Spatial Signal Processing (Beamforming)

What Is Beamforming?• Beamforming is spatial filtering, a means of transmitting

or receiving sound preferentially in some directions over others.

• Beamforming is exactly analogous to frequency domain analysis of time signals.

• In time/frequency filtering, the frequency content of a time signal is revealed by its Fourier transform.

• In beamforming, the angular (directional) spectrum of a signal is revealed by Fourier analysis of the way sound excites different parts of the set of transducers.

• Beamforming can be accomplished physically (shaping and moving a transducer), electrically (analog delay circuitry), or mathematically (digital signal processing).

Beamforming Requirements• Directivity – A beamformer is a spatial filter and can be

used to increase the signal-to-noise ratio by blocking most of the noise outside the directions of interest.

• Side lobe control – No filter is ideal. Must balance main lobe directivity and side lobe levels, which are related.

• Beam steering – A beamformer can be electronically steered, with some degradation in performance.

• Beamformer pattern function is frequency dependent: – Main lobe narrows with increasing frequency– For beamformers made of discrete hydrophones,

spatial aliasing (“grating lobes”) can occur when the the hydrophones are spaced a wavelength or greater apart.

A Simple Beamformer

α

plane wave signalwave fronts

d

h1

h2

0

plane wave has wavelength λ = c/f,

where f is the frequencyc is the speed of sound h1 h1 are two

omnidirectional hydrophones

Analysis of Simple Beamformer• Given a signal incident at the center C of the array:

• Then the signals at the two hydrophones are:

where

• The pattern function of the dipole is the normalized response of the dipole

as a function of angle:

)t(ie)t(R)t(s ω⋅=

αλ

πφ sind)( nn 1−=

⎟⎠⎞

⎜⎝⎛=+= α

λπα sindcos

sss)(b 21

)t(i)t(ii

iee)t(R)t(s φω⋅=

Beam Pattern of Simple Beamformer

-150 -100 -50 0 50 100 150-60

-50

-40

-30

-20

-10

0

α , degrees

Patt

ern

Loss

, dB

Pattern Loss vs. Angle of Incidence of Plane WaveFor Two Element Beamformer, λ/2 Element Spacing

0

-10

-20

-30

-40

-50

-180

-165

-150

-135

-120-105 -90 -75

-60

-45

-30

-15

0

15

30

45

607590105

120

135

150

165

Polar Plot of Pattern Loss For 2 Element Beamformerλ/2 Element Spacing

Beam Pattern of a 10 Element Array

-150 -100 -50 0 50 100 150-60

-50

-40

-30

-20

-10

0

α, degrees

Patt

ern

Loss

, dB

Pattern Loss vs. Angle of Incidence of Plane WaveFor Ten Element Beamformer, λ/2 Spacing 0

-10

-20

-30

-40

-50

-165

-150

-135

-120-105 -90 -75

-60

-45

-30

-15

0

15

30

45

607590105

120

135

150

165

180

Polar Plot of Pattern Loss For 10 Element Beamformerλ/2 Spacing

Beamforming – Amplitude Shading

• Amplitude shading is applied as a beamforming function.

• Each hydrophone signal is multiplied by a “shading weight”

• Effect on beam pattern:– Used to reduce side lobes– Results in main lobe broadening

Beam Pattern of a 10 Element Dolph-Chebychev Shaded Array

-80 -60 -40 -20 0 20 40 60 80-60

-50

-40

-30

-20

-10

0

α , degrees

Patt

ern

Loss

, dB

Comparison Beam Pattern Of A 10 Element Dolph-Chebychev BeamformerWith -40 dB Side Lobes And λ /2 Element Spacing With A Uniformly

Weighted 10 Element Beamformer

Dolph-ChebychevBeamformer

Uniform Beamformer

Analogy Between Spatial Filtering (Beamforming) and Time-Frequency Processing

Goals of Spatial Filtering:

1. Increase SNR for plane wave signals in ambient ocean noise.

2. Resolve (distinguish between) plane wave signals arriving from different directions.

3. Measure the direction from which plane wave signals are arriving.

Goals of Time-Frequency Processing:

