# 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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### 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","...
13 views

### Determining the time required to travel a distance (non linear)

From my experimental data, I have plots such as the attached. These plots are generally well fit (R2 > 0.99) by the following equation, $$y = \frac{a+bt}{1+ct+dt^2}$$ However, the constants a, b, c,...
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### Help with discrete-time population model; maximizing proportional harvesting

I'm working on a question that wishes me to determine the abstract maximum sustainable harvest for a proportional harvest of a discrete-time population model while also determining the risk of a ...
• 249
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### 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 $P$ ...
147 views

### State-space representation of SISO ARX models

Let, $y(k)=-a_1 y(k-1)-...-a_N y(k-N)+b_1u(k-1)+...+b_M u(k-M)$ be a discrete-time SISO ARX model with $N \geq M$. I want to convert this system to a state-space form, such as the controller canonical ...
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### 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 ...
19 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 ...
40 views

### Countable Markov Chain - Hitting Time

Would you have any book on countable markov chain with state space $\{1,2,...\}$ and hitting time to suggest? Most of the book I found are for finite Markov Chain. Otherwise, is there a way to link ...
17 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 ...
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### Is there a direction I can follow to express the equivalent difference equation?

In the Continuous Time Model let us define: \begin{equation} \Theta(t) = \displaystyle \int_{0}^{t} w(\tau) d\tau \end{equation} \begin{equation} x = cos\Theta \hspace {0.4 in} y = sin\Theta \...
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309 views

### Invariant curves induce invariant regions in discrete, 2D dynamical systems?

Consider a discrete dynamical system $x_{k+1} = f(x_k)$, where $f:\mathbb{R}^2 \rightarrow \mathbb{R}^2$, sufficiently smooth, and let $C \subseteq \mathbb{R}^2$ be an invariant, closed curve in the ...
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1 vote
221 views

### Prove that states accessible to an absorbing state are transient

Let $X$ be a Markov chain containing an absorbing state $s$ with which all other states are accessible, in the sense that $p_{is}(n) > 0$ for some $n = n(i)$. I need to prove that that all states ...
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### differential equation using matrix exponential not consistent solution

For the differential equation (physical friction) $\ddot x=-a\cdot \dot x$ The solution can be easily found using exponential ansatz and is $x(t)=c_1+c_2 \exp(-a\cdot t)$ Or expressing this using ...
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195 views

### Parseval's Identity Application

In this video, it is stated that Parseval's Identity "is how we go from discrete to continuous." However, I have not been able to find any material that expands on this use of Parseval's ...
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### Terminology for stability of equilibria in systems with discrete state-space

What is the proper terminology to describe the stability of a fixed point in a system with a discrete state-space? The states compose a high dimensional discrete torus, and the fixed point in question ...
27 views

### Identifying a raised sinusoid with uniformly spaced samples

You are given a model for an input signal of the form $y(t) = a \cos(\omega t) + b \sin(\omega t) + c$ where the constants $a,b,c$, and $\omega$ are unknown. You want to identify these unknowns from ...
66 views

### Maximum Sampling Period of a Second Oder Control System

When controlling a second-order system, the sampling period is suggested to be at least one tenth of the rise time of the step-response of the system (See "Real-Time Systems" By H.Kopetz, ...
119 views

### Vector Convolution?

I am working on a research problem which leads to the following optimization problem: \begin{equation} \hat{M} =\arg\max_M \|\sum_{k=0}^{M-1} {\mathbf y}_k \exp\left(-j 2\pi k \omega\right)\|^2 \;\;\;\...
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1 vote
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### How does the power of discrete-time maps propagate compared to the continuous-time system?

In continuous-time differential equations, if I have: $\quad \frac{d}{dt}x_1(t) = A_1\cdot x_1(t) + B_1$ $\quad \frac{d}{dt}x_2(t) = A_2\cdot x_2(t) + B_2$. the square of $x_1(t)$ and $x_2(t)$ can ...
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### Matrix to the infinite power shall be zero

Hello fellow people here. I found a problem, which started to haunt me: Defining the system: $$e_{k+1} = A e_{k}$$ With $A \in \mathbb{R}^{n \times n}, e_k \in \mathbb{R}^n, e_0 \text{ given}$ (...
1 vote
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### Is there a relationship between the retrieved spectrum width with the ratio between actual spectrum width and resolution?

I have a time domain data which when converted to frequency domain gives a Gaussian spectrum. If I have limited number of samples in time domain, my frequency domain resolution is bad. In that case, ...
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1 vote
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### Shortest path with jumps (dynamic Bayesian network)?

Suppose I have the following graph structure: It has the following properties: There are four states $\mathcal{S} = {q,s_1,s_2,s_3}$ where $q$ is some origin state where we start from (though it is ...
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I have come across the following formula of the inverse DFT: $x[n] = \frac{1}{N} \sum\limits_{k=0}^{N-1}X[k] e^{j \frac{2 \pi k n}{N}}$ The formula of the normal DFT seems intuitive to me as far as my ...