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# ROBINSON'S ENERGY DELAY THEOREM

We will now show that a minimum-phase wavelet has less energy delay than any other one-side wavelet with the same spectrum. More precisely, we will show that the energy summed from zero to any time t for the minimum-phase wavelet is greater than or equal to that of any other wavelet with the same spectrum. Refer to Figure 3-2.

 3-2 Figure 2 Percent of total energy in a filter between time and time t.

We will compare two wavelets and which are identical except for one zero, which is outside the unit circled for and inside for .We may write this as where b is bigger than s and P is arbitrary but of degree n. Next we tabulate the terms in question.

 t bp0 sp0 1 bp1 + sp0 sp1 + bp0 k bpk + spk -1 spk + bpk -1 n + 1 spn bpn (b2 - s2) (-pn2)

The difference, which is given in the right-hand column, is clearly always positive.

To prove that the minimum-phase wavelet delays energy the least, the preceding argument is repeated with each of the roots until they are all outside the unit circle.

## EXERCISES:

1. Do the foregoing minimum-energy-delay proof for complex-valued b, s, and P. [CAUTION:  Does or ?]

Next: THE TOEPLITZ METHOD Up: Spectral factorization Previous: ROOT METHOD
Stanford Exploration Project
10/30/1997