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 ...

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7 views

Mathematical model of a large AR system for sparse system identification

I am new to the topic of system identification and looking for a large model of Autoregressive (AR). I am working on Sparse system identification http://tbayes.eecs.umich.edu/yilun/sparselms (example) ...
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1answer
33 views

Is the following system LTI (finite speed of signal propagation)?

I wonder if the following acoustic system is i) considered to be linear time-invariant, and ii) if it is fully described by it's frequency response? A speaker produces waves at the input location of ...
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1answer
40 views

Relationship between Discrete Deconvolution and Toeplitz Matrices

I have 2 vectors, $a$ & $c$, both of length M. I know they are related by $a*b=c$. My goal is to recover $b$. Obviously $b=$deconv$(c,a)$. I am only interested in the first M elements of the ...
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32 views

Coefficient Cancelation on Particular Solution

I am stuck on a problem about finding the complete solution of a Discrete Linear System by using the parametrization $y_{c}(n) = y_{h}(n) + y_{p}(n)$ where $y_{h}(n)$ and $y_{p}(n)$ are, respectively, ...
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2answers
62 views

Impulse Response with non-delayed feedback

I've been trying to solve a Discrete LTI System problem where the overall impulse response of the following System, whose input and output sequences are, respectively, $x(n)$ and $y(n)$, has to be ...
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1answer
44 views

How to calculate the zero state response?

In the following question and answer I kind of get it how it is calculated at surface but I don't get it fully. Can someone please explain the answer in depth?
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2answers
86 views

Criteria to distinguish FIR and IIR filters from difference equation?

What criteria should be used to safely decide if it is an IIR filter or FIR filter from a difference equation? FIR filter are always stable, meaning all poles are inside the unit circle AND have ...
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2answers
92 views

Can I construct a linear system so that it results in a moving average?

I've just learned about linear systems and impulse response functions. I know that in a linear system consisting of $h_1[k]$ and $h_2[k]$, the impulse response of the system is $h_1[k] \ast h_2[k]$. ...
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1answer
167 views

The impulse response, $h(t)$, of a system

I have this differential equation $$ y''(t) + y'(t) + y(t) = x(t) - 3x'(t) $$ I found the system response of the system: $$ H(f) = \frac{1 - 6\pi fj}{(2\pi fj)^2 +2\pi fj+1} ...
3
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2answers
59 views

Deconvolution with an underdetermined system of equations?

If I have a measured signal $\mathbf{y}$ which is the result of the true signal $\mathbf{x}$ convolved with another function (linear and not circular convolution), I always seem to get an ...
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0answers
128 views

Solving linear system and find impulse response

I have this linear system that is defined by the differential equation $$ y''(t) + 4y'(t) + 5y(t) = 2x(t) + 3x'(t) $$ with x(t) the input and y(t) the output. I'm asked to find two ...
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26 views

Using output error model to model colored noise (using a prefilter)

I have a "black box" SISO system. I use zero input to measure pure noise on the output. After preprocessing (peakshave and removing DC) I try to model the noise using AR (and also ARMA). When ...
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12 views

Affine Non Autonomous State Space System

Normally We all know the state space model of the the form $$\dot{x} = Fx(t)+Gu(t)$$ $$y = Hx(t)+Ju(t)$$ However I came across a state space model which has the following form $$\dot{x} = ...
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1answer
228 views

Convolution of a signal by itself properties

I'm studying for a test and I can't really explain this question very well. If we convolve a signal by itself, will it have the same properties (time-invariant, linear, stable, casual)? I know how ...
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1answer
124 views

Are complex exponetials the only eigenfunctions of LTI systems

Can I have an example of eigenfunctions of Linear Time Invariant(LTI) which is not a complex exponential? This document: Eigenfunctions of LTI Systems says such eigenfuctions do exist. But I am not ...
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2answers
86 views

Is there a way to obtain the impulse response of a discrete system by just knowing it's response to the discrete unit step function?

In continuous time it was possible; $$\Large u(t)\rightarrow \Large{\Box}\rightarrow y(t)$$ $$\Large{\delta(t)=\frac{du(t)}{dt}\rightarrow \Box\rightarrow \frac{dy(t)}{dt}=h(t)}$$ Does the same apply ...
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1answer
83 views

What filter is better with -1/3 2/3 -1/3 or with -1/4 1/2 -1/4 coefficients?

