Filtering using Simple Mathematical Operations

The filtering effect of a mathematical operation can be characterized in the frequency domain by its frequency response function. The effect of a mathematical operator \( \mathcal{O}\left[ \ldots \right] \) can be determined in the frequency domain by converting the operation $$ y\left( t \right) = \mathcal{O}\left[ {x\left( t \right)} \right] $$ (16.1) into a convolution: $$ y\left( t \right) = x\left( t \right) * h\left( t \right), $$ (16.2) and subsequently determining the frequency response function H(f).