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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1answer
21 views

Does fading memory mean impulse response with finite support?

Given a continuous-time impulse response $h(t)$, bandlimited to $B$. The discrete-time $h[n]=h(n/(2B))$ is then a unique and perfect representation of $h(t)$ and a discrete-time system $h[n]$ is then ...
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0answers
49 views

Richardson-Lucy iterative deconvolution is producing erroneous deconvoluted images

I am working on producing deconvoluted image from degraded image where the process of degradation is known to be linear and space invariant. I have used Richardson-Lucy iterative deconvolution method ...
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2answers
47 views

Moving from deterministic signals to stochastic signals in s-domain (Power Spectral Density)

Assume we have the following system (coming from control systems theory, hence in s-domain) $ Y(s) = H_A (s) \cdot A(s) - H_B (s) \cdot B(s) $ I now wish to consider $a(t)$ and $b(t)$ as white noise ...
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42 views

Is the following statement true or false?

Is the following statement true or false? For the input into this system: $x(t)=\delta(t)x ( t ) = δ ( t )$ the output from the integrator will be: $y(t)=u(t)$
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2answers
91 views

Output of a stable LTI system

Let $\mathcal{L}$ be a stable LTI system. Is it true that if input has finite energy then output also has finite energy? I'm not sure about that. We know that $$\int_{-\infty}^{+\infty}|h(t)|dt\lt\...
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1answer
56 views

Find the length of the impulse response for the given output and input

Homework Question: Consider a signal $x[n]=\alpha e^{j \omega_{0} n}+\beta e^{j \omega_{1} n}+\gamma e^{j \omega_{2} n} .$ What is the length of impulse response $h[n]$ of a system (non-trivial) such ...
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23 views

Discrete Time Systems with cosine()

I am trying to see if y[n] = [cos(πn)]x[n] is casual, stable, linear and shift-invariant. I came up with the solution that it is not stable since it is not "...
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1answer
25 views

Testing for Linearity and Shift-Invariance

I started school a week and a half ago and we have gotten already a big chunk of material and I wanted to check if what I am currently doing is correct. Here below are the problems to work out that ...
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1answer
58 views

Why isn't option C correct answer for the given question?

W Please answer this question I am getting option c as my answer by putting input frequency of 3 radians in frequency response of system.
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1answer
50 views

Fourier transform of time division

I know that Fourier transform of $t^n f(t)= i^n \frac{d}{d\omega^n} F(\omega)$. But does this work when $n<0$? Is there any direct relation to compute the Fourier transform of $\frac{f(t)}{t}$?
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1answer
62 views

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 $t=0$ ...
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2answers
45 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 ...
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2answers
125 views

Derivation of the LMMSE (Linear Minimum Mean Squared Error) Estimate and the MMSE Under Gaussian Prior

I am learning estimation theory through Steven M. Kay - Fundamentals of Statistical Signal Processing, Volume 1: Estimation Theory. In the ...
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2answers
75 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 ...
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1answer
42 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) = \...
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1answer
36 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 ...
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1answer
119 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-...
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2answers
59 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 ...
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0answers
108 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-...
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1answer
34 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
42 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
1k views

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 ...
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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 ...
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1answer
41 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 ...
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2answers
154 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 ...
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1answer
29 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
32 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
28 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 ...
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0answers
84 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 ...
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1answer
82 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
38 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. ...
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1answer
46 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 ...
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1answer
46 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 ...
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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 $$w[n] = x[n] + aw[n − 1]$$ and $$y[n] = w[n] + bw[n − ...
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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 ...
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2answers
72 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: $$y(t)=\int_{-t}^{\infty}x(-3\tau)d\tau$$ ...
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1answer
22 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
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3answers
188 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 $x[n] = \cos[n\frac{\pi}{3}] + 2\cos[n\frac{\pi}{2} + \frac{\pi}{4}]$, ...
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2answers
139 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 ...
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6answers
431 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 ...
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1answer
116 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 ...
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2answers
225 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
29 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[...
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0answers
32 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 ...
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2answers
67 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 ...
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2answers
67 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 ...
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0answers
28 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 ...
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0answers
29 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 $\mathbf{B}$, ...
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1answer
66 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 (...
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3answers
51 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 ...

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