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This paper presents an acoustic noise cancelling technique using an inverse kepstrum system as an innovations-based whitening application for an adaptive finite impulse response (FIR) filter in beamforming structure. The inverse kepstrum method uses an innovations-whitened form from one acoustic path transfer function between a reference microphone sensor and a noise source so that the rear-end reference signal will then be a whitened sequence to a cascaded adaptive FIR filter in the beamforming structure. By using an inverse kepstrum filter as a whitening filter with the use of a delay filter, the cascaded adaptive FIR filter estimates only the numerator of the polynomial part from the ratio of overall combined transfer functions. The test results have shown that the adaptive FIR filter is more effective in beamforming structure than an adaptive noise cancelling (ANC) structure in terms of signal distortion in the desired signal and noise reduction in noise with nonminimum phase components. In addition, the inverse kepstrum method shows almost the same convergence level in estimate of noise statistics with the use of a smaller amount of adaptive FIR filter weights than the kepstrum method, hence it could provide better computational simplicity in processing. Furthermore, the rear-end inverse kepstrum method in beamforming structure has shown less signal distortion in the desired signal than the front-end kepstrum method and the front-end inverse kepstrum method in beamforming structure.

During the last five decades, noise cancelling and signal enhancing techniques have been developed. The techniques are fundamentally based on spectral subtraction, cepstrum and complex cepstrum methods using single-microphone sensor, and ANC and beamforming methods using multiple-microphone array sensors.

Since research on echo cancellation using adaptive filters and two-microphone sensors started in 1965, the adaptive filtering technique has been used as a solution tool for signal enhancing and noise cancelling schemes [

Pulsipher

Harrison

The beamforming technique has been introduced to maximize signal directivity, therefore it increases the performance in SNR. This method may require many microphones with accurate phase alignment among the microphones array and hence, a computational complexity in processing is expected. To increase the performance in SNR, the technique has been developed with the use of adaptive filter and also VAD [

The cepstrum processing technique [

The kepstrum [

In addition, the kepstrum has a distinction in the case of signal plus noise, where the logarithmic minimum phase transfer function becomes the minimum phase kepstrum spectral factor and it can be represented as a Kolmogorov [

By applying innovations-based whitening application in an ANC structure, it has been investigated in a simulation test, where it has been applied with the use of a FIR normalized least mean square (NLMS) algorithm for noise cancellation [

For the real-time processing using adaptive RLS filters, the innovations-based whitening application has been applied as a front-end application in beamforming structure with rear-end zero-model FIR RLS filters and it was found that it gives better noise cancelling performance than a pole-zero model IIR RLS filter in an ANC structure [

From the previous studies [

This section describes the analysis of cepstrum and complex cepstrum (kepstrum) with the relation of minimum phase kepstrum, and also for its whitening application (inverse kepstrum). It shows that the minimum phase kepstrum coefficients may be obtained from the logarithm of the minimum phase transfer function (Section 2.1) and also from the logarithm of the minimum phase spectral factor (Section 2.2). For the signal and noise, it shows that it can be represented as an output of normalized minimum phase spectral factor from innovations white noise input (Section 2.3). Based on this, it shows that logarithm of inverse minimum phase transfer function may be implemented as the innovations form of the normalized minimum phase kepstrum spectral factor for the whitening application (Section 2.4).

It is known that the causal transfer function can be expressed by Schwarz’s classical formula [^{jw}

This Equation gives the causal transfer function _{+}(_{R}_{M}_{M}

This indicates that the minimum phase transfer function may be expressed in terms of Schwarz’s formula and hence the minimum phase information can be recovered from Hilbert’s transform relation.

By defining that ^{jw}

For the

From

This shows that

From the power spectral density Φ(^{+}^{−}(

Let ^{−1}, then it follows that Φ(^{−1}) = ^{+}(^{−1})^{+}(

It follows that Φ(^{−1}). By defining ^{jw}

For the

As a result of the symmetry property of _{n}

Furthermore, since _{n}

From

In the case of random signal plus noise, it can also be represented as innovations-based form. From _{M}_{M}_{M}^{−1}) and _{M}^{−1}) are maximum phase counterparts.

