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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is y[n] = x[n] + n time invariant?
My steps were as follows:
$\ x_2[n] = x[n-k] $
$\ y[n-k] = x[n-k] + (n-k) $
and
$\ y_2[n] = x_2[n] + n = x[n-k]+(n-k)$
Does this mean that it is indeed time invariant?
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Does constant Q transform have linearity property in the transformed domain?
With FFT, I sometimes take advantage of the fact that I can pre-calculate a signals Fourier Transform, and then you can add the noise in the spectral domain:
$$ \mathscr{F} (x[n] + h[n]) = \mathscr{F}...
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2answers
41 views
LTI system output given input and frequency response
The question I'm trying to understand is as follows: A linear time-invariant continuous-time system has the frequency response function $$H(\omega)=\frac{1}{j\omega+1} $$
Compute the output response $...
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26 views
System memory, causality, stability
im new into systems and im supposed to solve if the system has memory, us causal, linear, stationery, BIBO stable...The problem is i have never had experience with this type of system where the actual ...
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1answer
38 views
Parallel LTI system
Imagine I know that a system comprises two LTI subsystems $H_a$ and $H_b$ connected in parallel. Also, suppose that I can measure the impulse response of the whole system, $H_a \| H_b$, and the ...
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50 views
Phase response of an LTI-system
I've got an LTI-system such as follows:
$$y'(t)+{2} y(t)={5} x(t-{3}), t>0,
\\ \\ y(0)=0 \ \text{ ja } x(t)=0, \ t<0.$$
From this i've already figured out the transfer function:
$$H(f)=\frac{5\...
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1answer
34 views
Is $y(t) = \cos(t) + x(t)$ a time-invariant system?
Is $y(t) = \cos(t) + x(t)$ a time-invariant system?
$y(t-k) = \cos(t-k) + x(t-k)$
But it isn't equal to $\cos(t) + x(t-k)$
So, would it be time-invariant?
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Are all LTI systems invertible? If not, what is a good counterexample?
I have been trying to figure this out for a while now. Everywhere I have looked I could easily find examples of invertible LTI systems, but I could not find any counterexamples. Can anybody shed some ...
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I somehow “proved” that given any LTI system, its transfer function has to be constant. What am I missing?
A transfer function is defined as the Laplace transform of the ratio of output to input.
Also, every LTI system has an eigenfunction. Given such eigenfunction as an input, the ratio of the output to ...
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31 views
Steady-State Output from Transfer Function
The progress I have made is as follows:
$\sin(t)$ is our signal therefore $\omega = 1 = 2\pi f$ and $f$ = $\frac{1}{2 \pi}$
Also, $f_s$ = 10Hz therefore T = $\frac{1}{f_s}$ = 0.1s
$H(z) = \frac{z}{...
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26 views
Show That a 2D Linear Transform $ T \left( \cdot \right) $ Is Homogeneous
By my understanding, a transform T is homogeneous if T[0] = 0.
Then to prove that a linear transformation is homogeneous we say that:
T[ax(n1, n2) + bx(n1, n2)] = aT[x(n1, n2)] + bT[x(n1, n2)]
What ...
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2answers
415 views
The spectrum of the sum of two periodic signals
Considering the continue, periodic signals $s_1(t)$ and $s_2(t)$, with the period $P_1$ and $P_2$ respectively.
Consider now a new signal $s$ the sum of the two pervious signals.
I come to know ...
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1answer
65 views
impulse response VS zero-input response
I am new in the field of systems and signals, and I have a rather basic for the majority of the group, question:
Can we find the impulse response function of homogeneous ODE, instead of its zero-...
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4answers
744 views
Can every type of linear filter be modelled by a convolution?
I have an input time series going through a filter that creates another time series as output.
If I assume in first approximation that my filter is linear, does it necessarily mean that I can model ...
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2answers
101 views
Convolution of Signal with a Gaussian Filter / Kernel
Following up on Analytical Solution for the Convolution of Signal with a Box Filter, I am now trying to convolve a Gaussian filter with the sine signal by hand.
My method is to use the definition of ...
2
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1answer
75 views
Initial Rest Condition for LCCDE causal LTI systems
I am self studying Alan Opennheim's course Signals and Systems. I am a math major and have no background in EE.
I understand that for a LCCDE system to be linear its auxiliary conditions must be 0.
