Questions tagged [linear-systems]
A linear system operates on inputs with only linear operators so the response to a complex input can be analysed as the sum of the response to a set of simpler inputs. This mathematical property makes the analysis of linear systems much simpler than non-linear systems where this summation or superposition does not hold. Linear systems are generally further classified as time invariant, meaning that there characteristics do not change over time.
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Continuous-time convolution of signals with negative amplitudes
While preparing for a mid-term exam, I encountered negative amplitudes for the first time while convolving two signals.
I've already solved the problem, but my result and results from others conflict ...
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Why lag compensator is preferred over PI for sinusoidal reference?
In this post, LJSilver mentioned that a PI compensator is not appropriate for a constantly changing reference, such as a sinusoidal waveform. In this scenario, a lag compensator is considered the ...
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how does steady-state error decrease as the pole of the compensator moves closer to the origin?
The steady-state error improves when the pole moves closer to the origin, as seen in lag or integral compensators with step input. Is there an intuitive explanation for this phenomenon?
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Function Transfer of Block Diagram Discrete System
I need to find out the transfer function in Z from this diagram, how can I extract this information from this diagram
Obs: Doing the math, my H_z gave, H_z = 1 + 0.5z^-1 + 2z^-2 + 2z^-3 + 0.5z^-4 + z^-...
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Is a PI-controller considered an LTI system?
Is a PI-controller considered an LTI system?
Intuitively it seems that the integral part would break the time-invariant requirement requirement, because the output depends on how wound up the ...
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Are complex exponentials the only eigenfunction for arbitrary LTI systems?
After reading a few posts, like this. I know that arbitrary LTI systems always have complex exponential eigenfunctions. And that for specific LTI systems you can also have other types of ...
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System Identification Using Sinusoidal Inputs
I have a system I would like to model using experimental data. I input several sinusoidal signal and measured the outputs. I can vary the frequency and the amplitude of the input signal:
Input 1: $A \...
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What is the adjoint of a linear operator and why is it useful?
The concept of linear operators and their adjoints arises frequently in some corners of signal processing, but is not particularly well documented, at least from a signal processing perspective (you ...
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random signals through LTI systems, why are these two signals joint wide sense stationary?
I’m trying to solve this problem but I don’t understand an assumption the solution makes:
The question:
let $\hat{W}$ be the best linear approximation of $W_t$ out of $Y_t$, find $\text{CoV}(W_4, \...
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Equipment to test my theory on time varying systems
How can I make a linearly time varying system to test my theory?
Can I ask for a PID microcontroller available from my university such that $K$ is gradually increased from 0 to $c$
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Sine as input to an LTI system
Everywhere in theory (books , online) there is this statement "For sinusoidal inputs, any LTI has a sinusoidal output with the gain of $|H(s)|$, the same frequency, and a phase shift equal to $\...
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If the convolution of two signals is a unit impulse, what does this tell us?
I have two discrete-time LTI systems whose transfer functions satisfy $h_1[n] * h_2[n]= \delta[n]$. We also know that system 1 is causal and stable. Does $h_1[n] * h_2[n]= \delta[n]$ tell us anything ...
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Why does convolution give the output of a passing a signal through a filter?
I have a rudimentary understanding of Convolution, the Convolution Theorem and why the output z(t) of an LTI system can be found using the convolution of input signal x(t) and the impulse response h(t)...
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How to recover the LTI system step response by the known output and input signals?
Having the input signal as a step-like pulse and the output as its distorted version after passing through the system:
is it possible to somehow recover the step response of the system?
In the Figure ...
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Why do they say that complex exponentials are eigenfunctions of LTI systems, when there are still transient responses?
Let $$\dot{x} = Ax+Bu$$ $$y = Cx + Du$$ be a linear ODE with $x(0)=0$. Here, I am assume $A$ is invertible.
As you can see, the relation $$H:u(.) \mapsto y(.),$$ where $(u(.),y(.))$ is a solution to ...
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How to find system output by its step response?
Inspired by this post, I tried to reproduce the procedure described in the answer in Python considering rectangular pulse:
...
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Magnitude spectrum of LTI system output signal
First year student so please excuse my lack of knowledge.
