Questions tagged [discrete-time]

For questions related to discrete time. Discrete time views values of variables as occurring at distinct, separate "points in time", or equivalently as being unchanged throughout each non-zero region of time ("time period")—that is, time is viewed as a discrete variable.

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Stability of fixed points in a 2D discrete time dynamical system

I am studying the stability of fixed points in the Henon map for a project, and I have a question regarding it. The Henon map we are studying is given by: \begin{align*} x_{n+1}&=1+y_n-\alpha x_n^...
Sarah Hadaidi's user avatar
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Find the slope at a given time value of this equation of a heart [closed]

The following equation draws the curve of a heart: $x(t)=12sin(t)-4sin(3t)$ $y(t)=13cos(t)-5cos(2t)-2cos(3t)-cos(4t)$ Image https://de.wikipedia.org/wiki/Benutzer:Georg-Johann I would like to find the ...
3jay1's user avatar
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24 views

find the position a mobile when $x'(t)=2 y'(t)$

A mobile is moving along the graph of $f(x)= \ln(x)$ , find the position of the mobile at the point in time when the rate of change of the abscissa is twice the rate of change of the ordinate. My ...
Antony Theo's user avatar
1 vote
1 answer
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In a discrete-time Markov chain, is the probability of not being able to get from a state $u$ to $v$ a rational number? [closed]

One can think about a discrete-time Markov chain as a weighted directed graph $D = (V, E)$ where $E \subseteq V \times (0, 1]$, as each vertex represents a state, and an edge $(u, v)$ with weight $w$ ...
sbh's user avatar
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When does an optimal input sequence for a discrete-time system exist?

Suppose an LTI discrete-time system is given by the equations $$ x_{k+1} = Ax_k + Bu_k,\\ y_{k} = Cx_k + Du_k $$ with $x_k\in\mathbb{R}^{m}$, $y_k\in\mathbb{R}^{n}$ and $u_k\in\mathbb{R}^{p}$ and $\...
Benjamin Tennyson's user avatar
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1 answer
27 views

Does a solution exist for this discrete time function with variable parameter?

I am interested in a simple parametric function: $$y(t) = \frac{a}{b} (1-e^{-b t})$$ Where $a$ and $b$ are real parameters greater than zero; $y(0)=0$; and $t$ is time in discrete units, e.g. months ...
FlamingGoose's user avatar
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Find Gain K and Time constant K of a system from the time response

There is a given system $\frac{K}{sT + 1}$ of order 1. The responses are in the image below and the 2 inputs are $u1(t) = 1(t)$ and $u_2(t) = \sqrt{2} \cdot \sin(\omega_2 t)$. How can I find the K and ...
sneha_jerin's user avatar
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64 views

Mapping transient characteristics from S domain to Z domain

The $s$ and $z$ domains are linked by the expression $z = e^{sT}$, where $s$ is the Laplace variable and $T$ is the sampling period. However, I have found no rigorous method that attempts to derive ...
Saeed's user avatar
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Final value of a recursion

Problem Given $p_1, \sigma > 0$, consider the following recursion \begin{equation*} p_{i}=(1-L_i)p_{i-1} \qquad i=2,\dots,k \end{equation*} where \begin{equation*} L_i \triangleq \frac{p_{i-1}}{p_{...
matteogost's user avatar
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72 views

Stability of discrete-time dynamical systems using Lyapunov stability where A is function of optimization variable

Hi I am trying to solve a constrained optimization problem using the Lyapunov stability. In the problem we aim to find $\beta$ such that $$\min_\beta ||\beta^TF-y|| \quad \text{s.t.}\quad A^{T}PA-P&...
geo200's user avatar
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Online algorithm for discrete time serie derivate

I'm looking for an algorithm to compute the numeric derivative $\dot x_i$ of a discrete time serie $(x_i, t_i)$. It runs in realtime, hence $\dot x_i$ can only depend on $x_{j<=i}$. The memory ...
piwicode's user avatar
2 votes
2 answers
82 views

Which stability criterion to use for LTI system with Gaussian noise?

