On Optimal Multi-user Beam Alignment in Millimeter Wave Wireless - - PowerPoint PPT Presentation

1/20

On Optimal Multi-user Beam Alignment in Millimeter Wave Wireless Systems

Abbas Khalili†, Shahram Shahsavari◇, Mohammad A. (Amir) Khojastepour☆ , Elza Erkip†

†New York University, ◇ University of Waterloo, ☆ NEC Laboratories America

ISIT 2020

Khalili, Shahsavari, Khojastepour, Erkip

2/20

Motivation

MmWave

Pros: {Large bandwidth Multi-Gbps data rate

Khalili, Shahsavari, Khojastepour, Erkip

2/20

Motivation

MmWave

Pros: {Large bandwidth Multi-Gbps data rate Cons: {High pathloss Intense shadowing

Khalili, Shahsavari, Khojastepour, Erkip

2/20

Motivation

MmWave

Pros: {Large bandwidth Multi-Gbps data rate Cons: {High pathloss Intense shadowing

⇒ Beamforming: Use narrow beams for communication

Khalili, Shahsavari, Khojastepour, Erkip

2/20

Motivation

MmWave channel is sparse with few spatial clusters

Khalili, Shahsavari, Khojastepour, Erkip

2/20

Motivation

MmWave channel is sparse with few spatial clusters Beamforming

Match the Tx pattern to the clusters’ angle of departure (AoD) Match the Rx pattern to the clusters’ angle of arrival (AoA)

Khalili, Shahsavari, Khojastepour, Erkip

3/20

Beam Alignment

Beam Alignment The AoD and AoA do not change rapidly

Khalili, Shahsavari, Khojastepour, Erkip

3/20

Beam Alignment

Beam Alignment The AoD and AoA do not change rapidly Use beams at the Tx and Rx to estimate the AoD and AoA

Khalili, Shahsavari, Khojastepour, Erkip

3/20

Beam Alignment

Beam Alignment The AoD and AoA do not change rapidly Use beams at the Tx and Rx to estimate the AoD and AoA Interactive beam alignment

Wait for feedback after each transmission

Non-interactive beam alignment

Sends all the packets without waiting for feedback

Khalili, Shahsavari, Khojastepour, Erkip

3/20

Beam Alignment

Beam Alignment The AoD and AoA do not change rapidly Use beams at the Tx and Rx to estimate the AoD and AoA Interactive beam alignment

Wait for feedback after each transmission

Non-interactive beam alignment

Sends all the packets without waiting for feedback

Analog beam alignment

use one RF-chain to perform beam alignment

Khalili, Shahsavari, Khojastepour, Erkip

4/20

Problem and Objective

This talk: Non-interactive analog beam alignment at base station when there are multiple users:

4/20

Problem and Objective

This talk: Non-interactive analog beam alignment at base station when there are multiple users: Objective: Minimize the average expected beamwidth of the users.

4/20

Problem and Objective

This talk: Non-interactive analog beam alignment at base station when there are multiple users: Objective: Minimize the average expected beamwidth of the users.

1 What is the fundamental limit?

4/20

Problem and Objective

This talk: Non-interactive analog beam alignment at base station when there are multiple users: Objective: Minimize the average expected beamwidth of the users.

1 What is the fundamental limit? 2 How to achieve this fundamental limit? Khalili, Shahsavari, Khojastepour, Erkip

5/20

Preview of the Results

Non-interactive analog beam alignment at the base station for two scenarios

Base station can use beams of any shapes Base station is constrained to use only contiguous beams

Khalili, Shahsavari, Khojastepour, Erkip

5/20

Preview of the Results

Non-interactive analog beam alignment at the base station for two scenarios

Base station can use beams of any shapes Base station is constrained to use only contiguous beams

Upper- and lower-bounds the minimum average expected beamwidth

Upper- and Lower-bounds meet for uniform priors

Khalili, Shahsavari, Khojastepour, Erkip

5/20

Preview of the Results

Non-interactive analog beam alignment at the base station for two scenarios

Base station can use beams of any shapes Base station is constrained to use only contiguous beams

