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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1answer
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Confusion in initial condition of differential equation using Laplace transform transform
I'm confused in solving linear constant coefficients differential equations (LCCDEs) by Laplace transform if initial conditions are given at time
just before $t=0$
just after $t=0$
exactly at $t=0$
...
1
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2answers
38 views
Stability of passive LTI system
Is a passive LTI system will always be a stable(bibo and lyapunov) system?
Or in other words
Passivity of a LTI system implies stability?
If above statements are true then there must be some ...
2
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0answers
53 views
linear minimum mean squared error estimate under Gaussian prior
I am learning estimation theory through Steven M.Kay's book Fundamentals Of Statistical Signal Processing--Estimation Theory. In the ...
1
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2answers
70 views
Output of marginally stable systems
In' b.p lathi's book of systems and signal it is written that
"marginally stable systems have one important application in the oscillator, which is a system that generates a signal on its own ...
2
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1answer
26 views
Determining a system's causality using its impulse response
I have the following input-output relation for a system:
$$y(t) = Odd Part Of [x(t)]$$
My question is: Is the system causal?
What my approach has been:
I expressed $y(t)$ alternatively as:
$$y(t) = \...
1
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1answer
34 views
Transients in marginally stable system
I know that if a system is asymptotically stable then after infinite time its output contains only characteristic of input signal and it is also known as steady state and there are no transient ...
0
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1answer
70 views
Time invariance of a summation sequence
The following sequence is given which is supposed to be time-variant:
$$y[n] = \sum_{k=n_0}^n x[k]$$
I'm having difficulties proving the time-variance or finding a counterexample for it being time-...
0
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2answers
50 views
How to compute transfer function $G(s) = \exp \left( - \sqrt{s} \right)$ in Matlab / Simulink?
How to compute transfer function
$$G(s) = \exp \left( - \sqrt{s} \right)$$
in Matlab / Simulink?
I am trying to calculate a PID controller for this function. This function describes heat transfer via ...
1
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0answers
99 views
Time-Invariant (non)autonomous Systems
Regarding potential distinctions between autonomous, non-autonomous, time-invariant and time-varying systems, I have found out opinions supporting that:
autonomous systems are time-invariant and non-...
0
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1answer
30 views
Compute output given input, transfer function and initial conditions
The problem statement is
Consider a causal LTI system whose transfer function $H(s)$ is given as $$H(s)=\frac{s+2}{(s+3)(s+4)}$$ Compute the output $y(t)$ for an input $x(t)=e^{-2t}u(t)$ when $y(0)=1$...
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1answer
38 views
Behaviour of integrator at steady state
I wanted to calculate response of integrator of sinusoidal input at steady state via these two methods as mention in image but these two methods give two different answers at steady state, so where ...
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5answers
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Are there any real world applications for complex-valued signals or impulse responses?
I was just curious...
$$x[n] {\longrightarrow} \boxed{h[n]} {\longrightarrow} y[n]$$
I've never seen a real world filter where the coefficient of $h[n]$ were complex, or where $x[n]$ was a complex ...
0
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1answer
26 views
Names of system functions in frequency domain
I was just trying to refresh my systems theory known from long ago, and I realized that I had forgetten the name of the basic functions. Specifically, what are the names of these functions ...
0
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1answer
35 views
Determine causality and stability from given filter structure
I have the diagram above. I found the transfer function below from it;
The question asks me to find out if the system is causal and stable, but didn't it have to indicate whether it was left-sided or ...
3
votes
2answers
148 views
Behaviour of System by Impulse Response
In a book by Alan Oppenheim, it is given that for a LTI system, its characteristics are completely determined by its impulse response. But if impulse response of a system is basically an input output ...
0
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1answer
16 views
Time invariance from convolution integral
I wanted to prove, time invariance property of system by convolution integral(i.e equal time shift in input result in equal shift in output) but as calculation shown in figure, I got unexpected result,...
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1answer
31 views
Determining if the system is linear and time invariant
The Fourier transforms of input $x[n]$ and output $y[n]$ of a discrete-time system are related by the following equation: $$Y(e^{j\omega}) + (e^{-j\omega})Y(e^{j\omega})= X(e^{j\omega}) - X^{'}(e^{j\...
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0answers
26 views
Existence of finite output of a system [closed]
1.If frequently response of a LTI system exist and finite then can we say that, for a periodic or non periodic input signal, output is also finite?