In the case of signal plus noise, the logarithm of each positive- and negative-sided transfer function becomes the kepstrum spectral factors of the z- transform spectral density and these are represented as a power series expansion. For the innovations-based inverse kepstrum approach, signal plus noise are represented as an output of normalized minimum phase spectral factor from the innovations white noise input. It may be applied to an optimum IIR Wiener filtering structure, where it has been defined by Kailath [

For the application of noise alone, it can be estimated during the absence of desired signal (

From the minimum phase kepstrum spectral factor of

By using the fact that minimum phase spectral factor allows its invertibility, it can be expressed as:

This shows that the inverse of the minimum phase spectral factor can be obtained from the kepstrum exponential by multiplying by minus one (−1). It can also be represented in a normalized form:

In the case of the application of additive noise alone, _{n}_{n}_{n}_{M}_{n}_{M}_{n}_{n}_{ɛ}

By taking logarithm of

From ^{+}(_{M}_{ɛ}

A normalized innovations-based inverse kepstrum is then represented from logarithmic minimum phase transfer function or logarithmic minimum phase spectral factor, which described as:

Therefore, it shows that logarithm of inverse minimum phase transfer function may be implemented as innovations form of normalized minimum phase kepstrum spectral factor (
_{ɛ}

For the whitening application, the inverse kepstrum is processed from the reference microphone _{n}_{n}_{n}

As a discrete estimate of the continuous power spectral density, it uses a modified weighted overlapped segment averaging (WOSA) algorithm and the auto-periodograms are processed from 50% overlapping Hanning-windowed FFTs in 2,048 frame size by using smoothing method

From _{1n} − _{2n}), we can get the kepstrum coefficients (_{n}) for kepstrum processing. On the other hand, inverse kepstrum coefficients (
_{2n}) from the reference microphone. The processing difference between inverse kepstrum method and kepstrum method can be found in

This indicates that inverse kepstrum processing requires only a negative sign of the kepstrum coefficients, which can be obtained for the inverse of acoustic path transfer function from the reference microphone input

On the other hand, it is compared with the kepstrum method, which uses the ratio of the acoustic transfer function between two-microphone inputs as

Both the kepstrum and inverse kepstrum coefficients are transformed into the corresponding impulse response using recursive formula [

For the cascaded adaptive FIR filter, the NLMS algorithm [_{n}^{2} is input power and the value of 0.0001 is used to prevent zero division.

For the comparison of processing among: (i) inverse kepstrum coefficients, (ii) kepstrum coefficients and (iii) NLMS algorithm based adaptive FIR filter weights, computational complexity in floating point operations per second (FLOPS) can be compared as shown in

The inverse kepstrum method uses the whitening application from one acoustic path transfer function, _{2}(_{1}(_{2}(

In the ANC structure, identification as an unknown system is represented as the ratio of acoustic transfer functions, _{1}(_{2}(

Secondly, the inverse kepstrum is applied in front of the adaptive FIR filter as shown in _{2}(_{1}(

Assuming that each acoustic path transfer functions between two microphones and noise source are as given in

As a simple example, we assume that one transfer function from acoustic path transfer functions is nonminimum phase term, such as:
_{1}(

The unknown system is then described as the ratio of transfer functions and it is represented as:

For the operation of the inverse kepstrum filter _{1}(

This indicates that the front-end inverse kepstrum _{1}(_{2}(

This indicates that the front-end kepstrum estimates the minimum phase term only from the ratio of overall transfer function, where nonminimum phase term is reflected to the minimum phase term by a reciprocal polynomial as ^{−n} H^{−1}). The cascaded adaptive FIR filter works then as an all-pass filter. From

Now let us check with inverse of overall transfer function, such that:
_{2}(

This indicates that the transfer function now has a nonminimum phase term in the denominator polynomial from the ratio of overall transfer functions.

