Test parameters
1. Introduction
Adaptive filters are often involved in many applications, such as system identification, channel estimation, echo and noise cancellation in telecommunication systems. In this context, the Least Mean Square (LMS) algorithm is used to adapt a Finite Impulse Response (FIR) filter with a relatively low computation complexity and good performance. However, this solution suffers from significantly degraded performance with colored interfering signals, due to the large eigenvalue spread of the autocorrelation matrix of the input signal (Vaseghi, 2008). Furthermore, as the length of the filter is increased, the convergence rate of the algorithm decreases, and the computational requirements increase. This can be a problem in acoustic applications such as noise cancellation, which demand long adaptive filters to model the noise path. These issues are particularly important in hands free communications, where processing power must be kept as low as possible (Johnson et al., 2004). Several solutions have been proposed in literature to overcome or at least reduce these problems. A possible solution to reduce the complexity problem has been to use adaptive Infinite Impulse Response (IIR) filters, such that an effectively long impulse response can be achieved with relatively few filter coefficients (Martinez & Nakano 2008). The complexity advantages of adaptive IIR filters are well known. However, adaptive IIR filters have the well known problems of instability, local minima and phase distortion and they are not widely welcomed. An alternative approach to reduce the computational complexity of long adaptive FIR filters is to incorporate block updating strategies and frequency domain adaptive filtering (Narasimha 2007; Wasfy & Ranganathan, 2008). These techniques reduce the computational complexity, because the filter output and the adaptive weights are computed only after a large block of data has been accumulated. However, the application of such approaches introduces degradation in the performance, including a substantial signal path delay corresponding to one block length, as well as a reduction in the stable range of the algorithm step size. Therefore for nonstationary signals, the tracking performance of the block algorithms generally becomes worse (Lin et al., 2008).
As far as speed of convergence is concerned, it has been suggested to use the Recursive Least Square (RLS) algorithm to speed up the adaptive process (Hoge et al., 2008).The convergence rate of the RLS algorithm is independent of the eigenvalue spread. Unfortunately, the drawbacks that are associated with RLS algorithm including its O(N^{2}) computational requirements, which are still too high for many applications, where high speed is required, or when a large number of inexpensive units must be built. The Affine Projection Algorithm (APA) (Diniz, 2008; Choi & Bae, 2007) shows a better convergence behavior, but the computational complexity increases with the factor P in relation to LMS, where P denotes the order of the APA.
As a result, adaptive filtering using subband processing becomes an attractive option for many adaptive systems. Subband adaptive filtering belongs to two fields of digital signal processing, namely, adaptive filtering and multirate signal processing. This approach uses filter banks to split the input broadband signal into a number of frequency bands, each serving as an independent input to an adaptive filter. The subband decomposition is aimed to reduce the update rate, and the length of the adaptive filters, hopefully, resulting in a much lower computational complexity. Furthermore, subband signals are usually downsampled in a multirate system. This leads to a whitening of the input signals and therefore an improved convergence behavior of the adaptive filter system is expected. The objectives of this chapter are: to develop subband adaptive structures which can improve the performance of the conventional adaptive noise cancellation schemes, to investigate the application of subband adaptive filtering to the problem of background noise cancellation from speech signals, and to offer a design with fast convergence, low computational requirement, and acceptable delay. The chapter is organized as follows. In addition to this introduction section, section 2 describes the use of Quadrature Mirror Filter (QMF) banks in adaptive noise cancellation. The effect of aliasing is analyzed and the performance of the noise canceller is examined under various noise environments. To overcome problems incorporated with QMF subband noise canceller system, an improved version is presented in section 3. The system is based on using twofold oversampled filter banks to reduce aliasing distortion, while a moderate order prototype filter is optimized for minimum amplitude distortion. Section 4 offers a solution with reduced computational complexity. The new scheme is based on using polyphase allpass IIR filter banks at the analysis stage, while the synthesis filter bank is optimized such that an inherent phase correction is made at the output of the noise canceller. Finally, section 5 concludes the chapter.
