nicolò michelusi ([email protected]), muddassar hussain · international conference on...

1
Nicolò Michelusi ([email protected]), Muddassar Hussain Based on a paper to appear at 56 th Allerton conference on Communication 2018 1 2 3 4 5 6 7 8 9 10 11 12 NSF funded project: NSF EARS CNS-1642982: "Real-time Control of Dense, Mobile, Millimeter Wave Networks Using a Programmable Architecture" (2016-2019), Nicolò Michelusi (PI), David Love (co-PI), James Krogmeier (co-PI), Purdue University; Alex Sprintson (co-PI), TAMU; Chris Anderson (co-PI), USNA M. Hussain, N. Michelusi “Coded Energy-Efficient Beam-Alignment in Millimeter Wave Networks”, to appear on 56th Annual Allerton Conference on Communication, Control, and Computing M. Hussain, N. Michelusi “Robust Energy-Efficient Beam-Alignment through Error Correction Coding”, submitted to ITA Workshop 2019 CNS-1642982 c = [1, 0] u = [1, 1] BA Design Objective mm-Wave Beam-Alignment Summary ECC Interpretation of BA Effects of BA Errors Error Correction -20 -15 -10 -5 0 5 10 15 20 Optimization Problem q High capacity à Narrow beams Ø Beam alignment (BA) q Side lobes & noise BA errors, outages q We need error-robust BA ) How to design a BA scheme robust to BA errors? q Single isotropic UE (AoD: ) Ø Generalized in the paper q Received signal: BA Beacon H 1 : y k = h k s 2P k |B k | s k + n k , (alignment), H 0 : y k = n k , (mis-alignment). Beam Alignment Feedback Data Communication B 1 B L T F T DT u Simplified Model q Optimal Detector: Ø Error free case: Ø In practice: Ø by tuning and q Over entire BA phase: Ø q Assumption: i.i.d Ø pmf: th p fa = p md = p e Weight of |s H k y k | 2 ks k k 2 H 0 7 H 1 th c k = χ(2 B k ) u = c e 2 {0, 1} L p(e)= L W (e) p W (e) e (1 - p e ) L-W (e) e φ s [energy/rad] u k = c k e k 2 {0, 1} u =[u 1 ,...,u L ] e Detection q Definitions: Ø Support of : Ø Set of all valid : Ø Set of all non-empty : U c , \ L k=1 B c k k c U c G , {U c : c 2 C} partition -! -! U c , \ L k=1 B c k k B k = [ c2C :c k =1 U c C , {c 2 {0, 1} L : U c 6= ;} B 1 B 2 B 3 U [1,0,0] U [1,0,1] U [0,1,1] U [0,1,0] C = {[1, 0, 0], [1, 0, 1], [0, 1, 1], [0, 1, 0]} (G , C ) F , {B 1 , B 2 , B 3 } B 1 k = B k B 0 k =[-, ) \B k error Incorrect inference à Outage Data Communication U [0,1] U [0,0] B 1 error True: Detected: True: Detected: Beam Alignment B 2 c = [1,x] u = [1,x] U [1,0] U [1,1] C = {[0, 0], [0, 1], [1, 0], [1, 1]} True: Detected: Beam Alignment B 3 c = [1, 0, 0, x, x] u = [1, 1, 0, x, x] B 5 True: Detected: True: Detected: c = [1, 0, 0, 0,x] u = [1, 1, 0, 0,x] c = [1, 0, 0, 0, 0] u = [1, 1, 0, 0, 0] error error error U [1,0,0,0,0] U [1,0,1,1,1] U [0,1,1,1,0] U [0,1,0,0,1] 1 error corrected Data Communication C = n [1, 0, 0, 0, 0], [1, 0, 1, 1, 1], [0, 1, 1, 1, 0], [0, 1, 0, 0, 1] o Φ(·): {0, 1} L 7! C Decoding rule B 4 q Minimize avg power as a function of , q Outage constraint: q to achieve desired rate requirement for UE q Decoding rule: q Change of variable : F = {B 1 ,..., B L } Φ Φ(·): {0, 1} L 7! C φ d [energy/rad] ¯ P avg (F , Φ) , 1 T fr E " L X k=1 φ s |B k | + φ d |B comm (u)| # B comm (u)= \ L k=1 B Φ(u) k = U Φ(u) q Optimization: s.t. q Combinatorial and non-convex Ø Select , and independently q as ECC that can correct up to -bits of errors q as min distance decoding for q q Convex Upper bound ( ) Ø can be made arbitrarily tight by reducing C G " Φ(·) C C G = arg min G ¯ P avg (G , C , Φ) min G ,C ,Φ ¯ P avg (G , C , Φ) Φ(·) P(W (e) > ") Optimal Solution q New Objective: q Optimization: q Water-filling Solution: Ø Higher à Frequent scanning W (c) ! = arg min !0 ˆ P avg (! ) s.t. X c2C ! c =2! c = ⇡φ s P(W (e) ")φ d [λ - W (c)] + Dual var : water level is partition of G [-, ) U [0,1] U [0,0] U [1,0] U [1,1] ( U c = \ L k=1 B c k k ) ¯ P avg (G , C , Φ) , 1 T fr E " L X k=1 φ s |[ d2C :d k =1 U d | + φ d U Φ(u) # ! c , |U c | Numerical Results P avg [dBm] Analog beams used Noise psd = -173 dBm Frame duration = 20ms Slot = 10us Bandwidth = 0.5 GHz Distance = 10m Pathloss exp. = 2 Mis-detection prob. = 0.3 False-alarm prob. = 0.3 #Antennas = 64 (BS) Spectral Efficiency [bps/Hz] uncoded (7,4) Hamming exhaustive Uncoded cannot correct error(s) Exhaustive can correct 1 error with 50% prob. (7,4) Hamming can correct 1 error with 100% prob. -1.5 dB -3 dB q Energy-efficient coded BA Ø Convex optimization based design Ø Water-filling solution Ø Superior performance Ø 3dB gain over exhaustive search Ø 4.5dB gain over uncoded BA q 1 BS, M users (UEs) q L beam-alignment slot Ø BS uses f 1 , f 2 , …, f L Ø UEs uses g 1 , g 2 , …, g L Ø i.i.d fading across the slots q L-bits collision free feedback from UEs q TDMA in data transmission M 1 d 1 Beam Alignment Feedback Data Communication d M (f 1 , g 1 ) (f L , g L ) T F 1 M u 1 u M Model P ( 62 U Φ(u) ) p out P ( 62 U Φ(u) ) p out ! Φ(ce) 2ˆ P avg (! ) , φ s T fr X c2C ! c W (c) + φ d 2T fr X c2C n ! 2 c P(W (e) ")+2⇡! c P(W (e) > ") o ¯ P avg (! )= φ s T fr X c2C ! c W (c) + φ d 2T fr X c2C n ! 2 c P(W (e) ")+ X e2{0,1} L :W (e)>! Φ(ce) ! c p(e) o

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Page 1: Nicolò Michelusi (michelus@purdue.edu), Muddassar Hussain · International Conference on Communications, November 2015, pp. 236– 240. Track with Basic Transmission Losses on Holloway

Nicolò Michelusi([email protected]),Muddassar HussainBasedonapapertoappearat56th AllertonconferenceonCommunication2018

1 2 3

4 5 6

7 8 9

10 11 12

NSF funded project: NSF EARS CNS-1642982: "Real-time Control of Dense, Mobile,Millimeter Wave Networks Using a Programmable Architecture" (2016-2019),Nicolò Michelusi (PI), David Love (co-PI), James Krogmeier (co-PI), PurdueUniversity; Alex Sprintson (co-PI), TAMU; Chris Anderson (co-PI), USNAM. Hussain, N. Michelusi “Coded Energy-Efficient Beam-Alignment in MillimeterWave Networks”, to appear on 56th Annual Allerton Conference onCommunication, Control, and Computing

M. Hussain, N. Michelusi “Robust Energy-Efficient Beam-Alignment through ErrorCorrection Coding”, submitted to ITA Workshop 2019

CNS-1642982

c = [1, 0]

u = [1, 1]<latexit sha1_base64="MRTHuvMgSbqNzvfxZvk5onlntyc=">AAACHnicbVDLSsNAFJ34rPEVdelmsAgupGS0oBuh4MZlBfuAJJTJZNIOnTyYmQgl9EPc+SfuRBfixoX+jZM0grYeGDiccy/nzvFTzqSy7S9jaXlldW29tmFubm3v7Fp7+12ZZILQDkl4Ivo+lpSzmHYUU5z2U0Fx5HPa88fXhd+7p0KyJL5Tk5R6ER7GLGQEKy0NrHM3wmrkh5DAKwed2p7papg/alaquVsG5YIGUzT1Blbdbtgl4CJBFamDCu2B9eEGCckiGivCsZQOslPl5VgoRjidmm4maYrJGA+po2mMIyq9vMycwmOtBDBMhH6xgqX6eyPHkZSTyNeTxdFy3ivE/zwnU+Gll7M4zRSNySwozDhUCSyaggETlCg+0QQTwfStkIywwETpPk3T1DWg+U8vku5ZA9kNdNust5pVITVwCI7ACUDgArTADWiDDiDgATyBF/BqPBrPxpvxPhtdMqqdA/AHxuc3oI2f3g==</latexit><latexit sha1_base64="MRTHuvMgSbqNzvfxZvk5onlntyc=">AAACHnicbVDLSsNAFJ34rPEVdelmsAgupGS0oBuh4MZlBfuAJJTJZNIOnTyYmQgl9EPc+SfuRBfixoX+jZM0grYeGDiccy/nzvFTzqSy7S9jaXlldW29tmFubm3v7Fp7+12ZZILQDkl4Ivo+lpSzmHYUU5z2U0Fx5HPa88fXhd+7p0KyJL5Tk5R6ER7GLGQEKy0NrHM3wmrkh5DAKwed2p7papg/alaquVsG5YIGUzT1Blbdbtgl4CJBFamDCu2B9eEGCckiGivCsZQOslPl5VgoRjidmm4maYrJGA+po2mMIyq9vMycwmOtBDBMhH6xgqX6eyPHkZSTyNeTxdFy3ivE/zwnU+Gll7M4zRSNySwozDhUCSyaggETlCg+0QQTwfStkIywwETpPk3T1DWg+U8vku5ZA9kNdNust5pVITVwCI7ACUDgArTADWiDDiDgATyBF/BqPBrPxpvxPhtdMqqdA/AHxuc3oI2f3g==</latexit><latexit sha1_base64="MRTHuvMgSbqNzvfxZvk5onlntyc=">AAACHnicbVDLSsNAFJ34rPEVdelmsAgupGS0oBuh4MZlBfuAJJTJZNIOnTyYmQgl9EPc+SfuRBfixoX+jZM0grYeGDiccy/nzvFTzqSy7S9jaXlldW29tmFubm3v7Fp7+12ZZILQDkl4Ivo+lpSzmHYUU5z2U0Fx5HPa88fXhd+7p0KyJL5Tk5R6ER7GLGQEKy0NrHM3wmrkh5DAKwed2p7papg/alaquVsG5YIGUzT1Blbdbtgl4CJBFamDCu2B9eEGCckiGivCsZQOslPl5VgoRjidmm4maYrJGA+po2mMIyq9vMycwmOtBDBMhH6xgqX6eyPHkZSTyNeTxdFy3ivE/zwnU+Gll7M4zRSNySwozDhUCSyaggETlCg+0QQTwfStkIywwETpPk3T1DWg+U8vku5ZA9kNdNust5pVITVwCI7ACUDgArTADWiDDiDgATyBF/BqPBrPxpvxPhtdMqqdA/AHxuc3oI2f3g==</latexit><latexit sha1_base64="MRTHuvMgSbqNzvfxZvk5onlntyc=">AAACHnicbVDLSsNAFJ34rPEVdelmsAgupGS0oBuh4MZlBfuAJJTJZNIOnTyYmQgl9EPc+SfuRBfixoX+jZM0grYeGDiccy/nzvFTzqSy7S9jaXlldW29tmFubm3v7Fp7+12ZZILQDkl4Ivo+lpSzmHYUU5z2U0Fx5HPa88fXhd+7p0KyJL5Tk5R6ER7GLGQEKy0NrHM3wmrkh5DAKwed2p7papg/alaquVsG5YIGUzT1Blbdbtgl4CJBFamDCu2B9eEGCckiGivCsZQOslPl5VgoRjidmm4maYrJGA+po2mMIyq9vMycwmOtBDBMhH6xgqX6eyPHkZSTyNeTxdFy3ivE/zwnU+Gll7M4zRSNySwozDhUCSyaggETlCg+0QQTwfStkIywwETpPk3T1DWg+U8vku5ZA9kNdNust5pVITVwCI7ACUDgArTADWiDDiDgATyBF/BqPBrPxpvxPhtdMqqdA/AHxuc3oI2f3g==</latexit>

BADesignObjective

mm-WaveBeam-Alignment

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src: 00:02:03:04:05:06dst: 00:20:30:40:50:60type: VLAN (0x8100)

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src: 10000dst: 5060

m / a / c / w / md / gInstructions

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Supplementaryslidesandfigures

Lon

Lat

54.5 60.9 67.3 73.8 80.2 86.6 93 99.4 106 112 119

TXRickoverHall

MichelsonHall

Benches

(dB)

(a) Track with Basic Transmission Losses on Holloway Road

(b) Path Loss for the Modified ITU Model

Fig. 1. Improve a statistical channel model by considering building blockages.(a) Path loss values on Holloway road illustrate the shadowing effect ofbuildings. (b) After being shifted by the diffraction losses, the ITU modelclosely follows the measurement results.

cordingly, 3-sigma ranges for both the original and shifted ITUmodels are shown and root mean square errors (RMSEs) arecomputed separately according to the mean of each model.As we can see, the modified ITU predictions follow the mea-surement data much better than the original ones, providing aRMSE improvement of 11.79 dB. Another observation is, fordistances below 280 m, the original ITU model overestimatedthe path loss by around 20 dB. This may be caused by somestrong reflection path(s). Also, in the same distance range, theKED model overestimated the attenuation caused by RickoverHall, which was probably caused by the fact that the blockagehappened at the southern vertical edge of the building, whichcorresponds to a short obstructing screen but the KED modelapplies for a screen with infinite height. Still, the KED modelhelped identify the path loss peak below 260 m.

IV. FOLIAGE ANALYSIS

The effect of foliage for modeling mm-wave channel isa vital consideration for suburban environments as scatteringand absorption at these frequencies can significantly attenuatethe signal. In our measurement campaign, eleven sites hadpartial or total obstruction of the LoS signal from foliage,

0 2 4 6 8 10 12 14 16 18

Vegetation Depth (m)

-20

-10

0

10

20

30

40

50

Att

en

uati

on

(d

B)

Specific Attenuation:Slope of Fitted line = 0.0662 dB/m

Measurements

Fitted

Weissberger

ITU-R

FITU-R

COST235

Fig. 2. Comparison of computed and measured path loss at 28 GHz.

ranging from a single tree to a small grove of trees. Ourmeasurement results were compared against four well knownempirical models [6] that are valid in this frequency range:COST235, Weissberger, ITU-R and FITU-R models.

Fig. 2 illustrates our measured excess vegetation attenuationversus vegetation depth as well as existing model predictions.From the figure, we observe that the mean measured valueof foliage attenuation (0.07 dB/m) is significantly less thanthat of model-predicted values. Our measurements, however,demonstrate a significant amount of multipath energy arrivingat the receiver, likely being scattered from other objects inthe environment. As a result, we recorded a greater signalstrength than what would be predicted by these simple single-path attenuation models.

V. CONCLUSION

In this paper, we illustrate two measurement- and geometry-based techniques for improving existing statistical mm-wavechannel models. Our approach is suitable for a holistic,network-level model that utilizes side information and theresults could be updated in real-time. Our techniques demon-strate a modest, but significant, overall improvement in prop-agation modeling accuracy.