The unknown system can be estimated from the ratio of transfer functions and it is represented as:

The inverse kepstrum and cascaded adaptive FIR filter are estimated as:

This indicates that the estimates,

In a beamforming structure, identification as an unknown system is represented as the ratio of combined acoustic transfer functions, _{1}(_{2}(_{1}(_{2}(

Based on this, an inverse kepstrum filter is applied in front of the adaptive filter as a rear-end application from the sum-and-subtract function from the beamforming structure as shown in _{1}(_{2}(_{1}(_{2}(_{1}(_{2}(_{1}(_{2}(

To compare with the operation in an ANC structure, the same components of the acoustic transfer functions are used as

From the unknown system, we have the numerator polynomial part from the ratio of overall combined transfer functions and it is represented as an overall transfer function and the denominator polynomial part works as a delay filter, described in

For the operation of inverse kepstrum filter _{1}(

This indicates that the rear-end inverse kepstrum _{1}(_{2}(

This indicates that the rear-end kepstrum estimates the minimum phase term only from the polynomial numerator part from the ratio of combined overall transfer functions, where the nonminimum phase term is reflected to the minimum phase term by the reciprocal polynomial as ^{−n} H^{−1}). The cascaded adaptive FIR filter estimates the remaining term from the polynomial numerator of the ratio of combined overall transfer functions. Based on this, the unknown system

Let us now check with the inverse of the overall transfer function,

Inverse kepstrum and cascaded adaptive FIR filter estimates as:

On the other hand, kepstrum and cascaded adaptive FIR filter estimates as:

It shows that the estimates,

Experiments were implemented in both simulation tests on pc software and real tests using real nonstationary noise in a room environment. According to the main three considerations (signal distortion in the desired signal, noise reduction performance in noise with nonminimum phase components and convergence level in estimates of noise statistics with the use of a small amount of adaptive FIR filter weights), the performances achieved when using an inverse kepstrum filter were verified in both the ANC and beamforming structures. Furthermore, the rear-end application of the inverse kepstrum method in beamforming structure was also compared with two front-end applications of the kepstrum method [

The methodology is based on the fact that the coefficients (kepstrum and inverse kepstrum) and weights (adaptive FIR filter) are continuously updated during noise alone period to estimate noise statistics. When the desired signal is applied to noise, the last updated coefficients and weights are frozen and applied to the desired signal and noise. For a precise test, it is programmed to stop updating coefficients and weights, and then these are applied to the desired signal and noise period. To check the strength in amplitude and distortion status in desired signal, simple three sine waveforms are added and used as the desired signal for both simulation and real tests. For the test using real noise, we use a nonstationary music sound, tuned to a certain radio station. For the use of an adaptive FIR filter, an NLMS algorithm has been used with the use of step size

For the simulation test, the acoustic transfer functions of

The first test is to verify the noise cancelling performance in the ANC structure by applying three adaptive FIR filter weights for the noise characteristic with nonminimum phase component in the polynomial numerator

From the simulation test based on the block diagram (

For the second test, the acoustic transfer function ^{−3} to reduce signal distortion in the ANC structure. On the other hand, it is found that there is no need to set the delay in the beamforming structure. From the test result, by using a small amount of three adaptive FIR filter weights, it is shown that the performance in the beamforming structure provides no signal distortion in the desired signal without any delay adjustment and also better noise reduction in noise with nonminimum phase term than the performance in the ANC structure, as shown in

From the above two test results in terms of noise reduction performance in reverberant noise with nonminimum phase (

The objective is to verify the performance of the inverse kepstrum method in ANC and beamforming structures. For the test in the ANC structure, the acoustic transfer functions of

From the test result, it is analyzed that in the ANC structure, the application of an inverse kepstrum filter works well with an adaptive FIR filter in terms of convergence with a smaller amount in adaptive filter weights, rather than when the adaptive FIR filter is used with application of a kepstrum filter From the test results between (c)–(ii) and (d)–(ii) in

For the test in beamforming structure, acoustic transfer functions of

Based on this, by using two inverse kepstrum coefficients, it can be verified that it can also be approximated to

From the test result, it is also analyzed in a beamforming structure and shows that application of inverse kepstrum filter works well with the adaptive FIR filter in terms of convergence with much smaller amount in adaptive filter weights, rather than when the adaptive FIR filter is used with application of a kepstrum filter. From the test results between (b)–(ii) and (d)–(ii) in

From the comparison results between inverse kepstrum filter and kepstrum filter to adaptive FIR filter in terms of convergence in noise statistics and its pole-zero placements, it is found that the application of the inverse kepstrum filter could give a convergence benefit with the use of a small amount in adaptive FIR filter weights in the ANC structure [

