# Rainbow-link: Beam-Alignment-Free and Grant-Free mmW Multiple Access using
True-Time-Delay Array

Ruifu Li, , Han Yan, , and Danijela Cabric, Ruifu Li, Han Yan, and Danijela
Cabric are with the Electrical and Computer Engineering Department, University
of California, Los Angeles, Los Angeles, CA 90095 (e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>danijela@ee.ucla.edu).This work is supported by NSF under
grant 1718742 and 1955672. This work was also supported in part by the
ComSenTer and CONIX Research Centers, two of six centers in JUMP, a
Semiconductor Research Corporation (SRC) program sponsored by DARPA.

###### Abstract

The millimeter-wave (mmW) communications is a key enabling technology in 5G to
provide ultra-high throughput. Current mmW technologies rely on analog phased
arrays to realize beamforming gain and overcome high path loss. However, due
to a limited number of simultaneous beams that can be created with
analog/hybrid phased antenna arrays, the overheads of beam training and beam
scheduling become a bottleneck for emerging networks that need to support a
large number of users and low latency applications. This paper introduces
rainbow-link, a novel multiple access protocol, that can achieve low latency
and massive connectivity by exploiting wide bandwidth at mmW frequencies and
novel analog true-time-delay array architecture with frequency dependent
beamforming capability. In the proposed design, the network infrastructure is
equipped with the true-time-delay array to simultaneously steer different
frequency resource blocks towards distinct directions covering the entire cell
sector. Users or devices, equipped with a narrowband receiver and either a
single antenna or small phased antenna array, connect to the network based on
their angular positions by selecting frequency resources within their rainbow
beam allocation. Rainbow-link is combined with a contention-based grant-free
access to eliminate the explicit beam training and user scheduling. The
proposed design and analysis show that rainbow-link grant-free access is a
potential candidate for latency-critical use cases within massive
connectivity. Our results show that, given less than $10^{-5}$ probability of
packet loss, a rainbow-link cell, over 1 GHz bandwidth using 64 element
antenna array, attains sub-millisecond user-plane latency and Mbps user rates
with an approximate $400\text{\,}\mathrm{m}$ line-of-sight coverage and a
density of up to 5 active single antenna users per second per m2.

Index Terms: frequency division multiple access, millimeter wave networks, low
latency, critical massive machine type communications, true-time-delay array.

##  I Introduction

ONE objective of the 5G evolution is to further diversify its performance and
support applications with improved data rates, latency, and number of
connected devices. For example, in the emerging Industry 4.0 [1], it is
estimated that communications of industrial/intelligent wireless sensor
networks (IWSN) in a small cell with up to 1 device per square-meter
connection density require 10 to 100 Mbps data rate, with 5 to 30 ms latency,
and medium device power consumption and cost [2, 3].  These stringent
requirements coincide with all three use categories in 5G specified as
enhanced mobile broadband (eMBB), ultra-reliable low-latency communication
(URLLC) and massive machine-type communication (mMTC). Specifically, the eMBB
is tailored for high peak rate and throughput. General requirements for URLLC
are sub 1ms user plane latency and packet error rate as low as $10^{-5}$ [4].
For mMTC, the general requirement, as suggested by its name, is to provide
massive wireless connectivity (e.g. beyond $10^{5}$ links) to machine-type
devices in a given area [5].

The millimeter-wave (mmW) communications is a key technology in 5G. Due to
abundant bandwidth and antenna array beamforming, mmW communications is the
enabler of eMBB. It is also envisioned as a promising candidate for the future
high-end IWSN [6]. However, the current mmW solutions have many disadvantages
in terms of latency, connection density, power and cost for the industry
internet of things (IoT) applications. Firstly, mmW systems rely on
beamforming to overcome severe propagation loss. To keep reasonable cost and
power consumption, current systems utilize phased antenna arrays at both base
station (BS) and user equipment (UE) for beamforming. The overhead associated
with beam alignment and beam scheduling using analog arrays is a latency
bottleneck. Secondly, the analog phased antenna arrays can connect only a
limited number of devices based on the number of beams and the number of radio
frequency (RF) chains. Each beamformed transmission occupies the entire
bandwidth, therefore frequency domain multiple access cannot be supported. As
a result, phased antenna array based mmW networks cannot support a high number
of connected devices. Lastly, cost and power consumption of wide-band mmW
devices are high which presents a major limitation for IoT applications.

Meeting diversified service requirements with reasonable cost and power
consumption is inherently hard. Specifically, to tackle URLLC requirement, key
5G NR features such as grant-free multiple access must be employed [7]. In
this regime, the network capacity is often limited by insufficient radio
resource elements. [8] evaluates URLLC use cases with 10MHz band at 4GHz
carrier frequency where the capacity attains 10 users per cell. Alternatively,
in mmW band large unlicensed band resources are available. However, due to the
need of beamforming, these resources cannot be shared between a large number
of spatially separated users [9]. To date, most works that focus on latency in
mmW systems emphasize on reducing the initial access and beam alignment
latency [10]. Latency due to resource scheduling among multiple users is
highlighted by work [11], where optimized frame structure and fully digital
arrays were proposed as a solution. The ability of a digital array to connect
with users in all directions is useful, however its power consumption and cost
is overwhelmingly high. Therefore, to address aforementioned weaknesses of
existing solutions for mMTC and URLLC, rethinking of array architectures and
multiple access schemes is required.

In this work, we leverage a new beamforming technique referred to as rainbow
beam [12], that is enabled by a novel True Time Delay (TTD) array architecture
[13]. Rainbow beams exhibit beam patterns that cover the entire angular space
by uniquely mapping different frequencies, e.g. orthogonal frequency-division
multiplexing (OFDM) subcarrier (SC), onto specific directions. As indicated in
Figure 1, rainbow beams are particularly attractive in wideband systems as
different SCs and corresponding beamforming directions can be allocated to a
large number of users. As long as the users select their appropriate frequency
resources, no beam training or scheduling is needed. Based on this feature, in
our design a grant free mmW multiple access protocol is combined with rainbow
beamforming. It has a potential to provide low user-plane latency for a
massive amount of users in IoT applications.

Figure 1:  Due to narrow angular coverage of analog beams, BS with phased
array can only serve a limited number of scheduled users. On the other hand,
TTD array can connect with users in the entire spatial sector without any
scheduling. With phased array, the entire band is pointed to a certain
direction, while with TTD array, each beam is pointed to a specific frequency
subband (represented by colors).

The idea of utilizing frequency dependent beamforming to enhance coverage of
directional beams (therefore provide multiple access) has been studied in sub-
terahertz communication system [14], [15], [16]. However, previous studies
largely considered broadband, high-rate communications, as opposed to IoT use
cases. Differing from most mmW networks, in our design the multiple access
based on rainbow beams is regarded as an enabling solution for latency-
critical mMTC [17]. On one hand it provides wide angular beam coverage in a
power efficient manner by spatially spreading SCs; on the other hand it
operates in mmW band where sufficient bandwidth resources are available to
reduce system overhead. Our main contributions are the following:

  * •

We propose a rainbow-beam based frequency domain multiple access which does
not require explicit beam-training and supports grant-free random access.

  * •

We propose a detailed design for radio interface, downlink (DL)
synchronization procedure, and random access protocol.

  * •

We analyze synchronization performance and probability of collision (packet
loss) and their impact on latency for given network design parameters.111 Our
proposed scheme inherently has only two sources of latency due to
synchronization and collision with other users that share the same frequency
and beam resources.

  * •

Our analytical study is supported by simulations that evaluate the proposed
multiple access scheme in terms of latency, reliability and effective rates.

The rest of the paper is organized as follows. In Section II, we introduce the
system model. The proposed rainbow link protocol and radio interface design
are included in Section III. The analysis of DL synchronization and uplink
(UL) grant-free transmission are presented in Section IV. Section V presents
simulation results. Discussion and future work suggestions are in Section VI.
The paper is concluded in Section VII.

Notations: Scalars, vectors, and matrices are denoted by non-bold, bold lower-
case, and bold upper-case letters, respectively, e.g. $h$, $\mathbf{h}$ and
$\mathbf{H}$. The element in $i$-th row and $j$-th column in matrix
$\mathbf{H}$ is denoted by $[\mathbf{H}]_{i,j}$. Transpose and Hermitian
transpose are denoted by $(.)^{\text{T}}$ and $(.)^{\text{H}}$, respectively.
The $l_{2}$-norm of a vector $\mathbf{h}$ is denoted by $||\mathbf{h}||$.
$\text{diag}(\mathbf{A})$ aligns diagonal elements of $\mathbf{A}$ into a
vector, and $\text{diag}(\mathbf{a})$ aligns vector $\mathbf{a}$ into a
diagonal matrix. The $i$-th element in set $\mathcal{S}$ is denoted as
$[\mathcal{S}]_{i}$.