REFERENCES

[1] T. S. Rappaport, et al., “Millimeter wave mobile communications for 5gcellular: It will work!,” IEEE Access, vol. 1, pp. 335–349, 2013.

[2] G. R. MacCartney and T. S. Rappaport, “73 ghz millimeter wavepropagation measurements for outdoor urban mobile and backhaul com-munications in new york city,” in IEEE International Conference onCommunications (ICC), 2014. IEEE, 2014, pp. 4862–4867.

[3] T. S. Rappaport, G. R. MacCartney, S. Sun, H. Yan, and S. Deng,“Small-scale, local area, and transitional millimeter wave propagation for5g communications,” IEEE Transactions on Antennas and Propagation,vol. PP, no. 99, pp. 1–1, 2017.

[4] Y. Zhang, et al., “28-ghz channel measurements and modeling for subur-ban environments,” Department of Electrical and Computer Engineering,Purdue University, West Lafayette, Indiana, Tech. Rep. TR-ECE-17-07,November 2017.

[5] ITU, “Propagation data and prediction methods for the planning of short-range outdoor radiocommunication systems and radio local area networksin the frequency range 300mhz and 100ghz,” Tech. Rep., June 2017.[Online]. Available: https://www.itu.int/rec/R-REC-P.1411-9-201706-I/en

[6] H. M. Rahim, C. Y. Leow, and T. A. Rahman, “Millimeter wavepropagation through foliage: Comparison of models,” in IEEE MalaysiaInternational Conference on Communications, November 2015, pp. 236–240.

TrackwithBasicTransmissionLossesonHollowayRoad

Lon

Lat

54.5 60.9 67.3 73.8 80.2 86.6 93 99.4 106 112 119

TXRickoverHall

MichelsonHall

Benches

(dB)

(a) Track with Basic Transmission Losses on Holloway Road

(b) Path Loss for the Modified ITU Model

Fig. 1. Improve a statistical channel model by considering building blockages.(a) Path loss values on Holloway road illustrate the shadowing effect ofbuildings. (b) After being shifted by the diffraction losses, the ITU modelclosely follows the measurement results.

cordingly, 3-sigma ranges for both the original and shifted ITUmodels are shown and root mean square errors (RMSEs) arecomputed separately according to the mean of each model.As we can see, the modified ITU predictions follow the mea-surement data much better than the original ones, providing aRMSE improvement of 11.79 dB. Another observation is, fordistances below 280 m, the original ITU model overestimatedthe path loss by around 20 dB. This may be caused by somestrong reflection path(s). Also, in the same distance range, theKED model overestimated the attenuation caused by RickoverHall, which was probably caused by the fact that the blockagehappened at the southern vertical edge of the building, whichcorresponds to a short obstructing screen but the KED modelapplies for a screen with infinite height. Still, the KED modelhelped identify the path loss peak below 260 m.

IV. FOLIAGE ANALYSIS

The effect of foliage for modeling mm-wave channel isa vital consideration for suburban environments as scatteringand absorption at these frequencies can significantly attenuatethe signal. In our measurement campaign, eleven sites hadpartial or total obstruction of the LoS signal from foliage,

0 2 4 6 8 10 12 14 16 18

Vegetation Depth (m)

-20

-10

0

10

20

30

40

50

Att

en

uati

on

(d

B)

Specific Attenuation:Slope of Fitted line = 0.0662 dB/m

Measurements

Fitted

Weissberger

ITU-R

FITU-R

COST235

Fig. 2. Comparison of computed and measured path loss at 28 GHz.

ranging from a single tree to a small grove of trees. Ourmeasurement results were compared against four well knownempirical models [6] that are valid in this frequency range:COST235, Weissberger, ITU-R and FITU-R models.

Fig. 2 illustrates our measured excess vegetation attenuationversus vegetation depth as well as existing model predictions.From the figure, we observe that the mean measured valueof foliage attenuation (0.07 dB/m) is significantly less thanthat of model-predicted values. Our measurements, however,demonstrate a significant amount of multipath energy arrivingat the receiver, likely being scattered from other objects inthe environment. As a result, we recorded a greater signalstrength than what would be predicted by these simple single-path attenuation models.

V. CONCLUSION

In this paper, we illustrate two measurement- and geometry-based techniques for improving existing statistical mm-wavechannel models. Our approach is suitable for a holistic,network-level model that utilizes side information and theresults could be updated in real-time. Our techniques demon-strate a modest, but significant, overall improvement in prop-agation modeling accuracy.

REFERENCES

[1] T. S. Rappaport, et al., “Millimeter wave mobile communications for 5gcellular: It will work!,” IEEE Access, vol. 1, pp. 335–349, 2013.

[2] G. R. MacCartney and T. S. Rappaport, “73 ghz millimeter wavepropagation measurements for outdoor urban mobile and backhaul com-munications in new york city,” in IEEE International Conference onCommunications (ICC), 2014. IEEE, 2014, pp. 4862–4867.

[3] T. S. Rappaport, G. R. MacCartney, S. Sun, H. Yan, and S. Deng,“Small-scale, local area, and transitional millimeter wave propagation for5g communications,” IEEE Transactions on Antennas and Propagation,vol. PP, no. 99, pp. 1–1, 2017.

[4] Y. Zhang, et al., “28-ghz channel measurements and modeling for subur-ban environments,” Department of Electrical and Computer Engineering,Purdue University, West Lafayette, Indiana, Tech. Rep. TR-ECE-17-07,November 2017.

[5] ITU, “Propagation data and prediction methods for the planning of short-range outdoor radiocommunication systems and radio local area networksin the frequency range 300mhz and 100ghz,” Tech. Rep., June 2017.[Online]. Available: https://www.itu.int/rec/R-REC-P.1411-9-201706-I/en

[6] H. M. Rahim, C. Y. Leow, and T. A. Rahman, “Millimeter wavepropagation through foliage: Comparison of models,” in IEEE MalaysiaInternational Conference on Communications, November 2015, pp. 236–240.

Improveastatisticalchannelmodelbyconsideringbuildingblockages.(a)PathlossvaluesonHollowayroadillustratetheshadowingeffectofbuildings.(b)Afterbeingshiftedbythediffractionlosses,theITUmodelcloselyfollowsthemeasurementresults.

Lon

Lat

54.5 60.9 67.3 73.8 80.2 86.6 93 99.4 106 112 119

TXRickoverHall

MichelsonHall

Benches

(dB)

(a) Track with Basic Transmission Losses on Holloway Road

(b) Path Loss for the Modified ITU Model

Fig. 1. Improve a statistical channel model by considering building blockages.(a) Path loss values on Holloway road illustrate the shadowing effect ofbuildings. (b) After being shifted by the diffraction losses, the ITU modelclosely follows the measurement results.

cordingly, 3-sigma ranges for both the original and shifted ITUmodels are shown and root mean square errors (RMSEs) arecomputed separately according to the mean of each model.As we can see, the modified ITU predictions follow the mea-surement data much better than the original ones, providing aRMSE improvement of 11.79 dB. Another observation is, fordistances below 280 m, the original ITU model overestimatedthe path loss by around 20 dB. This may be caused by somestrong reflection path(s). Also, in the same distance range, theKED model overestimated the attenuation caused by RickoverHall, which was probably caused by the fact that the blockagehappened at the southern vertical edge of the building, whichcorresponds to a short obstructing screen but the KED modelapplies for a screen with infinite height. Still, the KED modelhelped identify the path loss peak below 260 m.

IV. FOLIAGE ANALYSIS

The effect of foliage for modeling mm-wave channel isa vital consideration for suburban environments as scatteringand absorption at these frequencies can significantly attenuatethe signal. In our measurement campaign, eleven sites hadpartial or total obstruction of the LoS signal from foliage,

0 2 4 6 8 10 12 14 16 18

Vegetation Depth (m)

-20

-10

0

10

20

30

40

50

Att

en

uati

on

(d

B)

Specific Attenuation:Slope of Fitted line = 0.0662 dB/m

Measurements

Fitted

Weissberger

ITU-R

FITU-R

COST235

Fig. 2. Comparison of computed and measured path loss at 28 GHz.

ranging from a single tree to a small grove of trees. Ourmeasurement results were compared against four well knownempirical models [6] that are valid in this frequency range:COST235, Weissberger, ITU-R and FITU-R models.

Fig. 2 illustrates our measured excess vegetation attenuationversus vegetation depth as well as existing model predictions.From the figure, we observe that the mean measured valueof foliage attenuation (0.07 dB/m) is significantly less thanthat of model-predicted values. Our measurements, however,demonstrate a significant amount of multipath energy arrivingat the receiver, likely being scattered from other objects inthe environment. As a result, we recorded a greater signalstrength than what would be predicted by these simple single-path attenuation models.

V. CONCLUSION

In this paper, we illustrate two measurement- and geometry-based techniques for improving existing statistical mm-wavechannel models. Our approach is suitable for a holistic,network-level model that utilizes side information and theresults could be updated in real-time. Our techniques demon-strate a modest, but significant, overall improvement in prop-agation modeling accuracy.

REFERENCES

[1] T. S. Rappaport, et al., “Millimeter wave mobile communications for 5gcellular: It will work!,” IEEE Access, vol. 1, pp. 335–349, 2013.

[2] G. R. MacCartney and T. S. Rappaport, “73 ghz millimeter wavepropagation measurements for outdoor urban mobile and backhaul com-munications in new york city,” in IEEE International Conference onCommunications (ICC), 2014. IEEE, 2014, pp. 4862–4867.

[3] T. S. Rappaport, G. R. MacCartney, S. Sun, H. Yan, and S. Deng,“Small-scale, local area, and transitional millimeter wave propagation for5g communications,” IEEE Transactions on Antennas and Propagation,vol. PP, no. 99, pp. 1–1, 2017.

[4] Y. Zhang, et al., “28-ghz channel measurements and modeling for subur-ban environments,” Department of Electrical and Computer Engineering,Purdue University, West Lafayette, Indiana, Tech. Rep. TR-ECE-17-07,November 2017.

[5] ITU, “Propagation data and prediction methods for the planning of short-range outdoor radiocommunication systems and radio local area networksin the frequency range 300mhz and 100ghz,” Tech. Rep., June 2017.[Online]. Available: https://www.itu.int/rec/R-REC-P.1411-9-201706-I/en

[6] H. M. Rahim, C. Y. Leow, and T. A. Rahman, “Millimeter wavepropagation through foliage: Comparison of models,” in IEEE MalaysiaInternational Conference on Communications, November 2015, pp. 236–240.

PNSequenceGenerator

WindFreakSynthNV

400 MHz 0 dBm

WindFreakSynthNV

2.5 GHz +7 dBm

28-GHzUpconverter

MiniCircuitsSBLP-933+

DC–900 MHz

10 dB MiniCircuitsZFL-1000+

10–1000 MHz17 dB Gain

3 dB 3 dB MiniCircuitsVBF-2435+

2340–2530 MHz

WR-28Horn Antenna+22 dBi Gain

LO

IF RF

(a) Transmitter

WR-28

Horn Antenna

+22 dBi Gain

28-GHzDownconverter

MiniCircuits

VBF-2435+

2340–2530 MHz

IF

RF LO

MiniCircuits

SHP-900

>900 MHz

MiniCircuits

VBF-2435+

2340–2530 MHz

USRPB200

Laptop for Storage

PNSequenceGenerator

WindFreak

SynthNV

399.94 MHz 0 dBm

MiniCircuits

SBLP-933+

DC–900 MHz

9 dB MiniCircuits

ZFL-1000+

10–1000 MHz

3 dB

17 dB

Gain

(b) Receiver

Fig. 1. Block diagrams for the 28-GHz broadband sliding correlator channel sounder. Model numbers are labeled for some commercially available parts.

A. Measurement Equipment

A custom-designed broadband sliding correlator channelsounder [13] was used to record propagation data. Figure 1presents block diagrams for the channel sounder. At both theTX and the RX sides, horn antennas with a nominal +22 dBigain and 15� half-power beamwidth (HPBW) were employed,and pseudo-random noise (PN) generators produced the samePN sequences with a chip sequence length of 2047 [1].

At the TX, the PN probing signal was generated with achip rate of 400 megachips per second (Mcps). It was firstmodulated to a 2.5-GHz intermediate frequency (IF) and thenconverted to RF of 28 GHz by mixing it with a 25.5-GHz localoscillator (LO) at the upconverter. At the RX, the receivedsignal was first downconverted from 28 GHz to 2.5 GHz andthen cross-correlated with the identical PN sequence generatedwith a slightly slower clock rate of 399.94 MHz, similar to thesetup in [1]. A Universal Software Radio Peripheral (USRP)B200 was utilized to record the resulting in-phase (I) andquadrature (Q) signal components. It also regularly sampledthe RX’s location with the help of an on-board GPS disciplinedoscillator (TCXO version).

Table I summarizes the key parameters for the channelsounder. Note that the estimation for the maximum measurablepath loss is based on the following. (1) The RX low-noiseamplifier has a noise figure of 2.4 dB, so we assume a worst-case RX noise figure of 6 dB. (2) The RX detection bandwidthis 60 kHz. (3) The estimated minimum signal-to-noise ratio(SNR) for a detectable signal is 5 dB.

B. Measurement Setup and Procedure

Three types of measurements have been performed forlarge-scale path loss, emulated single-input and multiple-output (SIMO), and continuous tracks. The TX was installed at

TABLE IBROADBAND SLIDING CORRELATOR CHANNEL SOUNDER

SPECIFICATIONS

Carrier Frequency 28 GHz

Chip Sequence Length 2047

RF Bandwidth (First Null) 800 MHz

TX Chip Rate 400 Mcps

Temporal Resolution 2.5 ns

RX Chip Rate 399.94 Mcps

TX Power 23 dBm

TX/RX Antenna Gain 22 dBi

Measured TX/RX Azimuth HPBW 10.1�

Measured TX/RX Elevation HPBW 11.5�

Maximum Measurable Path Loss 182 dB

a height of 90 feet (27.4 m) to emulate a microcell deployment.The RX was moved around campus by an electric car or atwo-layer platform trolley to obtain the measurements, whichis illustrated by the photographs in Figure 2.

We used compasses and digital levels to achieve beamalignment prior to the measurements at each RX location.Aside from the USRP output files and GPS samples, theazimuths and elevations of both the TX and RX antennaswere recorded manually. This allowed us to reconstruct thegeometry relationship of the antennas and extract preciseantenna gains. The USRP gain was manually adjusted tomaximize the SNR at the RX.1

1The Python code for automatically carrying out the measure-ments at each location using GNU Radio, together with the MAT-LAB code for post-processing the collected data, are available athttps://github.com/YaguangZhang/EarsMeasurementCampaignCode.git

Blockdiagramsforthe28-GHzbroadbandslidingcorrelatorchannelsounder.Modelnumbersarelabeledforsomecommerciallyavailableparts.

