Starting from the early 80's, a number of new approaches to overcome this problem was proposed.
To the best of my knowledge, they can be all be related to five different methods, that are:

1.   Ridge regression method (Provencher and Glockner, 1981)
2.   Singular value decomposition (Hennesy and Johnson, 1981)
3.   Convex constraints analysis (Perczel et al., 1991)
4.   Principal component factor analysis (Pancoska and Keiderling, 1991)
5.   Backpropagation neural network (Bohm et al., 1992)

The original singular value decomposition method, in particular, was integrated with many other procedures, like the variable selection procedure (VarSelect), the locally linearized model (LL), the cluster analysis (CA), the self-consistent method (SELCON).

All these methods do not rely on the evaluation of "pure" secondary structure spectra. In this respect, the Bk basis are not to be intended anymore as the spectra of "pure" secondary structure elements, but instead as a set of mathematical functions whose linear combination give rise to your spectrum. Depending on the particular case you are dealing with, they can look like very much as a spectrum of a 100% helical protein or another "pure" secondary structure or they cannot.
A second common feature of these methods is the extraction of that set of Bk that best fit your spectrum, so that in one passage you obtain the basis set and also the deconvolution results. This is mostly obtained by cutting out from your reference set a number of proteins which either are not providing any significant information to the evaluation of the Bk set or which results in a Bk set with poor fitting to your spectrum.
Because you build up a different Bk set for each deconvolution, and this Bk set is chosen so to improve the fitting to your spectrum, the problem presented in the preceding page is solved.

The relation between the calculated Bk set combination and the secondary structure content of your protein, the way how this set is calculated and the significance of the set itself are all dependent on the particular deconvolution software you choose (and, of course, on the protein reference database).

A more detailed introduction to the different procedures reported above is in this book (also one of the best on circular dichroism itself).

Here, I will only give you some hint on the deconvolution of protein CD spectra:
1- try to include in your reference database some protein related to your;
2- try to use a starting reference database as larger as possible;
3- do not rely on a single deconvolution method, but instead compare results from more methods.

Some detailed protocols on how to run the software corresponding to the outlined deconvolution routines is given in the protocol section of this manual.