It is known that the causal transfer function can be expressed by Schwarz’s classical formula [23] as: (1) H + ( z ) = 1 2 π ∫ 0 2 π H R ( λ ) ( 1 + e j λ z − 1 1 − e j λ z − 1 ) d λ , | z | < 1where λ as the integration variable and z = re^jw.

This Equation gives the causal transfer function H₊(z) whose real part on the unit circle is H_R(w). Based on this, the phase information can be recovered by Hilbert’s transform relation. The logarithm of minimum phase transfer function logH_M(z) can be written as: (2) log H M ( z ) = 1 2 π ∫ − π π log | H M ( λ ) | ( 1 + e j λ z − 1 1 − e j λ z − 1 ) d λ ,       | z | < 1where log |H_M(λ)| is the magnitude part of minimum phase logarithmic transfer function.

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 z = e^jw, it can be described as magnitude and phase term: (3) log H M ( z ) = log H M ( e jw ) = log | H M ( e jw ) | + j arg [ log H M ( e jw ) ]

For the N -point discrete form: (4) log H M ( 2 π N k ) = [ c 0 + ∑ n = 1 N − 1 ( 1 2 c − n e j 2 π N kn + 1 2 c n e − j 2 π N kn ) ] + j [ ∑ n = 1 N − 1 − ( 1 2 c − n e j 2 π N kn − 1 2 c n e − j 2 π N kn ) ]

From Equation (4), the magnitude of logarithmic minimum phase transfer function is: (5) log | H M ( e j 2 π N k ) | = [ c 0 + ∑ n = 1 N − 1 ( 1 2 c − n e j 2 π N kn + 1 2 c n e − j 2 π N kn ) ]and the phase of the logarithmic minimum phase transfer function is: (6) arg   [ H M ( e j 2 π N k ) ] = j [ ∑ n = 1 N − 1 − ( 1 2 c − n e j 2 π N kn − 1 2 c n e − j 2 π N kn ) ]

This shows that Equation (5) shows an even cepstrum function, hence minimum phase kepstrum coefficients can be processed in the cepstrum domain by multiplying by two (2) in the positive time series, except for the first zeroth coefficient in the time series.

From the power spectral density Φ(z) it can be represented as causal spectral factor H⁺(z) and anticausal counterpart H⁻(z) as: (7) Φ ( z ) = H + ( z ) H − ( z ) = H + ( z ) H + ( z − 1 )

Let z = z⁻¹, then it follows that Φ(z⁻¹) = H⁺(z⁻¹)H⁺(z)

It follows that Φ(z) = Φ(z⁻¹). By defining Z = e^jw, we now have a logarithmic power spectrum as: (8) Φ ( w ) = Φ ( − w ) (9) Φ ( w ) = | H + ( e jw ) | 2 (10) log   Φ ( w ) = 2   log | H + ( e jw ) |

For the N -point discrete form: (11) K ( 2 π N k ) = log   Φ ( 2 π N k ) = 2   log | H + ( e j 2 π N k ) | = 2 [ k 0 + ∑ n = 1 N − 1 ( 1 2 k − n e j 2 π N kn + 1 2 k n e − j 2 π N kn ) ]

As a result of the symmetry property of k_n, it can be expressed as: (12) K + ( 2 π N k ) = [ 2 k 0 + ∑ n = 1 N − 1 ( k n e − j 2 π N kn ) ]

Furthermore, since k_n are real, only a half portion in length can be considered as: (13) K + ( 2 π N k ) = 2 k 0 + ∑ n = 0 ( N / 2 ) − 1 k n e − j 2 π N kn

From Equation (13), it shows that the kepstrum coefficients can be processed in the causal kepstrum domain by halving the first zeroth coefficient with the remaining coefficients truncated in size to (N / 2) −1. By truncating in size less than (N / 2) −1, kepstrum now becomes complex cepstrum, which is an approximation of the theoretical mathematical construct [19].

