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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questions
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Linearity of the system with Pearson correlation sliding window
I have a sampling system that consists of linear and non-linear components(analog filters, ADC, CPU, and so on). I made some HW changes and I want to verify that the linearity was preserved.
My method:...
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0answers
85 views
Find moments of a discrete process using $\mathcal{Z}$-Transform method
For a linear process relating a variable $Y_i$ to random, independent variables $X_i$ using the equation:
\begin{equation}
Y_i = aY_{i-1} + (1-a)X_i
\end{equation}
which has the solution:
\begin{...
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3answers
104 views
Are there any examples of "Causal" Continuous-Time Linear-Time-Invariant Systems (CT-LTI) with output given by finite-duration functions?
Are there any examples of "Causal" Continuous-Time Linear-Time-Invariant Systems (CT-LTI) with output given by finite-duration functions $y(t)$??
With output I am meaning that $y(t)$ is such ...
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1answer
52 views
How to find the impulse response from the following input/output relation
Suppose we have a system defined with input $x(t)$, and output $y(t)$, related by
$$
y(t) = \sum_i \alpha_i(t)x(t-\tau_i(t)) \tag 1
$$
This is a linear system, (according to the textbook), so we can ...
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1answer
60 views
Is the filter $1/(1-s)$ anti-causal?
The filter with the response function
$$
H(s) = \frac{1}{1 - s}
$$
Produces a positive phase shift and a negative group delay for all frequencies
Is it anti-causal? Is there a way to deduce such ...
1
vote
1answer
31 views
Time Invariance of Continuous Time System
I am trying to check whether the system given by the following input-output relation is LSI or not?
$$y(t) = \cos(x(t)-x(0))$$
My work:
The given relation: $$y(t) = \cos(x(t)-x(0))$$
If the input ...
0
votes
1answer
56 views
State-space model and/or TF for a $K$-th order linear system given the signal sequence
The discrete-time signal $x_n$ is a sum of $K$ discrete-time complex damped exponential:
\begin{equation}
x_n = \sum_{k=1}^{K} \underbrace{(a_k e^{j\phi_k})}_{c_k}
{\underbrace{e^{\{(- \...
-1
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1answer
62 views
Is this system linear or non linear? $y[n]=cos(\frac{5\pi}{8}*n+\frac{\pi}{4})$ [closed]
$y[n]=cos(\frac{5\pi}{8}*n+\frac{\pi}{4})$
If I let $(\frac{5\pi}{8}*n+\frac{\pi}{4})=x[n]$, I get the system as non linear. But one of my teacher says it is linear, whereas another teacher says it is ...
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vote
1answer
170 views
Can we tell if a system is linear and time-invariant from its frequency response?
Given a system with a known frequency response in the S-domain. Is there a way to find whether the system is linear and time invariant?
My current understanding is that we need to take the inverse ...
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0answers
35 views
What are the characteristics of a smartphone speaker?
What are the characteristics of a smartphone speaker? Whether it is linear or non-linear?
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1answer
30 views
FInd the amplitude function to a system
I'm studying a course in signal analysis and have come across an exercise where I find the analytical part a bit tricky.
I am to find the amplitude function $|H( f )|$ for a system with the impulse ...
0
votes
1answer
54 views
Filtering a sum of cosines
The block diagram below represents a linear modulation system operating at the frequency of $1000 Hz$, $f_C = 1000 Hz$, transmitting the message $m(t) = 2\cos(400πt)$
At point B, i got the signal: $$...
0
votes
1answer
64 views
How to find a system without an input $x[n]$ is linear or non linear
The question is this:
$$ y[n] = \cos\left(\frac{5\pi}{8}n + \frac{\pi}{4}\right)$$
This is what my teacher said when I asked him for help-:
In any system, inputs are not given, then we have to assume ...
2
votes
1answer
100 views
Why aren't the integrator and the differentiator inverse systems?
I have a statement that leads to a paradox, but I'm incapable of finding the part where I'm wrong.
The integrator system
$$x(t) \mapsto y(t)=\int_{-\infty}^{t}{x(\tau) \, {\rm d} \tau}$$
is a linear ...
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0answers
84 views
How to update point spread function of blind deconbolution by conjugate gradient?
There is an unblurred image $g$ and a blurred image $x$.
Their relationship is expressed by the following formula using $psf$(point spread fucntion, size is $5×5$ kernel).
$g = x \otimes psf\tag 1$
...
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2answers
70 views
What is the importance of obtaining a linear signal?
I am a computer scientist who has started doing some work in the electrical engineering space – in particular, photonics. While reading about interferometric systems, I have noticed that there seems ...
1
vote
1answer
43 views
Identify out signal from in signal and impulse response
Hi there I am studying a course in signal processing and systems.
