9 votes
Accepted

Minimum Phase - All Pass Decomposition For Large Linear Phase Filters

Computing the polynomial coefficients from the roots of the polynomial is a potentially ill-conditioned problem. However, it turns out that the order of the roots supplied to the ...
user avatar
  • 80.4k
5 votes

Not able to reach minimum phase using Hilbert transform

More taps. You don't have anywhere near enough taps for a filter that steep. Start large with 8192 or so cut to desired accuracy, if needed Due to the low number of tabs you are seeing the effect of "...
user avatar
  • 32.7k
4 votes
Accepted

Minimum phase FIR method

You made a minimum-phase filter but with a different magnitude response than the original linear phase filter. What you have to do to keep the magnitude the same is to reflect the zeros outside the ...
user avatar
  • 80.4k
4 votes
Accepted

What is the easiest, most straight-forward way to prove this about minimum-phase filters?

The Hilbert transform $\mathcal{H}\left\{f(\omega)\right\}$ with $$f(\omega)=-\frac12\log(1+\omega^2)\tag{1}$$ can be calculated in the following way. First, note that $$\frac{df(\omega)}{d\omega}=-...
user avatar
  • 80.4k
4 votes
Accepted

Hilbert transformer and minimum-phase

as a related aside question i posted this question about minimum-phase filters and the phase-magnitude relationship. let $N$ be the FFT size you will use. (often $N$ is a power of two, but it doesn'...
user avatar
4 votes
Accepted

Getting phase response from magnitude. How to develop and solve this Hilbert transform?

The problem with your example is that there's a zero right on the imaginary axis, making the system not strictly minimum-phase, and inversion with a causal and stable filter is not possible. So let's ...
user avatar
  • 80.4k
3 votes

Prove that the filter is stable, causal and minimum phase

In general, the algebraic expression of a transfer function alone doesn't uniquely describe a single system. For the example in your question with two complex conjugate poles in the left-half plane, ...
user avatar
  • 80.4k
3 votes

Minimum Phase - All Pass Decomposition For Large Linear Phase Filters

This paper describes a method using real cepstrum to calculate the minimum phase signal. I'll show the general idea. Definition of complex cepstrum and real cepstrum The Fourier transform of the ...
user avatar
  • 2,628
3 votes

Minimum Phase - All Pass Decomposition For Large Linear Phase Filters

In IEEE Signal Processing Magazine- Nov, 2003, the article "Factoring very-high-degree polynomials" discusses an accurate root finding algorithm. It describes the Lindsay-Fox algorithm. Here'...
user avatar
  • 2,751
3 votes

Minimum Phase - All Pass Decomposition For Large Linear Phase Filters

There's a chapter in Lyon's Streamlining DSP book on "Designing Nonstandard Filters with Differential Evolution" which combines gradient stochastic descent with a genetic algorithm. The ...
user avatar
  • 34k
3 votes

Phase correction algorithm: minimisation phase error

I commend you for using an intuitive algorithm. However, there are already established algorithms with far better performance. Phase recovery algorithms work by filtering the error signal down to zero....
user avatar
3 votes

Definition of minimum-phase system

one thing about a non-minimum phase system (with a rational transfer function), is that it can be thought of as the series concatenation (or cascade) of a minimum-phase system, having identical ...
user avatar
3 votes
Accepted

Transform minimum phase FIR into linear phase FIR

It is generally impossible to transform a given minimum-phase FIR system into a linear phase FIR system with the same magnitude response. There is one special case for which this is possible, and that ...
user avatar
  • 80.4k
2 votes
Accepted

Given Gain function, how to design a causal, stable and minimum phase IIR filter?

If $H(\omega)=e^{\alpha(\omega)+j\phi(\omega)}$ is a minimum phase frequency response, then the attenuation $\alpha(\omega)$ and the phase $\phi(\omega)$ are related by the following Hilbert transform ...
user avatar
  • 80.4k
2 votes
Accepted

How to prove these two definitions of the minimum phase transfer function are same?

To your second definition it should be added that you only consider causal transfer functions, because it is not difficult to find a smaller phase lag with a non-causal system: A minimum-phase ...
user avatar
  • 80.4k
2 votes

When is the sum of two (parallel) minimum-phase filters also minimum-phase?

