Adaptive noise cancelling

• Comment: See also Adaptive filter AngusW🐶🐶F (bark • sniff) 23:26, 18 January 2023 (UTC)

• Comment: This will need in-depth coverage in independent sources before being considered, we have no interest in your own work as sources. Theroadislong (talk) 09:58, 7 December 2021 (UTC)

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• Comment: Active noise control is a lightweight article entirely about control of acoustic noise and does not go into the theory or the math. Active noise control would be an application of Adaptive noise cancelling, so there is some merit in more discussion of active noise cancelling. However, given the substantial overlap with Adaptive filter, I believe that the material offered in this draft should be added to Adaptive filter. Part of the material should go in the application section and part should be used to improve the theory and history sections. The additional references would also be appreciated. Note: Adaptive noise cancelling is not unorthodox. Constant314 (talk) 23:23, 21 January 2023 (UTC)

Adaptive noise cancelling is an unorthodox signal processing technique that is highly effective in suppressing additive interference or noise corrupting a received target signal at the main or primary sensor in certain common situations where the interference is known and is accessible but unavoidable and where the target signal and the interference are unrelated, that is, uncorrelated. Examples of such situations include:

• a microphone attempting to receive a speech near machinery or other noise sources in the environment, such as an aircraft cockpit
• a naval ship towing sonar array where the ship's own noise can overwhelm a much weaker detected target signal
• a doctor obtaining a fetal electrocardiogram (ECG) where the presence of the mother's stronger ECG which represents an unavoidable interference.

Conventional signal processing techniques rely on filtering the received signal, consisting of the target signal corrupted by the added interference, so as to minimise the effect of the interference. Maximising the signal-to-noise ratio) is the objective of optimal filters such as the Wiener Filter. In contrast adaptive noise cancelling relies on a second sensor located near the source of the known interference to obtain a relatively 'pure' version of the interference free from the target signal and other interference. This second version of the interference and the sensor receiving it are called the reference.

Adaptive noise cancelling uses a self- adjusting adaptive filter to automatically transform the reference signal into an optimal estimate of the interference corrupting the target signal before subtracting it from the received signal thereby cancelling (or minimising) the effect of the interference at the noise canceller output. The adaptive filter adjusts itself continuously aiming to minimise the residual interference affecting the target signal at its output. The power of the adaptive noise cancelling concept is that it requires no detailed a priori knowledge of the target signal or the interference. The adaptive algorithm that optimises the filter relies only on ongoing sampling of the reference input and the noise canceller output.

Adaptive noise cancelling can be effective even when the target signal and the interference are similar in nature and the interference is considerably stronger than the target signal. The key requirement however is that the target signal and the interference are unrelated, that is uncorrelated. Meeting this requirement is normally not an issue in situations where adaptive noise cancelling is used.

Adaptive Noise Cancelling Configuration and Concept

The adaptive noise canceller configuration diagram below shows the target signal s(t) present at the primary sensor and the interference or noise source n(t) and its manifestations n_p(t) and  n_r(t) at the primary and reference sensors respectively.

As n_p(t) and n_r(t) are the manifestations of the same interference source in different locations, these will usually differ significantly in an unpredictable fashion due to different transmission paths through the environment to the two sensors. So the reference n_r(t) cannot be used directly to cancel or reduce the interference corrupting the target signal. It must first be appropriately processed to generate ñ_p(t), the optimal estimate of the version of the interference present at the primary sensor, before it can be used to minimise by subtraction the overall effect of the interference at the noise canceller output.

An adaptive noise canceller is based on a self-optimising adaptive filter that has a variable transform function shaped by adjustable parameters called weights. The adaptive filter transforms the reference n_r(t) into an optimal estimate ñ_p(t) of the interference n_p(t) corrupting the target signal and ‘cancelling’ the latter by subtraction, whilst leaving the target signal unchanged. So the output of the adaptive noise canceller shown below is: z(t) = s(t)+n_p(t)-ñ_p(t).

Adaptive filter configured as Noise Canceller

The power of the adaptive noise cancelling approach stems from the fact that the algorithm driving the iterative adjustment of weights in an adaptive filter, (for example Least-Mean-Square Filter) is a simple fully automatic iterative process that relies only on ongoing sequence of measurements of the noise canceller output z(t) and the weight inputs, which, in the case of the usual a tapped delay line adaptive filters is simply a sequence of samples of the reference signal r(t) = n_r(t).

