Arash Behboodi, Daniel Dijkman
Qualcomm Technologies Netherlands B.V.
Qualcomm AI Research
May 25, 2022
@QCOMResearch
Bringing AI research to wireless
communication and sensing
Qualcomm AI Research is an initiative of Qualcomm Technologies, Inc
3
Arash
Behboodi
Sr. Staff Manager,
Engineering
Qualcomm AI Research
Our presenters
How ML and wireless
complement one another
How ML is improving
communications
How ML is enabling
RF sensing
3
1
2
4
Agenda
Daniel
Dijkman
Principal Engineer,
Qualcomm AI Research
5 Questions?
Future AI for wireless
research directions
4
Wireless
Strengths
ML
Design with real world priors, fast
and flexible models
Accurate prediction in complex tasks
Accurate modeling of
generative process
Sensing and perception
Strengths
Wireless and ML have complementary strengths
Design driven by tractable mathematical
models
Interpretable solutions
Good generalization under different
deployment conditions
Simple model adaptation
5
AI for wireless is here today
World’s only
modem-RF system
for all 5G bands from
0.6-41 GHz
World’s 1st
Modem-RF
5G AI Processor
0.6 GHz
6 GHz
24 GHz
41 GHz
10 Gigabit 5G
Snapdragon® X70 5G Modem-RF System
Snapdragon, Qualcomm mmWave Module and Qualcomm 5G AI Processor are products of Qualcomm Technologies, Inc. and/or its subsidiaries.
.
6
Qualcomm 5G technology is licensed by Qualcomm Incorporated. Qualcomm
5G products and Qualcomm Cloud AI 100 Platform are products of Qualcomm
Technologies, Inc. and/or its subsidiaries.
MWCB 2022
Enabling AI/ML for air
interface evolution
Cross-node machine learning for channel
state feedback (CSF)
Using end-to-end over-the-air (OTA) testbed in San Diego
that operates in 3.5 GHz band over 100 MHz, utilizing
Qualcomm® Cloud AI 100 platform and Snapdragon®
Modem-RF system
Showing reduced communication overhead that leads to
improved throughput, latency, and capacity
Cross-node machine learning for beam
management
Using end-to-end over-the-air (OTA) testbed in San Diego
that operates in 28 GHz band capable of 800 MHz
bandwidth, utilizing Qualcomm® Cloud AI 100 platform and
Snapdragon® Modem-RF system
Bringing more efficient beam management to increase
usable capacity and extend device battery life
7
Channel estimation Radio resource allocation
Power saving
Vehicular communications
Positioning
Security
Device non-linearity Contextual awareness
Environmental sensing
MIMO detection
Full duplex
TCP optimization Beam management
and optimization
Spectrum sensing
AI research areas
to enhance 5G
8
Out-of-domain
generalization
Feasibility of
supervised learning
Adaptability of
ML models
Examples:
Unseen dopplers and channel conditions
Example:
Wireless fingerprinting for localization in dynamic environments
Examples:
Different antenna configurations and channel conditions
Challenges
in applying AI
to wireless
9
How ML is improving
communications
How ML is enabling
RF sensing
Our fundamental AI research
is fueling wireless innovation
ML
research
Wireless
technology
Generative
modeling
Neural
augmentation
Self-supervised
learning
Unsupervised
learning
For channel
representation and
simulation
For channel
estimation receiver
algorithms
For active
positioning
For RF sensing and
passive positioning
10
Machine learning is
enhancing wireless
communication
Machine learning design based on
wireless domain knowledge provides
superior gains
Channel modeling
Using generative modeling to provide a more accurate
channel representation and improve communications
Communication design
Using neural augmentation to enhance a
Kalman filter for improved communications
11
The wireless channel is complex
and includes useful information
Reflections change the transmitted signal
and have multiple effects
Line of sight
Reflected by floor
Reflected by
human target
Reflected by wall
Transmitter
Receiver
12
Neural models help address the challenges of classical channel models
Environment,
Antenna,
UE/gNB location,
Doppler,
Carrier frequency,
…
Channel
𝒉
Modeling physical propagation effects on wireless signals
Classical channel models
Data-driven neural channel models
Standard channel models:
3GPP TDL/CDL, WINNER, ray tracing
Neural channel models
Pro: Accurately match complex
field data distribution
Pro: Fast sampling for prototyping purposes
Pro: Works with simple traces
Con: Interpretability
Con: Cumbersome field measurements
Con: Hard-coded assumptions
Con: Limited scenarios, slow to prototype
13
Channel impulse response and channel frequency response
include all paths between sender and receiver
Channel impulse response Channel frequency response
Fourier transform
Attenuation
Sum over all paths
Channel frequency
response for
subcarrier i
Frequency of
subcarrier i
Path delay
𝐶𝐼𝑅(𝑡) = (
!
"
𝐻#𝛿 𝑡 − 𝜏#
Attenuation
Impulse
response Path delay
𝐶𝐹𝑅𝑖 = (
!
"
𝐻# . 𝑒$%&'(!)"
14
Data-driven neural channel models offer key benefits
Pro: it can model variable number
of antenna inputs and outputs
Pro: interpretable samples:
the channel between TX-𝑗 and RX-𝑖
(Unknown) Channel
𝒙[𝒎]
Transmitted signal Received signal
𝒚[𝒏]
Given I/O measurements
Learn parameters of channel model: 𝑦 = ℎ!(𝑥)
Channel ℎ!(𝑥)
𝒙 𝒚
Channel
𝒙 ∗ 𝒉
15
Neural augmentation enhances classical channel models