Upper- and lower-bounds the minimum average expected beamwidth

Upper- and Lower-bounds meet for uniform priors

Schemes that achieve the upper-bounds and optimal performance

Khalili, Shahsavari, Khojastepour, Erkip

5/20

Preview of the Results

Non-interactive analog beam alignment at the base station for two scenarios

Base station can use beams of any shapes Base station is constrained to use only contiguous beams

Upper- and lower-bounds the minimum average expected beamwidth

Upper- and Lower-bounds meet for uniform priors

Schemes that achieve the upper-bounds and optimal performance 2× reduction of average expected beamwidth compared to exhaustive search used in the standards

Khalili, Shahsavari, Khojastepour, Erkip

6/20

Literature

Chiu, Ronquillo, Javidi JSAC 2019.

Single-user analog interactive beam alignment Adaptive and sequential alignment algorithm Search time of the proposed algorithm asymptotically matches the performance of the noiseless bisection

Khalili, Shahsavari, Khojastepour, Erkip

6/20

Literature

Chiu, Ronquillo, Javidi JSAC 2019.

Single-user analog interactive beam alignment Adaptive and sequential alignment algorithm Search time of the proposed algorithm asymptotically matches the performance of the noiseless bisection

Hassan, Michelusi ITA 2018.

Two user analog interactive beam alignment Minimize the power consumption during data transmission Bisection search algorithm is optimal

Khalili, Shahsavari, Khojastepour, Erkip

6/20

Literature

Chiu, Ronquillo, Javidi JSAC 2019.

Single-user analog interactive beam alignment Adaptive and sequential alignment algorithm Search time of the proposed algorithm asymptotically matches the performance of the noiseless bisection

Hassan, Michelusi ITA 2018.

Two user analog interactive beam alignment Minimize the power consumption during data transmission Bisection search algorithm is optimal

Shahsavari, Khojastepour, Erkip PIMRC 2019.

Single-user analog interactive beam alignment Dynamic programming scheme that maximizes system throughput given the total number of time-slots

Khalili, Shahsavari, Khojastepour, Erkip

6/20

Literature

Chiu, Ronquillo, Javidi JSAC 2019.

Single-user analog interactive beam alignment Adaptive and sequential alignment algorithm Search time of the proposed algorithm asymptotically matches the performance of the noiseless bisection

Hassan, Michelusi ITA 2018.

Two user analog interactive beam alignment Minimize the power consumption during data transmission Bisection search algorithm is optimal

Shahsavari, Khojastepour, Erkip PIMRC 2019.

Single-user analog interactive beam alignment Dynamic programming scheme that maximizes system throughput given the total number of time-slots

This work:

Multi-user non-interactive analog beam alignment Bounds on the minimum average expected beamwidth Achievablity schemes

Khalili, Shahsavari, Khojastepour, Erkip

7/20

System Assumptions

N-user downlink each with a single link

Khalili, Shahsavari, Khojastepour, Erkip

7/20

System Assumptions

N-user downlink each with a single link Omnidirectional reception at users

Khalili, Shahsavari, Khojastepour, Erkip

7/20

System Assumptions

N-user downlink each with a single link Omnidirectional reception at users Analog beamforming at base station

Khalili, Shahsavari, Khojastepour, Erkip

7/20

System Assumptions

N-user downlink each with a single link Omnidirectional reception at users Analog beamforming at base station Ideal beams

θ1 θ2 θ3 θ4 ACR = (θ1, θ2] ∪ (θ3, θ4] ACR stands for angular coverage region

Khalili, Shahsavari, Khojastepour, Erkip

7/20

System Assumptions

N-user downlink each with a single link Omnidirectional reception at users Analog beamforming at base station Ideal beams

θ1 θ2 θ3 θ4 ACR = (θ1, θ2] ∪ (θ3, θ4] ACR stands for angular coverage region

Base station transmits beam alignment packets through b scanning beams

Khalili, Shahsavari, Khojastepour, Erkip

7/20

System Assumptions

N-user downlink each with a single link Omnidirectional reception at users Analog beamforming at base station Ideal beams

