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I'm specifically looking at something like Matlab's firpm function (I know there are C implementations of Remez available online in several sources).

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The firpm page on the MATLAB documentation, under "Extended Capabilities", says the function is compatible with C/C++ code generation with the MATLAB Coder.

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I did not see that, thanks. The page also says that all inputs must be constants. When I tried to use Matlab Coder on a function that took no inputs and used firpm with static, locally defined inputs for Fs, order etc. the C code that was generated only had empty functions.
Try to return the output of firpm from your entry-point function. You should see the answer hard-coded into the generated code. The idea is that if you use this value within the generated code, it will be preserved. If it is unused, Coder will optimize it out.

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