# Causal and minimum phase

I have couple of questions?

1. Is there a relationship between a FIR filter being minimum phase/mixed phase and being causal?
2. Is there a relationship between a FIR filter being minimum phase/mixed phase and its group delay. I understand that a symmetric/non symmetric filter has a constant phase and fixed group delay. But, what if filter is FIR and non-symmetric?Doe symmetry affect the min/max/mixed phase ?
• yeah, there is a lot in this. i might recommend splitting the question into one about causality, minimum-phase, and group delay and another one about the use of the LMS adaptive filter. – robert bristow-johnson Apr 15 '15 at 18:57
• @robert I put everything together, because if the LMS adaption was wrong because of incorrect sampling of data due to relationship between causality and how the impulse response looked with respect to symmetry and zeros, then, I could do something about it. So, my first 2 questions were related. – user915783 Apr 15 '15 at 19:20

A minimum-phase system is always causal and stable by definition, so in the case of discrete-time systems with rational transfer functions, all poles and zeros are inside the unit circle of the complex $z$-plane. This is why a minimum-phase system can be inverted by a causal and stable system. Because of this definition, it does not make sense to compare non-causal systems and their phase lag or group delay to the corresponding properties of minimum-phase systems. Minimum-phase systems have the smallest possible phase lag of all causal systems with the same magnitude response.

The group delay is a function of frequency. For symmetric/anti-symmetric impulse responses, the group delay is constant. This is probably what you meant in your question. Such linear-phase (constant group delay) systems can never be minimum-phase (unless their impulse response is just a unit impulse at $n=0$). Minimum-phase systems have a frequency dependent group delay, but at any frequency, their group delay is smaller than the group delay of any other causal system with the same magnitude response.

There's lots to your question, so this will only be a partial answer until we clear up some things.

Minimum Phase

A minimum phase filter is one that has all its poles and zeros inside the unit circle.

Alternatively, a minimum phase filter is one that is causal and stable and the inverse of which is also causal and stable.

Is there a relationship between a FIR filter being minimum phase/mixed phase and being causal?

By definition, a minimum phase system is causal.

To answer the second part (about mixed phase filters), we need to understand what causality is.

Causal

A filter is causal if the output does not depend on future inputs.

Is there a relationship between a FIR filter being minimum phase/mixed phase and being causal?

If your FIR filter can be written as: $$H(z) = \sum_{p=-M}^{+N} h_p z^{p}$$ then $H(z)$ is causal if $N \le 0$ (i.e. there are only zero or negative powers of $z$).

Constant Phase

You make the statement:

I understand that a symmetric/non symmetric filter has a constant phase

which is completely incorrect.

The only FIR filters with constant phase are the ones where $N=0$ and $M=0$ so that $H(z) = h_0$ (i.e. a constant).

Symmetric FIR filters and anti-symmetric FIR filters can have constant group delay (and generalized linear phase).

Group Delay

The group delay of a minimum phase FIR filter will be a function of frequency, because minimum phase FIR filters cannot have constant group delay (except in the trivial case of $H(z) = h_0$ noted above.

Placement of zeros

I get filter which has zeros outside the unit circle, and impulse response which is non symmetric. I am told that this is typical of non-causal filter.

It is perfectly possible for a causal FIR filter to have zeroes outside the unit circle.

Break

I think your real question at the bottom is about system identification, but I needed to clarify the misconceptions in your build up first.

Let me know if this makes sense, and I'll try to address the rest of your question later.

• I have couple of more comments. If the Filter is FIR ( hence stable), then as per the definitions you quoted, if its minimum phase( zeros inside the unit circle), it is guaranteed causal. But you also mention that It is perfectly possible for a causal FIR filter to have zeroes outside the unit circle. How these 2 reconcile? or is it that a minimum phase FIR will be causal and stable , but the converse may not be true? – user915783 Apr 15 '15 at 19:16
• Also, you mention that minimum phase filter cannot have a constant group delay, while symmetric FIR filter have constant group delay. Do these 2 statements mean that minimum phase filters cannot be symmetric and symmetry has nothing to do with causality? Also, what would be some of the characteristics of the impulse response for FIR filter that would indicate its causality? – user915783 Apr 15 '15 at 19:17
• Response to comment #1: A causal FIR filter can have zeros outside the unit circle. It just won't be minimum phase. Yes, a minimum phase FIR filter will be, by definition, causal and stable but just because an FIR filter is causal and stable does not mean it is minimum phase. Does that reconcile it? – Peter K. Apr 15 '15 at 19:29
• Response to comment #2: Yes, a minimum phase filter cannot be symmetric (except for the boring case of $H(z) = h_0$, a constant). The characteristic of the impulse response is that it has no non-zero coefficients for negative time (the impulse response does not start before the putative impulse). – Peter K. Apr 15 '15 at 19:31
• @PeterK.: Yes, now the system is causal. The only cosmetic problem now is that $h_p$ is not the impulse response, but the time-reversed impulse response. I usually write $\sum_{n=N_1}^{N_2}h[n]z^{-n}$, requiring $N_1\ge 0$ for causality (and of course $N_2\ge N_1$). – Matt L. Apr 16 '15 at 7:29