pulse from $TX_{1}$ by $s(t)$, the received signal at $RX_{1}$ turns
out to be

$\displaystyle
r(t)=a(t)*\left(s(t)+s(t-\frac{D}{c}(3-\cos(\theta_{d}))\right)$ (12)
$\displaystyle\Rightarrow R(f)=\mathcal{F}\\{r(t)\\}=a(f)S(f)(1+e^{-j2\pi
f\frac{D}{c}(3-\cos(\theta_{d})})=a(f)S(f)e^{-j\pi
f\frac{D}{c}(3-\cos(\theta_{d}))}\cos(\pi
f\frac{D}{c}(3-\cos(\theta_{d})))\,.$

Using (2), $\boldsymbol{E}_{r}(f,\theta_{i},\theta_{d})$ can be expressed as

$\displaystyle\boldsymbol{E}_{r}(f,\theta_{i},\theta_{d})=|4a(f)S(f)|^{2}\left|\cos(\pi
f\frac{D}{c}(1-\cos(\theta_{i})))\cos(\pi
f\frac{D}{c}(3-\cos(\theta_{d})))\right|^{2}$ (13)
$\displaystyle=|2a(f)S(f)|^{2}\left(\left(1+\cos(2\pi
f\frac{D}{c}(1-\cos(\theta_{i})))\right)\left(1+\cos(2\pi
f\frac{D}{c}(3-\cos(\theta_{d})))\right)\right)$
$\displaystyle=|2a(f)S(f)|^{2}(1+\cos(2\pi
f\frac{D}{c}(1-\cos(\theta_{i})))+\cos(2\pi f\frac{D}{c}(3-\cos(\theta_{d})))$
$\displaystyle+\frac{1}{2}\cos(2\pi
f\frac{D}{c}(4-\cos(\theta_{i})-\cos(\theta_{d})))+\frac{1}{2}\cos(2\pi
f\frac{D}{c}(2-\cos(\theta_{d})+\cos(\theta_{i}))))\,.$

Based on (13), $\boldsymbol{E}_{r}(f,\theta_{i},\theta_{d})$ is a summation of
four harmonics with frequencies equal to $\frac{D}{c}(1-\cos(\theta_{i}))$,
$\frac{D}{c}(3-\cos(\theta_{d}))$,
$\frac{D}{c}(4-\cos(\theta_{i})-\cos(\theta_{d}))$, and
$\frac{D}{c}(2-\cos(\theta_{d})+\cos(\theta_{i}))$.

$\boldsymbol{Lemma\>2.}$ For any AOA $\theta_{i}$ and AOD $\theta_{d}$ we have

$\displaystyle\frac{D}{c}(4-\cos(\theta_{i})-\cos(\theta_{d}))\geq\frac{D}{c}(3-\cos(\theta_{d}))\geq\frac{D}{c}(1-\cos(\theta_{i})),\frac{D}{c}(2-\cos(\theta_{d})+\cos(\theta_{i}))\geq
0\,.$ (14)

Proof. Assume $\alpha=\frac{D}{c}(3-\cos(\theta_{d}))$ and
$\beta=\frac{D}{c}(1-\cos(\theta_{i}))$,
$\forall{\theta_{i}},\forall{\theta_{j}}$,
$\frac{4D}{c}\geq\alpha\geq\frac{2D}{c}$ and $\frac{2D}{c}\geq\beta\geq 0$.
Thus we have