2. Adaptive noise cancellation using QMF banks
In this section, a subband adaptive noise canceller system is presented. The system is based on using critically sampled QMF banks in the analysis and synthesis stages. A suband version of the LMS algorithm is used to control a FIR filter in the individual branches so as to reduce the noise in the input noisy signal.
2.1. The QMF bank
The design of
The downsampling operation has a modulation effect on signals and filters, therefore the input to the system is expressed as follows;
where.^{ T } is a transpose operation. Similarly, the analysis filter bank is expressed as,
The output of the analysis stage is expressed as,
The total inputoutput relationship is expressed as,
The right hand side term of equation (4) is the aliasing term. The presence of aliasing causes a frequency shift of
By direct substitution into Equation (4), we see that the aliasing terms go to zero, leaving
In frequency domain, replacing
Therefore, the objective is to determine
2.2. Efficient implementation of the QMF bank
An efficient implementation of the preceding twochannel QMF bank is obtained using polyphase decomposition and the noble identities (Milic, 2009). Thus, the analysis and synthesis filter banks can be redrawn as in Fig.2. The downsamplers are now to the left of the polyphase components of
2.3. Distortion elimination in QMF banks
Let the inputoutput transfer function be
which represents the distortion caused by the QMF bank.
where
Systems which are alias free and satisfy (12) are called perfect reconstruction (PR) systems. For any pair of analysis filter, the choice of synthesis filters according to (7) and (8) eliminates aliasing distortion, the distortion can be expressed as,
The transfer function of the system in (14) can be expressed in terms of polyphase components as,
Since
For our purpose of adaptive noise cancellation, frequency responses are required to be more selective than (16). So, under the constraint of (13), perfect reconstruction is not possible. However, it is possible to minimize amplitude distortion by optimization procedures. The coefficients of
can be minimized by optimizing the coefficients of
2.4. Adaptive noise cancellation using QMF banks
A schematic of the twoband noise canceller structure is shown at Fig.3, this is a two sensor scheme, it consists of three sections: analysis which contains analysis filters
The system output
In the adaptive section of the twoband noise canceller, a modified version of the LMS algorithm for subband adaptation is used as follows;
where
for
where
2.5. The Mband case
The twoband noise canceller can be extended so as to divide the input broadband signal into
where c is a constant,
The reconstruction delay of a paraunitary filter bank is fixed by the prototype filter order,
The matrix in (24) contains the filters and their modulated versions (by the
2.6. Results of the subband noise canceller using QMF banks
2.6.1. Filter bank setting and distortion calculation
The analysis filter banks are generated by a cosine modulation function. A single prototype filter is used to produce the subfilters in the critically sampled case. Aliasing error is the parameter that most affect adaptive filtering process in subbands, and the residual noise at the system’s output can be very high if aliasing is not properly controlled. Fig.5 gives a describing picture about aliasing distortion. In this figure, settings of prototype filter order are used for each case to investigate the effect of aliasing on filter banks. It is clear from Fig.5, that aliasing can be severe for low order prototype filters. Furthermore, as the number of subbands is increased, aliasing insertion is also increased. However, for low number of subbands e.g. 2 subbabds, low order filters can be afforded with success equivalent to high order ones.
2.6.2. Noise cancellation tests
Initially, the twoband noise canceller model is tested using a variable frequency sine wave contaminated with zero mean, unit variance white Gaussian noise. This noise is propagating through a noise path
Parameter  Value 
Noise path length  92 
Adaptive filter length  46 
Step size 
0.02 
Sampling frequency  8kHz 
Input (first test)  Variable frequency sinusoid 
Noise (first test)  Gaussian white noise with zero mean and unit variance 
Input (second test )  Speech of a woman 
Noise ( second test)  Machinery noise 
In a second experiment, a speech of a woman, sampled at 8 kHz, is used for testing. Machinery noise as an environmental noise is used to corrupt the speech signal. Convergence behavior using mean square error plots are used as a measure of performance. These plots are smoothed with 200 point moving average filter and displayed as shown in Fig.6 for the case of variable frequency sine wave corrupted by white Gaussian noise, and in Fig.7 for the case speech input corrupted by machinery noise.