##  II System Model

Figure 2: The left figure shows a typical true-time-delay mmW arrays. The
middle figure illustrates the rainbow beam pattern of the base station with
$N_{\text{B}}=64$ antennas. The BS is in origin point of polar plot and gray
crosses represent connected user equipment.

In this section, we introduce the system model of mmW communication with TTD
beamforming. All important notations are summarized in Table I.

TABLE I: Nomenclature Symbol | Explanations
---|---
Symbols related to system model
$b$ |  Index of SC
$\mathbf{w}_{b}$ |  beamformer in the BS for $b$-th SC
$u$ | Index of UE
$\mathbf{v}_{u}$ | freq independent beamformer at the UE
$\mathcal{B}_{u}$ |  SC set a narrowband UE can access
$N_{\text{B}}$, $N_{\text{U}}$ | Number of antenna in BS and UE
$\mathrm{BW}$ | total bandwidth of the network
$M$ | Number of multi-path components
$\mathbf{H}_{u,m}$ | m-th narrowband channel for the u-th UE
$\mathbf{a}_{\text{B}}\left(\theta_{u}\right)$, $\mathbf{a}_{\text{U}}\left(\phi_{u}\right)$ | Spatial responses of BS and UE
$\phi_{u},\theta_{u}$, $g_{u}$, $\tau_{u}$ |  Path AoA/AoD/gain/delay of $u$-th user
Symbols related to multiple access
$n$ | Number of repetitions over resource blocks
$K$ | Narrowband cardinality of resource blocks
$B$ |  Total number of SC
$U$ | Number of active users per frame
$p$ | Rate of user activation per frame
$g_{\mathrm{group}}$ |  Number of SCs bundled into a resource block

We consider a mmW time-division duplex (TDD) network with center frequency
$f_{c}$ and total bandwidth $\mathrm{BW}$. The OFDM waveform with cyclic
prefix (CP) is used with a total number of $B$ SCs.

The network consists of a massive number of quasi-static mmW UEs that are
served by a BS. We assume each UE, a machine type device, is equipped with a
phased array with $N_{\text{U}}$ antenna. Furthermore, we assume each UE has
much narrower bandwidth than the BS to further reduce power and cost. As a
result, each user can only access a subset of SCs through orthogonal
frequency-division multiple access (OFDMA). The indices of those SCs
accessible by user $u$ are denoted by set $\mathcal{B}_{u}$ whose cardinality
satisfies $|\mathcal{B}_{u}|\ll B$, i.e., the SCs that a user can access is
much smaller than the total number of SCs in the broadband. In practical OFDM
based system, adjacent SCs are typically bundled into blocks for frequency
resource utilization. This mechanism is controlled by $g_{\mathrm{group}}$
which denotes how many SCs are consolidated into a resource block (RB). To be
clear on notations, we use $|\mathcal{B}_{u}|$ for the number of SCs and $K$
for the number of RBs in $\mathcal{B}_{u}$, i.e.,
$K=\left|{B}_{u}\right|/g_{\mathrm{group}}$ .

We assume the BS has a linear array with $N_{\text{B}}$ elements. The antennas
are critically spaced, i.e., half of wavelength that associates with $f_{c}$.
In this work we focus on a sparse geometric channel of $M$ multipath
components222For mmW communication, typically $M\leq 4$, where

the angle of arrival (AoA), angle of departure (AoD), and complex path gain of
the $u$-th UE and $m$-th multipath component are denoted as $\theta_{u,m}$,
$\phi_{u,m}$, and $g_{u,m}$, respectively. $\theta_{u,m}$ and $\phi_{u,m}$ are
assumed to be uniform randomly distributed in region $-\pi/2$ to $\pi/2$. The
channel $\mathbf{H}_{u,b}$, for $u$-th user and $b$-th SC is

$\displaystyle\mathbf{H}_{u,b}=\frac{1}{\sqrt{N_{\text{B}}N_{U}}}\sum\limits_{m=1}^{M}\alpha_{u,b,m}\mathbf{a}_{\text{U}}(\phi_{u,m})\mathbf{a}^{\text{H}}_{\text{B}}(\theta_{u,m})$
(1)

where $\alpha_{u,b,m}=g_{u,m}\mathrm{exp}(j2\pi b\tau_{u,m}\mathrm{BW}/B)$ is
the SC-wise complex gain, with $g_{u,m}$ and $\tau_{u,m}$ representing the
multipath gain and delay with respect to the first antenna element as a
reference. We denote $\mathbf{a}_{\text{B}}(\phi)$ and
$\mathbf{a}_{\text{U}}(\theta)$ as the narrowband333The narrowband array
response model holds true when the propagation delay across the array aperture
is less than the sampling duration [12]. For typical mmW BS array aperture
$\leq 0.32$m, the propagation delay across the aperture is up to 1 ns, which
is less than 2.5ns, the sampling duration of a 400MHz system. array response
vectors,
i.e.,$[\mathbf{a}_{\text{U}}(\theta)]_{n}=\mathrm{exp}\left[j\pi(n-1)\sin(\theta)\right]$
and
$[\mathbf{a}_{\text{B}}(\phi)]_{n}=\mathrm{exp}\left[j\pi(n-1)\sin(\phi)\right]$.
Amplitude of $g_{u,m}$ is characterized by the free space path loss model
[18].

In general, channel with multiple taps would be frequency selective. However,
since all UEs operate over a small portion of broadband bandwidth, we assume
the channel is approximately flat. We also assume the BS is equipped with a
reconfigurable TTD array with a single RF chain for its transceiver as shown
in Figure 2. In our previous work, we have shown that TTD array can realize
frequency dependent beams so that each OFDM SC is mapped to a particular
beamforming direction [12]. Delay taps in the array are set to be uniformly
spaced with inter-element delay spacing of $\Delta\tau$, regardless of whether
the delays are introduced in baseband or RF. The full-range rainbow beam444It
is noted that this rainbow beam operation has strict requirement on the delay
range of the circuit blocks that introduce delay. However, with the increased
bandwidth in mmW and sub-THz, as well as the recent break-through of circuits
[19], such regime is attainable. can be achieved with tap
$\Delta\tau=1/\mathrm{BW}$ such that the equivalent analog combiner for the
$b$-th SC is given by

$\displaystyle[\mathbf{w}_{b}]_{i}=\mathrm{exp}\left[j2\pi b(i-1)/B\right].$
(2)

Inspecting (2), it is clear that beamformer $\mathbf{w}_{b}$ steers SC $b$
onto a unique directions with respect to the rest of SCs. On the other hand,
any steering vector, i.e., $\mathbf{a}_{\text{B}}(\phi)$ for an arbitrary
$\phi$, is also synthesized at a certain SC. Hence it is referred to as full
coverage rainbow beamforming. For a certain user, we define $b^{\star}_{u}$ as
the index of SC whose encoded spatial direction is the closest to the AoA
$\theta_{u}$ of a certain path. This particular SC is also denoted as anchor
SC. In other words, the beamforming gain at anchor SC is the highest among all
subcarriers.

To better present our proposed network protocol, we briefly introduce the
signal model in the DL and UL. In the DL, we denote the transmit OFDM symbol
at the $b$-th SC as $S^{(\text{DL})}[b]$. Given an ideal bandlimited filter at
the UE defined by its narrowband $\mathcal{B}_{u}$, the received frequency
domain signal is

$\displaystyle
R_{u}^{(\text{DL})}[b]=\underbrace{\mathbf{v}_{u}^{\text{H}}\mathbf{H}_{u,b}\mathbf{w}_{b}}_{\beta_{u,b}}S^{(\text{DL})}[b]+z^{(\text{DL})}[b],\text{for}~{}b\in\mathcal{B}_{u}.$
(3)

In the above equation, $\mathbf{v}_{u}$ represents the precoder of the $u$-th
user (which is not frequency dependent) and scalar $\beta_{u,b}$ characterizes
the post-beamforming gain of the $b$-th SC. $z^{(\text{DL})}[b]$ refers to the
noise at the $b$-th SC. .

The received signal at the BS in the UL is expressed as

$\displaystyle
R^{(\text{UL})}[b]=\sum_{u=1}^{U}\underbrace{\mathbf{w}_{b}^{\text{H}}\mathbf{H}^{\text{H}}_{u,b}\mathbf{v}_{u}}_{\beta^{*}_{u,b}}S^{(\text{UL})}_{u}[b]+z^{(\text{UL})}[b]$
(4)

where the post-beam channel gain in the UL $\beta^{*}_{u,b}$ is the conjugate
of the one in the DL as long as the same SC and UE-side beamformer
$\mathbf{v}_{u}$ is used.

Since each SC is mapped to a spatial direction, UE can measure the receive
power on SCs during DL boardcasting and identify the segment of SCs with the
highest received power as the segment aligned with its AoA. Note that although
the channel gain is assumed to be flat across the narrowband
$\mathcal{B}_{u}$, post-beamforming gain $\beta_{u,b}$ is still frequency
dependent ($b$ dependent) due to TTD beamforming precoder $\mathbf{w}_{b}$ at
the BS side. This unique feature of rainbow-beam with TTD array causes
beamforming gain loss on different SCs. Even if a UE has broadband sampling
capability, there will only be certain SCs that it can use. This design
consideration is addressed in Section IV .