2.if responses(output) of LTI
system for ...
2
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0answers
82 views
Find a LTI system such that $\mathcal{T}\{\frac{\sin t}{t}\} = \frac{\sin 2t}{t}$
Let
$$x(t) = \frac{\sin t}{t} \qquad\text{and}\qquad y(t) = \frac{\sin 2t}{t}$$
Is it possible to find a LTI system such that $\mathcal{T}\{x(t)\} = y(t)$?
If not, what's the reason for ...
0
votes
1answer
64 views
Trouble showing Time Invariance of recursive system
The system is described with the following recursive differences equation:
$$y[n]-4y[n-1]+4y[n-2]=20x[n]+10x[n-1]$$
now lets say the input is delayed by k, then:
$$y[n]-4y[n-1]+4y[n-2]=20x[n-k]+10x[...
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2answers
36 views
maximum output of a LTI system within linear limits
In book , 'fundamental of signal and system'by M.J ROBERT it is written that
Since no practical system can ever produce an unbounded response, strictly speaking, all practical systems are stable. ...
1
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1answer
44 views
Finding the system output by convolution
pretty new to this concept, so do bear with me.
A linear dynamic system is described by the following differential equation:
Transfer function H(s) is calculated to be =
I've already found the ...
0
votes
1answer
29 views
Is Differentiation as a system, is an invertible system?
is the following system invertible?
as I understand it, invertible means finding an inverse function which should return back the original input from an output of the given system.
if so I ...
3
votes
1answer
32 views
Simplfiying a system output equation
I have a problem getting the final (simplified) version of the system's (in the figure below) output equation y[n]:
For this system, I know that $$w[n] = x[n] + aw[n ā 1]$$ and $$y[n] = w[n] + bw[n ā ...
0
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1answer
13 views
Implementing an L-System conversion from R2 to R1 for symmetric, square matrices
I have been reading for some time about L-Systems, and specifically the Hilbert Space filling curve. I am interested in writing a function to convert upper-triangular matrix coordinates into an ...
1
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2answers
45 views
Determining time-invariance of a system
I have a question about determining time-invariance of a linear system. We are given this system and we need to determine if it is time-invariant or not:
$$y(t)=\int_{-t}^{\infty}x(-3\tau)d\tau$$
...
1
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1answer
21 views
Linear response function for a system with derivative: $U=L \frac{d I}{dt}$, expressing $U=f(I)$
I have a super basic questions. I am a not really into signal processing (more about physics), but I would like to understand an aspect of linear response function (I think the question fits for this ...
3
votes
3answers
123 views
Determining output of a LTI system
Consider following LTI system $$y[n] - 2y[n-1] = x[n]$$ where $x[n]$ is the input to the system and $y[n]$ is the output. Let $x[n] = \cos[n\frac{\pi}{3}] + 2\cos[n\frac{\pi}{2} + \frac{\pi}{4}]$, ...
4
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2answers
133 views
Running Integral of sine and cosine functions
In typical signal processing course we were taught that the integral of signal $x(t)$ is given by
$$y(t) = \int_{-\infty}^{t}x(\tau) d\tau$$
How can we use this definition to evaluate the integrals of ...
3
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6answers
411 views
Design a LTI system which returns DC value of the input signal
Suppose that $h(t)$ is the impulse response of LTI system. The input signal $x(t)$ is periodic with period $T$. Determine $h(t)$ so that the output signal $y(t)$ only be the DC component of $x(t)$. Is ...
0
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1answer
54 views
Initial rest condition for the linear constant-coefficient differential equations
Suppose that system has the input/output relation as follows
$$\sum_{k=0}^{N}a_k \frac{d^ky(t)}{dt^k} = \sum_{k=0}^{M}b_k \frac{d^kx(t)}{dt^k}$$
Where $a_k, b_k \in \mathbb{R}$.
Obviously we need ...
5
votes
2answers
184 views
Negative group delay and envelope advance
I am having a doubt reading about delays in signal processing.
Let there be an input to a LTI system with frequency response $H(f)$, given signal $x(t) = a(t)\cos(2\pi f_ot)$, where $a(t)$ is a ...
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1answer
28 views
LTI system impulse response [closed]
I am learning signal processing and need some help with this question
$h[n]=-x[n]+2x[n-1]-3x[n-3]$
Given the following input signal $x[n]$ calculate y[n] the output of the system $x[0]=2, x[1]=0, x[...