This might (and hopefully will) be a very simple question but I'm quite stumped after doing some research: consider the noise-free discrete-time LTI dynamics given by $ \boldsymbol{x}(k+1) = \...
Bart Wolleswinkel's user avatar
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Find vectors a,b of causal discrete time filter, that correspond to moving average filter

I came across the following problem in a signals graduate class: Given the following general formula for a discrete causal filter, let's name (1): $y[n] + \sum_{j=1}^{M} a_j*y[n-j] = \sum_{i=0}^{N} ...
spiros's user avatar
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Topological conjugacy of the logistic map at different parameter values

I am wondering whether the dynamical systems generated by the discrete 1 dimensional map $g(x;p) = px(1-x)$ (the logistic map) at different values of $p$ are topologically conjugate. Of course, this ...
its_all_a_DS's user avatar
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51 views

Do other resources exist on Rate-Independent Systems?

Are Rate Independent System new in the world of mathematics? I can't tell because I certainly am new to mathematics. I couldn't find any more resources than this book: Rate-Independent Systems: Theory ...
Shadman Sakib's user avatar
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21 views

Correlation between spectral coefficients of a random process when the sample space is finite

This question has been bothering me for quite some time. Consider a complex random process in time domain $s$, whose Fourier transform is $S$. In many papers, I see that they assume: $$ \mathbb{E}[S_p ...
CfourPiO's user avatar
3 votes
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94 views

Determine the height $h(t)$ of the water level as a function of time $t$

Water flows out of a filled cylinder of base area $G$ through a small hole at the bottom. Let $h_0$ denote the initial height of the water at time $t_0 = 0$. Determine the height $h(t)$ of the water ...
Euler007's user avatar
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Discrete-Time Fourier Transform: Why are signals at frequencies near even multiples of $\pi$ slowly varying, but quickly varying near odd multiples?

In Signals & Systems: Second Edition by Alan V. Oppenheim and Alan S. Willsky with S. Hamid Nawab, on page 362, it is stated that (with referenced figure shown below): Signals at frequencies near ...
William's user avatar
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Deriving the Function which Minimizes the Sum of a Difference Equation given Constraints

We have a continuous variable $s$ which changes over discrete time $t$ according to the difference equation: $$ s_t = s_{t-1} + \lambda - \frac{Kn_t}{N} $$ where $0 \leq s \leq 1$, $n = 1, 2, ..., N$, ...
Colin M. Lynch's user avatar
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81 views

How can an exponential Fourier Series model a real function f(x) if the series contains imaginary numbers??

I'm an trying to model an arch using Fourier series. since I don't have the exact function of the arch, I want to take a sequence of points along the arch at fixed intervals and find the Fourier ...
Kian's user avatar
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How to take the conjugate of a single point in a purely real discrete time series (e.g. ECG). As seen in the auto-Wigner Distribution

Im trying to implement the Wigner distribution and I'm stuck on how to take the conjugate of a single point in a function. I can understand finding the conjugate of a function, and the conjugate of a ...
Taha Abbasi Hashemi's user avatar
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1 answer
66 views

Discrete Time LQR and Lyapunov Equations

Consider the following discrete time LQR system, where the dynamics are dictated by $$x_{t+1} = A x_t + B u_t.$$ and costs are $$C(K) = \sum_t x_t^\top Q x_t + u_t^\top R u_t.$$ For simplicity assume ...
Seth Devlin's user avatar
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31 views

Fourier pair of combining time shifting and multiplication in time

There are 2 known properties in DTFT: Time shift corresponds to multiplication with complex exponential $$x[n-k] \rightarrow e^{-j\omega k}\cdot X_{2\pi}(\omega) $$ Multiplication in time ...
vakula85's user avatar
2 votes
1 answer
112 views

Sufficient conditions for stabilizability of [A,B]

Let $\mathbf{A}$ and $\mathbf{B}$ be $n\times n$ and $m\times n$ matrices, respectively, the pair $(\mathbf{A},\mathbf{B})$ is stabilizable if there exists an $n\times m$ matrix $\mathbf{K}$ such that ...
user971821's user avatar
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Finding mutual information in discrete linear partial observation stochastic process

