SparsityandRankExploita2onforTime-VaryingLeakedOFDMUWAChannelEs2ma2on

Thanksto:ONRN00014-15-1-2550,NSFCNS-1213128,NSFCCF-1410009,AFOSRFA9550-12-1-0215,NSFCPS-1446901,theRoyalAcademyofEngineering,theFulbright

FoundaHonandtheLeverhulmeTrust

AmrEl-Nakeeb&UrbashiMitraUniversityofSouthernCalifornia

Anissueofscale

q  Theoceanisabigplaceo  Smaller,moderatenodedeploymentsinconstrainedenvironments

o  Intheocean…

q  Costo  Selfdevelopment/packagingo  Large-scaledeploymento  Economiesofscale

q  Acommsarespecialo  But,resultscanbeleveragedforotherapplicaHons

2

TimeVaria2on

q  HmevariaHono  Mobility,internalwaves,changesin

fineverHcalstructureofwater,small-scaleturbulence,surfacemoHon

q  Resampling:Yerramalli&M,JOE’11Beygi&M,SPLe[ers’13,Xu,Tang,Leus&MTSP’13

delaysvaryingasafuncHonofHme

fromMilicaStojanovic(Northeastern)

fromPaulvanWalree(NorwegianDefenceResearchEstablishment)

3

Pop quizLet A :=

)(1, 0)

T, (0, 1)

T, (≠1, 0)

T, (0, ≠1)

T*

, and let x := (≠ 1

5

, 1)

T . What isÎxÎA?

ANS: ÎxÎA =

6

5

.

x =

�� 1

51

�

conv(A)

x1

x2

Mathematics of Data: From Theory to Computation | Prof. Volkan Cevher, volkan.cevher@epfl.ch Slide 31/ 42

Pop quizLet A :=

)(1, 0)

T, (0, 1)

T, (≠1, 0)

T, (0, ≠1)

T*

, and let x := (≠ 1

5

, 1)

T . What isÎxÎA?

ANS: ÎxÎA =

6

5

.

x =

�� 1

51

�

conv(A)

x1

x2

Mathematics of Data: From Theory to Computation | Prof. Volkan Cevher, volkan.cevher@epfl.ch Slide 31/ 42

x =

Lÿ

l=0cl al where al œ K, cl Ø 0

ÎxÎK = inf {t > 0 : x œ t conv (K)} (1)

= inf

Iÿ

l

cl : x =

Lÿ

l=0clal, cl Ø 0, al œ K

J(2)

x =

1

5

◊ (≠1, 0)

T+ 1 ◊ (0, 1)

T

1

Whatcanthemathbuyyou?q  Noveltools–willtheywork?

o  SparseapproximaHon/compressedsensingo  LowrankmatrixcompleHono  Atomicnormdenoising

q  WhatistherightabstracHon/model?o  MulHpatho  Dopplerscalingo  Robust?Modelmismatch?

channel delay

Dop

pler

0 1 2 3 4 5 6 7 8 90

1

channel delay

abs(

h)

SUBMITTED TO .... 3

B. Received SignalIn general, the signal after passing through a linear time-

varying (LTV) channel can be written as,

y(t) =

+1Z

�1

h(t, ⌧)s(t� ⌧)d⌧ + n(t), (5)

where n(t) is assumed to be additive, white Gaussian noise.Thus the received signal, y(t), can be represented as

y(t) =PX

p=1

KX

k=1

hpskej2⇡fk((1+ap)t�⌧p)

+ n(t). (6)

Remark 1. Suppose the frequencies fk lie in [�W,W ], namelyB = 2W = K�f . Let call the first term of y(t) in Eq. (6) asx(t), i.e.,

y(t) = x(t) + n(t), and

x(t) =

PX

p=1

KX

k=1

hpskej2⇡fk((1+ap)t�⌧p). (7)

By taking regularly spaced Nyquist samples at t 2 �

i2W |i 2

[Ns]

, where [Ns] = {0, 1, 2, ..., Ns � 1} and Ns denotes thetotal number of Nyquist samples, we observe

x[i] =PX

p=1

KX

k=1

hpske�j2⇡fk⌧pe

jh2⇡

fk2W (ap+1)

ii,

where fk2W 2 [0, 1]. Therefore after a trivial translation of the

frequency domain, we can map fk2W =

fmin

K +

kK to the new

normalized frequency for x[i] as ˆfk =

kK . Note that hereafter

we do not define a new variable for normalized frequencyˆfk, and for simplicity in notation we consider fk =

kK as the

normalized frequency.

