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# Filtered-X LMS Algorithms (Adaptive Filter Toolkit)

LabVIEW 2013 Adaptive Filter Toolkit Help

Edition Date: June 2013

Part Number: 372357B-01

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In some adaptive filter applications, such as active noise control, you must take the secondary path into consideration. A secondary path is the path from the output of the adaptive filter to the error signal. The secondary path causes phase shifts or delays in signal transmission. Conventional least mean squares (LMS) algorithms cannot compensate for the effect of the secondary path. For those applications, you can use the filtered-x LMS algorithms to create adaptive filters.

## Filtered-X LMS

The following figure shows the diagram of an adaptive filter that you create by using the filtered-x LMS algorithm.

 where x(n) is the input signal to a linear filter at time n y(n) is the corresponding output signal d(n) is another input signal to the adaptive filter S(z) is the impulse response of the secondary path ys(n) is the signal from the secondary path e(n) is the error signal that denotes the superposition of d(n) and ys(n) is the estimation of S(z) fx(n) is the resulting output signal from

An LMS algorithm adjusts the coefficients of the linear filter iteratively to minimize the power of e(n).

 Note  In real-world applications, superposition typically occurs in the space domain. You cannot acquire d(n) or ys(n) separately. You can acquire only the superposition signal e(n) and then use the filtered-x adaptive filter to minimize the power of e(n).

Notice the following differences between the previous figure and the diagram of a typical adaptive filter.

• A secondary path exists in the diagram of a filtered-x adaptive filter. To compensate for the effects of the secondary path, you must estimate the impulse response of the secondary path and take this estimate into consideration. The diagram of a typical adaptive filter does not involve the secondary path.
• In the diagram of the filtered-x adaptive filter, you measure e(n) directly. In the diagram of a typical adaptive filter, you measure d(n) and y(n) separately and then compute e(n) by using the following equation: e(n) = d(n)–y(n).
• The input signals to a filtered-x adaptive filter are x(n) and e(n). The input signals to a typical adaptive filter are x(n) and d(n).

The filtered-x LMS algorithm performs the following operations to update the coefficients of an adaptive filter.

1. Calculates the output signal y(n) from the adaptive filter.
2. Filters the input signal x(n) with and generates fx(n).
3. Updates the filter coefficients by using the following equation:

where μ is the step size of the adaptive filter and is the filter coefficients vector.

Use the AFT Create FIR Filtered-X LMS VI to create an adaptive filter with the filtered-x LMS algorithm.

## Normalized Filtered-X LMS

The normalized filtered-x LMS algorithm is a modified form of the filtered-x LMS algorithm. The filtered-x LMS algorithm combines the filtered-x and normalized LMS algorithms. The normalized filtered-x algorithm updates the coefficients of an adaptive filter by using the following equation:

Use the AFT Create FIR Normalized Filtered-X LMS VI to create an adaptive filter with the normalized filtered-x LMS algorithm.