I was given an excecise which I am having a great deal of trouble solving.
I am allowed to solved using matlab, but for top marks I ...
0
votes
1answer
58 views
Definition of frequency response
I studied various signal processing materials for a long time, and I have a question.
Considering an LTI filter, one can define its frequency response by evaluating its transfer function $H(z)$ on the ...
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vote
0answers
55 views
Mixing Kalman filter and least-squares
I'm not sure it is the right department. I try my chance
I am wondering if there is a way to make a hybrid formulation of a least-square problem and a Kalman filter. Let me explain what I mean:
The (...
0
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1answer
54 views
Zeros and poles from transfer function
I have a transfer function
$$ H(z) = \frac{Y(z)}{X(z)} = 1 - 0.5z^{-1} \text{.}$$
I'm interested in zeros and poles.
I know I need to adjust the function to
$$ H(z) = \frac{\prod_i(z-n_i)}{\prod_i(z-...
0
votes
2answers
167 views
Difference equation to FIR filter coefficients
I have a difference equation
$$y[n] = y[n-1] + 0.5x[n] - 0.5x[n-2] \text{.}$$
According to this answer, the filter should have finite impulse response. I just can't figure out how to get the FIR ...
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0answers
26 views
time invariance $ y (t) = x(\frac{t}{3})$
check time variance of this system
$$y (t) = x(\frac{t}{3})$$
my solution
$$y_1(t) = x_1(t/3)\\
y_2(t) = x_2(t)= x_1(\dfrac{t}{3}-t_o)$$
where
$$ x_2(t) = x_1(t-t_o)\\
y_1(t - t_o) = x_1(\dfrac{t-...
0
votes
1answer
46 views
How to prove that this system is an invertible system or not?
How could i go throw proving that this system $y(t)=\int_{-\infty}^{t}e^{-(t-\tau)}x(\tau)d\tau$ is invertible system or not ?
2
votes
1answer
86 views
Linear system stability criteria
Suppose we have a closed-loop system $H(s)=\frac{A(s)}{1+A(s)f(s)}=\frac{A(s)}{1+T(s)}$. I've seen the stability of the system stated a couple ways:
If $H(s)$ has any poles in the RHP, then it is ...
1
vote
1answer
62 views
When is a LTI IIR $H(z)$ system minimum phase
EDIT: I failed to mention that the system's inverse also needs to be causal and stable.
I cannot wrap my mind around on how when a system and its inverse are both causal and stable and LTI IIR it is ...
0
votes
0answers
23 views
Please explain the following Linear Prediction estimate graph
I am reading this doc https://librosa.org/doc/latest/generated/librosa.lpc.html#librosa.lpc?
I can't understand the graph for linear prediction coefficients. Please help explain it as I a new to this ...
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vote
0answers
41 views
Signal separation
This is probably a blind signal separation problem of sorts, but it seems like it should be easier than I am finding it.
Let’s say I have N time series, each of length [M x 1] that are a superposition ...
2
votes
1answer
156 views
Why is particular solution zero for an impulse excitation signal?
We were being taught the impulse response for a series RC Circuit- consisting of simply one resistance, one capacitor, and an impulse excitation all in series.
I get that the homogeneous part of the ...
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2answers
203 views
Where to start with DSP?
I have a DSP exam coming up this summer and I got an abysmal mark on an assignment earlier this year. Obviously, the lecture slides are not enough for me to understand DSP, so I am wondering where can ...
2
votes
1answer
221 views
Impulse response for an LTI system
I'm new to signal processing and working my way through a textbook. There is an exercise where a causal LTI system is given that responds to a rectangular pulse.
I have an exercise where a causal LTI ...
1
vote
2answers
115 views
Under what conditions is the convolution of an input signal with the system's impulse response periodic?
I'm currently solving the following convolution problem from Oppenheimer's book:
In the solution, it was stated that "$x(t)$ periodic implies $y(t)$ is periodic"
So I wondered if it's ...
0
votes
1answer
38 views
How do i know it is will be periodic
(system is LTI and Casual)
1)If a periodic signal is applied to the input of this system.
Does output always have to be periodic ?
2)What conditions are required for this system to be linear?
thanks ...
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votes
1answer
84 views
Output of a stable LTI system (discrete)
Consider an LTI (linear and time invariant) system that is BIBO (bounded input
bounded output) stable and is such that $x[n] = 0$ for all $n < 0$ (note: this is sometimes referred to
as a relaxed ...
3
votes
1answer
63 views
Understanding LTI systems graphically
I'm currently working on this problem from Oppenheimer's book:
Given $x_{1}(t)$ and $y_{1}(t)$ I should figure out $y_{2}(t)$ and $y_{3}(t)$ given that the system is LTI.
