Computational Intelligence, SS08
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Adaptive Transversal Filters

In a transversal filter of length $ N$, as depicted in fig. 1, at each time $ n$ the output sample $ y[n]$ is computed by a weighted sum of the current and delayed input samples $ x[n], x[n-1],\ldots$

$\displaystyle y[n]= \sum_{k=0}^{N-1}c_k^{\ast}[n] x[n-k].$ (1)


Here, the $ c_k[n]$ are time dependent filter coefficients (we use the complex conjugated coefficients $ c_k^{\ast}[n]$ so that the derivation of the adaption algorithm is valid for complex signals, too). This equation re-written in vector form, using $ \mathbf{x}[n]= \bigl[x[n],x[n-1],\ldots,x[n-N+1] \bigr]^{\mathsf T}$, the tap-input vector at time $ n$, and $ \mathbf{c}[n]= \bigl[ c_0[n], c_1[n], \ldots, c_{N-1}[n] \bigr]^{\mathsf T}$, the coefficient vector at time $ n$, is
$\displaystyle y[n]= \mathbf{c}^{\mathsf H}[n]\mathbf{x}[n].$ (2)


Both $ \mathbf{x}[n]$ and $ \mathbf{c}[n]$ are column vectors of length $ N$, $ \mathbf{c}^{\mathsf H}[n]= \left(\mathbf{c}^{\ast}\right)^{\mathsf T}[n]$ is the hermitian of vector $ \mathbf{c}[n]$ (each element is conjugated $ ^{\ast}$, and the column vector is transposed $ ^{\mathsf T}$ into a row vector).
Figure 1: Transversal filter with time dependent coefficients
\includegraphics[scale=1]{transversalfilter}

In the special case of the coefficients $ \mathbf{c}[n]$ not depending on time $ n$: $ \mathbf{c}[n]= \mathbf{c}$ the transversal filter structure is an FIR1 filter of length $ N$. Here, we will, however, focus on the case that the filter coefficients are variable, and are adapted by an adaptation algorithm.