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

plane wave signal wave fronts h1

α d 0 plane wave has wavelength λ = c/f, h2 where f is the frequency c 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: ⋅= e)t(R)t(s ω )t(i

• Then the signals at the two hydrophones are: ω φ ⋅= )t(i i )t(i i ee)t(R)t(s

where πd φ −= 1)( n sin α n λ • The pattern function of the dipole is the normalized response of the dipole as a function of angle:

+ ss ⎛ πd ⎞ α )(b = 21 = cos ⎜ sin α ⎟ s ⎝ λ ⎠ Beam Pattern of Simple Beamformer

Pattern Loss vs. Angle of Incidence of Plane Wave Polar Plot of Pattern Loss For 2 Element Beamformer For Two Element Beamformer, λ/2 Element Spacing λ/2 Element Spacing

0 105 900 75 120 -10 60 135 45 -10 -20

150 -30 30 -20 -40 165 15 -50

-30 -180 0

-165 -15

Pattern Loss, dB Pattern Loss, -40 -150 -30

-50 -135 -45 -120 -60 -60 -105 -90 -75 -150 -100 -50 0 50 100 150 α , degrees Beam Pattern of a 10 Element Array

Polar Plot of Pattern Loss For 10 Element Beamformer Pattern Loss vs. Angle of Incidence of Plane Wave λ/2 Spacing λ For Ten Element Beamformer, /2 Spacing 105 90 0 75 0 120 -10 60

135 -20 45 -10 150 -30 30

-20 -40 165 15 -50

-30 180 0

-165 -15 Pattern Loss, dB -40

-150 -30 -50 -135 -45

-120 -60 -60 -150 -100 -50 0 50 100 150 -105 -90 -75 α, degrees 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

Comparison Beam Pattern Of A 10 Element Dolph-Chebychev Beamformer With -40 dB Side Lobes And λ /2 Element Spacing With A Uniformly Weighted 10 Element Beamformer 0

Dolph-Chebychev Uniform Beamformer -10 Beamformer

-20

-30

Pattern Loss, dB Pattern Loss, -40

-50

-60 -80 -60 -40 -20 0 20 40 60 80 α , degrees Analogy Between Spatial Filtering (Beamforming) and Time- Frequency Processing

Goals of Spatial Filtering: Goals of Time-Frequency Processing:

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

2. Resolve (distinguish between) plane 2. Resolve narrowband signals at different wave signals arriving from different frequencies. directions. 3. Measure the frequency of narrowband 3. Measure the direction from which signals. plane wave signals are arriving. Time-Frequency Filtering and Beamforming

Sine wave at f0 Sine wave at f1 Broadband noise spectrum Narrowband

filter at f0

f1 f0 Frequency

Plane wave at ψ0 Plane wave at ψ1 Ambient noise angular density Narrow spatial

filter at ψ0

ψ1 ψ0 Spatial angle ψ SNR Calculation: Time-Frequency Filtering

Define

2δα ≡− 0 )ff( Signal power spectral density (W/Hz) )f(N ≡ Noise power spectral density (W/Hz) )f(H 2 ≡ Filter power response

Signal Power Is:

∞ 2 P = 2 δα− 2 = α 2 )f(Hdf)f(H)ff( (watts) s ∫ 0 0 ∞− SNR Calculation: Time-Frequency Filtering (Cont’d)

If we assume ⎧ β β 2 ⎪ ≤−≤− Idealized = 1, ff 0 )f(H ⎨ 2 2 rectangular ⎪0, otherwise filter with ⎩ bandwidth β

Then the noise power is:

∞ N is the ⋅= 2 = Ndf)f(H)f(NP β (watts) 0 N ∫ 0 noise level ∞− in band

And SNR is: P α 2 SNR s == N NP 0 β SNR Calculation: Spatial Filtering

Define

2 ψψδα≡− 0 )( Signal power angular density (W/steradian) ψ )(N ≡ Noise power angular density (W/steradian) ψ )(G 2 ≡ Spatial filter angular power response

Signal Power Is:

2 P = 2 − 2 = 2 ψαψψψδα)(Gdf)(G)( (watts) s ∫ 0 0 4π SNR Calculation: Time-Frequency Filtering (Cont’d)

If we assume

⎧ ψ β ψ β Idealized 2 ⎪ ψψ≤−≤− ψ = 1, 0 “cookie cutter” )(G ⎨ 2 2 ⎪ 0, otherwise beam pattern ⎩ with width ψβ

Then the noise power is:

Kis the ⋅= 2 = ψΩψψKd)(G)(NP (watts) N ∫ β noise intensity 4π in beam

And SNR is: P α 2 SNR s == P N ψ β K Array Gain and Directivity Calculations

Define Assume

SNRArray Gain Array Gain = • plane wave signal SNROH • arbitrary noise distribution

For the omnidirectional hydrophone,

Ω )(G 2 = 1 for all Ω

Then 2 − 2 d)(G)( ΩΩΩΩδα P ∫ 0 OH α 2 s == 4π = SNROH P Ω ⋅ 2 d)(G)(N ΩΩ d)(N ΩΩ N ∫ OH ∫ 4π 4π Array Gain and Directivity Calculations (Cont’d)

