Acoustic Source Loc.

-

- Beamforming Basics

I Steering Vector

Delay - Sum

- MUSIC

- G- CC / T Do A Spectrum- Blind Channel Estimation

IcfRx

Assume far field,

i.e.,

all µ . . /rays a parallel at Rx . -

¥. - -

- -ET

, Tz Tz . . . TN.: Path difference of nfnf -d

TN with T,

= @- Ddcoso

@ This causes a phase diff . between TN and T,

on ( N - 1) DWSO z-

a-.

^.

@ say transmitted signal = cos Zaftreuivedyfggnal=

T.eywsfaft.li#doso.2a)

@ at 0 denote Clay.za )

Also,

note than wso = Re{ ejo }

.

'

. ylt ) = My Re { eitaft.

eieD0 }

;,Re{eioft Eyeing }2e{ ejtft

.

| - e' INO

an~

( ¥0 ) }

@ As N increases , side lobes get smaller,

main lobesharper

:#I#N= 2 N=4 N=8

@ Also note than the same R× at direction 0,

experiences different gain / SNR with

increasing N .

@ Beam Rotation

¥##@ now

,how do you rotate the main lobe of

the beam towards Q so Rx gets Max SNR .

@ Bani idea : Add a phase shift to each

Ti such that the

dcoxscf .

2h phase difference

gets compensated .

: T,

transmits cos @aft )Tz transmits Cos @aft - dco¥2h)Tz " Cos ( Zaft - 2dcosyB=)

:

÷N' ' cos (2aft-i@Dd_qB.2.a)

@ Revived signalYH ) =Re{emftp.yneilmmxddo.eiswytzdejp

= Re { eithft

MyeIln¥.2ad( coso -

copy

when a =D , we get ylt ) - Re { ejtnft }

which Is the Max possible signal power .

@ Why Max at o =p ?

Recall,

YLH = Re { Fy,

cos ( zhft + 2×1 .d( coso . wsBD}

when Q =p ,then an Cos 2 aft coherently

add up; else its incoherent .

@ this in caned -DELAY-SVMB#MFoRM1NG_because each Tx is getting umvhi plied bya phase factor ( ej C it B) which one

then an adding up at the Rx .

@ Ang6ofAmivd'(A# TO

#40R,Rz- - -

-Rn. ,

Ypp-

wszhft u ft-

|

YR•,Gsthftt 0 ej0left;;¥±f" leg

lay

ejln. Dg

YRn.ie0

HE.tl#:Iendnn

for multiple sources :

*#Ro 14 Rz . - - - Rn .

YR ,e

°

ej 02

d;dt¥i%;::→.tt#dejln - DO ,

in - in

Array Steering matrix ( m ) source

output signals .

@ so how do you pull out individual signals ?

Ee " ' e. '. .

.dk#Tfggrng..)=e*e*i..em*Tien.

= Large value ifAOA is ialons 01 ,

steering along an directions O,

weof AOA spectrum

,al a)

a.as=

MUSIC- m antenna

se = As theon samples

+ \A. strewing matrix

. . . . . .

In i :Lja . .

]+ !!m

I 2. . . .

d ~

T Gunman

multipathH

sen = ( As + n ) ( As + myH

= ( As ) ( As )"

t Asn" mfs)

"- inn

't

= As SHAH+ Asn 't-1ns HAH +n④

R×× = A RssAH

t Asn "t n SHAH + Run-

why Efren" ) -

- htt

Ef Asn't ) t EJNSHAHJ

mxmfmtxd

) I dxdlldxm) t

MX me

mxm diagonal matrix

but rank -- d rank = m

MUSIC :

-

⑦ Ima→ AoA spectrum

'

d'

→-

③ ifI39. - - - . .MP

Im×,

-_ [ M Nz Nz - -. am )

T

③ Model : Xm×,

= Am xd. Sdx ,

+hmm

③ Amid =

( Algo, Alya-

. -

Agha)where Aoi =

-

Lj duos Oi

l:÷÷÷:::band Sdn

=/ ÷:)si = source signal

③ nma

=

IFgm) n ; is noise at antenna i

⑨ EIXXH) -

-

EffAs + on ) C As + n )

"

]=ELLA.

Stn ) ( SHAH + NH ) ]= Ef As SHAH + A

.sn#+msHAH+mnHJ--ELAssHAHJto t O + E

fun 't )

R×× = A Rss AH + r ? I

- --

mxm r ? IAM xd . Rssdxd A "d×m m×m

- -

this matrix isQmxm Rank ( I ) .

- M

and Rank(9) = d

⑦ def ( A. Rss .A

't

) = O since Osman hasrank dem

.

.

'

.det ( R×× - r

' I ) = O = det ( AR ss AH)

This implies : TZ is a eigenvalue .

Also,

This is minimum eigenvalue .

