Convolution with Complex-Valued Functions: Applications and Limitations

[mnb96]

Hello,

given two functions f and g the operation of convolution $f\ast g$ finds many applications in many different branches of science. However, in such applications, it is typically assumed that one of the two functions (the convolution kernel) is a real scalar field, although the mathematical definition does not impose such constraint.

My question is: does the convolution of, say, two complex-valued functions has any known application? For instance when f,g are of the kind:
$f:\mathbb{R}^2 \rightarrow \mathbb{C}$, and
$g:\mathbb{R}^2 \rightarrow \mathbb{C}$ ?

Note that such functions could represent, for instance, vector fields on the Cartesian plane.

 

[jasonRF]

Yes, complex valued functions are used all the time in signal and image processing. These days the domain is usually discrete since most things are digital, but even for that case it is often useful to do analytical modelling in the continuous domain.

jason

EDIT: should have specified more, perhaps. for complex valued signals/images, one method of detecting known patterns (including phase) is to use a matched filter, which is a complex valued filter, so both the signal/image and the filter are complex. The matched filter is in some sense "optimal" for white Gaussian noise interference. These are used a lot in communications systems, image formation, etc.

 

[mnb96]

thanks jason,

in fact, it seems the only application I can find is the one where one wants to match a complex signal within another complex signal.

I expected also some physical application, perhaps in fluid mechanics or in other branches of physics, where vector fields are extensively used; but so far I haven't found anything related to convolutions between vector valued functions (i.e. convolution of vector fields).

 

[olivermsun]

One application in fluid mechanics is in spectral energy transfers, e.g., triad interactions. The energy transfer integral (which arises from the advection term in the equations of motion) is often expressed as an integral over wave vector space, e.g. $S(\mathbf{k}) = \int d^3 k_1 \int d^3 k_2 \; A(\mathbf{k}_1) \, B(\mathbf{k}_2) \, \Gamma(\mathbf{k}, \mathbf{k}_1, \mathbf{k}_2) \ldots \delta(\mathbf{k}-\mathbf{k}_1-\mathbf{k}_2)$
where $(\mathbf{k}, \mathbf{k}_1, \mathbf{k}_2)$ are 3-d wave vectors, $\Gamma$ is a (scalar) interaction cross-section, and the Dirac delta functions express the resonance condition $\mathbf{k} = \mathbf{k}_1 + \mathbf{k}_2$. In this case the double integral can be rewritten as a convolution using the fact $\mathbf{k}_2 = \mathbf{k} - \mathbf{k}_1$; this can be very useful for evaluating the integrals. If one is dealing with definite phases (instead of assuming random phase) or vector quantities (like velocity) then the integrands can also be complex- or vector-valued.

I think you can come up with more examples just by thinking of any other situations where we deal with products of transforms...

 

[mnb96]

Thanks olivermsun,

I was not aware of that kind of application. Your observation essentially answers my original question, although I am not familiar at all with the concepts that you mentioned in your response.

At this point I am wondering if there is any application of convolution between vector valued functions that could be more easily visualized. More or less like the case of convolution to simulate heat diffusion over time.