Moving-Average Filters and Spectral Zeros

Finite Filters and Frequency Response

A finite moving-average filter forms each output by combining a fixed number of present and past input values. Its coefficients determine which oscillations pass through strongly and which are attenuated or eliminated. The transfer function packages this frequency-dependent effect into a single expression; its zeros identify frequencies at which the filter removes an input component entirely.

Work on a probability space \((\Omega,\mathcal{F},\mathbb{P})\). Let \((X_n)_{n\in\mathbb{Z}}\) be a real-valued, weakly stationary process with state space \(\mathbb{R}\), mean \(\mu_X\), and autocovariance \(\gamma_X(h)=\operatorname{Cov}(X_{n+h},X_n)\). Time is discrete and indexed by all integers so that past inputs are defined at every \(n\). Use the Fourier convention \(f_X(\omega)=\frac{1}{2\pi}\sum_{h\in\mathbb{Z}}\gamma_X(h)e^{-ih\omega}\) when the covariance sequence has a spectral density. Frequencies are taken in \([-\pi,\pi]\), with the endpoints representing the same frequency.

The term “moving average” here describes a finite linear filter, not an assumption that the inputs are independent. If the input has a spectral density, the filter multiplies that density by the squared magnitude of its frequency response.

Proof. Since the sum defining \(Y_n\) is finite, its mean is \(\mathbb{E}[Y_n]=\sum_{j=0}^{q}a_j\mathbb{E}[X_{n-j}]=(\sum_{j=0}^{q}a_j)\mu_X\), which does not depend on \(n\). Its covariance at lag \(h\) is also independent of \(n\), and direct expansion gives \(\gamma_Y(h)=\sum_{j=0}^{q}\sum_{k=0}^{q}a_ja_k\gamma_X(h-j+k)\). Taking the Fourier transform, each shifted covariance sequence contributes the factor \(e^{-i(j-k)\omega}\): with \(r=h-j+k\), we have \(h=r+j-k\), so \(e^{-ih\omega}=e^{-ir\omega}e^{-ij\omega}e^{ik\omega}\). Therefore \(f_Y(\omega)=\sum_{j,k}a_ja_ke^{-ij\omega}e^{ik\omega}f_X(\omega)=H(\omega)\overline{H(\omega)}f_X(\omega)=|H(\omega)|^2f_X(\omega)\). The finite sums introduce no convergence issue, and the formula proves the claimed spectral density relation. \(\square\)

For white noise \((\varepsilon_n)\) with mean zero and variance \(\sigma^2\), the covariance is \(\gamma_\varepsilon(0)=\sigma^2\) and \(\gamma_\varepsilon(h)=0\) for \(h\ne0\). Its spectral density is the constant \(\sigma^2/(2\pi)\). Thus a finite moving average of white noise has \(f_Y(\omega)=\frac{\sigma^2}{2\pi}|A(e^{-i\omega})|^2\). The filter’s zeros can be found algebraically: solve \(A(z)=0\), then keep precisely the roots \(z\) with \(|z|=1\). Each such root corresponds to a frequency satisfying \(z=e^{-i\omega}\).

Worked Examples: Zeros and Suppression

What a Spectral Zero Means

The filter’s frequency response describes its action on an oscillation. At frequency \(\omega\), the output amplitude is multiplied by \(H(\omega)\), and the output power is multiplied by \(|H(\omega)|^2\). A zero therefore removes that frequency from the filtered signal. More generally, the finite-filter formula says that \(f_Y(\omega)=|H(\omega)|^2f_X(\omega)\) almost everywhere. If the input spectrum has positive density near a zero, the filter creates a notch there; if the input already has no spectral power there, the output cannot reveal whether the filter would have suppressed it.

Root multiplicity controls the local shape of the notch. Suppose \(A(z)=(z-z_0)^mC(z)\), where \(|z_0|=1\), \(C(z_0)\ne0\), and \(m\ge1\). Write \(z_0=e^{-i\omega_0}\). For small real \(\delta\), the differentiability of \(e^{-i\omega}\) gives \(e^{-i(\omega_0+\delta)}-z_0=-iz_0\delta+o(\delta)\). Hence \(|A(e^{-i(\omega_0+\delta)})|^2\) is proportional to \(|\delta|^{2m}\) to leading order. If \(f_X\) is continuous and strictly positive at \(\omega_0\), the output density has the same order of vanishing. In particular, a simple root produces quadratic power suppression, while a double root produces fourth-order power suppression.

This conclusion follows from the factorization: the \(m\) repeated factors each contribute a magnitude of order \(|\delta|\), and squaring the total amplitude gives order \(|\delta|^{2m}\). The condition \(C(z_0)\ne0\) ensures the remaining factor does not introduce another zero at that point; if it did, the multiplicity would be larger.

Proof. By definition \(H(\omega_0)=A(e^{-i\omega_0})\). Thus \(H(\omega_0)=0\) exactly when \(e^{-i\omega_0}\) is a root of \(A\), proving both directions of the criterion. The spectral formula gives \(f_Y(\omega_0)=|H(\omega_0)|^2f_X(\omega_0)=0\) when the density values are defined there and \(H(\omega_0)=0\). Conversely, if \(f_X(\omega_0)>0\) and \(f_Y(\omega_0)=0\), the same formula forces \(|H(\omega_0)|^2=0\), and therefore \(H(\omega_0)=0\). For densities defined only up to changes on sets of measure zero, these pointwise statements apply to a chosen continuous version when one exists; the almost-everywhere spectral identity itself is the general statement. \(\square\)

Common Mistakes and Practical Checks

A frequent error is to confuse white noise with independent observations. White noise, as used here, means zero mean, constant variance, and zero covariance at nonzero lags; it does not by itself assert independence. For the three-point average, one might incorrectly assume independence from the white-noise assumption. Even if the three terms were independent, the variance would be \(3\sigma^2/9=\sigma^2/3\); here, pairwise uncorrelatedness alone gives that same result. The correct computation expands the variance and uses the vanishing cross-covariances, rather than asserting independence.

Another common mistake is to locate roots of \(A(z)\) without checking whether they lie on the unit circle. Only a unit-circle root corresponds to a real frequency, since \(e^{-i\omega}\) always has modulus \(1\). Also distinguish a transfer-function zero from a zero in the input spectrum: the output spectrum is the product of both effects. For a real-coefficient filter, nonreal roots occur in conjugate pairs, so their corresponding frequencies occur as a pair modulo \(2\pi\); a root at \(-1\) is the boundary frequency represented by both \(\pi\) and \(-\pi\).

A reliable calculation follows this order: write the coefficients and polynomial, solve for roots, retain unit-circle roots, evaluate \(|H(\omega)|^2\), and multiply by the input spectrum. Then check at least one feature independently, such as the output variance from the coefficients or the covariance sequence. These checks catch missing factors of \(2\pi\), sign errors in the exponential convention, and incorrect variance assumptions.