Convolution in constant frequency planes

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  

$${k} = {h \exp(\tau_{1}-\tau)}$$ (8)

where ${\tau} = {\log({{t}\over{T_{c}}})}$.

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:  

$${\hat{Q_{1}}(\omega_{1},m,k)} = {\int_{-h}^{h} Q_{1}(\omega_{1},m+b,h = \sqrt{k^2+b^2})db},$$ (9)

where $\hat{Q_{1}}$ and Q₁ denote the output and input data, $\omega_{1}$is the frequency variable, m and h denote midpoint and offset, respectively. k represents the DMO offset of equation (1).