I have one basic question maybe is not to hard for you but I am a bit confused. Let our system be like this: \begin{align} X_{k+1} &= A_k X_k + W_k \\ Y_k &= C_k X_k + V_k \end{align} where $...
Kasi's user avatar
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Unity game ATB system with a minimum turn timer

I'm trying to make a spin on the Active Time Battle mechanic (ATB) most commonly seen in the Final Fantasy series. ATB is a timing mechanic, whereby units in combat will each have a meter that fills ...
Adam Ratcliffe's user avatar
2 votes
0 answers
68 views

Prove iterations of quadratic maps tends to $-\infty$

So far, I have to read about quadratic map in "Introduction to Dynamical Systems" by Brin and Stuck. The quadratic map is $$q_{\mu}(x) = \mu x(1-x), \quad \mu > 0.$$ Now, I need to show ...
FactorY's user avatar
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2 votes
2 answers
292 views

Stability of discrete-time dynamical systems using Lyapunov stability

I am studying the use of LMIs as an analysis tool for discrete-time dynamical systems. Consider the autonomous discrete-time system given by $$ x_{_{k+1}} = A x_{_k} \tag{1} \label{sys} $$ where $ x \...
AdamsK's user avatar
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Time Discretisation of a System of PDEs

Suppose we have an arbitrary system of PDEs $$ \partial_t u - D_1 \Delta u + a(v) u = f(t)$$ $$\partial_t v - D_2 \Delta v + b(u) v = g(t) $$ We want to discretise the system in time. We use the ...
Thede's user avatar
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1 vote
1 answer
59 views

Combined Markov Chain of N identical Markov Chains

I have a markov chain from https://ieeexplore.ieee.org/document/9766097 as follows The state transition matrix is as below The analysis pertains to packet drop rate for ONE device with certain ...
Yukti Kaura's user avatar
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44 views

Can anyone explain how the $\text{(magnitude)}^2$ can be calculated with multiplying $\frac{1}{1-ae^{jw}}$ from this DTFT?

Can anyone explain how the $(\text{magnitude})^2$ can be calculated with multiplying $\frac{1}{1-ae^{jw}}$ from this DTFT? Provided here. Kindly click in to see the equation
AYW's user avatar
  • 1
11 votes
4 answers
255 views

Determine $f(x)$ knowing $f(x)+f(x+\varepsilon)$

I encountered a problem at work that, in my opinion, has a fundamental mathematical reasoning to determine its solvability. Due to an unwitting software configuration, my associate recorded the audio ...
nickh's user avatar
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0 answers
24 views

Average time of random walk's maximum/minimum

I'm interested in a formula to calculate the average time (step count) at which a discrete one-dimensional random walk with $n$ steps (of lengths -1 and +1) either builds its maximum or minimum ...
Anti's user avatar
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0 answers
38 views

How should I interpret parameters of the SARIMA model in time series analysis?

I am a bit confused as to why the SARIMA model requires four parameters beyond the ARMIA model just to remove the seasonal component from a time series. Obviously $m$ is required to specify the ...
user10478's user avatar
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0 answers
26 views

Are non-constant polynomial means a special case of seasonality?

In this video, it is said that an otherwise-stationary time series with non-constant linear mean is analyzed by taking the first difference of the time series to produce a new, stationary time series. ...
user10478's user avatar
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4 votes
2 answers
182 views

Girsanov theorem for discrete-time stochastic processes

I am reading Buehler et al. (2022) "Learning to Trade II: Deep Hedging" and the slide on p. 44 states Fun fact: in discrete time, we can change also the volatility of a process by changing ...
p.vitzliputzli's user avatar
0 votes
1 answer
67 views

Limiting distribution of a Markov Chain with period 3

Question: Fix $\alpha \in (0,1)$. Let $(Y_n)_{n=0,1,2,...}$ be a Markov Chain with state space $S=\{1,2,3,4,5\}$ and transition matrix $$P= \begin{bmatrix} 0 & \alpha & 1 - \alpha & 0 &...
Balkys's user avatar
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In search of vector autoregression models supporting uniform bounds on coordinate-wise derivatives