$\alpha\geq\beta\geq 0.$

Therefore

$\alpha+\beta\geq\alpha\geq\beta,\alpha-\beta\geq 0.$

Hence, Lemma 2 holds.$\square$

Applying Fourier transform on $\boldsymbol{E}_{r}(f,\theta_{i},\theta_{d})$
similar to (5), the four frequencies of the four harmonics of the
$\boldsymbol{E}_{r}(f,\theta_{i},\theta_{d})$ are measurable. Referring to
Lemma 2, the two largest frequencies of
$\boldsymbol{E}_{r}(f,\theta_{i},\theta_{d})$ are
$\frac{D}{c}(4-\cos(\theta_{i})-\cos(\theta_{d}))$ and
$\frac{D}{c}(3-\cos(\theta_{d}))$, respectively. Using
$\frac{D}{c}(3-\cos(\theta_{d}))$, $\theta_{d}$ can be uniquely estimated if
$\theta_{d}\in[0,\pi]$. Next, using
$\frac{D}{c}(4-\cos(\theta_{i})-\cos(\theta_{d}))$, $\theta_{i}$ can be
uniquely estimated if $\theta_{i}\in[0,\pi]$. Referring to [18], using THz
rainbow technique $\theta_{d}$ is measurable in a very limited range
($[\theta_{i}-40^{o},\theta_{i}+20^{o}]$). However, leveraging the SSH
introduced in this section $\theta_{d}$ is measurable in the whole range of
$0^{o}$ to $180^{o}$. Referring to (13), here we applied SSH both at the RX
(the $\cos(\pi f\frac{D}{c}(1-\cos(\theta_{i})))$ term) and the TX (the
$\cos(\pi f\frac{D}{c}(3-\cos(\theta_{d})))$ term) to have enough information
to estimate both AoD and DoA. In this work, we have mainly made use of two
antennas and one delay line to from spectrum shapers. Nonetheless, future work
can go further by introducing more complex antenna and delay line schemes for
more advanced applications and superior performances.

##  IV Cramer Rao Lower Bound of Error

In this section, we analyze the effect of noise on the performance of the
system. Specifically, we derive Cramer Rao lower bound (CRB) of error for DoA
estimation for LoS scenario (Section II-A) and CRB for AoD and DoA estimation
(Section III).

###  IV-A CRB of DoA Estimation

Referring to Section II-A, we assume received signals at $RX_{1}$ and $RX_{2}$
are added by a white Gaussian noise with noise power $\frac{N_{0}}{2}$. We
have

$\displaystyle r_{1}(t)=a(t)*s(t)+n_{1}(t)$ $\displaystyle
r_{2}(t)=a(t)*s(t-\delta t_{i})+n_{2}(t)$ (15)

where $n_{1}(t)$ and $n_{2}(t)$ are independent white Gaussian noises with
power $\frac{N_{0}}{2}$. The received signals at the THz-TDS receiver equals
to

$\displaystyle r_{1}(t)+r_{2}(t-\frac{D}{c})=a(t)*(s(t)+s(t-\frac{D}{c}-\delta
t_{i}))+n_{1}(t)+n_{2}(t)\,.$ (16)

We define $n(t)=n_{1}(t)+n_{2}(t)$ which is a white Gaussian noise with power
$N_{0}$. The observation (denoted by $z(f)$) is the spectrum of the received
signal equal to

$\displaystyle z(f)=|a(f)s(f)\left(1+e^{j2\pi
f\frac{D}{c}(1-\cos(\theta_{i}))}\right)+n(f)|\,.$ (17)

$n(f)$ is a white Gaussian noise process with power $N_{0}$ and
$\boldsymbol{x}=(\theta_{i},a(f))$ denotes the vector of unknowns. Referring
to [24], the likelihood function of the observation is a Rice distribution
with the following parameters

$\displaystyle
p(z(f);\boldsymbol{x})=\frac{z(f)}{\frac{N_{0}}{2}}e^{-\left(\frac{z(f)^{2}+4a(f)^{2}s(f)^{2}\cos^{2}(\pi
f\frac{2D}{c}\sin^{2}(\frac{\theta_{i}}{2}))}{N_{0}}\right)}I_{0}(\frac{2z(f)|a(f)s(f)\cos(\pi
f\frac{2D}{c}\sin^{2}(\frac{\theta_{i}}{2}))|}{\frac{N_{0}}{2}})$ (18)

where $I_{0}()$ is the modified Bessel function of the first kind with order
zero, and $p(z(f);\boldsymbol{x})$ is the probability distribution function of
$z(f)$ parametrized by the $\boldsymbol{x}$. The covariance matrix of any
unbiased estimator $\hat{\boldsymbol{x}}$ is bounded by [25]

