6

Before I go into the specific issues, I'd like to note that there is a minor confusion with respect to M not representing the AR process order. More specifically an AR process of order M=4 would use the M previous values of y1 rather than M+1 values (in the loop with index j). So, with M=4 your implementation is really that of an AR process of order $p = 5$. ...


4

Impulse Response is basically the FIR coefficients of the system. Namely, a system $ H $ with an impulse response given by $ f [n] $ and a Filter $ F $ with an FIR representation of $ {f[0], f[1], \cdots, f[n]} $ are equivalent. Now, systems with Feedback are equivalent of both FIR and IIR (AR) filters. But given infinite length of FIR model any LTI ...


4

it doesn't tell you new information, but because the abs value function is non-analytic (i.e. not all of its derivatives are continuous), and the magnitude-squared is, then the latter can be manipulated mathematically in ways that the former cannot. one important property of the power spectrum is that it is the Fourier transform of the autocorrelation ...


4

The arma approach is to model the current output of the system as the sum of past outputs and past inputs explicitly. The assumption of gaussian model for the noise statistics still can be used for the unpredictable signal which cannot be modeled as arma. From system's frequency response spectrum modeling aspect: an ar model is able to model only the ...


3

From the definition of the process you know that $$x_{n+1}=x_n-0.2x_{n-1}+w_{n+1}\tag{1}$$ Since $w_n$ is white you can't predict it, so the best linear predictor for the given process is the filter $$P(z)=1-0.2z^{-1}\tag{2}$$ which is a first order filter. It estimates the future sample $x_{n+1}$ by computing $$\hat{x}_{n+1}=x_n-0.2x_{n-1}\tag{3}$$ ...


3

LPC reduces to AR modelling only if the stochastic time process is stationary (does not change distribution parameters over time) and ergodic (average over time is equivalent to mean of ensemble average). This connection between auto-regressive coefficients and the autocovariance of the process is described by the Yule-Walker-Equations (mentioned in the ...


3

As suggested by Marcus Müller, interpolation in the time domain could be a solution. I never had to perform such a task, and the outcomes may depend in the nonuniformity of your sampling. I propose a use of the Wiener-Khinchin theorem: The Fourier transform of the autocorrelation function is the power spectrum, or equivalently, the autocorrelation is ...


3

Which one you choose is quite arbitrary: it depends on which region of convergence (ROC) you choose. As Example 2 and Example 3 of that wikipedia page shows, if you want the signal to be causal, then choose the first one. If you want the signal to be anti-causal, choose the second. Another way to think about it is what happens when $|z| = 1$. On the unit ...


3

The simplest way to approximate an AR-2 process in Matlab / Octave is the following: N = 1024; % number of process samples. a = [1, -0.9, 0.2]; % denominator coefficients, p = 2. b = [1.0]; % numerator coefficient. x = filter(b,a, randn(1,N)); % generate N sample of AR-2 x[n]. Note: an AR process requires a ...


3

Have a look at my Fast Gaussian Blur Project at GitHub. You will find there implementation of IIR Approximation of Gaussian Blur which implements the following papaers: Recursive Gabor Filtering. Recursive Implementation of the Gaussian Filter. Boundary Conditions for Young - van Vliet Recursive Filtering. The idea is pretty straight forward.


2

The PDF (likelihood function) of the multi-tone estimation problem is: $$ {\cal L}({\bf x}; {\bf a}, {\bf f},{\bf \phi}, {\sigma}^2) = \frac{1}{(2\pi \sigma^2)^{\frac{N}{2}}} \exp\left( - \frac{1}{2\sigma^2} \sum_{n=0}^{N-1} ( x(n) - \sum_{p=1}^{P} {a}_p \cos(2\pi {f}_p + {\phi}_p) )^2 \right) $$ The paper by Djuric quotes the AIC and MDL values as: The ...


2

Let's try to make your system into a state space representation with state: $$ \mathbf{x}(t) = \left [ \begin{array}{c} x_1(t)\\ x_2(t)\\ x_1(t-1)\\ x_2(t-1)\\ \end{array} \right ] $$ so that $$ \mathbf{x}(t) = \mathbf{A} \mathbf{x}(t-1) + \mathbf{y}(t) $$ where $$ \mathbf{A} = \left [ \begin{array}{cccc} a_{11} & a_{12} & a_{13} & a_{14}\\ a_{...


2

It has been about one month since you asked your question, so it is likely you've addressed it and are comfortable with the answer. Regardless, my reply is that the implementation of the Levinson-Durbin algorithm in Matlab (i.e. the "levinson" command) is coded to estimate only an autoregressive process. The comment section of the "levinson" script-file ...


2

Given the strong correlation of the downward trend in each cycle, I offer the following that would offer an improved prediction over one linear regression, but not be significantly more complicated (but still would like to see what a precise "best estimator" would be): Do a linear regression to determine the general upward trend. This would be the Income - ...


2

Auto Regressive Model means the current output is a linear combination of previous outputs and driving noise: $$ y \left[ n \right] = \sum_{k = 1}^{p} {a}_{k} y \left[ n - k \right] + v \left[ k \right] $$ As you can see, the current value $ y \left[ n \right] $ depends on $ m $ values before it. The parameter $ p $ is the order of the model while $ v \...