The simulation test results suggest that the inverse kepstrum should achieve more noise reduction without signal distortion in the desired signal by using small amount of adaptive FIR filter for the real tests in a realistic reverberant environment. Therefore, the inverse kepstrum has been tested in a beamforming structure [

Furthermore, the performance [

As shown in

For real-time processing in a realistic reverberant environment, the kepstrum method and inverse kepstrum method have been applied to ANC and beamforming structures, hence its performance on modified application [

Firstly, the reverberant nature of most rooms gives rise to nonminimum phase components in the acoustic transfer function [

For an accurate discrepancy measurement in a desired signal, three added simple sine waveforms (disregarding whether it is narrowband or wideband in this test) have arbitrarily been used as a sample of a desired signal so that the distorted amount in a desired signal could be measured by calculating the average discrepancy in dB to check the consistency level of the signal strengths in dB from the original desired signal. For the precise testing in estimate of coefficients and weights, instead of using automatic VAD, it has been programmed to stop and make the last updated coefficients and weights to be frozen on demand of the application, and then it is applied to the desired signal and real noise period.

From the test results on main three considerations using the above mentioned methodology, it can be summarized that for the application of adaptive FIR filter to noise cancelling scheme, it is found that adaptive filter works better in a beamforming structure than the ANC structure in terms of signal distortion in desired signal, noise reduction in noise with nonminimum phase component (

For the rear-end application of the inverse kepstrum to beamforming structure, the inverse kepstrum method gives better convergence with a much smaller amount of adaptive FIR filter weights than the kepstrum method (

An adaptive FIR filter has shown a performance distinction between ANC and beamforming structures in terms of signal distortion in the desired signal and nonminimum phase in noise, where the beamforming structure provides better performance than the ANC structure on application of an adaptive FIR filter. Based on this, the innovations-based inverse kepstrum method has been applied to the beamforming structure, and its performance has also been compared with the kepstrum method and inverse kepstrum method as front-end applications in the beamforming structure. The simulation and real tests show that the innovations-based whitening inverse kepstrum method has provided a promising result in rear-end beamforming structure in terms of signal distortion in the desired signal, noise reduction in noise with nonminimum phase and convergence level in estimation of noise statistics with the use of a small amount of adaptive FIR filter weights. Furthermore, the inverse kepstrum method provides the computational simplicity in processing so that it could be a benefit to a practical real-time adaptive noise cancelling scheme. Further analysis and investigation of the inverse kepstrum method will be performed using ECG (electrocardiogram) signals in biomedical signal processing for noise cancelling schemes.

Author acknowledges that this work has been supported by UTM RMC grant (vote number 77523) and RU GUP grant (vote number Q.J130000.2636.02J61). The author thanks the anonymous reviewers for their helpful suggestions and comments.

Equivalence of outputs based on inputs of:

Representation of noise as innovations-based whitening form.

Periodogram estimate from each input microphone sensors.

Block diagram for the comparison between inverse kepstrum processing (−_{2n}) and kepstrum processing (_{1n} − _{2n}).

The conversion procedure from inverse kepstrum (
_{n}) to impulse response (_{n}_{n}

Analysis of identification of acoustic path transfer functions by an adaptive FIR filter in the ANC structure during the periods of

Application of

Block diagram for the operation comparison between an inverse kepstrum filter and a kepstrum filter as front-end application to an ANC structure.

Analysis of identification of acoustic path transfer functions by adaptive FIR filter in beamforming structure during the periods of

Rear-end application of

Block diagram for the operation comparison of an inverse kepstrum filter and a kepstrum filter as rear-end applications to a beamforming structure.