In the case of random signal plus noise, it can also be represented as innovations-based form. From Figure 1(upper part), it shows an equivalent relation of output from the inputs between white noise and innovations white noise as Equation (14): (14) Φ xx ( z ) = Φ ww ( z ) + Φ ss ( z ) = N M ( z ) N M ( z − 1 ) σ v 2 + S M ( z ) S M ( z − 1 ) σ ξ 2 = H n + ( z ) H n − ( z ) σ i 2where σ v 2, σ ξ 2 and σ i 2 are the variances of the additive white noise, white noise input and white noise innovations process, respectively. N_M(z) and S_M(z) are coloured minimum phase transfer functions and N_M(z⁻¹) and S_M(z⁻¹) are maximum phase counterparts. H n + ( z ) is the normalized minimum phase spectral factor, which has all its zeros inside |z| = 1 whilst H n − ( z ) is the counterpart, which has its zeros outside |z| = 1.

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 [18] as combination of two cascaded filters, a front-end whitening filter to generate the white innovations process and cascaded spectral shaping filter to provide spectral shaping function for the input signal.

For the application of noise alone, it can be estimated during the absence of desired signal (Figure 1(lower part)). Therefore, the additive noise now can be represented as innovations-based form, as shown in Figure 2.

From the minimum phase kepstrum spectral factor of Equation (15): (15) K + ( z ) = log H + ( z ) = k 0 + k 1 z − 1 + k 2 z − 2 + … …its exponentiation becomes a causal spectral factor as: (16) H + ( z ) = exp [ K + ( z ) ] = exp ( k 0 + k 1 z − 1 + k 2 z − 2 + … )

By using the fact that minimum phase spectral factor allows its invertibility, it can be expressed as: (17) [ H + ( z ) ] − 1 = exp [ − K + ( z ) ] = exp [ − ( k 0 + k 1 z − 1 + k 2 z − 2 + … ) ]

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: (18) H + ( z ) = exp [ k 0 ] exp [ k n + ( z ) ] = exp [ k 0 ] H n + ( z ) (19) log H + ( z ) = k 0 + log H n + ( z )

In the case of the application of additive noise alone, i_n becomes v_n. From the fact that x_n = H_M(z)v_n and x n = H n + ( z )   i n, the relationship between minimum phase transfer function H_M(z) and normalized minimum phase spectral factor H n + ( z ) is described as: (20) H M ( z ) = H n + ( z )   i n = H n + ( z ) v n = H n + ( z ) σ ɛwhere v_n, i_n and σ_ɛ are white noise, innovations process and standard deviation of normalized innovations sequence respectively.

By taking logarithm of Equation (20): (21) log H M ( z ) = log H n + ( z ) + log σ ɛ

From Equations (19) and (21) and with the fact that log H⁺(z) = log H_M(z): (22) k 0 = log σ ɛ (23) σ ɛ = exp [ k 0 ]   and   σ ɛ 2 = exp [ 2 k 0 ]where σ_ɛ and σ ɛ 2 are standard deviation and variance of normalized innovations sequence respectively.

A normalized innovations-based inverse kepstrum is then represented from logarithmic minimum phase transfer function or logarithmic minimum phase spectral factor, which described as: (24) log [ H M ( z ) ] − 1 = log [ H n + ( z ) ] − 1 − log σ ɛ = − K n + ( z ) − log σ ɛ

Therefore, it shows that logarithm of inverse minimum phase transfer function may be implemented as innovations form of normalized minimum phase kepstrum spectral factor ( log [ H M ( z ) ] − 1 = − K n + ( z ), assuming that standard deviation σ_ɛ = 1).

In the ANC structure, identification as an unknown system is represented as the ratio of acoustic transfer functions, H(z) = H₁(z)/H₂(z) as shown in Figure 7, where an ordinary adaptive FIR filter may be used to estimate the noise statistics during noise alone period. Its estimate is then applied to the signal and noise period to cancel the noise by using the estimated noise statistics with the condition that the desired signal should be only retained with no distortion.

Secondly, the inverse kepstrum is applied in front of the adaptive FIR filter as shown in Figure 8(a), where the inverse kepstrum filter estimates a denominator part, 1/H₂(z) and the cascaded adaptive FIR filter estimates a numerator part, H₁(z) from the ratio of overall transfer function. It is compared with the kepstrum method as shown in Figure 8(b), where the kepstrum filter estimates the minimum phase term only from the ratio of overall transfer function and the cascaded adaptive FIR filter estimates remaining term and hence it works as an all-pass filter.

