(Color online) A conventional pulse with a Gaussian envelope, sampled at twice the center frequency in (a). The zeros of the Z-transform of the sampled pulse is shown in (b). Because some zeros lie in and around the unit circle, the inverse is unstable and unbounded.
(Color online) An asymmetric pulse formed by multiplying the Gaussian envelope with a geometric series is shown in (a). The zeros of the Z-transform are retracted into the unit circle as shown in (b). This leads to a stable inverse filter.
(Color online) Another asymmetric pulse with the form of a Gaussian multiplied by the square root of t is shown in (a). The zeros of its Z-transform are shown in (b) indicating the availability of a stable inverse filter shown in (c).
(Color online) An unmodulated asymmetric envelope proposed for the transverse beampattern is shown in (a). Its Z-transform zeros are shown in (b) and corresponding stable inverse filter in (c).
A gray-scale plot of the two dimensional (2D) function that is separable and asymmetric in both vertical and horizontal directions. The vertical (axial) function is given by the function shown in Fig. 3(a), and the horizontal (transverse) function is the function shown in Fig. 4(a).
A set of random reflectors with patches of zeros in the shape of letters (a) undergoes 2D convolution with the pulse of Fig. 5. The resulting speckle envelope is shown in (b), and the letters “UR” are not visible due to the distribution of the speckle statistics. After inverse filtering in the vertical direction, the results are improved in (c), and after horizontal inverse filtering the final result is given in (d); 5% rms white noise was added to the convolution result before inverse filtering, so the operations are well conditioned in the presence of modest additive noise.
(Color online) Histograms showing the probability density functions for the original scatterers (a), the result of convolution with a pulse (b), and after inverse filtering (c). The standard Rayleigh speckle statistics are demonstrated in (b), whereas the original Gaussian statistics are restored in (c).
(Color online) A sampled pulse (a) and its Z-transform zeros (b) indicating the availability of a stable inverse filter.
(Color online) The same pulse sampled at twice the sampling rate of Fig. 8, shown in (a), and its Z-transform zeros in (b). A movement toward the unit circle is evident. Another doubling of the sampling rate results in (c) and (d). The trend is toward the unit circle and instability.
The effect of inexact reconstruction parameters. A region of reflectors is shown in (a). Imaging with a pulse using a geometric parameter of 0.7 yields a conventional speckle image in (b). Reconstruction with horizontal (exact) and vertical (inexact, geometric parameter 0.6 used) are shown in (c) and (d), respectively. The resulting inverse filtered image is a reasonable reconstruction of the original scatterers, despite the use of an erroneous model.
(Color online) Two functions in the spatial domain, a Rayleigh and a Gaussian, representing asymmetric and symmetric pulse envelopes, are given in (a). Their Fourier Transform magnitudes are given in (b). A somewhat greater distribution of the higher spatial frequencies is required to generate an asymmetric beam envelope.
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