On the stability of two-dimensional continuous systems

A necessary and sufficient approach is presented to check bounded-input-bounded-output stability of 2-D continuous systems. The approach works by obtaining the impulse response through the inverse 2-D Laplace transformation. It is shown that by using one-variable reactance transformation (or spectral transformation) to obtain the 2-D continuous transfer function, stability is, in general, not preserved. The effect of double bilinear transformation from the 2-D continuous transfer function to the corresponding discrete one is discussed. It is shown by an example that the impulse response of both continuous and discrete transfer functions is not preserved. It is further shown that the double bilinear transformation can transform a stable digital 2-D filter into an unstable continuous 2-D filter