- Email: [email protected]
An oversampled subband adaptive filter without cross adaptive filters
SIGNAL
PROCESSING ELSEVIER
Signal Processing 64 (1998)93-101
An oversampled subband adaptive filter without cross adaptive filters M. Harteneckav*, J.M. PBez-Borrallob, R.W. Stewart” aSignal Processing Division, Department. of Electrical and Electronic Engineering, University of Strathclyde, Glasgow, Gl lxW, Scotland, b ETSI Telecomunicacibn, Universidad Politkica
UK de Madrid, Ciudad, Universitaria s/n, 28040 Madrid, Spain
Received 9 October 1997
Abstract In this paper we propose a subband adaptive filter architecture that does not require cross-adaptive filters between subbands to compensate for the ‘inband’ alias caused by the subsampling process. The architecture uses real-valued oversampled filter banks with different subsampling ratios in each channel which reduce the ‘inband’ alias significantly. In the paper we describe the design algorithm for the filter banks and show simulation results in a system identification set-up to demonstrate the performance. 0 1998 Elsevier Science B.V. All rights reserved. Zusammenfassung Dieser Artikel stellt eine Frequenzteilband Filterarchitektur fiir adaptive Filter vor, die keine adaptiven Filter zwischen verschiedenen Teilblndern benlitigt, urn ‘Zwischenband-Aliasing’ zu unterdriicken. Die Architektur verwendet reellwertige, iiberabgetastete Filterblnke mit verschiedenen Unterabtastfaktoren in jedem Teilband, welche das ‘Zwischenband-Aliasing’ signifikant reduzieren. In diesem Artikel beschreiben wir den Filterentwurfsalgorithmus und prlsentieren Simulationsergebnisse in einer Systemidentifikationsumgebung, urn die Leistungsfihigkeit unseres Filterentwurfs zu verdeutlichen. 0 1998 Elsevier Science B.V. All rights reserved.
Nous proposons dans cet article une architecture de filtrage adaptatif en sous-bandes qui ne requiert pas de filtres inter-adaptatifs entre les sous-bandes pour compenser le repliement ‘dans la bande’ caustr par le processus de sous&hantillonnage. Cette architecture utilise des banes de filtres sur-ichantillonnts g valeurs rCelles avec des rapports de sous-kchantillonnage diffirents dans chaque canal qui riduisent significativement le repliement ‘dans la bande’. Nous dkcrivons dans cet article l’algorithme de conception des banes de filtres et montrons des r&sultats de simulation dans un contexte d’identification de systkmes pour mettre en Cvidence les performances. 0 1998 Elsevier Science B.V. All rights reserved. Keywords:
Subband adaptive filter; Oversampled filter bank; Filter bank design
*Corresponding author. Tel.: +44 141 548 2808; fax: +44 141 552 2487; e-mail: [email protected]. 0165-1684/98/$19.000 1998 Elsevier Science B.V. All rights reserved. PffSO165-1684(97)00179-5
94
M. Harteneck et al. 1 Signal Processing 64 (1998) 93-101
1. Introduction
Problem description
As shown in [S] subband adaptive filters require cross-band adaptive filters if the subband signals are contaminated by ‘inband’ aliasing and therefore a second source of information about the contaminated part of the subband signal is necessary to make the comparison between the subband input signal and the subband desired signal. The most frequent source of additional information is the adjacent subband which carries the same frequency band due to an overlap of the corresponding analysis filters. ‘Inband’ aliasing within a single channel i of a filter bank does not occur if the real valued analysis filter Hi(Z) satisfies H,(Z)‘Hi(ZW$)
M
0,
V’1= 1, *a*7Si - 1,
(1)
where Si is the subsampling ratio of this channel and ws = eWW is a modulation factor [S]. The frequency domain interpretation of Eq. (1) is that the analysis filter Hi(z) should not overlap with any of its modulated versions Hi(ZWk,). Another interpretation of Eq. (1) is that the channel i will not be affected by aliasing due to the subsampling process if the passband of its analysis filter Hi(z) does not include any of the normalized frequencies 2nk/Si, where k = 1, . . . , Si - 1.
95
M. Harteneck et al. / Signal Processing 64 (1998) 93-101
Therefore, a real-valued filter bank with the same subsampling ratio S for each channel and without ‘inband’ aliasing would require the same spectral zeros in each band, i.e. at the frequencies 2nk/S. However, another requirement for a filter bank is that it yields the perfect or near-perfect reconstruction property, i.e. the output signal Z(k) is approximately the same as the input signal x(k) plus a fixed delay A (g(k) z x(k - A)), and therefore common zeros in all analysis filters Hi(Z) ruled out information is at these cies [13]. frequencies 4 2.2.
subband adaptive
HI(z)
Fig.
F$z)
3.
96
M. Harteneck et al. / Signal Processing 64 (1998) 93-101 20 0 _
,..