This question is motivated by the desire to build mathematical models that forecast vector-valued discrete time series while guaranteeing a kind of "continuity" via uniform bounds on the ...
Bilal Khan's user avatar
2 votes
0 answers
69 views

Discrete time Fourier transform (DTFT)

I tried solving the following exercise but I am not sure about the result. Could you help me? The signal $x(t)= 12\text{sinc}^2(4t) + 4 \cos(4 \pi t)$ is sampled with a rate $f_c = 3$. It then becomes ...
Maghreb_1911's user avatar
2 votes
1 answer
68 views

What is the equivalent formulation for the Logistic Growth model in a population dynamics/branching processes setting?

I am reading a recently released book entitled Branching Processes: Variation, Growth, and Extinction of Populations by Haccou, Jagers, and Vatutin. One key concern of the book is the connection ...
krishnab's user avatar
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1 vote
1 answer
63 views

Discrete State Space eigenvalues: approach unity as loop rate increases?

Say I have a stable discrete State Space control system (feedback system) of the standard formulation: x[k+1] = Ad*x[k] + Bd + u [k]. Why do the |discrete eigenvalues| approach the unit circle (z = 1) ...
J B's user avatar
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1 vote
1 answer
27 views

Is this the proper approach to Discrete Time Fourier Transform of a derivative of a signal?

So in continuous time, the relationship is: $$\mathcal{F}(\frac{dx(t)}{dt}) \leftrightarrow j\omega X(e^{j\omega})$$ And from: $$\mathcal{F}(x(t)) = X(e^{j\omega})$$ We can recover $F(x(t))$ from the ...
Researcher R's user avatar
1 vote
1 answer
116 views

Should discrete and continuous time models give the same results?

Today I thought a lot about very simple population models, and there are still a few things that bug me. Consider a simple discrete exponential growth function: $$ n(t+1) = n(t) + n(t) b - n(t) d = n (...
Mirko's user avatar
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1 vote
1 answer
55 views

Monte Carlo simulation for economic growth

internet. For the institution I work for, I need to investigate economic data. What I was asked for is that based on any historical economic data of the country, I need to build "base","...
blknt3191's user avatar
-1 votes
2 answers
93 views

Lyapunov criteria for discrete linear system with noise

Consider constant model matrix $A$ and $B$, the Lyapunov criteria for system $x_{k+1}=Ax_k+Bu_k$ with state feedback input $u_k=Kx_k$ (K is designed matrix) is $P-(A+BK)P(A+BK)^\top>0$, where $P$ ...
Jeremy's user avatar
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0 votes
1 answer
159 views

frequency and periodicity

This is really a basic question: Let's say we have a message that is sent 2 times each second (periodic message). Thus, we can say that each 0.5s we send a message. But I am trying to understand it in ...
tonyjk's user avatar
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0 answers
25 views

A put option and a call option with identical exercise price are both marketable or neither is.

I am doing Exercise 1.13 in Introduction to Mathematical Finance: Discrete Time Models by Pliska. Exercise 1.13 Suppose the interest rate $r$ is a scalar, and let $c$ and $p$ denote the prices of a ...
atlantic0cean's user avatar
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0 answers
32 views

Properties of convolution for time discrete signals: Why is signal convolution with "delta" signal true?

I'm trying to explain to myself why this property is true; signal convolution with "delta" function = signal However, I don't understand why? I know that the Diracs delta function has a ...
begin's user avatar
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2 votes
1 answer
180 views

How to find appropriate discretization method for a continuous time domain state space model?

I have a dsp algorithm which is based on the below given state space model in the continuous-time domain $$ \begin{bmatrix} \frac{\mathrm{d}\hat{\psi}_{r_{\alpha}}}{\mathrm{d}t} \\ \frac{\mathrm{d}\...
Steve's user avatar
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1 vote
0 answers
281 views

How to discretize a continuous time domain state space model?

Let's say I have a state space model of a linear dynamic system in the continuous time domain in following form $$\dot{\mathbf{x}} = \mathbf{A}\cdot\mathbf{x} + \mathbf{B}\cdot\mathbf{u}$$ My goal is ...
Steve's user avatar
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