ECCInterpretationofBA EffectsofBAErrors

ErrorCorrection

-20-15-10-505101520

OptimizationProblem

v Incollaborationwith:

v TexasA&MUniversityv AlexSprintson,Professor

q Highcapacityà Narrowbeams

Ø Beamalignment(BA)

q Sidelobes&noise

BAerrors,outages

q Weneederror-robustBA

)

HowtodesignaBAschemerobusttoBAerrors?

q SingleisotropicUE(AoD:)

Ø Generalizedinthepaper

q Receivedsignal: BA Beacon

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H1 : yk = hk

s2⇡Pk

|Bk|sk + nk, (alignment),

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Beam Alignment<latexit sha1_base64="9geXDYHTXfXVQnPnM1ljW2zFhoU=">AAAB/nicbVDLSgMxFM3UV62vqrhyEyxCV2VGBF1W3bisYB/QDiWT3rahmcyQ3BHLUPBX3LhQxK3f4c6/MW1noa0HAodz7uHenCCWwqDrfju5ldW19Y38ZmFre2d3r7h/0DBRojnUeSQj3QqYASkU1FGghFasgYWBhGYwupn6zQfQRkTqHscx+CEbKNEXnKGVusWjDsIjptc2Qq+kGKgQFE66xZJbcWegy8TLSIlkqHWLX51exJNpmEtmTNtzY/RTplFwCZNCJzEQMz5iA2hbqlgIxk9n50/oqVV6tB9p+xTSmfo7kbLQmHEY2MmQ4dAselPxP6+dYP/ST4WKEwTF54v6iaQY0WkXtCc0cJRjSxjXwt5K+ZBpxtE2VrAleItfXiaNs4rnVry781K1nNWRJ8fkhJSJRy5IldySGqkTTlLyTF7Jm/PkvDjvzsd8NOdkmUPyB87nD2u8lbA=</latexit><latexit sha1_base64="9geXDYHTXfXVQnPnM1ljW2zFhoU=">AAAB/nicbVDLSgMxFM3UV62vqrhyEyxCV2VGBF1W3bisYB/QDiWT3rahmcyQ3BHLUPBX3LhQxK3f4c6/MW1noa0HAodz7uHenCCWwqDrfju5ldW19Y38ZmFre2d3r7h/0DBRojnUeSQj3QqYASkU1FGghFasgYWBhGYwupn6zQfQRkTqHscx+CEbKNEXnKGVusWjDsIjptc2Qq+kGKgQFE66xZJbcWegy8TLSIlkqHWLX51exJNpmEtmTNtzY/RTplFwCZNCJzEQMz5iA2hbqlgIxk9n50/oqVV6tB9p+xTSmfo7kbLQmHEY2MmQ4dAselPxP6+dYP/ST4WKEwTF54v6iaQY0WkXtCc0cJRjSxjXwt5K+ZBpxtE2VrAleItfXiaNs4rnVry781K1nNWRJ8fkhJSJRy5IldySGqkTTlLyTF7Jm/PkvDjvzsd8NOdkmUPyB87nD2u8lbA=</latexit><latexit sha1_base64="9geXDYHTXfXVQnPnM1ljW2zFhoU=">AAAB/nicbVDLSgMxFM3UV62vqrhyEyxCV2VGBF1W3bisYB/QDiWT3rahmcyQ3BHLUPBX3LhQxK3f4c6/MW1noa0HAodz7uHenCCWwqDrfju5ldW19Y38ZmFre2d3r7h/0DBRojnUeSQj3QqYASkU1FGghFasgYWBhGYwupn6zQfQRkTqHscx+CEbKNEXnKGVusWjDsIjptc2Qq+kGKgQFE66xZJbcWegy8TLSIlkqHWLX51exJNpmEtmTNtzY/RTplFwCZNCJzEQMz5iA2hbqlgIxk9n50/oqVV6tB9p+xTSmfo7kbLQmHEY2MmQ4dAselPxP6+dYP/ST4WKEwTF54v6iaQY0WkXtCc0cJRjSxjXwt5K+ZBpxtE2VrAleItfXiaNs4rnVry781K1nNWRJ8fkhJSJRy5IldySGqkTTlLyTF7Jm/PkvDjvzsd8NOdkmUPyB87nD2u8lbA=</latexit><latexit sha1_base64="9geXDYHTXfXVQnPnM1ljW2zFhoU=">AAAB/nicbVDLSgMxFM3UV62vqrhyEyxCV2VGBF1W3bisYB/QDiWT3rahmcyQ3BHLUPBX3LhQxK3f4c6/MW1noa0HAodz7uHenCCWwqDrfju5ldW19Y38ZmFre2d3r7h/0DBRojnUeSQj3QqYASkU1FGghFasgYWBhGYwupn6zQfQRkTqHscx+CEbKNEXnKGVusWjDsIjptc2Qq+kGKgQFE66xZJbcWegy8TLSIlkqHWLX51exJNpmEtmTNtzY/RTplFwCZNCJzEQMz5iA2hbqlgIxk9n50/oqVV6tB9p+xTSmfo7kbLQmHEY2MmQ4dAselPxP6+dYP/ST4WKEwTF54v6iaQY0WkXtCc0cJRjSxjXwt5K+ZBpxtE2VrAleItfXiaNs4rnVry781K1nNWRJ8fkhJSJRy5IldySGqkTTlLyTF7Jm/PkvDjvzsd8NOdkmUPyB87nD2u8lbA=</latexit>

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SimplifiedModel

q OptimalDetector:

Ø Errorfreecase:

Ø Inpractice:

Ø bytuningand

q OverentireBAphase:

Ø

q Assumption:i.i.d

Ø pmf:

⌧th<latexit sha1_base64="OgvXU5C07afturCRzMUCpmH9I3k=">AAAB/3icbVDLSsNAFJ3UV62vaJduBovgQkoiBV0W3LisYB/QhDCZTtuhM0mYuRFC6MIvcaluxK0f4sK/cdJmoa0HBg7n3Ms9c8JEcA2O821VNja3tnequ7W9/YPDI/v4pKfjVFHWpbGI1SAkmgkesS5wEGyQKEZkKFg/nN0Wfv+RKc3j6AGyhPmSTCI+5pSAkQK77gFJg9yTBKZK5jCdz3FgN5ymswBeJ25JGqhEJ7C/vFFMU8kioIJoPXSdBPycKOBUsHnNSzVLCJ2RCRsaGhHJtJ8vws/xuVFGeBwr8yLAC/X3Rk6k1pkML0NphoucetUuxP+8YQrjGz/nUZICi+jy1jgVGGJclIFHXDEKIjOEUMVNXEynRBEKprKa6cFd/fU66V01Xafp3rca7VbZSBWdojN0gVx0jdroDnVQF1GUoWf0it6sJ+vFerc+lqMVq9ypoz+wPn8AYm6WPg==</latexit><latexit sha1_base64="OgvXU5C07afturCRzMUCpmH9I3k=">AAAB/3icbVDLSsNAFJ3UV62vaJduBovgQkoiBV0W3LisYB/QhDCZTtuhM0mYuRFC6MIvcaluxK0f4sK/cdJmoa0HBg7n3Ms9c8JEcA2O821VNja3tnequ7W9/YPDI/v4pKfjVFHWpbGI1SAkmgkesS5wEGyQKEZkKFg/nN0Wfv+RKc3j6AGyhPmSTCI+5pSAkQK77gFJg9yTBKZK5jCdz3FgN5ymswBeJ25JGqhEJ7C/vFFMU8kioIJoPXSdBPycKOBUsHnNSzVLCJ2RCRsaGhHJtJ8vws/xuVFGeBwr8yLAC/X3Rk6k1pkML0NphoucetUuxP+8YQrjGz/nUZICi+jy1jgVGGJclIFHXDEKIjOEUMVNXEynRBEKprKa6cFd/fU66V01Xafp3rca7VbZSBWdojN0gVx0jdroDnVQF1GUoWf0it6sJ+vFerc+lqMVq9ypoz+wPn8AYm6WPg==</latexit><latexit sha1_base64="OgvXU5C07afturCRzMUCpmH9I3k=">AAAB/3icbVDLSsNAFJ3UV62vaJduBovgQkoiBV0W3LisYB/QhDCZTtuhM0mYuRFC6MIvcaluxK0f4sK/cdJmoa0HBg7n3Ms9c8JEcA2O821VNja3tnequ7W9/YPDI/v4pKfjVFHWpbGI1SAkmgkesS5wEGyQKEZkKFg/nN0Wfv+RKc3j6AGyhPmSTCI+5pSAkQK77gFJg9yTBKZK5jCdz3FgN5ymswBeJ25JGqhEJ7C/vFFMU8kioIJoPXSdBPycKOBUsHnNSzVLCJ2RCRsaGhHJtJ8vws/xuVFGeBwr8yLAC/X3Rk6k1pkML0NphoucetUuxP+8YQrjGz/nUZICi+jy1jgVGGJclIFHXDEKIjOEUMVNXEynRBEKprKa6cFd/fU66V01Xafp3rca7VbZSBWdojN0gVx0jdroDnVQF1GUoWf0it6sJ+vFerc+lqMVq9ypoz+wPn8AYm6WPg==</latexit><latexit sha1_base64="OgvXU5C07afturCRzMUCpmH9I3k=">AAAB/3icbVDLSsNAFJ3UV62vaJduBovgQkoiBV0W3LisYB/QhDCZTtuhM0mYuRFC6MIvcaluxK0f4sK/cdJmoa0HBg7n3Ms9c8JEcA2O821VNja3tnequ7W9/YPDI/v4pKfjVFHWpbGI1SAkmgkesS5wEGyQKEZkKFg/nN0Wfv+RKc3j6AGyhPmSTCI+5pSAkQK77gFJg9yTBKZK5jCdz3FgN5ymswBeJ25JGqhEJ7C/vFFMU8kioIJoPXSdBPycKOBUsHnNSzVLCJ2RCRsaGhHJtJ8vws/xuVFGeBwr8yLAC/X3Rk6k1pkML0NphoucetUuxP+8YQrjGz/nUZICi+jy1jgVGGJclIFHXDEKIjOEUMVNXEynRBEKprKa6cFd/fU66V01Xafp3rca7VbZSBWdojN0gVx0jdroDnVQF1GUoWf0it6sJ+vFerc+lqMVq9ypoz+wPn8AYm6WPg==</latexit>

pfa = pmd = pe<latexit sha1_base64="UlBEUhB7CT4G0E4VQi+VtFjDf4g=">AAACEXicbVDLSgMxFM3UV62vUZdugqXgQsqMFHRTKLhxWcE+oB2GTJppQ5PMkGSEMswXuPRLXKobcevahX9jpp1FbT0QODnnXu69J4gZVdpxfqzSxubW9k55t7K3f3B4ZB+fdFWUSEw6OGKR7AdIEUYF6WiqGenHkiAeMNILpre533skUtFIPOhZTDyOxoKGFCNtJN+uxX465EhPJE9DlGXNpT8fZRlswtgnvl116s4ccJ24BamCAm3f/h6OIpxwIjRmSKmB68TaS5HUFDOSVYaJIjHCUzQmA0MF4kR56fycDNaMMoJhJM0TGs7V5Y4UcaVmPLgMuCnOl1Wrdi7+5w0SHd54KRVxoonAi1lhwqCOYB4PHFFJsGYzQxCW1KwL8QRJhLUJsWJycFevXifdq7rr1N37RrXVKBIpgzNwDi6AC65BC9yBNugADJ7AC3gD79az9Wp9WJ+L0pJV9JyCP7C+fgFzJp3q</latexit><latexit sha1_base64="UlBEUhB7CT4G0E4VQi+VtFjDf4g=">AAACEXicbVDLSgMxFM3UV62vUZdugqXgQsqMFHRTKLhxWcE+oB2GTJppQ5PMkGSEMswXuPRLXKobcevahX9jpp1FbT0QODnnXu69J4gZVdpxfqzSxubW9k55t7K3f3B4ZB+fdFWUSEw6OGKR7AdIEUYF6WiqGenHkiAeMNILpre533skUtFIPOhZTDyOxoKGFCNtJN+uxX465EhPJE9DlGXNpT8fZRlswtgnvl116s4ccJ24BamCAm3f/h6OIpxwIjRmSKmB68TaS5HUFDOSVYaJIjHCUzQmA0MF4kR56fycDNaMMoJhJM0TGs7V5Y4UcaVmPLgMuCnOl1Wrdi7+5w0SHd54KRVxoonAi1lhwqCOYB4PHFFJsGYzQxCW1KwL8QRJhLUJsWJycFevXifdq7rr1N37RrXVKBIpgzNwDi6AC65BC9yBNugADJ7AC3gD79az9Wp9WJ+L0pJV9JyCP7C+fgFzJp3q</latexit><latexit sha1_base64="UlBEUhB7CT4G0E4VQi+VtFjDf4g=">AAACEXicbVDLSgMxFM3UV62vUZdugqXgQsqMFHRTKLhxWcE+oB2GTJppQ5PMkGSEMswXuPRLXKobcevahX9jpp1FbT0QODnnXu69J4gZVdpxfqzSxubW9k55t7K3f3B4ZB+fdFWUSEw6OGKR7AdIEUYF6WiqGenHkiAeMNILpre533skUtFIPOhZTDyOxoKGFCNtJN+uxX465EhPJE9DlGXNpT8fZRlswtgnvl116s4ccJ24BamCAm3f/h6OIpxwIjRmSKmB68TaS5HUFDOSVYaJIjHCUzQmA0MF4kR56fycDNaMMoJhJM0TGs7V5Y4UcaVmPLgMuCnOl1Wrdi7+5w0SHd54KRVxoonAi1lhwqCOYB4PHFFJsGYzQxCW1KwL8QRJhLUJsWJycFevXifdq7rr1N37RrXVKBIpgzNwDi6AC65BC9yBNugADJ7AC3gD79az9Wp9WJ+L0pJV9JyCP7C+fgFzJp3q</latexit><latexit sha1_base64="UlBEUhB7CT4G0E4VQi+VtFjDf4g=">AAACEXicbVDLSgMxFM3UV62vUZdugqXgQsqMFHRTKLhxWcE+oB2GTJppQ5PMkGSEMswXuPRLXKobcevahX9jpp1FbT0QODnnXu69J4gZVdpxfqzSxubW9k55t7K3f3B4ZB+fdFWUSEw6OGKR7AdIEUYF6WiqGenHkiAeMNILpre533skUtFIPOhZTDyOxoKGFCNtJN+uxX465EhPJE9DlGXNpT8fZRlswtgnvl116s4ccJ24BamCAm3f/h6OIpxwIjRmSKmB68TaS5HUFDOSVYaJIjHCUzQmA0MF4kR56fycDNaMMoJhJM0TGs7V5Y4UcaVmPLgMuCnOl1Wrdi7+5w0SHd54KRVxoonAi1lhwqCOYB4PHFFJsGYzQxCW1KwL8QRJhLUJsWJycFevXifdq7rr1N37RrXVKBIpgzNwDi6AC65BC9yBNugADJ7AC3gD79az9Wp9WJ+L0pJV9JyCP7C+fgFzJp3q</latexit>