$\displaystyle
E\\{(\hat{\boldsymbol{x}}-{\boldsymbol{x}})(\hat{\boldsymbol{x}}-{\boldsymbol{x}})^{H}\\}\geq\boldsymbol{J}^{-1}\,,$
(19)

where

$\displaystyle\boldsymbol{J}({\boldsymbol{x}})=\sum_{k=1}^{M_{f}}E\left\\{\left(\frac{\partial\ln{p(z(f_{k});\boldsymbol{x})}}{\partial\boldsymbol{x}}\right)\left(\frac{\partial\ln{p(z(f_{k});\boldsymbol{x})}}{\partial\boldsymbol{x}}\right)^{T}\right\\}\,.$
(20)

Here $M_{f}$ denotes the number of frequency samples of $z(f)$ in the relevant
band, and $f_{1},\dots,f_{M_{f}}$ are the frequencies at which $z(f)$ is
sampled by the THz-TDS technique. The analytical closed-form derivation of
$\boldsymbol{J}$ and consequently CRB for $\theta_{i}$ estimation is not
feasible because of the Bessel function. In Section V we will resort to a
numerical approximation of CRB for the sake of noise analysis of SSH and
comparison with the state-of-the-art.

###  IV-B CRB of AoD Estimation

Based on the discussion in Section III, referring to (12), the observation at
the RX would be in the form of

$\displaystyle z(f)=|a(f)s(f)\left((1+e^{j2\pi
f\frac{D}{c}(1-\cos(\theta_{i}))})(1+e^{j2\pi
f\frac{D}{c}(3-\cos(\theta_{d})})\right)+n(f)|\,.$ (21)

Defining $\boldsymbol{x}=(\theta_{i},\theta_{d},a(f))$ as the vector of
unknown parameters, and given $n(f)$ is white Gaussian noise, the likelihood
function turns out to be

$\displaystyle
p(z(f);\boldsymbol{x})=\frac{z(f)}{\frac{N_{0}}{2}}e^{-\left(\frac{z(f)^{2}+16a(f)^{2}s(f)^{2}\cos^{2}(\pi
f\frac{2D}{c}\sin^{2}(\frac{\theta_{i}}{2}))\cos^{2}(\pi
f\frac{2D}{c}(2+\sin^{2}(\frac{\theta_{d}}{2})))}{N_{0}}\right)}\times$
$\displaystyle I_{0}\left(\frac{4z(f)|a(f)s(f)\cos(\pi
f\frac{2D}{c}\sin^{2}(\frac{\theta_{i}}{2}))\cos(\pi
f\frac{2D}{c}(2+\sin^{2}(\frac{\theta_{d}}{2})))|}{\frac{N_{0}}{2}}\right)\,,$
(22)

where $\times$ is multiplication. Then using (20), $\boldsymbol{J}$ can be
calculated and CRB for $\theta_{i}$ and $\theta_{d}$ can be derived using
numerical techniques.

##  V Simulation Results

###  V-A Noise Analysis

In this section, the performance of our proposed link discovery technique is
studied in the presence of noise. Further, the performance of the proposed
technique is compared with the performance of the state-of-the-art lens array
and ULA using digital beamforming. The performance of ULA using digital
beamforming has been extensively studied in the literature [26, 27].
Implementing digital beamforming for ULA is exorbitant and massively complex
especially in mmwave bands [6]. On the other hand, Lens array (LA) is a more
recent technique utilizes a lens as a passive phase shifter to pursue
beamforming without the heavy network of phase shifters required in ULA arrays
[28]. Throughout this section, signal to noise ratio (SNR) is referred to the
average power of signal to noise variance ($\frac{N_{0}}{2}$) at each antenna
of the array (whether ULA, LA, or SSH).

