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This paper proposes an analytical design method for two-dimensional square-shaped IIR filters. The designed 2D filters are adjustable since their bandwidth and orientation are specified by parameters appearing explicitly in the filter matrices. The design relies on a zero-phase low-pass 1D prototype filter. To this filter a frequency transformation is next applied, which yields a 2D filter with the desired square shape in the frequency plane. The proposed method combines the analytical approach with numerical approximations. Since the prototype transfer function is factorized into partial functions, the 2D filter also will be described by a factorized transfer function, which is an advantage in implementation.

Various design methods for 2D filters, both FIR and IIR, have been proposed by many researchers [

Diamond filters have been used as antialiasing filters in the conversion between signals sampled on the rectangular sampling grid and the quincunx sampling grid. Different issues related to design methods for diamond filters were studied in [

In this paper an analytical design method is proposed for 2D adjustable zero-phase square-shaped filters, a larger class of filters which may be regarded as a generalization of the common diamond filter. Starting from a 1D prototype filter with factorized transfer function, the corresponding 2D filters are obtained by a particular 1D to 2D frequency mapping. The 2D filter will have a factorized transfer function, which is a useful feature in implementation. This work mainly focuses on presenting the proposed method and describes in detail the design steps. Several design examples are also provided. The typical applications of diamond filters were not approached here, as they are extensively treated in many other works. Image processing applications of the square-shaped filters with arbitrary bandwidth and orientation proposed here will be approached in further work.

A particular case of a square-shaped filter is the standard diamond filter; its shape in the frequency plane is shown in Figure

(a) diamond filter; (b) wide-band oriented filter; (c), (d) wide-band oriented filters with orientations forming an angle

In this work a more general case is approached, that is, a 2D diamond-type filter with a square shape in the frequency plane but with arbitrary side length and axis inclination angle, as shown in Figure

The square-shaped filter in Figure

The frequency response of

A more general filter belonging to this class is a rhomboidal filter, as shown in Figure

An analog filter of order

Let us consider a Butterworth low-pass filter

We look for a rational expression of the magnitude

Let us first find the approximation for the function (

The frequency response

(a) Zero-phase low-pass prototype filter characteristic for

Each of the rational expressions

As specified earlier, the variable

Thus the 1D prototype is parametric depending on

The main issue in this section is to find the transfer function of the desired 2D square filter

From a 1D prototype filter

In our case, since the rational prototype function

The currently used method to obtain a discrete filter from an analog prototype is applying the bilinear transform. If the sample interval takes the value

Even if this method is straightforward, the designed 2D filter, corresponding to the transfer function in

We obtain the following Chebyshev-Padé approximation on the range

Plots of mapping (

where the variable vectors are

The numerator matrix

The denominator matrix

The 2D directional IIR filter will have a

To summarize, the design steps for the adjustable square-shaped filter are the following.

Once adopted an 1D low-pass filter prototype of the form given by (

For a specified orientation angle

For a desired filter bandwidth

The matrices

The matrices

Following the previous steps, the design of a 2D square-shaped filter using this method is straightforward. The bandwidth

The first design example is the diamond-type filter whose frequency response

(a) Frequency response of a square-shaped filter with orientation

In Figures

(a) Frequency response of a square-shaped filter with orientation

(a) Frequency response of a square-shaped filter with orientation

A more general filter of this kind can also be derived, namely, a rhomboidal-shaped filter, in which the two component filters are not in a right angle to each other. Furthermore, a directional filter like the ones in Figure

Even if this analytical design method may not yield optimal filters in terms of complexity, as is generally the case with techniques based on numerical optimization, the advantage of having parametric, closed form expressions for the filter matrices is the possibility of tuning or adjusting the 2D filter for a new set of specifications—in our case bandwidth and orientation—without the need to resume the design process every time.

Stability of the proposed filters is also an important issue. The stability problem for 2D filters is much more difficult than for 1D filters. There exist various stability criteria [

The main goal of the paper was to present in detail this analytical design method and some relevant design examples for these filters. Since they can select adjustable square or rhomboidal regions in the frequency plane, they may have interesting applications in image processing. Their applications in image filtering will be investigated in future work on this topic.

An analytical design method for recursive zero-phase square-shaped filters was proposed. The design is based on analog zero-phase low-pass prototypes with specified parameters. As 1D prototype, a Butterworth filter was used, and, using an efficient rational approximation, a zero-phase filter small pass-band ripple was derived. The resulted 2D filter is parametric or adjustable in the sense that the specified parameters—bandwidth and orientation angle—appear explicitly in the final filter matrices. The method includes the bilinear transform applied along the two axes and uses a frequency pre-warping to compensate for distortions. This leads to a particular frequency mapping, which is applied to the 1D prototype in order to obtain the 2D filter. This design technique is relatively simple, efficient and versatile, in the sense that, by changing the specifications, the new 2D filter matrices result directly, without the need to resume the entire design process all the way from the start.