Apart from the availability of a suitable reference signal the only other essential requirement is that the target signal and the corrupting noise source are unrelated, that is uncorrelated, so that for all values of $\tau$, where the bar represents time averaging.

Adaptive noise cancelling does not require detailed a priori knowledge of the interference or the target signal.  However, the physical characteristics of the adaptive filter must be generally suitable for producing an adjustable frequency response or transfer function that will transform the reference signal n_r(t) into a close estimate of the corrupting interference, ñ_p(t), through the iterative adjustment of the filter weights.

Genesis

Adaptive noise cancelling evolved from the pioneering work on adaptive systems, adaptive filtering and signal processing carried out at the Information Systems Laboratories in the School of Engineering at Stanford University during the 1960’s and 70's under the leadership of Professor Bernard Widrow.  Adaptive filters incorporate` adjustable parameters called weights, controlled by iterative adaptive algorithms, to produce a desired transfer function.

Adaptive filters were originally conceived to produce the optimal filters prescribed by optimal filter theory during a training phase by adjusting the filter weights according to an iterative adaptive algorithm such as the Least-Means-Square (LMS) algorithm. During the training phase, the filter is presented with a known input and a training signal called a desired response. The filter weights are adjusted by the adaptive algorithm, which is designed to minimise the error, the difference between the adaptive filter output and the desired response.

At the completion of the training phase the adaptive filter has been optimised to process a certain class of target signals in the presence of interference with assumed frequency spectrums. In its normal operating phase such an optimised adaptive filter would then be used passively to process received signals to improve the signal-to noise ratio at the filter output. The theory and analysis of adaptive filters is largely based on this concept, model and terminology and took place before the introduction of the adaptive noise cancelling concept around 1970.

Adaptive noise cancelling is an innovation that represents a fundamentally different configuration and application of adaptive filtering in those common situations where a reference signal is available by:

1. using the training mode of the filter as the operational mode
2. using the reference as the adaptive filter input
3. using the primary input, the noise canceller input containing the target signal and interference as the desired response and
4. using the error as the noise canceller output, that is, the difference between the primary input and the output of the adaptive filter.

Whilst the discussion of adaptive noise cancelling reflects the above terminology, previously developed adaptive filter theory continues to apply.

Since the adaptation process will aim to minimise the error signal, it follows that in the noise canceller configuration the adaptation process will tend to minimise the overall signal power at the noise canceller output. So the adaptive filtering of the reference actually strives to suppress the overall signal power at the noise canceller output.

This counterintuitive concept can be understood by keeping in mind that the target signal s(t) and the interference n(t) are uncorrelated. So in aiming to minimise the error, using a reference as input, which is related only to the interference, the best the adaptive filter can do is to generate the optimal estimate of the interference at the primary sensor ñ_p(t). This will minimise the overall effect of the interference at the noise canceller output whilst leaving the target signal s(t) unchanged.

The iterative adaptive algorithms used in adaptive filtering require only an ongoing sequence of sampling measurements at the weight inputs and the error. As digital adaptive filters consists of tapped delay lines, the operation of an adaptive noise canceller requires only on an ongoing sequence of sampling measurements of the reference and the noise canceller output.

The noise cancelling approach and the proof of the concept, the first striking demonstrations that general broadband interference can be eliminated from a target signal in practical situations using adaptive noise cancelling, were set out and demonstrated in 1971-72 at the Stanford Information Systems Laboratory by Professor Widrow and John Kaunitz, an Australian doctoral student, and documented in the latter’s PhD dissertation Adaptive Filtering of Broadband signals as Applied to Noise Cancelling (1972) (also available here). The work was also published as a Stanford Electronics Labs report by Kaunitz and Widrow, Noise Cancelling Filter Study (1973). The initial demonstration of the noise cancelling concept for eliminating broadband interference was carried out by means of a prototype hybrid adaptive signal processor designed and built by Kaunitz and described in a Stanford Information Systems Laboratory report General Purpose Hybrid Adaptive Signal Processor (1971).

However, a 1975 paper published in the Proceedings of the IEEE by Widrow et al., Adaptive Noise Cancelling: Principles and Applications , is now the generally referenced introductory publication in the field that sets out the basic concepts of adaptive noise cancelling and summarises subsequent early work and applications. Earlier unpublished efforts to eliminate interference using a second input are also mentioned. This paper remains the main reference for the fundamental concept of noise cancelling and to date has been cited by over 2800 scientific paper and 380 patents. The topic is also covered by a number of more recent books.

