Extra on Noise-Canceling Headphones: Adaptive Controllers in Lively Noise Management Programs

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With extra individuals working from dwelling, noise-canceling headphones have gotten more and more widespread. In a earlier article, we mentioned that active noise control (ANC) requires an adaptive algorithm to adjust the filter coefficients and optimize noise attenuation. This is because of the truth that the noise traits in addition to the system response can fluctuate with time.

This text, which builds on important ideas on ANC from the primary article, will assess the adaptive controller of an ANC system in larger element. 


Gradient Descent Algorithm

An ANC system makes an attempt to seek out optimum filter weights by minimizing the imply sq. worth of the sound that’s picked up by the error microphone. 


Duct-acoustic feedforward ANC system. Picture used courtesy of Alina Mirza

The imply sq. worth of error, known as error floor in the remainder of the article, is a multivariable operate of the filter coefficients. For instance, with a two-tap filter, the error floor is a bowl-like operate as depicted under:


The error surface is a bowl-like function

Picture courtesy of Scott D. Snyder

The adaptive algorithm ought to discover the filter weights akin to the underside of this bowl. A commonly-used method to realize this can be a gradient descent algorithm. This optimization algorithm begins with an preliminary guess of the optimum filter weights and updates them iteratively to seek out the optimum values.

The algorithm calculates the partial by-product (or gradient) of the error floor with respect to a filter weight to determine how the preliminary worth of that weight needs to be up to date. You possibly can higher perceive this mechanism by contemplating a single-variable error operate akin to f(x)= x2 as depicted under:


Single-variable error function


The minimal of this error operate happens at x=0. If our present location is x=4, the by-product of f(x) (which is identical because the slope of the pink line) is a optimistic worth. On this case, we should always lower the present worth to lower f(x). Nonetheless, with a present weight worth of x=-4, the by-product of f(x) is damaging (the slope of the cyan line).

On this case, we should always improve x to lower f(x). Therefore, whether or not we should always improve or lower x could be decided by the by-product of f(x). This may be prolonged to a multivariable operate; we solely want to exchange by-product with partial by-product. Within the context of the gradient descent algorithm, this partial by-product of the multivariable operate is known as a gradient.

Primarily based on this dialogue, we are able to use the next equation to iteratively replace the weights:


wi, new = wi, previous – μ x (partial by-product of the error floor w.r.t wi


Right here, μ is the convergence coefficient and specifies the proportion of the damaging of the gradient that’s added to the present weight worth in every iteration.


The Cancelation Path

In an ANC system, the output of the adaptive filter is transformed to an analog sign after which to a sound wave on the output of the loudspeaker. This sound wave goes by way of the acoustic path between the loudspeaker and the error microphone. Then, it’s picked up by the error microphone and transformed to a digital sign.

The adaptive algorithm truly receives this digital sign as an enter. The trail from the output of the digital filter to the enter of the adaptive algorithm is normally known as the “cancelation path.”

If we mannequin the cancelation path by a switch operate S(z), we are able to mannequin the ANC system by the next block diagram: 


Block diagram of an ANC system modeling the cancelation path by a transfer function S(z)

Picture courtesy of Sen M. Kuo

Summarizing, the error sign for the LMS algorithm is derived from the output of the adaptive filter modified by S(z). That is in distinction to what now we have in different frequent adaptive filter purposes. Information of S(z) is required to calculate the gradient for the optimization algorithm. Apart from, published reviews of ANC have proven that the system depicted above might be typically unstable.

This concern could be resolved by putting an estimate of S(z) between the reference sign x(n) and the load replace of the LMS algorithm. That is illustrated under the place S(z) represents an estimate of the cancelation path switch operate S(z).


Filtered-X LMS algorithm

Picture courtesy of Sen M. Kuo


Since x(n) is filtered earlier than being utilized to the least imply sq. (LMS) block, this algorithm is named a filtered-X LMS algorithm within the literature. 


Cancelation Path Modeling

The cancelation path switch operate S(z) is estimated by using a second loop of adaptive filtering as proven under.  


Employing a second loop of adaptive filtering

Picture courtesy of Scott D. Snyder


An applicable sign (modeling sign) is utilized to each the cancelation path and its “Mannequin.” The “LMS algorithm” screens the error sign and makes an attempt to reduce it by adjusting the filter weights of the “Mannequin.” When the error sign is minimized, the “Mannequin” response will get nearer to that of the cancelation path.

A duplicate of the obtained mannequin might be used to filter the reference sign of the ANC system as mentioned within the earlier part. This provides us the next block diagram:  


Filtering the reference signal of the ANC system
Picture courtesy of Colin N. Hansen


These previous two articles have reviewed the fundamentals of ANC typically and adaptive controllers in ANC programs particularly. As noise cancelation turns into a staple in additional client headphone units, it is doubtless that extra electrical engineers will see how these rules come into play on the circuit degree. 

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