0 < mu < 2/lambda_max, where lambda_max is the maximum eigenvalue of the covariance matrix R=E[uu^H], where u is the tap input vector.  If you don't have this matrix than lambda_max < Tr[R] = Msigma^2, where sigma^2 is the variance (or power) of the input and M is the number of taps (assuming you are using a tapped delay line and not an antenna array or something like that).

May be I do not understand your question correctly.

But if you say about step in LMS probably you use the adaptive (online) variant of LMS - such as adaptive linear neuron (AdaLiN). For AdaLiN speed of correction of weights (synapses) choose based on trace of input signals autocorrelation matrix.  

For details I used books of Simon Haykin and  Stanislaw Osowski on neuronets.

In a recent project we used a LMS to realize a cognitive filter. While studying the literature, including the great book from Haykin, on possible constraints and an estimation on the relationship between convergence and step size, I found the work of Hassibi et al. also very interesting. Our original paper is also attached to this answer.

Article A Cognitive Filter to Automatically Determine Scintillation ...

Article H∞ bounds for least-squares estimators

Conference Paper LMS is H∞ optimal

Use some form of a line search method: Backtracking line search (Armijo rule) or exact line search (Bracketing)

The book of Widrow and Walach is a key review in the area because Adaline neural networks are based on the LMS. Here is the complete reference:

B. Widrow and E. Walach, Adaptive Inverse Control. Upper Saddle River: Prentice Hall Press, 1996.

And another interesting issue comes from

Optimal Step Size of the Adaptive Multichannel LMS Algorithm for Blind SIMO Identification, Yiteng Huang, Jacob Benesty, Jingdong Chen, IEEE SIGNAL PROCESSING LETTERS, VOL. 12, NO. 3, MARCH 2005

Which can be found here

http://externe.emt.inrs.ca/users/benesty/papers/spl_mar2005.pdf

https://www.researchgate.net/publication/3343203_Optimal_step_size_of_the_adaptive_multichannel_LMS_algorithm_for_blind_SIMO_identification

http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=1415774

Article Optimal step size of the adaptive multichannel LMS algorithm...

Please read the following paper. We solved the problem for correlated rayleigh fading channel with multipath. 

1) Least Mean Square Aided Adaptive Detection in Hybrid Direct-Sequence Time-Hopping Ultrawide Bandwidth Systems by Ahmed, Q.Z. ; Wei Liu ; Lie-Liang Yang

the paper is on ieeexplore.

I could send you the code as well if you could tell me the coding language. I hope that helps

0 < mu < 2/lambda_max, where lambda_max is the maximum eigenvalue of the covariance matrix R=E[uu^H], where u is the tap input vector.  If you don't have this matrix than lambda_max < Tr[R] = Msigma^2, where sigma^2 is the variance (or power) of the input and M is the number of taps (assuming you are using a tapped delay line and not an antenna array or something like that).

The step size controls the speed and the steady state error of the convergence. If the step size increases, the speed increases and the error increases. For small error we use small value of step size but it will longer to converge. 

Forthermore, if you use  large step size value, this could give unstable system.

Regards

Step-size is one of important parameters for LMS algorithm or other adaptive gradient descend algorithms. If the step-size is satisfied the necessary condition, then the algorithm will be stable. However, under different noise environments, step-size may set as very small to ensure the algorithm stable. In addition, standard LMS algorithm may not work well in special environment, such as impulsive noise. Hence, only considering the only one step--size may not good choice to solve above problems.

As stated in the earlier comments, to maintain the stability property the step-size of the adaptive algorithm (equalizer here) should be lower than 2/lambda_max, where lambda_max is the maximum eigenvalue of the covariance matrix R=E[uu^H], where u is the tap input vector. This upper bound is only a first approximation and it shoud be in reality lower by a factor of 1/10 in order to ensure stability whatever the experimental conditions are.

In your experiment, the properties of this covariance matrix R may change when varying the parameters of the Rayleigh multipath channel and in some configuration the fixed step-size is greater than the maximum allowed value. That is the reason why "...in some coefficients of Rayleigh multipath the training is bad".

To ensure stability, several algorithms are usually proposed :

1- use of of Normalized algorithm (i.e. NLMS) instead of the standard LMS. Then you will be independent of the energy variations in the tap input vector u.

2- use of standard approaches for computing the variable step-size such as (main relevant papers):

Regards

As already told, the step size represents a compromise between the speed of adaptation and the noise in steady state. It is possible to detect steady state and decrease step size when algorithm is in steady state. This will decrease the steady state noise.

For the jointly wide sense stationary process, the LMS algorithm converges in the mean if 0

Assoc Prof. Dr. AHMET RIZANER is an expert in this field. I think he can very easily answer this question.

I simply want to commend Pascal Scalart for his excellent tutorial

response to the problems being experienced by Jiajun Zhu. I have

copied his comments for future use by my team at Analog Devices.

Thank you, Pascal.

I feel the you may also try the GA algorithm with the boundary from [0 0 0.......0] to [max(randn(1,70)) ......] to find out min(sum(MSE)). having no. of tap = no. of variable . After convergence of the problem check value of mu should be in the range 0 < mu < 2/lambda_max.

LMS/NLMS based derivation of Equalizer is given in detail in the Rappaport Book on Wireless Communications. Hope that will be a good point to start with. 

how to find the stepsize required for convergence of w(n) within limited iterations?

example:

finding LMS step size so that wn converges in the mean to w = [0:8 􀀀 0:9]T within about 200 iterations.

Does anyone know the matlab code for variable step size LMS algorithm?

Hi Zhu,

I have been working with multipath Rayleigh Channel and have been using LMS for equalization my paper on 

Least Mean Square Aided Adaptive Detection in Hybrid Direct-Sequence Time-Hopping Ultrawide Bandwidth Systems 

will help. If you still have problems with the convergence do let me know and I can help you from there and then.

Regards

Qasim