Proof of Concept Demonstrations

The first practical demonstration of the adaptive noise cancelling concept, typical of general practical situations involving broadband signals, was carried out in 1971 at the Stanford School of Electrical Engineering. The ambient noise from the output of a microphone used by a speaker in a very noisy room was largely eliminated using adaptive noise cancellation. A triangular signal, representing a typical broadband signal, emitted by a loudspeaker situated in the room, was used as the interfering noise source. A second microphone situated near this loudspeaker served to provide the reference input. The output of the noise canceller was channeled to the earphones of a listener outside the room.

The adaptive filter used in these experiments was a hybrid adaptive filter consisting of RC-filter circuits as preprocessors providing inputs to digitally controlled analogue amplifiers as weights. These interfaced to a small HP2126 digital computer that ran a version of the LMS algorithm.

The experimental arrangement used by Kaunitz in the photo below shows the loudspeaker emitting the interference, the two microphones used to provide the primary and reference signals, the equipment rack (third from left) containing the hybrid adaptive filter and the digital interface and the HP 2116B minicomputer on the right of the picture.

Adaptive Noise Cancelling Demonstration by John Kaunitz at the Adaptive Systems Laboratory, Stanford University in 1971

The noise canceller effectively reduced the ambient noise overlaying the speech signal from an initially almost overwhelming level to barely audible and successfully re-adapted to the change in frequency of the triangular noise source and to changes in the environment when people moved around in the room. Recordings of these demonstrations are still available here and here.

The second application of the original noise canceller was to process ECGs from heart transplant animals studied by the pioneering heart transplant team at the Stanford Medical Centre at the time led by Dr Norman Shumway. Data was provided by Drs Eugene Dong and Walter B Cannon in the form of a multi-track magnetic tape recording.

In heart transplant recipients the part of the heart stem that contains the recipient’s pacemaker (called the sinoatrial or SA node) remains in place and continues to fire controlled by the brain and the nervous system. Normally the pacemaker controls the rate at which the heart is beating by triggering the atrioventricular (AV) nodes thus controlling heart rate to respond to the demands of the body. (See diagram below). In normal patients this represents a feedback loop, but in transplant patients the connection between the remnant SA node and the implanted AV nodes is not reestablished and the remnant pacemaker and the implanted heart beat independently, at differing rates.

The behavior of the remnant pacemaker in the open loop situation of a transplant patient was of considerable interest to researchers, but studying the ECG of the pacemaker (the p-wave) was made difficult because the weaker signal from the pacemaker was swamped by the signal from the implanted heart even when a bipolar catheter sensor (primary sensor) is inserted through the jugular vein close to the SA-node. (See the third trace from top in the diagram below). The noise cancelling arrangement to eliminate the effect of the donor heart from the ECG of the p-wave is shown below.

A reference signal was obtained through a limb-to-limb ECG of the patient (See top trace in the diagram below), which provided the main ECG of the donor heart largely free from the pacemaker p-wave. Adaptive noise cancelling was used to transform the reference into an estimate of the donor heart signal present at the primary input (see second trace from top) and used to substantially reduce the effect of the donor heart from the primary ECG (third trace), providing a substantially cleaned up version of the p-wave at the noise canceller output (see bottom trace) suitable for further study and analysis.

Extracting Remnant Pacemaker Signal from Heart transplant ECG

Applications

Adaptive noise cancelling techniques are relevant in a wide range of situations, including the following:

• Eliminating ambient noise from the speech signals of a microphone situated in a noisy environment by using a second microphone situated near the noise source as the reference signal.
• Eliminating the self-noise of a naval ship towing a sonar array searching for a target signal, by using a reference signal of the towing ship’s own noise which can be readily obtained.
• Some noise cancelling headphones utilise adaptive noise cancelling techniques. The effects of ambient ambient noise which penetrates inside the earphone can be minimised by using the version of the ambient noise from a small microphone situated on the headset as the reference signal.
• Another original demonstration of adaptive noise cancelling was the extraction of the remnant recipient pacemaker signal from a heart transplant animal from an ECG which also included the stronger ECG signal of the donor heart. A limb-to-limb ECG was used as the reference signal which was a version of the donor heart ECG in a relatively pure form.
• Similarly fetal electrocardiograms are received in the presence of the mother’s stronger ECG and can be extracted using adaptive noise cancelling to reduce the effect of the mother’s ECG.
• Adaptive noise cancelling has also been used to eliminate patient motion artifacts during general ECG measurements
• Adaptive noise cancelling techniques can also been used in the context of Active Noise Control to reduce acoustic noise in a physical space
• Adaptive noise cancelling has also been used in rail surface defect detection.
• Elimination of ambient noise by adaptive noise cancelling in the process of measuring lightning electric field signals
• Cancelling noise in underground mine powerline carrier communication
• Reducing the effect of noise in speech recognition systems
• Improving beam control for the linear collider at the SLAC (Stanford Linear Accelerator Centre)

In these situations a suitable reference signal can be readily obtained by placing a sensor near the source of the interference or by obtaining by other means an independent version of the inference (e.g. a version of the interfering ECG free from the target signal).