Weight of

|sHkyk|2

kskk2H0

7H1

⌧th<latexit sha1_base64="SVkCoMZ/NX5oCqgMnzqGu0s8k1E=">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</latexit><latexit sha1_base64="SVkCoMZ/NX5oCqgMnzqGu0s8k1E=">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</latexit><latexit sha1_base64="SVkCoMZ/NX5oCqgMnzqGu0s8k1E=">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</latexit><latexit sha1_base64="SVkCoMZ/NX5oCqgMnzqGu0s8k1E=">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</latexit>

ck = �(✓ 2 Bk)<latexit sha1_base64="C+XrgVPcYH/V50yV3bP7ATtXV8c=">AAACEXicbVBNS8NAEN34WetX1KOXxVKoICURQS9CUQSPFewHNCFsttt26WYTdidCCf0FHv0lHtWLePXswX/jts1BWx8MPN6bYWZemAiuwXG+raXlldW19cJGcXNre2fX3ttv6jhVlDVoLGLVDolmgkvWAA6CtRPFSBQK1gqH1xO/9cCU5rG8h1HC/Ij0Je9xSsBIgV2mQTYc40vs0QGveDBgQLDHJfYiAgNKBL4KhseBXXKqzhR4kbg5KaEc9cD+8roxTSMmgQqidcd1EvAzooBTwcZFL9UsIXRI+qxjqCQR0342fWeMy0bp4l6sTEnAU/X3REYirUdReBJGpnlyp563J+J/XieF3oWfcZmkwCSd7eqlAkOMJ/HgLleMghgZQqji5lxMB0QRCibEosnBnf96kTRPq65Tde/OSrWbPJECOkRHqIJcdI5q6BbVUQNR9Iie0St6s56sF+vd+pi1Lln5zAH6A+vzBzmRm/c=</latexit><latexit sha1_base64="C+XrgVPcYH/V50yV3bP7ATtXV8c=">AAACEXicbVBNS8NAEN34WetX1KOXxVKoICURQS9CUQSPFewHNCFsttt26WYTdidCCf0FHv0lHtWLePXswX/jts1BWx8MPN6bYWZemAiuwXG+raXlldW19cJGcXNre2fX3ttv6jhVlDVoLGLVDolmgkvWAA6CtRPFSBQK1gqH1xO/9cCU5rG8h1HC/Ij0Je9xSsBIgV2mQTYc40vs0QGveDBgQLDHJfYiAgNKBL4KhseBXXKqzhR4kbg5KaEc9cD+8roxTSMmgQqidcd1EvAzooBTwcZFL9UsIXRI+qxjqCQR0342fWeMy0bp4l6sTEnAU/X3REYirUdReBJGpnlyp563J+J/XieF3oWfcZmkwCSd7eqlAkOMJ/HgLleMghgZQqji5lxMB0QRCibEosnBnf96kTRPq65Tde/OSrWbPJECOkRHqIJcdI5q6BbVUQNR9Iie0St6s56sF+vd+pi1Lln5zAH6A+vzBzmRm/c=</latexit><latexit sha1_base64="C+XrgVPcYH/V50yV3bP7ATtXV8c=">AAACEXicbVBNS8NAEN34WetX1KOXxVKoICURQS9CUQSPFewHNCFsttt26WYTdidCCf0FHv0lHtWLePXswX/jts1BWx8MPN6bYWZemAiuwXG+raXlldW19cJGcXNre2fX3ttv6jhVlDVoLGLVDolmgkvWAA6CtRPFSBQK1gqH1xO/9cCU5rG8h1HC/Ij0Je9xSsBIgV2mQTYc40vs0QGveDBgQLDHJfYiAgNKBL4KhseBXXKqzhR4kbg5KaEc9cD+8roxTSMmgQqidcd1EvAzooBTwcZFL9UsIXRI+qxjqCQR0342fWeMy0bp4l6sTEnAU/X3REYirUdReBJGpnlyp563J+J/XieF3oWfcZmkwCSd7eqlAkOMJ/HgLleMghgZQqji5lxMB0QRCibEosnBnf96kTRPq65Tde/OSrWbPJECOkRHqIJcdI5q6BbVUQNR9Iie0St6s56sF+vd+pi1Lln5zAH6A+vzBzmRm/c=</latexit><latexit sha1_base64="C+XrgVPcYH/V50yV3bP7ATtXV8c=">AAACEXicbVBNS8NAEN34WetX1KOXxVKoICURQS9CUQSPFewHNCFsttt26WYTdidCCf0FHv0lHtWLePXswX/jts1BWx8MPN6bYWZemAiuwXG+raXlldW19cJGcXNre2fX3ttv6jhVlDVoLGLVDolmgkvWAA6CtRPFSBQK1gqH1xO/9cCU5rG8h1HC/Ij0Je9xSsBIgV2mQTYc40vs0QGveDBgQLDHJfYiAgNKBL4KhseBXXKqzhR4kbg5KaEc9cD+8roxTSMmgQqidcd1EvAzooBTwcZFL9UsIXRI+qxjqCQR0342fWeMy0bp4l6sTEnAU/X3REYirUdReBJGpnlyp563J+J/XieF3oWfcZmkwCSd7eqlAkOMJ/HgLleMghgZQqji5lxMB0QRCibEosnBnf96kTRPq65Tde/OSrWbPJECOkRHqIJcdI5q6BbVUQNR9Iie0St6s56sF+vd+pi1Lln5zAH6A+vzBzmRm/c=</latexit>

u = c� e 2 {0, 1}L<latexit sha1_base64="TxDuU0lUJITPaSsihsLpjLeRICw=">AAACIXicbZDLSgMxFIYzXmu9jbp0EyyCi1JmRNGNUBDBhYsK9gKdsWTSTBuaZIYkI5ShT+LSJ3GpbkRwIb6MmXYUbf0h8PGfc8g5fxAzqrTjfFhz8wuLS8uFleLq2vrGpr213VBRIjGp44hFshUgRRgVpK6pZqQVS4J4wEgzGJxn9eYdkYpG4kYPY+Jz1BM0pBhpY3XsY48j3Q9CmMAz+M0YelHMEvVjEOhRAb3UKbvQG91edeySU3HGgrPg5lACuWod+93rRjjhRGjMkFJt14m1nyKpKWZkVPQSRWKEB6hH2gYF4kT56fi8Edw3TheGkTRPaDh2f0+kiCs15EE54KY5W1lNlzPzv1o70eGpn1IRJ5oIPPkrTBjUEczigl0qCdZsaABhSc26EPeRRFibUIsmB3f66lloHFZcp+JeH5WqF3kiBbAL9sABcMEJqIJLUAN1gME9eATP4MV6sJ6sV+tt0jpn5TM74I+szy8pOKGa</latexit><latexit sha1_base64="TxDuU0lUJITPaSsihsLpjLeRICw=">AAACIXicbZDLSgMxFIYzXmu9jbp0EyyCi1JmRNGNUBDBhYsK9gKdsWTSTBuaZIYkI5ShT+LSJ3GpbkRwIb6MmXYUbf0h8PGfc8g5fxAzqrTjfFhz8wuLS8uFleLq2vrGpr213VBRIjGp44hFshUgRRgVpK6pZqQVS4J4wEgzGJxn9eYdkYpG4kYPY+Jz1BM0pBhpY3XsY48j3Q9CmMAz+M0YelHMEvVjEOhRAb3UKbvQG91edeySU3HGgrPg5lACuWod+93rRjjhRGjMkFJt14m1nyKpKWZkVPQSRWKEB6hH2gYF4kT56fi8Edw3TheGkTRPaDh2f0+kiCs15EE54KY5W1lNlzPzv1o70eGpn1IRJ5oIPPkrTBjUEczigl0qCdZsaABhSc26EPeRRFibUIsmB3f66lloHFZcp+JeH5WqF3kiBbAL9sABcMEJqIJLUAN1gME9eATP4MV6sJ6sV+tt0jpn5TM74I+szy8pOKGa</latexit><latexit sha1_base64="TxDuU0lUJITPaSsihsLpjLeRICw=">AAACIXicbZDLSgMxFIYzXmu9jbp0EyyCi1JmRNGNUBDBhYsK9gKdsWTSTBuaZIYkI5ShT+LSJ3GpbkRwIb6MmXYUbf0h8PGfc8g5fxAzqrTjfFhz8wuLS8uFleLq2vrGpr213VBRIjGp44hFshUgRRgVpK6pZqQVS4J4wEgzGJxn9eYdkYpG4kYPY+Jz1BM0pBhpY3XsY48j3Q9CmMAz+M0YelHMEvVjEOhRAb3UKbvQG91edeySU3HGgrPg5lACuWod+93rRjjhRGjMkFJt14m1nyKpKWZkVPQSRWKEB6hH2gYF4kT56fi8Edw3TheGkTRPaDh2f0+kiCs15EE54KY5W1lNlzPzv1o70eGpn1IRJ5oIPPkrTBjUEczigl0qCdZsaABhSc26EPeRRFibUIsmB3f66lloHFZcp+JeH5WqF3kiBbAL9sABcMEJqIJLUAN1gME9eATP4MV6sJ6sV+tt0jpn5TM74I+szy8pOKGa</latexit><latexit sha1_base64="TxDuU0lUJITPaSsihsLpjLeRICw=">AAACIXicbZDLSgMxFIYzXmu9jbp0EyyCi1JmRNGNUBDBhYsK9gKdsWTSTBuaZIYkI5ShT+LSJ3GpbkRwIb6MmXYUbf0h8PGfc8g5fxAzqrTjfFhz8wuLS8uFleLq2vrGpr213VBRIjGp44hFshUgRRgVpK6pZqQVS4J4wEgzGJxn9eYdkYpG4kYPY+Jz1BM0pBhpY3XsY48j3Q9CmMAz+M0YelHMEvVjEOhRAb3UKbvQG91edeySU3HGgrPg5lACuWod+93rRjjhRGjMkFJt14m1nyKpKWZkVPQSRWKEB6hH2gYF4kT56fi8Edw3TheGkTRPaDh2f0+kiCs15EE54KY5W1lNlzPzv1o70eGpn1IRJ5oIPPkrTBjUEczigl0qCdZsaABhSc26EPeRRFibUIsmB3f66lloHFZcp+JeH5WqF3kiBbAL9sABcMEJqIJLUAN1gME9eATP4MV6sJ6sV+tt0jpn5TM74I+szy8pOKGa</latexit>

p(e) =

✓L

W (e)

◆pW (e)e (1� pe)

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uk = ck � ek 2 {0, 1}<latexit sha1_base64="lRs98oQ/purla7LVFqKrbldhrzI=">AAACi3icbVFdaxQxFM2OX3W0dquPvgQXwYeyzKhQKRWKdsEXsaLbFjbDkGTv7IbNx5Bk1DWdV3+Nr/pf/DdmdvfBtl5IOJxzLrk5l9VSOJ9lf3rJjZu3bt/Zupveu7/9YKe/+/DUmcZyGHMjjT1n1IEUGsZeeAnntQWqmIQztnjb6WdfwDph9Ge/rKFQdKZFJTj1kSr7uCkX+DXm8Samlo3D0EGhScDZXk7asj/Ihtmq8HWQb8AAbeqk3O19IlPDGwXac0mdm+RZ7YtArRdcQpuSxkFN+YLOYBKhpgpcEVZfafHTyExxZWw82uMV+29HoMq5pWJ7TEWzon7ursod+T9t0vjqVRGErhsPmq/fqhqJvcFdNHgqLHAvlxFQbkUcF/M5tZT7GGBKNHzlRimqp4E48R3aSV4EIqHy+AIPckysmM073F72sriOtbcbiVOJ35Qh+tvOeAwxIwvvo/KhBku9sYGMvtXt2s1YGLVtmqZxDfnV0K+D0+fDPBvmH18OjkabhWyhx+gJeoZytI+O0Dt0gsaIox/oJ/qFfifbyYvkIDlcW5PepucRulTJ6C/y+MTa</latexit><latexit sha1_base64="lRs98oQ/purla7LVFqKrbldhrzI=">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</latexit><latexit sha1_base64="lRs98oQ/purla7LVFqKrbldhrzI=">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</latexit><latexit sha1_base64="lRs98oQ/purla7LVFqKrbldhrzI=">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</latexit>

u = [u1, . . . , uL]<latexit sha1_base64="l9Y6fbbe6EaHeHc5En/P0Ggm8j4=">AAACh3icbVHLbtQwFPWEVxteU1jCwmKExGI0xBWi3VQqj5FYgCiCaSslUWQ7NzNW/YhsBxhCNnwNW/gb/gZnZha05UqWjs45V/f6XFZL4XyS/BlEV65eu35jazu+eev2nbvDnXvHzjSWw4wbaewpow6k0DDzwks4rS1QxSScsLNXvX7yGawTRn/yyxpyRedaVIJTH6hi+DBT1C9YhRt8gNOmIONMlsa7cVO8zYvhKJkkq8KXAdmAEdrUUbEz+JiVhjcKtOeSOpeSpPZ5S60XXEIXZ42DmvIzOoc0QE0VuLxdfaPDjwNT4srY8LTHK/bfjpYq55aKjZkK5n5vd1Huyf9paeOr/bwVum48aL6eVTUSe4P7WHApLHAvlwFQbkVYF/MFtZT7EF6cafjCjVJUl23mxDfoUpK3mYTK4+94RHBmxXzR4+68l4VTrL39SpxK/LJog7/rja8hZGThXVDe12CpN7bNpl/rbu1mrJ12XRzH4QzkYuiXwfHuhCQT8uHZ6HC6OcgWeoAeoSeIoD10iN6gIzRDHP1AP9Ev9Dvajp5Gz6P9tTUabHruo3MVvfgLMPzDsg==</latexit><latexit sha1_base64="l9Y6fbbe6EaHeHc5En/P0Ggm8j4=">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</latexit><latexit sha1_base64="l9Y6fbbe6EaHeHc5En/P0Ggm8j4=">AAACh3icbVHLbtQwFPWEVxteU1jCwmKExGI0xBWi3VQqj5FYgCiCaSslUWQ7NzNW/YhsBxhCNnwNW/gb/gZnZha05UqWjs45V/f6XFZL4XyS/BlEV65eu35jazu+eev2nbvDnXvHzjSWw4wbaewpow6k0DDzwks4rS1QxSScsLNXvX7yGawTRn/yyxpyRedaVIJTH6hi+DBT1C9YhRt8gNOmIONMlsa7cVO8zYvhKJkkq8KXAdmAEdrUUbEz+JiVhjcKtOeSOpeSpPZ5S60XXEIXZ42DmvIzOoc0QE0VuLxdfaPDjwNT4srY8LTHK/bfjpYq55aKjZkK5n5vd1Huyf9paeOr/bwVum48aL6eVTUSe4P7WHApLHAvlwFQbkVYF/MFtZT7EF6cafjCjVJUl23mxDfoUpK3mYTK4+94RHBmxXzR4+68l4VTrL39SpxK/LJog7/rja8hZGThXVDe12CpN7bNpl/rbu1mrJ12XRzH4QzkYuiXwfHuhCQT8uHZ6HC6OcgWeoAeoSeIoD10iN6gIzRDHP1AP9Ev9Dvajp5Gz6P9tTUabHruo3MVvfgLMPzDsg==</latexit><latexit sha1_base64="l9Y6fbbe6EaHeHc5En/P0Ggm8j4=">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</latexit>

e<latexit sha1_base64="JWHQv8ZeLx1BG3Qo1nwo5hA5qLA=">AAAB8nicbVDLSsNAFL2prxpfVZduBovgqiQi6LIggssK9iFtKJPppB06mYSZG6GUfoU70YW49XPc+DdO2iy09cDA4Zx7mXNPmEph0PO+ndLa+sbmVnnb3dnd2z+oHB61TJJpxpsskYnuhNRwKRRvokDJO6nmNA4lb4fjm9xvP3FtRKIecJLyIKZDJSLBKFrpsRdTHIUR4f1K1at5c5BV4hekCgUa/cpXb5CwLOYKmaTGdH0vxWBKNQom+cztZYanlI3pkHctVTTmJpjOA8/ImVUGJEq0fQrJXP29MaWxMZM4tJN5QLPs5eJ/XjfD6DqYCpVmyBVbfBRlkmBC8uvJQGjOUE4soUwLm5WwEdWUoe3IdV1bg7989CppXdR8r+bfX1brt0UhZTiBUzgHH66gDnfQgCYwiOEZXuHNQefFeXc+FqMlp9g5hj9wPn8AfHqQSg==</latexit><latexit sha1_base64="JWHQv8ZeLx1BG3Qo1nwo5hA5qLA=">AAAB8nicbVDLSsNAFL2prxpfVZduBovgqiQi6LIggssK9iFtKJPppB06mYSZG6GUfoU70YW49XPc+DdO2iy09cDA4Zx7mXNPmEph0PO+ndLa+sbmVnnb3dnd2z+oHB61TJJpxpsskYnuhNRwKRRvokDJO6nmNA4lb4fjm9xvP3FtRKIecJLyIKZDJSLBKFrpsRdTHIUR4f1K1at5c5BV4hekCgUa/cpXb5CwLOYKmaTGdH0vxWBKNQom+cztZYanlI3pkHctVTTmJpjOA8/ImVUGJEq0fQrJXP29MaWxMZM4tJN5QLPs5eJ/XjfD6DqYCpVmyBVbfBRlkmBC8uvJQGjOUE4soUwLm5WwEdWUoe3IdV1bg7989CppXdR8r+bfX1brt0UhZTiBUzgHH66gDnfQgCYwiOEZXuHNQefFeXc+FqMlp9g5hj9wPn8AfHqQSg==</latexit><latexit sha1_base64="JWHQv8ZeLx1BG3Qo1nwo5hA5qLA=">AAAB8nicbVDLSsNAFL2prxpfVZduBovgqiQi6LIggssK9iFtKJPppB06mYSZG6GUfoU70YW49XPc+DdO2iy09cDA4Zx7mXNPmEph0PO+ndLa+sbmVnnb3dnd2z+oHB61TJJpxpsskYnuhNRwKRRvokDJO6nmNA4lb4fjm9xvP3FtRKIecJLyIKZDJSLBKFrpsRdTHIUR4f1K1at5c5BV4hekCgUa/cpXb5CwLOYKmaTGdH0vxWBKNQom+cztZYanlI3pkHctVTTmJpjOA8/ImVUGJEq0fQrJXP29MaWxMZM4tJN5QLPs5eJ/XjfD6DqYCpVmyBVbfBRlkmBC8uvJQGjOUE4soUwLm5WwEdWUoe3IdV1bg7989CppXdR8r+bfX1brt0UhZTiBUzgHH66gDnfQgCYwiOEZXuHNQefFeXc+FqMlp9g5hj9wPn8AfHqQSg==</latexit><latexit sha1_base64="JWHQv8ZeLx1BG3Qo1nwo5hA5qLA=">AAAB8nicbVDLSsNAFL2prxpfVZduBovgqiQi6LIggssK9iFtKJPppB06mYSZG6GUfoU70YW49XPc+DdO2iy09cDA4Zx7mXNPmEph0PO+ndLa+sbmVnnb3dnd2z+oHB61TJJpxpsskYnuhNRwKRRvokDJO6nmNA4lb4fjm9xvP3FtRKIecJLyIKZDJSLBKFrpsRdTHIUR4f1K1at5c5BV4hekCgUa/cpXb5CwLOYKmaTGdH0vxWBKNQom+cztZYanlI3pkHctVTTmJpjOA8/ImVUGJEq0fQrJXP29MaWxMZM4tJN5QLPs5eJ/XjfD6DqYCpVmyBVbfBRlkmBC8uvJQGjOUE4soUwLm5WwEdWUoe3IdV1bg7989CppXdR8r+bfX1brt0UhZTiBUzgHH66gDnfQgCYwiOEZXuHNQefFeXc+FqMlp9g5hj9wPn8AfHqQSg==</latexit>