My progress so far:
I'm ...
0
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3answers
207 views
How do I check for linearity for the following piecewise-defined system?
The problem at hand:
Where I'm currently stuck:
I'm not entirely sure about how to move on from this point, I'm trying to find the superposition of the responses of the two individual signals so I ...
0
votes
2answers
50 views
Physically, what does the usage of two variables mean for convolution
My intuition of convolution is that it is just a way to depict multiplication of two signals where each signal is made up of various frequencies and phases.
Since it isn't easy to find the value of $\...
1
vote
1answer
116 views
Scaling the input vs scaling the impulse response for an LTI system
Two different cases:
We pass $x(t)$ to an LTI system with impulse response $h(2t)$ and get the output $y(t)$.
We pass $x(2t)$ to an LTI system with impulse response $h(t)$ and get the output $z(t)$.
...
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0answers
30 views
Stable and causal system
How many stable and causal systems with the same magnitude response are there?
I know this relates to an all pass system for two rational transfer functions but am not sure about the specifics of this
0
votes
2answers
37 views
Z-transform of an affine function
What is the transfer function of the system described by the following affine input ($x$)-output ($y$) relationship:
$$
y[n] = \alpha x[n] + \beta.
$$
Using the Z-transform we find:
$$
Y[z] = \alpha X[...
0
votes
0answers
80 views
convolution of two exponential signals with imaginary numbers
I can solve problems without imaginary numbers, but when exponential contain imaginary numbers, I can't solve the problem.
For example, $x(t)=e^{3jt}+e^{4jt}$, $y(t)=(e^{-3t}-e^{-4t})u(t)$ (where $j$ ...
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1answer
78 views
Unit impulse response of a cascade interconnection of three discrete-time systems
I am nearly at the end of finishing a problem in my textbook but I couldn't understand something in the answer;
I did everything to the point I found the overall response of the system in terms of $...
0
votes
1answer
48 views
Calculate impulse response when output contributes in input
I have an exercise in which I need to find the impulse response for this given system:
$$y(n)=\frac{1}{2}y(n−1)+x(n−1)+x(n)$$
As per my knowledge, I need to find the homogenous solution. My homogenous ...
4
votes
1answer
338 views
How do I get a faster system response?
I have this model in simulink (the graph is my output):
The step input has amplitude 0.5 m/s, and it steps up after 0.1 seconds. The gain $K_p=5$.
The saturation block is to keep the voltage between -...
2
votes
1answer
62 views
Unit impulse response $h[n]$ of a discrete time system with multiple characteristic roots 0
I am trying to obtain the unit impulse response of a system in the form:
$$y[n+N]+...+a_{N-1}y[n+1]+a_{N}y[n]=b_0x[n+N]+...+b_{N-1}x[n+1]+b_{N}x[n]$$
$$Q[E]y[n]=P[E]x[n]$$
(where E is the unit advance ...
0
votes
1answer
63 views
Output of a linear time-invariant(LTI) system
I am very confused about one of the questions I received during an exam. How do I solve this question?
2
votes
1answer
72 views
Finding the impulse response given response to another signal
I was trying to solve this question :
I respresented $x(t) = u(t+1)-u(t-1)$
writing the convolution as $[u(t+1)-u(t-1)]*h(t) = y(t)$
I then used the property of differentiation to convert from the ...
1
vote
1answer
79 views
Initial rest condition applied on $x(t)$ vs $h(t)$
Define the LTI system $\mathcal{H} : x\mapsto y$
Define the convolution for continuous-time system :
$$
(x*h)(t)=\int_{-\infty}^{\infty}x(\tau)h(t-\tau)\;\text{d}\tau
$$
The initial rest condition ...
1
vote
1answer
66 views
Impulse Response of real coefficient, LTI System
I'm trying to obtain the impulse response $h[n]$ of a system whose frequency response is $H(e^{j\omega})=R(\omega)e^{-25j\omega}$. I believed that $h[n]=h[n-25]$, would be the correct answer, however ...
1
vote
1answer
34 views
understanding echo cancellation model
so, here is a very simple model.
The CCDE is given, but I was trying to derive it on my own and now I am stuck.
first of all, the reverse y[n] when goes through the delayed system, it turns to y[n-M]....
0
votes
1answer
49 views
Applying Superposition Property : $x^{2}[n]$ vs $x[n^{2}]$
If we consider the mapping $\mathcal{H} : x[n]\mapsto y[n]$ and define the following output signal $y_{1}[n]:=\mathcal{H}\{x[n]\}:=x^{2}[n]$, then one can easily verify that such system is non-linear ...