<latexit sha1_base64="I4JWqw7AzzrXhInhpG2C9Lys824=">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</latexit>
z ⇠ N(0, I)
<latexit sha1_base64="WNmCx7JwwE/UQQITqr2CY+kD70E=">AAAx8XiclVtZc9vIEdZuro1y7SaPeUFFdu1mS1ZJ9tYmj6v7oiTqPkyvCwSbICxcxgxBySzmX+QhL0kqr/kzec2/Sc8MBt0DQt6Kqmyhv2+mMcfX000Q6udxJOTq6n8/+fRHP/7JT3/22c8Xf/HLX/36N59/8dsrkY2LAC6DLM6Km74vII5SuJSRjOEmL8BP+jFc9+83FX9dQiGiLL2Qjzm8SfwwjYZR4EuEXvfKB+9rr1cke4tvP19aXVnVP978xVp1sbRQ/XTffrH6n94gC8YJpDKIfSFer63m8s3UL2QUxDBb7I0F5H5w74fwGi9TPwHxZqrHPPOeIzLwhlmB/1LpaZT3mPqJEI9JH1smvhyJJqfANu71WA7//GYapflYQhqYGw3HsSczTy2AN4gKCGT8iBd+UEQ4Vi8Y+YUfSFymxcXn6sc73r72jtYv9ryt7Z394/2L/ZPjc09Ti20DWcbfahpiuZ/M0Id35Bf3nsD74CILLxt6gZ+bazXjAoZQFFEaqkENojISttkwCscF4IRSmARZkvjpYNpDMIahnE2nPUi8rzp4/cfZbK5NgPsAhW21qa22dkUUjmpnZ8poayWz3La5yPK2Fv1MyiyxjTa0Ndeumrdvm/lPtejbFv2nWgS2RfBUi4FtMVAtcBv2cHaxmqHne9hebToMMVIGHq5N4vrAawXOXq+9QS/9obe0ppyglx29KWbXUFKw7MXZBIoXAcbdymIPXeplheHS2tRs4F96aE21g7buONxI+vGKt4NiEBIDRu29UDuGvPG4Yz3uND1qWk4ye8+ll9VdhWcbeTijynhpe7wf+wPqsvRq6Zu5bst1H3v1irv6Rk/n3Kj6o8uByjeDr0LAWY8WB3ZBTO9z2/u8pfeZ7aUDepLVUbZSL4y5u9ArU8fgE0vTdDgqAJoumb+lV/MeadWY71fzvv3UA9wE1bllyeC9mbNt8vSkHT/jPIfCU36Mm+3KzXabm3Wv8Ce07g1nL1688MssGnhjoY6maOjlmRARZhLjOo99DJ3K/9OjU6dpjpHUMkfFmO5Vm/97kpWjzdrR5g86wjmnIegz2LQVxoeG6xGhVCxtXb148aRMcHR+HGaYPUZJyzyRM6OrG310oszV3EzXrav1FldW8PZ+OIna18cPgwun0/oPdppb1BwPrWrmTH0KNcNVVx/bFNO/qd5u3b/r9rczrW+Ao1bXTw64EhxEsRJrrC7wQMcG6qryN4yzrNC0vjK8vqwaINVPpmvNbCMLDITZtKcyf+DH061mg9KPowFv8NZcF8nUULM5lyBkewfNzOoZQS5UjstFFGdplZ/O0EWWeKVfRD5Gq9U3SH+qPD/INCsS9Pqsh9CzmV3OokH7xPRdpk9M4DIBMQOXGRADLgPEDF1mSEzoMiExI5cZERO5TETMO5d5R8y9y9wTE7tMPNMyLhIvEhixWGMPHtVhZ3Zw2Xs3FtIbZOmX0lOFLsrxUZ08zsZ4SeU7dX2ndNfMZTJicpfJiXnvMu+JKVymIEa4jCBGuowkZuwyY2JKlymJmbjMhJgHl3kg5tFlHon54DIfZqbMswGAmTmrj/eyCpKpCaX+kIVNPW45MlFiW2ib8YzjcJ9gFhtlQDALjHJAMIuKEghmIVEOCWbxUIYEs2AoRwSzSCjHBLMwKN8RzGKgvCeYBUAZExwzOCE4YTBbaL7CGcFMzGVOMFNy+Z5gJuOyIJhpuBQEC76pBMv2NeHSLQlmui0nBDPRlg8EM8WWjwQzuZYfCLZa3Y5BfWDWn/aKFt2CEV3ruQxGea0nMxj5tZ7NYDTYejqDEWLr+QxGja0nNBhJtp7RYHTZekoj9+Q5DUahrSc1GJm2ntVgtNo8rS2XuFzCuSdPYjDSbT2Lwei39TQGI+LW8xiMkltPZDBybj2TwWi69VQGI+zWcxmMultPZjASbz2bwei89XQGI/bW8xmM4p8+oTEWiiioK5RkneJjncIm2SB4g8GbBG8yeIvgLQZvE7zN4B2Cdxi8S/Aug/cI3mPwPsH7DD4g+IDBhwQfMrhDcIfBRwQfMfiY4GMGnxB8wuAuwV0GnxJ8yuAzgs8YfE7wOYMvCL5g8CXBlwy+IviKwdcEXzP4huAbBt8SfMvgO4Lvnj5eXdGBUR3T6DrTr5Ye4zY4t+lym5zbcrktzm273Dbndlxuh3O7LrfLuT2X2+Pcvsvtc+7A5Q44d+hyh5zruFyHc0cud8S5Y5c75tyJy51wrutyXc6dutwp585c7oxz5y53zrkLl7vg3KXLXXLuyuWuOHftctecu3G5G87dutwt5+5czsr+ipcQ5QfQnyPws+tq3bfMUpjaz7MWS8YG6iWUNOqaWOFuPVxWMEP6BqE6RFchiFD1oWsPRKjmKKuRUKWh6wxEqL7Q1QUiVFXomgIRqiV0JYEIVRC6fkCE6gZdNSBC1YKuFRCJ2ToYhCoDXRcgkrL1MwhVAboGQIRyv878iFDG1/keEcrzOssjItiCG4RyelltC9uU0iCUv3X2RoSyts7ZiFCu1pkaEcrQOj8j0laNumVo6cf5SO23/l0rsOxX4tC6sCB91KInExUV+0l/oHqYCyKyBEKF698Ea0kqOVoAHSKC/xMkojBRXfVvgq1wK9HWE5lO+finSqzWQrEGZKFQB2xSUyVQa6FAh2ShOEOyUJgjsnC4bKwoyHdkoRjv2dpMlQjrmU+VAK2Fi8lWEcWXsSWZKtFZC0X3niwUXMFWaqqEVi/QVInMWrjQbJlRYCVZKK4JWSisB7JQVI9koaA+zKrvvDDPPhhc51jUGeVWnVkRoYyq8ykilEd1FkWEsqfOnYhQztQZExHKlDpPIkL5UWdHRCgr6pyICOVCnQkRoQyo8x8ilPd01kOEsp3OdYhQjtMZDhHKbDqvIUL5TGczRCiL6RyGCOUunbkQoYyl8xUilKd0lkKEspPOTYhQTtIZCRHKRDoPIUL5R2cfRCjr6JyDCOUanWkQuWM7SHmhz9NC0h1VB3EPr9jq2dBXTKcK/3pyVQwr7tzEsVbRBaRCfRO8BUHsF4CiGq2rEwjvaIo9MYzUo1JIg2wQpSE688exQsSwvk5mU6Ge8p6DfMpBP4sHP+Sm/zDDIGw+qU2F/o7Q5M3Kn35KXU1NmvoyFUz9csNipH+5aTGKALllMYoBuW0xigK5YzGKA7lrMYoEuWcxigW5bzGKBnlgMYoHeWgxigjZsRjFhDyyGEWFPLYYxYU8sRhFhuxajGJDnlqMokOeWYziQ55bjCJEXliMYkReWoyiRF5ZjOJEXluMIkXeWIxiRd5ajKJF3lnMVGQo5N3Cz0eGDe3n3MD5uBFuMJh0EW4ymKQRbjGY1BFuM5gEEu4wmDQS7jKYZBLuMZiUEu4zmMQSHjCY9BIeMpgkE3YYTKoJjxhMwgmPGUzaCU8YTPIJuwwmBYWnDCYRhWcMJh2F5wwmKYUXDCY1hZcMJkGFVwwmTYXXDCZZhTcMJmWFtwwmcYV3DLYVPx5tVakm6qcofSYusUEoaUtsEkrSEluEamU997b0NxljAZ7vCZAe3jqGgbe97PUh8BUuR5HwJtk4HiCEFnhCf++BteS48NSrO1mMjtT7LvCQY22pv8y1X6nv0B1JnWKXUBKn2COUtCn2CSVpigNCSZnikFASpugQSroUR4SSLMUxoaRKcUIoiVJ0CSVNilNCSZLijFBSpDgnlAQpLgglPYpLQkmO4opQUqO4JpTEKG4IJS2KW0JJiuKO0PqRS4p1H+iPEL552FIVgUAVQMct/lV5uE4WSnWDLJToJlkozS2y8LDbJgtFtEMWimeXLBTNHlkoln2yUCQHZKE4DslCUXTIQjEckYUiOCYLN/+ELNz0Llm42adk4SafkYWbe04WbuoFWbiZl2ThJl6RhZt3TRZu2g1ZuFm3ZOEm3bH7VZVWVWWpLQO+ZdJUXHikqPjVb+NhEBt02ZtEcpSNpYfljjfBlJZD4RZEQBWRUw1Vt5e1BnTDuUIQdLkEjXoJdMEEjYoJdMkEjZoJdNEEjaoJdNkEjboJdOEEjcoJdOkEjdoJdPEEjeoJdPkEjfoJdAEFjQoKdAkFjRoKdBEFjSoKdBkFjToKdCEFjUoKdCkFjVoKdDEFjWoKdDkFjXoKdEEFjYoKdEkFjZoKdFEFjaoKdFkFjboKdGEFjcoKdGkFjdoKdHEFjeoKdHkFjfoKdIEFrMLCTwqYcmQxBm+cDqCIH9VLSwNf+l4IKRSYbZQdCVR6f6xSjyvbXDWdTfO3016RTLWhE5/yCkkeFRGmPKd//e5g/1GnO/0aiLoJ5seGb/uGyMiX+EndvYXTsstbdmdtg0myAcQfm4huUM/EWHP3qRp1P9Yol1E8gKplTxv16OseeEzILBj5Qr04649lpj9BQeGMsPECa27a1GOsuswPYABOO2O2tCuQwEPHtjMmaiHQz9DcxrGfx34As/qNmk4FzLznXnXtLq/bf3tWZ7zt5kA6gr2002myZzOe2xunZpKzNW6QcTGzz91cooBwVj9Ja1KBpDkqKxpGUDRdi2woE/+BWlqg2Q6TRaZfYjIP2ea95PFYzf6DehLgsoedGX+D6bAzt4FXfkEjUEbTv8RffoF7X2Ss5fncBmxmJdHKUAK9zuJh4SfqgdRokhVYoAr/UXjPOt+/fKbe3tEvnY9T8xaqyHH/hX577FkP4pi1sQ9En3sbmAAx5FP13yPGOyTqLTZVBRunrLV6hTQbhzpn6qI4krCs3YvMG2Sg3E2i+yiHQeSvNF5BzookVg/vZ9PO96uzFjJLQXFrbZyc6H4v27hcMXkLo7XQ+b4XpUP52Ayd3C/Uw2E8NnwVLOeAZ63wQ/Ci1EuzqqCX8LDibY4yoZYnUwVgMPK28LNvCl8Kr59l9yuLzuOck1ydzlnxNWq8CPUA8HdvWV19rKE6J01DvGp3qdWKzfT/T7S4QEFdqDf8YpA9v49xFmeTfgH+/aL6G4i15l88zF9cvVxZ+3Zl9fSbpe++rf4a4rOF3y/8YeGrhbWFPy18t7C30F24XAgWsoW/Lvx94R9rYu1va/9c+5dp+uknVZ/fLTg/a//+H/wVUUM=</latexit>
