| Filter Design Toolbox | |
Inverse System Identification
By placing the unknown system in series with your adaptive filter, your filter becomes the inverse of the unknown system when e(k) gets very small. As shown in the figure the process requires a delay inserted in the desired signal d(k) path to keep the data at the summation synchronized. Adding the delay keeps the system causal.
Figure 3-4: Determining an Inverse Response to an Unknown System
Without the delay element, the adaptive filter algorithm tries to match the output from the adaptive filter (y(k)) to input data (x(k)) that has not yet reached the adaptive elements because it is passing through the unknown system. In essence, the filter ends up trying to look ahead in time. As hard as it tries, the filter can never adapt: e(k) never reaches a very small value and your adaptive filter never compensates for the unknown system response. And it never provides a true inverse response to the unknown system. Including a delay equal to the delay caused by the unknown system prevents this condition.
Plain old telephone systems (POTS) commonly use inverse system identification to compensate for the copper transmission medium. When you send data or voice over telephone lines, the copper wires behave like a filter, having a response that rolls off at higher frequencies (or data rates) and possibly having other anomalies as well. Adding an adaptive filter which has a response that is the inverse of the wire response, adapting in real time, removes the rolloff and the anomalies, increasing the available frequency range and data rate for the telephone system.
| | System Identification | Noise Cancellation (or Interference Cancellation) | |
