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A large class of acoustic noise sources has an underlying periodic process that generates a periodic noise component, and thus their acoustic noise can in general be modeled as the sum of a periodic signal and a randomly fluctuating signal (usually a broadband background noise). Active control of periodic noise (i.e., for a mixture of sinusoids) is more effective than that of random noise. For mixtures of sinusoids in a background broadband random noise, conventional FXLMS-based single filter method does not reach the maximum achievable Noise Attenuation Level (

Active noise control (ANC) is a technique of cancelling acoustic noise by generating an appropriate antinoise signal using loudspeakers, and directing it towards the region where noise cancellation is required. Rapid progress in digital signal processors (DSPs) and sensor technologies as well as new real-time adaptive control algorithm designs [

The feedforward FXLMS algorithm.

This idea, which was initially introduced in [

Active noise control of a mixture of sinusoids with or without harmonic relation is a well-studied topic. An indepth study of the convergence properties of the FXLMS algorithm when the noise to be controlled is a harmonically related mixture of sinusoids is presented in [

The purpose of this paper is to introduce a new active noise control (ANC) architecture for effective active noise control of sinusoidal mixtures in a random background noise. The key idea is that for sinusoidal signals in random noise, it is possible to separate the sinusoidal and random parts and generate separate anti-noise signals in a parallel fashion. Since periodic signals can be expressed as a mixture of sinusoids, the proposed method is also applicable to periodic signals with an additive random noise. The effectiveness of noise attenuation is expressed interms of the Noise Attenuation Level (NAL), which measures the relative level of attenuated noise with respect to the actual noise. The notion of, and an exact formula for, maximum achievable Noise Attenuation Level (

Given a feedforward ANC setup (one reference microphone and one error microphone setup) and a stationary acoustic noise input, there exists a maximum achievable noise attenuation level (

In the active noise control of a mixture of sinusoids and a random noise with a single noise cancelling filter (or control filter), the adaptive algorithm finds an optimum filter that is neither equal to the optimum filter when sinusoids alone are present nor equal to the optimum filter when random noise alone is present. Alternatively, assuming that the sinusoidal and random parts can be separated, better cancellation can be achieved by formulating two mean square errors (MSEs) criteria-one for the sinusoidal component and the other for the random component.

For the periodic signal part, perfect cancellation can be achieved by equalizing the nonminimum phase secondary path with a noncausal equalizer in a causal manner since for periodic signals, the future values can be determined from the knowledge of the period or alternatively from the frequencies, amplitudes, and phases of the constituent sinusoids. For the random part the best cancellation achieveable is given by

For a mixture of sinusoids and random noise, it is possible to separate the periodic and random components and have a parallel ANC architecture. This architecture results in two separate control filters, each of which is optimum for the respective component and hence the overall system can reach

The paper is organized as follows. The notational framework and various definitions are introduced in “Notations and Definitions” section, towards the end of the paper. Section

The key idea of this paper is that for sinusoidal plus random noise mixtures, separating the sinusoids and random noise and generating antinoise signals separately improves the performance of the ANC system. The motivation for this approach lies in the observation that the optimum control (cancelling) filters for the periodic part and the random part are not the same. Assuming perfect separation of

The best noise cancellation is achieved when

Let

Let

The expression for MMSE owing to statistical independence is then

It should also be noted that when a single adaptive filter is used, the adaptive algorithm tends to cancel the dominating signal component, that is, the tones more effectively. As a result, the random component may be attenuated to a lesser degree, unaffected, or even reinforced. In fact, the adaptive filter in the single-filter case shows a non-Wiener behavior [

The parallel feedforward ANC is formulated, as two subproblems, as follows.

Given an acoustic noise signal

Design an ANC system that has two parallel adaptive noise cancelling filters

The proposed scheme: active noise control employing separation of sinusoidal and random parts of the reference signal.

The optimum filter

In the proposed approach, the first sub-problem is addressed in three steps: a frequency analyzer block which estimates the sinusoidal frequencies, a synthesizer block which generates a reference signal based on the estimated frequencies, and an adaptive noise canceller block which separates

The proposed PFANC method can be divided into three major parts.

Estimation of sinusoidal parameters

Separation of sinusoidal and random components of the noise signal.

Generation of antinoise signals corresponding to the sinusoidal and random signal components.

This part estimates the sinusoidal parameters from the measured input noise signal

The above estimation problem is an important and a well-studied one [

The first step is the detection of

The following quantities need to be defined, before introducing the technique.

In general for any quantity

However in practice, the covariance matrix

The minimum is found by computing the

Plot of

The estimation of

For completeness of discussion, the steps in frequency estimation using the ESPRIT method are as follows.