Detectionq Definitions:

Ø Supportof :

Ø Setofallvalid:

Ø Setofallnon-empty:

Uc , \Lk=1B

ckk

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partition

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Uc , \Lk=1B

ckk

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✓<latexit sha1_base64="e6gN3ZJ6qevNzNhoPBZR6Oeirrw=">AAACb3icbVFdaxNBFJ2sH63rV6sPPggyGAQFCbsi1MfSGvBFrGjSQnYJM5O7ydj5WGbu2sZx/4Ov+s/8Gf4DZ5M82NYLA4dzzuXeOZfXSnrMst+95Nr1Gze3tm+lt+/cvXd/Z/fB2NvGCRgJq6w74cyDkgZGKFHBSe2Aaa7gmJ8edvrxV3BeWvMZlzWUms2NrKRgGKlxgQtANt3pZ4NsVfQqyDegTzZ1NN3tfSpmVjQaDArFvJ/kWY1lYA6lUNCmReOhZuKUzWESoWEafBlW67b0WWRmtLIuPoN0xf7bEZj2fqn5S66jWTNc+MtyR/5PmzRYvSmDNHWDYMR6VtUoipZ236cz6UCgWkbAhJNxXSoWzDGBMaS0MHAmrNbMzELh5TdoJ3kZCgUV0u+0n9PCyfmiw+1FL4+Rr73dSoIpejAN0d92xrcQM3LwPiofanAMrQvF8Lxu127Ow7Bt0zSNZ8gvh34VjF8N8myQf3zd3x9uDrJNHpOn5DnJyR7ZJ+/IERkRQb6QH+Qn+dX7kzxKniR0bU16m56H5EIlL/4CpF+8fw==</latexit><latexit sha1_base64="e6gN3ZJ6qevNzNhoPBZR6Oeirrw=">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</latexit><latexit sha1_base64="e6gN3ZJ6qevNzNhoPBZR6Oeirrw=">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</latexit><latexit sha1_base64="e6gN3ZJ6qevNzNhoPBZR6Oeirrw=">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</latexit>

B1k = Bk

B0k = [�⇡,⇡) \ Bk

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error

Incorrect inference à Outage

Data Communication

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error

True:

Detected:

True:

Detected:

Beam Alignment

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c = [1, x]

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U[1,0]<latexit sha1_base64="z0CaRV3QZLfWqC1xO/PWzx1sJgA=">AAAB/XicbVDLSsNAFL2prxpfsS7dDBbBhZRECrosuHFZwbRCGsJkOmmHTh7MTMQSin/iTnQhbv0RN/6NkzYLbT0wcDjnXu6ZE2acSWXb30ZtbX1jc6u+be7s7u0fWIeNnkxzQahLUp6K+xBLyllCXcUUp/eZoDgOOe2Hk+vS7z9QIVma3KlpRv0YjxIWMYKVlgKrMYixGhPMkRsUnnNu+7PAatotew60SpyKNKFCN7C+BsOU5DFNFOFYSs+xM+UXWChGOJ2Zg1zSDJMJHlFP0wTHVPrFPPsMnWpliKJU6JcoNFd/bxQ4lnIah3qyTCqXvVL8z/NyFV35BUuyXNGELA5FOUcqRWURaMgEJYpPNcFEMJ0VkTEWmChdl2maugZn+dOrpHfRcuyWc9tudtpVIXU4hhM4AwcuoQM30AUXCDzCM7zCm/FkvBjvxsditGZUO0fwB8bnD+bnk7g=</latexit><latexit sha1_base64="z0CaRV3QZLfWqC1xO/PWzx1sJgA=">AAAB/XicbVDLSsNAFL2prxpfsS7dDBbBhZRECrosuHFZwbRCGsJkOmmHTh7MTMQSin/iTnQhbv0RN/6NkzYLbT0wcDjnXu6ZE2acSWXb30ZtbX1jc6u+be7s7u0fWIeNnkxzQahLUp6K+xBLyllCXcUUp/eZoDgOOe2Hk+vS7z9QIVma3KlpRv0YjxIWMYKVlgKrMYixGhPMkRsUnnNu+7PAatotew60SpyKNKFCN7C+BsOU5DFNFOFYSs+xM+UXWChGOJ2Zg1zSDJMJHlFP0wTHVPrFPPsMnWpliKJU6JcoNFd/bxQ4lnIah3qyTCqXvVL8z/NyFV35BUuyXNGELA5FOUcqRWURaMgEJYpPNcFEMJ0VkTEWmChdl2maugZn+dOrpHfRcuyWc9tudtpVIXU4hhM4AwcuoQM30AUXCDzCM7zCm/FkvBjvxsditGZUO0fwB8bnD+bnk7g=</latexit><latexit sha1_base64="z0CaRV3QZLfWqC1xO/PWzx1sJgA=">AAAB/XicbVDLSsNAFL2prxpfsS7dDBbBhZRECrosuHFZwbRCGsJkOmmHTh7MTMQSin/iTnQhbv0RN/6NkzYLbT0wcDjnXu6ZE2acSWXb30ZtbX1jc6u+be7s7u0fWIeNnkxzQahLUp6K+xBLyllCXcUUp/eZoDgOOe2Hk+vS7z9QIVma3KlpRv0YjxIWMYKVlgKrMYixGhPMkRsUnnNu+7PAatotew60SpyKNKFCN7C+BsOU5DFNFOFYSs+xM+UXWChGOJ2Zg1zSDJMJHlFP0wTHVPrFPPsMnWpliKJU6JcoNFd/bxQ4lnIah3qyTCqXvVL8z/NyFV35BUuyXNGELA5FOUcqRWURaMgEJYpPNcFEMJ0VkTEWmChdl2maugZn+dOrpHfRcuyWc9tudtpVIXU4hhM4AwcuoQM30AUXCDzCM7zCm/FkvBjvxsditGZUO0fwB8bnD+bnk7g=</latexit><latexit sha1_base64="z0CaRV3QZLfWqC1xO/PWzx1sJgA=">AAAB/XicbVDLSsNAFL2prxpfsS7dDBbBhZRECrosuHFZwbRCGsJkOmmHTh7MTMQSin/iTnQhbv0RN/6NkzYLbT0wcDjnXu6ZE2acSWXb30ZtbX1jc6u+be7s7u0fWIeNnkxzQahLUp6K+xBLyllCXcUUp/eZoDgOOe2Hk+vS7z9QIVma3KlpRv0YjxIWMYKVlgKrMYixGhPMkRsUnnNu+7PAatotew60SpyKNKFCN7C+BsOU5DFNFOFYSs+xM+UXWChGOJ2Zg1zSDJMJHlFP0wTHVPrFPPsMnWpliKJU6JcoNFd/bxQ4lnIah3qyTCqXvVL8z/NyFV35BUuyXNGELA5FOUcqRWURaMgEJYpPNcFEMJ0VkTEWmChdl2maugZn+dOrpHfRcuyWc9tudtpVIXU4hhM4AwcuoQM30AUXCDzCM7zCm/FkvBjvxsditGZUO0fwB8bnD+bnk7g=</latexit>

U[1,1]<latexit sha1_base64="lGpzAG+/aFjSc6SrF4NUTdlyjJo=">AAAB/XicbVBPS8MwHE3nv1n/1Xn0EhyCBxmNDPQ48OJxgt2ErpQ0S7ewNC1JKo4y/CbeRA/i1S/ixW9juvWgmw8Cj/d+P34vL8o4U9p1v63a2vrG5lZ9297Z3ds/cA4bPZXmklCPpDyV9xFWlDNBPc00p/eZpDiJOO1Hk+vS7z9QqVgq7vQ0o0GCR4LFjGBtpNBpDBKsxwRz6IWFj85RMAudptty54CrBFWkCSp0Q+drMExJnlChCcdK+cjNdFBgqRnhdGYPckUzTCZ4RH1DBU6oCop59hk8NcoQxqk0T2g4V39vFDhRappEZrJMqpa9UvzP83MdXwUFE1muqSCLQ3HOoU5hWQQcMkmJ5lNDMJHMZIVkjCUm2tRl27apAS1/epX0LlrIbaHbdrPTrgqpg2NwAs4AApegA25AF3iAgEfwDF7Bm/VkvVjv1sditGZVO0fgD6zPH+hvk7k=</latexit><latexit sha1_base64="lGpzAG+/aFjSc6SrF4NUTdlyjJo=">AAAB/XicbVBPS8MwHE3nv1n/1Xn0EhyCBxmNDPQ48OJxgt2ErpQ0S7ewNC1JKo4y/CbeRA/i1S/ixW9juvWgmw8Cj/d+P34vL8o4U9p1v63a2vrG5lZ9297Z3ds/cA4bPZXmklCPpDyV9xFWlDNBPc00p/eZpDiJOO1Hk+vS7z9QqVgq7vQ0o0GCR4LFjGBtpNBpDBKsxwRz6IWFj85RMAudptty54CrBFWkCSp0Q+drMExJnlChCcdK+cjNdFBgqRnhdGYPckUzTCZ4RH1DBU6oCop59hk8NcoQxqk0T2g4V39vFDhRappEZrJMqpa9UvzP83MdXwUFE1muqSCLQ3HOoU5hWQQcMkmJ5lNDMJHMZIVkjCUm2tRl27apAS1/epX0LlrIbaHbdrPTrgqpg2NwAs4AApegA25AF3iAgEfwDF7Bm/VkvVjv1sditGZVO0fgD6zPH+hvk7k=</latexit><latexit sha1_base64="lGpzAG+/aFjSc6SrF4NUTdlyjJo=">AAAB/XicbVBPS8MwHE3nv1n/1Xn0EhyCBxmNDPQ48OJxgt2ErpQ0S7ewNC1JKo4y/CbeRA/i1S/ixW9juvWgmw8Cj/d+P34vL8o4U9p1v63a2vrG5lZ9297Z3ds/cA4bPZXmklCPpDyV9xFWlDNBPc00p/eZpDiJOO1Hk+vS7z9QqVgq7vQ0o0GCR4LFjGBtpNBpDBKsxwRz6IWFj85RMAudptty54CrBFWkCSp0Q+drMExJnlChCcdK+cjNdFBgqRnhdGYPckUzTCZ4RH1DBU6oCop59hk8NcoQxqk0T2g4V39vFDhRappEZrJMqpa9UvzP83MdXwUFE1muqSCLQ3HOoU5hWQQcMkmJ5lNDMJHMZIVkjCUm2tRl27apAS1/epX0LlrIbaHbdrPTrgqpg2NwAs4AApegA25AF3iAgEfwDF7Bm/VkvVjv1sditGZVO0fgD6zPH+hvk7k=</latexit><latexit sha1_base64="lGpzAG+/aFjSc6SrF4NUTdlyjJo=">AAAB/XicbVBPS8MwHE3nv1n/1Xn0EhyCBxmNDPQ48OJxgt2ErpQ0S7ewNC1JKo4y/CbeRA/i1S/ixW9juvWgmw8Cj/d+P34vL8o4U9p1v63a2vrG5lZ9297Z3ds/cA4bPZXmklCPpDyV9xFWlDNBPc00p/eZpDiJOO1Hk+vS7z9QqVgq7vQ0o0GCR4LFjGBtpNBpDBKsxwRz6IWFj85RMAudptty54CrBFWkCSp0Q+drMExJnlChCcdK+cjNdFBgqRnhdGYPckUzTCZ4RH1DBU6oCop59hk8NcoQxqk0T2g4V39vFDhRappEZrJMqpa9UvzP83MdXwUFE1muqSCLQ3HOoU5hWQQcMkmJ5lNDMJHMZIVkjCUm2tRl27apAS1/epX0LlrIbaHbdrPTrgqpg2NwAs4AApegA25AF3iAgEfwDF7Bm/VkvVjv1sditGZVO0fgD6zPH+hvk7k=</latexit>

