What is the position of all zeros of a minimum phase, Type 1 Linear Phase FIR Filter?

Question

Let me write down all the facts that I know of.

In context of the z plane:

1. Minimum phase system: All zeros and poles of such a system lie inside the unit circle.
2. Linear phase FIR filter: For every zero ($ z = z_0 $), its inverse ($ z = \frac{1}{z_0} $) is also a zero of this filter.
3. Real value coefficients FIR filter: For every zero, the complex conjugates are also zeros of the filter.

The first two facts match with the question but the third fact is unmentioned in the question.

Let $z_0 = re^{j\theta}$.

Then via fact 2: $z_0^{-1} = r^{-1}e^{-j\theta}$

From fact 1: $r < 1$

Hence if $z_0$ is inside the unit circle, then $z_0^{-1}$ must lay outside the unit circle. But this is not permissible.

Thus is it correct to say that such a filter has no zeros at all?

P.S It is mentioned that the filter is also low pass. So far, I failed to find this information of any use.

Answer 1

Your facts are correct and all make sense. The conjugate reciprocal of a pole or zero inside the unit circle will be outside of a unit circle.

What I think you may have missed is that a linear phase filter is NOT a minimum phase filter and therefore does not meet the first requirement. A linear phase filter will have zeros inside and outside the unit circle at conjugate reciprocal locations, and/or exactly on the unit circle. A minimum phase filter must have all poles and zeros strictly inside the unit circle.

Not all filters have real coefficients, but if the coefficients are real for any type of filter then as given in the third point any complex poles or zeros will appear in conjugate pairs.

There are also some important facts worth studying (and discussed on other posts here) specific to the four types of FIR filters (Type 1, Type 2, Type 3 and Type 4) with respect to their zeros.