##  III Rainbow Link Medium Access Control

In a conventional mmW UL access, a 4-step random access procedure is employed.
The BS broadcasts synchronization and beam training pilots during DL. The UEs
independently conduct synchronization, measure the received signal strength of
BS’s beams, and send feedback through the random access channel. After the BS
receives feedback, it applies user scheduling and resource allocation scheme
by reserving dedicated time-frequency resources for UEs before sending them
access grant. UEs then use the scheduled resources to complete UL
transmission. In this scheme a non-trivial latency is expected not only in the
grant request, but also in the resource scheduling because the served UEs in
each time slot are limited to a small angular region covered by a narrow
analog beam from the BS.

The rainbow link on the other hand, leverages the frequency-dependent beam
steering capability of TTD array which can simultaneously connect multiple
UEs. We propose to use a fixed beam configuration given by beamforming
precoder/combiner in (2) for both DL synchronization and UL access, i.e., on
the BS’s side no beam switching is required. Due to large coverage of rainbow
beam in angular domain, UEs always have beamforming gain to and from BS
without explicit beamforming training or feedback. Elimination of beamforming
feedback significantly cuts the overhead on beam maintenance and resource
scheduling which then makes the 2-step contention based random access
possible.

This section presents the proposed protocol for multiple access with rainbow
link. We start by discussing details of DL synchronization and UL grant-free
transmission. Based on the discussion, we then provide the design of radio
interface as well as relevant performance metrics that are analyzed in
Sections IV and V.

###  III-A Rainbow beam DL synchronization

Given a wideband OFDM transmission from the BS, the synchronization of the
narrowband UEs require estimation of the correct timing offset and fast
Fourier transform (FFT) window for decoding. Effectively, a narrowband
receiver at the UE side needs to select a set of SCs $\mathcal{B}_{u}$ that
are then used for UL transmission. Since rainbow beam maps each SC to a
specific spatial direction, UEs should select SCs that are mapped to its AoD
so that they leverage the maximum beamforming gain from BS and improve DL for
the detection of the synchronization signal. Due to narrowband receiver
processing at the UE, the synchronization involves following challenges:

  * •

UEs should synchronize using a received signal that contains only a fraction
of the entire OFDM preamble.

  * •

UEs needs to locate (in the frequency domain) a segment of SCs with the
minimum beamforming gain loss as its operating band $\mathcal{B}_{u}$.

To tackle the first challenge, we propose to load synchronization sequences on
all SCs. Since the AoD of a UE in DL channel might land on any SC across the
wideband, we employ pseudorandom noise (PN) sequences instead of Zadoff-Chu
sequences that are used in 4G Narrowband Internet of Things (NB-IoT). PN
sequences have a low peak to average power ratio, zero auto-correlation with
its time-shifted version, and do not require fixed narrowband reception for
completeness [20]. The preamble sequence is assumed to be known at UEs.

The second challenge is essentially dependent on the UE’s ability to maintain
beam alignment with the BS. With no prior information about which segment it
should select, a UE would have to traverse the entire broadband signal and
repeatedly search for $b^{\star}_{u}$, the anchor SC with the highest
beamforming gain. Undoubtedly, traversing the entire broadband frequencies
(with low sampling rate) would cause prohibitive overhead. We assume that UEs
are quasi-static and thus the AoDs change slowly. In this case, using the
previously selected segment as a prior, UEs can keep track of changes in AoDs.
This simplification eliminates the extra overhead of UEs searching for
$b^{\star}_{u}$.

As illustrated in Figure 2, synchronization sequences are loaded onto
different SCs in the DL broadcast. The DL synchronization signal contains
frequency multiplexed narrowband beacons555This concept is similar to NB-IoT
where a narrowband synchronization sequence is loaded in resource block. What
is unique in rainbow link is that each anchor carrier is coded with a unique
analog beam.. In order to participate in contention UEs are required to
successfully synchronize. The synchronization algorithm at the UE involves the
following steps:

  * •

Perform down conversion by multiplying the broadband signal with
$\exp\left(-1\mathrm{j}2\pi tb^{\star}_{u}/B\right)$, filtering, and sampling
to obtain a narrowband baseband signal. $b^{\star}_{u}$ here refers to the
anchor SC from the previous connection.

  * •

Locate the correct FFT window by estimating the integer and fractional timing
offset.

  * •

Identify the new anchor SC $b^{\star}_{u}$ with the highest signal strength.
The updated $\mathcal{B}_{u}$ for UL transmission is determined accordingly.

###  III-B Rainbow Beam UL Grant-free Transmission

The proposed work cycles and the corresponding timing diagram of the UL grant-
free protocol is shown in Figure 3. When their packets are generated, each UE
activates and synchronizes in the nearest DL time frame where BS broadcasts
synchronization signal, after which UE transmits in an arrive-and-go manner.
Namely, UE does not send a scheduling request for access grant as in grant-
based access. For the UL transmission it utilizes multiple RBs in the narrow
band centered666From a geometric argument, putting the anchor SC in the center
of $\mathcal{B}_{u}$ intuitively ensures that there is good channel gain. at
the detected anchor SC, i.e.,
$\mathcal{B}_{u}=[b^{\star}_{u}-\left|\mathcal{B}_{u}\right|/2,b^{\star}_{u}+\left|\mathcal{B}_{u}\right|/2-1]$.
UEs are sending packets that contain preamble and data so that BS can extract
UE’s identity and other necessary control information.

Naturally in a grant-free access, multiple packets transmitted in the same
time-frequency resource block from different UEs would cause a collision and
packet loss. For the sake of tractable analysis, we assume decoding on SCs
without collision always succeeds, and decoding on SCs with collision always
fails. In order to improve the packet loss rate, we propose a simple
repetition coding. With abundant time-frequency resource blocks in mmW
spectrum, UEs can perform repetitive transmissions on multiple resource blocks
[21]. In our proposed scheme, each UE repeats its transmission on $n$ RBs in
its narrow band of $K$ RBs. The $n$ replicas offer repetition coding and
diversity gain to reduce packet loss. Making a large number of repetitions
alleviates the loss of packets for a certain user. But the contention also
becomes harsh when other UEs do so. In this work we assume a fixed number of
repetition $n$ for all users and study the optimal design of $n$. Notice that
the optimal $n$ will also be influenced by the notion of grouping. When
multiple SCs are consolidated as a RB, UEs load more useful symbols into a
packet but have less resources to avoid collisions. We will analyze these
trade offs in Section IV.

---
(a) BS & UE Work Cycles
(b) User Plane Latency
Figure 3: The uplink grant-free access with proactive repetition scheme
similar to [22, 23]. The abbreviation P, Cont, ACK, and Sync denote packet
arrival, contention attempts, ACK and synchronization attempts, respectively.
A short guard period counts for propagation and processing delay. In the
diagram, UE succeeded in the first synchronization attempt and the second
contention attempt

###  III-C Radio Interface Design

In order to analyze rates, coverage, and latency of the proposed system, this
section briefly discusses radio interface design. Due to its popularity,
reliability, and flexibility for contention based grant-free random access
[23], we adopt a frame slotted Aloha. Both UL and DL transmissions are slotted
in terms of frames777The propose protocol utilizes subset of time-frequency
resource, similar to the interleaved NB-IoT resource units with LTE [24], and
we focus on relevant UL/DL only. and users can transmit without grant request
or allocated resources from BS. Critical specifications in radio interface for
the proposed system include [25]:

  * •

Subcarrier spacing and OFDM symbol duration.

  * •

Cyclic prefix to accommodate combined channel delay spread and the largest
delay tap of TTD array

  * •

Number of UL & DL TTIs

  * •

Length of UL & DL preamble

  * •

Payload in DL frames

In URLLC and mMTC use cases, short packets are commonly used [26, 27]. In that
sense, large spacing between SCs shall be considered. A natural choice is then
to adopt a numerology scaled from the current 5G and 4G networks. As pointed
out by [28], in current standards, SC separation (SCS) is of the form:
$2^{k}\times 15\mathrm{kHz}$ for integer $k$. Specifications with $k>2$ cannot
be implemented in sub-6GHz spectrum. To get a rough understanding of the
system, we then use the $k=5$ case as listed in Table II. Here the length of
CP is about $250$ns which is sufficient to accommodate the combination of
largest delay of TTD element888The largest delay is $N_{\mathrm{B}}$ samples
as specified by size of the array. $256$ samples could accommodate a $128$
element array. and the root mean square delay spread of mmW frequencies in
most outdoor environment [18].

Each frame would contain multiple transmission time interval (TTI)s. Frame
length is a critical design to accommodate various control data and payloads.
With the grant-free multiple access, the structure of DL TTI in the proposed
scheme is relatively simple. The detailed frame structure is specified in
Table III.