Assuming that each acoustic path transfer functions between two microphones and noise source are as given in (29), it is represented as the unknown system of (30). Its estimates are to be analyzed as shown in Figure 9, where the operation of the inverse kepstrum is to be compared with the kepstrum filter as a front-end application to the cascaded adaptive FIR filter.

As a simple example, we assume that one transfer function from acoustic path transfer functions is nonminimum phase term, such as: (29) H 1 ( z ) = 1 + 2 z − 1     and  H 2 ( z ) = 1 + 0.2 z − 1where H₁(z) is nonminimum phase transfer function.

The unknown system is then described as the ratio of transfer functions and it is represented as: (30) H ( z ) = ( 1 + 2 z − 1 ) / ( 1 + 0.2 z − 1 ) = 1 + 1.8 z − 1 − 0.36 z − 2 + 0.072 z − 3 − 0.0144 z − 4 … …where it can be estimated by ordinary adaptive FIR filter.

For the operation of the inverse kepstrum filter K′(z) and adaptive FIR filter W₁(z), each one is estimated as: (31) K ′ ( z ) = 1 / ( 1 + 0.2 z − 1 )     W 1 ( z ) = 1 + 2 z − 1

This indicates that the front-end inverse kepstrum K′(z) estimates the denominator part and the cascaded adaptive FIR filter W₁(z) estimates the numerator part from the ratio of overall transfer functions. On the other hand, for the operation of the kepstrum filter K(z) and adaptive FIR filter W₂(z), each one is estimated as: (32) K ( z ) = ( 1 + 0.5 z − 1 ) / ( 1 + 0.2 z − 1 )     W 2 ( z ) = ( 1 + 2 z − 1 ) / ( 1 + 0.5 z − 1 )

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 z⁻ⁿ H(z⁻¹). The cascaded adaptive FIR filter works then as an all-pass filter. From (31) and (32), we now have found that both methods show the same result as (30).

Now let us check with inverse of overall transfer function, such that: (33) H 1 ( z ) = 1 + 0.2 z − 1   and  H 2 ( z ) = 1 + 2 z − 1where H₂(z) is nonminimum phase transfer function.

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: (34) H ( z ) = ( 1 + 0.2 z − 1 ) / ( 1 + 2 z − 1 ) = [ ( 1 + 0.2 z − 1 ) / ( 1 + 0.5 z − 1 ) ] [ ( 1 + 0.5 z − 1 ) / ( 1 + 2 z − 1 ) ] = ( 1 + 0.2 z − 1 ) / ( 1 + 0.5 z − 1 ) = 1 − 0.3 z − 1 + 0.15 z − 2 + … …where it can also be estimated by adaptive FIR filter. The Equation (34) indicates that the nonminimum phase term of denominator part is converted to a minimum phase term for the operation of an adaptive FIR filter so that the cascaded remaining part of Equation (34) always works as an all-pass filter.

The inverse kepstrum and cascaded adaptive FIR filter are estimated as: (35) K ′ ( z ) = 1 / ( 1 + 0.5 z − 1 )     W 1 ( z ) = 1 + 0.2 z − 1and the kepstrum and cascaded adaptive FIR filter are estimated as: (36) K ( z ) = ( 1 + 0.2 z − 1 ) / ( 1 + 0.5 z − 1 )       W 2 ( z ) = 1

This indicates that the estimates, Equations (35) and (36), from both methods are the same as Equation (34), which is the inverse of the overall transfer function. The nonminimum phase component from the polynomial numerator and denominator of the overall transfer function may frequently occur in reverberant environments [26] and causes a fluctuation in the spectrum due to the difference between Equations (30) and (34). To deal with this problem, it indicates that neither the kepstrum nor the inverse kepstrum methods provide a solution in the ANC structure. Alternatively, we may solve this problem by swapping the position of the two microphones, but it is not a practical solution in a realistic environment.

In a beamforming structure, identification as an unknown system is represented as the ratio of combined acoustic transfer functions, H(z) = 0.5(H₁(z) + H₂(z))/0.5(H₁(z) − H₂(z)) as shown in Figure 10, where an ordinary adaptive FIR filter may also be used to estimate the ratio of combined overall transfer functions. Its estimate may then be applied to the signal and noise period to cancel the noise by using estimated noise statistics.