...
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2answers
111 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
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1answer
93 views
transient and steady-state response of first order system
Considering this general 1st order transfer function
$$
H(z) = \frac{b_0 + b_1z^{-1}}{1-az^{-1}}
$$
How to find (analytically) the transient and steady-state responses?
With steady-state response I ...
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2answers
48 views
Are Chebyshev Type 1 filters very unstable at very high orders?
I am currently experimenting with a digital implementation of a Chebyshev Type 1 filter of mine, for audio purposes. I have written the code myself so I was wondering if the behavior I've noticed is ...
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28 views
Determining if a 2D system is LSI
After a long time away from school, I'm a bit rusty and struggling with this question:
Determine if the following discrete system is LSI: $y(m,n) = mn*x(m+n) + mn*x(m-n)$
So here's what I've done ...
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1answer
29 views
Invariance of $T[x[n]]=\sum_{k=n-1}^{n+2}x[k]$
I have to test whether the following system is invariant or not: $T[x[n]]=\sum_{k=n-1}^{n+2}x[k]$, so I want to verify that, if $y[n]=T[x[n]]$, then $y[n-N]=T[x[n-N]]$.
$$T[x[n-N]]=\sum_{k=n-1}^{n+2}x[...
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1answer
75 views
Analytical Solution for the Convolution of Signal with a Box Filter
I have an exercise in which I am trying to filter an input signal $y(x) = \sin(x)$. Ideally, I would like to apply a box filter to this signal.
Previously, I successfully convolved the input signal $...
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2answers
99 views
Why does knowing the impulse response allow you to determine the output for any LTI system?
Going over some notes on LTI systems and I’m quite confused about how knowing the impulse response of a system allows us to recover the systems’ response to any signal. In the textbook screenshot ...
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1answer
46 views
Representing a continuous LTI system as a discrete one
I am aware that there are different ways to represent a continuous time system in discrete domain (e.g. bilinear transform, impulse invariance transform).
But my problem is as follows: Given an ...
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2answers
60 views
Cross-correlation of filtered random processes
I have a wide-sense-stationary (WSS) process $\{x(t)\}$ and two linear filters with impulse functions $h_1$ and $h_2$.
Let $\delta(\omega)$ be the power spectrum of $\{x(t)\}$ and $$H_1:\omega\...
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1answer
30 views
Difference between the two forms of input-output relationships of an LTI system
$$y(t)=f(t)*h(t)\tag{1}$$
$$y(t)=H(s)e^{st}\tag{2}$$
$$H(s)=\int_{-\infty}^{\infty}h(t)e^{-st}dt\tag{3}$$
Let $f(t)$ in Eqn $(1)$ be $e^{st}$. In many worked out examples, I have found that the two ...
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1answer
37 views
how do I derive the system equation for a simple delay with feedback?
I am a software engineer, and just learning digital signal processing formally, though I've hacked around before a fair amount.
I'm implementing a delay audio VST and I'm trying to wrap my head ...
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1answer
31 views
Controllability's dependence on B matrix
Assuming the LTI system:
$\dot{x}=Ax+Bu\\$ , where $x\in R^n$
I understand that
the $B$ matrix is usually given, and it is up to us to derive the optimal $u$ given an initial state of the system ...
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1answer
14 views
External/Internal Description of a System
When an LTI system is NOT completely controllable/observable, does the linear differential equation governing the system give the internal or the external description of a system?
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1answer
149 views
Unit Impulse : Bounded or Unbounded?
As the title suggests, I have a confusion here. In a Systems text I am going through, it mentions of the Unit Impulse as an unbounded signal. Yes, the unit impulse "height" is unbounded, but the "...
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4answers
82 views
Determining LTI system response to a scale change
We know that for an LTI system, if $y(t)$ is the output for $x(t)$ then the response for $x(t-2)$ will be $y(t-2)$ and so on.
But my question is what will be the system response for the input $x(-2t)$...
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2answers
43 views
What is the difference between a controller and a compensator?
I recently revisited some texts on control theory. In some textbooks the term compensator is used, seemingly interchangeably with controller (see e.g. The Control Handbook by Levine, chapter 14 on ...
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4answers
169 views
LTI system and initial conditions
I suspect this is rather obvious once explained; I must be misunderstanding something.