x ⇤ H
<latexit sha1_base64="2Ycnm71GsE0Asodgv6cqakgfFoI=">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</latexit>
x <latexit sha1_base64="8QKN12NklYvijUd/F07kRcnO2go=">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</latexit>
y
Classical channel models
Generative model
Generative models can generate channel
impulse response from complex distributions
𝐇: channel impulse response
Random seed
Generator
𝐇: channel
impulse response
G
16
<latexit sha1_base64="WNmCx7JwwE/UQQITqr2CY+kD70E=">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</latexit>
x ⇤ H
GAN: generative adversarial networks
Generative MIMO channel model based on GAN learns the channel
<latexit sha1_base64="MLeIhl3YmqH6bXjTkkCdxsBWUgs=">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</latexit>
hi,j(⌧) 2 CL
<latexit sha1_base64="I4JWqw7AzzrXhInhpG2C9Lys824=">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</latexit>
z ⇠ N(0, I)
Generator Discriminator
Transmit-receive
antenna index
Transmit-receive
antenna index channel
Spatial
correlation
Real/fake
The generator learns to generate the channel
Channel
output
The discriminator teaches the generator to learn channel distribution
17
MIMO-GAN learns
the communication
channel precisely
18
MIMO-GAN matches power and delay profile of ground truth channels
The method learns to find the channel function based on only input-output traces
MAE: mean absolute error
Orekondy et al., ICC 2022, https://arxiv.org/abs/2203.08588
TABLE 1: Power and delay statistics of MIMO-GAN and ground-truth (GT) channels
Total Power (dB) Average Delay (𝝁𝒔) RMS Delay Spread (𝝁𝒔)
TDL-A
MIMO-GAN 4.648 0.2643 0.2862
GT 4.628 0.2641 0.2897
MAE -18.69 3.57 x 10—3
3.57 x 10—3
TDL-B
MIMO-GAN 4.735 0.2276 0.2954
GT 4.688 0.2285 0.2987
MAE -14.99 3.37 x 10—3
3.37 x 10—3
19
19
Communication channels are hard to accurately estimate
A more accurate channel estimate at all the time steps for
different dynamics enables more efficient communications
Regular pilot symbols are
transmitted to get periodic
noisy observations (𝑜") of
the ground truth channel
Complex
channel vector
Mobile device
trajectory
Communication
channel
Channel states
Noisy observations
(pilots)
𝑜!
ℎ!
𝑜"
𝑜#
𝑜$
𝑜%
ℎ%
ℎ"
ℎ#
ℎ$
𝑡! 𝑡" 𝑡# 𝑡$
𝑡%
Time 𝑡!
Time 𝑡"
Time 𝑡#
ℎ&
20
Classical Kalman filters lose accuracy over different dynamics
Con: Optimal Kalman filter parameters
vary with Doppler values
Con: A single Kalman filter should not
be used for all the Doppler values
Pro: Kalman filter can work with arbitrary
SNR and pilot patterns
Pro: Kalman filter is interpretable
KF(𝜃)
Kalman state 𝑆$%" Estimated
channel $
ℎ$
Observation 𝑜$
Kalman tracks the channel
Time 𝑡$,
velocity 𝑣$
Time 𝑡%,
velocity 𝑣%
Time 𝑡&,
velocity 𝑣&
KF(𝜃#) KF(𝜃$) KF(𝜃%)
21
Standalone ML solutions for channel tracking have limitations
Con: Cannot naturally deal with sporadically
available observations (pilots) as input
Con: Have non-interpretable hidden states
Con: Do not generalize to different
configurations (pilot patterns, SNR)
Pro: Learn complex dynamics
LSTM-based channel tracking
22
Neural augmentation of Kalman filters offers the best of both worlds
Kalman filter parameters
RNN provides Kalman parameters at time 𝑡
Interpretability
Out-of-domain
generalization
Robustness
Expressive power
RNN Update
23
Neural-augmented KF closely matches the ground truth channel
24
Neural-augmented Kalman filter generalizes to unseen cases
Neural-augmented Kalman filter (NA-KF) outperforms LSTM and Kalman filter over unseen pilot ratio*
* Averaged error computed over high Dopplers
Kumar Pratik et al. https://arxiv.org/abs/2109.12561. Globecom 2021
When trained over the whole data, NA-KF performs
as good as or better than Kalman without knowledge
of the exact dynamics
LSTM breaks down on unseen pilot ratio
NA-KF generalizes across scenarios with
unseen Dopplers and pilot patterns
Seen pilot ratio / Doppler Unseen pilot ratio Seen pilot ratio / Doppler Unseen pilot ratio Unseen pilot ratio / Doppler
Kalman Filter
LSTM
Neural augmented Kalman
Channel
tracking
gain
(-NMSE
in
dB)
0
5
10
15
20
25
Channel
tracking
gain
(-NMSE
in
dB)
0
5
10
15
20
25 Kalman Filter
LSTM
Neural augmented Kalman
25
Machine learning
is enabling RF
sensing
Detect gestures, movements, and
objects by monitoring signal reflection
patterns, enabling new use cases
Active positioning
A communications device along with nearby
access points are used for positioning
Passive positioning
Access points alone are used to track the
environment and determine positioning
26
5G / Wi-Fi positioning is useful indoors
and assists GPS outdoors
Active positioning
with RF sensing has a
variety of use-cases
Indoor navigation Vehicular navigation
Asset tracking
AGV tracking
27
TRP: Transmission/reception point; SRS: Sounding reference signal; PRS: Positioning reference signal
• Access points (TRPs)
have known locations
and are synchronized
• A reference signal
(SRS or PRS) is
exchanged between
phone and access points
• The location of the
phone is determined by
analyzing the Channel
Impulse Response (CIR)
5G can
provide (indoor)
positioning
services
X-Y-Z
location?