Calculate the sample covariance matrix using (

Calculate the eigenvalue decomposition of

Let

The frequencies are estimated by solving the system of linear equations

The estimated frequencies are then used to generate a reference periodic signal

This block essentially estimates

This block generates the antinoise (i.e., the cancelling) signal in two steps via two parallel ANC systems,

The update equation for the weight vector

It should be noted that the adaptation methods presented here for the implementation of

The goal of this section is to sketch the method for calculating

(a) Feedforward ANC system (b) interchanging the

In the three subsections that follow, the expressions for

In linear estimation problems involving stationary signals, Wiener filtering provides the best estimate of a linear time-invariant (LTI) system based on (minimum mean square error) MMSE criterion [

In order to derive a measurable expression for

The expression for

Let

Assuming the signals in the system have real positive (power spectrum density) PSD satisfying Paley-Wiener condition [

It should be noted that any stable transfer function can also be written as a product of a stable minimum-phase and a stable all-pass system. In this case

By definition,

Equation (

When using numerical software, it should be noted that even though the form of

It should be noted that

When

Perfect nonminimum phase equalization for periodic signals.

When

The noise attenuation level

Due to linearity,

In case of a single tone (

As a summary, the procedure described in the paper is as follows.

Given a feedforward ANC set up as shown in Figure

If

If

Implement the PFANC scheme.

Simulation of the proposed method, using FXLMS for adaptation, was implemented by synthesizing

Magnitude responses of the primary and secondary paths used in the simulation.

Each experiment was done on a 100000 sample long

Performance comparison of the proposed separation-based parallel ANC and conventional single-filter ANC for different tones and SNR combination.

Number | Number of sinusoids, | Proposed PFANC NAL (dB) | Single-Filter ANC NAL (dB) | ||
---|---|---|---|---|---|

1 | 1 | 10 | 25.57 | 24.61 | 26.38 |

2 | 1 | 20 | 33.69 | 20.16 | 36.33 |

3 | 1 | 30 | 44.56 | 25.97 | 46.24 |

4 | 1 | 40 | 53.03 | 35.60 | 56.24 |

5 | 2 | 10 | 31.83 | 33.21 | 34.20 |

6 | 2 | 20 | 42.88 | 32.20 | 44.09 |

7 | 2 | 30 | 52.34 | 35.91 | 54.05 |

8 | 2 | 40 | 58.00 | 44.64 | 64.00 |

9 | 3 | 10 | 31.96 | 31.86 | 33.00 |

10 | 3 | 20 | 41.99 | 30.69 | 43.00 |

11 | 3 | 30 | 50.03 | 35.34 | 53.80 |

12 | 3 | 40 | 55.86 | 44.43 | 64.00 |

13 | 4 | 10 | 31.54 | 31.43 | 33.00 |

14 | 4 | 20 | 40.36 | 30.5 | 42.67 |

15 | 4 | 30 | 48.32 | 35.49 | 53.64 |

16 | 4 | 40 | 58.40 | 44.35 | 63.00 |

21 | 7 | 10 | 31.45 | 30.05 | 31.72 |

22 | 7 | 20 | 41.48 | 41.87 | 42.05 |

23 | 7 | 30 | 49.03 | 44.65 | 54.02 |

24 | 7 | 40 | 57.40 | 47.65 | 62.95 |

17 | 8 | 10 | 31.45 | 30.64 | 30.64 |

18 | 8 | 20 | 44.40 | 42.14 | 43.25 |

19 | 8 | 30 | 42.14 | 29.52 | 46.79 |

20 | 8 | 40 | 55.40 | 47.06 | 63.00 |

Experiments 21 through 24 were carried out using a signal derived from functional magnetic resonance imaging (fMRI) acoustic noise [

(a) Comparison of the conventional FXLMS with single filter and the proposed PFANC for 8 tones in white noise. (b) Comparison of the conventional FXLMS with single filter and the proposed PFANC for 10 tones in white noise.

The real-time Active Noise Control setup. The transparent semicircular structure mimics the MRI cavity. Recorded noise from actual MRI room is used to drive the speakers to recreate MRI environment. The area around the manikin’s ears is where cancellation is desired.

In this paper the focus has been on active noise control of acoustic signals that have a periodic/harmonic structure along with a random noise. The key idea stems from the observation that the periodic component and the random component may need different noise cancelling filters for the best performance. This leads to the idea of

Some important and recurring signals, blocks, and notation that are used in the paper are listed in the Notations and Definitions section. Other definitions are made as they are introduced.

It gives the signal energy. For a signal

The subscript “

The version of acoustic noise recorded by a reference microphone placed somewhere in the noisy area. This measurement is known as the reference signal in ANC literature.

The periodic component of noise which can be written as a sum of sinusoids, that is,

The random noise component. It is assumed to be zero-mean and stationary, where

The period of the periodic component.

The (signal-to-noise ratio) SNR of periodic component to the random component, that is,

The individual SNR of sinusoidal components, that is,

The measurement from error microphone. The cancellation is achieved around the error microphone. The error signal corresponding to a particular component will have a subscript to denote that. For example, the component of error signal corresponding to

The stable linear transfer function representing the

The stable linear transfer function representing the

An estimate of

The noise signal after passing through

The main adaptive noise cancelling (control) filter which generates the appropriate anti-noise signal using the error and reference input signals.

The control signal generated by

The maximum achieveable Noise Attenuation Level possible for a given ANC. algorithm/strategy. The

Noise Attenuation Level (NAL) of an ANC system which has the reference input

This study was supported by the VA IDIQ contract number VA549-P-0027 awarded and administered by the Dallas, TX VA Medical Center. The content of this paper does not necessarily reflect the position or the policy of the U.S. government, and no official endorsement should be inferred.