# 4.4: Linear Filtering

- Page ID
- 849

A linear filter uses specific coefficients \((\psi_s\colon s\in\mathbb{Z})\), called the impulse response function, to transform a weakly stationary input series \((X_t\colon t\in\mathbb{Z})\) into an output series \((Y_t\colon t\in\mathbb{Z})\) via

\[ Y_t=\sum_{s=-\infty}^\infty\psi_sX_{t-s}, \qquad t\in\mathbb{Z}, \]

where \(\sum_{s=-\infty}^\infty|\psi_s|<\infty\). Then, the frequency response function

\[ \Psi(\omega)=\sum_{s=-\infty}^\infty\psi_s\exp(-2\pi i\omega s) \]

is well defined. Note that the two-point moving average of Example 4.2.2 and the differenced sequence \(\nabla X_t\) are examples of linear filters. On the other hand, *any *\/ causal ARMA process can be identified as a linear filter applied to a white noise sequence. Implicitly this concept was already used to compute the spectral densities in Exampels 4.2.2 and 4.2.3. To investigate this in further detail, let \(\gamma_X(h)\) and \(\gamma_Y(h)\) denote the ACVF of the input process \((X_t\colon t\in\mathbb{Z})\) and the output process \((Y_t\colon t\in\mathbb{Z})\), respectively, and denote by \(f_X(\omega)\) and \(f_Y(\omega)\) the corresponding spectral densities. The following is the main result in this section.

**Theorem 4.4.1.**

*Under the assumptions made in this section, it holds that* \(f_Y(\omega)=|\Psi(\omega)|^2f_X(\omega)\).

**Proof. **First note that

\(\gamma_Y(h)=E\big[(Y_{t+h}-\mu_Y)(Y_t-\mu_Y)]\\[.2cm]\)

\(=\sum_{r=-\infty}^\infty\sum_{s=-\infty}^\infty\psi_r\psi_s\gamma(h-r+s)\\[.2cm]\)

\(=\sum_{r=-\infty}^\infty\sum_{s=-\infty}^\infty\psi_r\psi_s\int_{-1/2}^{1/2}\exp(2\pi i\omega(h-r+s))f_X(\omega)d\omega\\[.2cm]\)

\(=\int_{-1/2}^{1/2}\Big(\sum_{r=-\infty}^\infty\psi_r\exp(-2\pi i\omega r)\Big)\Big(\sum_{s=-\infty}^\infty\psi_s\exp(2\pi i\omega s)\Big)\exp(2\pi i\omega h)f_X(\omega)d\omega\\[.2cm]\)

\(=\int_{-1/2}^{1/2}\exp(2\pi i\omega h)|\Psi(\omega)|^2f_X(\omega)d\omega.\)

Now identify \(f_Y(\omega)=|\Psi(\omega)|^2f_X(\omega)\), which is the assertion of the theorem.

Theorem 4.4.1 suggests a way to compute the spectral density of a causal ARMA process. To this end, let \((Y_t\colon t\in\mathbb{Z})\) be such a causal ARMA(*p,q*) process satisfying \(Y_t=\psi(B)Z_t\), where \((Z_t\colon t\in\mathbb{Z})\sim\mbox{WN}(0,\sigma^2)\) and

\[ \psi(z)=\dfrac{\theta(z)}{\phi(z)}=\sum_{s=0}^\infty\psi_sz^s,\qquad |z|\leq 1. \]

with \(\theta(z)\) and \(\phi(z)\) being the moving average and autoregressive polynomial, respectively. Note that the \((\psi_s\colon s\in\mathbb{N}_0)\) can be viewed as a special impulse response function.

**Corollary 4.4.1.**

If \((Y_t\colon t\in\mathbb{Z})\) be a causal ARMA(*p,q*)\) process. Then, its spectral density is given by

\[ f_Y(\omega)=\sigma^2\dfrac{|\theta(e^{-2\pi i\omega})|^2}{|\phi(e^{-2\pi i\omega})|^2}. \]

**Proof.** Apply Theorem 4.4.1 with input sequence \((Z_t\colon t\in\mathbb{Z})\). Then \(f_Z(\omega)=\sigma^2\), and moreover the frequency response function is

\[ \Psi(\omega)=\sum_{s=0}^\infty\psi_s\exp(-2\pi i\omega s)=\psi(e^{-2\pi i\omega})=\dfrac{\theta(e^{-2\pi i\omega})}{\phi(e^{-2\pi i\omega})}. \]

Since \(f_Y(\omega)=|\Psi(\omega)|^2f_X(\omega)\), the proof is complete.

Corollary 4.4.1 gives an easy approach to define parametric spectral density estimates for causal ARMA(*p,q*) processes by simply replacing the population quantities by appropriate sample counterparts. This gives the spectral density estimator

\[ \hat f(\omega)=\hat\sigma_n^2\dfrac{|\hat\theta(e^{-2\pi i\omega})|^2}{|\hat\phi(e^{-2\pi i\omega})|^2}. \]

Now any of the estimation techniques discussed in Section 3.5 may be applied when computing \(\hat f(\omega)\).

## Contributers

Integrated by Brett Nakano (statistics, UC Davis)