Comparison in spectra between direct transfer function

Simulation test based on 3 adaptive FIR filter weights:

Snapshot of pole-zero placement:

Snap shot of pole-zero placement:

Real test in a room environment: (I) —waveforms of

Front-end application of:

Comparison of computational complexity in FLOPS on (i) inverse kepstrum processing, (ii) kepstrum processing and (iii) NLMS algorithm [

(i) Inverse kepstrum | 1 × WOSA Periodogram |
_{2}(5.12 / Δ_{2}^{1/3}(log ^{2} |

Total Computation ( |
0.057 × 10^{6} | |

(ii) Kepstrum | 2 × WOSA Periodogram |
2_{2}(5.12 / Δ_{2}^{1/3}(log ^{2} |

Total Computation ( |
0.08 × 10^{6} | |

(iii) NLMS | Real multiplication (A) |
3^{2} + 2 |

Total Computation ( |
(A) 0.12 × 10^{6} (B) 2.4 × 10^{6} |

Note: Total computation:

(*) is based on N: 2048 frame size,

(**) is based on 200 NLMS weights.

Weights and coefficients arrays showing each estimate output from the simulation test based on the block diagram in

(a) Adaptive filter weights (one zero) | |||||||

1 | 1.712 | – | – | – | – | – | |

(b) Adaptive filter weights (three zeroes) | |||||||

1 | 1.800 | −0.359 | 0.069 | – | – | – | |

(c) (i) Inverse kepstrum coefficients (two poles) and | |||||||

(i) | 1 | −0.194 | 0.018 | – | – | – | – |

(ii) | 1 | 1.992 | – | – | – | – | |

(iii) | 1 | 1.797 | −0.369 | 0.037 | – | – | – |

(d) (i) Kepstrum coefficients (two poles) and | |||||||

(i) | 1 | 0.299 | 0.044 | – | – | – | – |

(ii) | 1 | 1.501 | −0.846 | 0.234 | – | – | – |

(iii) | 1 | 1.800 | −0.352 | 0.048 | 0.032 | 0.010 | – |

Note: iii=i*ii, where * indicates convolution.

Weights and coefficients arrays showing each estimate output from the simulation test based on the block diagram in

(a) Adaptive filter weights (one zero) | |||||||

1 | 1.101 | – | – | – | – | – | |

(b) (i) Inverse kepstrum coefficients (two poles) and | |||||||

(i) | 1 | 0.004 | 0.000 | – | – | – | – |

(ii) | 1 | 1.092 | – | – | – | – | – |

(iii) | 1 | 1.097 | 0.005 | 0.000 | – | – | – |

(c) (i) Kepstrum coefficients (two poles) and | |||||||

(i) | 1 | 0.953 | 0.045 | – | – | – | – |

(ii) | 1 | −0.150 | – | – | – | – | – |

(iii) | 1 | 0.803 | 0.311 | −0.068 | – | – | – |

(d) (i) Kepstrum coefficients (two poles) and | |||||||

(i) | 1 | 0.898 | 0.403 | – | – | – | – |

(ii) | 1 | 0.200 | −0.582 | 0.443 | −0.163 | −0.031 | 0.092 |

(iii) | 1 | 1.098 | −0.003 | 0.002 | ∼0.000 | ∼0.000 | ∼0.000 |

Note: (A) iii=i*ii, where * indicates convolution.

(B) (d) (ii) and (iii) shows only seven weights estimates in arrays.

Comparison of average discrepancy on application of front-end kepstrum [

Frequency (Hz) | 500 | 550 | 700 | D | 500 | 550 | 700 | D | 500 | 550 | 700 | D |

Trial 1 | 30 | 29.5 | 29.5 | 0.25 | 20 | 20.5 | 21 | 0.5 | 25 | 25 | 24.5 | 0.25 |

Trial 2 | 29.5 | 30 | 30 | 0.25 | 21 | 20 | 20 | 0.5 | 25 | 25 | 25 | 0 |

Trial 3 | 29.5 | 30 | 29.5 | 0.25 | 20 | 20 | 20.5 | 0.25 | 25 | 25 | 25.5 | 0.25 |

Trial 4 | 30 | 29.5 | 30 | 0.25 | 21 | 20 | 21 | 0.5 | 25 | 25 | 25 | 0 |

Trial 5 | 29.5 | 29.5 | 30 | 0.25 | 21 | 20.5 | 20 | 0.5 | 25 | 24.5 | 25 | 0.25 |

Average D (dB) | 0.25 | 0.45 | 0.15 |

Note: D indicates discrepancy in dB from signal strength of desired signal in frequency domain.