9807

Separable and Nonseparable 2D Sequences

This Demonstration shows a separable 2D sinusoidal sequence (left) versus the nonseparable one (right) displayed as 16×16 images, with heights represented by colors.
,
where and are horizontal and vertical directions, respectively, and and are the horizontal and vertical angular frequencies. This sequence is called separable because it is obtained as a product of two one-dimensional sequences. Thus, if any of the sliders are set to 0, the sequences will be 0 (constant image). For any combinations of sliders, the resulting pattern will be rectangular.
The right figure (nonseparable) is generated using
.
This sequence is called nonseparable because it cannot be obtained as a product of two one-dimensional sequences. If one of the sliders is set to 0 and the other to any nonzero value, the resulting image will show a pattern only in one dimension. If both sliders are set to a nonzero value, the resulting pattern will never be rectangular.
The fact that the separable sequence can only achieve rectangular space patterns, while the nonseparable one offers more flexibility, is often used in discrete-time signal processing to design filters.

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Reference
[1] M. Vetterli, J. Kovačević, and V. K. Goyal, Signal Processing: Foundations, Cambridge: Cambridge University Press, forthcoming. www.fourierandwavelets.org.
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