# 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^-...
1 vote
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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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### 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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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 $\... 7 votes 2 answers 1k views ### If the convolution of two signals is a unit impulse, what does this tell us? I have two discrete-time LTI systems whose transfer functions satisfy$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 ... 1 vote 2 answers 173 views ### 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)... 1 vote 0 answers 268 views ### 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 ... 2 votes 1 answer 77 views ### 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 ... 0 votes 1 answer 72 views ### 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: ... 0 votes 1 answer 48 views ### 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. ... 2 votes 1 answer 131 views ### 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(... 3 votes 1 answer 171 views ### 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] ... 1 vote 1 answer 53 views ### 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 ... 2 votes 1 answer 71 views ### 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 ... 0 votes 2 answers 39 views ### 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 ... 0 votes 0 answers 34 views ### 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: $$y(m,n)=\sum_{i=-1}^{i=1}\sum_{... 0 votes 0 answers 43 views ### linear time-invariant system output question? Subtract the output of a linear time-invariant system whose shock response is h \left[ n \right] = \left( \frac{1}{4} \right)^{n} \left( u \left[ n \right] - u \left[ n - 2 \right] \right), for ... 1 vote 2 answers 284 views ### Calculating transfer function of a linear time varying system? If we excite a LTI system with the Dirac delta \delta(t), the system outputs the impulse response h(t). For a LTI system, it doesn't matter when we excite the system with the Dirac delta, we will ... 0 votes 1 answer 34 views ### Max input of a system given it's transfer function and an assumed step change (beginner) I have an exercise that gives me the following transfer function$$ \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 ... 3 votes 5 answers 764 views ### 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 ... 0 votes 0 answers 25 views ### 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? 1 vote 1 answer 110 views ### 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 ... 1 vote 1 answer 87 views ### 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... 0 votes 0 answers 62 views ### LLTV Systems breakdown(2) In this question I proved that for a linear linearly time varying systemS$such that if$t_{2} = t_{1}+t_{0}$then$h(t,t_{2}) = h(t,t_{1})+h(t,t_{0})\rightarrowh(t,t_{0}) = g(t_{0})h(t) $where$...
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 ...