However, it causes large estimation errors.Meanwhile, low estimation errors as well as slow convergence rates are resulted when the projectionorder 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 x”(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 w” (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, (”I) is a diagonal matrix with ” 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 ” is used,
where ” lies within the following range. 0 < ” < 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 ”i. The characteristic equation of R is set as follows.
where I an identity matrix, and ”j is given by the diagonal matrix below.
where, ”1, ”2 , ‘, ”M are the eigenvalues of R. The eigenvalue spread of R is calculated as follows,
where max(”i) and min(”i) 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 missadjustment.
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.