Questions tagged [discrete-time]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
1
vote
0answers
16 views

Mean time to go from state i to state j in following discrete-time Markov chain (DTMC)

Let the following discrete-time Markov chain (DTMC) be defined with its transition probabilities as follows: $$ \begin{aligned} &p_{0,0}=\frac{b_{0}}{b_{0}+b_{1}}, \quad p_{i, i+1}=\frac{b_{i+1}}{...
1
vote
0answers
29 views

Planar discrete dynamical systems' omega limit sets

This is just curiosity. Given a discrete time dynamical system $x_{n+1}=f(x_n)$ on the plane, $f:X\rightarrow X$ for $X$ a (relatively) compact region of $\mathbb{R}^2$, $f$ continuous on $X$. Suppose ...
0
votes
0answers
24 views

Continuous-index process, discrete-index process, stochastic integration, and linearization

I work with discrete-time continuous-state stochastic processes and log-linearized models. For instance, for $X_t$, I write \begin{align} X_t = e^{x_t} = e^{\overline{x}} + e^{\overline{x}}(x_t - \...
0
votes
0answers
19 views

Adaptive stepping of a discrete time system

I have a discrete time system of the form $$\mathbf{y}[k+1] = f(\mathbf{y}[k])$$ that I want to step until a specific condition $$g(\mathbf{y}[K]) < 0,$$ usually $K\sim1000$. Evaluations of $f$ are ...
0
votes
0answers
11 views

Is there a special subcategory/name for machine learning applied to real time data streams?

My real world problem is roughly this: An array of position sensors (it is in fact a UWB - ultrawideband radio chip) produces a stream of real-world coordinate tuples (say, [X|Y|Z]) for one specific ...
2
votes
0answers
41 views

Parseval's theorem for the Number Theoretic Transform

What is Parseval's theorem for the discrete Fourier transform (aka Number Theoretic Transform)? Namely, I have $\alpha$ a principal $n$th root of unity in a finite field $\mathbb{F}$ (say $\mathbb{Z}...
1
vote
1answer
28 views

Intuition / Interpretation of Balance Equations (Markov Chains)

I'm currently studying a module on Stochastic Processes and I'm quite confused as to the interpretation of Global/Local Balanced Equations for Discrete Markov Chains. From what I've learnt, Global ...
0
votes
0answers
8 views

Best way to determine cross-correlation between two time series of a source in two different spectral regions

I am working with x-ray and ultraviolent time series of astronomical sources, and want to study the correlation between the two for any given source. Namely, I want to examine whether there is any ...
1
vote
2answers
21 views

average proportions over time

Let us say I have a number of events per week over some time periods. During each week a proportion of these events are important (important_events_per_week/number_of_events_per_week). The ...
1
vote
0answers
18 views

calculate average percentage of measurements over time

Let us say I have weekly measurements over time and I have a target measure for each week. Is it too naive to calcuate the relative difference: ...
0
votes
0answers
22 views

Joint Distribution of a Time Series with lag = 2 ? Am I wrong or the book?

I try to reproduce the calculation of mutual information I(x;y) from the book "Nonlinear Analysis for Human Movement Variability" p319. Specifically calculating the joint distribution of the ...
0
votes
1answer
26 views

DTMC problem: Two machine

In a certain manufacturing system, there are 2 machines $M_1$ and $M_2$: $M_1$ is a fast and high precision machine whereas $M_2$ is a slow and low precision machine. $M_2$ is employed only when $M_1$ ...
0
votes
2answers
30 views

How to find the time for a treatment when counting the difference of pills taken?

The problem is as follows: Louis took three and a half type $A$ pills every twelve hours and a half type $B$ pills every $6$ hours. He did this until the difference in the number of pills taken was $...
0
votes
0answers
12 views

Determine the partial fraction convolution output y(ω) signal using fourier transform.

$x(t)=e^{-t}u(t)$ $H(\omega)\ =\frac{2+j\omega}{3+j4\omega-w^2}\ $ 2. $ x(t)\ =\ e^{-2t}\ u(t)\ $ $H(T)\ =\ e^{-4t}\ u(t)$ Can you guys help how to solve this two items in step-by-step. Youtube videos ...
2
votes
1answer
104 views

Stability of a discrete-time closed-loop

I modeled a dynamic system like so: $$ \dot{y} = au, $$ i.e. as an integrator. Every $T = 0.1$ seconds, the measurement is updated and remains constant in between. This motivated me to model it as a ...
0
votes
0answers
33 views

Discrete time embedded Markov chain

Let $(X_n)_{n \in \mathbb{N}}$ be a (time-homogeneous) Markov chain with countable state space $I$ and transition matrix $P$ and assume $X_0=i \in I$. Define the stopping times $T_0:=\inf\{n>0:X_n ...
0
votes
1answer
364 views

Expectation and variance of the maximum of k discrete, uniform random variables

Can anyone help me with this? I need to know if I'm on the right track. I have k discrete, uniformly, identically distributed, independent random variables on the interval [1,...,n]. I want to learn ...
-1
votes
1answer
24 views

What is the derivative of a delayed function in discrete time? [closed]

How do you calculate such derivative? $$\frac{d\ \ x_{1}\left(k+1\right)\ }{d\ \ x_{1}(k)}$$ Sorry for the probably silly question but I have been having some serious trouble with this
0
votes
2answers
19 views

How to multiply time in different denominations?