θ1 θ2 θ3 θ4 ACR = (θ1, θ2] ∪ (θ3, θ4] ACR stands for angular coverage region

Base station transmits beam alignment packets through b scanning beams Users’ feedback determines the indices of received beam alignment packets

Khalili, Shahsavari, Khojastepour, Erkip

7/20

System Assumptions

N-user downlink each with a single link Omnidirectional reception at users Analog beamforming at base station Ideal beams

θ1 θ2 θ3 θ4 ACR = (θ1, θ2] ∪ (θ3, θ4] ACR stands for angular coverage region

Base station transmits beam alignment packets through b scanning beams Users’ feedback determines the indices of received beam alignment packets No noise at the users’ side and no noise on the feedback

Khalili, Shahsavari, Khojastepour, Erkip

8/20

Objective

Objective: {Φ∗

j }b j=1 = arg min {Φj}b

j=1

N

∑

i=1

wiE[∣Beam(Ψi)∣] wi,∑N

i=1 wi = 1 ∶ Importance or quality of service

Φj ∶ Scanning beam used at the jth time slot Ψi ∼ fΨi(ψ) ∶ Angle of departure of the ith user ∣Beam(Ψi)∣ ∶ Width of the beam allocated to the ith user that includes its AoD

Khalili, Shahsavari, Khojastepour, Erkip

9/20

Multi-user to Single-user

E[∣Beam(Ψi)∣] =

M

∑

k=1

∣uk∣∫ψ∈uk fΨi(ψ)dψ, fΨi(ψ) ∶ The prior on the AoD of the ith user. {uk}M

k=1 ∶ Partition of (0,2π] created by scanning beams.

u1 u2 u4 u3 Φ1 Φ2

Khalili, Shahsavari, Khojastepour, Erkip

9/20

Multi-user to Single-user

E[∣Beam(Ψi)∣] =

M

∑

k=1

∣uk∣∫ψ∈uk fΨi(ψ)dψ, fΨi(ψ) ∶ The prior on the AoD of the ith user. {uk}M

k=1 ∶ Partition of (0,2π] created by scanning beams.

Lemma: Equivalent Single-user Problem {Φ∗

j }b j=1 = arg min {Φj}b

j=1

N

∑

i=1

wjE[∣Beam(Ψi)∣] , Multi-user = arg min

{Φj}b

j=1

E[∣Beam(Ψ)∣], , Single-user where Ψ ∼ ∑N

i=1 wifΨi(ψ)

Khalili, Shahsavari, Khojastepour, Erkip

10/20

No Constraints on the Scanning Beams

let U

∗ = min{Φj}b

j=1 ∑N

i=1 wiE[∣Beam(Ψi)∣]

Khalili, Shahsavari, Khojastepour, Erkip

10/20

No Constraints on the Scanning Beams

let U

∗ = min{Φj}b

j=1 ∑N

i=1 wiE[∣Beam(Ψi)∣]

Theorem: Minimum Average Expected Beamwidth (Non- contiguous) 2h(Ψ) 2b ≤ U

∗ ≤ 2π

2b , where h(Ψ) is the differential entropy of random variable Ψ.

Khalili, Shahsavari, Khojastepour, Erkip

10/20

No Constraints on the Scanning Beams

let U

∗ = min{Φj}b

j=1 ∑N

i=1 wiE[∣Beam(Ψi)∣]

Theorem: Minimum Average Expected Beamwidth (Non- contiguous) 2h(Ψ) 2b ≤ U

∗ ≤ 2π

2b , where h(Ψ) is the differential entropy of random variable Ψ. The upper-bound does not depend on the prior is achievable

Khalili, Shahsavari, Khojastepour, Erkip

10/20

No Constraints on the Scanning Beams

let U

∗ = min{Φj}b

j=1 ∑N

i=1 wiE[∣Beam(Ψi)∣]