_
(i)
20
._
0 @ g
-20
$j
-40
E H
-60 -80 -100
0.2
0
Normal~zd Frequ~~
(2fffs)
0.8
1
Fig. 2. Analysis filters (i) H,(z), (ii) H,(z) and (iii) H,(z) for the structure as shown in Fig. 1.
20
g E f
1
-20
-
-4o-60
-
-eo-
\
-100 -120
Y I 0.2 Normak:d Frequeg
0
0.8 (2f/fs)
1
Fig. 3. Power spectrum of the signal in channel 0 or 2 after subsampling by 2.
2.3. Oversampling complexity
ratio and computational
I
I
0.8 Nomlalz:d Frecf"Ey(2f/fs)
1
Fig. 4. Power spectrum of the signal in channel 1 after subsampling by 3.
(3) and for adaptive algorithms with a computational complexity of O(N’) to
(4)
This type of filter bank is an oversampled filter bank as more data is produced in the subband signal in comparison to the fullband signal. The oversampling ratio (OSR) can be computed as OSR=;i;$,
I, 0.2
To calculate the computational complexity of the proposed structure it is assumed that the calculations necessary to perform the subband decomposition are negligible in comparison to the calculations necessary for the adaptive filters and that the adaptive filter in the ith subband needs a length of (l/S& to represent the same equivalent fullband length 1,. The computational complexity for adaptive algorithms with a computational complexity of O(N) therefore equates to
O$j .s
-120 I0
(2)
where K is the number of channels and St is the subsampling ratio in the ith channel.
Table 1 shows the computational complexities C1 and Cz and the oversampling ratio OSR of the proposed structure for various oversampled filter banks and for a fullband adaptive filter. The filter banks, which are compared, are a 2-3-2 filter bank [6], where two channels are subsampled by 2 and one by 3, which is shown in Fig. 2, a 2-7-2 filter bank, where two channels are subsampled by 2 and one by 7 and a filter bank consisting of nine channels with different subsampling ratios [6].
91
M. Harteneck et al. 1 Signal Processing 64 (1998) 93-101
Table 1 Computational complexity of proposed structure Filter bank
CI
C2
OSR
Fullband
100% 1,
100% 1.”
100%
2-3-2 2-l-2 9 Channel
61%.1, 52% I, 31.1%. 1,
28.7%. 1: 25.3%. 1: 8.4%. 1:
133% 114% 178%
multiple of the subsampling ratios and therefore the approximation used in Eq. (5) to calculate the reconstruction error from only the impulse response of a single impulse sent at time 0 is valid only if the filter bank does not contain ‘inband alias. This however is ensured due to the appropriate choice of the stopbands of the analysis filters. The design algorithm [lo] iteratively minimizes Eq. (5) by minimizing at each iteration 1, a quadratic approximation of Eq. (5), which is given as 21,-l
(UT-,&U,
-
6(i - A))2,
3. Filter bank design To reduce the complexity of the design problem it is assumed that the analysis filters Hi(z) and the synthesis filters F,(z) are related a priori, e.g. FL(z) = Hi(z-‘) or that the analysis filters and synthesis filters are derived from the same prototypes (e.g. cosine modulation [13]). The filter design algorithm is based on an Iterative Least Squares Algorithm [lo], where the performance criterion which is minimized is defined as
where one vector u is substituted by the solution of the previous iteration ul_ 1 and so a quadratic performance criterium is achieved which can be solved easily. After convergence to a stationary point u’, Eqs. (5) and (6) represent the same performance criterion. The iteration is stopped when the change from Us_1 to uI is below a certain threshold.
UT~
+
y
C
(6)
i=O
4. Simulations
21,-l 5 =
=
u:Pu~ + y C
tl
(UTSiU
-
6(i
-
d))‘,
(5)
i=O
where y is a weighting factor, 6( .) is a Kronecker impulse and A is an a priori chosen delay. The first term represents the stopband energy of the analysis filters or prototypes. The matrix P is thereby obtained through the eigenjilter method [9] and u denotes a vector containing all the adjustable coefficients, which could be the filter parameters of the analysis filters or the coefficients of prototype filters. The second term calculates the reconstruction error of an impulse sent into the filter bank at time index 0. The term UTSiU represents the convolution operation of the filter bank to obtain the output at time i. The filter length of the analysis and synthesis filters is denoted by 1, and Si represents the necessary mapping to calculate the ith output of the filter bank [ll]. Note that the filter bank used in Fig. 1 is a cyclic time variant system due to the subsampling process with a periodicity of the least common
Floating point computer simulations were carried out using the proposed structure with least squares (LS) and least mean square (LMS) algorithms in a system identification setup where the unknown systems were IIR and FIR models of acoustic transfer functions. The algorithms, the proposed structure is compared to, are a fullband adaptive filter and a critically sampled two-band structure as proposed in [S] which needs adaptive crossfilters. The filter bank used for the two-band structure was a QMF bank using the 32C filter from [3]. The filter bank used for the proposed structure consists of the three analysis filters shown in Fig. 2 which have 151 taps each and yield ‘inband aliasing levels of - 115 dB, - 102 dB and - 112 dB for the first, second and third channel, respectively. The synthesis filters are the time-reversed analysis filters which gives the filter bank an overall delay of 15 1 taps and a reconstruction error of - 125dB.
