delay sum beamforming

Copyright 2013 Aaron Lanterman

Delay-and-Sum Beamforming for Plane Waves

Prof. Aaron D. Lanterman

School of Electrical & Computer Engineering

Georgia Institute of Technology AL: 404-385-2548

ECE 6279: Spatial Array Processing

Fall 2013 Lecture 6


Where We Are in J&D

Lecture material drawn from: Secs. 4.1, 4.1.2, 4.2.1 (up to but

not including Point Focusing part on p. 123), 4.2.3

Next lecture: Secs. 4.1.1, 4.1.3, 4.2.1 (Point

Focusing part on p. 123)


Integrating Across Apertures Heres one way aperture smoothing

functions show up Typically integrate across the aperture

z(t) = w(x) f (x, t)

dx

f (x, t) = exp{ j( 0t k 0 x)}

Input a monochromatic plane wave to the system

z(t) = exp( j 0t) w(x)exp( jk 0 x)

dxW (

k 0 )


Delay-and-Sum Beamforming

Array of M sensors at positions For convenience, put the phase center at

the origin

x0xM1

xmm=0

M1

=0

z(t) wmym (t m )m=0

M1

Delay-and-sum beamforming


Beamforming for Plane Waves

z(t) = wmym (t m )m=0

M1

f (x, t) = s(t 0 x) 0 =

0 / c

ym (t) = s(t 0

xm )

= wmm=0

M1

s(t m 0

xm )


When Things Line Up

z(t) = wmm=0

M1

s(t m 0

xm )

If we pick m =

0

xm = 0

xmc

then we get the signal back!

z(t) = wmm=0

M1

s(t)= s(t) wmm=0

M1

$

%&

'

()


When They Dont

z(t) = wmm=0

M1

s(t m 0

xm ) More generally, if we pick

m =

xm =

xmc

then we get a degraded version of the signal

z(t) = wmm=0

M1

s(t + ( 0 ) xm )


Strategy for Parameter Estimation

Find parameter that maximizes energy in z(t) Radar and sonar: If you know c, sweep to find

direction of arrival Seismology: If you know , sweep c to find wave

speed (determines material properties)

z(t) wmym (t m )m=0

M1

m =

xm =

xmc


Monochromatic Plane Waves (1) f (x, t) = exp{ j 0 (t 0 x)}

Plane wave delay-and-sum beamformer response

z(t) = wmm=0

M1

s(t + ( 0 ) xm )

= wmm=0

M1

exp( j 0[t + ( 0 ) xm ])

s(t) = exp( j 0t)= s(t 0 x)where


Monochromatic Plane Waves (2)

= wmm=0

M1

exp( j 0 ( 0 ) xm )$

%&

'

()exp( j 0t)

= wmm=0

M1

exp( j( 0 k 0 ) xm )

$

%&

'

()exp( j 0t)

z(t) = wmm=0

M1

exp( j 0[t + ( 0 ) xm ])

k 0 = 0 0Recall


Monochromatic Plane Waves (3)

z(t) = wmm=0

M1

exp( j( 0 k 0 ) xm )

$

%&

'

()exp( j 0t)

=W ( 0 k 0 )exp( j 0t)

W (k ) = wm

m=0

M1

exp( jk xm )

where the aperture smoothing function is

Also called the array pattern


General Wavefields

f (x, t) = 1(2 )4 F(k,)exp{ j(t

k x)}d

k

d

Delay-and-sum beamformer focused on

z(t) = 1(2 )4 F(k,)W (

k )exp( jt)d

k

d


General Plane Waves (1)

z(t) = 1(2 )4 F(k,)W (

k )exp( jt)d

k

d

f (x, t) = s(t 0 x)F(

k,) = S()(2 )3(

k 0 )

=12 S()W ([

0 ])exp( jt)d

Z() = S()W ([ 0 ])


General Plane Waves (2)

Z() = S()W ([ 0 ]) =

0 If we pick

Z() = S()W (0)z(t) = s(t)W (0)

we get the original signal back! If we pick 0

we get a filtered version


Uniform Linear Array (1)

W (k ) = sin(Mkxd / 2)sin(kxd / 2)

For a linear uniform array from the last lecture

W ( 0 k 0 ) = sin(M[

0x kx0 ]d / 2)sin([ 0x kx0 ]d / 2)

z(t) =W ( 0 k 0 )exp( j 0t)

From earlier slide, the response of delay-and-sum beamformer (tuned to ) to a monochromatic plane wave is


Uniform Linear Array (2)

kx = 0x Using W (kx kx0 ) =

sin(M[kx kx0 ]d / 2)sin([kx kx0 ]d / 2)

W (kx kx0 ) =sin M

[sin 0 sin]d"

#$

%

&'

sin [sin 0 sin]d"

#$

%

&'

In terms of angles, let kx = (2 / )sin()


Beam Pattern (Boresight)


kx = 0

M =12 (notice the negative sign)


Beam Pattern (60)


kx = (2 / )sin() = / 3

M =12 (notice the negative sign)


Beam Pattern (Boresight)

sin M [sin 0 sin]d"

#$

%

&'

sin [sin 0 sin]d"

#$

%

&'

= 0

M =12


Beam Pattern (60)

sin M [sin 0 sin]d"

#$

%

&'

sin [sin 0 sin]d"

#$

%

&'

= / 3

M =12


Terminology

Beampattern: fix

Steered response: fix

func( 0,k 0 ) =W ( 0

k 0 )

func( ) =W ( 0 k 0 )

0,k 0

=

k / = k

/ k

k,

=W 0k

k 0

"

#$

%

&'

=W 0k

k 0

"

#$

%

&'

func( ) =W (k0[

0 ])k = k0

, = 0 :

k = k0

, = 0 : func(

0 ) =W (k0[

0 ])If

If

delay sum beamforming

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