TABLE II: Numerology  Specification | Value
---|---
Sampling Frequency[MHz] | 983.04
Number of Points in FFT | 2048
$\mathrm{BW}$ [MHz] | 1000
SC Spacing [kHz] | 480
Cyclic Prefix [Samples] | 256
Symbols per TTI | 13
Symbol Duration [$\mu s$] | 2.083
Symbol Duration with $CP$ [$\mu s$] | 2.344
TTI Duration [$\mu s$] | 30.469

Although BS transmits wideband signals, only a portion of the band is
accessible to each narrowband UE. Therefore both UL and DL reference signals
(preambles) in our design are much longer than the ones in [25] due to lack of
broadband receiving capabilities. We also assume open-loop power control with
a maximum UE transmit power of $23$ dBm [29] which allows for shorter UL
reference symbols than that of DL frame.

TABLE III: Frame Design Specification | Design Values
---|---
UL Symbols [TTI] | 2
DL Symbols [TTI] | 2
Frame Length [$\mu s$] | 125
Frame Length [symbols] | 52
DL Reference Signal [symbol] | 22
DL Control Signal [symbol] | 2
Guard Interval [symbol] | 2
UL Preamble [symbol] | 8
UL Payloads [symbol] | 18

The proposed numerology and frame design are used throughout the rest of the
paper. It is worth mentioning that this numerology is tailored to achieve
URLLC with large SC spacing and mini-slots. In general, depending on the
application, there are designs with more efficient control fields. However,
these investigations are beyond the scope of this paper.

###  III-D Performance Metrics

To simplify the discussion, we assume that a successful DL synchronization and
UL transmission without collision together lead to a successful packet
delivery. During each DL broadcasting, BS encodes feedback per SC to indicate
whether previous UL payload on that SC is successfully decoded. Synchronized
UEs would persistently transmit until their packets are delivered, while UEs
with unsuccessful synchronizations do not participate in contention. The
probability of successful synchronization is dependent on the receiver DL
preamble. For users that are within the rainbow link coverage, we analyze the
following performance metrics:

  * •

Access latency: defined as the amount of time between packet arrival and its
successful delivery.

  * •

Effective rate: defined as the ratio of the number of data symbols (payload)
in a packet and the time required to successfully delivery through contention.

Both metrics depend on several factors including SNR, user density, the number
of repetitions, and grouping of subcarriers into RBs.

The proposed system is expected to serve a large number of spatially separated
users with low latency. In the next section, we analyze DL synchronization and
UL contention, and mathematically formulate performance metrics.

##  IV Performance Analysis of Rainbow Link

In this section, we provide analysis of the proposed rainbow link access. For
the sake of tractable analysis and concise notation, we make two additional
assumptions. Firstly, we mainly focus on a single path channel, i.e., the
summation and the $m$-dependency in (1) are dropped. The underlying rationale
is that each UE can only access a narrow bandwidth $\mathcal{B}_{u}$,
therefore it is very likely that only a single path aligns in the spatial
direction encoded on SCs within $\mathcal{B}_{u}$ as illustrated in Figure 4.
Secondly, we omit the discussion of beamforming in the UE’s side.
Specifically, $N_{\text{U}}=1$ is used, and we drop beamformer $\mathbf{v}$
and array response $\mathbf{a}_{\text{U}}\left(\phi_{u}\right)$ of UE in the
analysis. The rationale behind is that in the term
$\beta_{u,b}=\mathbf{v}_{u}^{\text{H}}\mathbf{H}_{u,b}\mathbf{w}_{b}$, UE’s
beamformer serves merely as another factor that has no dependency on $b$. In
other words, a multi-antenna UE results merely in a higher beamforming gain
and is not related to any unique feature of rainbow beamforming999Presumably,
using multiple antennas on UE side also requires extra overhead for
beamforming training..

Figure 4: The narrowband operation of UE and rainbow beams naturally reject
multipath components. For signal at a given subcarrier, its multi-path
components propagate via a different AoD and hence are filtered out by UE due
to its limited bandwidth.

###  IV-A Analysis of DL synchronization

Synchronization based on OFDM waveform usually involves estimation of timing
offset and carrier frequency offset [30]. With TTD array, in addition to
timing offset UEs need to locate their anchor SCs. Here we employ a similar
analysis as in [30] of NB-IoT without addressing the carrier frequency offset.
In order to locate the anchor SC $b^{\star}_{u}$ with the highest gain during
DL synchronization, a user needs to find:

$\displaystyle b^{\star}_{u}=\mathrm{argmax}_{b}\left|\beta_{u,b}\right|^{2}$
(5)

$b^{\star}_{u}$ is then the anchor SC whose steering angle is the most aligned
with $\theta_{u}$. Plugging (1), (2) into (3) and simplifying:

$\displaystyle\begin{split}\left|\beta_{u,b}\right|^{2}=\frac{|\alpha_{u}|^{2}}{N_{\text{B}}N_{\text{U}}}\underbrace{\left|\frac{\sin\bigg{[}\frac{N_{\text{B}}\pi}{2}\left(\frac{2b}{B}-\sin\theta_{u}\right)\bigg{]}}{\sin\bigg{[}\frac{\pi}{2}\left(\frac{2b}{B}-\sin\theta_{u}\right)\bigg{]}}\right|^{2}}_{F_{N_{\text{B}}}\left(\pi\zeta/B\right)}\\\
\end{split}$ (6)

The expression above leads to Fejér kernel $F_{N_{\text{B}}}(\cdot)$ which is
defined as
$F_{N}(x)=\left|\sin\left(Nx/2\right)/\sin\left(x/2\right)\right|^{2}$.
$b^{\star}_{u}$ should then satisfy the following criterion:

$-1\leq 2b^{\star}_{u}-B\sin\theta_{u}\leq 1$

From (6), one can see that as the total number of SCs in UE’s band increases,
the edge SC $b^{\star}_{u}\pm\left|\mathcal{B}_{u}\right|/2$ have an
increasing gain loss compared with the anchor SC $b^{\star}_{u}$. This has two
implications. Firstly, to ensure a relatively flat beamforming gain across the
UE narrowband bandwidth, the total number of SCs in the broadband should be
much larger than number of antennas at the BS $N_{\text{B}}$, i.e.,
$N_{\text{B}}/B\ll 1$. This is because the gain difference in narrow band is
only relevant to the ratio $N_{\text{B}}/B$ as shown in (6). Secondly, due to
the gain loss, there are only limited number of SCs that can be included in
$\mathcal{B}_{u}$. For instance, a generic threshold
$\left|\mathcal{B}_{u}\right|\leq B/N_{\text{B}}$ ensures that the SC at the
edge of $\mathcal{B}_{u}$ has less than 3dB gain loss compared with the anchor
SC . Increasing number of antennas $N_{\text{B}}$ at the BS might result in
large gain difference which then limits SCs that a user can access. Notice
that these phase and gain differences do not affect UL data transmission since
they can be treated as part of the channel. With UL preamble sequence, BS
estimates the channel per SC and thus naturally compensates it.

The rest of the subsection then addresses the estimation of timing offset. Let
$L$ denote the number of symbols in DL preamble sequence. Let $l$ be the index
of samples in a OFDM symbol such that $-N_{CP}\leq l\leq|\mathcal{B}_{u}|$.
Then $l$-th sample of the $m$-th symbol in preamble sequence can be expressed
as:

$\begin{split}s^{(\text{DL})}_{u}[l;m]&=\sum\limits_{b\in\mathcal{B}_{u}}g_{u}\left[\mathbf{a}_{\text{B}}^{\text{H}}\left(\theta_{u}\right)\mathbf{w}_{b}\right]S_{u}^{(\text{DL})}[b;m]e^{1\mathrm{j}2\pi\cdot\frac{b}{\left|\mathcal{B}_{u}\right|}\left(l-D_{u}\right)}\\\
&+z[l;m]\\\ \end{split}$ (7)

where $D_{u}$ amounts to normalized timing offset that includes an integer and
a fractional part due to down sampling. In the equation above, only SCs within
the narrow band $\mathcal{B}_{u}$ are taken into account. This assumes that
UEs sample the wideband signal without aliasing. In practice, such assumption
can be achieved with a high quality low pass filter. With channel gain $g_{u}$
remaining the same for $\mathcal{B}_{u}$ during $L$ symbols, the received
frequency domain symbol is then:

$\begin{split}\tilde{S}^{(\text{DL})}_{u}[b;m]&=g_{u}\left[\mathbf{a}_{\text{B}}^{\text{H}}\left(\theta_{u}\right)\mathbf{w}_{b}\right]S_{u}^{(\text{DL})}[b;m]e^{1\mathrm{j}2\pi\cdot\frac{b}{\left|\mathcal{B}_{u}\right|}\left(-D_{u}\right)}\\\
&+\tilde{z}[b;m]\\\ \end{split}$ (8)