In the first simulation, we compare CRB of DoA estimation for a symmetric ULA
with half wavelength antenna spacing equipped with $N$ elements [29], an arc
LA with aperture and focal length equals to $L$ equipped with $M$ elements
[30]. Further we set $L=M\frac{\lambda}{2}$, where $\lambda$ is the wavelength
of the operating frequency of the LA. To emulate SSH, we assume the spectral
resolution of the THz-TDS system is 1.5 GHz, and the pulse relevant band is
[100GHz,1THz]. Thus the number of spectrum samples are 600. We set $D=5$ mm.
For the channel frequency response $a(f)$ we assume the range is 10 0m and the
air is dry222We simulate signal attenuation due to atmospheric gases using
MATLAB${}^{\textsuperscript{\textregistered}}$ gaspl function. (unless
otherwise mentioned). As Fig. 6 illustrates CRB of SSH is close to a ULA with
111 elements and LA with 201 elements when SNR is 20 dB, which is a
spectacular performance considering much simpler hardware architecture of SSH
compared to the two other counterparts. Regarding Fig. 6 (a) to (d), when SNR
decreases the performance of SSH deteriorates sharper than ULA and LA in the
CRB sense. Specifically, looking at Fig. 6 (d), when SNR = -10 dB, the CRB of
SSH is in proximity of a ULA with 7 elements and a LA with 15 elements. This
performance decline is due to the fact that large arrays can integrate the
received signal energy over the whole antenna elements, while SSH only
utilizes 2 antennas, thus does not benefit from the same advantage. Speaking
of hardware complexity, even in low SNR levels, SSH still has superior
performance compared to ULA and LA. As shown, SSH shows identical performance
to a far larger ULA and LA.

(a) SNR = 20dB

(b) SNR = 10dB

(c) SNR = 0dB

(d) SNR = -10dB

Figure 6: CRB of DoA estimation is compared to ULA, Lens array and spectrum
shaping techniques. Here $\lambda$ denotes wavelength.

In the next simulation we investigate the effect of harsh frequency selective
channel on the performance of the CRB of the SSH technique. The THz channel
can be extremely frequency selective because of attenuation of atmospheric
gases specially water vapor and this effect exacerbates as range increases
[31]. Here we assume that the weather is exceedingly humid and water vapor
density equals $10\frac{g}{m^{3}}$. We set SNR = 5 dB and D = 5 mm. Fig. 7
shows CRB of SSH for ranges equal to 10 m, 100 m, and 1000 m. As expected, as
range increases the SSH performance deteriorates, since a large range of
frequencies in the relevant bandwidth experiences severe attenuation.
Nevertheless, SSH still can achieve DoA estimation accuracy of better than
$2^{o}$ for DoA $\in[30^{o},145^{o}]$ for range equal to 1 km and in the
presence of intense water vapor, which quite acceptable for majority of
directional communication applications.

Figure 7: CRB of spectrum shaping in the presence of intensive water vapor in
the air for different ranges. Here $D=5mm$, SNR = 5 dB.

In the next simulation we study the effect of $D$, the antenna gap, on CRB of
the SSH technique. In this simulation we assume SNR = 0 dB. We simulate CRB
for D equals to 1 mm, 5 mm, and 10 mm. As Fig. 8 shows, SSH performs more
accurately as $D$ increases. Referring to (5), $D$ directly multiplies to
$f\cos(\theta_{i})$, thus increasing $D$ is equivalent of increasing the
relevant band. Therefore $D$ is one the major parameter can improve SSH
performance.

Figure 8: CRB of spectrum shaping for different values of antenna spacing $D$.
Here SNR = 0dB.

The next simulation studies the CRB of AoD and DoA estimation of Section III.
Here we assume SNR = 5 dB, and $D=5$ mm. Fig. 9 demonstrates that AoD and DoA
have almost identical CRBs. Further, AoD has negligible effect on CRB of DoA
(which means different values of AoD results in the same DoA CRB) and vise
versa.

(a) AoD

(b) DoA

Figure 9: CRB of AoD and DoA estimation considering antenna configuration of
Fig. 5. Here $D=5mm$ and SNR = 5 dB.

