To find the impulse response of the DMO operator in the different transform domains, we keep the impulse location m,h, and t fixed. The impulse response of the DMO operator is spread along the intersection of a cylinder satisfying equation (1) and a radial plane satisfying (2). Logarithmic stretching of the time axis transforms the linear plane of slope h/t into a curved surface with
(8) |
The Impulse response after Fourier transform is located on the same semicircles in all frequency planes. The amplitude along the semicircle varies according to the complex factor of equation (7). I spread out the impulse along the circle defined by equation (1), conducting a 2D convolution. The convolution operator varies only with offset, but is midpoint independent. The convolution is identical on all frequency planes:
(9) |