From reading various answers here (such as this one), I understand that an LTI system must have zero output for ...
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0answers
33 views
Does phase margin exists only for linear systems?
If possible, how to determine the phase margin for nonlinear systems?
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2answers
73 views
Why the following discrete type system,variable accumulator, is time invariant?
$y[n]=T{x[n]}=\displaystyle\sum_{k=n-n_{0}}^{n+n_{0}} x[k]$
it is some sort of moving summer which computes $n^{\text{th}}$ output sample by adding all samples lying within length $n_{0}$ around some ...
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0answers
51 views
Using Kalman filter vs Extended Kalman filter for differential drive robot with IMU
I have an IMU that provides me with a heading that is pretty accurate and accurate encoders on the wheels of my differential drive robot which provides me with pretty accurate velocity but has ...
0
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0answers
23 views
Inversion problem for big dataset
Suppose my goal is to solve for $x$ in $Ax = b$ using CG method and $A$ is Toeplitz. The problem is that $b$ is extremely large in size and I can't just read in the whole vector $b$ and then solve for ...
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0answers
48 views
System identification with limited bandwidth
Given a FIR filter $h[n]$. Its action can described as:
$$
\mathbf{y} = \mathbf{H} \mathbf{x} \\
\mathbf{y} = \mathbf{X} \mathbf{h}
$$
where $\mathbf{H}$ and $\mathbf{X}$ is a Toeplitz matrix.
If $h$...
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2answers
110 views
What is the difference between y[n] and y(k) (using square brackets)?
Can I modify y[n] = αy[n-1] + x[n] the same way I would do it with y(k) --> Y(z)?
I need H(z), but I don't know what is the difference between using (k), and (n). I often encountered [k] notation, ...
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1answer
51 views
A question about Fourier transformation
Hello, this is my first time actually asking in stackexchange. I am a computer engineering student and currently i am doing a linear system course (i don't really know how this is equivalent in ...
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3answers
422 views
How to Prove a System Is Invertible?
what i know is that for a system to be invertibel it should be one-one ,
but I am confused that if i am given a transfer function of a LTI system how can I prove or verify if it is invertible.
...
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1answer
36 views
Reconstructed output mismatch for LTI system
I have a system with measured input (u) and output (y).
I assume that this is an linear time-invariant (LTI) system and I want ...
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1answer
58 views
Computing response of LTI system in terms of its step response
I have solved a problem, Kindly help me in determing if I solved it correctly.
I will post the question and my own working below.
Problem
My working
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4answers
912 views
Is the first derivative operation on a signal a causal system?
Please help me sort this issue out.
Consider a system whose output $y(t)$ is the first derivative of the input signal $x(t)$.
We can write the first derivative of an input signal as follows:
$$y(t)...
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1answer
60 views
How to detect algebraic loop in a system
I came across a system (screen below) that looks like it should result in a algebraic loop, but when writing the equations using the Z-transform, it obviously does not (calculation below too). But I ...
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1answer
23 views
Why does the Y component in YCbCr represent luma/intensity?
As far as I know, the RGB to YCbCr conversion is just a linear transformation, yet the Y channel successfully captures the intensity value of each pixel. How?
In other words, where do the following ...
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1answer
39 views
confused in difference between calculations of convolution and multiplication?
How are two signals multiplied? And how is it different from convolving two signals??
I have read the above question on dsp stack exchange and i also tried it in my matlab , code is as follow:
...
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1answer
26 views
Is it possible to use k-means algorithm with just one vector
Suppose I have a vector $X = (x_1, x_2 , . . . ,x_n)$, $x_i$ is the maximum of $X$ and $x_k$ is the minimum. Is it possible to use k-means algorithm to cluster the values in vector $X$ into two ...
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196 views
Output of an LTI system given its transfer function and input
Given the transfer function $$T(s) = \frac{100}{1 + \frac{s}{10^{6}}}$$ and the input $$v_i(t) = 0.1 \sin(100t)$$ find the output, $v_o(t)$.
My approach was to use $v_o(t) = \mathcal{L^{-1}}\left\{T(...
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2answers
64 views
time invariance concept?
First of all ,please let me know?Is cosine time invariant or time variant?
If latter is the case then if a time varying input like cos is given to a time invariant system,how will it behave?
I have a ...
