15 votes
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If the convolution of two signals is a unit impulse, what does this tell us?

It tells us that the systems are inverses of each other. The DFT of $$h_1[n]*h_2[n]= \delta[n]$$ is $$H_1[k] \cdot H_2[k] = 1 $$ so we get $$H_2[k] = \frac{1}{H_1[k]}, H_1[k] = \frac{1}{H_2[k]}$$ In ...
Hilmar's user avatar
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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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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
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7 votes
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why exponential term neglected in equation?

The magnitude of that complex exponential is 1. Recall from complex algebra: any complex number can be expressed as $z = r e^{j \phi}$ where $|z|=r$ is its magnitude and $\arg z = \phi$ is the ...
Atul Ingle's user avatar
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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

Why can adaptive IIR filters result in unstable solutions?

Although what @Fat32 wrote is correct, I think the potential instability of IIR filters is not the main reason for the instability of an adaptive IIR filter. After all, we can calculate the poles in ...
Hooman's user avatar
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6 votes
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Allan Variance vs Autocorrelation - Advantages

My current work involves the design details of atomic clocks where we use the Allan Variance and Allan Deviation (ADEV) extensively. The primary point is that it can be used for non-stationary ...
Dan Boschen's user avatar
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5 votes
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causality of the system $y[n] = x(2n)$

No it does not satisfy the condition. Simply take an example: $$n = 1 \implies y[1] = x[2]$$ Hence the output value at the present time $n=1$ depends on a future value of the input at time $n=2$. ...
Fat32's user avatar
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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
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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
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Can a Fourier Transform exist even if the j$\omega$ axis is not in the Region of Convergence in it's Laplace Transform

You're right that the Laplace transform is not more general than the Fourier transform. They are just different. There are several (theoretically) important functions for which the Laplace transform ...
Matt L.'s user avatar
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5 votes
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Why can adaptive IIR filters result in unstable solutions?

The IIR filter doesn't have to be unstable, but it has the potential of being so; unlike the FIR case which doesn't have even the potential. One reason for the (potential) unstability of an IIR (...
Fat32's user avatar
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Oscillations in a first-order discrete-time linear systems

For continuous-time systems, a pole at location $s_0=\sigma_0+j\omega_0$ will create a time-domain contribution of the form $$e^{s_0t}=e^{\sigma_0t}e^{j\omega_0t}\tag{1}$$ which is a damped ...
Matt L.'s user avatar
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Stability of a system

For BIBO stability in the case of discrete time, there is a necessary and sufficient condition given by $\sum |h[n]| < \infty$ that is if the impulse response is absolute summable then the system ...
Arka Sadhu's user avatar
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
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4 votes
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Does "improper" imply that a system cannot be stable and causal?

An improper system cannot be causal and stable. If the order of the numerator is greater than the order of the denominator, you'll always have at least one pole at infinity. Consequently, not all ...
Matt L.'s user avatar
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How to conclude LTI, causality and BIBO stability of a system represented by a differential equation?

I take your equation: $\frac{d^2y}{dt^2}-3\frac{dy}{dt}+2y(t)=x(t)$ Laplace transform will be: $ s^2Y(S) - 3sY(S) +2Y(s) = X(s) $ Now I can find transfer function: $H(s) \triangleq \frac{Y(s)}{X(s)}$...
Andrea's user avatar
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Realization of a filter based on its transfer function

A transfer function is called realizable if it can be implemented by a causal and stable system. The given frequency response is continuous and doesn't have any impulses, so the corresponding system ...
Matt L.'s user avatar
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4 votes

Relation between order and stability in IIR filter

For direct implementation (without cascading), 6th order is already too high. Let's look at the 6th order low-pass Butterworth filter with sample rate 24000 Hz and cut-off requency 110 Hz: ...
igorinov's user avatar
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4 votes

Relation between causality and stability?

They are independent of each other. Continuous systems: For stability, the ROC (region of convergence) must include the jw-axis of the s-plane. Causal systems have a ROC which is a right-sided plane,...
Axel Mancino's user avatar
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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4 votes

Stabilizing the inverse transform of a system

Like you mentionned, you cannot cancel a right-half-plane zero (or a zero outside the unit circle) by placing a pole on it. A unstable pole in your compensator will make the command of your controller ...
Ben's user avatar
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4 votes
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Necessary Conditions for stability in z domain?

The one and only condition for BIBO stability of a 1D discrete-time system, in the z-domain, is that its transfer functions's ROC (region of convergence) should include the unit circle : $|z| =1$. ...
Fat32's user avatar
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Can I set a constraint on the first tap of an FIR filter such that its inverse is stable?

The Invertible FIR Filter A constraint based on the first coefficient alone is developed as follows: From Cauchy's argument principle any FIR filter that meets the following constraint will be ...
Dan Boschen's user avatar
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4 votes
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Finding if a system is stable

since the impulse response is $h[n] = \delta[ n - 1] + u [n+2]$ It's not. Your system isn't LTI so it doesn't have an impulse response (in the LTI sense) and you can't use an LTI stability criteria. ...
Hilmar's user avatar
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3 votes
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BIBO stability of $y(t)=\int_{-\infty}^{t}{x(\tau)d\tau}$

Let us try with another hints: could you imagine a bounded input signal which could result in a non bounded output? general suggestion whenever analyzing a system: try a few "simple to compute" ...
Laurent Duval's user avatar
3 votes

BIBO stability of $y(t)=\int_{-\infty}^{t}{x(\tau)d\tau}$

Since this is most likely homework, here is a hint. Write the integral you have displayed in the form $\int_{-\infty}^\infty x(\tau)h(t-\tau) d\tau$ where you get to choose what the function $h(\cdot)...
Dilip Sarwate's 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 ...
robert bristow-johnson's user avatar
3 votes

Discrete Time Fourier Transform (DTFT) for an unstable system (Ideal Low Pass Filter)

It is generally not true that the relation $Y(\omega)=X(\omega)H(\omega)$ is independent of the system's stability. For systems with a rational transfer function (i.e., systems that can be described ...
Matt L.'s user avatar
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3 votes

Absolute Integrable Sinc function

The question has an answer on Math SE. The argument goes as follows: \begin{eqnarray} \int_{-\infty}^\infty \left| \frac{\sin t}{\pi t} \right|dt &=& \frac{1}{\pi}\sum_{n=-\infty}^\infty \...
Atul Ingle's user avatar
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