1. Increase SNR for narrowband signals in broadband noise.

2. Resolve narrowband signals at different frequencies.

3. Measure the frequency of narrowband signals.

ψ1 ψ0 Spatial angle ψ

Ambient noiseangular density

Plane wave at ψ1Plane wave at ψ0

Narrow spatial filter at ψ0

f1 f0 Frequency

Broadbandnoise spectrum

Sine wave at f1Sine wave at f0

Narrowbandfilter at f0

Time-Frequency Filtering and Beamforming

SNR Calculation: Time-Frequency Filtering

response power Filter

(W/Hz) density spectral power Noise (W/Hz) density spectral power Signal

2

02

≡

≡≡−

)f(H

)f(N)ff(δα

Define

Signal Power Is:

(watts) 20

220

2 )f(Hdf)f(H)ff(Ps αδα =−= ∫∞

∞−

SNR Calculation: Time-Frequency Filtering (Cont’d)

⎪⎩

⎪⎨⎧ ≤−≤−=

otherwise 022

1 02

,ff,)f(H

ββIf we assume

Idealized rectangularfilter withbandwidth β

Then the noise power is:

(watts) 02 βNdf)f(H)f(NPN =⋅= ∫

∞

∞−

N0 is the noise levelin band

And SNR is:

βα0

2

NPPSNR

N

s ==

SNR Calculation: Spatial Filtering

response power angular filter Spatial

an)(W/steradi density angular power Noise an)(W/steradi density angular power Signal

2

02

≡

≡≡−

)(G

)(N)(

ψ

ψψψδα

Define

Signal Power Is:

(watts) 20

2

4

20

2 )(Gdf)(G)(Ps ψαψψψδαπ

=−= ∫

SNR Calculation: Time-Frequency Filtering (Cont’d)

⎪⎩

⎪⎨⎧

≤−≤−=otherwise 0

22 1 02

,,)(G

ββ ψψψ

ψψ

If we assume

Idealized “cookie cutter”beam pattern

with width ψβ

Then the noise power is:

(watts) 4

2 Kd)(G)(NPN βπ

ψΩψψ =⋅= ∫K is the noise intensityin beam

And SNR is:

KPPSNR

N

s

β

2

ψα==

Array Gain and Directivity Calculations

OH

Array

SNRSNR

Gain Array =

Define

Then

∫∫

∫=

⋅

−==

ππ

π

ΩΩα

ΩΩΩ

ΩΩΩΩδα

4

2

4

24

20

2

d)(Nd)(G)(N

d)(G)(

PPSNR

OH

OH

N

sOH

Assume

• plane wave signal• arbitrary noise distribution

ΩΩ all for 12 =)(G

For the omnidirectional hydrophone,

Array Gain and Directivity Calculations (Cont’d)

For the array, assume it is steered in the direction of Ω0 and that

∫∫

∫⋅

=⋅

−==

ππ

π

ΩΩΩα

ΩΩΩ

ΩΩΩΩδα

4

2

2

4

24

20

2

d)(G)(Nd)(G)(N

d)(G)(

PPSNR

arrayarray

array

N

sarray

Putting these together yields

1Ω2

02 =)(G

Then

∫

∫⋅

=

π

π

ΩΩΩ

ΩΩ

4

24

d)(G)(N

d)(NAG

array

Array Gain and Directivity Calculations (Cont’d)If the noise is isotropic (the same from every direction)

Then the Array Gain (AG) becomes the Directivity Index (DI), a performance indexFor the array that is independent of the noise field.

K)(N =Ω

∫=

π4

2ΩΩ

π4

d)(GDI

array

Array Gain and Directivity Index are usually expressed in decibels.

Line Hydrophone Spatial Response

ψ

plane wave signalwave fronts

L

L/2

-L/2

0

plane wave has wavelength λ = c/f,

where f is the frequencyc is the speed of sound

x

ψ

x1X1sin ψ

Line Hydrophone Spatial Response (Cont’d)

The received signal is

Let the hydrophone’s response or sensitivity at the point x be g(x).Then, the total hydrophone response is

xc

sinxts

)t(s

point at

origin theat

⎟⎠⎞

⎜⎝⎛ + ψ

dx)c

sinxt(s)x(g)t(s/L

/Lout ∫

−

+=2

2

ψ

Line Hydrophone Spatial Response (Cont’d)

Using properties of the Fourier Transform:

And:

