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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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590 views

What is $H_2$ and $H_{\infty}$ control? [closed]

I can create a Linear Quadratic Gaussian Integral(LQGI) controller very easy by using GNU Octave. LQGI is in the area of Optimal Control Theory. But there is something called Robust Control Theory. ...
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0answers
100 views

Check if I has right: Is this the Extended Kalman Filter

I have learn the Kalman-Buncy filter for the LQG controllers. I know that this is a signal processing forum and not robotics not math forum. But Extended Kalman Filters are daily discussed here. ...
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1answer
65 views

Extended Kalman Filter in mechanics, electronics and hydraulics?

Extended Kalman Filter is most used in GPS and navigation systems. But how much do I gain to switch to Extended Kalman Filter to the linjear Kalman Filter if I do LQGI controller for the industry? ...
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1answer
159 views

What is the spectrogram of an impulse response used for?

I saw that in acoustics an estimated impulse response of a room can be transformed using STFT/wavelets to a spectrogram as an analysis tool. However, I did not understand how the spectrogram is ...
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6answers
333 views

The Least Norm Solution of Under Determined Linear System

Suppose I have a matrix $$A= \begin{pmatrix} 1 & 0 & 1 & 0\\ 0 & 1 & 1& 0\\ \end{pmatrix} $$ where the variables are channel information like assume $X_1$, $X_2$, $X_3$ ...
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1answer
176 views

Can I assume system is LTI when given by DTFT of impulse response

I'm having hard time to grasp it probably because i don't fully understand it. I understand that when a system is given by $h(t)$ (in general $h(t-\tau)$) i can assume that it is a LTI system. So i ...
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3answers
474 views

Help in understanding the formula of Signal-to-Noise-Ratio (SNR) - Part 1

Question 1: Consider an Autoregressive model : \begin{align} y[n] &= y[n-1] + x[n]\\ z[n] &= y[n] + w[n] \end{align} where $y$ is the output observation, $x$ is a random input and $w$ is ...
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2answers
334 views

LTI system with Laplace transform

Given the input $$x(t)=u(t)$$ and the corresponding output signal measured as $$y(t)= 2 e^{-3t} u(t)$$ determine the impulse response $h(t)$. This what have done so far: $$ h(t)= \mathscr{L}^{-1} \...
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0answers
59 views

Linear Systems, Sparse Solutions, and $4 \times 4$ Sudoku Algorithm [closed]

I am unable to understand the paper Linear Systems, Sparse Solutions, and Sudoku. I have to form a $4 \times 4$ Sudoku using the algorithm in this paper. Can somebody please provide me the algorithm ...
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3answers
117 views

Linear difference equation [closed]

Give the equation $$y[n]-y[n-1] +\frac14y[n-2]= x[n]$$ With initial conditions $$y[0]=0\\ y[1]= 0$$ Find the step response. I have been able to get the solution $$y[n] =C_1(\frac12)^n +C_2(\frac12)^...
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1answer
301 views

Steady State of a Difference Equation using MATLAB

I'm given this difference equation, and I'm being asked to plot its response to an input $x[n] = 4\sin^3\left(0.25\pi n+\frac \pi 3\right)u(n)$: $$ y(n) = 0.9051y(n-1)-0.598y(n-2)+0.29y(n-3)-0.1958y(...
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1answer
197 views

Step response of a differentiating system

Consider I can find the step response of a system with the following method: We can easily find the step input of a system from its transfer function. Given a system with input $x(t)$, output $y(t)$ ...
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1answer
197 views

Under what conditions do the phase margin and Nyquist criteria give the same results?

When designing feedback systems, I often evaluate stability by thinking about phase margin: the closed loop system $$T(s) = \frac{L(s)}{1+L(s)}$$ is stable if $L(s)$ has positive phase margin, i.e., $...
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1answer
52 views

Notation of an LTI system consisting of LTI filters

I would like to find a reference for two notations of an LTI system consisting of LTI filters. In z-domain, the LTI system is given by $$ \mathbf{y}(z) = \mathbf{C}(z) \mathbf{s}(z) + \mathbf{D}(z) \...
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1answer
63 views

Is applying a linear operation to the input and output of a continuous linear operator OK?

The scientist and engineers guide to digital signal processing, Steven W. Smith, p. 134-135 states: [...] imagine a linear system receiving an input signal, $x[n]$. and generating an output signal, ...
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1answer
206 views

Is ideal LPF a LTI system? And is it time invariant?

Ideal filter time-domain impulse response is $$h(t) = 2B \operatorname{sinc}(2Bt)$$ It is not causal nor BIBO stable. Is it time invariant? And how can i prove it?
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2answers
725 views

Why does reversing the order of these two transfer functions give me different outputs?

