Z transforms and Fourier transforms are related by the relations and .A problem with these relations is that simple ratios of polynomials in Z do not translate to ratios of polynomials in and vice versa. The approximation
(42) |
(43) |
(44) |
(45) |
For a Z transform B(Z) to be minimum phase, any root Z_{0} of 0 = B(Z_{0}) should be outside the unit circle. Since and , it means that for a minimum phase should be negative. (In other words, is in the lower half-plane.) Thus it may be said that maps the exterior of the unit circle to the lower half-plane. By inspection of Figures 20 and 21, it is found that the bilinear approximation (42) or (43) also maps the exterior of the unit circle into the lower half-plane.
2-20
Figure 20 Some typical points in the Z-plane, the -plane, and the -plane. |
Thus, although the bilinear approximation is an approximation, it turns out to exactly preserve the minimum-phase property. This is very fortunate because if a stable differential equation is converted to a difference equation via (42), the resulting difference equation will be stable. (Many cases may be found where the approximation of a time derivative by multiplication with 1 - Z would convert a stable differential equation into an unstable difference equation.)
A handy way to remember (42) is that corresponds to time differentiation of a Fourier transform and (1 - Z) is the first differencing operator. The (1 + Z) in the denominator gets things ``centered" at Z^{1/2}
To see that the bilinear approximation is a low-frequency approximation, multiply top and bottom of (42) by Z^{-1/2}
(46) |
(47) |
2-22
Figure 22 The accuracy of the bilinear transformation approximation. |
From Figure 22 we see that the error will be only a few percent if we choose small enough so that . Readers familiar with the folding theorem will recall that it gives the less severe restraint . Clearly, the folding theorem is too generous for applications involving the bilinear transform.
Now, by way of example, let us take up the case of a pole at zero frequency. This is integration. For reasons which will presently be clear, we will consider the slightly different pole
(48) |
(49) |