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C H A P T E R 6 Adaptive Filtering Adaptive filtering is used in many applications including noise cancellation and system identification. In most cases, the coefficients of an FIR filter are modified according to an error signal in order to adapt to a desired signal. In this chapter, a system identification and a noise cancellation system are presented wherein an adaptive FIR filter is used. 6.1 System Identification In system identification, the behavior of an unknown system is modeled by accessing its input and output. An adaptive FIR filter can be used to adapt to the output of the unknown system based on the same input. As indicated in Figure 6-1 , the difference in the output of the system, d ½ n , and the output of the adaptive FIR filter, y ½ n , constitutes the error term, e ½ n , which is used to update the coefficients of the filter. Unknown System Adaptive FIR filter Input x [ n ] d [ n ] y [ n ] e [ n ] + + Figure 6-1: System identification system. 157

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The error term, or the difference between the outputs of the two systems, is used to update each coefficient of the FIR filter according to the equation (known as the least mean square, or LMS, algorithm [ 1 ]) h n ½ k ¼ h n ± 1 ½ k þ d e ½ n x ½ n ± k (6.1) where h ’s denote the unit sample response or FIR filter coefficients, and d denotes a step size. This adaptation causes the output y ½ n to approach d ½ n . A small step size will ensure convergence but result in a slow adaptation rate. A large step size, though faster, may lead to skipping over the solution. 6.2 Noise Cancellation A system for adaptive noise cancellation has two inputs consisting of a noise- corrupted signal and a noise source. Figure 6-2 illustrates an adaptive noise cancellation system. A desired signal s ½ n is corrupted by a noise signal v 1 ½ n , which originates from a noise source signal v 0 ½ n . Bear in mind that the original noise source signal gets altered as it passes through an environment or channel whose characteristics are unknown. For example, this alteration can be in the form of a lowpass filtering process. Consequently, the original noise signal v 0 ½ n cannot be simply subtracted from the noise-corrupted signal, as there exists an unknown dependency between the two noise signals, v 1 ½ n and v 0 ½ n . The adaptive filter is thus used to provide an estimate for the noise signal v 1 ½ n . The weights of the filter are adjusted in the same manner stated previously. The error term of this system is given by e ½ n ¼ s ½ n þ v 1 ½ n ± y ½ n (6.2) Channel Adaptive FIR filter + + e [ n ] n 1 [ n ] y [ n ] Noise n 0 [ n ] Signal s [ n ] Figure 6-2: Noise cancellation system. 158 Chapter 6
The error e ½ n approaches the signal s ½ n as the filter output adapts to the noise component of the input v 1 ½ n . To obtain an effective noise cancellation system, one should place the sensor for capturing the noise source adequately far from the signal source.

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