Theorem: Minimum Average Expected Beamwidth (Non- contiguous) 2h(Ψ) 2b ≤ U

∗ ≤ 2π

2b , where h(Ψ) is the differential entropy of random variable Ψ. The upper-bound does not depend on the prior is achievable For uniform distribution, upper- and lower-bounds meet

Khalili, Shahsavari, Khojastepour, Erkip

10/20

No Constraints on the Scanning Beams

let U

∗ = min{Φj}b

j=1 ∑N

i=1 wiE[∣Beam(Ψi)∣]

Theorem: Minimum Average Expected Beamwidth (Non- contiguous) 2h(Ψ) 2b ≤ U

∗ ≤ 2π

2b , where h(Ψ) is the differential entropy of random variable Ψ. The upper-bound does not depend on the prior is achievable For uniform distribution, upper- and lower-bounds meet Uniform distribution: No prior information on AoD

Khalili, Shahsavari, Khojastepour, Erkip

11/20

Achieving the Upper-bound

i = 1 i = 2 i = 3 i = 4

An example of the proposed BA strategy for b = 4

Khalili, Shahsavari, Khojastepour, Erkip

11/20

Achieving the Upper-bound

i = 1 i = 2 i = 3 i = 4

An example of the proposed BA strategy for b = 4

Partition (0,2π] into 2b equal width regions

Khalili, Shahsavari, Khojastepour, Erkip

11/20

Achieving the Upper-bound

i = 1 i = 2 i = 3 i = 4

An example of the proposed BA strategy for b = 4

Partition (0,2π] into 2b equal width regions First beam Φ1: Union of any half of the regions

Khalili, Shahsavari, Khojastepour, Erkip

11/20

Achieving the Upper-bound

i = 1 i = 2 i = 3 i = 4

An example of the proposed BA strategy for b = 4

Partition (0,2π] into 2b equal width regions First beam Φ1: Union of any half of the regions Second beam Φ2: Union of any half of the regions inside the first beams and any half of the regions outside the first beam.

Khalili, Shahsavari, Khojastepour, Erkip

11/20

Achieving the Upper-bound

i = 1 i = 2 i = 3 i = 4

An example of the proposed BA strategy for b = 4

Partition (0,2π] into 2b equal width regions First beam Φ1: Union of any half of the regions Second beam Φ2: Union of any half of the regions inside the first beams and any half of the regions outside the first beam. Third beam Φ3: Union of any half of the regions inside Φ1 ∩ Φ2, Φ1 ∩ Φ2, Φ1 ∩ Φ2 and Φ1 ∩ Φ2.

Khalili, Shahsavari, Khojastepour, Erkip

11/20

Achieving the Upper-bound

i = 1 i = 2 i = 3 i = 4

An example of the proposed BA strategy for b = 4

Partition (0,2π] into 2b equal width regions First beam Φ1: Union of any half of the regions Second beam Φ2: Union of any half of the regions inside the first beams and any half of the regions outside the first beam. Third beam Φ3: Union of any half of the regions inside Φ1 ∩ Φ2, Φ1 ∩ Φ2, Φ1 ∩ Φ2 and Φ1 ∩ Φ2. The process continuous in a similar fashion.

Khalili, Shahsavari, Khojastepour, Erkip

12/20

Optimal Scanning Beams Design

We can use the construction for the upper-bound to design b scanning beams to achieve any 2b contiguous URs.

Khalili, Shahsavari, Khojastepour, Erkip

12/20

Optimal Scanning Beams Design

We can use the construction for the upper-bound to design b scanning beams to achieve any 2b contiguous URs. ⇒ If the optimal URs are contiguous and we have their intervals, we have a way to design the scanning beams.

