The waveform in equation (4) often arises in practice
(as the 2-D Huygen's wavelet).
Because of the discontinuities on the left side of equation (4),
it is not easy to visualize.
Thinking again of the time derivative
as a convolution with the doublet ,we imagine the 2-D Huygen's wavelet as a positive impulse followed
by negative signal decaying as -*t ^{-3/2}*.
This decaying signal is sometimes called the ``Hankel tail.''
In the frequency domain
has a 90 degree phase angle and
has a 45 degree phase angle.

In practice, it is easiest to represent
and to apply the 2-D Huygen's wavelet in the frequency domain.
Subroutine `halfdif()` is provided for that purpose.
Instead of using which
has a discontinuity at the Nyquist frequency
and a noncausal time function,
I use the square root of a causal representation
of a finite difference, i.e. ,(see any reference on *Z*-transforms)
which is well behaved at the Nyquist frequency
and has the advantage that the modeling operator is causal
(vanishes when *t*<*t _{0}*).
(Fourier transform is done using subroutine

# Half order causal derivative. # subroutine halfdif( conj, n, x, y) integer n2, i, conj, n real omega, x(n), y(n) complex cz, cv(4096) n2=1; while(n2<n) n2=2*n2; if( n2 > 4096) call erexit('halfdif memory') do i= 1, n2 { cv(i) = 0.} do i= 1, n if( conj == 0) { cv(i) = x(i)} else { cv(i) = y(i)} call ftu( +1., n2, cv) do i= 1, n2 { omega = (i-1.) * 2.*3.14159265 / n2 cz = csqrt( 1. - cexp( cmplx( 0., omega))) if( conj != 0) cz = conjg( cz) cv(i) = cv(i) * cz } call ftu( -1., n2, cv) do i= 1, n if( conj == 0) { y(i) = cv(i)} else { x(i) = cv(i)} return; end

hankel
Impulse response (delayed) of finite difference operator of half order.
Twice applying this filter is equivalent to once applying (1,-1).
Figure 1 |

11/17/1997