# Over-the-Air Aggregation for Federated Learning: Waveform Superposition and
Prototype Validation

Huayan Guo, Yifan Zhu, Haoyu Ma, Vincent K. N. Lau,
Kaibin Huang, Xiaofan Li, Huabin Nong, and Mingyu Zhou
This work was supported in part by the Hong Kong Innovation and Technology
Fund (ITF) project GHP-016-18GD. The first three authors contributed equally.
H. Guo, Y. Zhu, H. Ma and V. K. N. Lau are with The Hong Kong University of
Science and Technology, Hong Kong (e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>eeknlau@ust.hk).K. Huang is
with The University of Hong Kong, Hong Kong (e-mail: huangkb@eee.hku.hk).X. Li
is with Jinan University, China (e-mail: lixiaofan@jnu.edu.cn).H. Nong is with
Shenzhen Institute of Radio Testing and Technology, China (e-mail:
nonghuabin@srtc.org.cn).M. Zhou is with Baicells Technologies Co. Ltd., China
(e-mail: 18611991947@163.com).

###### Abstract

In this paper, we develop an orthogonal-frequency-division-multiplexing
(OFDM)-based over-the-air (OTA) aggregation solution for wireless federated
learning (FL). In particular, the local gradients in massive IoT devices are
modulated by an analog waveform and are then transmitted using the same
wireless resources. To this end, achieving perfect waveform superposition is
the key challenge, which is difficult due to the existence of frame timing
offset (TO) and carrier frequency offset (CFO). In order to address these
issues, we propose a two-stage waveform pre-equalization technique with a
customized multiple access protocol that can estimate and then mitigate the TO
and CFO for the OTA aggregation. Based on the proposed solution, we develop a
hardware transceiver and application software to train a real-world FL task,
which learns a deep neural network to predict the received signal strength
with global positioning system information. Experiments verify that the
proposed OTA aggregation solution can achieve comparable performance to
offline learning procedures with high prediction accuracy.

###### Index Terms:

Over-the-air aggregation, federated learning, Internet of Things (IoT)

##  I Introduction

Wireless federated learning (FL) is an emerging technique that enables a deep
neural network (DNN) to be collaboratively trained by massive Internet of
Things (IoT) sensors with the coordination of an edge parameter server
connecting to an access point (AP), while the privacy-preserving raw dataset
is stored locally without exchange [1, 2, 3, 4, 5, 6]. In particular, the IoT
sensors iteratively update their local model weights/gradients locally based
on their own datasets according to a broadcast common global model, and then
synchronize a new global model via weights/gradients aggregation to the
parameter server. As a result, the uplink weights/gradients aggregation
becomes a primary bottleneck because massive sensors will be involved but the
radio resource is limited. Much work has been done in this area to reduce the
aggregation overhead based on conventional multiple access protocols that
allocate dedicated radio resource to difference sensors. In [7, 8, 9],
compression technology is utilized to alleviate the aggregation burden on
weights/gradients exchanges. In [10, 11, 12, 13, 14, 15, 16, 17], user
scheduling algorithms are designed to reduce the number of sensors
participating in every aggregation iteration. In essence, all these solutions
deal with the trade-offs between aggregation overhead and accuracy in FL.
Fixing the target learning accuracy, the aggregation overhead generally grows
proportionally to the number of sensors, which usually is very large, and thus
it still imposes a huge requirement on the radio resource.

Recently, a new technique named over-the-air (OTA) aggregation was proposed to
address this scalability issue based on the fact that the FL aggregation only
concerns the summation of the local weights/gradients instead of the
individual updates. Specifically, all sensors transmit the local updates with
analog uncoded modulation in one common radio resource block by exploiting the
free aggregation property in the wireless channels. As a result, the increase
of involved sensors can be beneficial with a proper design since the total
power of the aggregated signals increases. Most existing OTA-FL investigations
focus on user scheduling and power control to minimize the mean squared error
(MSE) of the aggregated weights/gradients, and the modification on the
learning algorithm to make sure the FL problem can converge to the same
optimum as the noise-free case. In [18, 19, 20, 21], a channel-inversion
transmitter is proposed to extract an unbiased estimator of the updated
weights/gradients. In [22, 23], time-varying precoding is proposed to mitigate
the noise by exploiting the non-stationarity of the gradient updates. In [24],
joint channel-and-data-aware user scheduling is proposed with dynamic residual
feedback to guarantee the training convergence. In [25], a hierarchical OTA-FL
framework is proposed for the vertical data partitioning. The authors in [26,
27] prove that the training may converge even without transmit power control
and user scheduling by designing a dynamic learning rate.

Existing works on the OTA technique all suppose perfect superposition of the
analog waveforms coming from different sensors since a similar idea has
already been verified and applied for modulation-free remote state estimations
[28, 29, 30, 31, 32] and the over-the-air fusion of sensor measurements [33,
34, 35]. In the existing prototype validation works [36, 37], a time-domain
waveform design is adopted, and the waveform misalignment is the key issue
since the sensors experience different multi-path propagations and also suffer
different frame timing offsets (TOs). The misaligned waveforms not only cause
superposition distortion but also result in inter-symbol interference. In
order to alleviate this issue, the direct-sequence spread spectrum (DSSS)
technique is utilized in [36, 37], which multiplies each analog data symbol by
a pseudo-random sequence with good correlation characteristics. By setting a
long pseudo-random sequence ($N$), a high-quality waveform superposition can
be achieved, but the transmission efficiency is reduced to $1/N$. Fortunately,
this efficiency loss is negligible for remote state estimations and sensor
measurements since in these scenarios the size of the data to be transmitted
is small. However, it becomes a critical challenge for OTA-FL since DNNs
usually contain millions of weights, which leads to a substantial increase in
the resource demand.

Figure 1: Illustration of the over-the-air aggregation federated learning
system. Sensors train the local NN with local dataset and upload the
gradients. The gradients are aggregated in a wireless channel with an analog
waveform and are received by the access point (AP). The aggregated gradients
are used to update the global NN in the server.

We propose an effective waveform superposition solution with prototype
validation for the OTA-FL by utilizing the $N$-point orthogonal frequency-
division multiplexing (OFDM) transmission. In contrast to [36, 37], the
waveforms from different sensors can be aligned automatically in the
frequency-domain sub-carriers, while the time-domain misalignment issue is
addressed by the $L$-length cyclic prefixes. Therefore, the transmission
efficiency is improved to $(N-L)/N$ (while $N\gg L$). However, a new critical
issue occurs in the frequency domain. Specifically, in OFDM transmission, the
TO and the carrier frequency offset (CFO) act as frequency-domain
multiplicative phase noise varying over the sub-carriers. Since the sensors
suffer different TOs and CFOs, the multiplicative phase noises are different
among sensors, leading to failure of the coherent waveform superposition in
the frequency domain. To address this issue, we further propose a new protocol
and algorithm to estimate and pre-compensate the phase noise with reasonable
pilot overheads. The following summarizes the critical issues to be addressed
and the contributions of the paper.