Khalili, Shahsavari, Khojastepour, Erkip

13/20

Optimal Scanning Beams Design

Prior fΨ(⋅) is a monotone function in (0,2π]

Khalili, Shahsavari, Khojastepour, Erkip

13/20

Optimal Scanning Beams Design

Prior fΨ(⋅) is a monotone function in (0,2π] Proposition: Contiguity of Uncertainty Regions Optimal scanning beams result in contiguous partition

13/20

Optimal Scanning Beams Design

Prior fΨ(⋅) is a monotone function in (0,2π] Proposition: Contiguity of Uncertainty Regions Optimal scanning beams result in contiguous partition u∗

ℓ = (x∗ ℓ ,x∗ ℓ+1],ℓ ∈ [M]

arg min

xℓ,ℓ∈[M] M

∑

i=1

(xℓ+1 − xℓ)∫

xℓ+1 xℓ

fΨ(x)dx. where M is the cardinality of the partition created by optimal beams.

Khalili, Shahsavari, Khojastepour, Erkip

13/20

Optimal Scanning Beams Design

Prior fΨ(⋅) is a monotone function in (0,2π] Proposition: Contiguity of Uncertainty Regions Optimal scanning beams result in contiguous partition u∗

ℓ = (x∗ ℓ ,x∗ ℓ+1],ℓ ∈ [M]

arg min

xℓ,ℓ∈[M] M

∑

i=1

(xℓ+1 − xℓ)∫

xℓ+1 xℓ

fΨ(x)dx. where M is the cardinality of the partition created by optimal beams. Set M = 2b The scanning beams can be designed similar to the upper-bound construction

Khalili, Shahsavari, Khojastepour, Erkip

13/20

Optimal Scanning Beams Design

Prior fΨ(⋅) is a monotone function in (0,2π] Proposition: Contiguity of Uncertainty Regions Optimal scanning beams result in contiguous partition u∗

ℓ = (x∗ ℓ ,x∗ ℓ+1],ℓ ∈ [M]

arg min

xℓ,ℓ∈[M] M

∑

i=1

(xℓ+1 − xℓ)∫

xℓ+1 xℓ

fΨ(x)dx. where M is the cardinality of the partition created by optimal beams. Set M = 2b The scanning beams can be designed similar to the upper-bound construction There is a one-to-one mapping form any prior to a monotone function

Khalili, Shahsavari, Khojastepour, Erkip

14/20

Contiguous Scanning Beams

Non-contiguous beams might not be practical

Khalili, Shahsavari, Khojastepour, Erkip

14/20

Contiguous Scanning Beams

Non-contiguous beams might not be practical Proposition: Contiguity of Uncertainty Regions Optimal scanning beams result in contiguous partition

Khalili, Shahsavari, Khojastepour, Erkip

14/20

Contiguous Scanning Beams

Non-contiguous beams might not be practical Proposition: Contiguity of Uncertainty Regions Optimal scanning beams result in contiguous partition Optimal partition’s maximum cardinality is M = 2b

Khalili, Shahsavari, Khojastepour, Erkip

14/20

Contiguous Scanning Beams

Non-contiguous beams might not be practical Proposition: Contiguity of Uncertainty Regions Optimal scanning beams result in contiguous partition Optimal partition’s maximum cardinality is M = 2b contiguous scanning beams ⇒ contiguous data transmission beams

Khalili, Shahsavari, Khojastepour, Erkip

15/20

Contiguous Scanning Beams

let U

∗ = min{Φj}b

j=1 ∑N

i=1 wiE[∣Beam(Ψi)∣]

Theorem: Minimum Average Expected Beamwidth (Contigu-

• us)

2h(Ψ) 2b ≤ U

∗ ≤ π

b , where h(Ψ) is the differential entropy of random variable Ψ.