Professor asked me what filter is better -1/3 2/3 -1/3 or -1/4 1/2 -1/4, the answer was second. But what those coefficients means? It was during discussion of linear filter but i'm not sure was it ...
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1answer
120 views

Causal systems - Paradox? Confusion of definition?

For a system such that $y[n]=cos(x[n])$, I'd like to determine whether the system is causal or not. Definition 1: A system is causal if and only if the output $y[n]$ is a function of a linear ...
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1answer
81 views

Determining linearity, causality, memory, and time invariability from a picture of the signal

So, this is a picture of saccadic eye movement reflected by an EOG. How can I determine if the eye movement system is linear, causal, memoryless, and time invariant from this displayed waveform? ...
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61 views

Calculating complex frequency response and transforming a sinusoid question

I need help with a question from signal processing class. Question as follows: Question is in two parts Part 1: The ideal delay system is defined by $y[n] = x[n-n_{d}]$. First, derive the systems ...
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3answers
261 views

Sum of Sine and Cosine with Random Phase as LTI System

I have the following system: Where $ {H}_{1} \left( f \right) = {H}_{2} \left( f \right) $ and $ \theta \sim U[0, 2\pi]$ independent of any other factor in the system. Given the input is identical, ...
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1answer
65 views

Choosing inverse Z-transform equation, given that $|a|<1$

Given that $|a|<1$, then which of those inverse-Z-transform equations are we to use? I am leaning towards the first because (as I understand it), $z$ is merely a complex number that is ...
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45 views

Ideas on matrix factorizations and/or transformations for $\ell_1$ minimization

I am starting with a typical $\ell_1$ basis pursuit problem: $$ \min_{\mathbf{x}} \Vert \mathbf{x} \Vert_1 \quad \mathrm{s.t.} \quad \Vert \mathbf{ERx} - \mathbf{y} \Vert_2 \leq \epsilon, $$ where ...
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2answers
102 views

Situations where complexity is too big to exceed linearity and gaussianity

I'm studing about Kalman Filter and Particle Filter in multiple target tracking in computer vision (tracking pedestrians). Reading sientific papers I'm colliding with a lot of sentences like: The ...
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18 views

Panorama global registration

I'm trying to do global registration algorithm for panorama like described here. But I don't understand how they deal with false pairs. Also if expand the system u1*I*P1=u1*I => P1=I ...
3
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2answers
120 views

Frequency response of system with feedback and feedforward co-efficients?

For seismological application, I am playing with an elliptical filter, and once I have both sets of feedback and feedforward co-efficients, how do I make it so that I can plot the frequency response ...
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2answers
2k views

What are Lumped and Distributed Systems?

What is the exact difference between Lumped and Distributed systems? Also, Lumped systems are said to be ordinary differential equations while the latter is said to be partial differential equation. ...
3
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1answer
200 views

Impulse Response Question

For the differential equation $$\frac{d^2y(t)}{dt^2}+\frac{3dy(t)}{dt} + 2y = x(t),$$ I was able to find the frequency response as $$H(j\omega)= \frac{1}{-\omega^2+3j\omega+2}.$$ However, I am not ...
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1answer
93 views

How can frequency dispersion in a vibrating string be modelled using an LTI filter?

According to JOS on stiff-string synthesis, stiff strings (like on a piano) introduce inharmonicity (i.e. the harmonics of the tone are not all in tune) due to different frequency components of the ...
3
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1answer
122 views

How to perform model fitting for system identification

I am having a really hard time in understanding how to formulate a model say linear AR model to represent a communication channel or maybe any motion. I have the experimental data representing the ...
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1answer
233 views

Problem with the definition of Linearity

From the high school maths we know that y=mx+c is a linear equation. However, in DSP the linear system must satisfy Additivity properties which y=mx+c does not hold because of +c. So, is the ...
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1answer
76 views

Calculating a delayed response of a system

Suppose I have a system: $$y[n] = x[n] + x[n-1] + x[0]$$ To see if this system is time-variant, we can compare $H\{x[n-1]\}$ to $y[n-1]$ and check that they are different. I think the delayed (by ...
3
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1answer
384 views

Group Delay of a system

Say I have an LTI system whose impulse response h[n] is: $$ h[n] = \delta[n] + \delta[n-4] $$ I want to find the group delay of this system. I know the group delay ($grd$) is defined as: $$ ...
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1answer
150 views

Why is the state space representation, $dx/dt = A x(t) + B u(t)$ linear?

