By Daniel A. Murray
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Extra resources for A first course in infinitesimal calculus
The minimum support criterion for blind signal extraction. In: proc. of the int. conf. on Independent Component Analysis and Blind Signal Separation, Granada, Spain, pp. 57–64 (2004) 9. : SWM: A class of convex contrasts for source separation. In: proc. of the Int. Conf. on Acoustics, Speech and Signal Processing, Philadelphia (USA), vol. V, pp. 161–164 (2005) 10. : Criterion for blind simultaneous extraction of signals with clear boundaries. Electronics Letters 41(21), 1195–1196 (2005) 11. : Elements of Information Theory.
Indeed, complex valued BSS does not always sum up to an obvious generalization of the real valued case . We illustrate it in our context and show that, quite surprisingly, blind separation of the sources given by A3 can be achieved up to classical inderterminations of ICA. This is in contrast with the result in Equation (12) where additionnal indeterminacies appeared. First, we have: Lemma 4. Let y = gs where the vector of sources is deﬁned by A3. Assume that the vector (s1 , s2 , s3 ) takes all 43 possible values.
Let R(1, k+1, αk , βk ), for k = 1, . . , n−1, denote a class of planar rotation matrices, then u = e1 T R(1, n, αn−1 , βn−1 ) · · · R(1, 2, α1 , β1 ). Since the extraction solutions are non-diﬀerentiable points of the contrast function, we used the downhill simplex method of Nelder and Mead to optimize it in low dimensions . In high dimensions, an improved convergence is obtained when combining the previous optimization technique with numerical gradient and line-search methods. Each function evaluation requires the computation of the planar convex hull of a set of T outputs.
A first course in infinitesimal calculus by Daniel A. Murray