# Applications for the Maximum Phase Filter

A maximum phase digital filter has all zeros outside the unit circle, and has the maximum phase and therefore longest delay for a given magnitude response.

Besides the possibility of needing a filter with the longest delay possible, are there other practical applications and uses of the maximum phase filter?

I understand that we can deconvolve linear-phase filters into a minimum-phase filter and its reverse a maximum phase filter, but looking for further utility and interesting use cases for maximum-phase filters specifically.

we can deconvolve linear-phase filters into a minimum-phase filter and its reverse a maximum phase filter

We can formulate this more broadly. Any LTI system can be split into a cascade of it's minimum phase filter and an all pass (which is indeed a maximum phase filter). So, $$H(z) = H_{min}(z) \cdot A(z)$$

where $$H_m(z)$$ is the minimum phase filter that matches the magnitude $$|H(z)|$$ and $$A(z)$$ is an allpass , i.e.

$$A(z) = k \cdot \frac{\prod_{n=0}^{N-1}z-1/p_n^*}{\prod_{n=0}^{N-1}z-p_n}$$

The proof for this is straight forward: Let's assume that H(z) has a zero $$q_k$$ outdside the unit circle. We can simply factor this out as

$$H(z) = H_k(z) \cdot (z - q_k) = H_k(z) \cdot (z - q_k) \cdot \frac{z - 1/q_k^*}{z - 1/q_k^*}$$

$$= H_k(z) \cdot (z - 1/q_k^*) \cdot \frac{z - q_k}{z - 1/q_k^*} = H_{k,min}(z) \cdot A_k(z)$$

where $$H_{k,min}(z)$$ is the version of $$H(z)$$ where the zero $$q_k$$ has been made minimum phase by inverting it to $$1/q_k^*$$ with $$|1/q_k^*| < 1$$ and $$A_k(z)$$ is a first order allpass with a zero at $$q_k$$. Note that the magnitude stays the same, i.e. $$|H_{k,min}(z)| = |H(z)|$$

That type of splitting an LTI system into it's maximum and minimum phase part is can be quite useful.

Inverting: The minimum phase part is invertible and the inverse is also minimum phase. The inverse of the maximum phase is simply it's own time reverse. That's not directly invertible (since the inverse is non causal), but you easily "eyeball" how much bulk delay you need to add to get enough of the phase response correct.

Audio: Linear phase FIR are mostly useless in audio. Getting decent resolution at low frequencies requires many 1000s of taps and the computational complexity and the latency are often prohibitive. Equalizer or cross over design is often split into two steps: minimum phase filter to get the amplitude right and than allpass filters to do time/phase alignment as needed.

Building blocks: Allpass filters are very useful building blocks to build more complicated structures. For example you an build odd order Butterworth cross overs as sum + difference of two allpasses and as a bonus you get a signal that's phase matched to the perfect reconstruction path. Warped FIR filters are another useful topology where allpasses are quite handy. Beamformers need more or less independent control of amplitude and phase.

So in general in an application where you want to control phase and amplitude more or less independently, splitting into minimum (for amplitude) and maximum (for phase) can be quite useful.

• Great question. Great answer. May 27 '20 at 18:46
• @Hilmar Ah very good! The all pass (which reciprocal pole/zero pairs) would have all zeros outside the circle and be maximum phase. I didn't make that connection-- Thank you! May 27 '20 at 20:10