One thing puzzles me about the state space representation and that is why is it claimed that such systems are linear and time-invariant but particularly why are they claimed to be linear? The ...
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1answer
865 views

How to deduce a linear system's impulse response from a set of input/output signals?

I want to know how to solve those types of problems.. is it by inspection ? Consider the linear system below. When the inputs to the system $x_1[n]$, $x_2[n]$ and $x_3[n]$, the responses of the ...
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253 views

Concept of finding forced response from difference equation?

I know how the process works, first from difference equation make a homogeneous equation and then find natural response , then find a particular response (in the form of input) substitute it back in ...
3
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1answer
75 views

ARMA Process Representation as AR

Let's say we have an ARMA (Auto-regressive Moving Average Model) process where the transfer function is a minimum phase system (Namely Invertible). By Wold's Decomposition it is guaranteed to have MA ...
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1answer
189 views

Inverse Discrete Time Fourier Transform of a given Function using the definition

I would like to compute the Inverse Fourier Transform of a given function using the DTFT definition $$H(e^{jω})=\frac{1}{(2−e^{jω})(2−e^{−jω})}$$ But I have to use the definition of the DTFT ...
3
votes
1answer
187 views

How do I compute the impulse response of a robot?

I am trying to model a robot (my question in robotics.SE), so that I may train it using some reinforcement learning techniques. I was wondering if I can compute an impulse response of a four wheel ...
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1answer
259 views

Calculating the system output using frequency response

Given an input signal $$x(n)=\cos(6\pi n +\frac{\pi}{6})$$ and system $$y(n)=0.5x(n)-0.1x(n-1)$$. In this case, the coefficients of the difference equation are $a_0=1$, $b_0=0.5$, and $b_1=$. The ...
2
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1answer
246 views

Verifying Linear Time Invariance

I have a system of the form: $$T(x(n))=x(n)+3x(n-2)-5x(n-3)x(2n)$$ I claim that $$T(x(n-k))=x(n-k)+3x(n-k-2)-5x(n-k-3)x(2n-2k),$$ $$y(x(n-k)= x(n-k)+3x(n-k-2)-5x(n-k-3)x(2n-2k)$$ and ...
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5answers
1k views

Why do linear systems show sinusoidal fidelity?

I am looking for a proof for sinusoidal fidelity. In DSP we study a lot about linear systems. Linear systems are homogenous and additive. One more condition it satisifies is that if a signal is a sine ...
7
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1answer
384 views

Theory behind Linear Predictive Coding (LPC)

What is the theory behind LPC? Why are(were) certain implementations of LPC said to be more tolerant of transmission or encoding errors quantization than other compressed voice encoding schemes? ...
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1answer
175 views

Fitting new images from a SVD/PCA calculation

I'm trying to replicate the ideas from the Eigenface page on wikipedia. From a hundred sample images represented by a data matrix $\bf X$ (where each image flattened to a vector of length $n$, thus ...
4
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1answer
513 views

Finding the Wiener filter transfer function

I am trying to understand the mechanism of finding the transfer function for a Wiener function. ...
6
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1answer
284 views

How to combine a rotation matrix and a stretch matrix into a single matrix for easy Fourier Transform

For full disclosure, this is related to homework. I have to find the Fourier Transform of a function that I've boiled down to the following. I have a function $f(x,y)$ that I can think of as another ...
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2answers
702 views

Is Ideal LPF BIBO unstable?

In one of other discussions : How to find frequency response, stability, and causality of a linear system? I found a comment which was quite strong and definitely caught my attention. An ideal ...
6
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1answer
4k views

How to find frequency response, stability, and causality of a linear system?

I have the following transfer function: $$H(s)=\frac{s}{(s+1)(s+2)}$$ How can I find the gain and phase response of the above system? I know the first step has something to do with substituting $s = ...
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4answers
24k views

What is meant by a system's “impulse response” and “frequency response?”

Can anyone state the difference between frequency response and impulse response in simple English?
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1answer
1k views

How do I find a system's impulse response from its state-space repersentation using the state transition matrix?

Suppose we have a linear represented in the standard state space notation: $$ \dot{x}(t)=Ax(t)+Bu(t)$$ $$y(t) = Cx(t) + Du(t)$$ In order to get its impulse response, it is possible to take its ...