Now, Rxx .

e ; = Xi Lei ⇒ teigen Eq?

. : ( A Rss AH + r2 1) ei = Xi Zei

For minimum Xi = Amin,

⇐Rss A 'tt Amin 2) enifAmin Zemin⇒ A Rss AHemin O

since Rss is rank d,

and A has no

will space ,

ooo AH. emin= O

i. e, emin vectors I to the mixing matrix

columns.

H.

'

. For the correct Aoi , Aoi . emin = O

HFor wrong Aoi ,

Ace, emin should have

Some won - Zero

take

! AoA spectrum PLO ;) = 1-

HA"oi .

enoisellI I

I x m m x ( m - d)-

.

no .÷'ll liftedAo Az

" " mkhhn.

Oi

③ Questions around music

9 ① If sources are not independent Leg .

, multipath )where win the math break ?

Q ② If d > M, why does the math break ?

Q③ '

Why is The A matrix independent ?

A ③ Vandeven oud matrices are of the form

l÷÷÷÷÷÷÷÷÷u÷.

the proof is that The det ( Vandegrift ) = IT Gi - Lj)which is only 0 only when ai

= Lj

@ Generalized Cross Correlation ( Gcc ) :

÷"#\@ y.lt

)A #ym@ Yzltd

#

@ The Gcc algorithm :

T.FI/GTctsTDoa7- Spectrum

@ Ga pseudo code :

Input : Yilt ), yzlt )

i. Yi ( f ) = FFT (Y , ( t ) )

2 . Yzlf ) = FFT ( YZLTD3 '

For increasing values Of T

G ( T ) = g§g

Yi(f)Yz(f)d2N=| Y, (f) Yz* ( f ) )

4 . gtt ) = IFFT ( GCE ) )End

.

@ ^ gk )

•

r

t.tt#lnsLandTDOA

E

- The main peak says that The Los paths at

2 microphones differ by delay'

a '

.

- However,

Its not clear how otherechoes differ .

- The 1st.

order reflections from The walls

mayhave a very small difference ( r ) ,

or a large difference ( t ) .

- Example :

←±=:' :÷¥§.nu#_m '

Mz

- the path length can be viewed to be from an

image of The source behind the reflectingwall

.Further away the image ,

less in

The TDOA

@ Blindchannelestimation@ lowrider 2 microphones my and mz .

@

YET.

)@ - µ

@ At each mic. we record :

Y ,= Hroony .

×

Yz = Hroomz .

×

We do not know n,

and we do not know

any hroomi

@ Question Is : Can we get a better version ofwhat the user Is saying ,

i.e.,

some estimate

of in that Is better thany .

@ obune that

Y¥ = YIHz

: Y, Hz - YZH ,

= 0

Let's also assume H , ,Hz are sparse ,

i.e.,

L,

norm Of H, , Hz are small

. ( Recall,

L,

norm

counts no. of non . zero elements

. )to norm is better but NP complete .

Thus,

now we formulate an objective f ? as

www.t?ntni+EEttktYtfntIYt#fwhere HHH =

fund(• it = 1+2=0 is minimum but infeasible

sine HHHE = 1.

• The 7 regularization adds a penalty forH A

,11

,

and HHZH,

beaming larger .

Ideally ,we should have done Lo norm

,which would reduce the # of won . zero

elements in H ,and Hz .

MUSIC Loe ab cd-

ac bdd

abe ad

② .

- ④

f

\¥F⑦

man en

TRs, , Lse )

⑨ Autocorrelation ( seem, . nm ,)

1-whnhh.tt.

BTA CTB " to "

3)It .

FEI!.

,-

- -- -

- --

-,

.

A BA C Ba Dlc . ED Dec FEED .seGFEE

Autocorrelation

A.BACBn.tt/)crsBAEDcCBaTEoDcBGFeEDc-#.A BA ( BAA Do.rs/3AEDcCBaFEoDcB

-

A BA CBAA Do.rs/3AEDcCBafEoDcB

A BA ( BAA Do.rs/3AEDcCBaFEoDcB

A BA CBAA DCBBA EDI.BA ink

A BA CBAA DCBBA D¥E⇒Dc

cross Correlation ( Ho

hmrtipatn)

④ - → - ④

A"

B"

-

iEitiD"

B"

E'tFITEE "

o -

iii.I

D= O

③ Cross correlation ( am. , Mma ) wwwLij > = speaker i

,mic

. j11 11 11 21 11 21 11 U 11 21

yeABA CBAA Do.rs/3AEDcCBaFEoDCBthis

12 12 12 12 22 12 22

f-"

fjghaA BA C Ba DCBA En BA -

Eo

✓abcdrac bd

③ abc✓ad✓↳°④←④Delay ( 11,12 ) = •

deedCd

} crossDelay Cry ,

22) = ab

Delay ( 11,4 ) = ad

Delay ( 12,22 ) = be } auto