Cascaded Filters and End-to-End Spectral Calculations

From Individual Filters to a Cascade

Successive filters often arise naturally: a time series may first be smoothed, then differenced, or passed through a stable autoregressive filter before another transformation is applied. Calculating each stage separately can obscure how the stages interact. The central simplification is that, under suitable convergence conditions, the transfer functions multiply. The resulting end-to-end transfer function gives the output spectrum directly, while its coefficients provide an independent route to the output variance.

Work on a probability space \((\Omega,\mathcal{F},\mathbb{P})\), and let \((X_n)_{n\in\mathbb{Z}}\) be a real-valued weakly stationary process with finite second moment. Write its constant mean as \(\mu_X=\mathbb{E}[X_n]\), its autocovariance as \(\gamma_X(h)=\operatorname{Cov}(X_{n+h},X_n)\), and, when it has a spectral density, use the convention \(f_X(\omega)=\frac{1}{2\pi}\sum_{h\in\mathbb{Z}}\gamma_X(h)e^{-ih\omega}\), for \(-\pi\leq\omega\leq\pi\). Thus \(\gamma_X(0)=\int_{-\pi}^{\pi} f_X(\omega)\,d\omega\). The density describes covariance, not the mean; a nonzero mean is handled separately.

The absolute-summability condition ensures that the filter is well defined in mean square. In fact, the norm of a tail from indices \(r\) through \(s\) is at most \(\sqrt{\mathbb{E}[X_0^2]}\sum_{j=r}^s|a_j|\), by the triangle inequality in \(L^2\) and stationarity. The factor is the root second moment, not merely the standard deviation: \(\mathbb{E}[X_0^2]=\gamma_X(0)+\mu_X^2\). This distinction matters for nonzero-mean inputs. For example, if \(X_n=1\) deterministically, then \(\gamma_X(0)=0\), but a tail whose coefficient sum is \(1\) has mean-square norm \(1\).

Proof. Let \(U_n=\sum_{j\geq0}a_jX_{n-j}\), and filter \(U\) with coefficients \(b_k\). For finite sums, collecting the coefficient of \(X_{n-r}\) gives \(\sum_{k+j=r}b_ka_j\). These coefficients are absolutely summable because \(\sum_{r\geq0}\sum_{k+j=r}|b_k a_j|=(\sum_k|b_k|)(\sum_j|a_j|)<\infty\). The mean-square tail bound above therefore permits passage from finite sums to the infinite filters; the combined coefficients are \(c_r=\sum_{k+j=r}b_ka_j\). Absolute convergence also permits multiplying the two transfer-function series and collecting terms, giving \(C(\omega)=B(\omega)A(\omega)\). Taking expectations of the convergent sums gives \(\mu_Y=\mu_X(\sum_j a_j)(\sum_k b_k)=\mu_XA(0)B(0)\).

For a finite filter with coefficients \(c_j\), covariance propagation gives \(\gamma_Y(h)=\sum_{j,k}c_jc_k\gamma_X(h+j-k)\). Taking the Fourier series in \(h\), the shift \(h+j-k\) contributes the factor \(e^{ij\omega}e^{-ik\omega}\); summing over \(j,k\) gives \(|\sum_jc_je^{-ij\omega}|^2\). This proves the density formula for finite filters. For absolutely summable coefficients, the double sums converge absolutely because \(|\gamma_X(h)|\leq\gamma_X(0)\) and \(\sum_{j,k}|c_jc_k|<\infty\). Truncating the filters and taking the mean-square limit therefore gives the same covariance and density formula for the infinite filters. Finally, Fourier inversion for a process with spectral density gives \(\gamma_Y(0)=\int_{-\pi}^{\pi}f_Y(\omega)\,d\omega\). \(\square\)

In practice, the theorem supplies two useful checks. First multiply the transfer functions and simplify before calculating. Second, if the input is white noise, the output variance is also the sum of the squares of the combined coefficients, multiplied by the input variance. The examples use both approaches.

Stable Autoregressive Filters in a Cascade

A stable first-order autoregressive filter with parameter \(\phi\), where \(|\phi|<1\), has transfer function \(1/(1-\phi e^{-i\omega})\). Its causal coefficients are \(1,\phi,\phi^2,\ldots\), whose absolute sum is finite. It can therefore be combined with finite differencing or smoothing filters using the cascade rule. Stability is essential for this causal representation: the geometric series converges absolutely only when \(|\phi|<1\).

What the Cascade Calculation Does—and Does Not—Say

The transfer-function product is an end-to-end description of the linear transformation. It does not say that variances at intermediate stages can simply be multiplied. In general, the intermediate observations are dependent, and their covariance terms contribute to the final variance. The spectrum keeps these interactions through the frequency-by-frequency product of squared magnitudes.

Common mistake. In the first example, the differenced white noise has variance \(2\). A tempting but wrong calculation is to multiply \(2\) by the sum of squared weights of the three-point average, \(3(1/3)^2=1/3\), and report variance \(2/3\). That variance rule applies when the inputs to the weighted sum are uncorrelated with equal variance; the differenced observations are not uncorrelated. In fact, their lag-one covariance is \(-1\), since \(\operatorname{Cov}(\varepsilon_n-\varepsilon_{n-1},\varepsilon_{n-1}-\varepsilon_{n-2})=-1\). The three diagonal terms in the smoothed variance contribute \(3\cdot2=6\), and the two ordered lag-one pairs contribute \(2\cdot2\cdot(-1)=-4\); lag-two covariances are zero. Dividing the total \(6-4=2\) by \(9\) gives \(2/9\), not \(2/3\).

For a nonzero-mean input, the same cascade rule still applies to the mean: multiply the component gains at frequency zero and then multiply by \(\mu_X\). The covariance spectrum continues to use the centered process. Thus a filter with coefficient sum zero removes a constant mean, while a filter with nonzero coefficient sum generally preserves or rescales it. Keeping mean and covariance separate avoids treating a deterministic component as though it were variance.

For a reliable end-to-end calculation, write each component transfer function, multiply and simplify, form \(f_Y=|H|^2f_X\), and calculate the variance from the integral or from the combined coefficients. The coefficient route is especially convenient for white noise; the spectral route makes frequency-specific attenuation and spectral zeros visible.