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I am working on LMS algorithm,i am not getting exactly how we get desired response of the filter that is compared with the estimated output

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  • $\begingroup$ As for as I know, the desired signal for the LMS algorithm depends on application type. One example is given by the user (Matt L.) who commented above. The LMS algorithm find an iterative solution to the Wiener-Hopf equation. The LMS algorithm, may assume the model $y(n)=d(n)+w(n)$ for the (received or measured) data, where $d(n)$ is the desired signal and $w(n)$ is a random noise process. For what application you are using the LMS algorithm? Filtering, prediction, deconvolution or extrapolation? $\endgroup$ – Oliver Apr 8 '14 at 12:25
  • $\begingroup$ im using it for filtering(isi removal or for channel eualization) $\endgroup$ – sumaira Apr 9 '14 at 6:15
  • $\begingroup$ Please refer to the text book - "Statistical Digital Signal Processing and Modeling" by Monson H. Hayes. Pages 530-534. They contain what exactly you need . Also, refer page 506. I hope this helps you. $\endgroup$ – Oliver Apr 9 '14 at 6:39
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as the others have said you are best looking at practical examples of uses of LMS. once examples is noise cancellation. think of a system where we have two microphones, one mic is the source which contains speech and background noise. the other microphone will just contain noise. in this system we try to eliminate the noise from the speech. so the inputs to the LMS are our noise reference signal and the difference between the (speech + noise) - estimate of the noise, so in this case the LMS adapts to noise that appears at the source.

Another example is echo cancellation. here you using your landline, which goes through a hybrid circuit, the signal from the far end x(n) is mixed with the near end signal u(n), so the signal going to far end is x + u + v = d, where v is noise. at the far end the person does not want to hear their own voice. so at the far end the user will have access to x(n), so inputs to LMS are x(n) and e(n) where e = d - y. in this case y is our output of LMS, used to remove the x from signal.

in a comms system, as mentioned above the LMS is used to update the equaliser coeffs. in comms systems they use the training sequence (s[n]) which is known at both ends of the transmission system, this allows us to train the equaliser. this training sequence is used before speech transmision, and during speech transmission, and sometimes training is turned off completely if channel estimate is good. in this system the LMS has inputs r[n] (the recieved signal - the s[n] after passing through the channel), and e[n], which is from the s[n] - r[n], then you would update the equaliser coeffs f after doing f = f+mu*e*r.

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In a digital communication system you often use a known training sequence for an initial adaption of the filter taps. After this initialization, the output of the slicer is used as a desired signal. This mode of operation is called decision directed. This is just a (hopefully helpful) example. In general it depends on your application how you define your desired signal.

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