M. Harteneck et al. f Signal Processing
98
4.1.
64 (1998) 93-101
4.2. Same equivalent fullband length
Same computational complexity
In this set of simulations, the three algorithms were compared while maintaining the same computational complexity for each structure using LS adaptive algorithms. The unknown system is in this case an IIR model with an impulse response of about 200 taps [2]. The adaptive algorithm used was an RLS algorithm where the fullband adaptive filter had a length of 140 taps, the two-band structure [S] had 100 taps in the main adaptive filters and 40 taps in the cross adaptive filters and the proposed structure had adaptive filters with 130 taps in the channels subsampled by 2 and 87 taps in the channel subsampled by 3. The driving input signal x(k) was white pseudo noise. It can be seen in Fig. 5 that the proposed structure (d) outperforms the fullband (b) and the twoband structure (c). This however was expected as the equivalent fullband length 1, is 140 taps for the fullband structure, 200 taps for the twoband structure and 260 taps for the proposed structure and therefore a much better model could be found. Note the additional spectral peak and the degraded performance in the error signal of the two-band adaptive filter (c) which is caused by the aliasing produced by the filter bank.
In Fig. 6 the same simulation was performed, however the equivalent fullband length was kept the same for all structures. The fullband adaptive filter had 200 taps, the two-band structure 100 taps in its main adaptive filters and 40 taps in its cross adaptive filters and the proposed structure had adaptive filters with 100 taps in channel 0 and channel 2 and 66 taps in channel 1. It can be seen that the fullband structure and the proposed structure give the same performance whereas the two-band structure performs worse which is due to the aliasing encountered in the subbands. Here the near-perfect reconstruction property of the filter bank is quite important as it would show a higher minimum mean squared error in the simulation.
4.3. Time varying system echo-return-loss-enhancement Fig. 7 shows (ERLE) plots against time. ERLE is defined as the ratio of the power of the echo signal over the power of the residual signal and is a measure of the echo suppression of the algorithm. The ERLE is estimated over a time window of 100 samples. After 3333 samples the unknown system is
30 20 10 0 -10 -20 -30 -40 -50 -60 -70 0
0.8
0.2 Normally
Req”$&
1
@f/f@
Fig. 5. Residual error for same computational complexity: (a) desired signal d(k), (b) fullband QR algorithm, (c) two-band structure with crossterms, (d) proposed structure.
Fig. 6. Residual error for same equivalent fullband length: (a) desired signal d(k), (b) fullband QR algorithm, (c) two-band structure with crossterms, (d) proposed structure.
M. Harteneck et al. 1 Signal Processing 64 (1998) 93-101
99
Table 2 Noisy system identification results Echo level
232
25 B s
20
B !+
15
14.12dB 0.1 dB - 5.8 dB - 9.4dB - 11.9dB
10
-5
’
Fullband
SNR
ERLE
SNR
13.3dB 15.9dB 16.1dB 16.1dB 16.1dB
24.7 dB 16.1dB lO.OdB 6.7 dB 4.2dB
3.9 dB 9.9 dB 9.9 dB 9.9dB 9.9dB
ERLE 23.0 dB 10.0dB 4.0 dB 0.5 dB - 1.9dB
I 51
10002000300040005000600070005000900010000 Sample Fig. 7. Simulation results with system change after 3333 samples: (a) proposed structure, (b) fullband QR algorithm, (c) twoband structure with crossterms.
1
Q -10 -
-
-15
changed abruptly to a different IIR model which yields an impulse response of about 150 taps. The parameters for the adaptive structures were set such that the same equivalent fullband length I, of 100 taps was achieved and that the same memory lengths for the LS estimation were obtained. It can be seen from Fig. 7 that the fullband (b) and the proposed structure (a) behave the same in terms of achievable ERLE and convergence speed. The only difference is the delay caused by the filter bank. The two-band structure again suffers under the influence of the ‘inband’ aliasing and therefore gives a worse ERLE level when adapted. The convergence speed however is unaffected from this effect.
