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Accepted

### What is the difference between natural response and zero input response?

First it's important to realize that many authors use the terms zero-input response and natural response as synonyms. This convention is used in the corresponding wikipedia article, and for instance ...
• 90.4k
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### Who first understood the importance of poles?

If you consider poles of an integral transform domain to be important to the solution of differential equations: (as usual,) Euler did it first, 1753. One "importance" of poles is that they're part of ...
• 31.1k
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### Why don't unit circle poles lead to infinite amplitude response for Butterworth lowpass?

You've made an understandable mistake. You are probably looking at this picture: That is not the unit circle, and it isn't even in the $z$ domain. What you are looking at is the locations of the ...
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### Is the inverse of a causal system also causal?

It's not sufficient to only consider causality, you also need to check whether the inverse system is stable, otherwise it can't be implemented. If $G(z)$ has zeros on the unit circle, it cannot be ...
• 90.4k
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### Pole-zeros of a real-valued causal FIR system

Note the difference between the zeros at $0.3 \pi$ and at $0.8 \pi$. The first one is clearly a zero crossing, much like $abs(x)$ at $x=0$. At $\theta = 0.8 \pi$, however, the curve is tangent to ...
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• 90.4k
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### Poles and zeros of a transfer function

The "poles-inside-unit-circle" stability criterion only applies to causal systems. Your system is not causal because it uses one sample from the future owing to the $z$ term. The general technique to ...
• 4,134
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### Is this system causal or not?

Note that in this case you can see that the system is causal only from the given implementation. It's important to understand that you can't see it from the difference equation (if no initial ...
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### What are poles and zeros?

To add to the other good answers, I found the following graphics helpful in gaining a better intuitive understanding, more specifically to the poles and zeros of transfer functions. (UPDATE: I also ...
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### Why does a pole close to the unit circle result in an enhanced Q-factor?

Suppose some $H(q)=\frac{A(q)}{B(q)}$ where q is some complex variable and $A,B$ are functions of $q$. Whether in the s or z planes, to evaluate the magnitude of $H(q)$ at some $q$, you evalaute and ...
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### determining type of filter given its pole zero plot

In this answer I'll try to show you how to qualitatively evaluate a given pole-zero plot by just looking at it. Of course, this method has its limits, but for relatively simple pole-zero plots you can ...
• 90.4k
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### determining type of filter given its pole zero plot

You'd have to figure out the frequency response of the filter. Here are two methods. I prefer Method 2 because it's quick and dirty, and you don't really care about the exact gain values in the ...
• 4,134

### A question about the meaning of pole in time domain

Let $H(s)$ be a transfer function of the form $$H(s) = \frac{1}{s-p}$$ where $p$, which is a pole of $H(s)$, can be written as a complex number $a+jb$. Taking the inverse Laplace transform of $H(s)$ ...
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### Conjugate reciprocal pairs of zeros and poles in FIR design

Look up the complex conjugate root theorem which states that: If all the coefficients of a polynomial are real then its roots are either real or if there is a complex root, then its conjugate is ...
• 4,295

### Transposed Direct Form II VS Direct Form II IIR filters?

Compared to the Direct Form I, Direct Form-II has pros and cons. The advantage of DF-II is its more efficient usage of the delay lines. Although both use separate all-pole and all-zero sections, the ...
• 4,295
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### For a system to be causal, number of finite zeros <= number of finite poles. Why?

If the number of finite zeros is not greater than the number of finite poles then the transfer function is proper, i.e., the degree of the numerator polynomial is not greater than the degree of the ...
• 90.4k