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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8 votes
1 answer
493 views

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 ...
0 votes
1 answer
58 views

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 ...
0 votes
0 answers
59 views

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). ...
0 votes
1 answer
70 views

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
2 answers
1k views

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 ...
9 votes
2 answers
5k views

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 ...
0 votes
2 answers
60 views

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
1 answer
68 views

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
1 answer
112 views

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 ...
9 votes
7 answers
1k views

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 ...
5 votes
2 answers
13k views

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
0 answers
21 views

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 ...
0 votes
0 answers
16 views

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?
0 votes
0 answers
43 views

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
1 answer
48 views

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 ...
2 votes
1 answer
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
1 answer
63 views

Matrix form of Overlap-add

We know overlap-add of a en-framed signal can be done easily by following code ...
0 votes
1 answer
85 views

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 \...
0 votes
1 answer
36 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 ...
1 vote
0 answers
73 views

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, \...
4 votes
0 answers
221 views

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 ...
0 votes
0 answers
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 $c$
2 votes
3 answers
536 views

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 $\...
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
233 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
299 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 ...
1 vote
1 answer
943 views

Particular Solution to Difference Equation

I have a system given by $$y[n] - \frac{1}{4} y[n-1] - \frac{1}{8} y[n-2] =3x[n] $$ I want to solve for $y[n]$ for $x[n]=(\frac{1}{2})^nu[n]$. The complementary solution evaluates to $[k_1(\frac{1}{2})...
4 votes
2 answers
558 views

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 ...
2 votes
1 answer
78 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
82 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
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. ...
2 votes
1 answer
133 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
221 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] ...
7 votes
2 answers
580 views

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
1 answer
57 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
76 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
46 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
366 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 ...
3 votes
5 answers
781 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 ...
1 vote
1 answer
160 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 ...
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
88 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 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 $...
0 votes
0 answers
35 views

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 ...
1 vote
0 answers
72 views

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(...
1 vote
1 answer
98 views

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 -$\pi/4$ $...
0 votes
3 answers
228 views

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 -...
0 votes
1 answer
76 views

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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