4.4. Simulations with observation noise The next set of simulations was done to investigate the robustness towards noise and the achievable ERLE and SNR when compared to a fullband architecture. Table 2 shows the results of this set of simulations for different levels of echo where echo is defined as the ratio of power in the output from the unknown system to observation noise power. For the calculation of the SNR and the ERLE it was assumed that the observation noise is the signal of
-20 -
-25
-30 -35 -
40’
0
I 0.2
0.4 0.6 NormallzecJFrequency
0.8
1
Fig. 8. Power spectral density of input signal.
interest as it would be in an acoustic echo cancelation setup. It can be seen that generally the proposed structure outperforms a fullband implementation in terms of SNR and ERLE. The achieved improvement in ERLE is about 7dB and in SNR is about 6dB. The improvements are due to the reduced order of the adaptive filters and to the small reconstruction error of the used three channel filter bank.
4.5. Coloured input noise Fig. 9 shows the ERLE plots for an abrupt system change with a colored noise as driving input x(k) for (a) a fullband NLMS algorithm with 1000 taps and (b) the subband adaptive structure using NLMS algorithms in the subbands and with 500,
M. Harteneck et al. / Signal Processing 64 (1998) 93-101
100
a fullband implementation. In a noisy environment however, it outperforms a fullband implementation which is due to the lower filter order of the adaptive filters. If the driving input is colored, then the proposed structure shows a faster convergence then a fullband adaptive structure.
40 30 g
20
2
10
W
_~__....____.__..~._~.. ..__ .
0
Acknowledgements
-10
-20
1 ’ 20000
I 30000
40000a~Il
60000
70000
60000
Fig. 9. Echo-return-loss-enhancement for colored noise: (a) three-channel subband adaptive filter, (b) fullband adaptive filter.
334 and 500 taps respectively to obtain the same equivalent fullband length. The two different unknown systems were FIR models of room responses calculated according to [l]. The power spectral density of x(k) is shown in Fig. 8. The system change occurred at sample 20000 when both structures were at a steady state. It can be seen that the subband structure recovers its new steady state after about 20000 samples whereas the fullband adaptive filter has not recovered completely after 60 000 samples.
5. Conclusions In this paper we have described the development of a subband adaptive filter with oversampled realvalued filter banks. The filter banks have been designed in such a way that the ‘inband’ aliasing has been reduced significantly such that the use of adaptive crossfilters was not necessary any more. It has been shown that the proposed structure yields a lower computational complexity than a fullband implementation or traditional critically subsampled filter banks which is due to the fact that lower order adaptive filters can be used at a lower sampling frequency. In a noise-free environment, the proposed structure matches the minimum mean squared error of
Mr. Harteneck would like to acknowledge an ERASMUS grant and a grant from the University of Strathclyde which supported this work as part of an exchange visit to the ETSI Telecomunicacion, Universidad Politecnica de Madrid from October 1996 until March 1997. The authors also acknowledge the contributions of J.M. McWhirter and I.K. Proudler, DERA Malvern, for their input and future support for this work.
References Cl] J.B. Allen, D.A. Berkley, Image method for efficiently simulating small-room acoustics, J. Acoust. Sot. Amer. 65 (4) (1979) 943-950. [2] J. Chao, S. Kawabe, S. Tsujii, A new IIR adaptive echo canceller: GIVE, IEEE Trans. on Selected Areas in Commun. 12 (9) (1994) 153&1539. [3] R.E. Crochiere, L.R. Rabiner, Multirate Digital Signal Processing, Signal Processing Series, Prentice Hall, Englewood Cliffs, NJ, 1983. [4] P.L. de Leon II, D.M. Etter, Experimental results with increased bandwidth analysis filters in oversampled, subband acoustic echo cancelers, IEEE Signal Processing Letters 2 (1) (1995) l-3. [S] A. Gilloire, M. Vetterli, Adaptive filtering in subbands with critical sampling: Analysis, experiments, and application to acoustic echo cancellation, IEEE Trans. Signal Process. 40 (8) (1992) 1862-1875. [6] M. Harteneck, J.M. Paez-Borrallo, R.W. Stewart, A filterbank design for oversampled filter banks without aliasing in the subbands, 2nd UK Symposium on Applications of Time-Frequency and Time-Scale Methods (TFTS), Warwick, England, 27-29 August 1997, pp. 161-164. [7] M. Harteneck, J.M. Paez-Borrallo, R.W. Stewart, Filterbank design for oversampled filter banks without aliasing in the subbands, IEE Electronics Letters 33 (18) (1997) 1538-1539. [S] P.A. Naylor, J.E. Hart, Subband acoustic echo control using non-critical frequency sampling, in: Proc. of EUSIPCO-96, I, September 1996, pp. 37-40.
M. Harteneck
et al. / Signal Processing
[9] T.Q. Nguyen, Digital filter bank design quadratic-con-
strained formulation, IEEE Trans. Signal Process. 43 (9) (1995) 2103-2108. [lo] M. Rossi, J.-Y. Zhang, W. Steenaart, Iterative least squares design of perfect reconstruction QMF banks, in: Proc. of CCECE, May 1996. [1 l] G. Strang, T. Nguyen, Wavelets and Filter Banks, Wellesley-Cambridge Press, 1996.