Adaptive noise cancelling can be effective even when the target signal and the interference are similar in nature and the interference is considerably stronger than the target signal. Apart from the availability of a suitable reference signal the only other critical requirement is that the target signal and the corrupting noise source are unrelated, that is uncorrelated, so thatfor all values of $\tau$, where the bar represents time averaging.

Adaptive noise cancelling does not require detailed a priori knowledge of the interference or the target signal.  However, the characteristics of the adaptive filter must be generally suitable for producing an adjustable frequency response or transfer function that is able to transform the reference signal n_r(t) into an estimate of the corrupting interference, ñ_p(t), through the iterative adjustment of the filter weights. The interference in the above examples are usually irregular repetitive signals. Although the theory of adaptive filtering does not rely on this as an assumption, in practice this characteristic is very helpful as it limits the need for the adaptive filter to compensate for time shifts between the versions of the interference at the primary and reference sensors to appropriately compensating for phase shifts.

Principles of Operation

Adaptive noise cancelling is simply the application of an adaptive filter in a certain configuration. The body of theory and analysis previously developed for adaptive filtering therefore applies directly to adaptive noise cancelling.

Adaptive filters, are adjustable filter structures able to produce a range of transfer functions through adjustable parameters called weights which are adjusted using an iterative algorithms, such as the Least Means square LMS algorithm, to achieve the desired transfer function, as measured by a certain performance indicating cost function.

An adaptive filter structure will include some form of pre-processor that provides the memory of the filter input, as the basis for producing a range of desired transfer functions for various situations. Although the adaptive filter used in the original proof of concept demonstration was based on a RC pre-processor filter, the theory of adaptive filtering and practice using digital technology are based on the tapped delay line filter model, where successive recent samples of the filter input signal serve as the weight inputs and shifted along at each iteration. The weight outputs  are then additively combined to form the filter output.

Representing the sequence of inputs  x₁(t), x₂(t)... x_n(t) as a vector X(t) and the sequence of weights as vectors w₁, w₂, .... w_n as a vector W, the adaptive filter output can be represented in vector notation as:

y(t) = X^T(t)W = W^TX(t ) = $\sum _{i=1}^{n}$x_i w_i

This type of adaptive filter is called a linear combiner.

Adaptive filters were originally conceived as a method of producing optimal filters for various situations by adjusting the filter’s frequency response during a training process using a training signal called the desired response. So, as originally conceived, adaptive filters have two modes of operation: a training mode and the normal operating mode.

During the training phase the filter weights are adjusted using a control input signal and a desired response signal, which would maximize the signal to noise ratio at the filter output during normal operation, when the filter is presented with target signals s(t) in the presence of interference n(t) both with certain known frequency characteristics.

Optimised Adaptive Filter in Operating Mode

During the training phase the filter is adjusted or ‘trained’ to produce what is considered to be the desired optimum transfer function. This transfer function is produced by presenting the filter by a known input x(t) and a desired response d(t) and adjusting the weights by an iterative algorithm so that the filter output is an optimal estimate of the desired response. This configuration is deemed to be achieved where the mean squared error ξ(W) = ${\overline {e^{2}(t)}}$ = ${\overline {(d(t)-y(t))^{2}}}$, the time average of the difference between the desired response d(t) and the filter output y(t) is minimised.