2 2 = For the array, assume it is steered in the direction of Ω0 and that Ω0 )(G 1 Then

2 2 − ΩΩδα d)(G)( ΩΩ P ∫ 0 array α 2 s == 4π = SNRarray 2 2 PN Ω ⋅ d)(G)(N ΩΩ Ω ⋅ d)(G)(N ΩΩ ∫ array ∫ array 4π 4π

Putting these together yields ∫ d)(N ΩΩ = 4π AG 2 Ω ⋅ d)(G)(N ΩΩ ∫ array 4π Array Gain and Directivity Calculations (Cont’d) If the noise is isotropic (the same from every direction)

Ω = K)(N

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

= 4π DI 2 d)(G ΩΩ ∫ array 4π

Array Gain and Directivity Index are usually expressed in decibels. Line Hydrophone Spatial Response

x ψ L/2 ψ X1sin x1 plane wave signal wave fronts L 0

ψ

-L/2 plane wave has wavelength λ = c/f, where f is the frequency c is the speed of sound Line Hydrophone Spatial Response (Cont’d)

The received signal is )t(s theat origin ⎛ sinx ψ ⎞ ⎜ts + ⎟ at point x ⎝ c ⎠

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

/L 2 sinx ψ = t(s)x(g)t(s + dx) out ∫ − /L 2 c Line Hydrophone Spatial Response (Cont’d)

Using properties of the Fourier Transform: = ∫ − 2πfti dte)t(s)f(S And: ∞ sinx ψ − 2 sinfxi ψπ ∫ t(s + 2πfti = exp(dte) )f(S) ∞− c c

Or: ∞ sinx ψ 2 sinfxi ψπ t(s + = ∫ exp() + 2π df)f(S)fti c ∞− c Line Hydrophone Spatial Response (Cont’d)

Thus, the total hydrophone response can be written: ∞ /L 2 2 sinfxi ψπ = exp()x(g)t(s + 2π dfdx)f(S)fti out ∫∫ − /L 2 ∞− c

∞ /L 2 ⎡ 2 sinfxi ψπ ⎤ 2 πfti = ⎢ ∫∫ exp()x(g)f(S ⎥ dfedx) ∞− ⎣ − /L 2 c ⎦ ∞ sinf ψ ≡ ∫ (G)f(S 2 πfti dfe) ∞− c

Where /L 2 sinf ψ ≡ ⎡ sinf ψ ⎤ (G ∫ ⎢ 2π(iexp)x(g) ⎥dxx) c − /L 2 ⎣ c ⎦

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

An Example:

Unit Amplitude Plane Wave from direction ψ0:

= e)t(s 2 π 0 tfj −= δ 0 )ff()f(S The Line Hydrophone Response is: ∞ sinf ψ δ −= (G)ff()t(s 0 2 πfti dfe) out ∫ 0 ∞− c

⎛ sinf ψ ⎞ = G⎜ 0 0 ⎟e 2 π 0tfj ⎝ c ⎠

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

sinf sin ψψ u ≡ = c λ

instead of physical angle ψ. Uniform Aperture Function

Consider a uniform aperture function

⎧ 1 L ≤≤− L ⎪ , x ⎪ L 22 )x(g = ⎨ ⎪0 , otherwise ⎪ ⎩

The pattern function is:

∞ 1 /L 2 2π − ee 22 π /Luj/Luj 2 = ∫ 2πuxj dxe)x(g)u(G = ∫ e πuxj2 dx = ∞− L − /L 2 2πLuj /L 2 2π − ee 22 π /Luj/Luj 2 e 2πuxj π )Lusin( = = = Luj Luj Lu 2π 2π − /L 2 π ≡ sinc )Lu( Rectangular Aperture Function and Pattern Function Array Main Lobe Width (Beamwidth)

3 dB (Half-Power) Beamwidth

To find it, solve

2 ⎛ ψ ⎞ 1 G⎜ 3dB ⎟ = ⎝ 2 ⎠ 2

For ψ3dB Beamwidth Calculation Example: Uniform Weighting

⎛ πLu ⎞ sin⎜ 3dB ⎟ ⎝ 2 ⎠ = 1 πLu3dB 2 2 πLu 3dB = .391 2 ⋅⋅ = −1 ()= −1 ⎛ .3912 λ ⎞ ψ3dB λ 3dB sinusin ⎜ ⎟ ⎝ πL ⎠

− ⎛ ⎞ λλ = 1⎜.sin 885 ⎟ ≈ .885 radians ⎝ L ⎠ L λ ≈ 50 deg L

ofeffect the Note For λ 〈〈 .L Note the ofeffect increasing L. Line Array of Discrete Elements

L = Nd

x d

Aperture Function: Pattern Function: (N odd, uniform spacing)

