Prove The update rule w(n + 1) = w(n) + µe(n)x(n) It's the update rule, a fundamental expression in the Least Mean Squares (LMS) adaptive filter algorithm Note: w(n): The filter coefficients in iteration n. μ: The learning rate, which controls the convergence speed of the algorithm. It's a parameter that is adjusted according to the problem and conditions. • e(n): The error in iteration n, which is the difference between the desired output and the estimated output in that iteration. x(n): The input signal in iteration 7, which is the signal being filtered using the coefficients w to produce the estimated output.

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Prove
The update rule w(n + 1) = w(n) + µe(n)x(n)
It's the update rule, a fundamental expression in the Least Mean Squares (LMS) adaptive filter
algorithm
Note:
w(n): The filter coefficients in iteration n.
μ: The learning rate, which controls the convergence speed of the algorithm. It's a
parameter that is adjusted according to the problem and conditions.
• e(n): The error in iteration, which is the difference between the desired output and the
estimated output in that iteration.
• x(n): The input signal in iteration n, which is the signal being filtered using the coefficients
w to produce the estimated output.
Transcribed Image Text:Prove The update rule w(n + 1) = w(n) + µe(n)x(n) It's the update rule, a fundamental expression in the Least Mean Squares (LMS) adaptive filter algorithm Note: w(n): The filter coefficients in iteration n. μ: The learning rate, which controls the convergence speed of the algorithm. It's a parameter that is adjusted according to the problem and conditions. • e(n): The error in iteration, which is the difference between the desired output and the estimated output in that iteration. • x(n): The input signal in iteration n, which is the signal being filtered using the coefficients w to produce the estimated output.
Expert Solution
Step 1: The rule

The update rule provided i.e.

w left parenthesis n plus 1 right parenthesis space equals space w left parenthesis n right parenthesis space plus space mu space cross times e left parenthesis n right parenthesis space cross times space x left parenthesis n right parenthesis

is the Widrow-Hoff or LMS (Least Mean Squares) algorithm used for building adaptive filters or linear regression. The rule works by iteratively adjusting the weights or coefficients of the linear model and minimize the error e(n) which is the difference between the prediction and  the  actual.



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