TRP#1
Blocker
R
e
f
l
e
c
t
o
r
TRP#2
TRP#3
CIR
CIR
CIR
28
Testbed for indoor active positioning to prove technology
29
Pro: no labels required
Con: not very accurate in non-line-of-sight conditions
Con: doesn’t use multipath information
Pro: very accurate, uses multipath
Con: requires dense labels
Con: robustness issues
Position
Time difference of arrival
(TDOA)
ML-assisted RF fingerprinting
(RFFP)
RFFP TDOA
Current precise positioning methods
have limitations in accuracy or feasibility
CIR #1
CIR #2
30
Learn position and environment from multipath propagation
With enough unlabeled CSI samples, we can learn the geometry of the environment without labels
Reflector
Real Tx
Real Rx
access point
Virtual Rx
access point
Key idea 1
Multi-path components (e.g., from reflectors) can
help localize even with a single access point
Triangulation can use real and virtual access
points as reference.
Key idea 2
There is only one unique environment geometry
and access point location that can be compatible
with a collection of (unlabeled) CSI samples
31
Learn position and environment from multipath propagation
Neural SLAM demonstrates an end-to-end trainable network to learn positions and environment
that best reconstructs the CSI samples
*Inputs and outputs may be CSI or related features such as ToF and/or AoA
𝑯𝒖
NN
p
𝑝% 𝑝& 𝑝'
𝒑𝟎
Propagation model
Learnable virtual
AP locations
Forward pass →
Backward pass ←
Predicted
UE location
Input
feature*
Learnable
encoder network
Maps CSI to location
(Fixed)
Decoder network
Incorporates physics of reflections
Reflector
$
𝑯𝒖
Reconstructed
input*
𝒑𝟎 𝑝%
p
𝑯𝒖
32
32
Shreya Kadambi et al, ICC 2022, https://arxiv.org/abs/2203.08264
Neural RF SLAM achieves precise 3D positioning
Neural RF SLAM achieves ~43.4 cm accuracy for 90% of users using only
unlabeled CSI values from single anchor at 400 MHz bandwidth
3D ray tracing simulation
Neural RF
SLAM
33
33
• RF signals can be employed
as bi-static radar
• Any change in the environment
also affects the wireless channel
• Specifically, human motion,
gestures, respiration
• The signal propagation is complex
• Self-supervised and weakly
supervised machine learning
enables robust analysis
of the signal
RF sensing
is powered by
machine learning
34
RF sensing has a
variety of use cases
across industries
Home / enterprise / retail
automation and security
Consumer
electronics
Automotive
Healthcare
• Presence, positioning, tracking, activity classification
• Better privacy as compared with camera-based
• Works across walls
• Touchless control (Phone, TV, laptop)
• Proximity-based power save
• Baby presence alarm
• Presence-based setting
• Vitals, attention monitoring
• Contactless sleep monitoring, vitals, fall detection
35
WiCluster enables non-line-of-sight
passive positioning
No precise labels required
To initialize the system, the user is
guided by an app to provide room-level
labels (kitchen, living room, …)
The access points record the
corresponding CSI packets
WiCluster: making
passive positioning
deployable at scale
• Three to four commercial IEEE 802.11
access points (AP), 5 GHz band
• Circular array with 4cm radius
• Bandwidth: 80 MHz
• Packet rate: 90 Hz
36
36
Experiments in real environments to test feasibility of deployment
Environment #2
2D office,15m x 21m
Environment #1
2D office,14m x 20m
Environment #3
3D home
37
Ilia Karmanov et al. WiCluster: Passive Indoor 2D/3D Positioning using WiFi without Precise Labels, GlobeCom 2021
End-to-end training
Triplet loss to exploit temporal prior
Clustering loss to exploit spatial prior
• Cluster labels are updated after every epoch
The crossed softmax ensures that the
CSI-to-3D mapping is bijective
Zone loss is used for embedding
the 3D location into the floor plan
• Floor plan can be 2D or 3D
• Only requires a few labels
Cluster
Project
Cluster
Softmax
loss
Self-supervised
Weakly-supervised
WiCluster is first to do weakly-
supervised passive positioning
Input: CSI ResNet
Latent labels
Latent space
3D labels
3D locations
Triplet loss
Zone loss
Floor plan
Zone labels
38
WiCluster model performs well in a real-life scenario
39
WiCluster works in strong non-line-of-sight
Conference room with concrete walls:
strong non-line-of-sight
Ground truth Inference
1.13m
Offices
2.08m
Conference room
Mean error
Mean error
40
WiCluster works across multiple floors
96%accurate zone classification
+ approximate 3D location
41
Machine learning
design for wireless
communication and
RF sensing
Unsupervised learning
• Learning distributions and manifolds
is an approach to obtain features in an
unsupervised way
• Examples: WiCluster, Neural RF SLAM
• Other perspectives: self-supervised learning,
transfer learning
Adaptive models
• Models should be able to adapt to different
channel conditions and setups
• Examples: Hypernetwork Kalman, MIMO GAN
Generalization
• Designing ML models based on inductive bias,
gained from domain knowledge, or neural
augmentation can help generalization
• Examples: Hypernetwork Kalman, MIMO-GAN
Interpretability
• Neural augmentation helps interpretability
of modules in an ML model
• Examples: Hypernetwork Kalman, MIMO-GAN
42
Neural RF sensing and neural rendering offer synergistic capabilities
Analogous to the compelling capabilities of computer vision and computer graphics
Computer
vision
Computer
graphics
ML methods to recover
scenes/objects
Physical
world Image
ML methods
for rendering
Neural RF
SLAM/
sensing
Neural
rendering
Learning position and
environment from CSI
Environment
Channel State
Information (CSI)
Generate spatially consistent
CSI from the environment
43
AI is enhancing wireless
communications with generative
modeling and neural augmentation
AI is enhancing RF sensing
through self-supervised and
unsupervised learning to better
understand the environment
Qualcomm AI Research is
conducting leading research in
applying AI for RF sensing and
improved communications
Nothing in these materials is an offer to sell any of the components
or devices referenced herein.
©2018-2022 Qualcomm Technologies, Inc. and/or its affiliated
companies. All Rights Reserved.
Qualcomm is a trademark or registered trademark of Qualcomm
Incorporated. Other products and brand names may be trademarks or
registered trademarks of their respective owners.
References in this presentation to “Qualcomm” may mean Qualcomm
Incorporated, Qualcomm Technologies, Inc., and/or other subsidiaries or
business units within the Qualcomm corporate structure, as applicable.
Qualcomm Incorporated includes our licensing business, QTL, and the vast
majority of our patent portfolio. Qualcomm Technologies, Inc., a subsidiary
of Qualcomm Incorporated, operates, along with its subsidiaries,
substantially all of our engineering, research and development functions,
and substantially all of our products and services businesses, including our
QCT semiconductor business.
Follow us on:
For more information, visit us at:
qualcomm.com & qualcomm.com/blog
Thank you

Bringing AI research to wireless communication and sensing

  • 1.
