Two-dimensional filter

Two dimensional filters have seen substantial development effort due to their importance and high applicability across several domains.

This means that an arbitrary transfer function cannot generally be manipulated into a form required by a particular implementation.

Because the values of the output samples are fed back, the 2-D filter, like its 1-D counterpart, can be unstable.

Due to the rapid development of information science and computing technology, the theory of digital filters design and application has achieved leap-forward development.

The analog signals are continuous function of the independent variables, which can be one-dimensional, two-dimensional or multidimensional.

The resulting digital signal can be represented by a discrete sequence.

The design and implementation of filter is an important branch in signal analysis and processing technology.

Filters also play a main role in signal acquisition, transmission, processing and exchange.

Digital filters can be implemented numerically in software and have the advantages of high processing accuracy, steady system, little volume and light weight.

There is an important difference between the design of 1-D and 2-D digital filter problems.

In 1-D case, the design and the implementation of filters can be more easily considered separately.

The filter can first be designed and then, through the appropriate manipulations of the transfer function, the coefficients required by a particular network structure can be determined.

This means that an arbitrary multi-dimensional transfer function can generally not be manipulated into a form required by a particular implementation.

This has the effect of complicating the design problem and also limiting the number of practical implementations.

2-D FIR digital filter is achieved by a non-recursive algorithm structure while 2-D IIR digital filter is achieved by a recursive feedback algorithm structure.

[3] Another method to build up complicated 2-D IIR filters is by the parallel interconnection of subfilters.

and putting the sum in transfer function over a common denominator, we get the expanded form

The parallel form cannot be used to implement an arbitrary 2-D rational system function.

[4] Nevertheless, we can synthesize interesting 2-D IIR filters which can be implemented by a parallel architecture.

For example, the parallel form may be advantageous when designing a filter with multiple passband.

Many design techniques for 2-D IIR digital filters have been reported in the literature.

[1][2][3][4] In 2013, genetic algorithm had been successfully used to digital filter design for about a decade.

The figure below shows the proposed GA-Based design flow.

Filter coefficients are encoded in their CSD number representation.

Each coefficient has the pre-specified wordlength and maximum number of non-zero digits, which can be set to any desired values.

Mutation operation is the simple single bit flip.

After mutation, each coefficient in the offspring is checked upon CSD format.

Chromosomes failing the check are assigned fitness value 0.

The designed filter has non-separable numerator and separable denominator transfer function.

[6] The number restoration technique is used to ensure that the filter coefficients are represented in the pre-specified CSD format.