64 (1998) 93-101
101
[12] 0. Tanrikulu, B. Baykal, A.G Constantinides, J.A. Chambers, Residual signal in subband acoustic echo cancellers, in: Proc. of EUSIPCO 96, I, September 1996, pp. 21-24. [13] P.P. Vaidyanathan, Multirate Systems and Filter Banks, Prentice Hall, Englewood Cliffs, NJ, 1993. [14] Y. Yanada, H. Ochi, K. Kiya, A subband adaptive filter allowing maximally decimation, IEEE Journal on Selected Areas in Commun. 12 (9) (1994) lS48--1552.
PROCESSING ELSEVIER
Signal Processing 64 (1998)93-101
An oversampled subband adaptive filter without cross adaptive filters M. Harteneckav*, J.M. PBez-Borrallob, R.W. Stewart” aSignal Processing Division, Department. of Electrical and Electronic Engineering, University of Strathclyde, Glasgow, Gl lxW, Scotland, b ETSI Telecomunicacibn, Universidad Politkica
UK de Madrid, Ciudad, Universitaria s/n, 28040 Madrid, Spain
Received 9 October 1997
Abstract In this paper we propose a subband adaptive filter architecture that does not require cross-adaptive filters between subbands to compensate for the ‘inband’ alias caused by the subsampling process. The architecture uses real-valued oversampled filter banks with different subsampling ratios in each channel which reduce the ‘inband’ alias significantly. In the paper we describe the design algorithm for the filter banks and show simulation results in a system identification set-up to demonstrate the performance. 0 1998 Elsevier Science B.V. All rights reserved. Zusammenfassung Dieser Artikel stellt eine Frequenzteilband Filterarchitektur fiir adaptive Filter vor, die keine adaptiven Filter zwischen verschiedenen Teilblndern benlitigt, urn ‘Zwischenband-Aliasing’ zu unterdriicken. Die Architektur verwendet reellwertige, iiberabgetastete Filterblnke mit verschiedenen Unterabtastfaktoren in jedem Teilband, welche das ‘Zwischenband-Aliasing’ signifikant reduzieren. In diesem Artikel beschreiben wir den Filterentwurfsalgorithmus und prlsentieren Simulationsergebnisse in einer Systemidentifikationsumgebung, urn die Leistungsfihigkeit unseres Filterentwurfs zu verdeutlichen. 0 1998 Elsevier Science B.V. All rights reserved.
Nous proposons dans cet article une architecture de filtrage adaptatif en sous-bandes qui ne requiert pas de filtres inter-adaptatifs entre les sous-bandes pour compenser le repliement ‘dans la bande’ caustr par le processus de sous&hantillonnage. Cette architecture utilise des banes de filtres sur-ichantillonnts g valeurs rCelles avec des rapports de sous-kchantillonnage diffirents dans chaque canal qui riduisent significativement le repliement ‘dans la bande’. Nous dkcrivons dans cet article l’algorithme de conception des banes de filtres et montrons des r&sultats de simulation dans un contexte d’identification de systkmes pour mettre en Cvidence les performances. 0 1998 Elsevier Science B.V. All rights reserved. Keywords:
Subband adaptive filter; Oversampled filter bank; Filter bank design
*Corresponding author. Tel.: +44 141 548 2808; fax: +44 141 552 2487; e-mail: [email protected]. 0165-1684/98/$19.000 1998 Elsevier Science B.V. All rights reserved. PffSO165-1684(97)00179-5
94
M. Harteneck et al. 1 Signal Processing 64 (1998) 93-101
1. Introduction
Problem description
As shown in [S] subband adaptive filters require cross-band adaptive filters if the subband signals are contaminated by ‘inband’ aliasing and therefore a second source of information about the contaminated part of the subband signal is necessary to make the comparison between the subband input signal and the subband desired signal. The most frequent source of additional information is the adjacent subband which carries the same frequency band due to an overlap of the corresponding analysis filters. ‘Inband’ aliasing within a single channel i of a filter bank does not occur if the real valued analysis filter Hi(Z) satisfies H,(Z)‘Hi(ZW$)
M
0,
V’1= 1, *a*7Si - 1,
(1)
where Si is the subsampling ratio of this channel and ws = eWW is a modulation factor [S]. The frequency domain interpretation of Eq. (1) is that the analysis filter Hi(z) should not overlap with any of its modulated versions Hi(ZWk,). Another interpretation of Eq. (1) is that the channel i will not be affected by aliasing due to the subsampling process if the passband of its analysis filter Hi(z) does not include any of the normalized frequencies 2nk/Si, where k = 1, . . . , Si - 1.