2.7. Discussion
The use of the twoband QMF scheme, with near perfect reconstruction filter bank, should lead to approximately zero steady state error at the output of the noise cancellation scheme; this property has been experimentally verified as shown in Fig.6. The fullband adaptive filter performance as well as for a fourband critically sampled scheme are shown on the same graph for sake of comparison. The steady state error of the scheme with twoband QMF banks is very close to the error of the fullband filter, this demonstrate the perfect identification property. Those results show that the adaptive filtering process in subbands based on the feedback of the subbands errors is able to model perfectly a system. The subband plots exhibit faster initial parts; however, after the error has decayed by about 15 dB (4band) and 30 dB (2band), the convergence of the fourband scheme slows down dramatically. The errors go down to asymptotic values of about 30 dB (2band) and 20 dB (4band). The steady state error of the fourband system is well above the one of the fullband adaptive filter due to high level of aliasing inserted in the system. The improvement of the transient behavior of the fourband scheme was observed only at the start of convergence. The aliased components in the output error cannot be cancelled, unless cross adaptive filters are used to compensate for the overlapping regions between adjacent filters, this would lead to an even slower convergence and an increase in computational complexity of the system. Overall, the convergence performances of the twoband scheme are significantly better than that of the fourband scheme: in particular, the steady state error is much smaller. However, the convergence speed is not improved as such, in comparison with the fullband scheme. The overall convergence speed of the twoband scheme was not found significantly better than the one of the fullband adaptive filter. Nevertheless, such schemes would have the practical advantage of reduced computational complexity in comparison with the fullband adaptive filter.
3. Adaptive noise cancellation using optimized oversampled filter banks
Aliasing insertion in the critically sampled systems plays a major role in the performance degradation of subband adaptive filters. Filter banks can be designed aliasfree and perfectly reconstructed when certain conditions are met by the analysis and synthesis filters. However, any filtering operation in the subbands may cause a possible phase and amplitude change and thereby altering the perfect reconstruction property. In a recent study, Kim et al. (2008) have proposed a critically sampled structure to reduce aliasing effect. The interband aliasing in each subband is obtained by increasing the bandwidth of a linearphase FIR analysis filter, and then subtracted from the subband signal. This aliasing reduction technique introduces spectral dips in the subband signals. Therefore, extra filtering operation is required to reduce these dips.
In this section, an optimized 2fold oversampled
3.1. Problem formulation
The arrangement in Fig.3 is redrawn for the general case of
Distortions due the insertion of the analysis/synthesis filter bank are expressed as follows,
A critical sampling creates severe aliasing effect due to the transition region of the prototype filter. This has been discussed in section 2. When the downsampling factor decreases, the aliasing effect is gradually reduced. Optimizing the prototype filter by minimizing both
The objective is to find prototype filters
3.2. Prototype filter optimization
Recalling that, the objective here is to find prototype filter
For a lowpass prototype filter whose stopband stretches from
For
where
where vector
The optimum coefficients of the FIR filter are those that minimize the energyfunction
3.3. The adaptive process
The filter weight updating is performed using a subband version of the LMS algorithm that is expressed by the following;
The filter weights in each branch are adjusted using the subband error signal belonging to the same branch. To prevent the adaptive filter from oscillating or being too slow, the step size of the adaptation algorithm is made inversely proportional to the power in the subband signals such that
where
3.4. Polyphse implementation of the subband noise canceller
The implementation of DFT modulated filter banks can be done using polyphase decomposition of a single prototype filter and a Fast Fourier Transform (FFT). A DFT modulated analysis filter bank with subsequent
3.5. Results of the optimized 2fold oversampled noise canceller
The noise path used in these tests is an approximation of a small room impulse response modeled by a FIR processor of 512 taps. To measure the convergence behavior of the oversampled subband noise canceller, a variable frequency sinusoid was corrupted with white Gaussian noise. This noise was passed through the noise path, and then applied to the primary input of the noise canceller, with white Gaussian noise is applied to the reference input. Experimental parameters are listed in Table 2. Mean square error convergence is used as a measure of performance. Plots of MSE are produced and smoothed with a suitable moving average filter. A comparison is made with a conventional fullband system as well as with a recently developed critically sampled system (Kim et al 2008) as shown in Fig.11. The optimized system is denoted by (OS), the critically sampled system is denoted by (CS) and the fullband system is denoted by (FB). To test the behavior under environmental conditions, a speech signal is then applied to the primary input of the proposed noise canceller. The speech was in the form of Malay utterance “Kosong, Satu, Dua,Tiga” spoken by a woman. The speech was sampled at 16 kHz. Engine noise is used as a background interference to corrupt the above speech. Plots of MSE are produced as shown in Fig.12. In this figure, convergence plots of a fullband and critically sampled systems are also depicted for comparison.