As i understand i need to use convolution which is:
$$ y^{out}(t) = u^{in}(t) * h(t) $$
Or maybe my thoughts are wrong so please correct me.
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State space transformation
I have some governing equations of the form:
$$\begin{align}
\ddot \theta(t) &= \frac{MgL + mgl}{J} \theta(t) + \frac B J \dot x(t) - \frac \alpha J V + \frac {mg}{J} d - \frac{c_1}{J} \dot \theta(...
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How can I show that an LTI system can be expressed as a difference equation?
I'm in the process of re-learning DSP (not a subject I've visited since University) and in quite a few resources I see this general form of a DT-LTI difference equation:
$$y[n] + a_1y[n-1] + a_2y[n-2] ...
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Effect of BIBO-Instability on the frequency response of a ideal LPF
I recently came across this post stating that the continuous ideal LPF is BIBO-unstable since the impulse response is not absolutely integrable, and this post stating some examples.
I have been trying ...
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LTI system tradeoff between gain, bandwidth, and delay
For first-order LTI systems, the gain-bandwidth constant is often discussed. I've seen the claim that in general, gain and bandwidth don't directly trade off with each other as much as delay. For an ...
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Routh's stability condition
Assume we have a LTI system which has poles in the half left plane of the s domain.
Before I learnt Routh's stability condition I had imagined that this was enough to decide whether a LTI system was ...
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Verifying Linearity and Shift Invariance Under Summation
I am having some trouble working through verifying linearity and shift invariance when the transformation is under a summation.
The given transformation is as follows:
$$
y(m,n)=\sum_{i=-1}^{i=1}\sum_{...
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linear time-invariant system output question?
Subtract the output of a linear time-invariant system whose shock response is $h \left[ n \right] = \left( \frac{1}{4} \right)^{n} \left( u \left[ n \right] - u \left[ n - 2 \right] \right)$, for ...
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Calculating transfer function of a linear time varying system?
If we excite a LTI system with the Dirac delta $\delta(t)$, the system outputs the impulse response $h(t)$. For a LTI system, it doesn't matter when we excite the system with the Dirac delta, we will ...
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Max input of a system given it's transfer function and an assumed step change (beginner)
I have an exercise that gives me the following transfer function
$$
\frac{0.5}{s+0.5}
$$
and an assumed step change in the target of 20
I am asked to calculate the maximum input for the assumed step ...
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Non-causality of fractional delays
Given a physical system (e.g., loudspeaker and microphone) with DA and AD converters. Playing a single pulse from the loudspeaker, I will most likely receive the pulse at the microphone with a ...
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Proof for LTI Systems [duplicate]
Suppose we have a system $S$.If I know the impulse response $h(t)$ I can predict the output for any input $x(t)$ if a system is linear and time invariant.However how do we prove that statement?
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Impulse response of linear time-varying system
I am confused about linear time-varying system. For a time varying system, the output is given by
\begin{align}
y(t)=\int x(\tau) h_{\tau}(t) d\tau,
\end{align}
where $ h_{\tau}(t)$ is the output of ...
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Power of filtered Bernoulli process
I have some doubt about this exercise.
The Bernoulli random process $X(n)$ with means $p=0.5$ is sent in input to a LTI system with impulse response $h(n)= \cos(\frac{\pi n}{3}) R_3(n+1)$ , where $$...
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LLTV Systems breakdown(2)
In this question I proved that for a linear linearly time varying system $S$ such that if $t_{2} = t_{1}+t_{0}$ then $h(t,t_{2}) = h(t,t_{1})+h(t,t_{0})\rightarrow$ $h(t,t_{0}) = g(t_{0})h(t) $ where $...
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LLTV Systems linearity breakdown
Suppose we have a linear linearly time varying system $S$ such that the output depends on the input + the time at which the system becomes excited so $S(t,t_{0}) \rightarrow g(t_{0})S(t)$ and it is ...
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Linear linearly time varying systems Laplace transform
Suppose that for a system $S$ if we have $t_{2} = t_{1}+t_{0}\rightarrow h(t,t_{2}) =h(t,t_{1})+h(t,t_{0}) $ .Then if we take the double Laplace transform to$t,t_{2}$ we will get:
$$L_{t_{2}}(L_{t}(h(...