95
M. Harteneck et al. / Signal Processing 64 (1998) 93-101
Therefore, a real-valued filter bank with the same subsampling ratio S for each channel and without ‘inband’ aliasing would require the same spectral zeros in each band, i.e. at the frequencies 2nk/S. However, another requirement for a filter bank is that it yields the perfect or near-perfect reconstruction property, i.e. the output signal Z(k) is approximately the same as the input signal x(k) plus a fixed delay A (g(k) z x(k - A)), and therefore common zeros in all analysis filters Hi(Z) ruled out information is at these cies [13]. frequencies 4 2.2.
subband adaptive
HI(z)
Fig.
F$z)
3.
96
M. Harteneck et al. / Signal Processing 64 (1998) 93-101 20 0 _
,..
_
(i)
20
._
0 @ g
-20
$j
-40
E H
-60 -80 -100
0.2
0
Normal~zd Frequ~~
(2fffs)
0.8
1
Fig. 2. Analysis filters (i) H,(z), (ii) H,(z) and (iii) H,(z) for the structure as shown in Fig. 1.
20
g E f
1
-20
-
-4o-60
-
-eo-
\
-100 -120
Y I 0.2 Normak:d Frequeg
0
0.8 (2f/fs)
1
Fig. 3. Power spectrum of the signal in channel 0 or 2 after subsampling by 2.
2.3. Oversampling complexity
ratio and computational
I
I
0.8 Nomlalz:d Frecf"Ey(2f/fs)
1
Fig. 4. Power spectrum of the signal in channel 1 after subsampling by 3.
(3) and for adaptive algorithms with a computational complexity of O(N’) to
(4)
This type of filter bank is an oversampled filter bank as more data is produced in the subband signal in comparison to the fullband signal. The oversampling ratio (OSR) can be computed as OSR=;i;$,
I, 0.2
To calculate the computational complexity of the proposed structure it is assumed that the calculations necessary to perform the subband decomposition are negligible in comparison to the calculations necessary for the adaptive filters and that the adaptive filter in the ith subband needs a length of (l/S& to represent the same equivalent fullband length 1,. The computational complexity for adaptive algorithms with a computational complexity of O(N) therefore equates to
O$j .s
-120 I0
(2)
where K is the number of channels and St is the subsampling ratio in the ith channel.
Table 1 shows the computational complexities C1 and Cz and the oversampling ratio OSR of the proposed structure for various oversampled filter banks and for a fullband adaptive filter. The filter banks, which are compared, are a 2-3-2 filter bank [6], where two channels are subsampled by 2 and one by 3, which is shown in Fig. 2, a 2-7-2 filter bank, where two channels are subsampled by 2 and one by 7 and a filter bank consisting of nine channels with different subsampling ratios [6].
91
M. Harteneck et al. 1 Signal Processing 64 (1998) 93-101
Table 1 Computational complexity of proposed structure Filter bank
CI
C2
OSR
Fullband
100% 1,
100% 1.”
100%
2-3-2 2-l-2 9 Channel
61%.1, 52% I, 31.1%. 1,
28.7%. 1: 25.3%. 1: 8.4%. 1:
133% 114% 178%
multiple of the subsampling ratios and therefore the approximation used in Eq. (5) to calculate the reconstruction error from only the impulse response of a single impulse sent at time 0 is valid only if the filter bank does not contain ‘inband alias. This however is ensured due to the appropriate choice of the stopbands of the analysis filters. The design algorithm [lo] iteratively minimizes Eq. (5) by minimizing at each iteration 1, a quadratic approximation of Eq. (5), which is given as 21,-l
(UT-,&U,
-
6(i - A))2,
3. Filter bank design To reduce the complexity of the design problem it is assumed that the analysis filters Hi(z) and the synthesis filters F,(z) are related a priori, e.g. FL(z) = Hi(z-‘) or that the analysis filters and synthesis filters are derived from the same prototypes (e.g. cosine modulation [13]). The filter design algorithm is based on an Iterative Least Squares Algorithm [lo], where the performance criterion which is minimized is defined as
where one vector u is substituted by the solution of the previous iteration ul_ 1 and so a quadratic performance criterium is achieved which can be solved easily. After convergence to a stationary point u’, Eqs. (5) and (6) represent the same performance criterion. The iteration is stopped when the change from Us_1 to uI is below a certain threshold.