Parameter  Specification 
Acoustic noise path  FIR processor with 512 taps 
Adaptation algorithm type  Subband power normalized LMS 
Primary input (first test)  Variable frequency sinusoid 
Reference input (first test)  Additive white Gaussian noise 
Primary input (second test )  Malay utterance, sampled at 16 kHz 
Reference input ( second test)  Machinery noise 
3.6. Discussion
From Figure 11, it is clear that the MSE plot of the proposed oversampled subband noise canceller converges faster than the fullband. While the fullband system is converging slowly, the oversampled noise canceller approaches 25 dB noise reductions in about 2500 iterations. In an environment where the impulse response of the noise path is changing over a period of time shorter than the initial convergence period, initial convergence will most affect cancellation quality. On the other hand, the CS system developed using the method by (Kim et al. 2008) needs a longer transient time than that OS system. The FB canceller needs around 10000 iterations to reach approximately a similar noise reduction level. In case of speech and machinery noise (Fig12), it is clear that the FB system converges slowly with colored noise as the input to the adaptive filters. Tests performed in this part of the experiment proved that the proposed optimized OS noise canceller does have better performance than the conventional fullband model as well as a recently developed critically sampled system. However, for white noise interference, there is still some amount of residual error on steady state as it can be noticed from a close inspection of Fig.11.
4. Low complexity noise cancellation technique
In the last section, optimized oversampled filter banks are used in the subbandnoise cancellation system as an appropriate solution to avoid aliasing distortion associated with the critically sampled subband noise canceller. However, oversampled systems imply higher computational requirements than critically sampled ones.In addition, it has been shown in the previous section that oversampled FIR filter banks themselves color the input signal, which leads to under modeling and hence high residual noise at system’s output for white noise. Therefore, a cheaper implementation of the subband noise canceller that retains good noise reduction performance and low signal delay is sought in this section. The idea is centered on using allpass infinite impulse response filters. The filters can be good alternatives for FIR filters. Flat responseswith very small transition band, can be achieved with only few filter coefficients. Aliasing distortion in the analysis filter banks can be reduced to tolerable levels with lower expenses and acceptable delay. In literature, the use of allpass IIR filter banks for echo control has been treated by Naylor et al. (1998). One shortcoming of this treatment isthe spectral gaps produced as a result of using notch filtering to preprocess the subband signals at the analysis stage in an attempt to reduce the effect of nonlinearity onthe processed signal. The use of notch filters by Naylor et al. (1998) has also increased processing delay. In this section, an adaptive noise cancellation scheme that usesa combination of polyphase allpass filter banks at the analysis stage and an optimized FIR filter bank at the synthesis stage is developed and tested. The synthesis filtersare designed in such a way that inherent phase correction is made at the output of the noise canceller. The adaptive process is carried out as given by equations (34) (37). Details of the design of analysis and synthesis filter banks are described in the following subsections.
