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Digital Filters Design for Signal and Image Processing by Mohamed Najim

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9.3. Synthesizing with the two-dimensional windowing method

9.3.1. Principles of method

As we have seen in Chapter 8, the transformation that characterizes an invariant linear system is represented by the impulse response written hd(x, y), which satisfies the following formula:

images

where s(x,y) and w(x,y) respectively designate the two-dimensional input and output signals.

Depending on application requirements, we synthesize an impulse response that helps us to obtain a desired frequency response following typical shapes such as low-pass, high pass, cut-off band or passband filter. The correspondence between frequency domain representation and impulse response is assured by the 2-D Fourier transform.

Let us consider the truncated impulse response of the digital 2-D FIR filter defined by:

images

The quantities m and n represent the size of the impulse response in horizontal and vertical directions. To establish the link between discrete and continuous forms, equation (9.6) can be derived in a continuous spatial domain. We then have:

images

The quantities t1 and t2 represent the spatial boundaries associated with the non-null values of the impulse response h(x,y) and are directly proportional to m and ...

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