A note on digital filter synthesis

It is commonly assumed that digital filters with both poles and zeros in the complex z-plane can be synthesized using only recursive techniques while filters with zeros alone can be synthesized by either direct convolution or via the discrete Fourier transform (DFT). In this letter it is shown that no such restrictions hold and that both types of filters (those with zeros alone or those with both poles and zeros) can be synthesized using any of the three methods, namely, recursion, DFT, or direct convolution.