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Adaptive Filtering pp 9–60Cite as

Fundamentals of Adaptive Filtering

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Abstract

This chapter includes a brief review of deterministic and random signal representations. Due to the extent of those subjects, our review is limited to the concepts that are directly relevant to adaptive filtering. The properties of the correlation matrix of the input signal vector are investigated in some detail, since they play a key role in the statistical analysis of the adaptive filtering algorithms.

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Notes

  1. 1.

    The index k can also denote space in some applications.

  2. 2.

    An alternative and more accurate notation for the convolution summation would be \((x *h)(k)\) instead of \(x(k) *h(k)\), since in the latter the index k appears twice, whereas the resulting convolution is simply a function of k. We will keep the latter notation since it is more widely used.

  3. 3.

    Or equivalently, such that \(\mathrm{X}(k,\varrho ) \le y\).

  4. 4.

    The average signal power at a given sufficiently small frequency range, \(\Delta \omega \), around a center frequency \(\omega _{0}\) is approximately given by .

  5. 5.

    We can also change the order in which the \({\mathbf {{q}}}_{i}\)’s compose matrix \({\mathbf {{Q}}}\), but this fact is not relevant to the present discussion.

  6. 6.

    This property is valid for any square matrix, but for more general matrices the proof differs from the one presented here.

  7. 7.

    For non-Hermitian matrices, the maximum of the Rayleigh’s quotient may occur in a vector \({\mathbf {{w}}}\) not corresponding to an eigenvector, yet the quotient is utilized to quantify the size of a matrix [22].

  8. 8.

    Some books define \({\mathbf {{g}}}_{\mathbf {{w}}}\) as \(\left[ \frac{\partial \xi }{\partial {\mathbf {{w}}}}\right] ^{T}\), here we follow the notation more widely used in the subject matter.

  9. 9.

    It is valid for any \({\mathbf {{w}}}\).

  10. 10.

    The number of constraints should be less than \(N+1\).

  11. 11.

    Noise added to the reference signal originated from environment and/or thermal noise.

  12. 12.

    Assuming x(k) and v(k) are jointly WSS.

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  1. Universidade Federal do Rio de Janeiro, Niterói, Rio de Janeiro, Brazil

    Paulo S. R. Diniz

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Diniz, P.S.R. (2020). Fundamentals of Adaptive Filtering. In: Adaptive Filtering. Springer, Cham. https://doi.org/10.1007/978-3-030-29057-3_2

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  • DOI: https://doi.org/10.1007/978-3-030-29057-3_2

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