In (8), $\tilde{z}$ is the per SC noise in frequency domain. Synchronization
is essentially an estimation of $D_{u}$ from observed samples
$\tilde{S}^{(\text{DL})}_{u}[b;m]$. However, since $\theta_{u}$ is unknown,
the gain loss cannot be accurately modeled and causes degradation in
synchronization performances. We evaluate the following estimator for timing
offset and its synchronization performance:

$\displaystyle\begin{split}D_{u}^{*}&=\mathrm{arg}\max\limits_{D_{u}}\quad\left|J_{u}\left(D_{u}\right)\right|^{2}\\\
J_{u}\left(D_{u}\right)&=\sum\limits_{m,b}\left(\tilde{S}^{(\text{DL})}_{u}[b;m]\right)^{*}S_{u}^{(\text{DL})}[b;m]\times
e^{1\mathrm{j}2\pi\cdot\frac{b}{\left|\mathcal{B}_{u}\right|}D_{u}}\\\
\end{split}$ (9)

In (9) UEs merely perform correlation between
$\tilde{S}^{(\text{DL})}_{u}[b;m]$ $S_{u}^{(\text{DL})}[b;m]$ when estimating
$D^{*}_{u}$. As $\left|\mathcal{B}_{u}\right|$ increases, the summation in (9)
involves more symbols, but the correlation between the two sequences is also
more distorted. To conclude, in DL synchronization UEs suffer from gain loss
due to the rainbow beam pattern and can therefore only operate on a limited
number of SCs. This effect will be analyzed via numerical simulations.

We note that as a unique feature of frequency dependent rainbow beam pattern,
the synchronization does benefit from beamforming gain contributed by BS
antenna. The length of preamble $L$ should be designed accordingly to fulfill
high probability of detection and low synchronization error.

###  IV-B Analysis of packet loss rate

Next we analyze collisions and packet loss rate (PLR) for given radio
resources and user density. The purpose of the analysis is to mathematically
guide the design of various system parameters.

Since UEs are narrowband, there is no collision between UEs that contend in
non-overlapped bands. Given that all users choose a consecutive $K$ RBs, a
subband would suffer from collisions from users landing on a total number of
$2(K-1)+1=2K-1$ RBs. To tackle all the possibilities for a given number of
repetitions, a combinatorial approach is needed. In the analysis, we focus on
a specific user occupying bandwidth $\mathcal{B}_{u}$ and calculate the
density function of the number of occupied SCs in the band before this user
adds its replicas. From there, packet loss rate is simply the probability that
all replicas from this user are covered by already occupied SCs. Based on our
model, the packet loss rate
$P_{\mathrm{PLR}}\left(B,K,U,n,g_{\mathrm{group}}\right)$ is given in the
following proposition as a function of the number of SCs $B$, the number of
active users $U$, the number of RBs that a user can access $K$, the grouping
factor $g_{\mathrm{group}}$, and the number of repetitions $n$ for each user.

###### Proposition 1

With a uniform spatial distribution of users, an approximated expression of
PLR of the proposed rainbow link is

$\begin{split}P_{\mathrm{PLR}}=\sum\limits_{i=1}^{U-1}f\left(U-1,g_{\mathrm{group}}\frac{2K-1}{B},i\right)\sum\limits_{j=n}^{K}\left[\mathbf{T}^{i}\mathbf{p}_{0}\right]_{j}\frac{\mathrm{C}_{j}^{n}}{\mathrm{C}_{K}^{n}},\\\
\end{split}$ (10)

where $f\left(x,p,i\right)=\mathrm{C}_{x}^{i}\left(1-p\right)^{x-i}p^{i}$ is
the binomial probability distribution function, $\mathrm{C}_{x}^{i}$ is the
operator that evaluates $x$ choose $i$,
$\mathbf{p}_{0}=[1,0,\cdots,0]^{\text{T}}\in\mathbb{R}^{K+1}$ is the unit
vector, and the entry in $m_{1}$-th row and $m_{2}$-th column of the matrix
$\mathbf{T}\in\mathbb{R}^{(K+1)\times(K+1)}$ is given by

$\displaystyle\begin{split}\left[\mathbf{T}\right]_{m_{1},m_{2}}&=\frac{\mathrm{C}_{K-m_{1}}^{m_{2}-m_{1}}\mathrm{C}_{m_{1}}^{n-(m_{2}-m_{1})}}{(2K-1)\mathrm{C}_{K}^{n}}\\\
&+\sum\limits_{j=0}^{n}\sum\limits_{k=2}^{K}\frac{2\mathrm{C}_{K-k}^{n-j}\mathrm{C}_{K-m_{1}}^{m_{2}-m_{1}}\mathrm{C}_{m_{1}}^{j-(m_{2}-m_{1})}}{(2K-1)\mathrm{C}_{K}^{n}}\\\
\\\ \end{split}.$ (11)

###### Proof:

See Appendix -A. ∎

We note here that with all other parameters fixed, $P_{\mathrm{PLR}}$ as a
function of $n$ has a minimum. However, since $n$ is an integer and the
formula in (10) is complicated, we discard discussions on conditions of
optimal $n$. In practice, we can simply calculate $P_{\mathrm{PLR}}$
numerically for purposes of system design.

In general, increasing the number of SCs that a UE accesses can drastically
reduce packet loss rate. However, from a geometric perspective, the generic
argument is that $\left|\mathcal{B}_{u}\right|$ is limited by
$B/N_{\text{B}}$, the number of spatial directions in the main lobe of an
analog beam. This can also be seen from (8) that SCs outside of the main lobe
of $b^{\star}_{u}$ have large gain loss.

###  IV-C Latency and Rate Calculation

With both major causes of protocol overhead addressed, in this subsection we
discuss the performance metrics used in the evaluation of the system. Since
the system is designed to fulfill stringent latency requirements, we mainly
focus on the overall latency. Based on latency analysis, we then derive the
formulas for effective data rate of UEs.

As illustrated by Figure 3, the overall latency $T_{\mathrm{tot}}$ can be
computed as follows:

$\begin{split}&T_{\mathrm{tot}}=T_{\mathrm{sync}}+T_{\mathrm{cont}},\\\
&T_{\mathrm{sync}}=T_{\mathrm{activation}}+T_{\mathrm{failure}}+T_{\mathrm{DL}},\\\
&T_{\mathrm{cont}}=T_{\mathrm{packetloss}}+T_{\mathrm{DL}}.\\\ \end{split}$
(12)

Specifically, $T_{\mathrm{activation}}$ accounts for the random offset to the
nearest broadcasting when a UE activates. $T_{\mathrm{failure}}$ accounts for
unsuccessful synchronization attempts which might include multiple frames. The
extra $T_{\mathrm{DL}}$ is the overhead for successful synchronization. In
$T_{\mathrm{cont}}$, $T_{\mathrm{packetloss}}$ stands for packet loss in
contention and the extra $T_{\mathrm{DL}}$ stands for the overhead receiving
ACK from DL broadcasting.

The equation to calculate the effective rate is:

$R_{\mathrm{eff}}=\frac{N_{\mathrm{payloads}}}{T_{\mathrm{cont}}}=\frac{g_{\mathrm{group}}N_{\mathrm{packet}}}{T_{\mathrm{cont}}}$
(13)

where $N_{\mathrm{payloads}}$ is the total payload and $N_{\mathrm{packet}}$
is the payload per packet in terms of OFDM symbols.

The most non-trivial trade-off in our proposed design is controlled by
$g_{\mathrm{group}}$. On one hand, as indicated by the numerator in (13),
using a large number of SCs per RB (large $g_{\mathrm{group}}$) boosts the
data rate of UEs. On the other hand, large $g_{\mathrm{group}}$ also enlarges
the denominator in (13) through (12), (10). The interpretation is that given a
fixed $\left|\mathcal{B}_{u}\right|$, increasing $g_{\mathrm{group}}$ enhances
useful payloads but also reduces the number of RBs which means UEs become more
vulnerable to collisions.

Since the contention overhead depends on the number of users participating in
contention at that time, it is hard to evaluate it analytically. Simulation
results will analyze these performance metrics in the next section.

##  V Numerical and Simulation Results

In this section, we support our analysis in the previous sections with
numerical results. We start with a brief illustration on how certain system
parameters are chosen in the simulations.

  * •

Size of BS antenna $N_{\mathrm{B}}$: As discussed in Section IV,
$N_{\mathrm{B}}$ limits the size of the set $\mathcal{B}_{u}$, i.e., SC that
the $u$-th user uses. In simulation, the number of antenna is set to
$N_{\mathrm{B}}=64$. This enables the UEs to access at most
$\left|\mathcal{B}_{u}\right|\leq B/N_{\mathrm{B}}=32$ SCs in its narrow
baseband within 3dB gainloss with $B=2048$ as specified by the number of
points in Table II. The UE’s baseband bandwidth is then approximately 16MHz.

