Computational Intelligence, SS08
2 VO 442.070 + 1 RU 708.070 last updated:
Course Notes (Skriptum)
Online Tutorials
Introduction to Matlab
Neural Network Toolbox
OCR with ANNs
Adaptive Filters
VC dimension
Gaussian Statistics
PCA, ICA, Blind Source Separation
Hidden Markov Models
Mixtures of Gaussians
Automatic Speech Recognition
Practical Course Slides
Animated Algorithms
Interactive Tests
Key Definitions
Literature and Links

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

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.