    Arash Behboodi, DanielDijkman Qualcomm Technologies Netherlands B.V. Qualcomm AI Research May 25, 2022 @QCOMResearch Bringing AI research to wireless communication and sensing Qualcomm AI Research is an initiative of Qualcomm Technologies, Inc
  • 2.
    3 Arash Behboodi Sr. Staff Manager, Engineering QualcommAI Research Our presenters How ML and wireless complement one another How ML is improving communications How ML is enabling RF sensing 3 1 2 4 Agenda Daniel Dijkman Principal Engineer, Qualcomm AI Research 5 Questions? Future AI for wireless research directions
  • 3.
    4 Wireless Strengths ML Design with realworld priors, fast and flexible models Accurate prediction in complex tasks Accurate modeling of generative process Sensing and perception Strengths Wireless and ML have complementary strengths Design driven by tractable mathematical models Interpretable solutions Good generalization under different deployment conditions Simple model adaptation
  • 4.
    5 AI for wirelessis here today World’s only modem-RF system for all 5G bands from 0.6-41 GHz World’s 1st Modem-RF 5G AI Processor 0.6 GHz 6 GHz 24 GHz 41 GHz 10 Gigabit 5G Snapdragon® X70 5G Modem-RF System Snapdragon, Qualcomm mmWave Module and Qualcomm 5G AI Processor are products of Qualcomm Technologies, Inc. and/or its subsidiaries. .
  • 5.
    6 Qualcomm 5G technologyis licensed by Qualcomm Incorporated. Qualcomm 5G products and Qualcomm Cloud AI 100 Platform are products of Qualcomm Technologies, Inc. and/or its subsidiaries. MWCB 2022 Enabling AI/ML for air interface evolution Cross-node machine learning for channel state feedback (CSF) Using end-to-end over-the-air (OTA) testbed in San Diego that operates in 3.5 GHz band over 100 MHz, utilizing Qualcomm® Cloud AI 100 platform and Snapdragon® Modem-RF system Showing reduced communication overhead that leads to improved throughput, latency, and capacity Cross-node machine learning for beam management Using end-to-end over-the-air (OTA) testbed in San Diego that operates in 28 GHz band capable of 800 MHz bandwidth, utilizing Qualcomm® Cloud AI 100 platform and Snapdragon® Modem-RF system Bringing more efficient beam management to increase usable capacity and extend device battery life
  • 6.
    7 Channel estimation Radioresource allocation Power saving Vehicular communications Positioning Security Device non-linearity Contextual awareness Environmental sensing MIMO detection Full duplex TCP optimization Beam management and optimization Spectrum sensing AI research areas to enhance 5G
  • 7.
    8 Out-of-domain generalization Feasibility of supervised learning Adaptabilityof ML models Examples: Unseen dopplers and channel conditions Example: Wireless fingerprinting for localization in dynamic environments Examples: Different antenna configurations and channel conditions Challenges in applying AI to wireless
  • 8.
    9 How ML isimproving communications How ML is enabling RF sensing Our fundamental AI research is fueling wireless innovation ML research Wireless technology Generative modeling Neural augmentation Self-supervised learning Unsupervised learning For channel representation and simulation For channel estimation receiver algorithms For active positioning For RF sensing and passive positioning
  • 9.
    10 Machine learning is enhancingwireless communication Machine learning design based on wireless domain knowledge provides superior gains Channel modeling Using generative modeling to provide a more accurate channel representation and improve communications Communication design Using neural augmentation to enhance a Kalman filter for improved communications
  • 10.
    11 The wireless channelis complex and includes useful information Reflections change the transmitted signal and have multiple effects Line of sight Reflected by floor Reflected by human target Reflected by wall Transmitter Receiver
  • 11.
    12 Neural models helpaddress the challenges of classical channel models Environment, Antenna, UE/gNB location, Doppler, Carrier frequency, … Channel 𝒉 Modeling physical propagation effects on wireless signals Classical channel models Data-driven neural channel models Standard channel models: 3GPP TDL/CDL, WINNER, ray tracing Neural channel models Pro: Accurately match complex field data distribution Pro: Fast sampling for prototyping purposes Pro: Works with simple traces Con: Interpretability Con: Cumbersome field measurements Con: Hard-coded assumptions Con: Limited scenarios, slow to prototype
  • 12.
    13 Channel impulse responseand channel frequency response include all paths between sender and receiver Channel impulse response Channel frequency response Fourier transform Attenuation Sum over all paths Channel frequency response for subcarrier i Frequency of subcarrier i Path delay 𝐶𝐼𝑅(𝑡) = ( ! " 𝐻#𝛿 𝑡 − 𝜏# Attenuation Impulse response Path delay 𝐶𝐹𝑅𝑖 = ( ! " 𝐻# . 𝑒$%&'(!)"
  • 13.
    14 Data-driven neural channelmodels offer key benefits Pro: it can model variable number of antenna inputs and outputs Pro: interpretable samples: the channel between TX-𝑗 and RX-𝑖 (Unknown) Channel 𝒙[𝒎] Transmitted signal Received signal 𝒚[𝒏] Given I/O measurements Learn parameters of channel model: 𝑦 = ℎ!(𝑥) Channel ℎ!(𝑥) 𝒙 𝒚 Channel 𝒙 ∗ 𝒉
  • 14.
    15 Neural augmentation enhancesclassical channel models <latexit sha1_base64="I4JWqw7AzzrXhInhpG2C9Lys824=">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</latexit> z ⇠ N(0, I) <latexit sha1_base64="WNmCx7JwwE/UQQITqr2CY+kD70E=">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</latexit> x ⇤ H <latexit sha1_base64="2Ycnm71GsE0Asodgv6cqakgfFoI=">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</latexit> x <latexit sha1_base64="8QKN12NklYvijUd/F07kRcnO2go=">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</latexit> y Classical channel models Generative model Generative models can generate channel impulse response from complex distributions 𝐇: channel impulse response Random seed Generator 𝐇: channel impulse response G
  • 15.