)f(S)csinfxiexp(dte)

csinxt(s fti ψπ2ψ π2 =+ −

∞

∞−∫

dte)t(s)f(S fti∫ −= π2

df)f(S)fticsinfxiexp()

csinxt(s ∫

∞

∞−

+=+ π2ψπ2ψOr:

Line Hydrophone Spatial Response (Cont’d)

Thus, the total hydrophone response can be written:

Where

dfdx)f(S)fticsinfxiexp()x(g)t(s

/L

/Lout ∫∫

∞

∞−−

+= π2ψπ22

2

We call g(x) the aperture function and G((fsin ψ)/c) the pattern function.They are a Fourier Transform pair.

dfedx)c

sinfxiexp()x(g)f(S fti/L

/L

π22

2

ψπ2⎥⎦

⎤⎢⎣

⎡= ∫∫

−

∞

∞−

dfe)c

sinf(G)f(S fti π2ψ∫∞

∞−

≡

∫−

⎥⎦⎤

⎢⎣⎡≡

2

2

ψπ2ψ /L

/L

dxx)c

sinf(iexp)x(g)c

sinf(G

Response To Plane Wave

An Example:

Unit Amplitude Plane Wave from direction ψ0:

Note that the output is the input signal modulated by the value of the pattern function at ψ0

tfje)t(s 0π2=

dfe)c

sinf(G)ff()t(s ftiout

π200

ψδ∫∞

∞−

−=

tfjec

sinfG 0π200 ψ⎟⎠⎞

⎜⎝⎛=

The Line Hydrophone Response is:

)ff()f(S 0δ −=

Response To Plane Wave (Cont’d)

The pattern function is the same as the angular power response defined earlier.

Sometimes we use electrical angle u:

λψψ sin

csinfu =≡

instead of physical angle ψ.

Uniform Aperture Function

Consider a uniform aperture function

⎪⎪⎩

⎪⎪⎨

⎧ ≤≤−

=otherwise

,

LxL,L

)x(g0

221

The pattern function is:

)Lu(Lu

)Lusin(Luj

eLujee

Lujeedx

Ldxe)x(g)u(G

/L

/L

uxj/Luj/Luj

/Luj/Luj/L

/L

uxj

sinc

e uxj2

≡

==−=

−===

−

−

∞

∞−∫∫

ππ

π2π2

π21

2

2

π22π22π2

2π22π22

2

ππ2

Rectangular Aperture Function and Pattern Function

Array Main Lobe Width (Beamwidth)

21

2ψ

23 =⎟

⎠⎞

⎜⎝⎛ dBG

3 dB (Half-Power) Beamwidth

To find it, solve

For

dB3ψ

Beamwidth Calculation Example: Uniform Weighting

( ) ⎟⎠⎞

⎜⎝⎛ ⋅⋅==

=

−−

L.sinusin

.Lu

dBdB

dB

πλ3912λψ

3912

π

13

13

3

21

2π

2π

3

3

=⎟⎠⎞

⎜⎝⎛

dB

dB

Lu

Lusin

. increasing ofeffect the Note For

deg 50

radians 8858851

L.L

L

L.

L.sin

⟨⟨

≈

≈⎟⎠⎞

⎜⎝⎛= −

λ

λ

λλ

Line Array of Discrete Elements

( )n

N

nn xxδag(x) −= ∑

=1

( ) ( )ndxaxg

dNM

Mnn −=

+=

∑−=

δ

, spacing uniform and (Odd), 1 2M For

Aperture Function:

( ) dxexgG(u) πuxj 2∫∞

∞=

( ) dxendxa uxjn

M

Mn

π2δ −= ∫ ∑∞

∞−=

Pattern Function: (N odd, uniform spacing)

ndujn

M

Mnea π2∑

−==

x

L = Nd

d

Uniformly Weighted Discrete Line Array

πduj

n

er

n,N

a

2

:define yTemporaril

odd. spacing, uniform 1 Assume

≡

=

( ) 2121

2211//

/N/N

n

M

Mn

n

rrrr

Nar

NrG −

−

−= −−== ∑

then

( ) ( )( )udsinuNdsin

NuG

ππ1=

or

Pattern Function For Uniform Discrete Line Array

Notes On Pattern Function For Discrete Line Array

⎟⎠

⎞⎜⎝

⎛−λ1

λ1,

λ1

λ1 ≤≤− u

λ≥d

•u Only has physical significance over the range

•The region may include more than

one main lobe if , which causes ambiguity(called grating lobes).