We can express the sampled signal in Eq. (6) as y[i] =

x[i]+n[i], where the index i denotes the sample time. We canrewrite x[i] as

x[i] =LX

l=1

clzil , (8)

where L = PK denotes the number of exponential terms inthe summation, with l = (k � 1)P + p and 1 k K and1 p P . Hereafter, we use the following notation to showthe relationship between l and the pair (k, p)

cl = c(k,p) = hpe�j2⇡fk⌧psk, (9)

zl = z(k,p) = ej2⇡fk(1+ap). (10)

Note that this representation is unique, i.e for each l there isonly one pair (k, l) and vice versa.Remark 2. As seen from Eq. (9) the coefficients cl containsthe information about the channel attenuation gains hp anddelays ⌧p for 1 p P . Using Eq. (10), it is clear that zlonly depends on scale values ap and |zl| = 1, i.e. all the zl

0.2

0.4

0.6

0.8

1

30

210

60

240

90

270

120

300

150

330

180 0

Fig. 2. Illustration of zl for an OFDM signaling with K = 16 subcarriersand a typical UWA channel with P = 5 dominant paths. Doppler scales apare chosen randomly from [1� a, 1 + a] with a = 0.01.

for 1 l L are located on unit circle in the polar complexrepresentation plane.

Since ap are typically small, the value of fk(1+ap) for p =

1, . . . , P are very close to each other. In Fig. 2, we illustrate thezl generated by an OFDM signaling with K = 16 subcarriersand a typical UWA channel with P = 5 dominant paths. Wesee that by increasing the subcarrier frequency the Dopplereffect getting worse and destroying the orthogonality betweenthe subcarriers. Channel delay and scale parameters are chosenrandomly from their feasible support. For this example wehave considered amax = 0.01. It can be observed that the zlare clustered around each subcarrier frequency due to smallvalues of ap. In sequel we will see that this property can behelpful in MSML channel estimation problem.

III. CHANNEL ESTIMATION ALGORITHM

In this section we develop an algorithm to estimate thechannel parameters

P[p=1

{hp, ap, ⌧p}

using the receive signal samples, y[i] for i 2 {0, 1, 2, . . . , S},where S + 1 denotes the total number of available measure-ments. The MSML channel estimation problem is the paramet-ric estimation problem in (7), which also can be interpret asthe problem of retrieving the parameters of a sum of complexexponentials from noisy samples, given that we have someadditional information about the structure of signal x[i]. Thus,

4

q  ModifiedPronywithlow-rankdenoisingo  LowrankfromdirectlyexploiHngproperHesofUWAchannel(closeDoppler

values)

q  1024subcarriers,P=7pathsq  Dopplerscales<=0.01

Beygi&MTSP’15

RoomforProny

q  SA=directlyesHmatechannel(classicalsparse)

q  SDPonlylowrankq  ESPRIT/MUSIC–

terrible!

5

Separa2onofleakedchannelq  PracHcalsystemconstraints

o  Finitebandwidtho  finiteblocklengtho  losechannelsparsity

q  Leakedchannelmatrixisseparable(matrixfactorizaHon)

duetodelayduetoDoppler

A.Elnakeeb&M,ICASSP,April2018

7

Algorithm&ScalingLaw

q  Atomicnorm/SDP

o  BilinearLeakedChannelEsHmaHon(BLCE)

o  SDPyieldso  UsingparametricleakagefuncHonsdetermine

q  ScalingLawo  Iidrandomtraining(Rademacher)

o  MinimumDopplerseparaHonneeded

o  Numberofsamplesneeded

o  SDPrecoverswithprobability

Bhaskar,Tang&Recht,TSP12/13

Beygi&M,ICASSP3/17,Elnakeeb&MICASSP4/18

9

Performance:Synthe2cData

q  NormalizedMSE

q  MSEimproveswithnumberofsamples,degradeswithnumberofunknowns

q  BLCEoutperformsclassicalLASSOo  duetolostsparsityfromleakage

Bajwa,Haupt,Sayeed,Nowak,ProceedingsIEEE2010

10

Performance:MACE’10Data

q  BLCEneedsdelayandDopplerspreadso  NoncoherentdatademodulaHonàcoarsechannelesHmaHon(OFDM/scaling)

o  RunBLCEo  Re-esHmatedelayandDopplerspread-  Checkdifferences

o  DifferenHaldecoding-  MSEondata

Aval&Stojanovic“DifferenHallyCoherentMulHchannelDetecHonofAcousHcOFDMSignals”,IEEEJOE2015

11

FinalThoughts

q  Originalpaperforsinglecarrier(Beygi&MICASSP2017,Beygi,Elnakeeb&MTSP–underrevision)o  OFDM(ICASSP’18)o  MIMO(SPAWC’18submi[ed)o  CRBcomputaHon(SPAWC’18submi[ed)

-  ConstrainedCRBexploitssparsityofchannelviarankconstraint-  DecoupledleakageàdecoupledCRB(foriidtraining)

-  FordeterminisHctraining,howtoopHmize?(Dopplervsdelay)

q  Modelingo  Whatarethelimits?o  Mathdoeswork(someHmes!)o  DemocraHzaHonofdatasets(?!)

13