C = {[0, 0], [0, 1], [1, 0], [1, 1]}<latexit sha1_base64="LW9mPBtii5hGd7tEVYSZ53Bacow=">AAACJnicbZBLSwMxEMez9VXXV9Wjl2ARPJSykYKCCIVePFawD+guJZtm29Dsg2RWKEs/jDe/iTdRUMSbH8Vs24O2DmT48Z8ZJvP3Eyk0OM6XVVhb39jcKm7bO7t7+welw6O2jlPFeIvFMlZdn2ouRcRbIEDybqI4DX3JO/64kdc7D1xpEUf3MEm4F9JhJALBKBipX7p2dUilxG5IYcSoxA18g13JA3CznlNxvIrJxGQyY2IYu0oMR+BO+6WyU3VmgVeBLKCMFtHsl97dQczSkEfAJNW6R5wEvIwqEEzyqe2mmieUjemQ9wxGNOTay2ZHTvGZUQY4iJV5EeCZ+nsio6HWk9A3nfktermWi//VeikEV14moiQFHrH5oiCVGGKcO4YHQnEGcmKAMiXMXzEbUUUZGF9t2zY2kOWjV6F9USVOldzVyvXawpAiOkGn6BwRdInq6BY1UQsx9Iie0St6s56sF+vD+py3FqzFzDH6E9b3D5JboZ8=</latexit><latexit sha1_base64="LW9mPBtii5hGd7tEVYSZ53Bacow=">AAACJnicbZBLSwMxEMez9VXXV9Wjl2ARPJSykYKCCIVePFawD+guJZtm29Dsg2RWKEs/jDe/iTdRUMSbH8Vs24O2DmT48Z8ZJvP3Eyk0OM6XVVhb39jcKm7bO7t7+welw6O2jlPFeIvFMlZdn2ouRcRbIEDybqI4DX3JO/64kdc7D1xpEUf3MEm4F9JhJALBKBipX7p2dUilxG5IYcSoxA18g13JA3CznlNxvIrJxGQyY2IYu0oMR+BO+6WyU3VmgVeBLKCMFtHsl97dQczSkEfAJNW6R5wEvIwqEEzyqe2mmieUjemQ9wxGNOTay2ZHTvGZUQY4iJV5EeCZ+nsio6HWk9A3nfktermWi//VeikEV14moiQFHrH5oiCVGGKcO4YHQnEGcmKAMiXMXzEbUUUZGF9t2zY2kOWjV6F9USVOldzVyvXawpAiOkGn6BwRdInq6BY1UQsx9Iie0St6s56sF+vD+py3FqzFzDH6E9b3D5JboZ8=</latexit><latexit sha1_base64="LW9mPBtii5hGd7tEVYSZ53Bacow=">AAACJnicbZBLSwMxEMez9VXXV9Wjl2ARPJSykYKCCIVePFawD+guJZtm29Dsg2RWKEs/jDe/iTdRUMSbH8Vs24O2DmT48Z8ZJvP3Eyk0OM6XVVhb39jcKm7bO7t7+welw6O2jlPFeIvFMlZdn2ouRcRbIEDybqI4DX3JO/64kdc7D1xpEUf3MEm4F9JhJALBKBipX7p2dUilxG5IYcSoxA18g13JA3CznlNxvIrJxGQyY2IYu0oMR+BO+6WyU3VmgVeBLKCMFtHsl97dQczSkEfAJNW6R5wEvIwqEEzyqe2mmieUjemQ9wxGNOTay2ZHTvGZUQY4iJV5EeCZ+nsio6HWk9A3nfktermWi//VeikEV14moiQFHrH5oiCVGGKcO4YHQnEGcmKAMiXMXzEbUUUZGF9t2zY2kOWjV6F9USVOldzVyvXawpAiOkGn6BwRdInq6BY1UQsx9Iie0St6s56sF+vD+py3FqzFzDH6E9b3D5JboZ8=</latexit><latexit sha1_base64="LW9mPBtii5hGd7tEVYSZ53Bacow=">AAACJnicbZBLSwMxEMez9VXXV9Wjl2ARPJSykYKCCIVePFawD+guJZtm29Dsg2RWKEs/jDe/iTdRUMSbH8Vs24O2DmT48Z8ZJvP3Eyk0OM6XVVhb39jcKm7bO7t7+welw6O2jlPFeIvFMlZdn2ouRcRbIEDybqI4DX3JO/64kdc7D1xpEUf3MEm4F9JhJALBKBipX7p2dUilxG5IYcSoxA18g13JA3CznlNxvIrJxGQyY2IYu0oMR+BO+6WyU3VmgVeBLKCMFtHsl97dQczSkEfAJNW6R5wEvIwqEEzyqe2mmieUjemQ9wxGNOTay2ZHTvGZUQY4iJV5EeCZ+nsio6HWk9A3nfktermWi//VeikEV14moiQFHrH5oiCVGGKcO4YHQnEGcmKAMiXMXzEbUUUZGF9t2zY2kOWjV6F9USVOldzVyvXawpAiOkGn6BwRdInq6BY1UQsx9Iie0St6s56sF+vD+py3FqzFzDH6E9b3D5JboZ8=</latexit>

True:

Detected:

Beam Alignment

B3<latexit sha1_base64="3GZC7MHkjmVpvJNP5rrWpB4HlzY=">AAAB9XicbVDLSsNAFL2prxpfUZduBovgqiQq6LIogssK9gFNKJPppB06mYSZiVJCf8Od6ELc+jNu/BsnbRbaemDgcM693DMnTDlT2nW/rcrK6tr6RnXT3tre2d1z9g/aKskkoS2S8ER2Q6woZ4K2NNOcdlNJcRxy2gnHN4XfeaRSsUQ86ElKgxgPBYsYwdpIvh9jPSKYo+v+ed+puXV3BrRMvJLUoESz73z5g4RkMRWacKxUz3NTHeRYakY4ndp+pmiKyRgPac9QgWOqgnyWeYpOjDJAUSLNExrN1N8bOY6VmsShmSwyqkWvEP/zepmOroKciTTTVJD5oSjjSCeoKAANmKRE84khmEhmsiIywhITbWqybdvU4C1+epm0z+qeW/fuL2qN27KQKhzBMZyCB5fQgDtoQgsIpPAMr/BmPVkv1rv1MR+tWOXOIfyB9fkDPjSRPw==</latexit><latexit sha1_base64="3GZC7MHkjmVpvJNP5rrWpB4HlzY=">AAAB9XicbVDLSsNAFL2prxpfUZduBovgqiQq6LIogssK9gFNKJPppB06mYSZiVJCf8Od6ELc+jNu/BsnbRbaemDgcM693DMnTDlT2nW/rcrK6tr6RnXT3tre2d1z9g/aKskkoS2S8ER2Q6woZ4K2NNOcdlNJcRxy2gnHN4XfeaRSsUQ86ElKgxgPBYsYwdpIvh9jPSKYo+v+ed+puXV3BrRMvJLUoESz73z5g4RkMRWacKxUz3NTHeRYakY4ndp+pmiKyRgPac9QgWOqgnyWeYpOjDJAUSLNExrN1N8bOY6VmsShmSwyqkWvEP/zepmOroKciTTTVJD5oSjjSCeoKAANmKRE84khmEhmsiIywhITbWqybdvU4C1+epm0z+qeW/fuL2qN27KQKhzBMZyCB5fQgDtoQgsIpPAMr/BmPVkv1rv1MR+tWOXOIfyB9fkDPjSRPw==</latexit><latexit sha1_base64="3GZC7MHkjmVpvJNP5rrWpB4HlzY=">AAAB9XicbVDLSsNAFL2prxpfUZduBovgqiQq6LIogssK9gFNKJPppB06mYSZiVJCf8Od6ELc+jNu/BsnbRbaemDgcM693DMnTDlT2nW/rcrK6tr6RnXT3tre2d1z9g/aKskkoS2S8ER2Q6woZ4K2NNOcdlNJcRxy2gnHN4XfeaRSsUQ86ElKgxgPBYsYwdpIvh9jPSKYo+v+ed+puXV3BrRMvJLUoESz73z5g4RkMRWacKxUz3NTHeRYakY4ndp+pmiKyRgPac9QgWOqgnyWeYpOjDJAUSLNExrN1N8bOY6VmsShmSwyqkWvEP/zepmOroKciTTTVJD5oSjjSCeoKAANmKRE84khmEhmsiIywhITbWqybdvU4C1+epm0z+qeW/fuL2qN27KQKhzBMZyCB5fQgDtoQgsIpPAMr/BmPVkv1rv1MR+tWOXOIfyB9fkDPjSRPw==</latexit><latexit sha1_base64="3GZC7MHkjmVpvJNP5rrWpB4HlzY=">AAAB9XicbVDLSsNAFL2prxpfUZduBovgqiQq6LIogssK9gFNKJPppB06mYSZiVJCf8Od6ELc+jNu/BsnbRbaemDgcM693DMnTDlT2nW/rcrK6tr6RnXT3tre2d1z9g/aKskkoS2S8ER2Q6woZ4K2NNOcdlNJcRxy2gnHN4XfeaRSsUQ86ElKgxgPBYsYwdpIvh9jPSKYo+v+ed+puXV3BrRMvJLUoESz73z5g4RkMRWacKxUz3NTHeRYakY4ndp+pmiKyRgPac9QgWOqgnyWeYpOjDJAUSLNExrN1N8bOY6VmsShmSwyqkWvEP/zepmOroKciTTTVJD5oSjjSCeoKAANmKRE84khmEhmsiIywhITbWqybdvU4C1+epm0z+qeW/fuL2qN27KQKhzBMZyCB5fQgDtoQgsIpPAMr/BmPVkv1rv1MR+tWOXOIfyB9fkDPjSRPw==</latexit>

c = [1, 0, 0, x, x]

u = [1, 1, 0, x, x]<latexit sha1_base64="KLrQ6ia/G1X1GbLSpNXHgU1GGbw=">AAACKnicbVDLSsNAFJ3UV42vqEs3g0VwUUoiBd2IBTcuK9gHNKFMJpN26OTBzERaQj7HnX/iTuqiuPVDnKQRtPUMA4dz7uXee9yYUSFNc6FVNja3tnequ/re/sHhkXF80hVRwjHp4IhFvO8iQRgNSUdSyUg/5gQFLiM9d3Kf+71nwgWNwic5i4kToFFIfYqRVNLQuLMDJMeuDzG8HVh1U71pferotoL+4yWFl9rFuJQTL7OysnBo1MyGWQCuE6skNVCiPTTmthfhJCChxAwJMbDMWDop4pJiRjLdTgSJEZ6gERkoGqKACCctJmfwQike9COufihhof7uSFEgxCxwVWW+ulj1cvE/b5BI/8ZJaRgnkoR4OchPGJQRzFODHuUESzZTBGFO1a4QjxFHWKpsdV1XMVirR6+T7lXDMhvWY7PWapaBVMEZOAeXwALXoAUeQBt0AAYv4A3MwYf2qr1rC+1zWVrRyp5T8Afa1zcvUaOe</latexit><latexit sha1_base64="KLrQ6ia/G1X1GbLSpNXHgU1GGbw=">AAACKnicbVDLSsNAFJ3UV42vqEs3g0VwUUoiBd2IBTcuK9gHNKFMJpN26OTBzERaQj7HnX/iTuqiuPVDnKQRtPUMA4dz7uXee9yYUSFNc6FVNja3tnequ/re/sHhkXF80hVRwjHp4IhFvO8iQRgNSUdSyUg/5gQFLiM9d3Kf+71nwgWNwic5i4kToFFIfYqRVNLQuLMDJMeuDzG8HVh1U71pferotoL+4yWFl9rFuJQTL7OysnBo1MyGWQCuE6skNVCiPTTmthfhJCChxAwJMbDMWDop4pJiRjLdTgSJEZ6gERkoGqKACCctJmfwQike9COufihhof7uSFEgxCxwVWW+ulj1cvE/b5BI/8ZJaRgnkoR4OchPGJQRzFODHuUESzZTBGFO1a4QjxFHWKpsdV1XMVirR6+T7lXDMhvWY7PWapaBVMEZOAeXwALXoAUeQBt0AAYv4A3MwYf2qr1rC+1zWVrRyp5T8Afa1zcvUaOe</latexit><latexit sha1_base64="KLrQ6ia/G1X1GbLSpNXHgU1GGbw=">AAACKnicbVDLSsNAFJ3UV42vqEs3g0VwUUoiBd2IBTcuK9gHNKFMJpN26OTBzERaQj7HnX/iTuqiuPVDnKQRtPUMA4dz7uXee9yYUSFNc6FVNja3tnequ/re/sHhkXF80hVRwjHp4IhFvO8iQRgNSUdSyUg/5gQFLiM9d3Kf+71nwgWNwic5i4kToFFIfYqRVNLQuLMDJMeuDzG8HVh1U71pferotoL+4yWFl9rFuJQTL7OysnBo1MyGWQCuE6skNVCiPTTmthfhJCChxAwJMbDMWDop4pJiRjLdTgSJEZ6gERkoGqKACCctJmfwQike9COufihhof7uSFEgxCxwVWW+ulj1cvE/b5BI/8ZJaRgnkoR4OchPGJQRzFODHuUESzZTBGFO1a4QjxFHWKpsdV1XMVirR6+T7lXDMhvWY7PWapaBVMEZOAeXwALXoAUeQBt0AAYv4A3MwYf2qr1rC+1zWVrRyp5T8Afa1zcvUaOe</latexit><latexit sha1_base64="KLrQ6ia/G1X1GbLSpNXHgU1GGbw=">AAACKnicbVDLSsNAFJ3UV42vqEs3g0VwUUoiBd2IBTcuK9gHNKFMJpN26OTBzERaQj7HnX/iTuqiuPVDnKQRtPUMA4dz7uXee9yYUSFNc6FVNja3tnequ/re/sHhkXF80hVRwjHp4IhFvO8iQRgNSUdSyUg/5gQFLiM9d3Kf+71nwgWNwic5i4kToFFIfYqRVNLQuLMDJMeuDzG8HVh1U71pferotoL+4yWFl9rFuJQTL7OysnBo1MyGWQCuE6skNVCiPTTmthfhJCChxAwJMbDMWDop4pJiRjLdTgSJEZ6gERkoGqKACCctJmfwQike9COufihhof7uSFEgxCxwVWW+ulj1cvE/b5BI/8ZJaRgnkoR4OchPGJQRzFODHuUESzZTBGFO1a4QjxFHWKpsdV1XMVirR6+T7lXDMhvWY7PWapaBVMEZOAeXwALXoAUeQBt0AAYv4A3MwYf2qr1rC+1zWVrRyp5T8Afa1zcvUaOe</latexit>

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True:

Detected:

True:

Detected:

c = [1, 0, 0, 0, x]

u = [1, 1, 0, 0, x]<latexit sha1_base64="B6G10aZJ7917GLLjco/i+td9dws=">AAACKnicbVDLSsNAFJ3UV42vqEs3g0VwISWRgm7EghuXFewDklAmk0k7dPJgZiKWkM9x55+4k7oobv0QJ2kEbT3DwOGce7n3Hi9hVEjTnGu1tfWNza36tr6zu7d/YBwe9USccky6OGYxH3hIEEYj0pVUMjJIOEGhx0jfm9wVfv+JcEHj6FFOE+KGaBTRgGIklTQ0bp0QybEXQAxvbOvCLN+zqzsK+o+Xll7mlOMyTvzcyqvCodEwm2YJuEqsijRAhc7QmDl+jNOQRBIzJIRtmYl0M8QlxYzkupMKkiA8QSNiKxqhkAg3Kyfn8EwpPgxirn4kYan+7shQKMQ09FRlsbpY9grxP89OZXDtZjRKUkkivBgUpAzKGBapQZ9ygiWbKoIwp2pXiMeIIyxVtrquqxis5aNXSe+yaZlN66HVaLeqQOrgBJyCc2CBK9AG96ADugCDF/AGZuBDe9Xetbn2uSitaVXPMfgD7esbRqqjDg==</latexit><latexit sha1_base64="B6G10aZJ7917GLLjco/i+td9dws=">AAACKnicbVDLSsNAFJ3UV42vqEs3g0VwISWRgm7EghuXFewDklAmk0k7dPJgZiKWkM9x55+4k7oobv0QJ2kEbT3DwOGce7n3Hi9hVEjTnGu1tfWNza36tr6zu7d/YBwe9USccky6OGYxH3hIEEYj0pVUMjJIOEGhx0jfm9wVfv+JcEHj6FFOE+KGaBTRgGIklTQ0bp0QybEXQAxvbOvCLN+zqzsK+o+Xll7mlOMyTvzcyqvCodEwm2YJuEqsijRAhc7QmDl+jNOQRBIzJIRtmYl0M8QlxYzkupMKkiA8QSNiKxqhkAg3Kyfn8EwpPgxirn4kYan+7shQKMQ09FRlsbpY9grxP89OZXDtZjRKUkkivBgUpAzKGBapQZ9ygiWbKoIwp2pXiMeIIyxVtrquqxis5aNXSe+yaZlN66HVaLeqQOrgBJyCc2CBK9AG96ADugCDF/AGZuBDe9Xetbn2uSitaVXPMfgD7esbRqqjDg==</latexit><latexit sha1_base64="B6G10aZJ7917GLLjco/i+td9dws=">AAACKnicbVDLSsNAFJ3UV42vqEs3g0VwISWRgm7EghuXFewDklAmk0k7dPJgZiKWkM9x55+4k7oobv0QJ2kEbT3DwOGce7n3Hi9hVEjTnGu1tfWNza36tr6zu7d/YBwe9USccky6OGYxH3hIEEYj0pVUMjJIOEGhx0jfm9wVfv+JcEHj6FFOE+KGaBTRgGIklTQ0bp0QybEXQAxvbOvCLN+zqzsK+o+Xll7mlOMyTvzcyqvCodEwm2YJuEqsijRAhc7QmDl+jNOQRBIzJIRtmYl0M8QlxYzkupMKkiA8QSNiKxqhkAg3Kyfn8EwpPgxirn4kYan+7shQKMQ09FRlsbpY9grxP89OZXDtZjRKUkkivBgUpAzKGBapQZ9ygiWbKoIwp2pXiMeIIyxVtrquqxis5aNXSe+yaZlN66HVaLeqQOrgBJyCc2CBK9AG96ADugCDF/AGZuBDe9Xetbn2uSitaVXPMfgD7esbRqqjDg==</latexit><latexit sha1_base64="B6G10aZJ7917GLLjco/i+td9dws=">AAACKnicbVDLSsNAFJ3UV42vqEs3g0VwISWRgm7EghuXFewDklAmk0k7dPJgZiKWkM9x55+4k7oobv0QJ2kEbT3DwOGce7n3Hi9hVEjTnGu1tfWNza36tr6zu7d/YBwe9USccky6OGYxH3hIEEYj0pVUMjJIOEGhx0jfm9wVfv+JcEHj6FFOE+KGaBTRgGIklTQ0bp0QybEXQAxvbOvCLN+zqzsK+o+Xll7mlOMyTvzcyqvCodEwm2YJuEqsijRAhc7QmDl+jNOQRBIzJIRtmYl0M8QlxYzkupMKkiA8QSNiKxqhkAg3Kyfn8EwpPgxirn4kYan+7shQKMQ09FRlsbpY9grxP89OZXDtZjRKUkkivBgUpAzKGBapQZ9ygiWbKoIwp2pXiMeIIyxVtrquqxis5aNXSe+yaZlN66HVaLeqQOrgBJyCc2CBK9AG96ADugCDF/AGZuBDe9Xetbn2uSitaVXPMfgD7esbRqqjDg==</latexit>

