Higher-Order Filters

If one simply iterates $N$ times the procedure in Equation (5), which is equivalent to the definition of the first-order linear low-pass filter within the unit disk of the complex plane, one gets the corresponding higher-order filters in the complex-plane representation. For example, a second-order filter with range $2\epsilon$ can be obtained by applying the first-order filter twice, and results in the complex-plane definition

$$w_{2\epsilon}^{(2)}(z) = \left(\frac{-\mbox{\boldmath$\im... ...,{\rm e}^{-2\mbox{\boldmath$\imath$}\epsilon}\right) \right].$$

Note that only the second logarithmic primitive of $w(z)$ appears here, and hence that all singularities are softened by two degrees. A corresponding second-order filter with range $\epsilon$ can then be obtained by the exchange of $\epsilon$ by $\epsilon/2$,

$$w_{\epsilon}^{(2)}(z) = \left(\frac{-\mbox{\boldmath$\ima... ...\,{\rm e}^{-\mbox{\boldmath$\imath$}\epsilon}\right) \right].$$

It is now possible to define an $N^{\rm th}$ order filter, with range $N\epsilon$, by iterating this procedure repeatedly. If we look at it in terms of the singularities on the unit circle, the iteration corresponds to recursive singularity splitting as shown in Figure 2. In this diagram we can see the structure of the Pascal triangle, and the linear increase of the resulting range with $N$. Note that at the $N^{\rm th}$ iteration there are $N+1$ softened singularities within the interval $[-N\epsilon,N\epsilon]$ of the variable $\left (\theta -\theta '\right )$. It is important to observe that, while the singularities become progressively softer as one goes down the diagram, it is also the case that more and more singularities are superposed at the same point, particularly near the central vertical line of the triangle. From the structure of the Pascal triangle the coefficients of the superposition are easily obtained, so that from this diagram it is not too difficult to obtain the expression for the $N^{\rm th}$ order filter, with range $N\epsilon$, which turns out to be

$$w_{N\epsilon}^{(N)}(z) = \left(\frac{-\mbox{\boldmath$\im... ...ft(z\,{\rm e}^{\mbox{\boldmath$\imath$}[N-2n]\epsilon}\right).$$

In this case the range of the changes introduced in the real functions by the order-$N$ filter has the value $N\epsilon$. This means that, given a fixed value of $\epsilon$, the iteration of the first-order filter cannot be done indefinitely inside the periodic interval $[-\pi ,\pi ]$ without the range eventually becoming larger than the period. However, one may reduce the resulting range back to $\epsilon$ by simply using for the construction the linear filter with range $\epsilon/N$, resulting in

$$w_{\epsilon}^{(N)}(z) = \left(\frac{-\mbox{\boldmath$\ima... ...(z\,{\rm e}^{\mbox{\boldmath$\imath$}[1-2n/N]\epsilon}\right).$$

With this renormalization of the parameter $\epsilon$ it is now possible to do the iteration of the first-order filter indefinitely inside the periodic interval $[-\pi ,\pi ]$, keeping the range constant, and therefore to define filters of arbitrarily high orders. In this case a singularity at $\theta '$ on the unit circle will be split into $N+1$ singularities softened by $N$ degrees, homogeneously distributed within the interval $[-\epsilon ,\epsilon ]$ of the variable $\left (\theta -\theta '\right )$. One may even consider iterating the filter an infinite number of times in this way, keeping the range constant. However, this does not work quite as one might expect at first. A detailed discussion of this case can be found in Section 3.

Note that since these higher-order filters are obtained by the repeated application of the first-order one, they inherit from it many of its properties. For example, they are all the identity when applied to linear real functions on the unit circle [4], and they all maintain the periodicity of periodic functions [9]. Also, they all have the elements of the Fourier basis as eigenfunctions and hence they all commute with the second-derivative operator, as demonstrated in [3]. In terms of the DP Fourier series, if one considers the $N$-fold repeated application of the first-order filter to the original real function, since each instance of the first-order filter contributes the same factor to the coefficients, as shown in [3], one simply gets for the filtered real functions

$$\begin{eqnarray*} f_{N\epsilon,{\rm c}}^{(N)}(\theta) & = & \sum_{k=1}^{\inft... ...\frac{\sin(k\epsilon)}{(k\epsilon)} \right]^{N} \sin(k\theta). \end{eqnarray*}$$