I am writing some software that must do generalized time multiplication of various denominations. I can't wrap my head around how to generalize it. [edit] The use case is this. You have a clock ...
0
votes
0answers
9 views

Confusion about discrete time convolution.

Question : Calculate the convolution of $X(k)=\frac{1}{2}^{(k-1)}$ with itself , assume a causal system. What I've tried : $X(k) * X(k) = \sum_{p=-\infty}^{p=\infty} X(p).X(k-p) = 1 $ since the ...
0
votes
0answers
18 views

Going in Reverse From Z-Domain to a Difference Equation

I am attempting the task of beginning with a $z$-domain equation, and then reverse engineering a difference equations from it. The eventual goal is to solve the difference equation, and then go back ...
0
votes
1answer
27 views

Linear Difference Equation Solution

$a[n + 2] - ea[n + 1] = 3f[n]; f[n] = e^n, a[-1] = \frac{e^2 - 3}{e^3}$ Wolfram Alpha gives the solution $a[n] = 3e^{n - 2}n + e^n$, but also says that there is no solution. The solution it gives ...
1
vote
0answers
11 views

Discrete Time Convolution

I'm trying to solve a problem on convolution from Alan V.Oppenheim: Find the convolution output $y[n]$ for the following signals: $x[n]= u[n]$ and $h[n]=a^{n}u[-n-1], a>1 $ I started the evaluation:...
2
votes
1answer
77 views

Input in Differential Equations and Difference Equations

According to my understanding: In differential equations, the input is defined as the entire inhomogeneous part of the equation as it would be written on the right hand side. For example, the input ...
0
votes
0answers
17 views

What is the intuition behind Coppock curve?

Coppock curve is defined as the following: $$z_{k+1} = Coppock = WMA[10] \; of \; (ROC[14] + ROC[11])$$ where $ROC[14]$ is percentage rate of change of the current value of input $s$ with respect to ...
1
vote
1answer
17 views

The integral of the log spectral density for a infinite MA process.

Let us consider a MA($\infty$) process: $y_t= \sum_{j=0}^{\infty} b_j \varepsilon_{t-j} $ where $ \sum_{j=0}^{\infty} b_j^2 < \infty \>\>\>\>\>\> $ (1) Given $f(\lambda)$ as ...
0
votes
0answers
21 views

Comparing horse times across different terrains

I have a very large dataset containing horse racing results. What's intriguing to me is the variance across the time taken across the various race tracks. It has led me to look into causal factors. ...
0
votes
0answers
59 views

Fundamental Period of Harmonic Complex Exponentials

If I have a complex exponential in the discrete time domain $e^{j\Omega n}$ where $\Omega =\frac {2\pi k}{N}$.I understand that the signal is periodic as k varies but what I'm not able to grasp is ...
2
votes
1answer
47 views

Exponential Distribution Events occurring before another

I've been trying to solve this question, but my answer doesn't match the correct one. So I am trying to see where I am going wrong. Agent 1 has 2 tasks: Task 1 and Task 2. The time it takes Agent 1 to ...
2
votes
0answers
28 views

How to continuize a discrete time Markov chain?

I want to use an advanced Monte Carlo techniques to a discrete time Markov chain (DTMC), thus I need to continuize my DTMC. What is the best way to do that? I want to measure some observable along ...
0
votes
0answers
41 views

Closed form expression for stochastic discrete-time simulation

I have a model relating innovation to metabolism in a colony of ants. In this stochastic discrete time model, a colony of N ants are working to complete a single task. The number of active ants at any ...
0
votes
0answers
10 views

x(t) = A cos(Ωt+φ) is between −2.0 and +2.0 for 70% of its period. What is A?

I split the cosine into parts. 7.5% of period T from the maximum value of x(t) = A to x(t) = 2, then 35% of T to the value x(t) = -2, then 15% through the minimum and back up to x(t) = -2, then 35% up ...
0
votes
0answers
15 views

Orthogonal Polynomial series / orthogonal over finite extent / zero at origin / second derivative zero at limit of extent

I wish to observe trends in noisy data. Here are my assumptions regarding the physics of the data: Time series of values (finite in extent: $ S(t)$ discretized to $S_i$) Sampling is uniform in time ...
2
votes
0answers
32 views

Stability of Discrete Time Controller with Quantized Outputs

I am simulating a fairly complex parallel buck converter power system with a discrete time controller which dictates the output voltage (1kHz sampling rate). I am trying to compute the loop gain of ...
0
votes
1answer
1k views

the slower grape crusher

Two Grape Crushers take 4 days to crush certain amount of grapes.If one of them crushed half the grapes and the other crushed other half , then they complete the job in 9 days. How many days will it ...
1
vote
0answers
26 views

Term for a function $f$ that is injective on some $f^N$ of its domain

I have a smooth function on a manifold, $f:X\to X$, with the following property. Although $f$ is not injective, there exists an integer $N\geq1$ such that the restriction of $f$ to $f^N(X)$ is ...
0
votes
0answers
13 views

Discrete time process with discrete state space when initial condition fixed.