Adaptive Filter in Training Mode

When the adaptive filter is a linear combiner the mean squared error is: ξ(W) = ${\overline {e^{2}(t)}}$=${\overline {(d(t)-X^{T}(t)W))^{2}}}$

Adaptive algorithms aim to minimise this mean squared error. The above  expression shows this to be a multi-dimensional paraboloid function of the weight vector with a single minimum that can be reached by gradient descent algorithms that adjust the weight vector opposite the gradient. Typically these iterative algorithms depend only on a series of measurements of the error signal and the weight inputs. For example the Least Mean Square (LMS) algorithm iteratively adjusts weights according to the formula:

W_k+1 = W_k + µe_kX_k

Where k represents the k^th step in the iteration process and µ is the adaptation constant that controls the rate and stability of the adaptation process and e_k and X_k are samples of the error and the input vector respectively. In noise cancelling terminology this becomes:

W_k+1 = W_k + µz_kR_k

The typical algorithms that adaptive noise cancelling relies on is thus a simple process that requires only an ongoing series of simultaneous measurements at the adaptive filter input and the error, or in noise cancelling terminology, the reference input r(t) and the noise canceller output z(t) . The LMS (Least Mean Square) and other gradient descent algorithms will converge to a set of weight values which will tend to minimise the mean squared error.

The fundamental innovation of adaptive noise cancelling is to use adaptive filters differently by:

1. using the training mode of the filter as the operational mode
2. using the primary signal containing the target signal as the desired response and
3. using the error as the noise canceller output, that is, the difference between the primary signal the output of the adaptive filter.

Since the adaptation process will aim to minimise the error signal, it follows that in the noise canceller configuration the adaptation process will tend to minimise the overall signal power at the noise canceller output.

To understand how counterintuitive concept works, it helps to consider the situation where the target signal is absent s(t)=0. In this case the output of the noise canceller is the difference between the interference at the primary sensor and the adaptive filter output and the mean squared error are:

So in this case the desired response of the adaptive filter is the interference n_p(t) present at the primary sensor and the adaptation process will simply tend to minimise the mean squared error by transforming the reference input n_r(t) into an estimate ñ_p(t) of the interference present at the primary sensor. So in the absence of a target signal the noise canceller will suppress the overall effect of the interference at the noise canceller output and make its output ‘quiet’ by minimising the average output power.

When the target signal is additionally present:

Since ñ_p(t) = X^TW = N_r^TW  and the fundamental assumption that s(t) is uncorrelated with n_p(t) and n_r(t) the middle terms drop out and we are left with:

Since the first term is independent of W, the adaptation process will result in minimising the second term which is the same as when the target signal is absent.

So, as long as the target signal and the interference are uncorrelated, the adaptation will result in producing, at the adaptive filter output an optimal estimate of the interference at the primary sensor, thus minimising the overall effect of the interference n(t) at the noise canceller output whilst leaving the target signal unchanged.

A comprehensive analysis of algorithms designed to optimise adaptive filters when applied to stochastic signals is presented by Widrow and Stearns in their book Adaptive Signal Processing. An analysis of noise cancelling where s(t) and n(t) are assumed to be bounded deterministic signals was presented by Kaunitz in his PhD dissertation.^[1]

Adaptive Noise Cancelling and Active Noise Cancelling

Adaptive Noise Cancelling is not to be confused with Active Noise Control. These terms refer to different areas of scientific investigation in two different disciplines and the term "noise" has a different meaning in the two contexts.

Active Noise Control is a method in acoustics to reduce unwanted sound in physical spaces and an area of research that preceded the development of adaptive noise cancelling . The term noise is used here with its common meaning of unwanted audible sound.

As explained above Adaptive Noise Cancelling is a technique used in communication and control to reduce the effect of additive interference corrupting an electric or electromagnetic target signal. In this context noise most refers to such interference and the two terms are used interchangeably. In the 1985 book by Widrow and Stearns the relevant chapter is in fact entitled "Adaptive Interference Cancelling". Although in retrospect this would have been the preferable terminology, "adaptive noise cancelling" is the term that prevailed and is now in common usage.

However, after its development in signal processing, the adaptive noise cancelling approach was also adopted in active noise control, for example in some (but not all), noise cancelling headphones. So the two areas in fact intersect. Nevertheless, active noise control is just one of the many applications of adaptive noise cancelling and, conversely adaptive noise cancelling is just a small part of the field of active noise control.

References

Main Sources

• J. Kaunitz, "Adaptive Filtering of Broadband Signals as Applied to Noise Cancelling," Stanford Electronics Laboratories, Stanford University, Stanford, California, Rep. SU-SEL-3-038, August 1972 (Ph.D. dissertation) OCLC 15201972
• B. Widrow, J. R. Glover JR, J. M. McCool, J. Kaunitz, C. S. Williams, R. H. Hearn, J. R. Zeidler, E. Dong JR. and R. C. Goodlin, "Adaptive Noise Cancelling: Principles and Applications," Proc. IEEE, Vol. 63, December 1975
• B. Widrow and S. D. Stearns, "Adaptive Signal Processing," Pearson Education, Inc.