The **GMT** programs * filter1d* (for tables of data indexed
to one independent variable) and

Impulse responses are shown here for the boxcar, cosine, and gaussian filters. Only the relative amplitudes of the filter weights shown; the values in the center of the window have been fixed equal to 1 for ease of plotting. In this way the same graph can serve to illustrate both the 1-d and 2-d impulse responses; in the 2-d case this plot is a diametrical cross-section through the filter weights (Figure J.1).

Although the impulse responses look the same in 1-d and 2-d, this is not true of the transfer functions; in 1-d the transfer function is the Fourier transform of the impulse response, while in 2-d it is the Hankel transform of the impulse response. These are shown in Figures J.2 and J.3, respectively. Note that in 1-d the boxcar transfer function has its first zero crossing at , while in 2-d it is around . The 1-d cosine transfer function has its first zero crossing at ; so a cosine filter needs to be twice as wide as a boxcar filter in order to zero the same lowest frequency. As a general rule, the cosine and gaussian filters are ``better'' in the sense that they do not have the ``side lobes'' (large-amplitude oscillations in the transfer function) that the boxcar filter has. However, they are correspondingly ``worse'' in the sense that they require more work (doubling the width to achieve the same cut-off wavelength).

One of the nice things about the gaussian filter is that its
transfer functions are the same in 1-d and 2-d. Another nice
property is that it has no negative side lobes. There are many
definitions of the gaussian filter in the literature (see page
7 of Bracewell^{J.1}). We
define equal to 1/6 of the filter width, and the impulse
response proportional to
. With this
definition, the transfer function is
and the wavelength at which the transfer function equals 0.5 is
about 5.34 , or about 0.89 of the filter width.