  * •

OFDM-based effective waveform superposition solution for OTA-FL: We propose a
novel waveform superposition solution for OTA-FL by utilizing the OFDM
modulation to address the waveform misalignment issue in existing works. By
exploiting the circular convolution property, the waveforms from different
sensors are aligned automatically in the frequency-domain sub-carriers with
the aid of time-domain cyclic prefixes. The transmission efficiency can be
greatly improved by setting a long OFDM symbol length.

  * •

Protocol and algorithm for phase error pre-compensation: After utilizing the
OFDM, the original time-domain symbol misalignment issue is transformed to the
phase noise effect in frequency domain. Inspired by broadband OFDM system
imperfection analysis [38], we further propose an efficient protocol to
estimate and to pre-compensate the phase noise by adding one more signaling
round before the consecutive OTA frames for data transmission. The additional
pilot overhead is scalable to the number of involved sensors and is also
negligible compared to the huge amount of data to be transmitted.

  * •

Real-world prototype design and experimental validation: We develop the first
real-world prototype to prove the concept of OTA-FL with the proposed waveform
superposition solution. The prototype is designed based on the Xilinx software
defined radio (SDR) solution, in which we have one AP and two IoT sensors. A
neural network (NN), which predicts the received signal strength (RSS) given
global positioning system (GPS) information, is trained. Experimental results
confirm that the OTA-FL training can be done successfully with the proposed
waveform superposition solution in practice.

##  II Signal Model and Transmission Frame

###  II-A Overview of the OTA-FL

In FL, $K$ sensors collaboratively train a global model, which minimizes the
training objective function, given as follows, with respect to the model
weight parameters $\bf w$:

$\displaystyle{\mathcal{P}}{\text{(A)}}\quad\min_{{\bf w}}$
$\displaystyle\sum_{k=1}^{K}\epsilon_{k}f({\bf w};{\cal D}_{k}),$

where $f(\cdot)$ denotes the learning model function, ${\cal D}_{k}$ denotes
the $k$-th user’s local dataset, and $\epsilon_{k}$ denotes the ratio of the
dataset size to the global dataset. In the $t$-th training round, the model
weight ${\bf w}_{t}$ is broadcast from the server to all the sensors. Each
sensor then calculates the local gradient:

${\bf g}_{k,t}=\epsilon_{k}\nabla f({\bf w}_{t};{\cal D}_{k}).$ (1)

Instead of updating ${\bf g}_{k,t}$ via the error-free channel by allocating
dedicated resource to each user, the AP aggregates all the gradients in a
common wireless resource, as shown in Fig. 1. If perfect waveform
superposition is achieved, the server may aggregate an unbiased estimation of
the true gradient by utilizing the channel-inversion transmitter:

$\displaystyle{\hat{\bf g}}_{t}=\sum_{k=1}^{K}{\bf g}_{k,t}+{\bf z}_{t},$ (2)

where ${\bf z}_{t}$ is the additive noise. Then, stochastic gradient decent
(SGD) is applied to update the model, which guarantees the training may
converge to a first-order optimum with a proper stepsize design:

${\bf w}_{t+1}={\bf w}_{t}-\eta_{t}{\hat{\bf g}}_{t},$ (3)

where $\eta_{t}$ is the stepsize.

Our key task is to effectively realize equation (2) in practice. In
particular, we need a symbol-level coherent superposition of the analog
waveforms from the sensors, which bears the information of local gradients
${\bf g}_{k,t}$ for all $k$ and $t$.

###  II-B Signal Model

Denote the digital baseband time-domain symbol by $x[m]$ for $m=1,2,\dots,M$.
The corresponding continuous transmitted signal
$x\left(t\right)=\sum_{m=1}^{M}x[m]g_{T}\left(t-mT_{s}\right)$ is constructed
by $\\{x[m]\\}$ with a pulse-shaping filter $g_{T}\left(t\right)$, where
$T_{s}$ denotes the baseband sampling period. For the conventional point-to-
point transmission, the received signal is given by

$\displaystyle r(t)=x(t)*h(t)e^{j2\pi\Delta ft}+w(t),$ (4)

where $*$ denotes the convolution operation,
$w\left(t\right)\sim\mathcal{CN}\left(0,\sigma^{2}\right)$ is the additive
white Gaussian noise, $\Delta f$ is the CFO, and $h(t)$ is the multipath
channel response, which is given by

$\displaystyle h(t)=\sum_{p=1}^{P}a_{p}\delta(t-\tau_{p}),$ (5)

where $a_{p}$ and $\tau_{p}$ is the channel gain and path delay of the $p$-th
path ($p=1,2,\dots,P$), respectively. Next, the received signal
$r\left(t\right)$ is down-sampled to discrete values with a TO (denoted by
$\Delta T$) due to the imperfection of the frame synchronization, yielding

$\displaystyle r[m]=$ $\displaystyle x(t)*h(t)\delta(t-mT_{s}-\Delta
T)e^{j2\pi\Delta ft}+w[m]$ $\displaystyle=$ $\displaystyle x(t)*h(t+\Delta
T)\delta(t-mT_{s})e^{j2\pi\Delta ft}+w[m].$ (6)

We further define the effective channel as follows:

$\displaystyle\bar{h}(t)=$ $\displaystyle h(t+\Delta T).$ (7)

For an $N$-point OFDM system, according to the Fourier transform relationship
between the delay domain and the frequency domain, the effective channel of
the $n$-th sub-carrier ($\tilde{h}[n]$) is given by

$\displaystyle\tilde{h}[n]$
$\displaystyle=\mathcal{F}\left[\bar{h}\left(t\right)\right]\delta\left(f-nf_{s}/N\right)$
$\displaystyle=e^{j2\pi nf_{s}\Delta T/N}\sum_{p}a_{p}e^{-j2\pi
nf_{s}\tau_{p}/N}$ $\displaystyle=e^{j2\pi nf_{s}\Delta T/N}\tilde{a}[n],$ (8)

where $\mathcal{F}\left[\cdot\right]$ denotes the Fourier transform,
$f_{s}=1/T_{s}$ is the baseband sampling rate, and $\tilde{a}[n]$ is the
multipath channel coefficient.

In the proposed prototype, the transmission frames consist of three basic sub-
frames.

####  II-B1 Sub-Frame for Frame Timing

The frame timing (FT) sub-frame is used to identify the starting point of a
frame, which is determined when a correlation peak is observed by the match
filter. In particular, the FT sequence adopts the up-sampled time-domain
differential encoded pseudo-random BPSK sequence as follows:

$\displaystyle x_{\text{FT}}\left[m+2\right]=x_{\text{FT}}\left[m\right]\oplus
q\left[m\right],$ (9)

for $m=1,\dots,M_{\text{FT}}$, where
$\text{mod}\left(M_{\text{FT}},2\right)=0$. Define
$x_{\text{FT}}\left[1\right]=x_{\text{FT}}\left[2\right]~{}=~{}1$, and
$\mathbf{q}=\left\\{q\left[1\right],q\left[1\right],\dots,q\left[\frac{M_{\text{FT}}}{2}\right],q\left[\frac{M_{\text{FT}}}{2}\right]\right\\}\in\\{-1,1\\}^{\left(M_{\text{FT}}\right)\times
1}$ is the pseudo-random BPSK sequence.

