Abstract: In this research, we present the steady state analysis of adaptive IIR notch filters based on the least mean p-power error criterion. We consider the cases when the sinusoidal signal is contaminated with white Gaussian noise and p = 3,4. We first derive two difference equations for the convergence of the mean and the Mean Square Error (MSE) of the adaptive filter`s notch coefficient and then give the steady state estimation bias and MSE. Stability conditions on the step size value are also derived. Simulation experiments are presented to confirm the validity of the obtained analytical results. It is shown that the notch coefficient steady state bias of the p-power algorithm for small step size values is independent of the step size value and is equal for p = 1,2,3 and 4. However, for larger step size values, the p-power algorithm with p = 3 provides the best performance in term of the MSE.
INTRODUCTION
Adaptive notch filters have been successfully used for detecting sinusoid signals in wide-band noise in several applications, such as digital communications, active noise control and biomedical signal processing and so on. Adaptive IIR notch filters have received considerable research interest due to their significantly low computational load and memory requirement when compared to their FIR counterpart with similar notch band-width. So far, several adaptive IIR notch filtering algorithms based on least Mean Square Error (MSE) criteria have been proposed (Stoica and Nehorai, 1988; Petraglia et al., 1994; Chicharo and Ng, 1990). However, MSE criteria do not always provide the best performance and accordingly there has been increased interest in developing adaptive algorithms based on Lp normed minimization after it has been proven successful in different signal processing applications. For adaptive IIR notch filtering, several Lp norm based algorithms have been proposed so far, such as the Sign Algorithm (SA) (Martin and Sun, 1986; Schroeder et al., 1991) and the p-power algorithm (Pei and Tseng, 1993). However, the question to be asked here is: for which value of p does the p-power algorithm provide the best performance? The answer to this question, as simulation experiments show, depends on the nature of additive noise. The SA algorithm seems to provide the best performance when the sinusoidal signal is contaminated with impulsive noise (Schroeder et al., 1991). However, for the case when the additive noise is Gaussian, Pei and Tseng (Pei and Tseng, 1993) have shown by intensive simulations that the performance of the p-power algorithm for p = 3 is better than that of the LMS algorithm (i.e., p = 2) and the SA (i.e., p = 1). For white Gaussian additive noise scenario, the performance analysis for p = 1 (Sign Algorithm) and for p = 2 (LMS algorithm) have been intensively studied in the literature (Stoica and Nehorai, 1988; Xiao et al., 2001; Xiao et al., 2003; Mvuma et al., 2006). However for p>2, no performance analysis has been presented so far.
In this research, we present the steady state analysis of the p-power algorithm (Pei and Tseng, 1993) for the cases when p = 3 and p = 4 and the additive noise is white Gaussian.
THE p-POWER ALGORITHM
One of the efficient IIR notch filter structures known so far is the adaptive notch filter with constrained poles and zeros (Nehorai, 1985). In this structure the zeros of the filter are constrained to lie on the unit circle at the sinusoid frequency, while the poles have to be inside the unit circle on the same angles and as close as possible to the zeros.
The transfer function of the second-order IIR notch filter with constrained poles and zeros is given by
(1) |
where X(z) and E(z) are, respectively the Z-transform of the filter’s input signal x(n) and output signal e(n), ρ is the pole radius of the adaptive filter which is restricted to the range [0,1) to insure stability of the IIR filter. The band-width of the notch filter is related to the pole radius ρ as follows:
(2) |
The parameter a in (1) is the filter notch coefficient; its true value is calculated by
(3) |
where ω0 is the frequency of the input sinusoidal signal x(n) . In this research, we consider the case when the input signal x(n) consists of a single sinusoid embedded in noise as defined in the following Eq.