In the next simulation the mean squared error (MSE) of minimum mean square
estimator (MMSE) for ULA, LA and SSH is simulated and compared. Assuming array
response in a noisy environment is denoted by $A(\theta)$ and the expected
response in an ideal noise-free environment is denoted by $A_{E}(\theta)$,
NR: rewrite this sentencefor SSH, $A_{E}(\theta)$ equals
$\boldsymbol{E}_{r}(f,\theta_{i})$, the MMSE estimator can be defined as

$\hat{\theta}=\min_{0^{o}\leq\theta\leq
180^{o}}|A(\theta)-A_{E}(\theta)|^{2}\,,$

where $\theta$ is DoA and $\hat{\theta}$ is the estimated DoA. Thus the MSE
can be defined as the average of squared error
($|\hat{\theta}-\theta_{0}|^{2}$ , where $\theta_{0}$ is the actual DoA). In
this simulation we compare the performance of a ULA with N = 60, an arc LA
with M = 80 and SSH with $D=5mm$. To calculate MSE we repeat the experiment
for 1000 times and average over all experiments’ results. As Fig. 10
illustrates ULA shows lower root MSE (RMSE) for negative SNRs in comparison
with SSH and LA. SSH performs slightly better than ULA and LA for SNR larger
than 3 dB. These results is in harmony with the outcome of CRB analysis that
SSH performance improves significantly as SNR increases.

Figure 10: RMSE of ULA, LA and spectrum shaping versus SNR. Spectrum shaping
performance surpasses large arrays in positive SNR, while spectrum shaping
attains a much simpler hardware structure in comparison to arrays.

Next, we study RMSE of SSH for different antenna spacing in the presence of
intense water vapor. We set SNR = 5 dB, DoA = $60^{o}$, and water vapor
density to $10\frac{g}{m^{3}}$. As Fig. 11 demonstrates SSH with wider antenna
spacing performs significantly better in the presence of intense water vapor.
Nonetheless, as range increases the performance of SSH regardless of antenna
spacing deterioration due to the harsh frequency selective attenuation of the
channel. Additionally, even in the presence of expensively harsh frequency
selective channel, SSH with 5 mm antenna spacing can achieve better than
$3^{o}$ accuracy at the range of 1800 m.

Figure 11: RMSE of DoA estimation in the presence of intense water vapor
($10\frac{g}{m^{3}}$)

versus range assuming SNR stays intact (5 dB). DoA equals $60^{o}$.

In the next experiment, we examine the performance of SSH in simultaneous AoD
and DoA estimation. We set D = 5 mm, and SNR = 5 dB. As Fig. 12 illustrates
and as expected, the RMSE of both AoD and DoA are mostly identical and improve
remarkably as SNR increases. This identical response is due to the almost
identical configuration at the TX and the RX and is in agreement with the
result of CRB analysis.

Figure 12: RMSE of DoA and AoD estimation using MMSE estimator. DoA equals
$60^{o}$ and SNR = 5 dB.

###  V-B Channel State Information Estimation Using Spectrum Shaping

Here we consider an indoor scenario where there are two paths between the TX
and the RX, one LoS and one NLoS. We set the DoA of the LoS path $60^{o}$ and
the DoA of NLoS path $100^{o}$. We assume the NLoS path is $0.5$ m longer than
the LoS path and the power of NLoS path is 6 dB less than that of the LoS
path. We suppose the TX is transmitting a quasi sinc pulse with flat bandwidth
between [0.1THz,1THz]. We set $D=5mm$. We suppose To find the DoAs of incoming
signals we simply apply the matched filter of the expected harmonic (derived
in (3)) on the output of THz-TDS
($\boldsymbol{E}_{r}(f,\theta_{1},\theta_{2})$) to form
$\boldsymbol{E}(\theta)$ given by

$\boldsymbol{E}(\theta)=\int_{0.1THz}^{1THz}\cos(2\pi
f\frac{2D}{c}\sin^{2}(\frac{\theta}{2}))\boldsymbol{E}_{r}(f,\theta_{1},\theta_{2})df\,.$
(23)

Applying the matched filter on $\boldsymbol{E}_{r}(f,\theta_{1},\theta_{2})$,
it only passes $\frac{2D}{c}\sin^{2}(\frac{\theta_{i}}{2})$ Fourier components
and filters out the third term in (8). Measuring $\boldsymbol{E}(\theta)$ for
the whole range of possible DoAs ($[0^{o},180^{o}]$), we can estimate DoAs and
their powers by finding local maximum of $\boldsymbol{E}(\theta)$.