Consider these two systems: \begin{align} &u\ {\longrightarrow}\boxed{s}{\longrightarrow}{\boxed{\frac 1s}}{\longrightarrow}\ y\\ &u\ {\longrightarrow}\boxed{\frac 1s}{\longrightarrow}{\boxed{...
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1answer
1k views

Determining if the given system is LTI

Problem Given the compound system below, with the input $x(t)=\operatorname{sinc(t)}$, the output of A is $y(t)=\operatorname{sinc(2t)}$ and the output of B is $z(t)=\operatorname{sinc(t)}$, ...
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2answers
475 views

Is a filter/control transfer function with positive phase “causal”?

In control we often use transfer functions with positive phase, i.e., a "lead filter" has transfer function $$G_c(s) = \frac{\alpha \tau s+1}{\tau s+1}$$ (with $\alpha>1$). Since the zero occurs ...
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1answer
147 views

Create distortion from basic linear (and non-linear if neccessary) DSP elements

I'm studying mechatronics and I'm intrested in DSP basics. My lecturer said that there are four basic linear DSP elements: Adder (and other mathematical operations) Amplification (shown on diagrams ...
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1answer
285 views

In the context of transfer functions, what is the relationship between the terms “proper”, “causal”, and “realizable”?

I am thinking about these terms in the context of linear control. A transfer function is proper if the degree of the numerator is not greater than the degree of the denominator. I've read often that ...
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2answers
445 views

Control design: under what conditions can closed-loop poles be placed arbitrarily?

Say we have a single-input linear system $\dot{\mathbf{x}} = A\mathbf{x}+Bu$. With full-state feedback ($u=-G\mathbf{x}$), it is straightforward to arbitrarily place the $n$ closed-loop poles (i.e., ...
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1answer
99 views

Why is it necessary to average signals with filters?

As someone who is new in working with filters, why do signals need to be averaged? What are the advantages, by performing filters on signals? As I know, there are different types of filters for ...
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2answers
157 views

Why do we assume zero mean noise in sensor data?

I am reading a paper on measuring respiratory patterns from video data. In defining the model, the authors formulate the problem mathematically as: $x_i(t)=h_i(t) \ast g(t) + n_i(t) $ Where $n_i(t)...
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0answers
62 views

identifying overshoots in given state space system with step inputs

Suppose I have the following state space system: $$ \dot{x}(t) = Ax(t) + Bu, \quad y(t) = Lx(t), \quad x(0) = x_0 $$ where $A$, $B$ and $L$ are real matrices, $u$ is a constant real vector (so that ...
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1answer
570 views

Differential equations and LTI systems [duplicate]

I've seen in many textbooks on Signals and Systems that an LTI (Linear Time-Invatiant) system can be described as a constant-coefficient linear differential equation, such as $$\sum_{k=1}^N a_k \frac{...
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1answer
848 views

Stability of a logarithmic function and system

A system such that $$y[n] = \log_{10}(|x[n]|) $$ was given as a stable one. I am not able to understand it though. When $x[n] = 0$ (which is bounded), the output $y[n]$ tends to get unbounded. So ...
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1answer
253 views

Linearity and shift-invariance of 2-D system on lattice

I know how to check a 1-D system for these conditions, but am confused about translating this to a 2-D system over a lattice. The system $H$ is define as: $$w[\mathbf x] = H\left\{u[\mathbf x]\right\}...
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1answer
1k views

Using the linear operator to check for time invariance of a differential equation?

I have a differential equation $$\frac{d^2y(t)}{dt^2}+y(t) = \frac{dx(t)}{dt} + x^2(t)$$ and I need to see if this system with input $x(t)$ and output $y(t)$ is time invariant and linear. I tried to ...
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3answers
722 views

Deconvolution and Polynomial division

I found this comment inside MATLAB's deconv.m function Deconvolution and polynomial division are the same operations as a digital filter's impulse response $B(z)/A(z)$. What does this statement ...
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0answers
51 views

Variance and Co-variance of a Linear Forecast

Consider a linear forecasting problem where all shocks $\{\epsilon_i\}_1^n$ are independently distributed with $\epsilon_i\sim N(0,\sigma_i^2)$ for all $i$. Suppose you want to forecast $\theta = \...
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3answers
361 views

BIBO stable LTI system frequency response for this input signal?

For an LTI system with bounded input and bounded output, I have the input $$x(t) = 5 + \cos(12t+\pi/4)$$ and output $$y(t) = 6\sin(12t)$$ It is said that the magnitude of the frequency response $|H(...
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0answers
74 views

DSP Homework Help - Proofs and Causality

I'm completing my last semester of university and for my last EE minor course, I had only one option available, an Audio and Speech processing class. Safe to say, it's difficult for me and would like ...
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2answers
4k views

What is the difference between a lag filter and “PI” control?

