Noll (2006) US Notes 3: page 1 Ultrasound Notes 3: Array Systems While early US systems used a single focus tranducer and mechanical sweeping of the transducer to different angles, nearly all modern US systems are array systems where focussing is not preset and beam steering is done through time delays associated with each element of the array. Transmit mode: Focussing and Beam Steering Typical array system shown here in transmit mode: where xn is the x coordinate of the nth element (0 is at the center of the array). For focussing at (z,xz=0) set the delay to: czxxzxnznn2)0,,(''2===ττ (Actually, this is a negative delay – to focus on-axis in the Fresnel zone, we require that the edge elements fire before the center element because they have farther to propagate.) In polar coordinates, for focussing at (r0,θ0): 00220002cossin),,(''crxcxrxnnnnθθθττ+−== A focal depth and direction (r0,θ0) is selected once when the pulse is transmitted. Once it leaves the transducer, it can no longer be changed.Noll (2006) US Notes 3: page 2 Receive mode: Delay-Sum Beamforming - Signals from each transducer are delayed and summed to produce a signal localized to the focal point, e.g. (r0,θ0): ∑=+=Nnnntvtv1)'()(τ - Signal from other angles are suppressed by destructive interference of the wavefronts when summed together (recall the each pulse is made up of modulated waves at a carrier frequency, f0. - Unlike the transmit case, the received data can be combined in many different ways. In otherwords, we can separately focus for any point in time (depth plane, r). Also we can focus to any angle at by reprocessing the data. Typically, “dynamic” focussing is used to focus for each depth plane – here τ’n changes for every depth r = ct/2. Sampling in Space At each point in time, the array is actually sampling the instantaneous pressure wave as a function of space. Aliasing is potentially a problem. A quick review of sampling:Noll (2006) US Notes 3: page 3 Aliasing can be prevented if less than π has accrued between neighboring samples. For our case of waves impinging upon the detector array, sampling becomes a bigger issues if the source of the waves is coming from a large angle, θ: The sampling requirement for this case can be written as the maximum difference in propagation delay between a point (r,θ) an two neighboring transducer elements cannot lead to phase accrual of greater than π: ()nrrxrxnn,, ),,(),,(01θπωθτθτ∀≤⋅−+ assuming the element spacing is d (e.g. xn+1 = xn + d), then:Noll (2006) US Notes 3: page 4 ()crdrdxdcrdxdxrrxxrrxrxnnnnnnn022202222012cos2cos2sin2cos)(sin)(2cossin),,(),,(ωθθθωθθθθωθτθτ⋅⎟⎟⎠⎞⎜⎜⎝⎛−−=⋅⎟⎟⎠⎞⎜⎜⎝⎛+−++−+−=⋅−+ The worst case is for plane waves, that is :∞→r ()πλπθωθωθτθτ≤⋅=⋅=⋅−+2sinsin),,(),,(001dcdrxrxnn and thus: 2sinλθ≤d and if we want to unambiguously distinguish between arrival directions (points of reflectivity) over a full π of angle, then: 2λ≤d In this case, the number of transducer elements, N, should be: Aperture) (Numerical242⋅=≥=λadaN The “Numerical Aperture” for an array is the size of the transducer in terms of the number of wavelengths (e.g. N.A. = 2a/λ). Comments: – We’ve just talked about the array sampling a pressure wave propagating towards the transducer. It is important to realize that sampling also occurs during transmit as well – in order to unambiguously transmit a wave in a particular direction (over π angles) , we also need to satisfy the above transducer spacing. – Below under discussion of the point spread function, we will see the effects of insufficient sampling by the array.Noll (2006) US Notes 3: page 5 Angular Sampling of the Object For this analysis, we will consider the far-field (or focal plane) solution: {}⎟⎠⎞⎜⎝⎛′′=ℑ⋅==λθθθλθsin)(cos),(sinSKxsrerpuikr As described above, we must transmit our beam at a particular angle (θi) and thus, this axis is discretized as well: λθiSsin locationsat sampled is )(⋅ This is a reverse sampling argument – we are sampling the FT of s(x) and thus the samples of S(u) must be positioned at least as close as half the maximum extent of s(x). Thus, if s(x) goes from [-a, a], then: Aperture Numerical12sin21sin=≤Δ≤⎥⎦⎤⎢⎣⎡Δaaλθλθ This is the solution for the receive only or a transmit only case. Recall that the actual beam function is: [][]222sin),(),,(),,(),(⎥⎦⎤⎢⎣⎡⎟⎠⎞⎜⎝⎛′′===λθθθωθωθSKrprhrhrBRT and this is the function that is used to sample the object in the angular direction. Since we must now sample the square of the FT of the space limited aperture function, we must know the maximum extent of the its IFT, which is s(x)* s(x). If if s(x) goes from [-a, a],Noll (2006) US Notes 3: page 6 then s(x)* s(x) is bounded by a function that extends of [-2a, 2a]. For example, if s(x) = rect(x/2a), then s(x)* s(x) = triangle(x/2a), which goes from -2a to 2a. Thus: N.A.214sin⋅=≤Δaλθ - How many beams to sample π? N.A.484/)1(1)(sin)min(sin)max(sin⋅==−−=Δ−λλθθθaa - Is sampling uniform in θ? No, sampling is uniform in sinθ. - Shouldn’t we just sample more finely than we need to? No.  It takes extra time (you can’t transmit the next pulse at a new angle until the last one is received).  It adds no new information. - What happens if we don’t sample finely enough? We might miss an object feature.Noll (2006) US Notes 3: page 7 Point Spread Function for Discrete Transducers For odd N: ⎟⎠⎞⎜⎝⎛⎥⎦⎤⎢⎣⎡⎟⎠⎞⎜⎝⎛⎟⎠⎞⎜⎝⎛=⎟⎠⎞⎜⎝⎛−=∑−−−wxDxdxdwndxxsNNrect*rectcomb1rect)(212)1( and the FT is: []∑∞−∞=⎟⎟⎠⎞⎜⎜⎝⎛⎟⎠⎞⎜⎝⎛−⋅=⋅=ndnuDwudDwDuDduwuwuSsinc )(sinc)(sinc *)(comb)(sinc )( Substituting u = sin θ/λ, in the far field (or focal plane) for an on-axis beam, we get: ∑∞−∞=⎟⎟⎠⎞⎜⎜⎝⎛⎟⎠⎞⎜⎝⎛−⋅=ndnawKrpλθλθθθsin2sinc )sin(sinc cos),( - the first sinc function provides additional weighting as a function of θ. This could reduce the angular “field of view” by restricting over what range of angles we can effectively transmit to/receive from. Let’s look at these effects for a Nyquist sampled array (d=λ/2). First we look at the effect of the