c = [1, 0, 0, 0, 0]

u = [1, 1, 0, 0, 0]<latexit sha1_base64="D/VfwcdaD5Q0tMynInJNsUXo7sw=">AAACKnicbVDLSsNAFJ34rPEVdelmsAgupCRS0I1YcOOygn1AEspkMmmHTiZhZiKU0M9x55+4k7oobv0QJ2kEbT3DwOGce7n3niBlVCrbnhtr6xubW9u1HXN3b//g0Do67sokE5h0cMIS0Q+QJIxy0lFUMdJPBUFxwEgvGN8Xfu+ZCEkT/qQmKfFjNOQ0ohgpLQ2sOy9GahREEMNb17m0F883PQ3zx8tKL/fKcbkg4dSZVoUDq2437BJwlTgVqYMK7YE188IEZzHhCjMkpevYqfJzJBTFjExNL5MkRXiMhsTVlKOYSD8vJ0/huVZCGCVCf65gqf7uyFEs5SQOdGWxulz2CvE/z81UdOPnlKeZIhwvBkUZgyqBRWowpIJgxSaaICyo3hXiERIIK52taZo6Bmf56FXSvWo4dsN5bNZbzSqQGjgFZ+ACOOAatMADaIMOwOAFvIEZ+DBejXdjbnwuSteMqucE/IHx9Q1fI6J+</latexit><latexit sha1_base64="D/VfwcdaD5Q0tMynInJNsUXo7sw=">AAACKnicbVDLSsNAFJ34rPEVdelmsAgupCRS0I1YcOOygn1AEspkMmmHTiZhZiKU0M9x55+4k7oobv0QJ2kEbT3DwOGce7n3niBlVCrbnhtr6xubW9u1HXN3b//g0Do67sokE5h0cMIS0Q+QJIxy0lFUMdJPBUFxwEgvGN8Xfu+ZCEkT/qQmKfFjNOQ0ohgpLQ2sOy9GahREEMNb17m0F883PQ3zx8tKL/fKcbkg4dSZVoUDq2437BJwlTgVqYMK7YE188IEZzHhCjMkpevYqfJzJBTFjExNL5MkRXiMhsTVlKOYSD8vJ0/huVZCGCVCf65gqf7uyFEs5SQOdGWxulz2CvE/z81UdOPnlKeZIhwvBkUZgyqBRWowpIJgxSaaICyo3hXiERIIK52taZo6Bmf56FXSvWo4dsN5bNZbzSqQGjgFZ+ACOOAatMADaIMOwOAFvIEZ+DBejXdjbnwuSteMqucE/IHx9Q1fI6J+</latexit><latexit sha1_base64="D/VfwcdaD5Q0tMynInJNsUXo7sw=">AAACKnicbVDLSsNAFJ34rPEVdelmsAgupCRS0I1YcOOygn1AEspkMmmHTiZhZiKU0M9x55+4k7oobv0QJ2kEbT3DwOGce7n3niBlVCrbnhtr6xubW9u1HXN3b//g0Do67sokE5h0cMIS0Q+QJIxy0lFUMdJPBUFxwEgvGN8Xfu+ZCEkT/qQmKfFjNOQ0ohgpLQ2sOy9GahREEMNb17m0F883PQ3zx8tKL/fKcbkg4dSZVoUDq2437BJwlTgVqYMK7YE188IEZzHhCjMkpevYqfJzJBTFjExNL5MkRXiMhsTVlKOYSD8vJ0/huVZCGCVCf65gqf7uyFEs5SQOdGWxulz2CvE/z81UdOPnlKeZIhwvBkUZgyqBRWowpIJgxSaaICyo3hXiERIIK52taZo6Bmf56FXSvWo4dsN5bNZbzSqQGjgFZ+ACOOAatMADaIMOwOAFvIEZ+DBejXdjbnwuSteMqucE/IHx9Q1fI6J+</latexit><latexit sha1_base64="D/VfwcdaD5Q0tMynInJNsUXo7sw=">AAACKnicbVDLSsNAFJ34rPEVdelmsAgupCRS0I1YcOOygn1AEspkMmmHTiZhZiKU0M9x55+4k7oobv0QJ2kEbT3DwOGce7n3niBlVCrbnhtr6xubW9u1HXN3b//g0Do67sokE5h0cMIS0Q+QJIxy0lFUMdJPBUFxwEgvGN8Xfu+ZCEkT/qQmKfFjNOQ0ohgpLQ2sOy9GahREEMNb17m0F883PQ3zx8tKL/fKcbkg4dSZVoUDq2437BJwlTgVqYMK7YE188IEZzHhCjMkpevYqfJzJBTFjExNL5MkRXiMhsTVlKOYSD8vJ0/huVZCGCVCf65gqf7uyFEs5SQOdGWxulz2CvE/z81UdOPnlKeZIhwvBkUZgyqBRWowpIJgxSaaICyo3hXiERIIK52taZo6Bmf56FXSvWo4dsN5bNZbzSqQGjgFZ+ACOOAatMADaIMOwOAFvIEZ+DBejXdjbnwuSteMqucE/IHx9Q1fI6J+</latexit>

error

error

error

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1 error corrected

Data Communication

C =n[1, 0, 0, 0, 0], [1, 0, 1, 1, 1],

[0, 1, 1, 1, 0], [0, 1, 0, 0, 1]o

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�(·) : {0, 1}L 7! C<latexit sha1_base64="ih5eS1oDiJMYk3VBVtHVuAvoNDM=">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</latexit><latexit sha1_base64="ih5eS1oDiJMYk3VBVtHVuAvoNDM=">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</latexit><latexit sha1_base64="ih5eS1oDiJMYk3VBVtHVuAvoNDM=">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</latexit><latexit sha1_base64="ih5eS1oDiJMYk3VBVtHVuAvoNDM=">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</latexit>

Decoding rule

B4<latexit sha1_base64="5qvbZIDIFgBACTjpFVFtOYqBzQ8=">AAAB9XicbVDLSsNAFL2prxpfVZduBovgqiRS0GVRBJcV7AOaUCbTSTt0MgkzEyWE/oY70YW49Wfc+DdO2iy09cDA4Zx7uWdOkHCmtON8W5W19Y3Nreq2vbO7t39QOzzqqjiVhHZIzGPZD7CinAna0Uxz2k8kxVHAaS+Y3hR+75FKxWLxoLOE+hEeCxYygrWRPC/CekIwR9fD5rBWdxrOHGiVuCWpQ4n2sPbljWKSRlRowrFSA9dJtJ9jqRnhdGZ7qaIJJlM8pgNDBY6o8vN55hk6M8oIhbE0T2g0V39v5DhSKosCM1lkVMteIf7nDVIdXvk5E0mqqSCLQ2HKkY5RUQAaMUmJ5pkhmEhmsiIywRITbWqybdvU4C5/epV0Lxqu03Dvm/XWbVlIFU7gFM7BhUtowR20oQMEEniGV3iznqwX6936WIxWrHLnGP7A+vwBP7qRQA==</latexit><latexit sha1_base64="5qvbZIDIFgBACTjpFVFtOYqBzQ8=">AAAB9XicbVDLSsNAFL2prxpfVZduBovgqiRS0GVRBJcV7AOaUCbTSTt0MgkzEyWE/oY70YW49Wfc+DdO2iy09cDA4Zx7uWdOkHCmtON8W5W19Y3Nreq2vbO7t39QOzzqqjiVhHZIzGPZD7CinAna0Uxz2k8kxVHAaS+Y3hR+75FKxWLxoLOE+hEeCxYygrWRPC/CekIwR9fD5rBWdxrOHGiVuCWpQ4n2sPbljWKSRlRowrFSA9dJtJ9jqRnhdGZ7qaIJJlM8pgNDBY6o8vN55hk6M8oIhbE0T2g0V39v5DhSKosCM1lkVMteIf7nDVIdXvk5E0mqqSCLQ2HKkY5RUQAaMUmJ5pkhmEhmsiIywRITbWqybdvU4C5/epV0Lxqu03Dvm/XWbVlIFU7gFM7BhUtowR20oQMEEniGV3iznqwX6936WIxWrHLnGP7A+vwBP7qRQA==</latexit><latexit sha1_base64="5qvbZIDIFgBACTjpFVFtOYqBzQ8=">AAAB9XicbVDLSsNAFL2prxpfVZduBovgqiRS0GVRBJcV7AOaUCbTSTt0MgkzEyWE/oY70YW49Wfc+DdO2iy09cDA4Zx7uWdOkHCmtON8W5W19Y3Nreq2vbO7t39QOzzqqjiVhHZIzGPZD7CinAna0Uxz2k8kxVHAaS+Y3hR+75FKxWLxoLOE+hEeCxYygrWRPC/CekIwR9fD5rBWdxrOHGiVuCWpQ4n2sPbljWKSRlRowrFSA9dJtJ9jqRnhdGZ7qaIJJlM8pgNDBY6o8vN55hk6M8oIhbE0T2g0V39v5DhSKosCM1lkVMteIf7nDVIdXvk5E0mqqSCLQ2HKkY5RUQAaMUmJ5pkhmEhmsiIywRITbWqybdvU4C5/epV0Lxqu03Dvm/XWbVlIFU7gFM7BhUtowR20oQMEEniGV3iznqwX6936WIxWrHLnGP7A+vwBP7qRQA==</latexit><latexit sha1_base64="5qvbZIDIFgBACTjpFVFtOYqBzQ8=">AAAB9XicbVDLSsNAFL2prxpfVZduBovgqiRS0GVRBJcV7AOaUCbTSTt0MgkzEyWE/oY70YW49Wfc+DdO2iy09cDA4Zx7uWdOkHCmtON8W5W19Y3Nreq2vbO7t39QOzzqqjiVhHZIzGPZD7CinAna0Uxz2k8kxVHAaS+Y3hR+75FKxWLxoLOE+hEeCxYygrWRPC/CekIwR9fD5rBWdxrOHGiVuCWpQ4n2sPbljWKSRlRowrFSA9dJtJ9jqRnhdGZ7qaIJJlM8pgNDBY6o8vN55hk6M8oIhbE0T2g0V39v5DhSKosCM1lkVMteIf7nDVIdXvk5E0mqqSCLQ2HKkY5RUQAaMUmJ5pkhmEhmsiIywRITbWqybdvU4C5/epV0Lxqu03Dvm/XWbVlIFU7gFM7BhUtowR20oQMEEniGV3iznqwX6936WIxWrHLnGP7A+vwBP7qRQA==</latexit>

q Minimizeavgpowerasafunctionof,

q Outageconstraint:

q toachievedesiredraterequirementforUE

q Decodingrule:

q Changeofvariable :

F = {B1, . . . ,BL}<latexit sha1_base64="xr8DH/ee/bzV1bj6h5RjJaN9IKc=">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</latexit><latexit sha1_base64="xr8DH/ee/bzV1bj6h5RjJaN9IKc=">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</latexit><latexit sha1_base64="xr8DH/ee/bzV1bj6h5RjJaN9IKc=">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</latexit><latexit sha1_base64="xr8DH/ee/bzV1bj6h5RjJaN9IKc=">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</latexit>

�<latexit sha1_base64="lK/UcGlbN7YtcRFRgMKRAfz/nz0=">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</latexit><latexit sha1_base64="lK/UcGlbN7YtcRFRgMKRAfz/nz0=">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</latexit><latexit sha1_base64="lK/UcGlbN7YtcRFRgMKRAfz/nz0=">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</latexit><latexit sha1_base64="lK/UcGlbN7YtcRFRgMKRAfz/nz0=">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</latexit>

�(·) : {0, 1}L 7! C<latexit sha1_base64="ih5eS1oDiJMYk3VBVtHVuAvoNDM=">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</latexit><latexit sha1_base64="ih5eS1oDiJMYk3VBVtHVuAvoNDM=">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</latexit><latexit sha1_base64="ih5eS1oDiJMYk3VBVtHVuAvoNDM=">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</latexit><latexit sha1_base64="ih5eS1oDiJMYk3VBVtHVuAvoNDM=">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</latexit>