UT~
+
y
C
(6)
i=O
4. Simulations
21,-l 5 =
=
u:Pu~ + y C
tl
(UTSiU
-
6(i
-
d))‘,
(5)
i=O
where y is a weighting factor, 6( .) is a Kronecker impulse and A is an a priori chosen delay. The first term represents the stopband energy of the analysis filters or prototypes. The matrix P is thereby obtained through the eigenjilter method [9] and u denotes a vector containing all the adjustable coefficients, which could be the filter parameters of the analysis filters or the coefficients of prototype filters. The second term calculates the reconstruction error of an impulse sent into the filter bank at time index 0. The term UTSiU represents the convolution operation of the filter bank to obtain the output at time i. The filter length of the analysis and synthesis filters is denoted by 1, and Si represents the necessary mapping to calculate the ith output of the filter bank [ll]. Note that the filter bank used in Fig. 1 is a cyclic time variant system due to the subsampling process with a periodicity of the least common
Floating point computer simulations were carried out using the proposed structure with least squares (LS) and least mean square (LMS) algorithms in a system identification setup where the unknown systems were IIR and FIR models of acoustic transfer functions. The algorithms, the proposed structure is compared to, are a fullband adaptive filter and a critically sampled two-band structure as proposed in [S] which needs adaptive crossfilters. The filter bank used for the two-band structure was a QMF bank using the 32C filter from [3]. The filter bank used for the proposed structure consists of the three analysis filters shown in Fig. 2 which have 151 taps each and yield ‘inband aliasing levels of - 115 dB, - 102 dB and - 112 dB for the first, second and third channel, respectively. The synthesis filters are the time-reversed analysis filters which gives the filter bank an overall delay of 15 1 taps and a reconstruction error of - 125dB.
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4.1.
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4.2. Same equivalent fullband length
Same computational complexity
In this set of simulations, the three algorithms were compared while maintaining the same computational complexity for each structure using LS adaptive algorithms. The unknown system is in this case an IIR model with an impulse response of about 200 taps [2]. The adaptive algorithm used was an RLS algorithm where the fullband adaptive filter had a length of 140 taps, the two-band structure [S] had 100 taps in the main adaptive filters and 40 taps in the cross adaptive filters and the proposed structure had adaptive filters with 130 taps in the channels subsampled by 2 and 87 taps in the channel subsampled by 3. The driving input signal x(k) was white pseudo noise. It can be seen in Fig. 5 that the proposed structure (d) outperforms the fullband (b) and the twoband structure (c). This however was expected as the equivalent fullband length 1, is 140 taps for the fullband structure, 200 taps for the twoband structure and 260 taps for the proposed structure and therefore a much better model could be found. Note the additional spectral peak and the degraded performance in the error signal of the two-band adaptive filter (c) which is caused by the aliasing produced by the filter bank.
In Fig. 6 the same simulation was performed, however the equivalent fullband length was kept the same for all structures. The fullband adaptive filter had 200 taps, the two-band structure 100 taps in its main adaptive filters and 40 taps in its cross adaptive filters and the proposed structure had adaptive filters with 100 taps in channel 0 and channel 2 and 66 taps in channel 1. It can be seen that the fullband structure and the proposed structure give the same performance whereas the two-band structure performs worse which is due to the aliasing encountered in the subbands. Here the near-perfect reconstruction property of the filter bank is quite important as it would show a higher minimum mean squared error in the simulation.
4.3. Time varying system echo-return-loss-enhancement Fig. 7 shows (ERLE) plots against time. ERLE is defined as the ratio of the power of the echo signal over the power of the residual signal and is a measure of the echo suppression of the algorithm. The ERLE is estimated over a time window of 100 samples. After 3333 samples the unknown system is
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Fig. 5. Residual error for same computational complexity: (a) desired signal d(k), (b) fullband QR algorithm, (c) two-band structure with crossterms, (d) proposed structure.
Fig. 6. Residual error for same equivalent fullband length: (a) desired signal d(k), (b) fullband QR algorithm, (c) two-band structure with crossterms, (d) proposed structure.
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Table 2 Noisy system identification results Echo level
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SNR
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SNR
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ERLE 23.0 dB 10.0dB 4.0 dB 0.5 dB - 1.9dB
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10002000300040005000600070005000900010000 Sample Fig. 7. Simulation results with system change after 3333 samples: (a) proposed structure, (b) fullband QR algorithm, (c) twoband structure with crossterms.
1
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-
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changed abruptly to a different IIR model which yields an impulse response of about 150 taps. The parameters for the adaptive structures were set such that the same equivalent fullband length I, of 100 taps was achieved and that the same memory lengths for the LS estimation were obtained. It can be seen from Fig. 7 that the fullband (b) and the proposed structure (a) behave the same in terms of achievable ERLE and convergence speed. The only difference is the delay caused by the filter bank. The two-band structure again suffers under the influence of the ‘inband’ aliasing and therefore gives a worse ERLE level when adapted. The convergence speed however is unaffected from this effect.
4.4. Simulations with observation noise The next set of simulations was done to investigate the robustness towards noise and the achievable ERLE and SNR when compared to a fullband architecture. Table 2 shows the results of this set of simulations for different levels of echo where echo is defined as the ratio of power in the output from the unknown system to observation noise power. For the calculation of the SNR and the ERLE it was assumed that the observation noise is the signal of
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Fig. 8. Power spectral density of input signal.
interest as it would be in an acoustic echo cancelation setup. It can be seen that generally the proposed structure outperforms a fullband implementation in terms of SNR and ERLE. The achieved improvement in ERLE is about 7dB and in SNR is about 6dB. The improvements are due to the reduced order of the adaptive filters and to the small reconstruction error of the used three channel filter bank.