####  II-B2  Sub-Frame for CFO Estimation

The CFO estimation sub-frame is generated by choosing one active sub-carrier
whose index is $n_{\text{CFO}}$:

$\displaystyle{x}_{\text{CFO}}\left[m\right]=\tilde{x}\left[n_{\text{CFO}}\right]e^{j2\pi
mn_{\text{CFO}}/N},~{}m=1,\dots,M_{\text{CFO}},$ (10)

where $M_{\text{CFO}}>N$ is the sequence length, and
$\tilde{x}\left[n_{\text{CFO}}\right]\in\mathbb{R}$ is the frequency-domain
pilot symbol.

####  II-B3 OFDM sequence

In this prototype, we adopt OFDM for both data transmission and OTA
aggregation. Suppose that we have already achieved a successful coarse CFO
compensation, while the residual CFO is given by $\Delta f_{r}=\Delta
f-\Delta{\hat{f}}$, where $\Delta{\hat{f}}$ is the estimated CFO. When $\kappa
N\Delta f_{r}T_{s}\ll 1$ where $\kappa$ is the number of OFDM symbols in one
frame, $\Delta f_{r}$ can be ignored. In particular, denote the symbol in the
$n$-th sub-carrier by $\tilde{s}\left[n\right]$. The received signal in the
$n$-th sub-carrier is approximated by

$\displaystyle\tilde{r}\left[n\right]=$ $\displaystyle e^{j2\pi\Delta
f_{r}t_{0}}\tilde{h}\left[n\right]\tilde{x}\left[n\right]+\tilde{w}\left[n\right],$
(11)

where $t_{0}$ is the time stamp of this received symbol, and $e^{j2\pi\Delta
f_{r}t_{0}}$ is the difference between $t=t_{0}$ and $t=0$ in the effective
channel.

###  II-C Phase Noise Issue for the Proposed OFDM-based OTA Aggregation

In order to realize the analog waveform superposition, a channel-inversion
transmitter is suggested in [18, 19, 20, 21] to pre-equalize the channel
coefficients of different sensors in each sub-carrier based on the downlink
estimated channel. Combining (II-B) with the residual CFO effect, the
effective downlink channel of the $k$-th sensor at time $t_{\text{DL}}^{(k)}$
is given by

$\displaystyle\tilde{h}_{\text{DL}}^{(k)}[n]=e^{j2\pi(\Delta
f_{r}^{(k)}t_{\text{DL}}^{(k)}+nf_{s}\Delta
T_{\text{DL}}^{(k)}/N)}\tilde{a}_{k}[n],$ (12)

where $\Delta f_{r}^{(k)}$ is the residual CFO of the $k$-th sensor and
$\Delta T_{\text{DL}}^{(k)}$ is the downlink TO. Note that the residual CFO of
the uplink channel is opposite to that of the downlink channel. Thus, the
effective uplink channel of the $k$-th sensor at time $t_{\text{UL}}^{(k)}$ is
given by

$\displaystyle\tilde{h}_{\text{UL}}^{(k)}[n]=e^{j2\pi(-\Delta
f_{r}^{(k)}t_{\text{UL}}^{(k)}+nf_{s}\Delta
T_{\text{UL}}^{(k)}/N)}\tilde{a}_{k}[n],$ (13)

where the $\Delta T_{\text{UL}}^{(k)}$ is the uplink TO. After pre-
equalization, the uplink received aggregated signal of the $n$-th sub-carrier
at the AP is given by

$\displaystyle\tilde{r}_{\text{OTA}}[n]=$
$\displaystyle\sum_{k}\tilde{h}_{\text{UL}}^{(k)}[n]\frac{\tilde{x}_{k}[n]}{\tilde{h}_{\text{DL}}^{(k)}[n]}+\tilde{w}[n]$
$\displaystyle=$ $\displaystyle\sum_{k}\underbrace{e^{j(\phi_{k}+2\pi
nf_{s}\tau_{k}/N)}}_{\text{effective OTA
channel}}\tilde{x}_{k}[n]+\tilde{w}[n],$ (14)

where $\phi_{k}=-2\pi\Delta
f_{r}^{(k)}(t_{\text{DL}}^{(k)}+t_{\text{UL}}^{(k)})$, and $\tau_{k}=\Delta
T_{\text{UL}}^{(k)}-\Delta T_{\text{DL}}^{(k)}$. It is observed that the OTA
waveform superposition is invalid due to the non-identical effective OTA
aggregation channel, which is caused by two issues, as shown in Fig. 2. First,
there is an unknown phase error $\phi_{k}$ due to the existence of the
residual CFO. Second, the imperfection of the frame timing causes a timing
offset difference $\tau_{k}$ in OTA computation for the $k$-th sensor, which
causes imperfect compensation of the channel response. Therefore, the $k$-th
sensor requires $\phi_{k}$ and $\tau_{k}$ to perform a proper pre-equalization
for OTA aggregation. Note that $\phi_{k}$ changes with $t_{\text{UL}}^{(k)}$
and $t_{\text{DL}}^{(k)}$, and $\tau_{k}$ may be different with a $\pm 1$
sample between two transmissions due to the synchronization offset.

Figure 2: Phase response of the effective OTA aggregation channel of $k$-th
sensor.

###  II-D Basic Transmission Frames

Figure 3: Three basic transmission frame structure. (a) is the Initialization
preamble frame, (b) is the digital transmission frame, and (c) is the OTA
frame.

In the proposed prototype, there are three types of transmission frame:

  * •

Preamble for initialization: The frame structure of the initialization
preamble is shown in Fig. 3(a). The initialization preamble consists of two
sub-frames: 1) the FT sub-frame, and 2) the CFO estimation sub-frame. In the
initialization preamble we use a long CFO sequence to perform CFO estimation
and adjustment at the beginning of the OTA procedure. In our prototype, the
length of the CFO in the preamble is $10^{6}$ (i.e., 65.1 ms with a 15.36 MHz
digital baseband sampling rate).

  * •

Digital transmission frame: The frame structure of the digital transmission
frame is shown in Fig. 3(b). Compared to the preamble for initialization, the
CFO sub-frame in the digital transmission frame is shorter, which is set to
the length of two OFDM symbols. In addition, we adopt an orthogonal pilot for
channel estimation, which consists of $K$ OFDM symbols. The orthogonal pilot
OFDM symbol contains 256 sub-carriers with a 32-length cyclic prefix, and it
adopts 4-QAM modulation. Data symbols are followed with orthogonal pilots,
which adopt 16-QAM.

  * •

OTA aggregation frame: The frame structure of the OTA aggregation frame is
shown in Fig. 3(c). In contrast to the digital transmission frame, the OTA
aggregation frame does not have a CFO sub-frame. The OTA aggregation frame
uses one common OFDM symbol for all sensors, and the analog data is modulated
with pulse amplitude modulation in each OFDM subcarrier.

##  III Protocol and Algorithms for the Waveform Superposition

In this section, we address the issues caused by the TO and the CFO that lead
to the failure of OTA aggregation by proposing a new MAC protocol and the
corresponding algorithms.

###  III-A Brief Introduction to the Proposed Protocol

Figure 4: Illustration of the proposed physical layer OTA protocol.

