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 • y: aer C of the arrt the cent aten a signal incidenGiv iω ( t ) s ( t )= R ( ⋅ t ) e • e:ophones ardryo h the twtThen the signals a ω φ =i ⋅ ( ti ) i ( t ) s ( t )i R ( t ) e e ewher πd φ =() −1 n sin α n λ • esponse ed of the dipole rern function of the dipole is the normalizttThe pa as a function of angle: s+ s ⎛ πd ⎞ b (α = ) 1 2 = 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 eringequency Filttion: Time-FrSNR Calcula fineDe α2 δ − ≡ Signal (power f0 f ) spectral density (W/Hz) Noise power N ( f≡ ) spectral density (W/Hz) Filter H ( f2 ≡ ) power response Is:er woSignal P ∞ 2 (P f=α f2 ) δ H− ( f )2 df= α 2 H(watts) ( f ) s ∫ 0 0 − ∞ d)tering (Conequency Filt’tion: Time-FrSNR Calcula e If wassume ⎧ β β 2 ⎪ − ≤ − ≤ ed Idealiz 1= , f f0 H ( f ) ⎨ 2 2 rectangular 0⎪ , otherwise er withfilt ⎩ bandwidt hβ er is:Then the noise pow ∞ N is the P N= ( f ) H ⋅ (2 f ) = β df(watts) N 0 N ∫ 0 elnoise lev − ∞ in band And SNR is: P α 2 SNR =s = PNN 0 β eringtial Filttion: SpaSNR Calcula fineDe α δ2 ψ− ψ ≡ Signal() 0 power angular(W/steradi an) density Noise N() ψpower≡ angular(W/steradi an)density Spatial G() filter ψ 2 ≡ angular power response Is:er woSignal P 2 αP( δ= 2 ψ )− G ψ ( ) ψ2 df= 2 α G ( ψ(watts) ) s ∫ 0 0 4π d)tering (Conequency Filt’tion: Time-FrSNR Calcula e If wassume ⎧ ψ β ψ β ed Idealiz 2 ⎪ − ≤ψ − ψ ≤ ψ1= , 0 ertookie cutc”“ G() ⎨ 2 2 0⎪ , otherwise ernttbeam pa ⎩ with width ψβ er is:Then the noise pow Ki e h t s P N=ψ ( ) ⋅G ψ (2 ) Ω = d ψ (watts) K N ∫ β enstitynoise in 4π in beam And SNR is: P α 2 SNR =s = P N ψ β K tionsctivity Calculae Gain and DiryaArr fineDe Assume SNRArray Array Gain= • signalevaplane w SNROH • ary noise ributiontarbitrdis ophone,dryctional er the ohomnidirF 1G()Ω 2 = forΩ all Then α δ(2 Ω− ) Ω G (2 Ω ) d Ω P ∫ 0 OH α 2 =s = 4π = SNROH P N (Ω )⋅ GΩΩ (2 ) d NΩΩ ( ) d N ∫ OH ∫ 4π 4π d)ttions (Conectivity Calcula Gain and Diry’aArr 2 2 = ction of eed in the direertsit is assume , yathe arrr oFΩ0 tand tha G()Ω0 1 Then 2 α(2 δ Ω− ) G Ω ΩΩ ( ) d P ∫ 0 array α 2 =s = 4π = SNRarray 2 2 PN N (Ω )⋅ GΩΩ ( )N d (Ω )⋅ GΩΩ ( ) d ∫ array ∫ array 4π 4π ther eyieldsgoing these ttPut N∫ ΩΩ ( ) d = 4π AG 2 N ( )Ω ⋅ GΩΩ ( ) d ∫ array 4π Array Gain and Directivity Calculations (Cont’d) If the noise is isotropic (the same from every direction) N()KΩ = 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 GΩΩ ( ) d ∫ 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 d)tesponse (Contial R’Spaophone drLine Hy ed signal isceiveThe r s ( t ) at the origin ⎛ x sinψ ⎞ s⎜ t+ ⎟ at x point ⎝ c ⎠ t the sponse et spoinor sensitivity arophone’dryt the Lehx be g(x). sponse is eophone rdryal htoThen, the t L/2 x sinψ s (t)= g(x)s(t+ ) dx out ∫ −L/2 c d)tesponse (Contial R’Spaophone drLine Hy orm:fansroperties urier Toof the FUsing pr S ( f )= s∫ ( ti− )2π ft e dt And: ∞ x sinψ − iπ fx2 sin ψ s∫ (+ t) ei dt2π ft = exp( ) S ( f ) − ∞ c c Or: ∞ x sinψ iπ fx2 sin ψ s (+ t )= ∫ exp( i ft+ 2 )π S ( f ) df c − ∞ c d)tesponse (Contial R’Spaophone drLine Hy n:etan be writsponse ceophone rdryal htothe tThus, ∞ L/2 iπ2 fx sin ψ s ( t )= g ( x ) exp(i ft+ )2π S ( f ) dfdx out ∫ ∫ −L/2 − ∞ c ∞ L/2 ⎡ iπ2 fx sin ψ ⎤ i2 π ft S (= f∫ )⎢ g ∫ ( x ) exp( ) dx⎥ e df − ∞ ⎣ −L/2 c ⎦ ∞ f sinψ S≡ ∫ ( f ) G () ei2 π ft df − ∞ c eWher L/2 f sinψ ≡ ⎡ f sinψ ⎤ G() g (∫ x ) exp⎢ 2π i) x ( ⎥ dx c −L/2 ⎣ c ⎦ all ceWg(x) function ethe apertur and G((fsin ψ)/c) ern functionttthe pa. .pairorm fansrurier Toe a Fy arThe evao Plane Wesponse TR ample:An Ex ction eom dir frevaUnit Amplitude Plane Wψ0: s ( t )= j2 eπ f0 t = − S ( f )δ ( f0 f ) sponse is:eophone RdrLine HyThe ∞ f sinψ s ( t )= (δ f − f ) G) (0 ei2 π ft df out ∫ 0 − ∞ c f⎛ sinψ ⎞ = G⎜ 0 0 ⎟ej2 π f0 t ⎝ c ⎠ the input si the output is t thaeNotalue of the y the ed bvtgnal modula tern function attpaψ0 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: f sinψsin ψ u ≡ = c λ instead of physical angle ψ. e Functionorm AperturUnif functionerm aperturoConsider a unif ⎧ 1 −L ≤L ≤ ⎪ , x ⎪ L 2 2 g ( x= )⎨ ⎪ 0 , otherwise ⎪ ⎩ ern function is:ttThe pa ∞ 1 L/2 je Lu2π /2− e j 2π Lu2 / G ( u )= g∫ ( xj2π ) ux e= ∫ej2 dxπ uxdx = − ∞ L −L/2 j2π Lu L/2 je Lu2π /2− e j 2π Lu2 / ej2π ux sin(π Lu ) = = = j Lu j Lu 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 tingigheWorm ample: Uniftion Exwidth CalculaBeam ⎛ πLu ⎞ sin⎜ 3dB ⎟ ⎝ 2 ⎠ = 1 πLu3dB 2 2 πLu 3dB 1= .