# Outline. Aperture function and aperture smoothing function. Aperture and Arrays. INF5410 Array signal processing. Ch. 3: Apertures and Arrays, part I

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1 INF541 Array signal processing. Ch. 3: Apertures and Arrays, part I Andreas Austeng Department of Informatics, University of Oslo February 1 Outline Finite Continuous Apetrures Aperture and Arrays Aperture function and aperture smooting function Classical resolution Geometrical optics Ambiguities & Aberrations Aperture and Arrays Study apertures: Examine the effect of sensors that gather signal energy over finite areas. Arrays: Group of sensors combined to produce single output. At m th sensor position, x m : Fields value: f ( x m, t). Sensors output: ym (t). If sensor is perfect (i.e. linear transf., infinite bandwidth, omni-directional): y m (t) = κ f ( x m, t), κ R (or C). Aperture function and aperture smoothing function Directional omni-directional If sensor has (significant) spatial extent, it will spatially integrate energy, i.e. it focus in a particular propagation direction. Example: Parabolic dish. Apertures ( point sources) are described by the aperture function, w( x). Spatial extend reflects size and shape Aperture weighting; relative weighting of the field within the aperture (also known as shading, tapering, apodization). Aperture smoothing function: W ( k) = w( x) exp(j k x)d x 4 5

2 Aperture smoothing function ( Sec ) Assume aperture in x x Given Aperture: w( x) Field: f ( x, t) linear sensor contribution from area δξ at ξ : w( ξ)f ( ξ, t)δξ contribution from sensor z(t) = aperture w( ξ)f ( ξ, t)δξ. 6 z( x, t) = ξ w( ξ x)f ( ξ)δξ = w( x) f ( x, t), (i.e. spatial correlation) should have been spatial convolution (space-time F.T.) Z ( k, w) = x t z( x, t)e jwt e j k x dt d x = W ( k) F( k, w) Must use symmetry assumption... = W ( k) F( k, w) This differs from Eq. (3.1)! 7 Assume a single plane wave, propagating in direction ζ, ζ = k /k f ( x, t) = s(t α x), α = ζ /c F( k, w) = S(w)δ( k w α ) (Sec..5.1) This prop. wave contains energy only along the line k = w α in wavenumber-frequency space. Subst. of F( k, w) = S(w)δ( k w α ) into Z ( k, w) = W ( k)f( k, w) gives Z ( k, w) = W ( k)s(w)δ( k w α ) i.e. the spectrum of the output signal (Z ( k, w)) is multiplied by a wavenumber-frequency-dependent gain W ( k). 8 9

3 Characterizing W ( k) Linear aperture: b(x) = 1, x D/ W ( k) = sin kx D/ k x / Rectangular aperture: w(x, y) = b 1 (x)b (y) W (k x, k y ) = W (x)w (y) W (k x, k y ) = Circular aperture: sin kx Dx / sin k y D y / k x / k y / o(x, y) = 1, x + y R O(k xy ) = πr k xy J 1 (k xy R) Linear aperture: 1. sidelobe at k x.86π/d W (k x.17d ML SL D.17 = dB Circular aperture: 1. SL at k xy 5.14/R ML dB. SL Projection-slice theorem W(k x ) W(k x ) *log 1 W(k x ) Hovedlobe Sidelober Hovedlobe Sidelober [π/d] 1 11 Classical resolution Spatial extent of w( x) determines the resolution with which two plane waves can be separated. Ideally, W ( k) = δ( k), i.e. infinite spatial extent! Rayleigh criterion: Two incoherent plane waves, propagating in two slightly different directions, are resolved if the mainlobe peak of one aperture smoothing function replica falls on the first zero of the other aperture smoothing function replica, i.e. half the mainlobe width. Classical resolution... Linear aperture of size D W (k x ) = sin(kx D/) sin(π sin θd/λ) k x / (= Dsinc(k x D/)) = π sin θ/λ -3 db width: θ 3dB.89λ/D -6 db width: θ 6dB 1.1λ/D Zero-to-zero distance: θ = λ/d Circular aperture of diameter D W (k xy ) = πd/ k xy J 1 (k xy D/) -3 db width: θ 3dB 1.λ/D -6 db width: θ 6dB 1.41λ/D Zero-to-zero distance: θ.44λ/d Rule-of-thumb; Angular resolution: θ = λ/d 1 13

4 Geometrical optics Validity: down to about a wavelength Near field-far field transition dr = D /λ for a maximum phase error of λ/8 over aperture f-number Ratio of range and aperture: f # = R/D Resolution Angular resolution: θ = λ/d Azimuth resolution: u = Rθ = f# λ Depth of focus Aperture is focused at range R. Phase error of λ/8 yields r = ±f# λ or DOF=f # λ (proportional to phase error)

5 Ultrasound imaging Near field/far field transition, D=8mm, f=3.5mhz λ = 154/ =.44mm and d R = D /R = 178mm All diagnostic ultrasound imaging occurs in the extreme near field! Azimuth resolution, D=8mm, f=7mhz λ =.mm and θ = λ/d =.45, i.e. about lines are required to scan ±45 Depth of focus, f # =, f=5mhz λ =.38mm and DOF = f # λ.5mm. Ultrasound requires T = /154 = 3.µs to travel the DOF. This is the minimum update rate for the delays in a dynamically focused system. Ambiguities & Aberrations Aperture ambiguities Due to symmetries Aberrations Deviation in the waveform from its intended form. In optics; due to deviation of a lens from its ideal shape. More generally; Turbulence in the medium, inhomogeneous medium or position errors in the aperture. Ok if small comp. to λ φ sin φ k represents two kind of information 1. k = π/λ: No. waves per meter. k/ k : the wave s direction of prop. If signal have only a narrow band of spectral components, (i.e. all w), we can replace k with w /c = π/λ. Example: Linear array along x-axis: W ( k sin φ) = sin kx D sin φ kx sin φ W ( π sin φ/λ ) = W (φ) = λ sin D π sin φ π sin φ, D = D/λ W (φ) = W (φ + π), i.e. periodic!! W (k) is not! Often W (u, v), u = sin φ cos θ, v = sin φ sin θ Co-array for continuous apertures c( χ) w( x)w( x + χ)d x, χ called lag and its domain lag space. Important when array processing algorithms employ the wave s spatiotemporal correlation function to characterize the wave s energy. Fourier transform of c( χ)(= W ( k) ) gives a smoothed estimate of the power spectrum S f ( k, w). 1

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