In order to estimate $\phi_{k}$ and $\tau_{k}$, we proposed a physical layer
protocol to jointly estimate and compensate the TO and CFO for the OTA
aggregation procedure. As shown in Fig. 4, the OTA aggregation physical layer
protocol can be considered as a two-stage handshake procedure. The first stage
is the pre-equlization stage, which compensates the initial CFO effect and
multi-path fading. The second stage is the online OTA aggregation stage,
consisting of multiple OTA aggregation sub-stages in which the instantaneous
effect of the TO and residual CFO are tracked and compensated. The details of
each step of the protocol are given as follows.

  * •

Pre-equlization Stage (Downlink): The AP broadcasts a digital transmission
frame to trigger the sensors. When the sensor $k$ receives the frame, it
starts a timer and record the current time $t^{(k)}_{\text{DL},0}$. Then, the
effective downlink channel $\tilde{h}_{\text{DL},0}^{(k)}$ is estimated with
pilot OFDM symbols.

  * •

Pre-equlization Stage (Uplink): The sensors feed back a pre-equalized digital
transmission frame in (II-C) as acknowledgment, and records the current time
$t^{(k)}_{\text{UL},0}$. With the orthogonal pilots, the AP estimates the
effective OTA aggregation channel $\\{\tilde{h}_{\text{OTA},0}^{(k)}\\}$ in
(II-C) to obtain the phase error $\\{\phi_{k,0}\\}$ and the timing offset
$\\{\tau_{k,0}\\}$ in this stage for each sensor.

  * •

Online OTA Aggregation Stage (Downlink): The AP broadcasts a digital
transmission frame to request the OTA aggregated data. Each sensor records the
time $t^{(k)}_{\text{DL},i}$ when receiving the frame, and estimates the
effective downlink channel $\tilde{h}_{\text{DL},i}^{(k)}$. Note that in the
$1$-st OTA aggregation operation, $\\{\phi_{k,0}\\}$ and $\\{\tau_{k,0}\\}$
are broadcast to the sensors by the AP.

  * •

Online OTA Aggregation Stage (Uplink): The timer records the current time
$t^{(k)}_{\text{UL},i}$. The downlink and uplink period between the previous
and the current OTA aggregation sub-stage, $\Delta
t_{\text{DL},i}^{(k)}=t^{(k)}_{\text{DL},i}-t^{(k)}_{\text{DL},i-1}$ and
$\Delta t_{\text{UL},i}^{(k)}=t^{(k)}_{\text{UL},i}-t^{(k)}_{\text{UL},i-1}$,
are calculated. Each user estimates $\phi_{k,i}$ and $\tau_{k,i}$ according to
the effective downlink channel $\tilde{h}_{\text{DL},i}^{(k)}$, the downlink
period $\Delta t_{\text{DL},i}^{(k)}$, and the uplink period $\Delta
t_{\text{UL},i}^{(k)}$. Then the OTA aggregation frame is transmitted after
pre-equalization with estimated $\phi_{k,i}$ and $\tau_{k,i}$ simultaneously,
which is shown in Section III-B.

###  III-B Phase Error Estimation and Compensation

In the proposed protocol, the OTA pre-equalization requires the estimated
phase noise $\\{\hat{\phi}_{k}\\}$ and the estimated timing offset
$\\{\hat{\tau}_{k}\\}$. Specifically, $\\{\hat{\phi}_{k,0}\\}$ and
$\\{\hat{\tau}_{k,0}\\}$ are estimated in the pre-equlization stage at the AP
side. As illustrated in Section II-C, $\hat{\phi}_{k}$ changes with
$t_{\text{DL}}$ and $t_{\text{UL}}$, such that in the $i$-th OTA aggregation
sub-stage uplink step it changes to $\phi_{k,i}=e^{-j2\pi\Delta
f_{r}^{(k)}(\Delta t_{\text{DL},i}^{(k)}+\Delta
t_{\text{UL},i}^{(k)})}\phi_{k,i-1}$. As a result, the residual CFO $\Delta
f_{r}^{(k)}$ needs to be estimated to perform correct OTA aggregation. We
first introduce the estimation method for the desired variables. The
estimation of the channel $\tilde{h}$ is denoted by $\hat{h}$.

####  III-B1 Estimation of $\\{\phi_{k,0}\\}$ at the AP

As shown in Fig. 2, $\phi_{k,0}$ reflects on the intercept of the effective
OTA aggregation channel phase response in the pre-equlization stage. Thus, it
can be estimated through the average phase of the effective OTA aggregation
channel as

$\displaystyle\hat{\phi}_{k,0}=\frac{1}{N-2}\sum_{n=1}^{N/2-1}(\angle\hat{h}_{\text{OTA},0}^{(k)}[-n]+\angle\hat{h}_{\text{OTA},0}^{(k)}[n]),$
(15)

where we restrict
$\angle\hat{h}_{\text{OTA},0}^{(k)}[n+1]-\angle\hat{h}_{\text{OTA},0}^{(k)}[n]\in(-\pi,\pi]$
to avoid the phase ambiguity.

####  III-B2 Estimation of $\\{\tau_{k,0}\\}$ at the AP

As shown in Fig. 2, $\tau_{k,0}$ reflects on the slope of the effective OTA
channel phase response in the pre-equlization stage. Thus, it can be estimated
by taking the average of the phase increment of the effective OTA aggregation
channel between the adjacent sub-carriers:

$\displaystyle\hat{\tau}_{k,0}=\frac{N}{2\pi
f_{s}}\angle\sum_{n=-N/2}^{N/2-2}\hat{h}_{\text{OTA},0}^{(k)*}[n]\cdot\hat{h}_{\text{OTA},0}^{(k)}[n+1].$
(16)

####  III-B3 Estimation of $\Delta f_{r}^{(k)}$ at the $k$-th Sensor

From (12), the relationship between the two effective downlink channels
$\tilde{h}_{\text{DL},i-1}^{(k)}$ and $\tilde{h}_{\text{DL},i}^{(k)}$ is given
by

$\displaystyle\tilde{h}_{\text{DL},i}^{(k)}[n]=e^{j2\pi\Delta
f_{r}^{(k)}\Delta t_{\text{DL}}^{(k)}}\tilde{h}_{\text{DL},i-1}^{(k)}[n].$
(17)

So the residual CFO $\Delta f_{r}^{(k)}$ can be estimated by

$\displaystyle\Delta\hat{f}_{r}^{(k)}=\frac{1}{2\pi\Delta
t_{\text{DL},i}^{(k)}}\angle\sum_{n=-N/2}^{N/2-1}\hat{h}_{\text{DL},i-1}^{(k)*}[n]\cdot\hat{h}_{\text{DL},i}^{(k)}[n].$
(18)

Note that when $|\Delta f_{r}^{(k)}\Delta t_{\text{DL},i}^{(k)}|\geq 1/2$,
there will be phase ambiguity, causing the error of $\Delta\hat{f}_{r}^{(k)}$.
So the residual CFO must satisfy

$\displaystyle|\Delta f_{r}^{(k)}|<\frac{1}{2\Delta t_{\text{DL},i}^{(k)}}.$
(19)