Khalili, Shahsavari, Khojastepour, Erkip

15/20

Contiguous Scanning Beams

let U

∗ = min{Φj}b

j=1 ∑N

i=1 wiE[∣Beam(Ψi)∣]

Theorem: Minimum Average Expected Beamwidth (Contigu-

• us)

2h(Ψ) 2b ≤ U

∗ ≤ π

b , where h(Ψ) is the differential entropy of random variable Ψ. The upper-bound does not depend on the prior is achievable

Khalili, Shahsavari, Khojastepour, Erkip

15/20

Contiguous Scanning Beams

let U

∗ = min{Φj}b

j=1 ∑N

i=1 wiE[∣Beam(Ψi)∣]

Theorem: Minimum Average Expected Beamwidth (Contigu-

• us)

2h(Ψ) 2b ≤ U

∗ ≤ π

b , where h(Ψ) is the differential entropy of random variable Ψ. The upper-bound does not depend on the prior is achievable For uniform distribution, upper- and lower-bounds meet

Khalili, Shahsavari, Khojastepour, Erkip

16/20

Achieving the Upper-bound

i = 1 i = 2 i = 3 i = 4

An example of the proposed BA scheme for b = 4

Khalili, Shahsavari, Khojastepour, Erkip

16/20

Achieving the Upper-bound

i = 1 i = 2 i = 3 i = 4

An example of the proposed BA scheme for b = 4

Partition (0,2π] into 2b equal width regions Assign indexes {1,2,⋯,2b} to the regions

Khalili, Shahsavari, Khojastepour, Erkip

16/20

Achieving the Upper-bound

i = 1 i = 2 i = 3 i = 4

An example of the proposed BA scheme for b = 4

Partition (0,2π] into 2b equal width regions Assign indexes {1,2,⋯,2b} to the regions First beam Φ1: Union of the regions {1,2,⋯,b}

Khalili, Shahsavari, Khojastepour, Erkip

16/20

Achieving the Upper-bound

i = 1 i = 2 i = 3 i = 4

An example of the proposed BA scheme for b = 4

Partition (0,2π] into 2b equal width regions Assign indexes {1,2,⋯,2b} to the regions First beam Φ1: Union of the regions {1,2,⋯,b} Second beam Φ2: Union of the regions {2,⋯,b,b + 1}

Khalili, Shahsavari, Khojastepour, Erkip

16/20

Achieving the Upper-bound

i = 1 i = 2 i = 3 i = 4

An example of the proposed BA scheme for b = 4

Partition (0,2π] into 2b equal width regions Assign indexes {1,2,⋯,2b} to the regions First beam Φ1: Union of the regions {1,2,⋯,b} Second beam Φ2: Union of the regions {2,⋯,b,b + 1} The process continuous in a similar fashion.

Khalili, Shahsavari, Khojastepour, Erkip

17/20

Optimal Scanning Beams Design

Theorem: Optimal Scanning Beams (Contiguous) Contiguous uncertainty regions {u∗}2b

{ℓ=1}

u∗

ℓ = (x∗ ℓ ,x∗ ℓ+1],ℓ ∈ [2b]

arg min

xℓ,ℓ∈[2b] 2b

∑

i=1

(xℓ+1 − xℓ)∫

xℓ+1 xℓ

fΨ(x)dx.

17/20

Optimal Scanning Beams Design

Theorem: Optimal Scanning Beams (Contiguous) Contiguous uncertainty regions {u∗}2b

{ℓ=1}

u∗

ℓ = (x∗ ℓ ,x∗ ℓ+1],ℓ ∈ [2b]

arg min

xℓ,ℓ∈[2b] 2b

∑

i=1

(xℓ+1 − xℓ)∫

xℓ+1 xℓ

fΨ(x)dx. Same construction as of the upper-bound gives us: φ∗

i = (x∗ j ,x∗ j+b],j ∈ [b]

Khalili, Shahsavari, Khojastepour, Erkip

18/20

Two User Scenario: An Example

Equal weights w1 = w2 = 0.5 Ψ1 ∼ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ uniform(0, π

2 ]

p = 0.9, uniform(π

2 ,2π]

p = 0.1. Ψ2 ∼ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ uniform(π, 3π

2 ]

p = 0.9, uniform((0,π] ∪ (3π

2 ,2π])

p = 0.1.