∞ N 2πuxj = ()− G(u) = () dxexg g(x) ∑ anδ xx n ∫∞ n=1 ∞ M = δ()− 2 πuxj dxendxa , spacing uniform and (Odd), 1 2M For N 2M += 1 (Odd), and uniform spacing d, ∫∞ ∑ n −= Mn M ()= δ − ( ) ∑ n ndxaxg M −= Mn = 2 πnduj ∑ n ea −= Mn Uniformly Weighted Discrete Line Array

1 Assume a = , uniform spacing, n odd. n N Temporarily :define r ≡ e 2πduj

then 1 M 1 /N 2 − rr − /N 2 ()rG = n =ar ∑ n / − − / 2121 N −= Mn N rr or 1 ()πuNdsin ()uG = N ()πudsin Pattern Function For Uniform Discrete Line Array Notes On Pattern Function For Discrete Line Array

⎛ 1 1 ⎞ •u Only has physical significance over the range ⎜ − ⎟ ⎝ λ, λ ⎠ 1 1 •The region u ≤≤− may include more than λ λ

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

General trade-offs in array design: 1)Want L large so that beamwidth is small and resolution is good 2)Want d ≤ λ to avoid grating lobes. 3)Since L=Nd, or N=L/d, increasing L and decreasing d Both cause N to increase, which costs more money Effects of Array Shading (Non-Uniform Aperture Function)

• Shading reduces sidelobe levels at the expense of widening the main lobe. • For other aperture functions: First sidelobe ψ Aperture 3dB ,degrees level, dB Rectangular 50λ/L -13.3 Circular 58λ/L -17.5 Parabolic 66λ/L -22.0 Triangular 73λ/L -26.5 Beam Steering

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(x) = ()− 2πuxj dueuG ∫∞ ∞ ′ )x(g = ()− − 2πuxj dueuuG ∫∞ 0 ∞ G(u- u ) − 0 = 2πuj 0 x () 2πpxj dpepGe ∫∞ = 2πuj 0 x g(x)e

Beam steering is accomplished by multiplying the non-steered aperture function by a unit amplitude complex exponential, which is just a delay whose value depends on x. Beam Steering For Discrete Arrays

The steered aperture function becomes

N π g(x) = ∑ δ ()− endx)nd(g 2 nduj 0 n=1 − π π −= e)d(g... 2 nduj 0 δ ()+− 0 δ + 2 nduj 0 δ ()++ ...dxe)d(g)x()(gdx

= −1 The steered physical angle is ψ0 λ 0 ).u(sin The phase shift at element n is equivalent to a time shift: 2 sinnd ψπ sinnd ψ 2πndu = 0 = 2πf 0 = 2 τπ .f 0 λ c n sinnd ψ where τ = 0 n c

Is the time shift which must be applied to the nth element. Effect Of Beam Steering On Main Lobe Width

Beam steering produces a projected aperture. 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

For a uniformly weighted, evenly spaced (d=λ/2), 8 element array, the pattern function is 1 [π − )uu(Ndsin ] )uu(G =− 0 0 []− N π 0 )uu(dsin

+ 2 =− − 2 = 1 To find the beamwidth, set G(u 0 0 u-G( )u)u 2 Which yields + =− − = -( ( π ( 0 π 0 -( .)uud)uud 1750

Using Using = λ 2 and = /sinu/d λψ the condition is

+ =− − = sin sin 0 0 in- ψψψψ .ssin 11140 Beam Steering: An Example (Cont’d)

As an example, take N=8, (d=λ/2).

= + − == :1 Case :1 sin ψ 0 0 sin sin ψψ .11140 + = 0 − = 0 = 0 ⇒ ψ .46 ψ .46 ψ 3dB .812

λ λ 50 :Note 50 50 === .512 0 ndL 4 Beam Steering: An Example (Cont’d)

= 0 = :2 Case :2 ψ0 45 ψ0 .sin 7070 ψ + = −1 =+ .)..(sin 954707011140 0 ψ − = −1 =− .)..(sin 636707011140 0 = 0 ψ3dB .318 (wider) not is beam But, beam is not symmetrica :l . 00 =− .9945954 0 0 0 =− .. 4863645 0 = 0 until lobe grating no is therethat Notice therethat is no grating lobe until ψ0 90 point at which point there is a grating lobe at ψ −= 900 λ optimal is whyis This whyis d = is optimal 2 Directivity Index For A Discrete Line Array

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

• Uniform weighting • = λ frequency theis f0 theis frequency at which d 2/ f ===• When ff 0 ),1( DI log10 N f0 >• When 0 , DIff oscillates N.log10about However, grating lobes ambiguity. cause cause ambiguity. <• When 0 , DIff drops at 3dBabout octave.per End fire beam broader but DI,higher has has DI,higher but broader beamwidth, hence poorer ldirectiona .resolution Array Gain: Discrete Line Array

:Model Noise :Model

⎧ φ,sinK 0 φ ≤≤ 900 0 φ )(N = ≈ 10( ).L ⎨ φ 0 φ ≤≤−− 0 B ⎩ B ),sin(KL 090 Array Gain: Discrete Line Array (Cont’d)

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