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13 votes
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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 ...
Matt L.'s user avatar
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12 votes
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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 ...
Marcus Müller's user avatar
12 votes
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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 ...
TimWescott's user avatar
  • 12.8k
11 votes

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 ...
Matt L.'s user avatar
  • 90.4k
11 votes
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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 ...
Juancho's user avatar
  • 5,026
10 votes

Filter design with zero - pole placement method

Here let me show you a simple procedure very similar to pole zero placement which will be helpful for your notch filter design. First, lets analyse the frequency response of a single zero and let $$ ...
Fat32's user avatar
  • 28.3k
10 votes
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Position of poles and Stability in $z$ domain

Short Answer: All the poles of a causal (right-sided) and stable LTI system must be inside the unit circle whereas all the poles of an acausal (left-sided) and stable LTI system must be outside the ...
Fat32's user avatar
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8 votes
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Why zeroes near the unit circle cause a dip in frequency response, while poles cause a peak?

To answer this you need to understand what is a pole and what is a zero of a transfer function. Let's look at a simple 2 poles 2 zeros filter (also called biquad filter) transfer function : $$ H(z) = ...
Florent's user avatar
  • 754
8 votes

Single pole IIR filter, fixed point design

One thing to consider when implementing an IIR filter, whatever the order, is quantization and limit cycles. Let me show you with a quick example with your original filter $y[n] = a*x[n]+(1-a)*y[n-1]...
Ben's user avatar
  • 3,777
7 votes
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Nyquist plot interpretation when curve hits the origin

First to clear up the OP's misunderstanding: the Nyquist Stability Criteria involves clockwise encirclements of -1, not the origin, and this would be the polar plot for the open-loop gain specifically....
Dan Boschen's user avatar
7 votes
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Stability of system with poles inside unit circle - conflict with differential equation

What you are missing is that this is about a discrete-time system, because we're talking about poles and zeros in the complex $z$-plane and about poles inside or outside the unit circle. So there is ...
Matt L.'s user avatar
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6 votes
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Laplace Transform of Cosine, Poles and Mapping to Frequency Domain

You're comparing the transforms of two different functions. You consider the Fourier transform of the function $x_1(t)=\cos(\omega_0 t)$, but you took the Laplace transform of the function $x_2(t)=\...
Matt L.'s user avatar
  • 90.4k
6 votes
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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 ...
Atul Ingle's user avatar
  • 4,134
6 votes
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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 ...
Matt L.'s user avatar
  • 90.4k
5 votes

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 ...
Dan Boschen's user avatar
5 votes
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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 ...
A_A's user avatar
  • 10.7k
5 votes

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 ...
Matt L.'s user avatar
  • 90.4k
5 votes
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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 ...
Atul Ingle's user avatar
  • 4,134
5 votes

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 $H(s)$ ...
anpar's user avatar
  • 957
5 votes
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BIBO Stability for system with no poles

Why does a system with no poles have a finite support? If a system doesn't have finite poles, then its transfer function is of the form: $$H(z) = \frac{Y(z)}{X(z)} = a_Nz^N+a_{N-1}z^{N-1}+...+a_1z+...
Tendero's user avatar
  • 5,020
5 votes
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Poles and zeros map

Note that the pole locations would be the same for all $4$ classic types of frequency-selective filters (low pass, high pass, band pass, band stop). It's the location of the zeros that determines ...
Matt L.'s user avatar
  • 90.4k
5 votes

Stability of system with poles inside unit circle - conflict with differential equation

You're conflating the discrete-time definition of a system with the continuous-time representation of a system. Your discrete-time $$Y(z)\cdot\big(z-\frac{1}{2}\big)=X(z)\cdot z$$ does not ...
Peter K.'s user avatar
  • 25.8k
5 votes

How to realize Poles and zeros at infinity??especially through transfer function?

It's actually quite straightforward: positive powers of $s$ (or, in discrete-time, $z$), correspond to poles at infinity. Negative powers give you zeros at infinity. Let's look at some examples. In ...
Matt L.'s user avatar
  • 90.4k
5 votes
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Poles and zeros form of a transfer function

The two expressions are generally not identical. In the special case $K=L$ they're equivalent, otherwise they differ by a (positive or negative) power of $z$: $$\frac {\displaystyle\prod_{k=1}^K (1 - \...
Matt L.'s user avatar
  • 90.4k
4 votes

Nyquist plot interpretation when curve hits the origin

The problem here is that for the given functions, the usual Nyquist contour (see figure below) results in Nyquist plots from which no decision can be made about stability. Nyquist contour ($M\to\...
Matt L.'s user avatar
  • 90.4k
4 votes
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What's the Q of a pole at the origin of the s-plane?

Note that the formula given in your question is valid for a system with a complex-conjugate pole pair $p$ and $p^*$ with $\text{Re}(p)\neq 0$. As you've correctly pointed out, if for $|p|>0$ the ...
Matt L.'s user avatar
  • 90.4k
4 votes

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 ...
msm's user avatar
  • 4,295
4 votes

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 ...
msm's user avatar
  • 4,295
4 votes
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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 ...
Matt L.'s user avatar
  • 90.4k
4 votes
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Difference between repeated poles and distinct poles?

Repeated poles simply means there are more than one pole at the same location. If a pole is not repeated then it is a Distinct Pole. Consider the simple case of a cascade of two integrators (in s) ...
Dan Boschen's user avatar

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