General trade-offs in array design:1)Want L large so that beamwidth is small and resolution is good2)Want λ≤d to avoid grating lobes.

3)Since L=Nd, or N=L/d, increasing L and decreasing dBoth cause N to increase, which costs more money

• Shading reduces sidelobe levels at the expense of widening the main lobe.

• For other aperture functions:

First sidelobe

Aperture ψ3dB ,degrees level, dB

Rectangular 50λ/L -13.3

Circular 58λ/L -17.5

Parabolic 66λ/L -22.0

Triangular 73λ/L -26.5

Effects of Array Shading (Non-Uniform Aperture Function)

Beam Steering

( )

( )( )

g(x)edpepGe

dueuuG)x(g

dueuGg(x)

xπuj

πpxjxπuj

πuxj

πuxj

0

0

2

22

20

2

==

−=′

=

−∞

∞

−∞

∞

−∞

∞

∫∫

∫

Want to shift the peak of the pattern function from u = 0 to u = u0.What is the aperture function needed to accomplish this?

G(u- u0)

Beam steering is accomplished by multiplying the non-steered aperturefunction by a unit amplitude complex exponential, which is just a delaywhose value depends on x.

Beam Steering For Discrete Arrays

).u(sin 01

0 λψ −=

( )( ) ( ) ...dxe)d(g)x()(gdxe)d(g...

endx)nd(gg(x)ndujnduj

ndujN

n

++++−−=

−=−

=∑

δδδ

δππ

π

00

0

22

2

1

0

The phase shift at element n is equivalent to a time shift:

The steered aperture function becomes

The steered physical angle is

.fc

sinndfsinndndu nτπ2ψπ2λ

ψπ2π2 000 ===

csinnd

n0ψτ where =

Is the time shift which must be applied to the nth element.

Effect Of Beam Steering On Main Lobe Width

Beam steering produces a projectedaperture. Since reducing the aperture increases the beam width, beam steering causes the width of the (steered) main lobe to increase. The lobe distorts (fattens) more on the side of the beam toward which the beam is being steered,

Beam Steering: An Example

21

00 ==− −+ 22u-G( G(u )u)u

[ ][ ])uu(dsin

)uu(NdsinN

)uu(G0

00 π

π1−−=−

For a uniformly weighted, evenly spaced (d=λ/2), 8 element array, the pattern function is

To find the beamwidth, set

1750ππ 00 .)uud)uud ==− −+ -( (

Which yields

is condition the and Using λψ2λ /sinu/d ==

11140ψψψψ 00 .ssinsin sin ==− −+ in-

Beam Steering: An Example (Cont’d)

As an example, take N=8, (d=λ/2).

0

03

000

512450λ50λ50

812ψ46ψ46ψ11140ψψ0ψ

.ndL

....sinsinsin

dB

===

===⇒

===−+

−+

:Note

:1 Case

Beam Steering: An Example (Cont’d)

optimal is whyis This

at lobe grating a is therepoint at which until lobe grating no is therethat Notice

:lsymmetricanot is beam But,(wider)

:2 Case

2λ

90ψ90ψ

48636459945954

318ψ636707011140ψ954707011140ψ

7070ψ45ψ

0

00

000

000

03

01

010

00

=

−=

==−=−

==−==+=

==

−−

−+

d

..

..

..)..(sin.)..(sin

.sin

dB

Directivity Index For A Discrete Line Array

DI for a discrete line array, broadside and endfire with N=9.

.resolution ldirectionapoorer hence beamwidth,broader but DI,higher has beam fire End octave.per 3dBabout at drops , When

ambiguity. cause lobes grating However, .log10about oscillates , When

log10 ),1( When

2/at which frequency theis weighting Uniform

0

0

00

0

DIff

NDIff

NDIffff

df

<•

>•

===•

=••

λ

Array Gain: Discrete Line Array

).L),sin(KL

,sinK)(N B

B

10( 090

900

:Model Noise

00

00

≈⎩⎨⎧

≤≤−−≤≤

=φφ

φφφ

Array Gain: Discrete Line Array (Cont’d)

Nine-element vertical line array gain versus elevation steeringangle in a surface-generated ambient noise field