4.1. Analysis filter bank design
The analysis prototype filter of the proposed system is constructed from second order allpass sections as shown in Fig.13. The transfer function of the prototype analysis filter is given by
where,
where α_{
k,n
} is the coefficient of the
Furthermore, to maintain the performance of the filters in fixed point implementation, it is advantageous to use cascaded first or secondorder sections (Mendel 1991). These filters can be used to produce multirate filter banks with high filtering quality (Milić 2009). Elliptic filters fall into this class of filters yielding very lowcomplexity analysis filters (Poucki et al. 2010).The two band analysis filter bank that is shown on the L.H.S. of Fig.1 can be modified to the form of the polyphase implementation(type1) as shown in Fig.14 and is given by
Filters
4.2. Analysis/synthesis matching
For phase correction at the noise canceller output, a relationship that relates analysis filters to synthesis filter is established as follows. The analysis prototype filter
where
where
where
where
4.3. Computational complexity and system delay analysis
The total computational complexity of the system can be calculated in three parts, analysis, adaptive and synthesis. The complexity of 8band analysis filter bank with eight coefficients prototype filter, and for tree implementation of three stages giving a total of 28 multiplication operations per unit sample by utilizing the noble identities. The complexity of the adaptive section is calculated as the fullband adaptive filter length
In the technique offered by Narasimha (2007) for example, the output is calculated only after the accumulation of
Kim (2008)  Narasimha (2007) 
Choi&Bai (2007) 
Proposed ( LC) 

Complexity  890  27136  2056  532 
Delay/samples  430  512  128  97 
4.4. Results and discussion of the low complexity noise canceller
The same input signals and noise path as in in previous section are used in testing the low complexity system. In the sequel, the following notations shall be used, LC for low complexity noise canceller, OS and FB stand for oversampled and fullband systems, respectively. It is shown in Fig. 18 that mean square error plots of the OS system levels off at 25 dB after a fast initial convergence. This due to the presence of colored components as discussed in the last section. Meanwhile, the MSE plot of the proposed LC noise canceller outperforms the MSE plot of the classical fullband system during initial convergence and exhibits comparable steady state performance with a little amount of residual noise. This is probably due to some non linearity which may not be fully equalized by the synthesis stage, since the synthesis filter bank is constructed by an approximation procedure. However, subjective tests showed that the effect on actual hearing is hardly noticed. It is obvious that the LC system reaches a steady state in approximately 4000 iterations. The fullband (FB) system needs more than 10000 iterations to reach the same noise cancellation level. On the other hand, the amount of residual noise has been reduced compared to the OS FIR/FIR noise canceller. Tests performed using actual speech and ambient interference (Fig. 19) proved that the proposed LC noise canceller does have an improved performance compared to OS scheme, as well as the FB canceller. The improvement in noise reduction on steady state ranges from 1520 dB compared to fullband case, as this is evident from Fig. 20. The improved results for the proposed LC system employing polyphase IIR analysis filter bank can be traced back to the steeper transition bands, nearly perfect reconstruction, good channel separation and very flat passband response, within each band. For an input speech sampled at 16 kHz, the adaptation time for the given channel and input signal is measured to be below 0.8 seconds. The convergence of the NLMS approaches above 80% in approximately 0.5 seconds. The LC noise canceller possesses the advantage of low number of multiplications required per input sample. To sum up, the proposed LC approach showed an improved performance for white and colored interference situations, proving usefulness of the method for noise cancellation.
5. Conclusion
Adaptive filter noise cancellation systems using subband processing are developed and tested in this chapter. Convergence and computational advantages are expected from using such a technique. Results obtained showed that; noise cancellation techniques using critically sampled filter banks have no convergence improvement, except for the case of twoband QMF decomposition, where the success was only moderate. Only computational advantages may be obtained in this case. An improved convergence behavior is obtained by using twofold oversampled DFT filter bank that is optimized for low amplitude distortion. The price to be paid is the increase in computational costs. Another limitation with this technique is the coloring effect of the filter bank when the background noise is white. The use of polyphase allpass IIR filters at the analysis stage with inherent phase compensation at the synthesis stage have reduced the computational complexity of the system and showed convergence advantages. This reduction in computational power can be utilized in using more subbands for high accuracy and lower convergence time required to model very long acoustic paths. Moreover, the low complexity system offered a lower delay than that offered by other techniques. A further improvement to the current work can be achieved by using a selective algorithm that can apply different adaptation algorithms for different frequency bands. Also, the use of other transforms can be investigated.
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