####  III-B4 Phase Noise Compensation in the OTA Aggregation Stage

With the above estimation results, the pre-equalization in the OTA aggregation
stage can be designed. First, $\hat{\phi}_{k,i}$ is given by

$\displaystyle\hat{\phi}_{k,i}=e^{-j2\pi\Delta\hat{f}_{r}^{(k)}(\Delta
t_{\text{DL},i}^{(k)}+\Delta t_{\text{UL},i}^{(k)})}\hat{\phi}_{k,i-1}.$ (20)

Then, the $\pm 1$ sample difference between $\hat{\tau}_{k,i}$ and
$\hat{\tau}_{k,i-1}$ can be obtained by comparing
$\tilde{h}_{\text{DL},i}^{(k)}$ and $\tilde{h}_{\text{DL},i-1}^{(k)}$.
Finally, the pre-equalized symbol at sensor $k$ in the $i$-th OTA aggregation
operation is given by

$\displaystyle\tilde{x}_{k,e}[n]=\frac{1}{e^{j(\hat{\phi}_{k,i}+2\pi
nf_{s}\hat{\tau}_{k,i}/N)}\hat{h}_{\text{DL},i}^{(k)}[n]}\tilde{x}_{k}[n],$
(21)

Combining (II-C), (17) and (21), the aggregated signal at the AP is given by

$\displaystyle\tilde{r}_{\text{OTA}}[n]=$
$\displaystyle\sum_{k}\tilde{h}_{\text{UL},i}^{(k)}[n]\tilde{x}_{k,e}[n]+\tilde{w}[n]$
$\displaystyle=$ $\displaystyle\sum_{k}\tilde{x}_{k}[n]+\tilde{w}[n],$ (22)

which satisfies the requirements of waveform superposition for the OTA
aggregation procedure.

###  III-C Carrier Frequency Offset Estimation

Effectiveness of the proposed protocol for waveform superposition requires
residual CFO $|\Delta f_{r}^{(k)}|<1/(2\Delta t_{\text{DL},i}^{(k)})$. Thus,
CFO compensation and residual CFO tracking is critical. To address this issue,
we propose a two-step method to compensate the CFO for the OTA aggregation
procedure: 1) initialization of coarse CFO correction, and 2) residual CFO
tracking. Specifically, initialization of CFO correction is performed
immediately after the prototype enters the OTA aggregation state, and residual
CFO tracking is performed to monitor the value of the residual CFO.

####  III-C1 Coarse CFO Estimation

At the system startup stage, an initialization of CFO correction is performed
to guarantee the residual CFO is within the requirement of the system.
Specifically, the CFO estimation is performed with the initialization preamble
with a maximum likelihood estimator. Given the received preamble
$r_{\text{Init}}\left[m\right]$, the estimated coarse CFO $\Delta\hat{f}_{c}$
is given by [39]

$\displaystyle\Delta\hat{f}_{c}=\frac{1}{2\pi
T_{s}L_{\text{SPAN}}}\angle\sum_{m=1}^{M_{\text{CFO}}}r_{\text{Init}}^{*}\left[m\right]r_{\text{Init}}\left[m+L_{\text{SPAN}}\right],$
(23)

where $T_{s}$ is the baseband sampling period, and $L_{\text{SPAN}}$ is the
length of the span, which satisfies that $\angle
r^{*}_{\text{Init}}\left[m+L_{\text{SPAN}}\right]{r}_{\text{Init}}\left[m\right]=0$.
As such, the coarse CFO estimation range is
$\Delta\hat{f}_{c}\in\left(-\frac{1}{2T_{s}L_{\text{SPAN}}},\frac{1}{2T_{s}L_{\text{SPAN}}}\right]$.

####  III-C2 CFO Tracking

The residual CFO is tracked to ensure that it satisfies the requirement of the
OTA aggregation protocol since the CFO usually varies slowly. Given the
received signal with residual CFO $\Delta f_{r}$ by (13), and that the channel
is unchanged during the OTA aggregation procedure, the received signal in the
$n$-th subcarrier at time $\text{t}_{p}$ is denoted as
$\tilde{r}\left[n\right]\left(\text{t}_{p}\right)$, which can be obtained by

$\displaystyle\tilde{r}\left[n\right]\left(\text{t}_{p}\right)=$
$\displaystyle e^{j2\pi\Delta
f_{r}\text{t}_{p}}\tilde{h}\left[n\right]\tilde{x}\left[n\right]+\tilde{w}\left[n\right].$
(24)

The estimated residual CFO $\Delta\hat{f}_{r}$ can be obtained by periodically
sending an identical pilot frame. Specifically, for $N$-point OFDM symbols,
the $\Delta\hat{f}_{r}$ at time $\text{t}_{p}$ is given by

$\displaystyle\Delta\hat{f}_{r}\left(\text{t}_{p}\right)$
$\displaystyle=\frac{1}{\left(\text{t}_{p}-\text{t}_{p-1}\right)N}\sum_{n=1}^{N}\angle\frac{\tilde{r}\left[n\right]\left(\text{t}_{p}\right)}{\tilde{r}\left[n\right]\left(\text{t}_{p-1}\right)}$
$\displaystyle=\Delta f_{r}+\phi_{w},~{}p=1,\dots,P,$ (25)

where $\phi_{w}\sim\mathcal{N}\left(0,\sigma^{2}_{\phi_{w}}\right)$ is the
phase noise caused by additive Gaussian noise. Given (14), the sequential
estimation of the CFO is denoted as
$\Delta\bar{f}_{r}\left(\text{t}_{p}\right)$, which is given by

$\displaystyle\Delta\bar{f}_{r}\left(\text{t}_{p}\right)=\frac{p-1}{p}\Delta\bar{f}_{r}\left(\text{t}_{p-1}\right)+\frac{1}{p}\Delta\hat{f}_{r}\left(\text{t}_{p}\right),~{}\forall
p.$ (26)

##  IV Developing Key Modules

Figure 5: Structure of the prototype system. There are two parts in the
system: 1) the PS ARM part, which runs the software application and driver,
and 2) the PL ARM part, which runs the physical layer transceiver modules. The
data exchange between PS-PL is handled by an AXI bus.

As shown in Fig. 5, the prototype system consists of two major parts: 1) the
programmable software (PS) ARM, and 2) the programmable logic (PL) FPGA. The
key signal processing modules are designed in PL, and these modules are
controlled by PS. The data exchange between PS-PL is performed with a physical
AXI bus.

###  IV-A Control Logic Designs in the ARM Platform

Figure 6: The state machine (SM) for control logic in the prototype. The
system initialization state is the IDLE state, and the state transition
depends on last state and input signals. The SM is divided into two parts:
part A is for the OTA and part B is for digital communication.

The control logic for the transceiver is implemented as a state machine (SM),
which is shown in Fig. 6. As the figure shows, each block represents a state,
and the state transition depends on the previous state and current input
signals. The SM can be divided into two parts. In part A, the SM provides the
control logic for OTA aggregation operation, and in part B, it provides the
control logic for digital communication. The SM part A is implemented
according to the proposed protocol.

