Questions tagged [discrete-time]

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Sampling period in difference equations?

I recently stumbled upon an article talking about discrete-time difference equations. The person talked about the existence of a "difference operator" defined as $$ \Delta f(x)=f(x+\Delta x)-f(x) $$ ...
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Kernel of an LTI system

For Linear Time-invariant systems mapping any complex sequence $x[n]$ to an output $y[n]$, the question is if there the kernel must be trivial, given that the LTI system does not map everything to the ...
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From discrete transfer function to state space model

I have been trying to make state space model from discrete function typed bellow. I have tried to solve it in many ways, but was unable to find appropriate model. What I have is this discrete funcion ...
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16 views

From Newton method to a time-dependent process

I don't know if my question is legitimate, or if it is meaningless : Newton's method consist in solving the non linear equation : $F(\mathbf{x})=0$ in proceeding to the iteration : $\mathbf{x}_{n+1}=\...
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43 views

How to solve this cubic Integral?

I tried to solve this integral but I have problem with the cubic term because I can't add $\sin(\frac{n{\pi}x}{L})$: $$\int_{0}^1 \Biggl[ A\Biggl(\sum_{n=1}^N q_n(t)\sin(\frac{n{\pi}x}{L})\Biggr)^3\...
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Tree shaped Markov Chain stationary distribution

I am working with a tree shaped DT Markov Chain transition matrix. I mean that there is no cycle in the associated graph and as in a tree you can only go down further the graph as time passes. I ...
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9 views

Using delay block in Simulink to account for flow rate in a fluid loop

I am trying to model a fairly simple cooling system loop where coolant flows over a battery to remove heat, then flows into a large reservoir where the coolant is mixed, coolant then flows out of the ...
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18 views

Issue w/ extracting coefficients from generating function using IDTFT

This q will make use of these 3 DTFT pairs... $$ \require{extpfeil}\Newextarrow{\xleftrightarrow}{15,15}{0x2194} \begin{array}{rcl} \alpha x_1[n] + \beta x_2[n] & \xleftrightarrow{\mathscr{F}} &...
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Proving DTFT pair as special case of another.

Consider these 2 basic discrete-time Fourier transform (DTFT) pairs... $$ \require{extpfeil}\Newextarrow{\xleftrightarrow}{15,15}{0x2194} \begin{array}{rcl} u[n] & \xleftrightarrow{\mathscr{F}} &...
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33 views

How is the noise gain function defined for higher order discrete piecewise white noise in a Newtonian system?

Background I have been trying to understand Kalman filters and implement them in a project I have. I have been following Roger Labbe's online book (https://nbviewer.jupyter.org/github/rlabbe/Kalman-...
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1answer
73 views

Determine Discrete-Time Fourier Transform of exponential or sine with time-shift?

Consider the following periodic and time-discrete signal: $$y[n] = \sin (\Omega_0 n+\phi)$$ How do I determine the discrete-time Fourier transform for $-\pi<\Omega<\pi$ for that signal? I ...
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Extract Fourier Transform from discrete time signal

I have two discrete time signals and I have to firstly determine their function form before determining their Fourier transforms and compare them. Here are the two signals As you can see, one is the ...
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1answer
56 views

Smoothness of a time series: relationship between ARMA model and signal derivatives

I have a discrete stochastic function $f(t_k)$ (in my case, it's a time signal related to atmospheric parameters). I know this function to have a certain smoothness. One way I can specify its ...
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2answers
16 views

How to translate from a 2x2 state-space difference equation to a 2nd-order difference equation

I have a state-space evolution equation of the form $$\begin{bmatrix}u_k\cr v_k\end{bmatrix} = \begin{bmatrix}1-a & c \cr b(a-1) & 1-bc\end{bmatrix} \begin{bmatrix}u_{k-1}\cr v_{k-1}\end{...
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what would be its solution in discrete as well as continous time

If I have an equation like ( assume everything as its suitable dimension and all are real matrices, $p,q$ are real constants) $(1)$ $x(k+1)= pf(x(k))+q(A\otimes B)x(k)$ $(2) \dot x(t)=pf(x(t))+q(A\...