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
346 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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2answers
130 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
526 views

Convolution of Signal with a Gaussian Filter / Kernel

Following up on Analytical Solution for the Convolution of Signal with a Box Filter, I am now trying to convolve a Gaussian filter with the sine signal by hand. My method is to use the definition of ...
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1answer
691 views

Solving a second order difference equation

I need to solve the following 2nd order difference equation $$3y[n]-y[n-1]+y[n-2]={\bigg(\frac{1}{2}\bigg)}^{n} u[n]$$ where $u[n]$ is Heaviside's step function. I started by trying to find the ...
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1answer
224 views

Overshoot computations not matching

Take the following transfer function of a 3rd order system: $$H(s)=\dfrac{2.302~s+0.3548}{s^3+0.739~s^2+3.223~s+0.3548}$$ with poles: ...
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1answer
3k views

Convolution of $\sin(\omega t)$ and $\cos(\omega t)$?

If $x(t)=\sin(\frac{\pi t}{4})$ and $y(t)=\cos(\frac{\pi t}{4})$ then i need to find the Convolution $$z(t)=x(t) \circledast y(t)$$ So convolution will be $$\begin{align} z(t) &=\int_{-\infty}^{...
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2answers
97 views

How to show that the signal $x_n = A\cos(\omega n)$ can be fully predicted by a system with two weights $w_1,w_2$

I am trying to solve the following exercise: Show that the signal $x_n = A\cos(\omega n)$ can be fully predicted by a system with two weights $w_1,w_2$ (i.e. $x_n = w_1 x_{n-1} + w_2 x_{n-2}$). ...
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1answer
254 views

System Identification with Periodic Signal Input

I want to find the ETFE (Empirical Transfer Function Estimate) of the system $G(e^{j\omega})$: Where $H(e^{j\omega})$ is some filter that zero-mean white Gaussian noise $e(k)$ passes through. Let's ...
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1answer
170 views

How to periodically estimate states of a LTI if the output is measured irregularly?

How can I periodically estimate the states of a discrete linear time-invariant system in the form $$\dot{\vec{x}}=\textbf{A}\vec{x}+\textbf{B}\vec{u}$$ $$\vec{y}=\textbf{C}\vec{x}+\textbf{D}\vec{u} $$...
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1answer
83 views

Representing a continuous LTI system as a discrete one

I am aware that there are different ways to represent a continuous time system in discrete domain (e.g. bilinear transform, impulse invariance transform). But my problem is as follows: Given an ...
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134 views

How to implement the RLS for matrices

I need to implement the RLS algorithm but it's for matrices instead for vectors, I have made the below code, but still something wrong is not working well, EDIT: The code should be done as below, ...
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3answers
6k views

Why $y[n] = x[-n]$ is not time-invariant?

I followed these steps, but the answer still says that this system is time-invariant let: $x_2[n] = x[n-k]$ $$\begin{align} y_2[n] &= x_2[-n] \\ &= x[-(n-k)] \\ &= x[k-n] \\ \end{align}...
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2answers
272 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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3answers
2k views

Why unit impulse function is used to find impulse response of an LTI system?

Hello i am working in digital image restoration field, recently i have studied concept of convolution, i studied that to find the impulse response/point-spread function of an LTI system, an unit ...
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3answers
499 views

Is this system linear?

I have this system: $$ y[n] = -\frac{1}{2} x[n+2] - y[n+1] $$ I have no idea how to prove if the system is linear or not, because, it depends on future outputs... Thanks for the help
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3answers
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Solving for impulse response h[n] given input-output pairs

This is a homework problem that I've worked on and I want to confirm some of my reasoning. I'm given two sets of input-output pairs of a particular system, $S$, that we know is linear time-invariant ...
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3answers
2k views

Why cosine is not an eigen signal?

According to this website: If the output of a system has the same type as its input signal, then the input signal is referred to as the eigen function of the system. but in this question it is ...
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2answers
202 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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2answers
2k views

How to modify an LTI differential equation to be acausal or anti-causal?

I'm trying to wrap my head around causality in LTI-systems. Considering continuous time only, I'm happy with the fact that the system is causal iff the impulse response function $h(t)=0$ for $t<0$. ...
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1answer
478 views

System response: LTI system for $x[n] = \sin(\frac{\pi n}{4})$

Consider the LTI system with frequency response $$H(e^{j\omega}) = \frac{1-e^{-j2\omega}}{1+\frac{1}{2}e^{-j4 \omega}}, -\pi < \omega < \pi$$ Determine the output $y[n]$ for all $n$ if the input ...
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6answers
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Why do we always characterize a LTI system by its impulse response?

Why do we always characterize a LTI system by its impulse response and not by another response, like the step response? What does the impulse response have that is so special?
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1answer
594 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 $$T(...
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2answers
81 views

Linear model of the modified error function

We can write the error function ($E(w)=1/2\sum_{n=1}^{N}\{y(x_n,w)-t_n\}^2$) as a linear model using its partial derivatives. Is it possible to do the same thing about the modified error function? ...
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1answer
725 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
2k 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 pole-...
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2answers
244 views

LTI system output given input and frequency response

The question I'm trying to understand is as follows: A linear time-invariant continuous-time system has the frequency response function $$H(\omega)=\frac{1}{j\omega+1} $$ Compute the output response $...
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2answers
117 views

Are Chebyshev Type 1 filters very unstable at very high orders?