###  IV-B Transceiver Module Designs in the FPGA

Figure 7: Physical layer transceiver structure. The upper link is the receiver
channel and the bottom link is the transmitter channel. Two enable signals are
controlled by the SM control logic for either digital communication purposes
or OTA purposes.

The proposed prototype performs real-time signal processing in the PL FPGA
with a customized transceiver module. The structure of the transceiver module
is shown in Fig. 7. As shown in the figure, the transceiver consists of a
transmitter chain and a receiver chain consisting of four key sub-modules: 1)
the frame detection and timing module, 2) the CFO estimation and compensation
module, 3) the channel estimation and compensation module, and 4) the data
pre-equalization module.

####  IV-B1 Frame Detection and Timing

Figure 8: Performance of frame timing. (a) The probability of correct
synchronization when a valid correlation peak is detected, and (b) The
probability of the correlation peak detection.

The frame detection and timing module finds the start point of a received
frame, which is realized with a differential decoder and correlator. To be
specific, the received samples are first differentially decoded, and then
cross correlation is performed with the known PR BPSK sequence as

$\displaystyle\hat{q}\left[m\right]$
$\displaystyle=\text{sign}\left(r\left[m\right]r^{*}\left[m+2\right]\right),$
(27) $\displaystyle\text{Corr}\left[m^{\prime}\right]$
$\displaystyle=\sum_{m=-\infty}^{\infty}\hat{q}\left[m\right]q\left[m-m^{\prime}\right],$
(28)

where $\text{Corr}\left[m^{\prime}\right]$ is the output result of the
correlation sequence. Under a noise-free channel, the maximum correlation
output value is equal to $M_{\text{FT}}-2$. The start point of a frame $m_{0}$
is obtained as

$\displaystyle
m_{0}=\text{arg}\max_{m^{\prime}}\left|\text{Corr}\left[m^{\prime}\right]\right|-M_{\text{FT}}+1.$
(29)

A threshold $\gamma_{th}$ is defined so that only
$\text{Corr}\left[m_{0}\right]\geq\gamma_{th}$ is denoted as a valid
correlation output, and we set $\gamma_{th}=\frac{M_{\text{FT}}-2}{2}$. As
shown in Fig. 8, a longer FT sequence achieves a higher probability of correct
frame timing and a higher probability of miss detection.

####  IV-B2 Carrier Frequency Offset Estimation and Correction

Figure 9: The CFO estimation performance. Coarse CFO estimation sequence
length is $10^{6}$. (a) The residual CFO after coarse CFO estimation and
compensation, and (b) The NMSE of residual CFO estimation with SNR = 0 dB.

After the start point of the received frame, the coarse CFO estimation
sequence can be obtained. As introduced in Section III-C, the coarse CFO
estimation is performed with a maximum likelihood estimator. The estimated
coarse CFO is compensated for both digital transmission and the OTA procedure.
To evaluate the performance of the coarse CFO estimation, we define the
normalized mean square error (NMSE) of the estimated coarse CFO as

$\displaystyle\text{NMSE}_{\text{CFO}}=\frac{\|\Delta\hat{f}_{c}-\Delta
f_{c}\|^{2}_{2}}{\|\Delta f_{c}\|^{2}_{2}}.$ (30)

As shown in Fig. 9(a) the residual CFO after coarse CFO estimation and
compensation decreases with an increasing SNR. With a SNR over 0 dB, the
residual CFO after coarse CFO compensation is within 10 Hz, while the
requirement of maximum residual CFO in our prototype is 500 Hz.

After the coarse CFO is estimated and compensated, the residual CFO is
estiamted and tracked sequentially by using the algorithm in Section III-C. As
shown in Fig. 9(b), the NMSE of $\Delta\bar{f}_{r}\left(\text{t}_{p}\right)$
is decreasing with respect to the number of estimations. Denoting the
sequential estimation of the residual CFO at time stamp $t_{p}$ as
$\Delta\bar{f}_{r}\left(\text{t}_{p}\right)$, the estimation period is given
as $\Delta\text{t}=\text{t}_{p}-\text{t}_{p-1},\forall p$. The system will
request a new coarse CFO correction if $\Delta\text{t}\Delta\bar{f}_{r}>1/2$.

####  IV-B3 Channel Estimation and OFDM Symbol Reconstruction

Figure 10: Constellation of the received OFDM symbols before and after channel
equalization. The received SNR is 30 dB. (a) The channel response is not
compensated, and (b) The channel response is estimated and compensated.

The channel estimation is performed with a least-square estimator in each sub-
carrier. The received pilot OFDM symbols are obtained with DFT as
$\tilde{\mathbf{r}}_{\text{pilot}}^{{}^{\prime}}=\text{DFT}\left(\mathbf{r}_{\text{pilot}}^{{}^{\prime}}\right)$.
Thus, the estimated channel coefficients in the $n$-th sub-carrier
$\hat{h}\left[n\right]$ is given by

$\displaystyle\hat{h}\left[n\right]=\frac{\tilde{r}_{\text{pilot}}^{{}^{\prime}}\left[n\right]}{\tilde{x}_{\text{pilot}}\left[n\right]},~{}n=1,\dots,N.$
(31)

The OFDM symbols $\hat{\mathbf{x}}_{\text{data}}$ are reconstructed with
estimated channel coefficients as
$\hat{x}_{\text{data}}\left[n\right]=\hat{h}^{-1}\left[n\right]\tilde{r}_{\text{data}}^{{}^{\prime}}\left[n\right],~{}n=1,\dots,N$.
As shown in Fig. 10, after the channel estimation and compensation, the
received 16-QAM OFDM symbols can be decoded correctly.

####  IV-B4 Data pre-equalization module

Figure 11: NMSE evaluation for $\mathcal{D}_{\text{OTA}}$. There are 200
measurements in this experiment and we evaluate the measruement results
with/without TO and CFO compensation. (a) shows the NMSE of each measurement
result, and (b) shows the cumulative probability function of the NMSE. It can
be seen that 90% of measurements can obtain an NMSE of less than 0.01 by using
the proposed protocol.

The data pre-equalization module is for OTA aggregation purposes, which
follows the method in Section III-B. In order to verify the performance of the
proposed OTA aggregation protocol and transceiver, we perform an ‘A+B’ test
for two random number sequences by using the OTA aggregation approach. In
particular, two IoT sensors transmit two independent identical distributed
(i.i.d.) random number sequences $\mathbf{d}_{A}$ and $\mathbf{d}_{B}$, and
the AP obtains the result $\mathbf{d}_{\text{OTA}}$ with the OTA aggregation
approach. Denoting the true result of $\mathbf{d}_{A}+\mathbf{d}_{B}$ as
$\mathbf{d}_{\text{true}}$, we evaluate the NMSE of the OTA aggregation
result, which is given by

$\displaystyle\text{NMSE}_{d}=\frac{\|\mathbf{d}_{\text{OTA}}-\mathbf{d}_{\text{true}}\|^{2}_{2}}{\|\mathbf{d}_{\text{true}}\|^{2}_{2}}.$
(32)

As shown in Fig. 11, with the proposed protocol, the NMSE for OTA aggregation
results improved significantly. The NMSE of all the results among 200 sets of
measurements are less than 0.05, and this shows that 90% of the results have
an NMSE of less than 0.01.

