= Fourier filtering of overlapping amplitude-modulation sidebands
The <modulation property of the Fourier transform> adds shifted copies of the message spectrum. If those copies overlap, their complex amplitudes add before taking a magnitude; a frequency filter then acts on this sum. For a real even triangular message spectrum proportional to $(1-|\omega|)_+$ and carrier frequency $1/2<\Omega<1$, the translated spectra overlap over $|\omega|<1-\Omega$. A filter retaining $|\omega|<\Omega$ therefore leaves a central plateau and sloping outer sections, rather than two disjoint triangular bands. Spectral <Dirac delta> lines at the cutoff need an explicitly specified filter gain: multiplying a <distribution> by a discontinuous mask is not defined by pointwise reasoning alone.
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