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 Figure 11(a), where the inverse kepstrum filter estimates the polynomial denominator part, 1/0.5(H₁(z) − H₂(z)) and the cascaded adaptive FIR filter estimates the polynomial numerator part, 0.5(H₁(z) + H₂(z)) from the ratio of combined overall transfer functions. It is compared with the kepstrum method as shown in Figure 11(b), where the kepstrum filter estimates the minimum phase term only from the numerator polynomial part, 0.5(H₁(z) + H₂(z)) and the cascaded adaptive FIR filter estimates the remaining part from the numerator polynomial part, 0.5(H₁(z) + H₂(z)), where this numerator polynomial part is to be an overall transfer function with a delay filter in the rear-end primary input, d n '.

To compare with the operation in an ANC structure, the same components of the acoustic transfer functions are used as Equation (29) and this is represented as an unknown system as in Equation (37). It is to be analyzed as shown in Figure 12, where the operation of the inverse kepstrum is to be compared with the kepstrum filter as a rear-end application to the sum-and-subtract function of the beamforming structure.

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 Equation (37): (37) H ( z ) = 0.5 [ ( 1 + 2 z − 1 ) + ( 1 + 0.2 z − 1 ) ] = 1 + 1.1 z − 1     with one sample delay ,   D − 1

For the operation of inverse kepstrum filter K′(z) and adaptive FIR filter W₁(z), each one is estimated as: (38) K ′ ( z ) = 1   and   W 1 ( z ) = 1 + 1.1 z − 1   with one sample delay ,   D − 1

This indicates that the rear-end inverse kepstrum K′(z) works as a whitening filter with one sample delay and a cascaded adaptive FIR filter W₁(z) estimated a numerator part from the ratio of combined overall transfer functions. On the other hand, for the operation of the kepstrum filter K(z) and the adaptive FIR filter W₂(z), each one is estimated as: (39) K ( z ) = 1 + 0.9 z − 1   and   W 2 ( z ) = 1 + 0.2 z − 1   with one sample delay ,   D − 1

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 z⁻ⁿ H(z⁻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 Equation (37) may be estimated by the operations (38) and (39) as rear-end applications of the inverse kepstrum and kepstrum methods to the sum-and-subtract function of the beamforming structure, respectively.

Let us now check with the inverse of the overall transfer function, (33), where nonminimum phase term in denominator polynomial is no longer exist and overall transfer function is now obtained from numerator polynomial part of the ratio of overall combined transfer function.

Inverse kepstrum and cascaded adaptive FIR filter estimates as: (40) K ′ ( z ) = 1   and  W 1 ( z ) = 1 + 1.1 z − 1   with one sample delay ,   D − 1where inverse kepstrum filter works as whitening filter and adaptive FIR filter estimates numerator polynomial part of the ratio of overall combined transfer function.

On the other hand, kepstrum and cascaded adaptive FIR filter estimates as: (41) K ( z ) = 1 + 0.9 z − 1   and  W 2 ( z ) = 1 + 0.2 z − 1   with one sample delay ,   D − 1where kepstrum filter works as minimum phase filter and adaptive FIR filter estimate remaining term from the numerator polynomial part of the ratio of overall combined transfer function.

It shows that the estimates, Equations (40) and (41) by both methods on the inverted transfer function are same as Equations (38) and (39), which are the estimates by the both methods on the direct transfer function. It indicates that both kepstrum and inverse kepstrum methods do provide a solution in beamforming structure, which may give a practical solution in a reverberant noise with nonminimum phase component from overall transfer function because it does not need to swap the two microphones position. The detailed analysis on nonminimum phase transfer function has been investigated between ANC and beamforming structures [27].

For the simulation test, the acoustic transfer functions of (29) are used as the unknown system, which has nonminimum phase component in noise. The desired signal, consisting of three frequencies (500 Hz, 550 Hz and 700 Hz), has arbitrarily been used as shown in Figure 14(a).

