INF5410 Array signal processing. Ch. 3: Apertures and Arrays

Transcription

1 INF5410 Array signal processing. Ch. 3: Apertures and Arrays Endrias G. Asgedom Department of Informatics, University of Oslo February 2012

2 Outline Finite Continuous Apetrures Apertures and Arrays Aperture function Classical resolution Geometrical optics Ambiguities & Aberrations Spatial sampling Sampling in one dimension Arrays of discrete sensors Regular arrays Grating lobes Element response Irregular arrays

3 Outline Finite Continuous Apetrures Apertures and Arrays Aperture function Classical resolution Geometrical optics Ambiguities & Aberrations Spatial sampling Sampling in one dimension Arrays of discrete sensors Regular arrays Grating lobes Element response Irregular arrays

4 Apertures and Arrays Aperture: a spatial region that transmits or receives propagating waves. Array: Group of sensors combined in a discrete space domain to produce a single output. To observe a wavefield at m th sensor position, x m, we distingush: Fields value: f ( x m, t). Sensors output: y m (t). e.g If sensor is perfect (i.e. linear transf., infinite bandwidth, omni-directional): y m (t) = κ f ( x m, t), κ R (or C).

5 Aperture function Our sensors gather a space-time wavefield only over a finite area. Omni-directional sensors: have no directional preference. E.g Sensors in a seismic exploration study. Directional sensors: have significant spatial extent. They spatially integrate energy over the aperture, i.e. they focus in a particular propagation direction. E.g Parabolic dish. They are described by the aperture function, w( x), which describes: Spatial extent reflects size and shape Aperture weighting: relative weighting of the field within the aperture (also known as shading, tapering, apodization).

6 Aperture smoothing function ( Sec ) Aperture function at ξ: w( ξ) Field recorded at ξ: f ( x ξ, t) contribution from the area δ ξ at ξ for a monochromatic wave with angular frequency ω is: w( ξ)f ( x ξ, ω)d ξ contribution of the full sensor z( x, ω) = aperture w( ξ)f ( x ξ, ω)dξ. z( x, ω) = w( x) f ( x, ω). Z ( k, ω) = W ( k)f( k, ω).

7 Aperture smoothing function... Aperture smoothing function: W ( k) = w( x) exp(j k. x)d x The wavenumber-frequency spectrum of the field: F( k, ω) = f ( x, t) exp(j k. x ωt)d xdt Assume a single plane wave, propagating in direction ζ 0, ζ 0 = k 0 /k f ( x, t) = s(t α 0 x), α 0 = ζ 0 /c F( k, ω) = S(ω)δ( k ω α 0 ) (Sec ) This prop. wave contains energy only along the line k = ω α 0 in wavenumber-frequency space.

8 Aperture smoothing function... Subst. of F( k, ω) = S(ω)δ( k ω α 0 ) into Z ( k, ω) = W ( k)f( k, ω) gives Z ( k, ω) = W ( k)s(ω)δ( k ω α 0 ) Z ( k, ω) = W ( k ω α 0 )S(ω) For k = ω α 0 Z (ω α 0, ω) = W (0)S(ω): The information from the signal s(t) is preserved. For k ω α 0 : The information from the signal s(t) gets filtered.

9 Aperture smoothing function... 1 Linear aperture: b(x) = 1, x D/2 sin kx D/2 k x /2 W ( k) = Sidelobe at k x0 2.86π/D W (k x D ML SL 13.3dB D = W(k x ) W(k x ) 20*log 10 W(k x ) Hovedlobe Sidelober Hovedlobe Sidelober [π/d]

10 Aperture smoothing function... Circular aperture: o(x, y) = 1, x 2 + y 2 R O(k xy ) = 2πR k xy J 1 (k xy R) SL at k xy /R ML dB. SL

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. W(k x )/D k x /[π/d]

12 Classical resolution... Linear aperture of size D W (k x ) = sin(kx D/2) sin(π sin θd/λ) k x /2 (= Dsinc(k x D/2)) = π sin θ/λ -3 db width: θ 3dB 0.89λ/D -6 db width: θ 6dB 1.21λ/D Zero-to-zero distance: θ0 0 = 2λ/D Circular aperture of diameter D W (k xy ) = 2πD/2 k xy J 1 (k xy D/2) -3 db width: θ 3dB 1.02λ/D -6 db width: θ 6dB 1.41λ/D Zero-to-zero distance: θ λ/D Rule-of-thumb; Angular resolution: θ = λ/d

13 Geometrical optics Validity: down to about a wavelength Near field-far field transition dr = D 2 /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# 2 λ or DOF=2f # 2 λ (proportional to phase error)

14 Geom.Opt: Near field/far field crossover (From Wright: Image Formation...)

15 Geom.Opt: Near field/far field crossover (From Wright: Image Formation...)

16 Geom.Opt: Near field/far field crossover (From Wright: Image Formation...)

17 Ultrasound imaging Near field/far field transition, D=28mm, f=3.5mhz λ = 1540/ = 0.44mm and d R = D 2 /R = 1782mm All diagnostic ultrasound imaging occurs in the extreme near field! Azimuth resolution, D=28mm, f=7mhz λ = 0.22mm and θ = λ/d = 0.45, i.e. about 200 lines are required to scan ±45 Depth of focus, f # = 2, f=5mhz λ = 0.308mm and DOF = 2f 2 # λ 2.5mm. Ultrasound requires T = /1540 = 3.2µs to travel the DOF. This is the minimum update rate for the delays in a dynamically focused system.