A lag filter/compensator has the form $$G_c(s) = \frac{s+z}{s+p}$$ with $-z < -p < 0$. In practice, the effect of lag compensation in feedback control is to increase the DC gain of the open-...
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1answer
87 views

Modeling an infinite delay system

If black holes swallow everything and assumes that the black holes do not get out the things from them, then can we consider a black hole as an infinite delay system? It takes the input and it is a ...
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1answer
337 views

Non invertibility of system $y[n]=x[n]-x[n-1]$ using transform method?

For the system to be invertible, we should have different outputs for different inputs. In terms of constant functions say, $$X_1[n]=3 \quad \forall n \in \mathbb{Z}$$ and $$X_2[n]=4 \quad \forall ...
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1answer
144 views

Fourier transform relationship

I am having trouble understanding the relationship between a frequency function and it's Inverse Fourier transform. The Frequency function is $$\frac{1+0.8(e^{-j 2\pi f}+e^{j 2\pi f})+0.64}{1+1.4\...
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1answer
162 views

What is the steady state response of a system of two exponentials to sinusoidal excitation?

What is the two exponential steady state response? Here is an example solution for sinusoidal excitation of a system having a single exponential response to a impulse excitation: Example: ...
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1answer
433 views

advantage of using impulse response function for LTI systems? [duplicate]

i have a discrete-time LTI system $L$ that takes input signal $x[n]$ and gives the output signal $y[n]$. since $L$ is LTI, $y[n]$ can be derived as a sum of shifted and scaled impulse responses of $L$....
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1answer
227 views

Impulse response (general form for linear systems as two-variables function $h(t,\tau)$) applied to Time-invariant systems

If a system is linear, then the operator $S$ mapping input signals into output signals - i.e. $y(t)=S\{u(t)\}(t)$ - is the integral of the input weighted by the impulse response: $$ y(t) = \int_{-\...
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1answer
338 views

Combined impulse responses

I'm looking for a simple way to show that you can form a single impulse response that is the equivalent $M$ other impulse responses, ie: $$ h\left(t\right)=h_0\left(t\right)\star h_1\left(t\right)...\...
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1answer
70 views

Is blending of colour lights a linear system?

Can we consider recombination process of different light colors a linear system/phenomenon? Like, when you shed two different light sources for example, a blue light and a yellow light on a surface or ...
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1answer
302 views

Causal system and Physical Systems

According to the Paley-Wiener criterion, a system is causal if satisfies: $$\int\limits_{-\infty }^{+\infty }{\frac{\ln (|H(f)|)}{1+{{f}^{2}}}}df<\infty$$ So I want to know This equation is ...
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2answers
92 views

convergence of Fourier transform of $e^{-t}\sin(2\pi ft)u(t)$

As you see Fourier transform function is being divergent for the first statement but it seems to converge. What is my fault? $$ \begin{align} \int\limits_{-\infty }^{+\infty }{{{e}^{-t}}\sin(2\pi ...
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1answer
368 views

How to shift input for testing shift invariance in a system?

As I know, when we want to test if a system is shift invariant, we define system as $g(x) = H(f(x))$ which $f(x)$ is the input function, $H$ is the system and $g(x)$ is the output function. Then we ...
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3answers
1k views

How to get the sum of unit steps and unit ramps from a discrete signal?

I have the discrete-time signal depicted below. This signal can be written as follows $$y[n] = \{0, 0, 2, 4, 6, 6 ,6,0,0,\ldots\}\,,$$ or as the sum of shifted impulses as: $$y[n] = 2 \delta [n-2]...
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3answers
1k views

Signals and systems : why do we study causal signals?

Till now I have read that causal signals are right sided and anti-causal, left sided. Why did we need to classify a signal with respect to its position? What is it's physical interpretation? ...
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1answer
443 views

Difference equations applied to DSP

I've found almost nothing about difference equations on the internet. Can you please recommend me something like books or pdf online that handle deep this topic? I'm searching also some exercises. ...
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1answer
133 views

Continuous-time frequency response question

Suppose the CTFT of continuous-time input $x_c(t)$ to an LTI system is $X_c(j\Omega)$ and that of its continuous-time output $y_c(t)$ is $Y_c(j\Omega)$. We have, $$X_c(j\Omega) = 0,\phantom{1}\text{...
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1answer
2k views

How to find ROC of system when input is two sided and output is one-sided

Given the $\mathcal Z$-transform of input $x[n]$ and output $y[n]$, how can I find the ROC of the system function $H(z) = Y(z)/X(z)$? I have $$X(z) = \frac{2z\left(z-\frac{10}{3}\right)}{\left(z-\frac ...