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 [Figure 11 I-(a)] in a room. To verify the performance in signal, the desired signal [Figure 17 I-(a)] consisting of three frequencies (500 Hz, 550 Hz and 700 Hz) has been used, and music sounds tuned to a radio station [Figure 17 I-(b)] has been used as real nonstationary noise in a room. From the test on the ANC structure, it is shown that the shape of the signal waveform has been distorted in time domain, as shown in Figure 17 I-(c). On the other hand, it is shown that almost the same shape of the signal waveform has been maintained in the beamforming structure as shown in Figures 17 I-(d). The performance has also been compared in spectra in frequency domain as shown in Figure 17 II-(e). In the beamforming structure, the performance has been compared between 200 adaptive FIR filter weights alone [Figures 17 II-(f)-(i)] and 32 inverse kepstrum coefficients with 50 cascaded adaptive FIR filter weights [Figure 17 II-(f)-(ii)], where the test results show that inverse the application of the kepstrum with reduced adaptive FIR filter weights gives even better noise reduction performance while maintaining the original desired signal. It indicates that the application of the inverse kepstrum could provide a benefit of more computational simplicity than a kepstrum in processing as well as the ordinary adaptive FIR filter (Table 1). To reduce the computational complexity, the alternative method could be considered by decreasing the window size but the problem has been found with the widening sidelobe on the desired signal. By using the inverse kepstrum, it has been found that increasing to 4096 the window size is acceptable for the real-time processing with the narrowing sidelobe on the desired signal [21].

Furthermore, the performance [Figure 17 II-(f)-(ii)] of the inverse kepstrum in rear-end beamforming [Figure 11(a)] has been compared with the front-end application of the inverse kepstrum [Figure 18(a)] and kepstrum [Figure 18(b) in a beamforming structure. With the use of a kepstrum (or inverse kepstrum) coefficients to 32 and a reduced size of adaptive filter weights to 50, it is now to verify the performance on signal distortion in a desired signal when the desired signal is applied to the noise estimates of coefficients and weights, which are abruptly frozen on the rapid change of noise statistics. To verify the performance, the average discrepancy between strengths of three frequencies after processing has been measured since each input frequency is applied with equal strengths in dB. For a trial test, the same desired signal is applied at random intervals to the frequently changing real noise environment, where the estimate of noise transfer function is randomly frozen accordingly. Figure 18 shows snapshot of a performance among the application of front-end kepstrum, front-end inverse kepstrum with comparison of application of rear-end inverse kepstrum in time and frequency domains.

As shown in Figure 18(c,e), the test results show that front-end inverse kepstrum provides better noise reduction but with attenuated amplitude in a desired signal. On the other hand, the front-end kepstrum [14] shows more strength in signal amplitude with almost same noise reduction as shown in Figure 18(d,f). However, it has been found from the test that both front-end methods [Figure 18(a,b)] are more vulnerable to signal distortion than the rear-end inverse kepstrum method in beamforming structure. As shown in Table 4, the level of signal distortion has been compared by measuring an average discrepancy of signal strength in dB from front-end kepstrum [14], front-end inverse kepstrum [22] and rear-end inverse kepstrum, where it has been calculated as 0.25 dB, 0.45 dB and 0.15 dB, respectively.

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 [4,6–8] to desired signal and adaptive filter has been investigated in detail in this paper.

Firstly, the reverberant nature of most rooms gives rise to nonminimum phase components in the acoustic transfer function [28] and its inverse is often required in a realistic reverberant environment [26], hence it might produce instability in processing. Therefore, the acoustic transfer function and its inverted transfer function have been tested in ANC and beamforming structures. Secondly, the acoustic noise transfer function changes rapidly and frequently in a realistic reverberant environment and the estimated noise statistics for the acoustic transfer function are abruptly frozen. It is then instantly applied during the desired signal period, which might cause a signal distortion. To verify the performance, a discrepancy in dB has been compared among the application of front-end kepstrum, front-end inverse kepstrum and rear-end inverse kepstrum in a beamforming structure. Thirdly, a fast convergence in adaptive filter application is essential in real-time processing so that a fast convergence with small amount of adaptive filter weights has been tested in both kepstrum method and inverse kepstrum method.

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 (Figure 14) and consistency of estimate in noise statistics on its inverted transfer function (Figure 13).

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 (Table 3), which contributes to better real-time processing (Table 1). It has also been found that the combination of rear-end inverse kepstrum and cascaded adaptive FIR filter gives better noise reduction performance with highly reduced adaptive filter weights than the sole application of an adaptive FIR filter (Figure 17). In addition, it gives less signal distortion in the desired signal than the front-end applications of kepstrum method and inverse kepstrum method (Figure 18, Table 4).