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.
127
questions
1
vote
1
answer
27
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","...
- 27
0
votes
0
answers
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,...
- 55
0
votes
0
answers
72
views
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
0
votes
2
answers
32
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 $P$ ...
- 1
0
votes
0
answers
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 ...
- 11
0
votes
1
answer
25
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 ...
- 3
0
votes
0
answers
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 ...
- 615
0
votes
0
answers
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 ...
0
votes
0
answers
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 ...
- 1
2
votes
1
answer
70
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
$$
\begin{bmatrix}
\frac{\mathrm{d}\hat{\psi}_{r_{\alpha}}}{\mathrm{d}t} \\
\frac{\mathrm{d}\...
- 259
1
vote
0
answers
49
views
How to discretize a continuous time domain state space model?
Let's say I have a state space model of a linear dynamic system in the continuous time domain in following form
$$\dot{\mathbf{x}} = \mathbf{A}\cdot\mathbf{x} + \mathbf{B}\cdot\mathbf{u}$$
My goal is ...
- 259
0
votes
0
answers
12
views
Periodicity of Sum of Two Real, Periodic Signals in Discrete-Time System
In a discrete-time system, if there are two discrete, real, periodic signals f and g such that:
f[n] = sin[(1000pi)n], n is all reals
g[n] = cos[2500n], n is all reals
then is their sum periodic as ...
- 31
1
vote
0
answers
42
views
Issue with IDTFT (Inverse Discrete Time Fourier Transform) of a periodic impulse train
I'm working through examples in Oppenheim and Willsky's Signals and Systems 2nd Edition, and I'm having trouble with taking the Discrete Time Inverse Fourier Transform of a periodic impulse train ...
- 43
0
votes
0
answers
24
views
Math functions that take fixed time to compute
I was wondering if there are any mathematical functions that take a known minimum amount of time to calculate. The closest thing I could find to this is Proof of Work algorithms used for blockchain, ...
- 133
2
votes
1
answer
64
views
Converting between continuous and discrete-time stochastic processes
I'm reading through Dixit and Pindyck's Investment under Uncertainty, where I found the following passage. First, they introduce the Ornstein-Uhlenbeck process
$$
dx = \eta (x - \bar{x})dt + \sigma ...
- 747
0
votes
0
answers
51
views
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 \...
- 51
0
votes
0
answers
13
views
How to find the mean for the autocorrelation function?
$m$
0
1
2
3
>3
$\phi_{xx}$
0.8
0.6
-0.2
0.45
0.16
How do I find the mean for the autocorrelation values as shown in the table? The correct answer is 0.4. The formula for autocorrelation is $\phi_{...
- 21
0
votes
0
answers
11
views
Is there a formula for a multiple discrete convolution?
Question related to Multiple Convolution closed form., but I wonder if there is a nice-looking, discrete version of this?
0
votes
0
answers
68
views
Describing thinning of Poisson processes in notation
How does one describe thinning in notation. For example a question says:
Claims arriving under A follow a Poisson process with a rate of 5 claims per day: $\lambda = 5$.
Claims arrive independently ...
- 25
0
votes
0
answers
22
views
How to convert a continuous time optimal control problem into to discrete time optimal control problem?
I have an optimal control problem and want to embed it into multi stage optimization problem by first discretizing it into discrete time optimal control problem...
Can I do it by just converting ...
1
vote
0
answers
12
views
Autoregressive Time Series in Continuous Time?
I have seen that AR(1) process in discrete time corresponds to Ornstein–Uhlenbeck process in continuous time, is there a general analog of AR(n) process in continuous time?
- 165
0
votes
0
answers
33
views
Changing the transition matrix to figure out time when a state is first entered
Consider a discrete-time Markov chain $(X_0,X_1,\dots)$ on a state space $S=\{1,2,3,4,5,6\}$ with transition probabilities given by the matrix:
$$\begin{pmatrix}
1/2 & 1/4 & 0 & 1/4 & ...
user999236
0
votes
0
answers
13
views
Invertibility of fundamental matrix in discrete time
The fundamental matrix $\Phi(t,s)$ corresponding to the linear time-varying ODE
$$\dot{X}(t) = A(t) X(t), \quad X(s) = I$$ is always invertible, since it obeys the semigroup property $\Phi(t,s)\Phi(s,...
- 2,449
1
vote
0
answers
23
views
How can I find the following convolution sum?
$$ h[n]= \left( \frac{8}{9} \right) ^n u[n-3] $$
And the function is:
$$x[n] =
\begin{cases}
2 & \text{if } 0 \leq n \leq9, \\ ...