(4) |
The additive noise v(n) in (4) is assumed to be zero mean white Gaussian noise
with variance
In the p-power algorithm (Pei and Tseng, 1993), the notch parameter a is adjusted so that the mean p-power error of the notch filter’s output signal defined as
(5) |
is minimized. Accordingly, the steepest descent adaptive algorithm used to update the filter’s notch coefficient a is given by
(6) |
where μ is the step size value and
with s(n) denoting the gradient signal calculated by
(7) |
For p = 1 and p = 2 , this algorithm reduces to the SA and LMS algorithm respectively. The performance analysis of this algorithm for p = 2 has been intensively studied by many authors (Stoica and Nehorai, 1988; Chicharo and Ng, 1990). However, for odd values of p, the analysis is more difficult due to the presence of the sign function in the update procedure. On the other hand, the analysis becomes more complicated for p>3 due to the need of calculating many terms with higher order moments of the output and gradient signals. Recently Xiao et al. (2003) have presented a detailed convergence analysis for the SA. In this research, we present the steady state analysis of the p-power algorithm for the cases when p = 3 and p = 4 using similar assumptions to those used in the analysis of the SA analysis (Xiao et al., 2003) and LMS (Chicharo and Ng, 1990).
STEADY STATE ANALYSIS
Define the estimation error
(8) |
Using (6), the update procedure for the estimation error δa (n) is given by
(9) |
for p = 3 and by
(10) |
for p = 4 , with μp = pμ.
At the steady state, the filter’s notch coefficient â(n) becomes close enough to its true value a0. Thus, using Taylor series expansion of the notch filter transfer function in the vicinity of a0, the output and gradient signal can be calculated by
(11) |
(12) |
where, for notation simplification, we here have defined
with v1(n) and v2(n) denoting the additive noise in the filter’s output and gradient signal respectively. φ and B are defined as
(13) |
with
The variance of v1(n) and v2(n) and their cross correlation R1,2, to be used later on in the analysis, can be calculated using the theory of residues and is found to be
(14) |
Through our analysis, the following assumptions are assumed to hold:
| A1: | The filter’s output signal e(n) is Gaussian distributed with mean value μe and variance which σ2e can be calculated directly from (11). |
| A2: | The output signal e(n) and the estimation error δa (n) are jointly Gaussian distributed. |
| A3: | The estimation error δa (n) is uncorrelated with the noise signals v1(n) and v2(n). |
| A4: | The output signal e(n) and the noise signals v1(n) and v2(n) are jointly Gaussian distributed. |
All these assumptions have been tested (Xiao et al., 2003) and proved to hold for small step size values.
Convergence of the mean: Using (11) and (12) in (9) ((10)), applying the expectation operator E and after long and complicated mathematical work based on assumptions A1-A4, we can find that the difference equation for the convergence of the mean is given by
(15) |
with
for p = 3 and
(16) |
for p = 4 .
In the calculation of Eq. (15), all the terms with higher
order moments are calculated using the Gaussian factoring theorem, the relations
between the higher order cumulants of random signals and their moments and the
property that higher order cumulants for Gaussian signals equal zero. The terms
of
Convergence of the MSE: Squaring both sides of (9) (10), using (11) and (12) and then averaging, we can, after long and complicated calculations, derive the following difference equation for the convergence of the MSE
(18) |
with
for p = 3 and
for p = 4.
Steady state bias and MSE: At the steady state, we have
(19) |
Using (19) in (15) and (18) and then solving the resulting two equations simultaneously, the following closed form expressions for the steady state bias and MSE can be obtained
(20) |
(21) |
SIMULATION RESULTS
To confirm the obtained analytical results, we have conducted several experiments. Figure 1 and 2 show comparison between simulated and theoretical steady state bias and MSE versus the sinusoidal signal frequency ω0 for p = 3,4 . It can be observed that the theoretical results match the simulations very well except in the neighborhood of ω0 = 0.5π.
Figure 3 and 4 show comparison between simulated and theoretical steady state bias and MSE versus the pole radius ρ for p = 3,4 . As expected, the bias and MSE decrease as the pole radius p increases.