�d [energy/rad]<latexit sha1_base64="7I6w/7+ja09sH5GRkvSaSz0fN1g=">AAACH3icbVDLSsNAFJ34tr6qLt0MFsGF1EQFXRZEcKlgVWhCmUxu2sHJJMzciCXkQ1z6JS7Vjbhx0b9x2mbh68DAmXPOZeaeMJPCoOsOnanpmdm5+YXF2tLyyupafX3j2qS55tDmqUz1bcgMSKGgjQIl3GYaWBJKuAnvTkf+zT1oI1J1hYMMgoT1lIgFZ2ilbv3Qz/qiW0Ql9SXE2Cl8hAcsQIHuDcpyv7prFpU2okWvj0G33nCb7hj0L/Eq0iAVLrr1Tz9KeZ6AQi6ZMR3PzTAomEbBJZQ1PzeQMX7HetCxVLEETFCMlyvpjlUiGqfaHoV0rH6fKFhizCAJ98LEhhOGffPbHon/eZ0c45OgECrLERSfvBXnkmJKR2XRSGjgKAeWMK6F/S7lfaYZR1tpzfbg/d76L7k+aHpu07s8arTOqkYWyBbZJrvEI8ekRc7JBWkTTh7JM3klb86T8+K8Ox+T6JRTzWySH3CGX8ISpJw=</latexit><latexit sha1_base64="7I6w/7+ja09sH5GRkvSaSz0fN1g=">AAACH3icbVDLSsNAFJ34tr6qLt0MFsGF1EQFXRZEcKlgVWhCmUxu2sHJJMzciCXkQ1z6JS7Vjbhx0b9x2mbh68DAmXPOZeaeMJPCoOsOnanpmdm5+YXF2tLyyupafX3j2qS55tDmqUz1bcgMSKGgjQIl3GYaWBJKuAnvTkf+zT1oI1J1hYMMgoT1lIgFZ2ilbv3Qz/qiW0Ql9SXE2Cl8hAcsQIHuDcpyv7prFpU2okWvj0G33nCb7hj0L/Eq0iAVLrr1Tz9KeZ6AQi6ZMR3PzTAomEbBJZQ1PzeQMX7HetCxVLEETFCMlyvpjlUiGqfaHoV0rH6fKFhizCAJ98LEhhOGffPbHon/eZ0c45OgECrLERSfvBXnkmJKR2XRSGjgKAeWMK6F/S7lfaYZR1tpzfbg/d76L7k+aHpu07s8arTOqkYWyBbZJrvEI8ekRc7JBWkTTh7JM3klb86T8+K8Ox+T6JRTzWySH3CGX8ISpJw=</latexit><latexit sha1_base64="7I6w/7+ja09sH5GRkvSaSz0fN1g=">AAACH3icbVDLSsNAFJ34tr6qLt0MFsGF1EQFXRZEcKlgVWhCmUxu2sHJJMzciCXkQ1z6JS7Vjbhx0b9x2mbh68DAmXPOZeaeMJPCoOsOnanpmdm5+YXF2tLyyupafX3j2qS55tDmqUz1bcgMSKGgjQIl3GYaWBJKuAnvTkf+zT1oI1J1hYMMgoT1lIgFZ2ilbv3Qz/qiW0Ql9SXE2Cl8hAcsQIHuDcpyv7prFpU2okWvj0G33nCb7hj0L/Eq0iAVLrr1Tz9KeZ6AQi6ZMR3PzTAomEbBJZQ1PzeQMX7HetCxVLEETFCMlyvpjlUiGqfaHoV0rH6fKFhizCAJ98LEhhOGffPbHon/eZ0c45OgECrLERSfvBXnkmJKR2XRSGjgKAeWMK6F/S7lfaYZR1tpzfbg/d76L7k+aHpu07s8arTOqkYWyBbZJrvEI8ekRc7JBWkTTh7JM3klb86T8+K8Ox+T6JRTzWySH3CGX8ISpJw=</latexit><latexit sha1_base64="7I6w/7+ja09sH5GRkvSaSz0fN1g=">AAACH3icbVDLSsNAFJ34tr6qLt0MFsGF1EQFXRZEcKlgVWhCmUxu2sHJJMzciCXkQ1z6JS7Vjbhx0b9x2mbh68DAmXPOZeaeMJPCoOsOnanpmdm5+YXF2tLyyupafX3j2qS55tDmqUz1bcgMSKGgjQIl3GYaWBJKuAnvTkf+zT1oI1J1hYMMgoT1lIgFZ2ilbv3Qz/qiW0Ql9SXE2Cl8hAcsQIHuDcpyv7prFpU2okWvj0G33nCb7hj0L/Eq0iAVLrr1Tz9KeZ6AQi6ZMR3PzTAomEbBJZQ1PzeQMX7HetCxVLEETFCMlyvpjlUiGqfaHoV0rH6fKFhizCAJ98LEhhOGffPbHon/eZ0c45OgECrLERSfvBXnkmJKR2XRSGjgKAeWMK6F/S7lfaYZR1tpzfbg/d76L7k+aHpu07s8arTOqkYWyBbZJrvEI8ekRc7JBWkTTh7JM3klb86T8+K8Ox+T6JRTzWySH3CGX8ISpJw=</latexit>

P̄avg(F ,�) , 1

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k=1

�s |Bk|+ �d |Bcomm(u)|#

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Bcomm(u) = \Lk=1B

�(u)k = U�(u)

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q Optimization:s.t.

q Combinatorialandnon-convex

Ø Select, andindependently

q asECCthatcancorrectupto-bitsoferrors

q asmindistancedecodingfor

q

q ConvexUpperbound()

Ø canbemadearbitrarilytightbyreducing

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C<latexit sha1_base64="gt5tHop5FFf9F7pCErzgoGD2DHg=">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</latexit><latexit sha1_base64="gt5tHop5FFf9F7pCErzgoGD2DHg=">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</latexit><latexit sha1_base64="gt5tHop5FFf9F7pCErzgoGD2DHg=">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</latexit><latexit sha1_base64="gt5tHop5FFf9F7pCErzgoGD2DHg=">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</latexit>

G⇤ = argminG

P̄avg(G, C,�)<latexit sha1_base64="JyQvQMYym5IXjmphteKueORoqMw=">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</latexit><latexit sha1_base64="JyQvQMYym5IXjmphteKueORoqMw=">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</latexit><latexit sha1_base64="JyQvQMYym5IXjmphteKueORoqMw=">AAACxXicbZHfbtMwGMXd8G+Efx1ccmNRIQ1UVck0CW6QJsY0bhBF0G2iLpXjfkmt2U5kO4ViLF6N1+AFuIVHwGkDYxufZOnonJ9l63xZJbixSfK9E126fOXqtY3r8Y2bt27f6W7ePTRlrRmMWClKfZxRA4IrGFluBRxXGqjMBBxlJ3tNfrQAbXip3tllBRNJC8VzzqgN1rT7nkhq54wKfPDhMX6GCdUFkVxN3WngMcmoxsPW09LRReH91inRx3/1Xp8M5xw/wvG020sGyWrwRZG2oofaGU43O2/JrGS1BGWZoMaM06SyE0e15UyAj0ltoKLshBYwDlJRCWbiViV4/DA4M5yXOhxl8cr994aj0pilzPqZDHDzXXM+bsz/ZePa5k8njquqtqDY+q28FtiWuCkVz7gGZsUyCMo0D9/FbE41ZTZUHxMFH1kpJVUzRwz/DH6cThwRkFv8BfdSTDQv5o32Z9ksLHLN/mn3+dQF3jfgCwgdaXgVktcVaGpL7cj+p8qv6Sxz+97HcbOG9HzpF8Xh9iBNBumbnd7uTruQDXQfPUBbKEVP0C56iYZohBj6hn6gn+hXdBDJyEaLNRp12jv30JmJvv4GFMbbmw==</latexit><latexit sha1_base64="JyQvQMYym5IXjmphteKueORoqMw=">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</latexit>

minG,C,�

P̄avg(G, C,�)<latexit sha1_base64="54PHPeoPCD1bc+kfuZDv9j9hwXM=">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</latexit><latexit sha1_base64="54PHPeoPCD1bc+kfuZDv9j9hwXM=">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</latexit><latexit sha1_base64="54PHPeoPCD1bc+kfuZDv9j9hwXM=">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</latexit><latexit sha1_base64="54PHPeoPCD1bc+kfuZDv9j9hwXM=">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</latexit>

�(·)<latexit sha1_base64="lupv35foC3lJ209JEP8EU8hzTug=">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</latexit><latexit sha1_base64="lupv35foC3lJ209JEP8EU8hzTug=">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</latexit><latexit sha1_base64="lupv35foC3lJ209JEP8EU8hzTug=">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</latexit><latexit sha1_base64="lupv35foC3lJ209JEP8EU8hzTug=">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</latexit>

P(W (e) > ") ⌧ 1<latexit sha1_base64="8aOlCwLbPP1KxpiYlHMm+Yv/Bsk=">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</latexit><latexit sha1_base64="8aOlCwLbPP1KxpiYlHMm+Yv/Bsk=">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</latexit><latexit sha1_base64="8aOlCwLbPP1KxpiYlHMm+Yv/Bsk=">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</latexit><latexit sha1_base64="8aOlCwLbPP1KxpiYlHMm+Yv/Bsk=">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</latexit>

OptimalSolutionq NewObjective:

q Optimization:

q Water-fillingSolution:

Ø Higher à FrequentscanningW (c)<latexit sha1_base64="XIyCFTvI1uEvmHpGuEVDlUjx99k=">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</latexit><latexit sha1_base64="XIyCFTvI1uEvmHpGuEVDlUjx99k=">AAACd3icbVFNbxMxFHSWQsvy0RaOHLCIQEFC0RohlWMFjcSlohWkqZRdItt5m1j1x8r2AqnZ/8EV/lV/Sm/1JjnQlidZGs3Mk0fzWCWF81l20UnubNy9t7l1P33w8NHj7Z3dJyfO1JbDkBtp7CmjDqTQMPTCSzitLFDFJIzY2cdWH30H64TRX/2igkLRmRal4NRH6tuolyvq56zE/HWaTna6WT9bDr4NyBp00XqOJrudL/nU8FqB9lxS58Ykq3wRqPWCS2jSvHZQUX5GZzCOUFMFrgjL2A1+GZkpLo2NT3u8ZP/dCFQ5t1DsDVPR3OZ0N+WW/J82rn35vghCV7UHzVd/lbXE3uC2BjwVFriXiwgotyLGxXxOLeU+lpXmGn5woxTV05A7cQ7NmBQhl1B6/At3Cc6tmM1b3Fz3slj9yttG4lTiD5MQ/U1rPIDYkYXDqHyuwFJvbMgHP6tm5WYsDJomXZ6B3Cz9Njh52ydZnxy/6+4P1gfZQs/QC9RDBO2hffQJHaEh4sii3+gP+tu5TJ4nr5Leypp01jtP0bVJyBUoo75q</latexit><latexit sha1_base64="XIyCFTvI1uEvmHpGuEVDlUjx99k=">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</latexit><latexit sha1_base64="XIyCFTvI1uEvmHpGuEVDlUjx99k=">AAACd3icbVFNbxMxFHSWQsvy0RaOHLCIQEFC0RohlWMFjcSlohWkqZRdItt5m1j1x8r2AqnZ/8EV/lV/Sm/1JjnQlidZGs3Mk0fzWCWF81l20UnubNy9t7l1P33w8NHj7Z3dJyfO1JbDkBtp7CmjDqTQMPTCSzitLFDFJIzY2cdWH30H64TRX/2igkLRmRal4NRH6tuolyvq56zE/HWaTna6WT9bDr4NyBp00XqOJrudL/nU8FqB9lxS58Ykq3wRqPWCS2jSvHZQUX5GZzCOUFMFrgjL2A1+GZkpLo2NT3u8ZP/dCFQ5t1DsDVPR3OZ0N+WW/J82rn35vghCV7UHzVd/lbXE3uC2BjwVFriXiwgotyLGxXxOLeU+lpXmGn5woxTV05A7cQ7NmBQhl1B6/At3Cc6tmM1b3Fz3slj9yttG4lTiD5MQ/U1rPIDYkYXDqHyuwFJvbMgHP6tm5WYsDJomXZ6B3Cz9Njh52ydZnxy/6+4P1gfZQs/QC9RDBO2hffQJHaEh4sii3+gP+tu5TJ4nr5Leypp01jtP0bVJyBUoo75q</latexit>

!⇤ = argmin!�0

P̂avg(!)

s.t.X

c2C!c = 2⇡

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!⇤c =

⇡�s

P(W (e) ")�d[��W (c)]+

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Dual var :water level

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NumericalResults

Pavg [dBm]

AnalogbeamsusedNoisepsd =-173dBmFrameduration=20msSlot=10usBandwidth=0.5GHzDistance=10mPathloss exp.=2Mis-detectionprob.=0.3False-alarmprob.=0.3#Antennas=64(BS)

SpectralEfficiency[bps/H

z]

uncoded

(7,4)Hamming

exhaustive

• Uncoded cannotcorrecterror(s)• Exhaustivecancorrect1error

with50%prob.• (7,4)Hammingcancorrect1

errorwith100%prob.

-1.5 dB -3 dB

q Energy-efficientcodedBA

Ø Convexoptimizationbaseddesign

Ø Water-fillingsolution

Ø Superiorperformance

Ø 3dBgainoverexhaustivesearch

Ø 4.5dBgainoveruncoded BA

q 1BS,M users(UEs)

q Lbeam-alignmentslot

Ø BSusesf1,f2, …,fLØ UEsusesg1,g2, …,gLØ i.i.d fadingacrosstheslots

q L-bitscollisionfreefeedbackfromUEs

q TDMAindatatransmission

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Beam Alignment<latexit sha1_base64="9geXDYHTXfXVQnPnM1ljW2zFhoU=">AAAB/nicbVDLSgMxFM3UV62vqrhyEyxCV2VGBF1W3bisYB/QDiWT3rahmcyQ3BHLUPBX3LhQxK3f4c6/MW1noa0HAodz7uHenCCWwqDrfju5ldW19Y38ZmFre2d3r7h/0DBRojnUeSQj3QqYASkU1FGghFasgYWBhGYwupn6zQfQRkTqHscx+CEbKNEXnKGVusWjDsIjptc2Qq+kGKgQFE66xZJbcWegy8TLSIlkqHWLX51exJNpmEtmTNtzY/RTplFwCZNCJzEQMz5iA2hbqlgIxk9n50/oqVV6tB9p+xTSmfo7kbLQmHEY2MmQ4dAselPxP6+dYP/ST4WKEwTF54v6iaQY0WkXtCc0cJRjSxjXwt5K+ZBpxtE2VrAleItfXiaNs4rnVry781K1nNWRJ8fkhJSJRy5IldySGqkTTlLyTF7Jm/PkvDjvzsd8NOdkmUPyB87nD2u8lbA=</latexit><latexit sha1_base64="9geXDYHTXfXVQnPnM1ljW2zFhoU=">AAAB/nicbVDLSgMxFM3UV62vqrhyEyxCV2VGBF1W3bisYB/QDiWT3rahmcyQ3BHLUPBX3LhQxK3f4c6/MW1noa0HAodz7uHenCCWwqDrfju5ldW19Y38ZmFre2d3r7h/0DBRojnUeSQj3QqYASkU1FGghFasgYWBhGYwupn6zQfQRkTqHscx+CEbKNEXnKGVusWjDsIjptc2Qq+kGKgQFE66xZJbcWegy8TLSIlkqHWLX51exJNpmEtmTNtzY/RTplFwCZNCJzEQMz5iA2hbqlgIxk9n50/oqVV6tB9p+xTSmfo7kbLQmHEY2MmQ4dAselPxP6+dYP/ST4WKEwTF54v6iaQY0WkXtCc0cJRjSxjXwt5K+ZBpxtE2VrAleItfXiaNs4rnVry781K1nNWRJ8fkhJSJRy5IldySGqkTTlLyTF7Jm/PkvDjvzsd8NOdkmUPyB87nD2u8lbA=</latexit><latexit sha1_base64="9geXDYHTXfXVQnPnM1ljW2zFhoU=">AAAB/nicbVDLSgMxFM3UV62vqrhyEyxCV2VGBF1W3bisYB/QDiWT3rahmcyQ3BHLUPBX3LhQxK3f4c6/MW1noa0HAodz7uHenCCWwqDrfju5ldW19Y38ZmFre2d3r7h/0DBRojnUeSQj3QqYASkU1FGghFasgYWBhGYwupn6zQfQRkTqHscx+CEbKNEXnKGVusWjDsIjptc2Qq+kGKgQFE66xZJbcWegy8TLSIlkqHWLX51exJNpmEtmTNtzY/RTplFwCZNCJzEQMz5iA2hbqlgIxk9n50/oqVV6tB9p+xTSmfo7kbLQmHEY2MmQ4dAselPxP6+dYP/ST4WKEwTF54v6iaQY0WkXtCc0cJRjSxjXwt5K+ZBpxtE2VrAleItfXiaNs4rnVry781K1nNWRJ8fkhJSJRy5IldySGqkTTlLyTF7Jm/PkvDjvzsd8NOdkmUPyB87nD2u8lbA=</latexit><latexit sha1_base64="9geXDYHTXfXVQnPnM1ljW2zFhoU=">AAAB/nicbVDLSgMxFM3UV62vqrhyEyxCV2VGBF1W3bisYB/QDiWT3rahmcyQ3BHLUPBX3LhQxK3f4c6/MW1noa0HAodz7uHenCCWwqDrfju5ldW19Y38ZmFre2d3r7h/0DBRojnUeSQj3QqYASkU1FGghFasgYWBhGYwupn6zQfQRkTqHscx+CEbKNEXnKGVusWjDsIjptc2Qq+kGKgQFE66xZJbcWegy8TLSIlkqHWLX51exJNpmEtmTNtzY/RTplFwCZNCJzEQMz5iA2hbqlgIxk9n50/oqVV6tB9p+xTSmfo7kbLQmHEY2MmQ4dAselPxP6+dYP/ST4WKEwTF54v6iaQY0WkXtCc0cJRjSxjXwt5K+ZBpxtE2VrAleItfXiaNs4rnVry781K1nNWRJ8fkhJSJRy5IldySGqkTTlLyTF7Jm/PkvDjvzsd8NOdkmUPyB87nD2u8lbA=</latexit>

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