0
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0answers
29 views
Is a LTI system with dead-time equivalent with a LTI system with no dead-time but delayed input signal?
I wonder if the system parameters would remain the same after all.
According to the definitions, the gain is the change of the output induced by the change of the input if we assume that the gain is ...
0
votes
1answer
54 views
Determine the haar wavelet function is linear or nonlinear
I'm looking for analytical justification of linearity or non linearity of the wavelet transform with the real Haar mother wavelet function.
I have googling already. But I can't find and understand ...
0
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2answers
63 views
Characterizing an unknown LTI system
What are the best approaches to characterize an unknown discrete time LTI system? I believe one of the approaches is to input a known input and measure the output to find a transfer function. What are ...
0
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0answers
24 views
Characterizing a non-LTI system
How should we characterize a non-LTI system? For example we have: $y[n]=x[3n]+x[2n]+x[n]$ which is clearly not LTI. Also, the impulse response will be $h[n]=3\delta[n]$ and if we take the DTFT of this ...
0
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0answers
27 views
Discretizing my continuous system in Matlab
I was hoping someone could help me with conducting my discretization in MATLAB.
I have a linear system with known data: sampling interval $T$, system matrix $\mathbf{A}$, input matrix $\mathbf{B}$, ...
0
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1answer
64 views
On the use of Permutation matrix to perform iFFT
I have a question about using the permutation matrix for performing $iFFT$ for such matrix and then reshape it row-wise and column-wise way.
Let's say that we have a random matrix $x$ whose size is (...
1
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3answers
41 views
System classification: unit-time delay
I'm reading a book on linear systems and I can't understand why the unit-time delay is a distributed system. This is the example given in the book:
I understand that the initial state of the system ...
0
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1answer
97 views
Autocorrelation of LTI system output
I'm having trouble showing the following relation:
The autocorrelation of a LTI system with impulse response $h[n]$, input $x[n]$ and output $y[n]=h[n]*x[n]$ is given as:
$r_{yy} = r_{hh}*r_{xx}$
...
0
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2answers
51 views
Filter response to sinusoidal input signal
This question is related to Does 46 dB gain of a filter for a frequency imply 200 times more amplitude? In the accepted answer it is mentioned that "For a sine wave input the amplitude of the steady ...
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1answer
24 views
DTFT of inverse of any function
In my book solution is given like this.
But i am solving like this , am i doing wrong??
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3answers
160 views
Estimating the Signal by Deconvolution with a Prior on the Filter Coefficients and the Signal Samples
Assume I have signal $y[n]$ which is a result of convolution between channel $h[n]$ and signal $x[n]$. which means:
$$y[n] = h[n] \ast x[n]$$ where $\ast$ is the convolution operation
The signal $...
1
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3answers
120 views
Whether $y(t) = x(t/3)$ is a memoryless system?
I am a newbie in signal and system, and feeling confused about the memoryless property, it says at the definition, if a system gives output only depends on present input, then its memoryless. So if at ...
1
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0answers
35 views
Trouble checking if a given difference equation corresponds to an LTI System
I'm having trouble on how to check if a given difference equation corresponds to an LTI system. For instance,
Does the following difference equation correspond to a LTI DT system ? Explain.
$$y[...
6
votes
3answers
285 views
Oscillations in a first-order discrete-time linear systems
According to my understanding, continuous first order systems do not exhibit oscillatory behavior because a first order system has only one energy storage element and thus oscillation is not possible. ...
2
votes
1answer
57 views
What are the properties of continuous-time improper systems?
I am trying to better understand the properties of improper systems $H(s) = \frac{b(s)}{a(s)}$, for which the order of the numerator $b(s)$ is greater than the order of the denominator $a(s)$ (in the ...
3
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1answer
68 views
Proving linearity and time-invariance
This question was asked in one of our quiz. Question is
Determine if the system with the input-output relationship given by
$$y(t) = \int_{t-10}^{t} \cos(\tau)x(\tau)d\tau$$
is linear and time ...
0
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1answer
66 views
Time invariance and linearity of recursive system
I am confused about the definition of linearity and time invariance of recursive system
Given $$y[n] = y[n-1] - y[n-2] - x[n]$$
To test time invariance, we shift the input $x[n]$ for shifting and ...