Just looking at it I can see it's related to control theory. If you make a contrived open-loop gain $$H(z) = \frac{A(z)D(z)}{B(z)C(z)}$$ and then wrap it with unity-gain feedback, you get $$G(z) = \...
user avatar
  • 8,666
2 votes

Minimum phase All-pass

An LTI system is said to be minimum-phase if the system and its inverse system are causal and stable. That's implying that all poles and zeros must be strictly inside the unit circle. An all-pass ...
user avatar
  • 2,628
2 votes

Homework Help: What does $h[0] = 1$ represent? What is $\ln \big| H(e^{j \omega})\big|$?

The transfer function of a discrete-time minimum phase system described by a linear difference equation with constant coefficients can be written as $$H(z)=h[0]\cdot\frac{\prod_{k=1}^M(1-c_kz^{-1})}{\...
user avatar
  • 80.4k
2 votes
Accepted

On what it means for a system to be minimum phase

The minimum phase system will have the minimum group delay for a given magnitude response. The phase response will have the least excursion over the frequency domain (due to all zeros being inside the ...
user avatar
  • 37.7k
1 vote

inverting unstable zeros

When you invert a transfer function $H(s) = n(s)/d(s)$, all zeros become poles and all poles become zeros since inverting means $H(s)^{-1} = d(s)/n(s)$ where $n(s)$ and $d(s)$ refer to the original ...
user avatar
  • 37.7k
1 vote

On what it means for a system to be minimum phase

Here is another way to think about. Let's look at a transfer function $H(z)$ that has one zero outside the unit circle at a location $z=q$ with $|q|>1$. We can write this as $$H(z)=H_1(z)\cdot (1- ...
user avatar
  • 32.7k
1 vote

Get minimum phase from function

Why is it that reflecting any poles or zeros of a rational function across the unit circle gives a minimum phase system? It doesn't. You are starting with a wrong assumption. Here's an example, it ...
user avatar
  • 32.7k
1 vote

minimum-phase phase via Hilbert transform returned values

I computed and compared the minimum phase HRIR and the original one. This is my final code: ...
user avatar
1 vote

Validity of an argument that two transfer functions are minimum-phase based on their ratio being minimum-phase

two transfer functions HL and HR can each be represented as a minimum-phase filter (MPF) plus a pure delay. That is generally not true and it's easy enough to disprove it by counter example. Let's ...
user avatar
  • 32.7k
1 vote
Accepted

What does nonnegative zero-phase response mean?

What they mean here is that the real-valued amplitude function of the linear phase FIR filter that you provide to the function must be non-negative, because it is interpreted as the desired squared ...
user avatar
  • 80.4k
1 vote

When is the sum of two (parallel) minimum-phase filters also minimum-phase?

I don't think you will have much luck there. Minimum phase means that all the roots of all polynomials $A,B,C,D$ are inside the unit circle. That means that the product of two polynomials will also ...
user avatar
  • 32.7k
1 vote
Accepted

Algorithm to Count Zeros Outside Unit Circle for FIR Filter

Here is one answer, if someone can improve on this I will select it as the "right" answer (also comments very welcome on obvious flaws with this approach): Given Cauchy's argument principle, the ...
user avatar
  • 37.7k
1 vote

How to create matched "minimum phase" for a system of parallel FIRs?

The Hilbert "transform" or relation define one phase response for a given magnitude response, so you can't get both matched and minimum phase in your case. Regards.
user avatar
1 vote
Accepted

Finding the transfer function of a discrete signal described by two equations

HINT: (because it's a homework problem) Apply the $\mathcal{Z}$-transform to both equations. Use the first equation to express $Q(z)$ in terms of $X(z)$, and plug that into the second equation to ...
user avatar
  • 80.4k
1 vote
Accepted

Under what conditions do the phase margin and Nyquist criteria give the same results?

As you have pointed out, the nyquist stability criterion is more general, moreover, is the only stability criterion. Nevertheless, the Bode plot give us the exact same information that the nyquist ...
user avatar
  • 268

Only top scored, non community-wiki answers of a minimum length are eligible