    16 <latexit sha1_base64="WNmCx7JwwE/UQQITqr2CY+kD70E=">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</latexit> x ⇤H GAN: generative adversarial networks Generative MIMO channel model based on GAN learns the channel <latexit sha1_base64="MLeIhl3YmqH6bXjTkkCdxsBWUgs=">AAA2dniclVtbc9w4dtZuNsnGm8ts8pCHVG2xVnbNbEpWWZ5ZZx9Xd8lqXSy1Lta0x8XuPs2mxZsBNFtyV+fP5DX5QfknecwBQPAcsElPraosEd8HHBLAd3AOCXhYJLFUr1797y9++Ve/+uu/+dtf/92z3/z9P/zjP33z23++kflMjOB6lCe5uBuGEpI4g2sVqwTuCgFhOkzgdviwq/nbEoSM86yvngr4kIZRFk/iUagQ+vjNv04/LuKNT8vvBiqc/SEYxFkwkLs/9T5+s/5q85X5CVYvtqqL9bXq5+Ljb/94OBjno1kKmRoloZQ/br0q1IdFKFQ8SmD5bDCTUISjhzCCH/EyC1OQHxamB8vgBSLjYJIL/JepwKC8xSJMpXxKh1gzDdVUNjkNtnE/ztTkTx8WcVbMFGQje6PJLAlUHujhCMaxgJFKnvAiHIkYnzUYTUMRjhQO2rNnL/RPcLZ/G5xu94+Cvf2D47Pj/vH52VVgqGdtD7KBf3U35MYwXaKN4DQUD4HE++CQyyCfBKOwsNe6xwImIEScRfqhxnEZS1dtEkczAdihDOajPE3DbLwYIJjARC0XiwGkwXc9vP7DcrlSZ4TzAMLV2jWltnoijqa1sUtdaKul8sLV6edFW41hrlSeuko7prRSr+p36KqFXTWGrsawq8bI1Rh11Ri7GmNdA6fhCHuX6B4GYYD19aTDBP1mHODYpL4NvNbg8setD2hlOAnWt7QRtHJgJsXOGkoKNoIkn4N4OUIv3Hw2QJNmWGGyvrWwE/ifAywtjIG25vi4sQqTzeAAxSAVOoyee6lnDHlr8cBZPGhaNLSa5+6e66+ru8rAVQqwR1XhtWvxeRaOqcn69+s/rDTbqNu4q++5qR9Md66sqr86HKh8+/CVC3jj0WLADYhtfeVaX7W0vnStjEPP89rLNuuBsXeXZmRqH+wYmqbBqQBommT21r9ftUijxmx/v2o7zALASdCNW4YMPts+uyrdnfbszIoCRKDtWDP7lZn9NjPbgQjnNO4NYy9fvgzLPB4HM6mXpngSFLmUMcYVa7pIQnSdyn730+nVtEBPaumjZmzzqs5f3MnK0G5taPdnDWGfswjMGmzrSmvDwPUToVQc7Uy9fNkpE3y6MIlyjB7TtKWfyNmnqyt9taPM1EpPt52p7RZTTvDuftiJ2tbXF4O+12j7ZxutDGqBi1bVc6Y+jdrH1VdfmxTbvqnei7r9hd/e9bS+AT61vu584EpwECdarIm+wAUdK+iryt4kyXNhaHNleXNZVUBqmC62mtFGCXSE5WKgI/8oTBZ7zQplmMRjXuGjvRbpwlLLFZMgVXsDwyzrHkEhdYwrZJzkWRWfLtFEngZlKOIQvdXpG1S40JYfVZaLFK0+HyD0fOmGUzTokJihzwyJGfnMiJixz4yJAZ8BYiY+MyEm8pmImKnPTImJfSYm5pPPfCLmwWceiEl8JlkaGYs0iCV6LGbc4ye92NkZ3Ag+zaQKxnn2rQp0ootyfNIrjzcxQVrZznzbGd0195mcmMJnCmI++8xnYoTPCGKkz0hilM8oYmY+MyOm9JmSmLnPzIl59JlHYp585omYLz7zZWnTPOcAGJnzenkvKydZWFcaTpjb1M+tptZLXA1TZjzjODwkmPlGOSKYOUY5Jph5RQkEM5coJwQzfygjgpkzlFOCmSeUM4KZG5SfCGY+UD4QzBygTAhOGJwSnDKYDTQf4ZxgJuayIJgpufxMMJNxKQhmGi4lwZJPKsGqfUy4dEuCmW7LOcFMtOUjwUyx5RPBTK7lF4KdVvcT0C/M5m1PtOgWrOha12WwymtdmcHKr3VtBqvB1tUZrBBb12ewamxdocFKsnWNBqvL1lUauc51GqxCW1dqsDJtXavBarW5Wjsu9bmUc50rMVjptq7FYPXbuhqDFXHregxWya0rMlg5t67JYDXduiqDFXbrugxW3a0rM1iJt67NYHXeujqDFXvr+gxW8d0rNPqCiEd1hpJuk39sk9ukOwTvMHiX4F0G7xG8x+B9gvcZfEDwAYMPCT5k8BHBRww+JviYwW8JfsvgE4JPGNwjuMfgU4JPGXxG8BmDzwk+Z/AFwRcMfkfwOwZfEnzJ4CuCrxjcJ7jP4GuCrxl8Q/ANg28JvmXwHcF3DH5P8HsG3xN83728+qIDqzqm0W2mXyM9xu1wbtfndjm353N7nNv3uX3OHfjcAecOfe6Qc0c+d8S5Y5875txbn3vLuROfO+Fcz+d6nDv1uVPOnfncGefOfe6ccxc+d8G5dz73jnOXPnfJuSufu+Jc3+f6nLv2uWvO3fjcDedufe6Wc3c+d8e59z73nnP3Pudkf8NTiPILmPcIfHd9Vbct8wwW7n3WYenMQoOUgkadE2vcz4fLCmbI0CKUh5gsBBHKPkzugQjlHGX1JJRpmDwDEcovTHaBCGUVJqdAhHIJk0kgQhmEyR8QobzBZA2IULZgcgVEEjYOFqHMwOQFiGRs/CxCWYDJARCh2G8iPyIU8U28R4TivInyiEg24BahmF5W08ImpbQIxW8TvRGhqG1iNiIUq02kRoQitInPiLRlo34aWoZJMdXzbf7WCiyHlTiMLhxIr1r0ZaKikjAdjnULe0FEnkKkcfOXYCNJLUcHoEFE8DdBMo5S3dT8JdgJtxJt3ZHFgj//QovVlVCsIyqhUMesUwstUFdCgU6ohOKMqITCnFIJH5c9KwryE5VQjA9sbBZahHXPF1qAroSDyUYRxZezIVlo0bkSiu4zlVBwgo3UQgutHqCFFpkr4UCzYUaBlVRCcc2phMJ6pBKK6olKKKgvy2rPC+Pso8VNjEWdUWw1kRURiqgmniJCcdREUUQoeprYiQjFTBMxEaFIaeIkIhQfTXREhKKiiYmIUCw0kRARioAm/iFCcc9EPUQo2plYhwjFOBPhEKHIZuIaIhTPTDRDhKKYiWGIUOwykQsRilgmXiFCccpEKUQoOpnYhAjFJBOREKFIZOIQIhR/TPRBhKKOiTmIUKwxkQaRezaDFBeGPCykF9NqIR7gFRs95/qa6VXuX3eu8mHNXVk/NirqQyb1TvAejJJQAIpquq1XILyjTfbkJNafSiEb5eM4i9BYOEs0Iif1dbpcSP2V9wpUl4Fhnox/zszwcYlO2PxSm0mzR2jjZmXPfKWuuqZsfplJpn614zDSv9p1GHmA2nMY+YDadxh5gTpwGPmBOnQYeYI6chj5gjp2GHmDeusw8gd14jDyCNVzGPmEOnUYeYU6cxj5hTp3GHmGunAY+YZ65zDyDnXpMPIPdeUw8hDVdxj5iLp2GHmJunEY+Ym6dRh5irpzGPmKeu8w8hZ17zCbkaGQD0VYTC0buffckfe6Ee0wmHQR7TKYpBHtMZjUEe0zmAQSHTCYNBIdMphkEh0xmJQSHTOYxBK9ZTDpJTphMEkm6jGYVBOdMpiEE50xmLQTnTOY5BNdMJgUFL1jMIkoumQw6Si6YjBJKeozmNQUXTOYBBXdMJg0Fd0ymGQV3TGYlBW9ZzCJK7pnsMv4cWmrUjVZf0UZMnHJHUJJW3KXUJKW3CPUKOtFsGd2MmYSgjCQoAK8dQLjYH8jGMIo1LiaxjKY57NkjBCWIJBm3wNzyZkI9NGdPEFD+rwLPBaYW5rNXLelfkB3JHXKQ0JJnPKIUNKmPCaUpCnfEkrKlCeEkjBlj1DSpTwllGQpzwglVcpzQkmU8oJQ0qR8RyhJUl4SSoqUV4SSIGWfUNKjvCaU5ChvCCU1yltCSYzyjlDSonxPKElR3hNaf3LJMO8D8woR2o8tVRIIlAH0/ORfp4fbVEKp7lAJJbpLJZTmHpVwsdunEorogEoonkMqoWiOqIRiOaYSiuQtlVAcJ1RCUfSohGI4pRKK4IxKOPnnVMJJv6ASTvY7KuEkX1IJJ/eKSjipfSrhZF5TCSfxhko4ebdUwkm7oxJO1nsq4STds/tVmVaVZekpAz5lymZcuKRo/zWn8dCJLboRzGM1zWcqwHQnmGNIK0D4CRFQRuRlQ9XtVa0BU3ElEQSTLkEjXwKTMEEjYwKTMkEjZwKTNEEjawKTNkEjbwKTOEEjcwKTOkEjdwKTPEEjewKTPkEjfwKTQEEjgwKTQkEjhwKTREEjiwKTRkEjjwKTSEEjkwKTSkEjlwKTTEEjmwKTTkEjnwKTUEEjowKTUkEjpwKTVEEjqwKTVkEjrwKTWEEjswKTWkEjtwKTXEEjuwKTXkEjvwKTYAHLsPBNAUOOEjMIZtkYRPKkDy2NQxUGEWQgMNrocixR6cOZDj2+bAtddbkoPi4GIl2Yggl82iqkRSxiDHle+/rs4PDJhDtzDETfBONjw7Y7ITINFb6p+7fwal7wmhfLtodJ8zEkX+uIqVD3xJZW7lNVuvhapULFyRiqmgNTqJ++boHLhMpH01Dqg7PhTOXmDQqE94SNA6yFrVM/Y9Vk9QHG4NWzxZZ6AglcdFw9W0QtjMw3NL9yEhZJOIJlfaKmVwHL4EVQXfvD67ffX9YRb7/5ID3JDu30muzlksf2xqqZFmyMG2Qilu67m08IiJb1l7QmNVLUR12KJzGIpmmZT1QaPlJNBzTrYbDIzSEm+5Ft1UqRzHTvv+gvAT570lvyE0wnvZUJvAkFPYEuNO0r/BMKnHuRs5pXKxOwm5dE64IW6G2eTESY6g9S03kuMEGV4ZMMnvd+ev1cn94xh85nmT2FKgucf2lOjz0fQJKwOu6D6ItgBwMgunymfz2hv0OqT7HpLNgaZbX1EdJ8FpmYaZLiWMGGMS/zYJyDNjePH+ICxnG42TiCnIs00R/vl4veT6+WLWSegea22jg1N+1et3GFZooWxmih99Mgzibqqek6RSj0x2FcNkLtLFeAa60MIwjiLMjyKqFX8LgZ7E5zqYcn1wngaBrs4btvBt/KYJjnD5vPvM8554VenXPx76hxEZkHwL+DDX31tYp6nbQV8ardpFErVjO/O2r0UVB9fcIvATUIh+hnST4fCggfdFDZnUll9wyn7myXDF4GfVwcprMyzEBOg3MBD3k2fmp6jD2hrn1KX64sRGN7hB/5vYP+ikvERB9XvA19+GSNww2XtG8Sp2HjFMZxSk3DvHGUZPucUqwwHzdJyp6+NFves5Zfmi3vbUvLJnnjHEPvnHKJbIU9q2jLz5vHwig5masG1a/ygObxR5yu5rlS0bnlKzo3fEXndq/o3OwVnVu9onOjV3Ru84rOTV7RucUrOjd4Ref2rujc3BWdW7uic2NXdG7ris5NXdG5pSs6N3RF53au6NzMFZ1buaJzI1d0buOKzk1c0bmFK9o2cD9+s77V/I9Wqxc3rze33mz+8O6H9T//qfpPWL9e+7e13699t7a19h9rf147WrtYu14brS3X/mvtv9f+54//9+Z3b168+dZW/eUvqjb/sub9vHn1/1MZHl4=</latexit> hi,j(⌧) 2 CL <latexit sha1_base64="I4JWqw7AzzrXhInhpG2C9Lys824=">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</latexit> z ⇠ N(0, I) Generator Discriminator Transmit-receive antenna index Transmit-receive antenna index channel Spatial correlation Real/fake The generator learns to generate the channel Channel output The discriminator teaches the generator to learn channel distribution
  • 16.
  • 17.
    18 MIMO-GAN matches powerand delay profile of ground truth channels The method learns to find the channel function based on only input-output traces MAE: mean absolute error Orekondy et al., ICC 2022, https://arxiv.org/abs/2203.08588 TABLE 1: Power and delay statistics of MIMO-GAN and ground-truth (GT) channels Total Power (dB) Average Delay (𝝁𝒔) RMS Delay Spread (𝝁𝒔) TDL-A MIMO-GAN 4.648 0.2643 0.2862 GT 4.628 0.2641 0.2897 MAE -18.69 3.57 x 10—3 3.57 x 10—3 TDL-B MIMO-GAN 4.735 0.2276 0.2954 GT 4.688 0.2285 0.2987 MAE -14.99 3.37 x 10—3 3.37 x 10—3
  • 18.
    19 19 Communication channels arehard to accurately estimate A more accurate channel estimate at all the time steps for different dynamics enables more efficient communications Regular pilot symbols are transmitted to get periodic noisy observations (𝑜") of the ground truth channel Complex channel vector Mobile device trajectory Communication channel Channel states Noisy observations (pilots) 𝑜! ℎ! 𝑜" 𝑜# 𝑜$ 𝑜% ℎ% ℎ" ℎ# ℎ$ 𝑡! 𝑡" 𝑡# 𝑡$ 𝑡% Time 𝑡! Time 𝑡" Time 𝑡# ℎ&
  • 19.
    20 Classical Kalman filterslose accuracy over different dynamics Con: Optimal Kalman filter parameters vary with Doppler values Con: A single Kalman filter should not be used for all the Doppler values Pro: Kalman filter can work with arbitrary SNR and pilot patterns Pro: Kalman filter is interpretable KF(𝜃) Kalman state 𝑆$%" Estimated channel $ ℎ$ Observation 𝑜$ Kalman tracks the channel Time 𝑡$, velocity 𝑣$ Time 𝑡%, velocity 𝑣% Time 𝑡&, velocity 𝑣& KF(𝜃#) KF(𝜃$) KF(𝜃%)
  • 20.
    21 Standalone ML solutionsfor channel tracking have limitations Con: Cannot naturally deal with sporadically available observations (pilots) as input Con: Have non-interpretable hidden states Con: Do not generalize to different configurations (pilot patterns, SNR) Pro: Learn complex dynamics LSTM-based channel tracking
  • 21.
    22 Neural augmentation ofKalman filters offers the best of both worlds Kalman filter parameters RNN provides Kalman parameters at time 𝑡 Interpretability Out-of-domain generalization Robustness Expressive power RNN Update
  • 22.
    23 Neural-augmented KF closelymatches the ground truth channel
  • 23.
    24 Neural-augmented Kalman filtergeneralizes to unseen cases Neural-augmented Kalman filter (NA-KF) outperforms LSTM and Kalman filter over unseen pilot ratio* * Averaged error computed over high Dopplers Kumar Pratik et al. https://arxiv.org/abs/2109.12561. Globecom 2021 When trained over the whole data, NA-KF performs as good as or better than Kalman without knowledge of the exact dynamics LSTM breaks down on unseen pilot ratio NA-KF generalizes across scenarios with unseen Dopplers and pilot patterns Seen pilot ratio / Doppler Unseen pilot ratio Seen pilot ratio / Doppler Unseen pilot ratio Unseen pilot ratio / Doppler Kalman Filter LSTM Neural augmented Kalman Channel tracking gain (-NMSE in dB) 0 5 10 15 20 25 Channel tracking gain (-NMSE in dB) 0 5 10 15 20 25 Kalman Filter LSTM Neural augmented Kalman
  • 24.
