page_title: Adaptive filter
section_title: Block diagram
hierarchical_section_title: Adaptive filter / Block diagram
caption_attribution_description: English: Adaptive filter showing the filter and the adaption process. Adaptive Filter. k = sample number, x = reference input, X = set of recent values of x, d = desired input, W = set of filter coefficients, ε = error output, f = filter impulse response, * = convolution, Σ = summation, upper box=linear filter, lower box=adaption algorithm.
context_page_description: An adaptive filter is a system with a linear filter that has a transfer function controlled by variable parameters and a means to adjust those parameters according to an optimization algorithm. Because of the complexity of the optimization algorithms, almost all adaptive filters are digital filters. Adaptive filters are required for some applications because some parameters of the desired processing operation are not known in advance or are changing. The closed loop adaptive filter uses feedback in the form of an error signal to refine its transfer function.
  Generally speaking, the closed loop adaptive process involves the use of a cost function, which is a criterion for optimum performance of the filter, to feed an algorithm, which determines how to modify filter transfer function to minimize the cost on the next iteration. The most common cost function is the mean square of the error signal.
  As the power of digital signal processors has increased, adaptive filters have become much more common and are now routinely used in devices such as mobile phones and other communication devices, camcorders and digital cameras, and medical monitoring equipment.
context_section_description: The idea behind a closed loop adaptive filter is that a variable filter is adjusted until the error (the difference between the filter output and the desired signal) is minimized.  The Least Mean Squares (LMS) filter and the Recursive Least Squares (RLS) filter are types of adaptive filter.
  There are two input signals to the adaptive filter:  and  which are sometimes called the primary input and the reference input respectively.  The adaption algorithm attempts to filter the reference input into a replica of the desired input by minimizing the residual signal, .  When the adaption is successful, the output of the filter  is effectively an estimate of the desired signal.
   which includes the desired signal plus undesired interference and
   which includes the signals that are correlated to some of the undesired interference in .
  k represents the discrete sample number.
  The filter is controlled by a set of L+1 coefficients or weights.
   represents the set or vector of weights, which control the filter at sample time k.
  where  refers to the 'th weight at k'th time.
   represents the change in the weights that occurs as a result of adjustments computed at sample time k.
  These changes will be applied after sample time k and before they are used at sample time k+1.
  The output is usually  but it could be  or it could even be the filter coefficients.(Widrow)
  The input signals are defined as follows:
  where:
  g = the desired signal,
  g' = a signal that is correlated with the desired signal g ,
  u = an undesired signal that is added to g , but not correlated with g or g'
  u' = a signal that is correlated with the undesired signal u, but not correlated with g or g',
  v = an undesired signal (typically random noise) not correlated with g, g', u, u' or v',
  v' = an undesired signal (typically random noise) not correlated with g, g', u, u' or v.
  The output signals are defined as follows:
  .
  where:
   = the output of the filter if the input was only g',
   = the output of the filter if the input was only u',
   = the output of the filter if the input was only v'.
caption: Adaptive Filter. k = sample number, x = reference input, X = set of recent values of x, d = desired input, W = set of filter coefficients, ε = error output, f = filter impulse response, * = convolution, Σ = summation, upper box=linear filter, lower box=adaption algorithm
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