Adaptive Filters and Applications - Digital Signal Processing: Fundamentals and Applications - page 496

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The objective is to minimize the filter instantaneous output power y ðnÞ . Once the output power

is minimized, the filter parameter q ¼ 2 pf =f s will converge to its corresponding frequency
f Hz. The LMS algorithm to minimize the instantaneous output power y

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ðnÞ is given as

qðn þ 1 Þ¼qðnÞ 2 myðnÞbðnÞ;

where the gradient function bðnÞ¼vyðnÞ=vqðnÞ can be derived as follows:
b n ¼ 2sin q n x n 1 2 r sin q n y n 1 þ 2 r cos q n b n 1 r b n 2

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m is the convergence factor which controls speed of algorithm convergence.

In this project, plot and verify the notch frequency response by setting f s ¼ 8,000 Hz,
f ¼ 1,000 Hz, and r ¼ 0 : 95. Then generate the sinusoid with a duration of 10 seconds,
frequency of 1,000 Hz, and amplitude of 1. Implement the adaptive algorithm using an initial
guess qð 0 Þ¼ 2 p 2000 =f s ¼ 0 : 5 p and plot the tracked frequency f ðnÞ¼qðnÞf s = 2 p for
tracking verification.

Notice that this particular notch filter only works for a single frequency tracking, since the
mean squares error function E½y

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ðnÞ has one global minimum (one best solution when the
LMS algorithm converges). Details of the adaptive notch filter can be found in Tan and Jiang
(2012). Notice that the general IIR adaptive filter suffers from local minima, that is, the LMS
algorithm converges to local minimum and the nonoptimal solution results.