4.5. Coloured input noise Fig. 9 shows the ERLE plots for an abrupt system change with a colored noise as driving input x(k) for (a) a fullband NLMS algorithm with 1000 taps and (b) the subband adaptive structure using NLMS algorithms in the subbands and with 500,
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a fullband implementation. In a noisy environment however, it outperforms a fullband implementation which is due to the lower filter order of the adaptive filters. If the driving input is colored, then the proposed structure shows a faster convergence then a fullband adaptive structure.
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-10
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Fig. 9. Echo-return-loss-enhancement for colored noise: (a) three-channel subband adaptive filter, (b) fullband adaptive filter.
334 and 500 taps respectively to obtain the same equivalent fullband length. The two different unknown systems were FIR models of room responses calculated according to [l]. The power spectral density of x(k) is shown in Fig. 8. The system change occurred at sample 20000 when both structures were at a steady state. It can be seen that the subband structure recovers its new steady state after about 20000 samples whereas the fullband adaptive filter has not recovered completely after 60 000 samples.
5. Conclusions In this paper we have described the development of a subband adaptive filter with oversampled realvalued filter banks. The filter banks have been designed in such a way that the ‘inband’ aliasing has been reduced significantly such that the use of adaptive crossfilters was not necessary any more. It has been shown that the proposed structure yields a lower computational complexity than a fullband implementation or traditional critically subsampled filter banks which is due to the fact that lower order adaptive filters can be used at a lower sampling frequency. In a noise-free environment, the proposed structure matches the minimum mean squared error of
Mr. Harteneck would like to acknowledge an ERASMUS grant and a grant from the University of Strathclyde which supported this work as part of an exchange visit to the ETSI Telecomunicacion, Universidad Politecnica de Madrid from October 1996 until March 1997. The authors also acknowledge the contributions of J.M. McWhirter and I.K. Proudler, DERA Malvern, for their input and future support for this work.
References Cl] J.B. Allen, D.A. Berkley, Image method for efficiently simulating small-room acoustics, J. Acoust. Sot. Amer. 65 (4) (1979) 943-950. [2] J. Chao, S. Kawabe, S. Tsujii, A new IIR adaptive echo canceller: GIVE, IEEE Trans. on Selected Areas in Commun. 12 (9) (1994) 153&1539. [3] R.E. Crochiere, L.R. Rabiner, Multirate Digital Signal Processing, Signal Processing Series, Prentice Hall, Englewood Cliffs, NJ, 1983. [4] P.L. de Leon II, D.M. Etter, Experimental results with increased bandwidth analysis filters in oversampled, subband acoustic echo cancelers, IEEE Signal Processing Letters 2 (1) (1995) l-3. [S] A. Gilloire, M. Vetterli, Adaptive filtering in subbands with critical sampling: Analysis, experiments, and application to acoustic echo cancellation, IEEE Trans. Signal Process. 40 (8) (1992) 1862-1875. [6] M. Harteneck, J.M. Paez-Borrallo, R.W. Stewart, A filterbank design for oversampled filter banks without aliasing in the subbands, 2nd UK Symposium on Applications of Time-Frequency and Time-Scale Methods (TFTS), Warwick, England, 27-29 August 1997, pp. 161-164. [7] M. Harteneck, J.M. Paez-Borrallo, R.W. Stewart, Filterbank design for oversampled filter banks without aliasing in the subbands, IEE Electronics Letters 33 (18) (1997) 1538-1539. [S] P.A. Naylor, J.E. Hart, Subband acoustic echo control using non-critical frequency sampling, in: Proc. of EUSIPCO-96, I, September 1996, pp. 37-40.
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strained formulation, IEEE Trans. Signal Process. 43 (9) (1995) 2103-2108. [lo] M. Rossi, J.-Y. Zhang, W. Steenaart, Iterative least squares design of perfect reconstruction QMF banks, in: Proc. of CCECE, May 1996. [1 l] G. Strang, T. Nguyen, Wavelets and Filter Banks, Wellesley-Cambridge Press, 1996.
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[12] 0. Tanrikulu, B. Baykal, A.G Constantinides, J.A. Chambers, Residual signal in subband acoustic echo cancellers, in: Proc. of EUSIPCO 96, I, September 1996, pp. 21-24. [13] P.P. Vaidyanathan, Multirate Systems and Filter Banks, Prentice Hall, Englewood Cliffs, NJ, 1993. [14] Y. Yanada, H. Ochi, K. Kiya, A subband adaptive filter allowing maximally decimation, IEEE Journal on Selected Areas in Commun. 12 (9) (1994) lS48--1552.