Results on a simple 1-D example

For the simple 1-D example under consideration, the filter coefficients are now estimated with the $\ell^1$ norm approximated with the Huber function. In Figure , I display the result of the adaptive subtraction when the $\ell^1$ norm is utilized to estimate a single shaping filter (equation () with a small $\epsilon$).

 

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Figure 5 (a) The signal estimated with the $\ell^1$ norm. (b) The noise estimated with the $\ell^1$ norm.

The estimated signal in Figure a resembles the true signal very well, and so does the estimated noise. It is easy to check that the energy (in a $\ell^1$ sense) in Figure a (e₁=2) is less than the energy (in a $\ell^1$ sense) in Figure a (e₁=3.2). Figure shows the shaping filter associated with the $\ell^1$ norm. This filter is a spike at lag=0. This simple 1-D example demonstrates that the $\ell^1$ norm should be utilized whenever significant amplitude differences exist between multiples and primaries. In the next section, a synthetic example where internal multiples are attenuated illustrates the properties of the $\ell^1$ subtraction.