##  V Prototype for OTA FL

With the proposed OTA aggregation protocol and the transceiver module, we can
design the prototype for OTA-FL. The proposed prototype consists of the
hardware platform and software applications. The hardware platform is used for
implementation of the proposed transceiver and OTA aggregation protocol, and
software applications are designed to perform the FL operation. In this
section, we introduce the design of the proposed prototype with one AP and two
sensors to learn an NN federatively for a toy task.

###  V-A Brief Introduction to the Design

Figure 12: System Setup. There are two sensors and one AP. Each sensor uses a
ZC706 FPGA for baseband signal processing and the AP uses a ZCU102 FPGA for
baseband signal processing. The Radio front end is AD9361 and we use a
circulator between transmitter and receiver for sharing antenna. Figure 13:
Application layer protocol for proposed OTA prototype. Communication between
host PC and AP uses TCP/IP.

The structure of the hardware platform is shown in Fig. 13. As shown in the
figure, the platform consists of an RF front-end (RFE) and a system-on-chip
(SoC) baseband processing unit. In the setup, the AD9361 is adopted as the
REF, the AP uses a Xilinx-ZCU102 SoC, and IoT sensors use a Xilinx-ZC706 SoC.
The system parameters are listed as follows:

  * •

The digital baseband sampling rate is 15.36 MHz;

  * •

The RF baseband sampling rate is 30.72 MHz;

  * •

The carrier frequency is 2.72 GHz, and the RF bandwidth is 40 MHz.

The software application for the FL operation follows an application layer
protocol, which is provided in Fig. 13. We summarize the application layer
protocol as follows.

  * •

Step 1: The host PC initializes the model weights, and sends the weights to
the AP. The host PC starts the FL training operation;

  * •

Step 2: The AP broadcasts the model weights to the sensors;

  * •

Step 3: The sensors perform local training, and perform OTA aggregation for
the gradients;

  * •

Step 4: The AP receives the OTA aggregation results and updates the model
weights;

  * •

Step 5: Return to Step 2 and repeat until convergence;

  * •

Step 6: The AP reports the final model weights to the host PC at the end of
the training.

###  V-B FL problem to be learnt

Figure 14: The RSS coverage map of the AP. The red markers denote the APs site
position. We consider 2 AP sites and take 2000 RSS measurements within a
circle with a 400 m radius from a center between two APs.

In order to verify the performance of the proposed prototype, we promote an FL
problem by training a fully connected NN that predicts the RSS given the GPS
information (latitude and longitude) of a sensor. As shown in Fig. 14, we set
the position of two AP sites (red markers) and generate the coverage map with
the MATLAB communication toolbox. With the radio coverage map, two sensors are
randomly placed within a circular area with 400 meter radius with a center
between two AP sites. The sensors take 2000 RSS measurements as training data.
Note that the RSS that is closer than 20 meters to the AP sites is not
measured. The prediction NN is trained with an OTA aggregation approach by
using the proposed prototype. The structure of the NN is shown in Table. I.

TABLE I: NN Structure for RSS Predition Operation Layer | Size of output
---|---
Input GPS | $2$
Fc 1 | Fully connected | $20$
ReLU | $20$
Fc 2 | Fully connected | $20$
ReLU | $20$
Fc 3 | Fully connected | $20$
Predicted RSS | $1$

###  V-C Finally testing

In this subsection, we use the proposed prototype to solve the promoted FL
problem with the OTA method. To evaluate the performance of the proposed
prototype, we show the training loss to demonstrate the convergence, and we
compare the predicted RSS with the ground truth. The training result is shown
in Fig. 15. We can see that our proposed OTA aggregation solution (the blue
curve) achieves a similar convergence speed to the offline training (the red
curve), in which the local gradients of the two sensors are aggregated
noiselessly. The normalized squared predication error on the test set is shown
in Fig. 16.

Figure 15: The training performance of the FL problem. The learning rate is
set by $\eta_{t}=\frac{2}{2000+t}$. Each sensor stores 1000 pieces of training
data. In each round, each sensor randomly picks 200 samples to calculate the
local gradient. Figure 16: The heatmap of the prediction error. The NMSE of
most positions is below $0.005$ with the exception of the positions that are
close to AP sites due to the lack of training data.

##  VI Conclusion

In this paper, we developed an OFDM-based over-the-air aggregation solution
for a real-world federated learning task. Specifically, we analyzed the effect
of frame timing offset and carrier frequency offset in the over-the-air
aggregation channel, and proposed a two-stage waveform pre-equalization
technique with a customized multiple access protocol to estimate and mitigate
the timing offset and carrier frequency offset for the over-the-air
aggregation. Based on the proposed protocol, we developed the prototype with a
hardware transceiver and corresponding application software to train a deep
neural network that predicts the radio signal strength with global positioning
system information. To verify the performance of the proposed prototype, we
performed experimental measurement and compared the learning results of over-
the-air aggregation with offline learning results. From the experimental
results, we can see that the proposed OFDM-based over-the-air aggregation
prototype is capable for real-world federated learning tasks.

