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| The '''Multidelay block frequency domain adaptive filter''' (MDF) algorithm is a block-based frequency domain implementation of the (normalised) [[Least mean squares filter|Least mean squares filter (LMS)]] algorithm.
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| == Introduction ==
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| The MDF algorithm is based on the fact that convolutions may be efficiently computed in the frequency domain (thanks to the [[Fast Fourier Transform]]). However, the algorithm differs from the [[Fast LMS algorithm]] in that block size it uses may be smaller than the filter length. If both are equal, then MDF reduces to the FLMS algorithm.
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| The advantages of MDF over the (N)LMS algorithm are:
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| * Lower algorithmic complexity
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| * Partial de-correlation of the input (which 'may' lead to faster convergence)
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| == Variable definitions ==
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| [[File:Lms filter.svg|444px|LMS filter]]
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| Let <math>N</math> be the length of the processing blocks, <math>K</math> be the number of blocks and <math>\mathbf{F}</math> denote the 2Nx2N Fourier transform matrix. The variables are defined as:
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| : <math>\underline{\mathbf{e}}(\ell) = \mathbf{F}\left[ \mathbf{0}_{1xN}, e(\ell N),\dots,e(\ell N-N-1) \right]^T</math>
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| : <math>\underline{\mathbf{x}}_k(\ell) = \mathrm{diag} \left\{ \mathbf{F}\left[ x((\ell -k+1) N),\dots,x((\ell -k-1) N-1) \right]^T \right\}</math>
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| : <math>\underline{\mathbf{X}}(\ell) = \left[ \underline{\mathbf{x}}_0(\ell), \underline{\mathbf{x}}_1(\ell), \dots, \underline{\mathbf{x}}_{K-1}(\ell) \right]</math>
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| : <math>\underline{\mathbf{d}}(\ell) = \mathbf{F}\left[ \mathbf{0}_{1xN}, d(\ell N),\dots,d(\ell N-N-1) \right]^T</math>
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| With normalisation matrices <math>\mathbf{G}_1</math> and <math>\mathbf{G}_2</math>:
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| : <math>\mathbf{G}_1 = \mathbf{F}\begin{bmatrix}
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| \mathbf{0}_{NxN} & \mathbf{0}_{NxN} \\
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| \mathbf{0}_{NxN} & \mathbf{I}_{NxN} \\
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| \end{bmatrix}\mathbf{F}^{-1}</math>
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| : <math>\tilde{\mathbf{G}}_2 = \mathbf{F}\begin{bmatrix}
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| \mathbf{I}_{NxN} & \mathbf{0}_{NxN} \\
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| \mathbf{0}_{NxN} & \mathbf{0}_{NxN} \\
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| \end{bmatrix}\mathbf{F}^{-1}</math>
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| : <math>\mathbf{G}_2 = \mathrm{diag} \left\{ \tilde{\mathbf{G}}_2, \tilde{\mathbf{G}}_2, \dots, \tilde{\mathbf{G}}_2 \right\} </math>
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| In practice, when multiplying a column vector <math>\mathbf{x}</math> by <math>\mathbf{G}_1</math>, we take the inverse FFT of <math>\mathbf{x}</math>, set the first <math>N</math> values in the result to zero and then take the FFT. This is meant to remove the effects of the circular convolution.
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| == Algorithm description ==
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| For each block, the MDF algorithm is computed as:
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| : <math> \underline{\hat{\mathbf{y}}}(\ell) = \mathbf{G}_1 \underline{\mathbf{X}}(\ell) \underline{\hat{\mathbf{h}}}(\ell-1) </math>
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| : <math> \underline{\mathbf{e}}(\ell) = \underline{\mathbf{d}}(\ell) - \underline{\hat{\mathbf{y}}}(\ell) </math>
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| : <math>\mathbf{\Phi}_\mathbf{xx} = \underline{\mathbf{X}}(\ell)\underline{\mathbf{X}}(\ell)^H</math>
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| : <math> \underline{\hat{\mathbf{h}}}(\ell) = \underline{\hat{\mathbf{h}}}(\ell-1) + \mu\mathbf{G}_2\mathbf{\Phi}_\mathbf{xx}^{-1}(\ell) \underline{\mathbf{X}}^H(\ell) \underline{\mathbf{e}}(\ell) </math>
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| It is worth noting that, while the algorithm is more easily expressed in matrix form, the actual implementation requires no matrix multiplications. For instance the normalisation matrix computation <math>\mathbf{\Phi}_\mathbf{xx} = \underline{\mathbf{X}}(\ell)\underline{\mathbf{X}}(\ell)^H</math> reduces to an element-wise vector multiplication because <math>\underline{\mathbf{X}}(\ell)</math> is block-diagonal. The same goes for other multiplications.
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| == References ==
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| * J.-S. Soo and K. Pang, “Multidelay block frequency domain adaptive filter,” ''IEEE Transactions on Acoustics, Speech and Signal Processing'', vol. 38, no. 2, pp. 373–376, 1990.
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| * H. Buchner, J. Benesty, W. Kellermann, "An Extended Multidelay Filter: Fast Low-Delay Algorithms for Very High-Order Adaptive Systems". ''Proc. IEEE [[International Conference on Acoustics, Speech, and Signal Processing]] (ICASSP)'', 2003.
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| * A free implementation of the MDF algorithm is available in [http://www.speex.org Speex] ([http://svn.xiph.org/trunk/speex/libspeex/mdf.c main source file])
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| == See also == | |
| * [[Adaptive filter]]
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| * [[Recursive least squares]]
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| * For statistical techniques relevant to LMS filter see [[Least squares]].
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| [[Category:Digital signal processing]]
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| [[Category:Filter theory]]
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