18 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 λ0.

19 φ sin φ k represents two kind of information 1. k = 2π/λ: No. waves per meter 2. 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 0 /c = 2π/λ 0. Example: Linear array along x-axis: W ( k sin φ) = sin kx D sin φ 2 kx sin φ 2 W ( 2π sin φ/λ 0 ) = W (φ) = λ 0 sin D π sin φ π sin φ, D = D/λ 0 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

20 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) 2 ) gives a smoothed estimate of the power spectrum S f ( k, w).

21 Outline Finite Continuous Apetrures Apertures and Arrays Aperture function Classical resolution Geometrical optics Ambiguities & Aberrations Spatial sampling Sampling in one dimension Arrays of discrete sensors Regular arrays Grating lobes Element response Irregular arrays

22 Periodic spatial sampling in one dimension Array: Consists of individual sensors that sample the environment spatially Each sensor could be an aperture or omni-directional transducer Spatial sampling introduces some complications (Nyquist sampling, folding,...) Question to be asked/answered: When can f (x, t 0 ) be reconstructed by {y m (t o )}? f (x, t) is the continuous signal and {y m (t)} is a sequence of temporal signals where y m (t) = f (md, t), d being the spatial sampling interval.

23 Periodic spatial sampling in one dimension... Sampling theorem (Nyquist): If a continuous-variable signal is band-limited to frequencies below k 0, then it can be periodically sampled without loss of information so long as the sampling period d π/k 0 = λ 0 /2.

24 Periodic spatial sampling in one dimension... Periodic sampling of one-dimensional signals can be straightforwardly extended to multidimensional signals. Rectangular / regular sampling not necessary for multidimensional signals.

25 Outline Finite Continuous Apetrures Apertures and Arrays Aperture function Classical resolution Geometrical optics Ambiguities & Aberrations Spatial sampling Sampling in one dimension Arrays of discrete sensors Regular arrays Grating lobes Element response Irregular arrays

26 Regular arrays Assume point sources (W tot = W array W el )). Easy to analyze and fast algorithms available (FFT).

27 Regular arrays; linear array Consider linear array; M equally spaced ideal sensor with inter-element spacing d along the x direction. The discrete aperture function, wm. The discrete aperture smoothing function, W (k): W (k) m w me jkmd Spatial aliasing given by d relative to λ.

28 Grating lobes Given an linear array of M sensors with element spacing d. W (k) = sin kmd/2 sin kd/2. Mainlobe given by D = Md. Gratinglobes (if any) given by d. Maximal response for φ = 0. Does it exist other φg with the same maximal response? k x = 2 π λ sin φ g ± 2 π d n sin φ g = ± λ d n. n = 1: No gratinglobes for λ/d > 1, i.e. d < λ. d = 4λ: sin φ g ± n 1/4 φ g = ±14.5, ±30, ±48.6, ±90.

29 Element response If the elements have finite size: W e ( k) = w( k)e j k x d x If linear array: Continuous aperture devided into M parts of size d Each single element: sin(kd/2) k/2 first zero at k = 2π/d Total response: W total ( k) = W e ( k) W a ( k), where W a ( k) is the array response when point sources are assumed.

30 Irregular arrays Discrete co-array function: c( χ) = (m 1,m 2 ) ϑ( χ) w m 1 wm 2, where ϑ( χ) denotes the set of indices (m 1, m 2 ) for which x m2 x m1 = χ. 0 c( χ) M = c( 0). Equals the inverse Fourier Transform of W ( k) 2 sample spacing in the lag-domain must be small enough to avoid aliasing in the spatial power spectrum. Redundant lag: The number of distinct baselines of a given length is grater than one.

31 Examples

32 Examples...

33 Irregular arrays Sparse arrays Underlying regular grid, all position not filled. Position fills to acquire a given co-array Non-redundant arrays with minimum number of gaps Maximal length redundant arrays with no gaps. Sparse array optimization Irregular arrays can give regular co-arrays...

34 Examples Non-redundant arrays == Minimum hole arrays == Golumb arrays 1101, , Redundant arrays == Minimum redundancy arrays 1101, ,

35 Random arrays W ( k) = M 1 m=0 ej k x m (assumes unity weights) E[W ( k)] = M 1 m=0 E[ej k x m ] = M p x ( x m )e j k x m d x = M Φ x ( k) i.e. Equals the array pattern of a continuous aperture where the probability density function plays the same role as the weighting function. var[w ( k)] = E[ W ( k) 2 ] (E[W ( k)]) 2 E[ W ( k) 2 ] = E[ M 1 m 1 =0 ej k x m1 M 1 m 2 =0 e j k x m2 ] = E[M 1 + m 1,m 1 m 2 e j k x m1 m 2 e j k x m2 ] Assumes uncorrelated x m (E[x y] = E[x] E[y]) E[ W ( k) 2 ] = M + (M 2 M) Φ x ( k) 2 var[w ( k)] = M M Φ x ( k) 2