## References

  * [1] J. Konec̆ný, B. McMahan, and D. Ramage, “Federated optimization: Distributed optimization beyond the datacenter,” _arXiv preprint arXiv:1511.03575_ , 2015.
  * [2] J. Konec̆ný, H. B. McMahan, D. Ramage, and P. Richt¨¢rik, “Federated optimization: Distributed machine learning for on-device intelligence,” _arXiv preprint arXiv:1610.02527_ , 2016.
  * [3] H. B. McMahan, E. Moore, D. Ramage, and S. Hampson, “Communication-efficient learning of deep networks from decentralized data,” _arXiv preprint arXiv:1602.05629_ , 2016.
  * [4] P. Kairouz, H. B. McMahan, B. Avent, A. Bellet, M. Bennis, A. N. Bhagoji, K. Bonawitz, Z. Charles, G. Cormode, and R. Cummings, “Advances and open problems in federated learning,” _arXiv preprint arXiv:1912.04977_ , 2019\.
  * [5] T. Li, A. K. Sahu, A. Talwalkar, and V. Smith, “Federated learning: Challenges, methods, and future directions,” _IEEE Signal Process. Mag._ , vol. 37, no. 3, pp. 50–60, 2020.
  * [6] Q. Yang, Y. Liu, T. Chen, and Y. Tong, “Federated machine learning: Concept and applications,” _ACM Trans. Intel. Syst. Tec. (TIST)_ , vol. 10, no. 2, p. 12, 2019.
  * [7] D. Alistarh, D. Grubic, J. Li, R. Tomioka, and M. Vojnovic, “QSGD: Communication-efficient sgd via gradient quantization and encoding,” in _Proc. Adv. Neural Inf. Process. Syst._ , vol. 30, 2017, pp. 1709–1720.
  * [8] J. Bernstein, Y.-X. Wang, K. Azizzadenesheli, and A. Anandkumar, “signSGD: Compressed optimisation for non-convex problems,” in _Proc. ICML_ , 2018, pp. 560–569.
  * [9] H. Wang, S. Sievert, Z. Charles, S. Liu, S. Wright, and D. Papailiopoulos, “Atomo: Communication-efficient learning via atomic sparsification,” _arXiv preprint arXiv:1806.04090_ , 2018.
  * [10] H. H. Yang, Z. Liu, T. Q. Quek, and H. V. Poor, “Scheduling policies for federated learning in wireless networks,” _IEEE Trans. Commun._ , vol. 68, no. 1, pp. 317–333, 2019.
  * [11] N. H. Tran, W. Bao, A. Zomaya, M. N. Nguyen, and C. S. Hong, “Federated learning over wireless networks: Optimization model design and analysis,” in _Proc. IEEE INFOCOM_. IEEE, 2019, pp. 1387–1395.
  * [12] W. Shi, S. Zhou, Z. Niu, M. Jiang, and L. Geng, “Joint device scheduling and resource allocation for latency constrained wireless federated learning,” _IEEE Trans. Wireless Commun._ , vol. 20, no. 1, pp. 453–467, 2020.
  * [13] S. Luo, X. Chen, Q. Wu, Z. Zhou, and S. Yu, “Hfel: Joint edge association and resource allocation for cost-efficient hierarchical federated edge learning,” _IEEE Trans. Wireless Commun._ , vol. 19, no. 10, pp. 6535–6548, 2020.
  * [14] M. Chen, Z. Yang, W. Saad, C. Yin, H. V. Poor, and S. Cui, “A joint learning and communications framework for federated learning over wireless networks,” _IEEE Trans. Wireless Commun._ , 2020.
  * [15] M. M. Amiria, D. Gündüzb, S. R. Kulkarni, and H. V. Poor, “Convergence of update aware device scheduling for federated learning at the wireless edge,” _IEEE Trans. Wireless Commun._ , 2021.
  * [16] M. Chen, H. V. Poor, W. Saad, and S. Cui, “Convergence time optimization for federated learning over wireless networks,” _IEEE Trans. Wireless Commun._ , 2020.
  * [17] J. Ren, Y. He, D. Wen, G. Yu, K. Huang, and D. Guo, “Scheduling for cellular federated edge learning with importance and channel awareness,” _IEEE Trans. Wireless Commun._ , vol. 19, no. 11, pp. 7690–7703, 2020.
  * [18] M. M. Amiri and D. Gündüz, “Federated learning over wireless fading channels,” _IEEE Trans. Wireless Commun._ , vol. 19, no. 5, pp. 3546–3557, 2020.
  * [19] K. Yang, T. Jiang, Y. Shi, and Z. Ding, “Federated learning via over-the-air computation,” _IEEE Trans. Wireless Commun._ , vol. 19, no. 3, pp. 2022–2035, 2020.
  * [20] G. Zhu, Y. Wang, and K. Huang, “Broadband analog aggregation for low-latency federated edge learning,” _IEEE Trans. Wireless Commun._ , vol. 19, no. 1, pp. 491–506, 2020.
  * [21] M. M. Amiri and D. Gündüz, “Machine learning at the wireless edge: Distributed stochastic gradient descent over-the-air,” _IEEE Trans. Signal Process._ , vol. 68, pp. 2155–2169, 2020.
  * [22] H. Guo, A. Liu, and V. K. N. Lau, “Analog gradient aggregation for federated learning over wireless networks: Customized design and convergence analysis,” _IEEE Internet Things J._ , vol. 8, no. 1, pp. 197–210, 2021\.
  * [23] T. Sery, N. Shlezinger, K. Cohen, and Y. C. Eldar, “Over-the-air federated learning from heterogeneous data,” _IEEE Trans. Signal Process._ , vol. 69, pp. 3796–3811, 2021.
  * [24] L. Su and V. K. N. Lau, “Data and channel-adaptive sensor scheduling for federated edge learning via over-the-air gradient aggregation,” _IEEE Internet Things J._ , 2021, Early Access.
  * [25] ——, “Hierarchical federated learning for hybrid data partitioning across multitype sensors,” _IEEE Internet Things J._ , vol. 8, no. 13, pp. 10 922–10 939, 2021.
  * [26] T. Sery and K. Cohen, “On analog gradient descent learning over multiple access fading channels,” _IEEE Trans. Signal Process._ , vol. 68, pp. 2897–2911, 2020.
  * [27] H. H. Yang, Z. Chen, T. Q. Quek, and H. V. Poor, “Revisiting analog over-the-air machine learning: The blessing and curse of interference,” _arXiv preprint arXiv:2107.11733_ , 2021.
  * [28] M. Tang, S. Cai, and V. K. N. Lau, “Over-the-air-aggregation with multiple shared channels and graph-based state estimation for industrial iot systems,” _IEEE Internet Things J._ , 2021, Early Access.
  * [29] S. Cai and V. K. N. Lau, “MSE tail analysis for remote state estimation of linear systems over multi-antenna random access channels,” _IEEE Trans. Autom. Control_ , vol. 65, no. 5, pp. 2046–2061, 2020.
  * [30] M. Tang, S. Cai, and V. K. N. Lau, “Remote state estimation with asynchronous mission-critical IoT sensors,” _IEEE J. Sel. Areas Commun._ , vol. 39, no. 3, pp. 835–850, 2021.
  * [31] S. Cai and V. K. N. Lau, “Cloud-assisted stabilization of large scale multi-agent systems by over-the-air-fusion of IoT sensors,” _IEEE Internet Things J._ , 2019.
  * [32] M. Gastpar, “Uncoded transmission is exactly optimal for a simple Gaussian sensor network,” _IEEE Trans. Inf. Theory_ , vol. 54, no. 11, pp. 5247–5251, 2008.
  * [33] L. Chen, X. Qin, and G. Wei, “A uniform-forcing transceiver design for over-the-air function computation,” _IEEE Wireless Commun. Lett._ , vol. 7, no. 6, pp. 942–945, 2018.
  * [34] D. Wen, G. Zhu, and K. Huang, “Reduced-dimension design of MIMO over-the-air computing for data aggregation in clustered IoT networks,” _IEEE Trans. Wireless Commun._ , vol. 18, no. 11, pp. 5255–5268, 2019.
  * [35] G. Zhu and K. Huang, “MIMO over-the-air computation for high-mobility multi-modal sensing,” _IEEE Internet Things J._ , vol. 6, no. 4, pp. 6089–6103, 2019.
  * [36] A. Kortke, M. Goldenbaum, and S. Stańczak, “Analog computation over the wireless channel: A proof of concept,” in _Proc. IEEE SENSORS_ , 2014, pp. 1224–1227.
  * [37] U. Altun, S. T. Basaran, H. Alakoca, and G. K. Kurt, “A testbed based verification of joint communication and computation systems,” in _Proc. 25th TELFOR_ , 2017, pp. 1–4.
  * [38] M. Speth, S. Fechtel, G. Fock, and H. Meyr, “Optimum receiver design for wireless broad-band systems using OFDM. I,” _IEEE Trans. Commun._ , vol. 47, no. 11, pp. 1668–1677, 1999.
  * [39] E. Sourour, H. El-Ghoroury, and D. McNeill, “Frequency offset estimation and correction in the IEEE 802.11a WLAN,” in _Proc. Veh. Technol. Conf_ , vol. 7, Sep. 2004, pp. 4923–4927.