8

I unfortunately don't know a whole lot about Kalman filters, but I think I can help you out with the state space stuff. In Example 1, the AR model is exactly your good old DSP recursive definition of output: $$ y_t = \alpha + \phi_1y_{t-1} + \phi_2y_{t-2} + \eta_t$$ In this case we write down the state-space model with direct correspondence with the above ...


5

A pitch estimator is commonly used to find the vocal periodicity. Common pitch estimators include cepstrum/cepstral analysis, harmonic product spectrum, and composite algorithms, such as YAAPT.


4

I think your best bet is the "YIN" pitch detector, described in this paper: http://audition.ens.fr/adc/pdf/2002_JASA_YIN.pdf. It's fairly simple, and performs very well. They present it in steps, or improvements upon the previous idea, and even just implementing the first few steps should be sufficient. Most pitch detectors that are actually in use are ...


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 ...


4

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 $p = 5$. ...


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

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

In any signal processing problem, there are usually two components: the signal model and the channel model. The signal model is the mathematical description of how your ideal signal, call it $s(t)$ is generated. The channel model is the mathematical description of how your channel corrupts or alters the signal. One aim in telecommunications is to look at ...


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 ...


2

The coefficients $a_i$ of an autoregressive (AR) model of a signal $x(n)$ enter the model definition in the following way: $$x(n)=\sum_{i=1}^Na_ix(n-i)+e(n)$$ where $e(n)$ is zero-mean white noise and $N$ is the model order. I.e. the signal is modeled as the output of a linear time-invariant filter with transfer function $$H(z)=\frac{1}{1-\sum_{i=1}^Na_iz^...


2

MA (Moving Average) filter is a FIR filter. In general it is given in this form - $$ Y(n) = \sum { w_i * X(n-i) } $$ In inherently it is always stable. AR (Auto regressive) filter is an IIR filter. In general, it is given that AF filter is in the form $$ Y(n) = X(n) * w_0 + \sum { w_i * Y(n-i) } $$ If you want to design a filter who's frequency ...


2

What appears to be happening is, as you start over-modeling, the error signal becomes less and less white. I modified your code to return the error signal (part of the residue term). This plot shows the off-zero-lag coefficients of the xcorr of the error for order = 2 (blue), 3 (red), and 4 (green). As you can see, the close-to-but-not-zero lag terms are ...


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

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 ...


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

Ok, The answer is as following, any ARMA process which is Minimum Phase system could be represented as MA or AR system. The MA part is easy by Wold's Decomposition. Since the filter in Wold's Decomposition is Invertible (Minimum Phase by itself), Shall be called $ H \left( z \right) $, It is easy to build the optimal linear predictor from it, $ P \left( z \...


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

ARMA models are useful when you need to model a Signal plus Noise situation where the signal is an AR process and the noise models sensor noise. The overall model is an ARMA model. See HL Van Trees, Detection, Estimation, and Modulation Theory, vol 4 Array Processing. He gives an example of a Spatial AR process sensed by noisy sensors. The overall model is ...


2

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 ...


2

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 ...


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 ...


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 ...


1

Your true AR model is order 4. If an AR(16) model did a good job, then 12 of its parameters should be zero. Because of noise effects / overfitting, those 12 parameters are NOT zero and, more than likely, the other matching parameters will be more inaccurate because of this. Hence the error in an AR(16) will be larger than that for an AR(4) fit.


1

If you drive your AR model with white noise, you can generate a time series that has the same power spectral density as the original time series. But it won't be the same time series.


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