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# 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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### In EKF should Kalman Gain converge to a specific value?

I have implemented an EKF using the standard predict and update equations in order to perform state estimation of a vehicle with multiple sensors. The model has process covariance matrix $Q$ and each ...
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### Why is $y(n)=Ax(n)+B$ not relaxed if $B\neq 0$?

I'm wondering why $y(n)=Ax(n)+B$ not relaxed if $B\neq 0$? The definition of a relaxed system as I've understood is that $y(n)$ does not depend on any $y$-values. So what is the definition of a ...
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### Inverse of a causal and stable system

Consider a discrete-time causal and stable LTI system $S_1$​. The inverse system $S_2$​ is defined as the system that takes the output of $S_1$​ as its input and provides the input of $S_1$​ as its ...
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### How does a time delay affect the difference equation of a LTI-system?

I'm right now working on my digital signal processing homework and among the exercises is a tough LTI-system in a canonical-like form, however right before the output is a time delay. Suppose the ...
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### DT System with input initially at rest

This is the very basics of Signals but my lecturer has provided barely any notes and I cannot find answers online (not just to this, but most of signals for some reason) Dealing with DT system: y[n]=y[...
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### is this system linear? causal?

$$y(t) = \int_{t_0}^t u(\tau)\, d\tau + y(t_0)$$ I have trouble determining whether this system is causal or not and linear or not. I think this system is causal because it integrates input signals ...
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### Does the impulse response of every stable system have finite energy?

I need to show that the impulse response of every stable system has finite energy. I have trouble solving this problem. Since impulse input signal is unbounded, I can't apply BIBO stability of system ...
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### If the input of the system depends on the future outputs then is the system non-causal?

While I'm aware of the fact that causality implies that the present output is only dependent on present and past inputs, something that is bugging me is what if the input is dependent on future ...
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### truncating a system and plotting it on bode

Let system G(s) be: $$G(s)=\sum_{i=0}^{10}\frac{(-1)^i}{(2i+1)^2}\frac{\omega_i}{s^2+2\zeta_i \omega_i s+\omega_i ^2}$$ $$\omega_i=\frac{(2i+1)\pi}{T}\, T=1\, \zeta_i=0.2$$ Its impulse response is an ...
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### Is heat equation considered as LTI system?

The heat equation $$\frac{\partial u}{\partial t} = \alpha \nabla^2 u + f(x,t) \tag 1$$ Here, $u$ represents the temperature field, $\alpha$ is the thermal diffusivity of the material, $\nabla^2$ is ...
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### Is $y(t) = y(t-4)+x(t-4)$ time invariant or not?

I want to check the time invariability of this recursively defined function $$y(t) = y(t-4)+x(t-4)$$ We can check time invariability of functions expressed in terms of x(t), but I couldn't find ...
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### Doubt on LTI systems (Zero input-Zero Output)

So, I have a problem where the input and output of a system are given : \begin{align} x(t) &= \sin(10t) \\ y(t) &=5\cos(10t+\frac{\pi}{6}) \end{align} So, I need to determine whether this ...
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### How to find impulse response for the given system?

How can I find the impulse response for the following system in time domain? I actually would like to find my mistake in my attempt. Below is what I have tried according to the answer given for this ...
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### Correlation between two signals

I am having an input signal x[n]=[1 2 3 4 5 6] and another signal y[n]=[2 3 4]. In order to check whether y[n] is present in the signal x[n],I did cross correlation and the result obtained was [ 0 0 0 ...
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### How to design a system that stretches input by factor of 2?

In general, How do we design a system that takes input x(t) and outputs x(t/2)? I know that if system outputs x(t/2) then the frequency domain of output signal has the form X(2f). But what should be ...
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### Linearity of a system to biomedical applications

Suppose $x(t)$ is temperature and $y(t)$ is sweat. If the following equation describes the system, answer if is it linear or not. $y(t) = [H(x)](t)$ . I would say that it isn't linear because we dont ...
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### Interpreting eigenvalues of non-normalized covariance matrix of time-series measurements

Cross-posted from physics stackexchange Summary: Eigenvalues of a "non-normalized" covariance matrix of time-series measurements from a linear system have units of Action (energy * time). ...
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### Linear System: Symmetric Under Time Reversal?

In class, my professor mentioned that "Linear systems must be symmetric under time reversal" in an off-handed way and did not clarify further. I assume this is true, but I'm not sure how one ...
4 votes
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### What math should I study to really understand signal processing?

I am reading an elementary book on signal processing - "Signals and Systems". It never struck until recently, the math involving signal processing seemingly has a lot more depth than the ...
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### Output of time-continuous linear system with a phase shifter as impulse response

I stumbled upon a false/true statement which goes: A time-continuous linear system, whose impulse response $c(t) = \frac{1}{\pi t}$ has a pole at the origin, always produces an output signal $y(t)$ ...
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### How to find time-varying impulse responses?

Given is a system that can be described as $y(t) = x(t)\cdot \sigma(t)$ with $\sigma(t) = \left\{\begin{array}{ll} 1, & t \geq 0 \\ 0, & t<0\end{array}\right. .$ The output of a ...
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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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### 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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