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Frequency response of an LTI system described by the diagram below
I am trying to solve the below problem:
To begin with the frequency response of the ideal low pass filter with $\omega_c= \pi/4$ is given by $$
H(\omega) =
\begin{cases}
1 & \text{if -$\pi/4$ $...
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is $y[n] = y[n - 4] + x[n - 4]$ time variant or invariant?
I am confused about a solution because there is feedback.
Let's introduce a delay parameter $k$ and rewrite the system equation as:
$$ y \left[ n \right] = y \left[ n - 4 - k \right] + x \left[ n - 4 -...
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Real & Imaginary part of the frequency response of LTI system
I am looking at the H(f) which is the frequency response of an LTI system. What kind of relationship should I expect between the real and imaginary part of this frequency response? Here is an example ...
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Feedback stabilizes unstable systems?
Can feedback stabilize a unstable system?I guess so since we may get rid of any poles in the positive complex plane by feedback but I am not sure.
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How can an impulse generate an output in the past time frame?
I am studying signal processing and currently doing signals & systems. While going through convolution and especially the impulse response , there are problems where LTI systems wherein the input ...
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Linear Linearly Time Varying Systems properties?
Yesterday I asked a question about LTV Systems and how the impulse response h(t,t0) is found.
I was thinking about it and realized there is a subsclass of LTV systems which I call Linear Linearly Time ...
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Impulse response of an LTI system
I want to find the impulse response of the linear time invariant system given by
$y''(t)+4y'(t)+3y(t)=x'(t)+2x(t)$
using time domain analysis.
As solved in the textbook "Continuous and Discrete ...
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Non-linear external effect in Kalman filter
Let's say I have a Kalman filter with this simple state model:
$$\begin{pmatrix} x^0_{k+1}\\ x^1_{k+1}\\ \end{pmatrix} = \begin{pmatrix} 1 & \Delta t\\ 0 & 1\\ \end{pmatrix} \begin{pmatrix} x^...
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How are LTI systems related to Toeplitz matrices?
I am having trouble understanding why the system matrix of an LTI system is Toeplitz. I am following an Edx online course by Professor Richard Baraniuk of Rice University, named discrete-time signals ...
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A response of an unstable system
I'm dealing with the following question :
The system realizes an accumulator, therefore its impulse response (h[n]) is just the unit step function multiplied by some factor . In the next subsections ...
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Initial conditions of state-space realization
In a discrete-time difference equation (ARX model), the output $y[k]$ is dependent on its past values and inputs, expressed as:
$y[k] = -a_1 y[k-1] - ... - a_N y[k-N] + b_1 u[k-1] + ... + b_M u[k-M]$, ...
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System Characterization: multiply imaginary component by scalar
I tried solving a test question that frankly stumped me. If you could explain to me the solution I’d be really grateful.
Given $a, b \in \mathcal{R}$ the system $R_v$ takes the complex input $a + bj$ ...
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Matrix form of Overlap-add
We know overlap-add of a en-framed signal can be done easily by following code
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Real coefficient linear phase FIR filter design based on least square
I want to build up linear phase filter by least square approach.
I have formulated a least square problem similar to Matt's blog. To approximate the desired complex response $F \times 1 $ vector $d$ ...
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Show that decomposition does not hold for non-linear system
The solution to an inhomogeneous differential equation can be split up into homogeneous solution and a particular solution (forced response).
Another way to split up the solution to an inhomogeneous ...
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Invertible system for the eigenfunction $x[n]=e^{j\omega n}$
I was doing some calculations in my LTI systems course and I stumbled in an interesting question I wasn't really sure how to answer so I'd appreciate any direction or solution you can give me:
I'm ...
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Finding transfer functions from a system of multiple inputs
If I have a system: $$sX(s) = AX(s)+BY(s)+CZ(s)$$ How would I find the transfer functions $\frac{X(s)}{Y(s)}$ and $\frac{X(s)}{Z(s)}$?
Do I simply disregard one of the inputs? I am quite confused and ...