  * •

Grouping factor $g_{\mathrm{group}}$: To keep the discussion simple, we assume
that for each user, the number of resource blocks $K$ in $\mathcal{B}_{u}$ is
an integer. This means $g_{\mathrm{group}}$ would be a factor of
$\left|\mathcal{B}_{u}\right|$. In simulations, we consider cases where
$g_{\mathrm{group}}=1,2,4$. Using $g_{\mathrm{group}}=8$ can greatly enhance
the rate of UEs but there will be only 4 RBs in $\mathcal{B}_{u}$ to avoid
collisions.

  * •

User active rates $p$: For the ease of implementation in our simulations, $p$
exclusively controls how many active users are added per frame. Specifically,
in each frame we consider $1000$ new links uniformly distributed in a semi-
circle covered by the rainbow beam pattern. Activations of these new UEs are
treated as independent events, i.e., $U$ as a random variable follows binomial
distribution $U\sim B\left(1000,p\right)$. These new users will contend with
persistent users added in a previous frame.

Given $N_{\mathrm{B}}$, specifications of link budget at different distances
are given in Table IV. The network is assumed to operate at
$60\text{\,}\mathrm{GHz}$ carrier frequency at which free space path loss
model in [18] is adopted with line-of-sight (LoS) pathloss exponent equal to
$2$. Specifically, DL noise power is calculated based on BS’s active bandwidth
of $1\text{\,}\mathrm{GHz}$, i.e., a UE’s receiving signal-to-noise ratio
(SNR) is independent of its active bandwidth.

TABLE IV: DL Link Budget Calculation Assuming 1GHz Effective Noise Bandwidth Parameter | Value
---|---
Distance [m] | 100 | 200 | 400
BS Configuration
Tx Power [dBm] | 20
Antenna Gain [dBm] | 18.1
Channel
Path loss [dB] | 108 | 114 | 120
Rx Power Configuration
Rx power w/o BF [dBm] | -69.9 | -75.9 | -81.9
Noise Figure | 12
Noise Power | -83.9
Shadow Fading | 4.2
Rx DL SNR [dB] | -2.2 | -8.2 | -14.2

With the radius of coverage, the activation rate $p$, and the current frame
length (specified in Table III), one can compute user capacity of the system.
For instance, a $400\text{\,}\mathrm{m}$ radius of coverage and $p=0.03$ gives
the following density on average:

$\displaystyle\begin{split}\rho&=\frac{\mathbb{E}\left[U\right]}{\pi\times
400^{2}\times 0.5}\times\frac{1}{T_{\mathrm{frame}}}\\\ &=\frac{1000\times
0.03}{\pi\times 400^{2}\times 0.5}\times\frac{1}{125\times 10^{-6}}\\\
&\approx 0.95\\\ \end{split}$ (14)

in unit of UE activation per $\text{\,}\mathrm{m}$2 per second. This density
$\rho$ serves as a generic metric that evaluates network capacity.

A perceivable drawback of the proposed system is the low spectral efficiency.
The drawback comes from the following facts:

  * •

UE packets are short such that good coding schemes are not applicable. Non-
payload portions cannot be ignored.

  * •

UEs trade its bandwidth for low rate of collisions.

Even if $g_{\mathrm{group}}=4$ is employed along with $16$\- quadrature
amplitude modulation (QAM) and $1/2$ coding scheme, maximum
$R_{\mathrm{eff}}$101010the maximum rate is when a UE always succeeds in its
first attempt of contention for a single UE is merely $\frac{4\times
18}{125\times 10^{-6}}\times 4\times\frac{2}{3}=1.536$ Mbps. Thus the proposed
systme is not intended for high rate applications.

In the evaluation we use the following parameters unless otherwise specified.
The BS has an array size of $N_{\text{B}}=64$ elements. The total number of SC
is $B=2048$ and UEs operate on
$\left|\mathcal{B}_{u}\right|=B/N_{\text{B}}=32$ SCs. A maximum UE transmit
power is set to $23\text{\,}\mathrm{dBm}$ which provides a coverage of $400$m
to achieve about $10\text{\,}\mathrm{dB}$ SNR for UL transmission. For users
near the cell edge, due to their very low SNR, multiple synchronization
attempts might be needed to ensure a satisfying detection performance. We
evaluate the probability of successful synchronization as a function of SNR
and its impact on access latency.

An overall histogram of DL SNR values for UEs randomly distributed in a semi-
circle of $400\text{\,}\mathrm{m}$ radius is given in Figure 5. This range of
SNR values are used throughout simulations.

Figure 5: 2D uniform user distribution in a semi-circle with $400$m radius.

###  V-A Synchronization Performance

Through Monte Carlo simulations, we explore probability of correct estimation
of $D_{u}^{*}$ as a function of broadband SNR. Figure 6 gives detailed
comparison of synchronization performance with and without including impacts
of the aforementioned gain loss and a multiple-path channel. As specified in
Table III, a PN sequence with length $L=22$ is employed as preamble.

Figure 6: DL detection with $L=22$ symbols, with $B=2048$ SCs and
$N_{\text{B}}=64$ antenna elements, $B/N_{\text{B}}$ = 32.

The results in Figure 6 clearly show that the gain loss has a non-negligible
impact. The more SCs are used for narrowband transmission, the larger the
impact of beamforming gain loss is. As $\left|\mathcal{B}_{u}\right|$
approaches $B/N_{\text{B}}$, the synchronization performance starts to
saturate.

With only $\left|\mathcal{B}_{u}\right|=8$ SCs in band, probability of
successful synchronization is not visibly impacted by the rainbow beam
pattern. This means that with $\left|\mathcal{B}_{u}\right|\leq
B/4N_{\text{B}}$ , i.e., only a small number of SCs included in
$\mathcal{B}_{u}$, the impact of rainbow beamforming is trivial. With 16 or 32
SCs in band, synchronization performance loss due to distortion becomes
noticeable. There is a $4\text{\,}\mathrm{dB}$ equivalent SNR loss due to
distortion when system uses $\left|\mathcal{B}_{u}\right|=32$ SCs, which is
effectively the threshold for $\left|\mathcal{B}_{u}\right|$ as discussed
previously. In such case, the detection performance is merely comparable with
the case of $\left|\mathcal{B}_{u}\right|=16$ when there is no distortion.

The implications of numerical results are the following. Firstly, the proposed
narrowband synchronization beacon is robust to both thermal noise and
frequency selectivity introduced by the rainbow beam. In a typical use case of
mmW IoT with large number of devices, high probability of DL synchronization
can be achieved. Secondly, synchronization performance would benefit greatly
from an accurate estimation of $b^{\star}_{u}$ where (9) can then be altered
to count for the gain loss coherently. Otherwise, our analysis suggests that
$\left|\mathcal{B}_{u}\right|\leq B/4N_{\text{B}}$ can be a generic threshold
for achieving negligible beamforming gainloss.

###  V-B Collision and Packet Loss Rate

Figure 7 presents both the simulated and theoretical packet loss rate given by
(10). In the evaluation, we focus on the two cases $g_{\mathrm{group}}=1,2$.

---
(a) Packet loss rate with UE’s bandwidth as wide as $K=32$ RBs.
(b) Packet loss rate with UE’s bandwidth as wide as $K=16$ RBs.
Figure 7: Packet loss rate $P_{\mathrm{PLR}}$ as function of UE’s repetition
transmission number $n$. Different curves are evaluated with different number
of active users $U$ in contention. The optimal $n$ drifts slowly to smaller
values as $U$ increases.

We have the following observation from the results. Firstly, the theoretical
results agree well with simulations for small and moderate $n$ values. This
confirms the correctness of the Proposition 1 where we derived PLR.
Admittedly, discrepancy occurs for large $n$ values because our derivation
omitted the cases when each user uses repetition transmission that occupies
the entire band. Secondly, the results indicate that for a given number of
active users, there is an optimal value of repetitions that minimizes
$P_{\mathrm{PLR}}$. The optimal value of $n$ decreases with an increasing
number of active users in contention. This makes sense since intuitively users
should avoid making too many repetitions when the network is crowded, so as to
leave resources for others. Lastly, compared to probability of failed
synchronizations, packet losses due to collisions among users have much
smaller probabilities in the given network settings. Results from Figure 7
show that even with $g_{\mathrm{group}}=2$ and with 20 active users
transmitting simultaneously, the packet loss rate is still below $10^{-3}$.
With multiple retransmissions, the system can then provide high reliability.

###  V-C Latency and Rates

Next we evaluate the total latency $T_{\mathrm{tot}}$ due to DL
synchronization and UL contention. Here we focus on the trade-off based on
$g_{\mathrm{group}}$ as specified in IV-C. We note here again that users are
generated with uniform distribution in a 2D-sector of $400\text{\,}\mathrm{m}$
radius and activation probability of $0.03$ per frame over $1000$.

