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ASSP Magazine, IEEE

Issue 3 • Date July 1988

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  • Householder transforms in signal processing

    Publication Year: 1988 , Page(s): 4 - 12
    Cited by:  Papers (29)
    Save to Project icon | Request Permissions | Click to expandQuick Abstract | PDF file iconPDF (3331 KB)  

    The author explores Householder transforms and their applications in signal processing. He shows that these transforms can be viewed as mirror-image reflections of a data vector about any desired hyperplane. The virtue of reflections is that they are covariance invariant, that is, they preserve the covariance matrix of the data. One can construct a finite sequence of such reflections that maps a block of data vectors into a lower rectangular matrix. If only the covariance eigenvalues need to be preserved, one can map into a bidiagonal matrix. The former sparse form is useful for inverting covariance matrices and the latter is useful in finding eigenvalues of covariance matrices.<> View full abstract»

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  • The Verhulst process

    Publication Year: 1988 , Page(s): 13 - 14
    Save to Project icon | Request Permissions | Click to expandQuick Abstract | PDF file iconPDF (164 KB)  

    In 1845, P.F. Verhulst proposed that, for any given niche, there is a maximum population size, X, that can be supported and that as the population approaches this maximum size, the rate should decrease. If the maximum population size is taken as one, Verhulst postulated that the growth, rate r at time n should be proportional to (1-X/sub n/), i.e., r= lambda (1-X/sub n/) where lambda >0. The dynamical law that describes the evolution of the population has been referred to as a Verhulst process. The dynamics of the Verhulst process complicated when lambda >2. For 2< lambda < square root 6, the process eventually becomes a periodic process that oscillates between the two values. As lambda increases further, the stable periodic fixed points of period N become unstable and are replaced with periodic fixed points of period 2N. These period doubling bifurcations continue with increasing frequency until, at a critical value of lambda =2.570, the process becomes aperiodic and breaks into chaotic behaviour. An example is shown for lambda =3.0.<> View full abstract»

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This Magazine ceased production in 1990. The current retitled publication is IEEE Signal Processing Magazine.

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