# 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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### 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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### 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 ...
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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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### Physical Meaning of Negative Group Delay for causal LTI systems

I have implemented in Matlab (with minor variations) the example 5.1.2 "Illustration of Effects of Group Delay and Attenuation" I found in Alan Oppenheim's Discrete-Time Signal Processing ...
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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 $y(t)$ ...
1 vote
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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 ...
1 vote
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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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### Intuition behind commutativity of convolution in LTI systems

Why is convolution commutative, as it seems to treat two signals in a different way in an LTI system? If you imagine $y[n] = x[n] \star h[n]$ with $x[n]$ being an input signal and $h[n]$ being the ...
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### Given the input and the output, how to determine the impulse response?

I would like to find the impulse response, $h[n]$, of an LTI system given the input $$x[n] = [1,-3,2]$$ and the output $$y[n] = [1,-1,-4,4]$$ I know that $y[t]=x[t]*h[t]$, but I am having hard ...
1 vote
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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 ...
110 views

### 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 ...
1 vote
63 views

We know overlap-add of a en-framed signal can be done easily by following code ...
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### How to solve Hilbert Transform with empirical discrete data in frequency domain?, from zero to infinity

I have a filter/LTI system frequency response in form of list of values in the frequency domain. I want to get the phase curve/data from magnitude data. Input data can have either linear spaced points ...
72 views

### 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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### Why doesn't the convolution of the impulse response match the system's output?

If you define an LTI system sys in scipy, you may conveniently feed an input x to it to get ...
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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: ...
54 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. ...
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### Why can adaptive IIR filters result in unstable solutions?

For adaptive filtering, both finite and infinite impulse response (FIR/IIR) filters can be utilized. As an advantage of FIR filters in this context, guaranteed stability is often mentioned, while IIR ...
1 vote
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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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### 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 -...