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Frequency domain and spectrum

MIDA1 Model Identification and Data Ana... Frequency Domain Analysis

The frequency domain is another way to obtain the weak (wide sense) characterization of a stationary stochastic process. Consider a stationary stochastic process $y(t)$: $\mathbb{E}[y(t)]=m_y \text{ } \forall t$ $\gamma_y(\tau)=\mathbb{E}[(y(t)-m_y)(y(t-\tau)...

Spectrum antitrasformation and relation with covariance

MIDA1 Model Identification and Data Ana... Frequency Domain Analysis

There exists an antitrasformation: $$\gamma_y(\tau)=\frac{1}{2\pi}\int_{-\pi}^{+\pi}{\Gamma_y(\omega)e^{j\omega\tau}d\omega}$$ Notice that the variance: $$\gamma_y(0)=\mathbb{E}[(y(t)-m_y)^2]=\frac{1}{2\pi}\int_{-\pi}^{+\pi}{\Gamma_y(\omega)d\omega}$$ so the v...

Wiener–Khinchin theorem

MIDA1 Model Identification and Data Ana... Frequency Domain Analysis

Spectrum of ARMA processes

MIDA1 Model Identification and Data Ana... Frequency Domain Analysis

Given the $ARMA$ process $y(t)=\frac{C(z)}{A(z)}e(t)$ with $e(t) \sim WN(0,\lambda^2)$ the spectrum is given by: $$\Gamma_y(\omega)=\mid\frac{C(e^{j\omega})}{A(e^{j\omega})}\mid^2\times\lambda^2$$ Since both $C(z)$ and $A(z)$ are polynomial the result is a rat...

Linear optimal prediction

MIDA1 Model Identification and Data Ana... Model prediction

Starting from an $ARMA$ process $y(t)=W(z)e(t)=\frac{C(z)}{A(z)}e(t)$, wehere $e(t) \sim WN(0,\lambda^2)$ (non zero mean case is an easy extension) we assume: $W(z)$ is asymptotically stable, and so $y(t)$ is well defnined $y(t)=\frac{C(z)}{A(z)}e(t)$ is the ...

Linear predictors from noise

MIDA1 Model Identification and Data Ana... Model prediction

Starting from an $ARMA$ process $y(t)=W(z)e(t)=\frac{C(z)}{A(z)}e(t)$, wehere $e(t) \sim WN(0,\lambda^2)$ we know that $y(t)$ is a steady state solution and $y(t) \sim MA(\infty)$, so: $$y(t) = w_0e(t)+w_1e(t-1)+...+w_ie(t-i)+...=\sum_{i=0}^{+\infty}{w_ie(t-i)...

Long k-step division method

MIDA1 Model Identification and Data Ana... Model prediction

The steady state solution of an $ARMA$ process can be obtained with a long k-step division of $C(z)$ and $A(z)$ seen as polynomial of $z$ performing only k-steps. $$\frac{C(z)}{A(z)}=E(z)+\frac{z^{-k}F(z)}{A(z)}$$ Where: $E(z)$ is the quotient $z^{-k}F(z)$ i...

Linear predictor from output

MIDA1 Model Identification and Data Ana... Model prediction

In the real world we cannot measur the white noise, the only aviable information is the values of the process up to time $t$. We need to construct the white noise underlyning the generation of the process from the values of the process itself up to time $t$. T...

Prediction of non zero mean ARMA

MIDA1 Model Identification and Data Ana... Model prediction

Starting from an $ARMA$ process $y(t)=W(z)e(t)=\frac{C(z)}{A(z)}e(t)$, wehere $e(t) \sim WN(\mu,\lambda^2)$ that is canonical and minimum phase we need to compute the unbiased processes: $$ \tilde{y}=y(t)-M_y\newline \tilde{e}(t)=e(t)-\mu $$ The resulting arma...

ARMAX predictors

MIDA1 Model Identification and Data Ana... Model prediction

Starting from an $ARMAX$ process $y(t)=\frac{B(z)}{A(z)}u(t-d)+\frac{C(z)}{A(z)}e(t)$, wehere $e(t) \sim WN(0,\lambda^2)$ and $\frac{C(z)}{A(z)}$ is canonical and minimum phase then we assume that the input signal $u(t-d)$ is either: completely known from $t=...