- 453
0
votes
0
answers
25
views
Does Tikhonovs' theorem for dynamical systems also hold in discrete time?
Tikhonov's theorem in dynamical systems tells us how we can 'separate timescales' if we have a system of two variables, where one changes much slower than the other.
That is, consider the following ...
- 1,337
4
votes
1
answer
198
views
Discrete-time dynamical systems with variable state space dimensions (or output space dimensions)
I am trying to figure out how to formalize a dynamical system whose state vector can change dimensions from one step to the next. For example, I have a process (a discrete-time dynamical system, if ...
- 103
1
vote
1
answer
53
views
Time invariance of diffusion
Let us assume that we have a diffusion problem in 1D that can be modelled with Brownian Motion. The diffusion is coefficient $D$, and the particle is starting at $x=0$.
Then, the probability that we ...
- 880
1
vote
1
answer
31
views
proving Lenglart's dominated property for some function
$\textbf{Definition:}$ $X$ is $L-$dominated by $Y$ if $\mathbb{E}(|X_T|) \leq \mathbb{E}(|Y_T|)$ for every bounded stopping time $T$. Define $\mathbb{N}:=\{1,2,3,...\}$, i.e. the natural number ...
- 367
0
votes
1
answer
35
views
Closed form expression for discrete-time sum of a cosinusoid
I'm facing a problem in digital signal processing and am wondering if there is a closed form expression for the sum
$$Y[n] = \sum_{k=0}^{n-1}\cos(\frac{2\pi k}{f}),$$
where $n$ < $f$.
In case I ...
- 3
1
vote
1
answer
255
views
Conditional probability for path calculation in a markov chain model
This question extends a previous question I asked with respect to the markov chain model. I post another question here since I'm trying to follow the Stack Exchange's guidelines.
So, based on the ...
- 207
0
votes
1
answer
64
views
Transitional probabilities in a markov chain model
Just for the sake of providing some context, I'm dealing with the following:
Alice's smartphone plan includes a voicemail service with a maximum storage capacity of two voice messages.
On a daily ...
- 207
0
votes
0
answers
106
views
LQR design for Augmented system not working.
I am trying to design an LQR for the system matrices given below. The augmented system is a 4-state, 2-input, and 1-output system. I also had to design an observer for the augmented system. The ...
- 1
0
votes
0
answers
13
views
Are these steady states of non linear dynamic system actually steady states?
I have the following non linear dynamic system in discrete time:
\begin{equation}
x_{t+1} = \frac{1}{1 + \exp\left(- \beta \left( 2 d \left(c + \frac{(1 - c)}{1 + a (1 - x_{t}) d}\right) - b - d \...
0
votes
1
answer
47
views
Discrete time positive martingale with non-trivial limit
Let $X_t$ be a positive discrete time martingale. (I.e., $X_t>0$ and $\mathbb{E}_t X_{t+1}=X_t$ for all $t\in\mathbb{Z}$.)
Then by Doob's supermartingale convergence theorem, there exists some non-...
- 673
0
votes
0
answers
18
views
How to use the solution of an homogeneous non linear DE when plugging a discretized input.
there's a certain class of non-linear (NL) DE (non linear in $f$)
$$
x' = f(x)+g(t),
$$
the homogeneous form of which admits an analytic solution. Suppose the solution to such an homogeneous form $x'=...
5
votes
2
answers
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 ...
- 5,039
1
vote
2
answers
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 ...
- 163
0
votes
1
answer
147
views
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 ...
- 146
3
votes
0
answers
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 ...
- 1,554
0
votes
1
answer
25
views
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 ...
- 68
0
votes
1
answer
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 ...
2
votes
1
answer
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, ...
- 87
0
votes
0
answers
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 \;\;\;\...
- 376
1
vote
1
answer
22
views
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 ...
- 599
0
votes
1
answer
54
views
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} $ (...
- 3
1
vote
0
answers
33
views
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, ...
- 113
1
vote
1
answer
50
views
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 ...
- 682
0
votes
0
answers
17
views
Intutive understanding for the formula of the inverse DFT
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 ...
- 137
1
vote
0
answers
34
views
Discrete-Time Systems and Connections Between Functions
I'm studying control theory from the book "Mathematical Control Theory" by Sontag. I have a question about discrete-time systems, which is defined on page 32.
Definition.
A system $\Sigma=(\...
- 500
2
votes
0
answers
51
views
How to find or approximate probability distribution from known values of the characteristic function?
If I have a known discrete values for the characteristic function (I know $a_n$ values for specific values of $\omega_n = \frac{2\pi i n}{d}$):
$$a_n = \phi_x\left(\omega = \frac{2\pi in}{d}\right),$$
...
- 21