However, it causes large estimation errors. Meanwhile, low estimation errors as well as slow convergence rates are resulted when the projection order and the step-size are set to small values [3]. These problems have been the focused on by many researches in recent years. An example of these algorithms is the fast APA proposed in [4].
The fast APA improves the convergence rate when the projection order is set to a fixed value with a small step size. Another example is the variable order APA [5], named as the evolutionary APA. The order of the projection of the latter is determined according to the output error and a predefined threshold. This algorithm was modified by using the dichotomous coordinate descent DCD technique [6]. The modified algorithm possesses less number of operations than the original one [6]. Other literature proposals have focused on varying the step-size of the APA, such as those found in [7] and [8]. Also, a dynamic selection APA is proposed in [9], which deals choosing the input vector. These algorithms showed an improved convergence performance, and a small estimation error with a lower computational power than the original APA version. However, the performances of most of these algorithms have only been compared to the conventional APA, and still possess larger estimation errors than the classical LMS. Moreover, employing an optimized or variable projection order APA has not been widely discussed in literature, because this would involve analytical solutions for optimization problems, which in turn leads to either impracticable solutions or very expensive to implement in real-time applications. In reference [10] a variable projection order APA is proposed. The technique is based on a mixed analytical and experimental procedure. However, the method uses extensive simulations and investigations to work out the projection order of the algorithm, which looks somehow complicated and needs a lot of background process to achieve the goal. Thus, this paper proposes a new technique to vary the projection order of the APA. The procedure is based on monitoring the variations in the characteristics of the noise via measuring the eigenvalue spread of the autocorrelation matrix of this noise. The proposed method presented in this paper uses an algorithmic procedure to vary the projection order of the algorithm, which is much simpler than the one found in [10]. The paper is organized as follows. In addition to this introduction section, section 2 presents details of the proposed technique in this paper. Section 3 displays the results of simulating the proposed algorithm and discussing the main aspects of these results, and finally section 4 concludes the paper.
2. Proposed Noise Cancellation Approach
A block diagram of a typical adaptive noise canceller with two inputs is shown in Figure 1, where s(n) is the signal of interest, x(n) is the input noise. The noise x(n) is being transmitted over an unknown path to be added with s(n) as a correlated noise xˆ(n) . The adaptive noise canceller ANC attempts to reduce the noise by subtracting the adaptive filter output y(n) from the desired input d(n). The error signal e(n) is used to alter the coefficients of a digital filter so as to reduce the noise in the useful signal s(n).
Basically, any type of adaptive algorithms can be used as the controlling algorithm. Adaptive algorithms such as the LMS and RLS can be used. However, these algorithms have their drawbacks in acoustical environments, as it has been made clear in the introduction section. Therefore, in this research paper, a modified affine projection algorithm is used to control the coefficients of a transversal finite impulse response FIR filter. The choice of the filter is based on the stability that offered by this type of filters.
The set of equation that are used to update the conventional APA are given by: where wˆ (n) is the adaptive filter weight vector at time n. The desired input d(n) and the input noise x(n) are given by the following vectors:
Here, P is the projection order of APA, (δI) is a diagonal matrix with δ is a small constant along the diagonal which is used to regularize the inverse matrix in the AP algorithm, e(n) is the error signal and T is a transpose notation. To control stability, convergence, and final error, a step size μ is used, where μ lies within the following range. 0 < μ < 2. (6)
Now, in the conventional APA operation, the projection order of the algorithm is set to a certain value and never changed during adaptation process .In this paper, the projection order is made variable during adaptation process. The algorithm is modified so that it changes its projection order according to the value of the eigenvalue spread of the noise autocorrelation matrix, which represents the characteristics of the noise. The target application here is voice communication in mobile systems. In such applications, the voice signal is often corrupted by several types of environmental noise. These types of noise are hard to eliminate using conventional methods. The eigenvalue spread of the noise is determined from its autocorrelation matrix known a R and expressed the following:
where E is an expectation operator, xH(n) is the Hermitian transposition of input noise vector x(n). From this matrix, the eigenvalue spread is determined from the ratio of the maximum to the minimum eigenvalues of R. The eigenvalues are represented by λi. The characteristic equation of R is set as follows.
where I an identity matrix, and λj is given by the diagonal matrix below.
where, λ1, λ2 , …, λM are the eigenvalues of R. The eigenvalue spread of R is calculated as follows, where max(λi) and min(λi) are the maximum and minimum eigenvalues respectively. Using frames of data from the input noise, the value of (R) is measured and the projection order is set to a certain value depending on the type of interference that can corrupt the communication signal, hence reducing the effect of different types of interference in the corrupted signal. The proposed technique in this paper is expected to give good noise cancellation performance at a moderate computational power, since high projection orders are not needed for long intervals of time during adaptation process. In some cases of noise such as white noise, the algorithm performs with complexity very much like the LMS.