Let $X_0\sim \mathcal{L}([0,1]^n)$ where $\mathcal{L}([0,1]^n)$ is some continuous distribution over $[0,1]$ and consider a finite set of doubly stochastic invertible $n\times n$ matrices $\mathcal{M}$...
0
votes
1answer
31 views

Discrete Dynamical System with Global Chaotic Attractor

I am looking for examples of dynamical systems in discrete time, with state space $\mathbb{R}^n$ (for some arbitraty $n$), with a chaotic attractor $A \subset \mathbb{R}^n$ with a global basin of ...
3
votes
0answers
66 views

What is the name of this interval operation?

I have encountered an basic operation on a set of intervals which I think should have a common name and be in textbooks. Unfortunately, my searches have turned up empty. Imagine you have a set of ...
0
votes
0answers
24 views

Euler-Maruyama discretisation of Markov (diffusion) process => Markov chain?

Suppose that we are given a diffusion process on a finite-dimensional manifold $M$ \begin{equation} dX_t = b(X_t,t)dt + \sigma(X_t,t)dW_t. \end{equation} My question is: Suppose that we discretize ...
0
votes
1answer
26 views

Does linear discrete-time controllability imply stabilizability

Does linear discrete-time controllability imply stabilizability? I feel like it should, since controllability is the ability to steer from any state $x(0)$ to another state $x(1)$ in finite time and ...
1
vote
1answer
35 views

Evans and Murthy: if $\sum_{i=0}^r a_iA^ib=0 , a_i > 0 , i = 0,1,\dots,r$ then $x$ can be expressed as a linear combination of $A^ib$

In the article of Evans and Murthy (1977) the following lemma is given: If $A,b$ satisfy the relationship $$\sum_{i=0}^r a_iA^ib=0 \quad \quad a_i > 0 \quad i = 0,1,\dots,r$$ then any vector $x$ ...
2
votes
0answers
24 views

Stable measure of spread of a function over a finite interval

Consider finite real functions on an interval $x \in [0, 1]$. Let's call an example function $f(x)$. It is known that such a function is a sum of non-negative slowly-varying part we will call signal, ...
0
votes
1answer
32 views

Prove that $\mathcal{F}\{\cos(\omega_0 n)\}=\pi\sum_{k=-\infty}^{+\infty}[\delta(\omega-\omega_0-2\pi k)+\delta(\omega+\omega_0-2\pi k)]$

$\mathcal{F}$ is the discrete Fourier transform. A suggestion is to use the inverse transform, so I applied it to both sides in the hope of obtaining an identity. Exchanging the integral with the ...
0
votes
1answer
90 views

Stability conditions of discrete dynamic systems

I have a system of 6 difference equations with three state variables. Let $X_t = [x_{1t}, x_{2t}, x_{3t}, x_{4t}, x_{5t}, x_{6t}]^T$. The general form of the problem is $g(X_{t+1}, X_t) = 0$. The log-...
0
votes
1answer
54 views

real dynamical systems

I am a bit confused on how to solve this system or generally these kind of systems $$x(t+1)= x(t) -2y(t)$$ $$y(t+1) = 2x(t) +y(t)$$ My confusion derives from that fact that I am working with $(t+1)$ ...
0
votes
0answers
16 views

Independent Variables in Convolution Derivation

When deriving the convolution representation of a radioactive dumping problem, a professor appears to use both $u$ and $t$ to represent time, as shown in this image: The final result is shown in this ...
0
votes
0answers
19 views

Multiplication with discrete delta function

What would be the outcome of multiplying the delta function of discrete time with another function e.g.: δ(n-m)•y(n) =? What would be the outcome of the multiplication between two different deltas?...
1
vote
1answer
55 views

A discrete-time system is positively stabilizable iff (A,B) is stabilizable and the eigenvalues are in the open unit disk

I'm currently busy with some research in positive controllability and positive stabilizability for discrete time systems. In an article of M.E. Evans and D.N.P. Murthy ("Controllability of discrete-...
0
votes
1answer
30 views

Necessary stability condition for a second order discrete time system $x(k+2) = Ax(k+1) + Bx(k)$

Let $x(k) \in \mathbb{R}^n$, $A \in \mathbb{R}^{n\times n}$, $B \in \mathbb{R}^{n\times n}$. Consider the following discrete time system: $$x(k+2) = Ax(k+1) + Bx(k)$$ where $x(1) = Ax(0)$ and $x(0) \...