My question is related to a theoretical part of the book: Digital Processing of Synthetic Aperture Radar Data, Algorithms and Implementation by I.G. Cumming et al. [Chapter 4. Equation 4.25 (for those who know the book)].
Basically what I undertand that is said there, is that if I have my transmit pulse waveform $s_{pul}(\tau)$ and some ground reflectivity function $g_r(\tau)$, "the reflected energy at any illumination instant is a convolution ..." as: $$ s_r(\tau) = g_r(\tau) \ast s_{pul}(\tau) $$
An example is given for a point target at a distance $R_a$ from the radar having a magnitude $A_0$ accounting for the RCS. The assumed reflectivity function is given as: $$ g_r(\tau) = A_0 \delta(\tau - \frac{2R_a}{c}) $$ which would yield the backscattered signal as: $$ s_r(\tau) = A_0 s_{pul}(\tau - 2\frac{2R_a}{c}) $$
So far so good, for this point target case. If I think in terms of the Fourier Transforms, the dirac-delta would correspond to a constant (and a phase slope for the shift), which is multiplied by the spectrum of my transmit signal. However, what if I assume a different ground reflectivity, lets say a bumpy surface modeled by a sinusoid function. Its FT would be more of a "peaky" signal in the frequency domain. Multiplying this with the FT of my transmit signal would be some sort of filtering, no?
I have the feeling that I am missing an important part here, or that the convolution equation may be a simplification, leaving out certain aspects that skew my understanding. E.g. what maybe confuses me is the dependency in time via $\tau$, but there is no spatial dependency. I know from electromagnetics courses that for scattered fields there is typically also some green's function involved, but there is always some dependency on the location afaik. Maybe a key phrase from the book is "... reflected signal at any illumination instant ...".
I would be happy for some links, reference or any hints to better understand this convolution equation.