---
(a) Statistics for $T_{\mathrm{sync}}$ compared with phased array
(b) Statics for $T_{\mathrm{tot}}$ with different sizes of RBs.
Figure 8: Simulated statistics of latency with active rate $p=0.03$.
$T_{\mathrm{cont}}$ exhibits a typical zig-zag pattern due to multiple
contention attempts.

Simulation results for user-plane latency are presented in Figure 8. In Figure
8a, a comparison on synchronization overhead between the proposed TTD array
and phased arrays with multiple RF chains is presented. In both cases, UEs
operate on only 32 SCs and BS has a 64-element array. To make a fair
comparison, in the case of phased array BS employs random beamforming
strategy. In each frame, it steers analog beams randomly to serve users. Thus
in both cases, there is no overhead due to control signaling or
scheduling111111Presumably, the number of beams that a BS can steer with the
phased array is limited by the number of RF chains. Since all RF chains share
the same band resources, BS should point them to distinct angular sectors to
cover as many users as possible. In our comparison all sectors have a equal
probability to be chosen. To our best knowledge, this is the only regime with
phased array that’s compatible with a grant-free multiple access scheme as
depicted in fig. 3. From the result, the performance of the proposed system
approaches that of a fully-digital array. As discussed in the previous
section, the degradation comes from beamforming gainloss due to frequency
dependent beam pattern. For conventional phased arrays, as the number of RF
chains decrease, the overall beam coverage is limited as depicted in Figure 1.
UEs start to experience prohibitively long synchronization overhead that
cannot satisfy requirements of latency-critical uses cases.

The Figure 8b shows that with $2$-grouping and active rate $p=0.03$,
$T_{\mathrm{sync}}$ dominates the overall latency while with $4$-grouping,
$T_{\mathrm{cont}}$ dominates the overall latency. The tail in contention
latency is due to persistent, spatially clustered users in certain narrow band
segment. Although in both cases the general URLLC requirement for reliability
greater than $1-10^{-5}$ at $1\text{\,}\mathrm{ms}$ latency is satisfied, in
$4$-grouping case the margin is small. This means that $2$-grouping can still
support more active users per frame (higher active probability $p$) while in
$4$-grouping case $p=0.03$ is nearly the maximum activity rate it can support
for URLLC.

In Figure 9, we show the effective rates for different grouping strategies.
For instance, subcarrier grouping with $g_{\mathrm{group}}=4$ achieves higher
rates than $g_{\mathrm{group}}=2$ yet users are more likely to collide. As
stated before, the proposed design is tailored for URLLC and for sporadic,
latency critical transmissions. Higher grouping ratio $g_{\mathrm{group}}$
might bring higher rates, but is very unlikely to satisfy URLLC requirement.

Figure 9: Rate of a single UE with active probability $p=0.03$ based on
contention delay $T_{\mathrm{cont}}$ shown in Figure 8. Actual rates in bits
per second depend on modulation and coding schemes.

In Figure 10, we demonstrate trade-off of $g_{\mathrm{group}}$ with respect to
network capacity. Specifically, we gradually increase $p$ to see what’s the
approximate maximum traffic that the network can support with
$g_{\mathrm{group}}=1$ and $g_{\mathrm{group}}=4$, respectively. The
saturation for $g_{\mathrm{group}}=1$ appears at about $p=0.165$ and as
mentioned earlier the saturation for $g_{\mathrm{group}}=4$ appears at about
$p=0.03$. Using Equation 10, we numerically calculate $P_{\mathrm{PLR}}$ and
find that the optimal number of repetitions $n$ are $5$ and $3$, respectively.
Although with $g_{\mathrm{group}}=1$, each user has only one forth of the data
rate, BS can serve 5 times as many users as that in the case of
$g_{\mathrm{group}}=4$.

---
(a) Nearly Saturated Latency Statistics
(b) Aggregated Throughput
Figure 10: Latency and rates with different active probabilities. The $p=0.03$
curve is the same as that in previous results

In both cases, the achievable latency for a reliability of $1-10^{5}$ is very
close to $1\text{\,}\mathrm{ms}$, as can be seen from Figure 10. Although in
the first case there is no grouping and rate of each user is small, the system
can still achieve a higher sum rate on average by serving more users. With
$p=0.165$, a $16$-QAM, and $1/4$ coding rate, the aggregated throughput can
reach $28\text{\,}\mathrm{Mbps}$ Mbps on BS side. In this case, the BS serves
on average $165$ active, low-rate users per frame. Thus the proposed system is
better utilized when handling massive multiple access with stringent latency
and reliability requirements.

##  VI Discussion and Future Work

In this section, we briefly discuss the implications of numerical results and
open research questions.

The narrow bandwidth and number of SCs used by the UE are critical parameters
for system performance. With the increased number of SCs that UE can access,
there is improvement in both synchronizations and contentions. On the other
hand, the physical nature of rainbow beamforming limits the number of SCs
(approximately $B/N_{\text{B}}$) that a user can access: only a fraction of
radio resources are aligned with the AoA of a UE and SCs at edge of the
segment $\mathcal{B}_{u}$ suffer severely from beamforming mismatch.

Based on the presented analysis, the proposed system has the following
performance:

  * •

Power and Coverage: The proposed rainbow link based network can achieve a
coverage up to $400\text{\,}\mathrm{m}$ when BS with $64$ TTD analog antenna
array serves single antenna UEs with the transmit power of
$23\text{\,}\mathrm{dBm}$ assuming line-of-sight condition.

  * •

Rates and user capacity: The BS can serve up to 5 active UEs per second per
$\text{\,}\mathrm{m}$2 with Mbps data rates depending on the grouping of SCs
per RB.

  * •

Latency and reliability: The UEs experience $1\text{\,}\mathrm{ms}$ latency
and reliability higher than $1-10^{-5}$ when the optimal number of repetitions
$n$ is used.

The greatest strength of the proposed system is its flexibility to support
combined URLLC and massive multiple access. Based on the proposed design of
numerology and multiple access protocol, the system is better suited to serve
a massive number of users than to provide high data rates. When conservative
grouping is used (in our case no grouping or 2-grouping), the proposed system
can eventually achieve a sum rate at BS higher than that of the case with
larger grouping.

The open research questions include but are not limited to the following:

Efficient Frame Design:  If no grouping is used, one frame has only $18$
symbols in its payload based on the proposed frame design. Typical URLLC
packets have 32 bytes, so either dense modulation scheme or more aggressive
grouping strategies would have to be used. In this work we assume $1:1$ UL DL
ratio. Alternative frame design, for instance, a frame length of
$250\text{\,}\mathrm{\SIUnitSymbolMicro s}$ with $1:3$ UL and DL ratio could
support higher data rate for each user with extra DL TTIs all contributing to
the payload.

Robustness for Non-LoS channels and user mobility: In the current discussion,
UEs are assumed to be in LoS channels. But the same discussion might not be
applicable for non-LoS channel conditions. For instance, when a UE suffers
from blockage, it loses track of its anchor SC and needs to switch to a
different path. As pointed out in III-A, the switching of $\mathcal{B}_{u}$
without any prior requires prohibitively long overheads. Thus the current
design is not robust to mobility of UEs and non-LoS channel conditions. These
questions need to be addressed for practical implementations of rainbow-link.

Dynamic Control of Random Access: Increasing bandwidth of the narrowband UE,
at the expense of the hardware cost, can greatly boost performance of the
proposed system. While we analyzed the system performance under uniform user
distribution, distributions of users based on clustered model could further
intensify contentions. In this case there might be no optimal number of
repetitions as the UEs would persistently contend with other UEs in the same
spatial sector. To overcome this effect, the BS can control the number of
repetitions that each user can make. If there are too many collisions, BS
would tell each UE to reduce its repetitions so as to make sure that at least
some UEs can transmit.

Alternative Multiple Access Protocol: The proposed system can benefit from non
deterministic repetition coding strategy. As it has been proposed in [31],
each user can use a random number of repetitions $n$ for each contention
period. This might increase system robustness to clustered scenario.
Unfortunately, the analysis used in [31] is not directly applicable due to its
assumption of sufficient length of code words. Alternatively, instead of
repetition coding, a non-orthogonal coding can be applied. Each user can
transmit data on all SCs in its band and multiply data symbols with a unique
code in frequency domain. Though collisions will happen on SCs, BS can still
decode with these unique code words. This is similar to [32] where NOMA is
employed for multiple access.

##  VII Conclusion

In this work we proposed a novel frequency domain multiple access, referred as
rainbow link, that exploits wideband spectrum at mmW to serve a large number
of narrowband users. By exploiting TTD array architecture at the BS, frequency
resources are mapped to specific spatial directions so that users can be
assigned a subset of SCs in OFDMA and leverage beamforming gain of the entire
array. Rainbow link enables a grant free multiple access and can support a
very large number of users with stringent latency requirements. With a single
RF chain and 64-element array, the BS can provide reliable access with
$400\text{\,}\mathrm{m}$ LoS coverage and sub-millisecond overhead for up to 5
UE activations per second per square meter. We believe the proposed rainbow-
link can be a candidate for future critical mMTC use cases.