2

ARMA divides the signal into two parts and that models the two parts. Financial time series are corrupted by different types of correlated and uncorrelated noises with definite functions that allow modeling and others more difficult as per having to use aproximations. In addition financial or economic times series may exhibit long memory processes (Black ...


2

it's a hand-wavy argument, but the idea is to derive the coefficients $\alpha_i$ so that the norm of the error $\Big\||\epsilon[n]|^2\Big\|$ is minimized. we assume that $x[n]$ is a stationary "random" process, we know all of the previous samples $x[n-i]$ and we want to make a good guess at $x[n]$ with a linear combination of the previous samples. we can ...


2

One natural way to do this is to use the the total $2N$ samples (two chunks of $N$ samples each) to estimate the parameters. Amplitude, frequency, and phase estimation are all textbook equations that you can look up so the problem here is the fact that they are all mixed together. If you are assuming that you don't know how many sinusoids are mixed, then you ...


1

With $$\begin{align}r_k=E\{x_nx_{n+k}\}&=E\{(v_n+\alpha x_{n-1}+\beta x_{n-2})x_{n+k}\}\\&=E\{v_nx_{n+k}\}+\alpha E\{x_{n-1}x_{n+k}\}+E\{x_{n-2}x_{n+k}\}\\&=\sigma_v^2+\alpha r_{k+1}+\beta r_{k+2}\tag{1}\end{align}$$ and with $$r_k=r_{-k}$$ you can obtain $3$ equations $$\begin{align}r_0&=\sigma_v^2+\alpha r_1+\beta r_2\\ r_1&=\alpha ...


1

We have \begin{equation} x[n] = v[n] + 0.75x[n-1]-0.25x[n-2] \end{equation} You started correct, you get \begin{equation} \begin{split} r_{xx}[k] = E\{(v[n] + 0.75x[n-1]-0.25x[n-2])(v[n-k] + 0.75x[n-k-1]-0.25x[n-k-2])\} \\ \end{split} \end{equation} You get 9 terms \begin{equation} \begin{split} r_{xx}[k] &= Ev[n]v[n-k] +0.75E x[n-k-1]v[n] - 0.25E ...


1

Given that you know the autocorrelation function, you can use the Yule-Walker equations for a generic AR model of order $p$. As you are interested in the case $p=2$, there's no need to complicate things that much. As stated in this paper, the equations for the case of an AR(2) are given by: $$a=\frac{r_1(1-r_2)}{1-r_1^2}$$ $$b=\frac{r_2-r_1^2}{1-r_1^2}$$ ...


1

Hi: As far as what you wrote in confusion 2): The AR expression should not have an $x[t]$ on both sides. For example, below is an AR(2): $x[t] = a1*x[t-1] + a2*x[t-2]+ e[t]$ Similarly, the MA expression should always have a coefficient of 1 on the $e[t]$ term. So, for example, an MA(2) would be: $x[t] = e[t] + a1*e[t-1] + a2*e[t-2]$ As Stanley said, ...


1

Question 1) How to generate colored noise in general? Produce white noise and filter it. Question 2) How to add the colored noise in the form of process noise to the Rossler system or any continuos time system This is a little bit more complicated. The addition of noise is a simple matter of $y = x + q$ where $q$ is the noise signal. The key problem ...


1

Hi: You're expression will be a 3 by 3 matrix with the 1's along the diagonal and the correlations in the respective places. $\begin{matrix} 1 & \sigma_{12} & \sigma_{13} \\ \sigma_{21} & 1 & \sigma_{23} \\ \sigma_{31} & \sigma_{32} & 1 \end{matrix}$ The 12 and 21 elements are $\rho$ because ...


1

This is a neat idea and could probably be made to work - it seems like a variant on time-domain reflectometry, which exploits correlations of this kind https://en.wikipedia.org/wiki/Time-domain_reflectometry. However, there's an important difference between your two use cases: we can normally inject a test signal onto an electrical transmission line, while ...


1

If you have access to Statistical Digital Signal Processing and Modeling by Monson H. Hayes (Wiley, 1996), see Section 4.7.3 - Moving Average Models for a more detailed description of characterizing moving average signals. In this section two methodologies are covered, spectral factorization and Durbin's method. Both of these methods have their own innate ...


1

Consider the signal $$s[n]=\frac{a^{|n|}}{1-a^2},\, |a|<1$$ Assuming $$x[n]=a^n u[n]\stackrel{\text{DTFT}}\longleftrightarrow\frac{1}{1-a e^{-j\omega}},$$ one can show that $$s[n]=x[n]*x[-n]$$ The proof is simple. For example, for $n>0$ we have $$\sum_{k=n}^{\infty}x[k]x[k-n]=\sum_{k=n}^{\infty}a^ka^{k-n}=\sum_{k=n}^{\infty}a^{2k-n}=a^n(1+a^2+a^4+\...


1

Your understanding is right. Just a few points... I think you meant Yule-Walker equations. It is better to distinguish $N$ from $n$. In an AR($N$) model we consider last $N$ terms to approximate the next term. An important interpretation that you missed was the fact that when we consider $$\epsilon=x[n]-\tilde{x}[n]=x[n]-\sum_ka_kx[n-k],$$ or in a more ...


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