3. Simulation Results and Discussion
In this section, initial results are obtained from simulating the interference canceller shown in Fig.1, using a model audio signal represented by a sinusoidal waveform, which simulates the voice communication signal. This signal is deliberately subjected to two sections of white and colored noise. The noise sections are concatenated alternately running for enough time to test the algorithm. The white noise possesses the least eigenvalue spread of 1.27, while the colored noise has a calculated value of 10.5. The colored section is generated by passing the white noise through a second order IIR filter. The resulting variable noise is simulating environmental noise which can change from one type to another producing varying characteristics noise. In real-life, acoustical noise can change from car engine noise to voice babbles to a broadband environmental noise..etc. The eigenvalue spread values of the input noise signal used in these simulations are calculated on a frame base. The calculation of the eigenvalue spread is repeated for each frame of data to observe the changes in the noise signals, hence giving a control command to change to an appropriate projection order. Two values for the projection order are taken,2 and 16. The algorithm is programmed so that if the spread lies below 5, then the order is set to 4. Otherwise, the order is set to 16. The starting order is made 16 so that the possibility of starting with highly colored noise i.e. with large eigenvalue spread is taken into account. This is only a taste of what the actual algorithm could be. The choice of these values is based on the fact that for low spread values, only low order APA is required, hence it resembles the simple LMS in its computational complexity, while the performance is equivalent to RLS algorithm. On the other hand, for high values of eigenvalue spread, the order is increased so that the algorithm can cope with colored noise that cannot be removed using the simple LMS algorithm. Therefore, an optimum noise filtering can be achieved this way. The performance of the noise canceller with the modified APA proposed in this paper is evaluated using mean square error MSE performance of the noise canceller. Figure 2 shows the MSE plots of the proposed method compared to the performance using conventional APA, LMS and LRLS algorithms. The experiments under conventional APA have been conducted with two constant orders 4, and 32 to appreciate the difference in performance compared to the modified APA proposed in this paper.
Fig.2 MSE performance of the propose APA compared to conventional algorithms.
Now, we try to interpret the behavior of the proposed algorithm compared to existing algorithms. It is clear from Fig.2 that the proposed noise canceller with the modified APA shows initial fast convergence, as good as the RLS which somehow appears with relatively large steady state error. Meanwhile, the LMS flattened very quickly after a short initial convergence. This means that the residual noise in the communication signal persists with large amount. This can be related to the inability of the LMS to cope with signal with large eigenvalue spread. The performance of the proposed APA is compared to two conventional APAs with two different orders 4 and 32. The step size is kept at 0.05 for all cases. It is evident from Fig.2 that the APA with order 4 converges slowly with very smooth steady state behavior, while the APA with order 32 converges very fast with large miss adjustment.
To some up, the proposed canceller with modified APA showed a fast convergence with a small miss-adjustment error. To confirm the success of the proposed method, we display the signal before and after filtering using various algorithms as shown in Fig.3. This figure depicts the signal before and after processing using proposed as well as conventional methods. It is clear from Fig.3 that the modified variable order APA outperforms the conventional LMS as well as the RLS. As far as the computational complexity is concerned, the proposed APA has a lower complexity than the RLS, as well as any APA with a high order such as the one used here for comparison which has an order of 32. The latter has reached a fast convergence with a higher steady state error. Furthermore, the APA with order of 4 has a lower computational power than the proposed algorithm at the expense of a very slow initial convergence, which is a very important issue in mobile and hand free communications.
4. Conclusions
An improved noise cancellation system is developed in this paper. The canceller is based on using variable order affine projection algorithm. The projection order was made to change algorithmically according to the noise characteristics. The results presented in this paper proved the validity of the method. In addition to the gain in computational power, the system has shown a better performance compared to the conventional adaptive filtering. A further extension of this work would include testing using real environmental interferences and more development on the algorithm.