As Fig. 13 illustrates, $\boldsymbol{E}(\theta)$ shows two distinct picks at
$60^{o}$ and $100^{o}$ with $6$ dB difference between their amplitudes. Thus,
$\boldsymbol{E}(\theta)$ is sufficient to estimate the DoAs and powers of the
incoming paths.

Figure 13: The output of applying the matched filter of expected received
harmonic response on $\boldsymbol{E}_{r}(f,\theta_{1},\theta_{2})$. The
$60^{o}$ and $100^{o}$ show the DoAs of two incoming signals to the antenna
pair, from a LoS path and a NLoS path, respectively.

Next, to find the the relative ToF and relative distance between the two paths
we can employ the spectrum of $\boldsymbol{E}_{r}(f,\theta_{1},\theta_{2})$.
Fig. 14 depicts $\boldsymbol{E}_{r}(f,\theta_{1},\theta_{2})$ and the absolute
value of the Fourier transform of it. Referring to (8), the third term of
$\boldsymbol{E}_{r}(f,\theta_{1},\theta_{2})$ shows a harmonic with
$T^{\prime}_{1}-T^{\prime}_{2}$ frequency. Considering
$\frac{2D}{c}\ll(T_{1}-T_{2})$, we have $T^{\prime}_{1}-T^{\prime}_{2}\approx
T_{1}-T_{2}$. Applying Fourier transform over
$\boldsymbol{E}_{r}(f,\theta_{1},\theta_{2})$ and adjusting the frequency by
multiplying it by $c$, we can observe a pick at $c(T_{1}-T_{2})$ which equals
to the relative distance between the two paths.

Figure 14: The spectrum and the spectrum of spectrum of the received signal at
THz-TDS receiver versus frequency (top) and frequency multiplied by light
speed (bottom), respectively assuming two paths are received at the receiver
with 0.5m difference in path length.

As Fig. 14 illustrates, a distinct pick at $0.5$ m is observable at the
spectrum of $\boldsymbol{E}_{r}(f,\theta_{1},\theta_{2})$, which is equal to
the relative distance between the two paths. Thus using the proposed
technique, we could estimate DoAs, powers, and relative distance between the
two paths. By obtaining this information, and exploiting the Cost 2100 MIMO
channel model [32], we can estimate the channel state information (CSI)
between the RX and the TX. Even in sub-6 GHz regime, estimating CSI in a
single shot has not been possible and requires exchanging multiple pilot
signals between the two side of the link [33]. Therefore, not only SSH can be
used to enable DoA estimation and beam selection for THz massive MIMO systems,
but it also can assist the system by providing CSI information in a single
shot.

###  V-C Discussion

As analytical developments and simulation results have shown, SSH is capable
of providing CSI between a BS and the user. Referring to [8], finding CSI (or
the knowledge of the dominant propagation paths) is sufficient to find the
best beam pair at TX and RX for the sake of maximizing communication rate.
Thus, SSH can potentially eliminate the need for exhaustive beam search for
finding the optimal beam pair. Accordingly, SSH utterly quashes the burden of
beam selection on the wireless communication and improves the system
throughput radically. Further, SSH takes advantage of a very simple structure
that is accessible via current fabrication technology, while fully digital or
hybrid ULA and LA counterparts require complex and costly structures to
conduct the same task. All in all, single-shot techniques and specifically SSH
can revolutionize beam management at THz band and help us realize the full
potential of ultra-fast communications feasible using THz wireless systems.

##  VI Conclusion

In this paper we have introduced a novel direction of arrival (DoA) estimation
technique for THz wireless systems. We have shown that the proposed technique
is capable of measuring line of sight (LoS) and multiple non-LoS DoAs
simultaneously in a single-shot measurement. We have introduced a modification
to the design that provides it with the capability of measuring the angle of
departure (AoD) of the signal from the TX. We have shown by simulation that
the proposed technique performs similar to a large ULA and LA in terms of
angular resolution in the presence of noise. Further, we have demonstrated
that we can increase the angular resolution of the technique by widening the
gap between its two antennas without any additional hardware adjustment.

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