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Recall the Weiner filter problem

x k , d k jointly wide sense stationary

Find W minimizing k 2 k d k y k d k i M 1 0 w i x k - i d k X k W k X k x k x k - 1 x k - M + 1 W k w 0 k w 1 k w M - 1 k The superscript denotes absolute time, and the subscript denotes time or a vector index.

the solution can be found by setting the gradient 0

k W k 2 2 k X k -2 d k X k W k X k 2 d k X k X k X k W -2 P 2 R W
W opt R P Alternatively, W opt can be found iteratively using a gradient descent technique W k + 1 W k k In practice, we don't know R and P exactly, and in an adaptive context they may be slowly varying with time.

To find the (approximate) Wiener filter, some approximations are necessary. As always, the key is to make the right approximations!

Approximate R and P :RLS methods, as discussed last time.
Approximate the gradient! k W k 2
Note that k 2 itself is a very noisy approximation to k 2 . We can get a noisy approximation to the gradient by finding the gradient of k 2 ! Widrow and Hoff first published the LMS algorithm, based on this clever idea, in 1960. k W k 2 2 k W d k W k X k 2 k X k 2 k X k This yields the LMS adaptive filter algorithm

The lms adaptive filter algorithm

  • y k W k X k i 0 M 1 w i k x k - i
  • k d k y k
  • W k + 1 W k k W k -2 k X k W k 2 k X k ( w i k + 1 w i k 2 k x k - i )
Got questions? Get instant answers now!

The LMS algorithm is often called a stochastic gradient algorithm, since k is a noisy gradient. This is by far the most commonly used adaptive filtering algorithm, because

  • it was the first
  • it is very simple
  • in practice it works well (except that sometimes it converges slowly)
  • it requires relatively litle computation
  • it updates the tap weights every sample, so it continually adapts the filter
  • it tracks slow changes in the signal statistics well

Computational cost of lms

To Compute y k k W k + 1 = Total
multiplies M 0 M 1 2 M 1
adds M 1 1 M 2 M

So the LMS algorithm is O M per sample. In fact, it is nicely balanced in that the filter computation and the adaptation require the sameamount of computation.

Note that the parameter plays a very important role in the LMS algorithm. It can also be varied with time, but usually a constant ("convergence weight facor") is used, chosen after experimentation for a givenapplication.

Tradeoffs

large : fast convergence, fast adaptivity

small : accurate W less misadjustment error, stability

Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
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can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
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Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
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Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
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A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
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Source:  OpenStax, Adaptive filters. OpenStax CNX. May 12, 2005 Download for free at http://cnx.org/content/col10280/1.1
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