    25 Machine learning is enablingRF sensing Detect gestures, movements, and objects by monitoring signal reflection patterns, enabling new use cases Active positioning A communications device along with nearby access points are used for positioning Passive positioning Access points alone are used to track the environment and determine positioning
  • 25.
    26 5G / Wi-Fipositioning is useful indoors and assists GPS outdoors Active positioning with RF sensing has a variety of use-cases Indoor navigation Vehicular navigation Asset tracking AGV tracking
  • 26.
    27 TRP: Transmission/reception point;SRS: Sounding reference signal; PRS: Positioning reference signal • Access points (TRPs) have known locations and are synchronized • A reference signal (SRS or PRS) is exchanged between phone and access points • The location of the phone is determined by analyzing the Channel Impulse Response (CIR) 5G can provide (indoor) positioning services X-Y-Z location? TRP#1 Blocker R e f l e c t o r TRP#2 TRP#3 CIR CIR CIR
  • 27.
    28 Testbed for indooractive positioning to prove technology
  • 28.
    29 Pro: no labelsrequired Con: not very accurate in non-line-of-sight conditions Con: doesn’t use multipath information Pro: very accurate, uses multipath Con: requires dense labels Con: robustness issues Position Time difference of arrival (TDOA) ML-assisted RF fingerprinting (RFFP) RFFP TDOA Current precise positioning methods have limitations in accuracy or feasibility CIR #1 CIR #2
  • 29.
    30 Learn position andenvironment from multipath propagation With enough unlabeled CSI samples, we can learn the geometry of the environment without labels Reflector Real Tx Real Rx access point Virtual Rx access point Key idea 1 Multi-path components (e.g., from reflectors) can help localize even with a single access point Triangulation can use real and virtual access points as reference. Key idea 2 There is only one unique environment geometry and access point location that can be compatible with a collection of (unlabeled) CSI samples
  • 30.
    31 Learn position andenvironment from multipath propagation Neural SLAM demonstrates an end-to-end trainable network to learn positions and environment that best reconstructs the CSI samples *Inputs and outputs may be CSI or related features such as ToF and/or AoA 𝑯𝒖 NN p 𝑝% 𝑝& 𝑝' 𝒑𝟎 Propagation model Learnable virtual AP locations Forward pass → Backward pass ← Predicted UE location Input feature* Learnable encoder network Maps CSI to location (Fixed) Decoder network Incorporates physics of reflections Reflector $ 𝑯𝒖 Reconstructed input* 𝒑𝟎 𝑝% p 𝑯𝒖
  • 31.
    32 32 Shreya Kadambi etal, ICC 2022, https://arxiv.org/abs/2203.08264 Neural RF SLAM achieves precise 3D positioning Neural RF SLAM achieves ~43.4 cm accuracy for 90% of users using only unlabeled CSI values from single anchor at 400 MHz bandwidth 3D ray tracing simulation Neural RF SLAM
  • 32.
    33 33 • RF signalscan be employed as bi-static radar • Any change in the environment also affects the wireless channel • Specifically, human motion, gestures, respiration • The signal propagation is complex • Self-supervised and weakly supervised machine learning enables robust analysis of the signal RF sensing is powered by machine learning
  • 33.
    34 RF sensing hasa variety of use cases across industries Home / enterprise / retail automation and security Consumer electronics Automotive Healthcare • Presence, positioning, tracking, activity classification • Better privacy as compared with camera-based • Works across walls • Touchless control (Phone, TV, laptop) • Proximity-based power save • Baby presence alarm • Presence-based setting • Vitals, attention monitoring • Contactless sleep monitoring, vitals, fall detection
  • 34.
    35 WiCluster enables non-line-of-sight passivepositioning No precise labels required To initialize the system, the user is guided by an app to provide room-level labels (kitchen, living room, …) The access points record the corresponding CSI packets WiCluster: making passive positioning deployable at scale • Three to four commercial IEEE 802.11 access points (AP), 5 GHz band • Circular array with 4cm radius • Bandwidth: 80 MHz • Packet rate: 90 Hz
  • 35.
    36 36 Experiments in realenvironments to test feasibility of deployment Environment #2 2D office,15m x 21m Environment #1 2D office,14m x 20m Environment #3 3D home
  • 36.
    37 Ilia Karmanov etal. WiCluster: Passive Indoor 2D/3D Positioning using WiFi without Precise Labels, GlobeCom 2021 End-to-end training Triplet loss to exploit temporal prior Clustering loss to exploit spatial prior • Cluster labels are updated after every epoch The crossed softmax ensures that the CSI-to-3D mapping is bijective Zone loss is used for embedding the 3D location into the floor plan • Floor plan can be 2D or 3D • Only requires a few labels Cluster Project Cluster Softmax loss Self-supervised Weakly-supervised WiCluster is first to do weakly- supervised passive positioning Input: CSI ResNet Latent labels Latent space 3D labels 3D locations Triplet loss Zone loss Floor plan Zone labels
  • 37.
    38 WiCluster model performswell in a real-life scenario
  • 38.
    39 WiCluster works instrong non-line-of-sight Conference room with concrete walls: strong non-line-of-sight Ground truth Inference 1.13m Offices 2.08m Conference room Mean error Mean error
  • 39.
    40 WiCluster works acrossmultiple floors 96%accurate zone classification + approximate 3D location
  • 40.
    41 Machine learning design forwireless communication and RF sensing Unsupervised learning • Learning distributions and manifolds is an approach to obtain features in an unsupervised way • Examples: WiCluster, Neural RF SLAM • Other perspectives: self-supervised learning, transfer learning Adaptive models • Models should be able to adapt to different channel conditions and setups • Examples: Hypernetwork Kalman, MIMO GAN Generalization • Designing ML models based on inductive bias, gained from domain knowledge, or neural augmentation can help generalization • Examples: Hypernetwork Kalman, MIMO-GAN Interpretability • Neural augmentation helps interpretability of modules in an ML model • Examples: Hypernetwork Kalman, MIMO-GAN
  • 41.
    42 Neural RF sensingand neural rendering offer synergistic capabilities Analogous to the compelling capabilities of computer vision and computer graphics Computer vision Computer graphics ML methods to recover scenes/objects Physical world Image ML methods for rendering Neural RF SLAM/ sensing Neural rendering Learning position and environment from CSI Environment Channel State Information (CSI) Generate spatially consistent CSI from the environment
  • 42.
    43 AI is enhancingwireless communications with generative modeling and neural augmentation AI is enhancing RF sensing through self-supervised and unsupervised learning to better understand the environment Qualcomm AI Research is conducting leading research in applying AI for RF sensing and improved communications
  • 43.
    Nothing in thesematerials is an offer to sell any of the components or devices referenced herein. ©2018-2022 Qualcomm Technologies, Inc. and/or its affiliated companies. All Rights Reserved. Qualcomm is a trademark or registered trademark of Qualcomm Incorporated. Other products and brand names may be trademarks or registered trademarks of their respective owners. References in this presentation to “Qualcomm” may mean Qualcomm Incorporated, Qualcomm Technologies, Inc., and/or other subsidiaries or business units within the Qualcomm corporate structure, as applicable. Qualcomm Incorporated includes our licensing business, QTL, and the vast majority of our patent portfolio. Qualcomm Technologies, Inc., a subsidiary of Qualcomm Incorporated, operates, along with its subsidiaries, substantially all of our engineering, research and development functions, and substantially all of our products and services businesses, including our QCT semiconductor business. Follow us on: For more information, visit us at: qualcomm.com & qualcomm.com/blog Thank you