Figure 5 and 6 show comparison between simulated and theoretical steady state bias and MSE versus the step size value μp. These two figures indicate that the p-power algorithm results in similar steady state bias for both p = 3 and p = 4 for sufficiently small step size values. In fact, for small step size values the second term in the numerator of (20) can be neglected and it can be verified that
(22) |
That is, for small step size, the steady state bias is independent of the step size value. Interestingly, a similar equation has been derived for the SA (i.e., p = 1) and the LMS (i.e., p = 2 ) (Xiao et al., 2003). That is, for small step size values, the p-power algorithm results in similar steady state bias for all values of p.
| Fig. 1: | Comparison between theoretical and simulated steady state
estimation bias versus sinusoidal frequency of the input signal ω0 (μp = 0.00005, ρ = 0.9, A = |
| Fig. 2: | Comparison between theoretical and simulated steady state
estimation MSE versus sinusoidal frequency of the input signal ω0 (μp = 0.00005, ρ = 0.9, A = |
However, as it can be observed from Fig. 5 and 6, the p-power algorithm with p = 3 performs better than that with p = 4 when the step size value μp>10-4 for the bias and for the MSE. It has been observed through many experimental results the convergence speed of the p–power algorithm with p = 3 is better than that with p = 1,2,4. Detailed comparison o f the performance of this algorithm for different values of p is out of scope of this research and will be presented elsewhere.
| Fig. 3: | Comparison between theoretical and simulated steady state
bias versus the pole radius ρ (μp = 0.00005, ω0 = 0.3π, A = |
| Fig. 4: | Comparison between theoretical and simulated steady state
MSE versus the pole radius ρ (μp = 0.00005, ω0 = 0.3π, A = |
STABILITY CONDITIONS ON THE STEP SIZE VALUE
After we have derived the two difference equations for the convergence of the Mean and MSE, it would be useful to search for the stability conditions on the step size value to insure the convergence of the adaptive algorithm.
| Fig. 5: | Comparison between theoretical and simulated steady state
bias versus the step size value μp (ρ = 0.00005, ω0 = 0.3π, A = |
| Fig. 6: | Comparison between theoretical and simulated steady state
MSE versus the step size value μp (ρ = 0.9, ω0 = 0.3π, A = |
Now, if the influence of the second term in (15) is ignored, we can find that the sufficient condition for the convergence of the mean is given by
(23) |
Assuming that the condition in (23) is satisfied, the sufficient condition for the convergence of the MSE can then be deduced from (18) as:
| Fig.7: | Comparisons between theoretical and simulated stability bounds versus the pole radius ρ (p = 3, ω0 = 0.3π, A = 1) |
(24) |
Figure 7 shows Comparisons between theoretical and simulated stability bounds versus the pole radius p for different values of SNR with p = 3.
CONCLUSIONS
We have presented the steady state analysis for an adaptive IIR notch filtering algorithm based on the Lp normed minimization for the case when p = 3, 4 and the sinusoidal signal is contaminated with additive Gaussian noise. Closed form expressions for the steady state estimation bias and MSE have been derived and the sufficient conditions on the step size value for the convergence of the mean and MSE have also been presented. Simulation results confirm the validity of the obtained theoretical results. It has been shown that for small step size values, the p-power algorithm provides similar bias and MSE for different values of p . However, for larger step size values the performance of the adaptive algorithm with p = 3 is better in terms of convergence speed as well as MSE.
ACKNOWLEDGMENT
This research was done when the author was a visiting researcher at the Electronic Engineering Department, Tohoku University, Japan. Most part of it was presented in ISCAS 2003 (Shadaydeh and Kawamata, 2003). The author would like to thank Professor Masayuki Kawamata at Tohoku University and Professor Yegui Xiao at Hiroshima Prefectural Women’s University for their invaluable comments on this work.