# Decomposition minimal phase & all pass

• Engineering
• Hidd
In summary, the conversation discusses the design of two transfer functions, G1 and G2, with poles and zeros outside the unit circle. The speaker is unsure how to move the zero S_01 = 1 to the unit circle and is seeking advice on the all-pass G2 transfer function. There is also a mention of non-causal digital filters and the difficulty of deriving all-pass and min-phase functions from G. Eventually, the conversation ends with the suggestion of using G(s) = (1-s)/(2 + 10s) to achieve an all-pass filter, while noting that all-pass filters can never be minimum phase.
Hidd
Thread moved from the technical forums to the schoolwork forums (this includes for non-homework revision studying)
Homework Statement
I have the following transfer function, and I would like to decompose it to a minimal phase G1 & all pass G2 transfer functions:

G(s) = (1-s) / (2 + 10s)

G(s) = G1 * G2
Relevant Equations
All-pass ==> magnitude of G2(jw) =1
Minimum phase ==> Re{Zeros,poles}<0
Hey everybody!

I have put G1 = (1-s)/(2-10s) & G2 = (2-10s)/ (2 +10s)
but than I read that all poles and zeroes should be inside the unit circle, and I don't know how to move the Zero S_01 = 1 to the unit circle

Last edited:
Is this a homework problem? Do you know what the all-pass G2 transfer function is?

it's when the magnitude of G2(jw) =1

Last edited:
DaveE
OK, but G1 isn't minimum phase since it has right-half plane poles and/or zeros.
Still, I doubt that G2 = (1 + s)/(1 - s) is allowed since it's not causal (RHP pole). Still it is technically correct, |G2|=1. Otherwise, I don't see how to get rid of the RHP zero in G without a RHP pole in G2.* Note that non-causal digital filters are used sometimes.

But it's been decades since I did this sort of filter design problem. It's not the sort of thing you'll ever see much in practice, IMO.

I think the unit circle comment relates to discrete time systems where z outside the unit circle is equivalent to s in the RHP.

edit: * Oops! confused the first time

Last edited:
That means that it's impossible to derive the all-pass & min-phase functions from G ?!

Hidd said:
That means that it's impossible to derive the all-pass & min-phase functions from G ?!
Oops, sorry I got some of my signs confused. Let's start over.
$$G(s) = \frac{(1-s)}{(2 + 10s)} = \frac{(1-s)}{(1+s)} ⋅ \frac{(1+s)}{(2 + 10s)}$$

Note that all-pass filter are never minimum phase, they can't be. You must have a RHP zero to cancel the magnitude response of the LHP pole (or the other way around if you don't care about causality & stability due to RHP poles).

berkeman and Hidd
Thnak you DaveE for your help

berkeman and DaveE

## What is decomposition minimal phase?

Decomposition minimal phase is a method used in signal processing to break down a system into smaller components that have minimal phase characteristics. This allows for easier analysis and manipulation of the system.

## What is the difference between minimal phase and all pass systems?

The main difference between minimal phase and all pass systems is that minimal phase systems have a unique impulse response, while all pass systems have a non-unique impulse response. This means that minimal phase systems have a one-to-one relationship between their input and output signals, while all pass systems do not.

## How are decomposition minimal phase and all pass related?

Decomposition minimal phase and all pass are related in that they both involve breaking down a system into smaller components. However, decomposition minimal phase focuses on finding the minimal phase components, while all pass focuses on finding the all pass components.

## What are the applications of decomposition minimal phase and all pass?

Decomposition minimal phase and all pass have various applications in signal processing, such as in audio and image processing, system identification, and control systems. They are also used in the design and analysis of filters and equalizers.

## What are the limitations of decomposition minimal phase and all pass?

One limitation of decomposition minimal phase and all pass is that they assume the system is linear and time-invariant. This may not always be the case in real-world systems. Additionally, these methods may not be suitable for systems with non-minimum phase components or systems with complex dynamics.

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