Khalili, Shahsavari, Khojastepour, Erkip

18/20

Two User Scenario: An Example

Equal weights w1 = w2 = 0.5 Ψ1 ∼ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ uniform(0, π

2 ]

p = 0.9, uniform(π

2 ,2π]

p = 0.1. Ψ2 ∼ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ uniform(π, 3π

2 ]

p = 0.9, uniform((0,π] ∪ (3π

2 ,2π])

p = 0.1.

i = 1 i = 2 i = 3 i = 4

Scanning beams and corresponding partitioning regions when b = 4.

Khalili, Shahsavari, Khojastepour, Erkip

18/20

Two User Scenario: An Example

Equal weights w1 = w2 = 0.5 Ψ1 ∼ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ uniform(0, π

2 ]

p = 0.9, uniform(π

2 ,2π]

p = 0.1. Ψ2 ∼ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ uniform(π, 3π

2 ]

p = 0.9, uniform((0,π] ∪ (3π

2 ,2π])

p = 0.1.

i = 1 i = 2 i = 3 i = 4

Scanning beams and corresponding partitioning regions when b = 4.

Narrower partition regions in more likely areas

Khalili, Shahsavari, Khojastepour, Erkip

19/20

Two User Scenario: An Example

Khalili, Shahsavari, Khojastepour, Erkip

19/20

Observations

Observations For b ≥ 3, the optimal average expected beamwidth is at least 2× smaller than that of the exhaustive search. The distance between upper- and lower-bounds reduces as b is

• increased. Therefore, for sufficiently large b one can use the

upper-bound construction instead of solving the optimization.

Khalili, Shahsavari, Khojastepour, Erkip

20/20

Conclusion

Non-interactive analog beam alignment at the base station

Khalili, Shahsavari, Khojastepour, Erkip

20/20

Conclusion

Non-interactive analog beam alignment at the base station Two scenarios

Base station can use beams of any shapes Base station is constrained to use only contiguous beams

Khalili, Shahsavari, Khojastepour, Erkip

20/20

Conclusion

Non-interactive analog beam alignment at the base station Two scenarios

Base station can use beams of any shapes Base station is constrained to use only contiguous beams

Upper- and lower-bounds the minimum average expected beamwidth

Upper- and Lower-bounds meet for uniform priors

Khalili, Shahsavari, Khojastepour, Erkip

20/20

Conclusion

Non-interactive analog beam alignment at the base station Two scenarios

Base station can use beams of any shapes Base station is constrained to use only contiguous beams

Upper- and lower-bounds the minimum average expected beamwidth

Upper- and Lower-bounds meet for uniform priors

Schemes that achieve the upper-bounds and optimal performance

Khalili, Shahsavari, Khojastepour, Erkip

20/20

Conclusion

Non-interactive analog beam alignment at the base station Two scenarios

Base station can use beams of any shapes Base station is constrained to use only contiguous beams

Upper- and lower-bounds the minimum average expected beamwidth

Upper- and Lower-bounds meet for uniform priors

Schemes that achieve the upper-bounds and optimal performance 2× reduction of average expected beamwidth compared to exhaustive search used in the standards

Khalili, Shahsavari, Khojastepour, Erkip

20/20

References

[1] A. Khalili, S. Shahsavari, M. A. A. Khojastepour, and E. Erkip, “On Optimal Multi-user Beam Alignment in Millimeter Wave Wireless Systems,” in Proc. IEEE International Symposium on Information Theory, (ISIT 2020). [2] S-E. Chiu, N. Ronquillo, and T. Javidi, “Active learning and CSIacquisition for mmWave initial alignment,” IEEE Journal on SelectedAreas in Communications, (JSAC 2019). [3] R. A. Hassan, and N. Michelusi, “Multi-user beam-alignment for millimeter-wave networks,” in Proc. IEEE Information Theory and Applications Workshop, (ITA 2018). [4] S. Shahsavari, M. A. A. Khojastepour, and E. Erkip “Beam Training Optimization in Millimeter-wave Systems under Beamwidth, Modulation and Coding Constraints,” in Proc. IEEE International Symposium on Personal, Indoor and Mobile Radio Communications, (PIMRC 2019).

Khalili, Shahsavari, Khojastepour, Erkip