A VLMS Based Pseudo-Fractional Optimum Order Estimation Algorithm Asutosh Kar Dept. of Electrical Engineering National Institute of Technology Hamirpur, Himachal Pradesh, India
Ravinder Nath Dept. of Electrical Engineering National Institute of Technology Hamirpur, Himachal Pradesh, India
[email protected]
[email protected]
Alaka Barik Dept. of Electrical Engineering National Institute of Technology Hamirpur, Himachal Pradesh, India
[email protected]
In principle minimum mean square error (MMSE) is a monotonic non increasing function of the filter order but it is not advisable to have a too long order filter as it introduce adaptation noise and extra complexity due to more taps[2],[3]. The first variable order estimation algorithm [4] proved that shorter filters have faster convergence than the longer ones and adjusting the order can improve the convergence of LMS [5], [6]. But a too short order filter results in inefficient model of the system and increases the mean square error (MSE) [7], [8]. In both [4], [5] variable order estimation is proposed in which order can be increased in one direction. The method proposed in [6] based on time constant is better than the previously proposed methods but the whole concentration was on improving the convergence rather than the optimum order estimation. The step size control for convergence has less effect on the order of the filter [9].The concept of optimum order for improving output SNR, power consumption and complexity reduction rather than accelerating the convergence of LMS was first proposed in [10].More relevant work based on segmented filter(SF) was proposed in [11],[12]. The tapped delay line (TDL) structure of adaptive filter is segmented and the difference of output error between last two consecutive segments controls the overall length of the filter. The gradient descent (GD) approach [13] finds the optimum order in the negative direction of the squared estimation error. The GD approach is smother and easier to implement than the SF approach but causes tap-length wandering problem in the higher range of optimum length.
ABSTRACT Most current adaptive filters fix the filter order at some compromise value resulting in too short and too long filters with issues like undermodelling and adaptation noise in time varying scenarios. The tap length learning algorithm dynamically adapt the filter order to the optimum value makes the variable order adaptive filter more efficient including smaller computational complexity, higher output SNR and lower power consumption. The optimum order best balance the complexity and steady state performance of the adaptive filter. Choice of parameter, noise level and convergence issues affect the performance up to a great extent in the existing dynamic order estimation algorithm. In this paper a variable step LMS (VLMS) based pseudo-fractional optimum order estimation algorithm has been proposed that improves the overall performance of the adaptive filter searching the optimum order dynamically with fast convergence. Simulations and results are provided to observe the analysis of the proposed algorithm.
Categories and Subject Descriptors H.4.3 [Information Systems Applications]: Communication Applications--computer conferencing, teleconferencing, and video conferencing; G.1.6 [Numerical Analysis]: Optimization-gradient methods.
General Terms
The fractional tap-length LMS algorithm (FT-LMS) was proposed [2], [14],[15] relaxing the constraint on filter order to be an integer value. It retains the advantages of both SF and GD and proves itself to be less complex than the previously proposed methods but it is highly affected by parameter choice and noise level. Convex combination of adaptive filter proposed in [16] found a solution to this issue but the parallel combination of adaptive filters for tap-length adaptation increases the overall complexity and creates synchronization problem.
Algorithms, Performance, Experimentation.
Keywords Adaptive filter, MMSE, Filter order.
1. INTRODUCTION The overall performance and complexity of the adaptive filter depends on its structure [1]. The no of taps is one of the most important structural parameters of the liner adaptive filter [1].
In this paper improved fractional order estimation is proposed with constrained and optimum use of the leaky factor to tackle the parameter variation and noise level in the FT-LMS algorithm. The modified weight update equation and variable error width in the proposed algorithm makes it independent of initialization and it finds faster convergence at high eigen value spread scenarios.
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The cost function for optimum order estimation has been defined in section 2. In section 3 VLMS based pseudo-fractional optimum order estimation algorithm has been shown with the basic algorithm and the principle it follows. The computer simulation setup and results are shown in section 4.
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less variable order estimation and requires no priori information about the steady state MSE for the previous order.
2. COST FUNCTION FOR OPTIMUM ORDER ESTIMATION The SF and GD approach carry their own advantages and are results in equivalent performance under some specific constraints [15].This concept can be utilized to find the cost function which is to be minimized for optimum order estimation incorporating the advantage of both SF and GD methods. In [2], [14] the optimum order is defined as P that satisfies,
DP
for all P ≥ P
3. PROPOSED ALGORITHM 3.1 BASIC ALGORITHM In the proposed pseudo fractional algorithm the constraint that is filter order should be an integer has been relaxed. The steady state MSE is not available in general and can be found by exponential averaging [2],[14] ,[15]
(1)
J ( n 1 ) (1 F ) e 2 ( n 1 ) F J ( n )
where DP J P1() J P () is defined as the difference between the converged MSE when the filter order is increased
where F is the smoothing constant which control the effective memory of the iterative process[1]. In the fractional order estimation the error width plays an important role. Large value of speed up the convergence but also brings a heavy computational complexity and lead to large bias in the optimum order estimation. Whereas small results in slow convergence and under model the system. provides the trade-off between the convergence rate and steady state bias [17],[18]. It also avoids the pseudo-optimum orders. The variable error width (n) can be obtained as [18]
from P-1 to P and δ is a very small positive number set pertaining to the system requirement and
min{ P | J P 1 J P }
(2)
is the cost function with respect to the filter order P. In many scenarios pseudo-optimum orders are observed [2], [14]. Let there exist a positive integer L that satisfies,
L P and D L
(3)
where S is smoothing factor whose value changes according to the change in the changing factor D P and the maximum and minimum value of .
From (1) it is clear that the performance index for searching P can be obtained as: (4)
S
where is a positive integer. If is larger than the maximum pseudo optimum width then the local minima can be escaped to find P . In this case the changing factor D P can be defined as D P J P J P . The algorithm proposed in [2], [14] converge to the optimum order in mean but suffers from slow convergence under various scenarios. If the initialization is imperfect then the algorithm diverge from the optimum order due to transient behavior of the order adaptation.
(P)
d ( n ) W P (1 : K ) X P (1 : K )
(P)
E eK
P 2
(5)
(P)
P
J
(P)
P
(6)
}
the
error width
changes
Lf 1 P(n) P(n) (12) 2 2 Pnf (n) Pnf (n1) [(eP(n) (n)) (eP(n)(n) (n)) ] 2 1 L (log f 10 Lf ) where Pnf is the filter order that can take fractional values. As the
With this the modified cost function for searching the optimum order can be defined as min { P | J
0.5)
(11)
The fractional tap-length algorithm proposed in [2],[14],[15], [17], [18] takes different values of the small leaky factor Lf
