Minimal multiplier realization of 2-D all-pass digital filters

Orthogonal realization of first-order allpass filters for two-dimensional signals

“…In particular, it follows that 11A [I < 1 is a sufficient condition for the absence of limit cycles (simply choose G = I in Inequality (15) The properties of our orthogonal realization can be compared with those of some alternative (non-orthogonal) realizations presented by Reddy, Roy and Hazra [1986] and by Manivannan and Eswaran [1988]. The realization in [Reddy, Roy and Hazra 1986] which leads to the same conclusion, viz., II A(m) 112 >-1.…”

“…The realization in [Reddy, Roy and Hazra 1986] which leads to the same conclusion, viz., II A(m) 112 >-1. Thus it appears that our orthogonal realization should have numerical advantages with respect to the nonorthogonal realizations in [Reddy, Roy and Hazra 1986] and [Manivannan and Eswaran 1988]. Furthermore, one can verify that for certain stable transfer functions these realizations give rise to state-space models for which the constraint (15) cannot be satisfied by any G > 0.…”

“…For instance, the realization of zw -0.9z + 0.09w H(z, w) = by the method of [Reddy, Roy and Hazra 1986] has a state-space model with a matrix A for which no G > 0 can be found to satisfy Inequality (15). This can be seen by letting by the method of [Manivannan and Eswaran 1988] there is no G > 0 such that Inequality (t5) is satisfied. However, since the condition (15) is only sufficient (but not necessary), we cannot conclude that these realizations will actually exhibit limit cycles.…”

“…The converse statement is established in Section 3 via the Givone-Roesser model associated with our orthogonal realization. The same model is used in Section 4 to study the finite-precision effects of our and of two alternative (non-orthogonal) realizations presented in [Reddy, Roy and Hazra 1986] and ] Manivannan and Eswaran 1988]. Section 5 contains concluding remarks.…”

“…However, the generalization to the 2-D case of the efficient 1-D orthogonal synthesis techniques appears to be a genuinely difficult (and still unsolved) probIemo Therefore, we consider in this paper onlyfirst-order allpassfitters, for which some nonorthogonal realizations have already been proposed [Joshi, Singh and Rai 1981;Ganapathy, Reddy and Reddy 1983;Reddy, Roy and Hazra 1986;Manivannan and Eswaran 1988].…”