I am currently experimenting with a digital implementation of a Chebyshev Type 1 filter of mine, for audio purposes. I have written the code myself so I was wondering if the behavior I've noticed is ...
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2answers
108 views

What Is the Derivative of the Function $ s \left( t \right) = \left( 1 - {e}^{\frac{-t}{RC}} \right) u \left( t \right) $?

What is the derivative of the following function $$ s \left( t \right) = \left( 1 - {e}^{\frac{-t}{RC}} \right) u \left( t \right) $$ with respect to $t$, where $u(t)$ is a unit step ...
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2answers
708 views

Linear Constant Coefficient Differential Equations: Zero-Input and Zero-State responses

The solution to a linear constant coefficient differential equation of the form $$\sum_{k = 0}^{N} a_k y^{(k)} (t) = \sum_{k = 0}^{M} b_k x^{(k)} (t)$$ can be written as $y(t) = y_{ZI} (t) + y_{ZS} (t)...
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3answers
1k 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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2answers
240 views

What is the difference between y[n] and y(k) (using square brackets)?

Can I modify y[n] = αy[n-1] + x[n] the same way I would do it with y(k) --> Y(z)? I need H(z), but I don't know what is the difference between using (k), and (n). I often encountered [k] notation, ...
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3answers
414 views

Causal unstable system turn into stable anticausal?

I would appreciate it very much if someone would be able to provide some clarity, help or comment on this problem. I have been reading several papers on time series identification such as https://www....
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1answer
421 views

Signals and systems book with linear algebra approach

I am currently taking a course on discrete-time signals and systems. I am using B. P. Lathi's Signal Processing and Linear Systems, which I don't like at all, as it doesn't draw any paralells to ...
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2answers
241 views

Is system $h(t) = u(t)-u(t-1)$ LTI or not?

$$y(t) = \int_{0}^{\infty} x(\tau)h(t-\tau) \ \mathrm{d}\tau$$ where $h(t) = u(t)-u(t-1)$. I have some problem with understanding whether system LTI or LTV just looking the impulse response of the ...
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1answer
245 views

Is the sum of discrete sinusoids an eigenfunction of an LTI system?

Does $$x[n]=e^{j\omega n}+e^{2j\omega n}$$ represent an eigenfunction of an LTI system?
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1answer
815 views

Question on convolution

I am stuck at question number 2.8 This is how I have gone about solving it: I have calculated $y(t)$ by convolving $x(t)$ with $h(t)$ using the fact that $x(t)$ convolved with an impulse at $t=t_0$ ...
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1answer
939 views

Exponential decaying step response in LTI System

I'm attempting to better understand the relationship between step responses, impulse responses, and convolutions. Say that I have a system where if I apply a constant input, my output decays from a ...
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2answers
7k 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
8k views

Compute step response from impulse response of continuous-time LTI system

The impulse response of the LTI system is $$h(t)=e^{-4t} u(t)$$ The expression for the step response is $$\frac14 \left(1-e^{-4t}\right)u(t)$$ My question is how $u(t)$ appears in the answer.
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1answer
2k views

How does one calculate a pole-zero plot?

To my understanding, pole-zero plots are used to analyze or visualize transfer functions. Suppose there is some very simple system, for example a simple low-pass filter (so it is linear and time-...
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1answer
92 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 evaluated ...
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1answer
44 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
155 views

Frequency response of a system given its block diagram

Given the following block diagram, I am asked to find the frequency response $H(f)$ of the system. This is what I have done: The output of the first block is $j2\pi fX(f)$, and after going through $...
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1answer
33 views

Show That a 2D Linear Transform $ T \left( \cdot \right) $ Is Homogeneous

By my understanding, a transform T is homogeneous if T[0] = 0. Then to prove that a linear transformation is homogeneous we say that: T[ax(n1, n2) + bx(n1, n2)] = aT[x(n1, n2)] + bT[x(n1, n2)] What ...
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1answer
80 views

How to detect algebraic loop in a system

I came across a system (screen below) that looks like it should result in a algebraic loop, but when writing the equations using the Z-transform, it obviously does not (calculation below too). But I ...
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2answers
214 views

Frequency response of marginally stable LTI systems

The frequency response of a system is defined as: $$\int_0^\infty{h(t)e^{-j\omega t}dt}$$ where $h(t)$ is the impulse response. But in marginally stable systems, $h(t)$ does not decay so the integral ...
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1answer
819 views

Covariance matrix, Q, for a Kalman filter given the stochastic differential equation for the state of the system?

Given that I have a stochastic differential equation describing the motion of my system like so: $$ \ddot{x}(t) + \Omega_0^2x(t) - C\dfrac{dW(t)}{dt} = 0$$ Where $\Omega_0$ and $C$ are constants. I ...
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1answer
1k views

Is this system invertible or not?

Prove that the following system is invertible. $$y(t) = \mathcal{T}\{x(t)\} = \int_{-\infty}^{3t} x(\tau) \,\mathrm d \tau$$ Answer: yes, the system is invertible. I need some hint here, not the full ...
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1answer
133 views

Analyzing a particular discrete-time LTI system for input signal $x[n]=(1/3)^n$ for *all* $n$

I'm considering the following problem from some course notes. Suppose the following is known about a discrete-time LTI system: Given the input $x[n]=(1/3)^n$ for all $n$, the system ...
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1answer
669 views

Lowpass filter impulse response in frequency domain

A Linear Time-Invariant system with impulse response $h_1[n]$ is an ideal lowpass filter with cutoff frequency $\omega_c =\pi/2$. The frequency response of the system is $H_1\left(e^{j\omega}\right)$. ...

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