User:Kaunitzj/sandbox: Difference between revisions
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Genesis[edit] |
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Adaptive noise cancelling evolved from the pioneering work on adaptive systems, adaptive filtering and [[signal processing]] carried out at the Information Systems Laboratories in the School of Engineering at Stanford University during the 1960’s and 70's under the leadership of Professor [[Bernard Widrow]]. [[Adaptive filter]]<nowiki/>s incorporate` adjustable parameters called ''weights,'' controlled by iterative ''[[adaptive algorithms]]'', to produce a desired transfer function. |
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Adaptive filters were originally conceived to produce the optimal filters prescribed by optimal filter theory during a ''training phase'' by adjusting the filter weights according to an iterative adaptive algorithm such as the [[Least mean squares filter|Least-Means-Square]] (LMS) algorithm. During the training phase, the filter is presented with a known input and a training signal called a ''desired response.'' The filter weights are adjusted according to the adaptive algorithm, which is designed to minimise the ''mean-squared''-''error, ξ,'' the common performance measure of the difference between the adaptive filter output and the desired response,. |
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[[File:Formula for mean squared error.png|none|thumb|361x361px]] |
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The above expression shows this to be a multi-dimensional paraboloid function of the weight vector with a single minimum that can be reached by ''gradient descent algorithms'' that adjust the weight vector opposite the gradient. Typically these iterative algorithms depend only on a series of measurements of the error signal and the weight inputs. For example the [[Least mean Squares filter|Least Mean Square (LMS) algorithm]]<sup>[21][10]</sup> iteratively adjusts weights according to the formula: |
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W<sub>k+1</sub> = W<sub>k</sub> + µe<sub>k</sub>X<sub>k</sub> |
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