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How can I choose the step size $\mu$, when I'm comparing different algorithms such as LMS, NLMS and transform domain adaptive filters, regarding their convergence speed, to get a fair comparison between them for mean-squared-error learning curves?

Is it necessary for a fair comparison to use the same step size for all algorithms since in most of the literature $\mu$ is not specified for the individual algorithms?

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For fair comparison of one algorithm to another, the value of step size does not need to be same. You can adjust the step sizes of both algorithm so that the mean-squared-error learning curves base floor gets same and in this way you will be able to differentiate the performance of algorithm. For base floor I mean the value at which the mean squared error curves coincides with each other at the end of iterations or data samples. You can adjust the step size by hit and trial method.

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