This will of course imply that the filtered DP Fourier series converge significantly faster than the original ones, and to significantly smoother functions. In this case the range of the changes introduced in the real functions has the value $N\epsilon$. Once more one may reduce the resulting range back to $\epsilon$, using the linear filter with range $\epsilon/N$, thus leading to

$$\begin{eqnarray*} f_{\epsilon,{\rm c}}^{(N)}(\theta) & = & \sum_{k=1}^{\infty... ...c{\sin(k\epsilon/N)}{(k\epsilon/N)} \right]^{N} \sin(k\theta). \end{eqnarray*}$$

This modification changes only the range of the alterations introduced in the real functions by the order-$N$ filter, and not the level of smoothness of the resulting filtered functions, which depends only on $N$.

Since they are themselves real functions defined on a circle of radius $\rho\leq 1$ in the complex plane, centered at the origin, the kernels of the order-$N$ filters can also be represented by inner analytic functions within the corresponding open disk. This is a simple extension of the structure we developed in the earlier papers [1] and [2]. If $z_{1}=\rho_{1}\exp(\mbox{\boldmath$\imath$}\theta_{1})$ is a point on the circle $\rho_{1}\leq 1$ and $z=\rho\exp(\mbox{\boldmath$\imath$}\theta)$ a point inside the corresponding disk, the kernels of constant range $\epsilon$ can be written as the real parts of the complex kernels

$$\kappa_{\epsilon}^{(N)}\!\left(z,z_{1}\right) = \frac{1}{... ...psilon/N)} \right]^{N} \left( \frac{z}{z_{1}} \right)^{k},$$

where it should be noted that the coefficients are real. Except for the constant term this is the Taylor series of an inner analytic function inside the disk of radius $\rho_{1}$, rotated by the angle $\theta_{1}$. If we take the limit $\rho\to\rho_{1}$ we get

$$\begin{eqnarray*} \kappa_{\epsilon}^{(N)}(\theta-\theta_{1}) & = & \frac{1}{2... ...lon/N)}{(k\epsilon/N)} \right]^{N} \sin[k(\theta-\theta_{1})], \end{eqnarray*}$$

and therefore we have

$$\Re\!\left[\kappa_{\epsilon}^{(N)}(\theta-\theta_{1})\right] = K_{\epsilon}^{(N)}(\theta-\theta_{1}).$$

Note that using this complex-plane representation it is easy to prove that the kernels of the order-$N$ filters have unit integral. We consider the integral over the circle $C_{1}$ of radius $\rho_{1}$, that appears in the Cauchy integral formula for $\kappa_{\epsilon}^{(N)}(z,z_{1})$ around $z=0$,

$$\frac{1}{2\pi\mbox{\boldmath$\imath$}} \oint_{C_{1}}dz\, ... ...\epsilon}^{(N)}(z,z_{1}) = \kappa_{\epsilon}^{(N)}(0,z_{1}).$$

Since we have the value $\kappa_{\epsilon}^{(N)}(0,z_{1})=1/(2\pi)$, we get

$$\frac{1}{\mbox{\boldmath$\imath$}} \oint_{C_{1}}dz\, \frac{1}{z}\, \kappa_{\epsilon}^{(N)}(z,z_{1}) = 1.$$

If we now write the integral explicitly over the circle, with $z=\rho_{1}\exp(\mbox{\boldmath$\imath$}\theta)$ and $dz=\mbox{\boldmath$\imath$}zd\theta$, we get

$$\int_{-\pi}^{\pi}d\theta\, \kappa_{\epsilon}^{(N)}(\theta-\theta_{1}) = 1.$$

Finally, if we consider explicitly the real and imaginary parts we get

$$\begin{eqnarray*} \int_{-\pi}^{\pi}d\theta\, \left\{ \Re\!\left[\kappa_{\epsi... ...eft[\kappa_{\epsilon}^{(N)}(\theta-\theta_{1})\right] & = & 0. \end{eqnarray*}$$

Since the real part is $K_{\epsilon}^{(N)}(\theta-\theta_{1})$, the result follows,

$$\int_{-\pi}^{\pi}d\theta\, K_{\epsilon}^{(N)}(\theta-\theta_{1}) = 1,$$

for all $N$.