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2 The Sum of Two Ellipsoids Recall that we can parameterize an ellipsoid in Rn in terms of its center c [ Rn and a symmetric non-negative definite configuration matrix Q [ RnÂn as E(c, Q) ¼ {Q1=2 u þ c j kuk 1}, where Q1=2 is any matrix square root satisfying Q1=2 (Q1=2 )T ¼ Q. Let x [ E 1 ¼ E(c1 , Q1 ) and y [ E 2 ¼ E(c2 , Q2 ). The range of values of the geometrical (or Minkowski) sum z ¼ x þ y is contained in the ellipsoid E ¼ Eðc1 þ c2 , Q( p)Þ (1:64) Q( p) ¼ (1 þ pÀ1 )Q1 þ (1 þ p)Q2 ; (1:65) for all p .

6 UNCERTAINTY ELLIPSOID CALCULUS 39 where F1 ¼ In 0 ! 0 , In F2 ¼ In In ! 0 , 0 and ! ÀIn , 0 0 F3 ¼ In 0 F4 ¼ 0 ! In : In The multiplications associated with matrices F1 , . . , F4 are achieved with a reordering of the calculations. 73). 67). 7 An Improved Approximation We now make use of two facts that generally lead to tighter approximations. 73); hence, it is productive to reduce the number of these terms. Consider a Givens rotation matrix of the form: 2 cos u1 6 6 6 6 T ¼6 6 À sin u1 6 6 4 ..

They may cause a substantial performance degradation of adaptive beamformers because they limit the training sample size and may lead to interference undernulling. When such nonstationarity effects are combined with the effect of the presence of the desired signal in the training cell, the aforementioned degradation can become much stronger than in the case of signal-free beamformer training data [56]. 2 BACKGROUND AND TRADITIONAL APPROACHES 51 One typical example of negative effects of nonstationarity is the case when the interfering sources move rapidly.

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