October 2006
Intermediate to advanced
369 pages
8h 11m
English
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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:
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:
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:
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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