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

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.

• The question title is meaningless. There is no (non-trivial) minimum-phase filter that has a linear phase. Feb 26 at 12:56