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###  -A Generic Formula for Set Covering Distribution

For the simplicity of notation, in the appendix we use $B$ for the total
number of RBs so that it integrates $g_{\mathrm{group}}$. Before the detailed
of derivation of (10), we first rewrite it in a more intuitive manner:

$\begin{split}P_{\mathrm{PLR}}(U,K,B,n)&=\sum\limits_{i=1}^{U-1}f\left(U-1,\frac{2K-1}{B},i\right)\\\
&\times P_{K,n}(i),\\\ \end{split}$ (15)

$P_{K,n}(i)$ is probability of packet loss of a user given $i$ other users are
transmitting on nearby SCs in its narrow baseband $\mathcal{B}_{u}$.
Specifically, nearby SCs actually refer to the $\left(2K-1\right)$ SCs whose
narrow band segments have intersection with $\mathcal{B}_{u}$. Since the
user’s choice of $n$ SCs are completely random, $P_{K,n}(i)$ can be decomposed
in a straight forward manner:

$\displaystyle\begin{split}P_{K,n}(i)&=\sum\limits_{j=0}^{K}P(j\
\mathrm{among}\ K\ \mathrm{\acs{SC}s}\ \mathrm{\ occupied\ by\ }i\
\mathrm{users})\\\ &\times\mathrm{C}_{j}^{n}\left/\mathrm{C}_{K}^{n}\right.\\\
\end{split}$ (16)

Essentially, packet loss results in the case when all $n$ choices of the user
fall on occupied SCs. Therefore, once we know the probability distribution of
the number of occupied SCs, packet loss rate can be thoroughly calculated. The
strategy here is to characterize occupancy of the selected narrow band as a
Markov chain where each competing user causes transition of states in terms of
number of occupied SCs. Let $\mathbf{p}_{i}\in\mathbb{R}^{K+1}$ be the vector
characterizing state of the narrow band with $i$ competing users, i.e.,
$\left[\mathbf{p}_{i}\right]_{j}=P\left(j\ \mathrm{among}\ K\ \acs{SC}s\
\mathrm{\ occupied\ by\ }i\ \mathrm{users}\right)$. Correspondingly, let
$\mathbf{T}\in\mathbb{R}^{(K+1)\times(K+1)}$ be the transition matrix created
by a single competing users. Specifically,

$\displaystyle\begin{split}\left[\mathbf{T}\right]_{m_{1},m_{2}}&=P\left(\mathrm{new\
m_{2}-m_{1}\ occupied\ \acs{SC}s}\right)\\\
&=\sum\limits_{j=0}^{n}\sum\limits_{k=1}^{K}P\left(\mathrm{user\ occupies\
}\mathrm{band\ with}\ k\ \mathrm{overlap}\right)\\\ &\times
P_{k}\left(\mathrm{user\ adds}\ j\ \mathrm{repetitions\ in\ band}\right)\\\
&\times P\left(j\ \mathrm{repetitions\ add}\ m_{2}-m_{1}\ \mathrm{new}\
\mathrm{occupations}\right)\\\ \end{split}$ (17)

Since any segment of joint $K$ SCs can be chosen as a narrow band, competing
users can have different numbers of overlapping SCs with the user of interest.
As a consequence, for a competing user, its $n$ repetitions might fall only
partially on those overlapped SCs. Then, those repetitions (on overlapped SCs)
contributed by a competing user may or may not land on occupied SCs in
$\mathcal{B}_{u}$. All these probabilities are given by:

  * •

Overlapping configurations:

$\begin{split}P\left(\mathrm{user\ occupies\ }\mathrm{band\ with}\ k\
\mathrm{overlap}\right)&=\\\ \left\\{\begin{matrix}2/(2K-1),&1\leq k\leq
K-1\\\ 1/(2K-1),&k=K\\\ \end{matrix}\right.&\\\ \end{split}$

  * •

Probability that a user with a band that has $K$ overlapped blocks with
$\mathcal{B}_{u}$ contributes to $j$ repetitions:

$P_{k}\left(\mathrm{user\ adds}\ j\ \mathrm{repetitions\ in\
band}\right)=\frac{\mathrm{C}_{K-k}^{n-j}\mathrm{C}_{k}^{j}}{\mathrm{C}_{K}^{n}}$

  * •

Probability for newly added repetitions to cause transition:

$\begin{split}&P\left(j\ \mathrm{repetitions\ add}\ m_{2}-m_{1}\ \mathrm{new}\
\mathrm{occupations}\right)=\\\
&\qquad\frac{\mathrm{C}_{K-m_{1}}^{m_{2}-m_{1}}\mathrm{C}_{m_{1}}^{j-(m_{2}-m_{1})}}{\mathrm{C}_{K}^{j}}\\\
\end{split}$

This formula is essentially a generic approximation. To our best knowledge,
the exact formula here can not be tracked analytically. The main reason is
that probabilities for different configurations of $j$ repetitions in
$\mathcal{B}_{u}$ are not even. Competing users with a highly overlapped band
might generate some configurations that other users cannot. This means that
state of the band can not be fully characterized by the number of occupied SCs
alone.

Since a complete treatment is computationally infeasible, we here assume that
all configurations for $j$ repetitions are equivalent. This assumption works
well for small and medium value of $n$ and introduces deviation for $n$
approaching $K$.

With the transition matrix, calculation of $\mathbf{p}_{i}$ is relatively
simple:

$\displaystyle\mathbf{p}_{i}=\mathbf{T}^{i}\mathbf{p}_{0}$ (18)

And then:

$\displaystyle
P_{K,n}(i)=\sum\limits_{j=0}^{K}\left[\mathbf{T}^{i}\mathbf{p}_{0}\right]_{j}\mathrm{C}_{j}^{n}\left/\mathrm{C}_{K}^{n}\right.$
(19)

|  Ruifu Li received the B.Sc. degree from the University of Wisconsin-Madison
in 2020. He is currently working towards his M.S. degree in electrical and
computer engineering at University of California, Los Angeles. His research
interests include signal processing and wireless communications.
---|---
|  Han Yan received the B.E. degree from Zhejiang University, Hangzhou, China,
in 2013, and the M.S. and Ph.D. degrees in electrical and computer engineering
from UCLA in 2015 and 2020, respectively. He has broad research interests in
signal processing and communication system design for millimeter-wave mobile
networks, cooperative unmanned aerial vehicles networks, and dynamic spectrum
sharing radios. Dr. Yan was a recipient of the UCLA Dissertation Year
Fellowship in 2018, Qualcomm Innovation Fellowship in 2019, UCLA ECE
Distinguished Ph.D Dissertation Award in 2020, and best paper award at the
2020 ACM mmNets workshop.
---|---
|  Danijela Cabric is Professor in Electrical and Computer Engineering at
University of California, Los Angeles. She earned MS degree in Electrical
Engineering in 2001, UCLA and Ph.D. in Electrical Engineering in 2007, UC
Berkeley, Dr. Cabric received the Samueli Fellowship in 2008, the Okawa
Foundation Research Grant in 2009, Hellman Fellowship in 2012, the National
Science Foundation Faculty Early Career Development (CAREER) Award in 2012,
and the Qualcomm Faculty award in 2020 and 2021. She served as an Associate
Editor of IEEE Transactions of Cognitive Communications and Networking, IEEE
Transactions of Wireless Communications, IEEE Transactions on Mobile Computing
and IEEE Signal Processing Magazine, and IEEE ComSoc Distinguished Lecturer.
Her research interests are millimeter-wave communications, distributed
communications and sensing for Internet of Things, and machine learning for
wireless networks co-existence and security. She is an IEEE Fellow.
---|---
  *[IWSN]: industrial/intelligent wireless sensor networks
  *[eMBB]: enhanced mobile broadband
  *[URLLC]: ultra-reliable low-latency communication
  *[mMTC]: massive machine-type communication
  *[mmW]: millimeter-wave
  *[IoT]: internet of things
  *[BS]: base station
  *[UE]: user equipment
  *[RF]: radio frequency
  *[TTD]: True Time Delay
  *[OFDM]: orthogonal frequency-division multiplexing
  *[SC]: subcarrier
  *[DL]: downlink
  *[UL]: uplink
  *[CP]: cyclic prefix
  *[OFDMA]: orthogonal frequency-division multiple access
  *[RB]: resource block
  *[AoA]: angle of arrival
  *[AoD]: angle of departure
  *[FFT]: fast Fourier transform
  *[PN]: pseudorandom noise
  *[NB-IoT]: Narrowband Internet of Things
  *[ACK]: acknowledgement
  *[TTI]: transmission time interval
  *[PLR]: packet loss rate
  *[LoS]: line-of-sight
  *[SNR]: signal-to-noise ratio
  *[QAM]: quadrature amplitude modulation