Trivial and non-trivial zeros

I am new to DSP, and I'm self studying. Could someone please explain to me what do we mean by trivial and non-trivial zeros?

• Could you give a reference that uses these terms? – Matt L. Nov 22 '18 at 11:49

I wasn't familiar with that term in the context of signal processing. (Instead, I've seen the term being used in the context of the Riemann zeta function.) But I've found a document and this book where the term is used in a DSP context. The (obvious) definition is that trivial poles and zeros are the ones at the origin $$z=0$$ and at infinity $$|z|=\infty$$. They're called trivial because they don't affect the magnitude of the corresponding frequency response. Multiplying a given $$\mathcal{Z}$$-transform with $$z$$ (i.e. adding a pole at infinity and a zero at the origin) just advances the corresponding sequence by one sample, and multiplying by $$z^{-1}$$ (i.e., adding a pole at the origin and a zero at infinity) delays the corresponding sequence by one sample.

As an example, note that a causal FIR filter has as many poles as zeros, but all of them are at the origin $$z=0$$, i.e., they can't be used to shape the magnitude response. The same is true for an all-pole filter: all its zeros are at the origin, so they don't help to create a stop band (which would be one possible function of non-trivial zeros).

I think it's important to point out that the term trivial zero (or pole) is not a standard term used in DSP - as far as I know - but it appears to be an idiosyncratic use of a few authors.

• the typical way in the literature to refer to something idiosyncratic , “so called trivial zeros” – Stanley Pawlukiewicz Nov 22 '18 at 15:52