Fast RLS Algorithms Running On Roughly Quantized Signals

w(n) = R?1 (n)rdx (n) xx
where
(2)
n?k d(k)x(k):
Rxx(n) =
n X k=1
n?k x(k)xt (k);
rdx (n)
n X k=1
(3)
From (2), a recurrence relation for the adaptive lter coe cients can be obtained 1]:
E xq(k)xq (k + m)] = E x(k)x(k + m)] + E xq(k)y(k + m)] = E x(k)y(k + m)]
2
(m);
(5) (6)
where (m) is the Kronecker delta and 2 is the amount of bias introduced in the zero-lag value of the autocorrelation function and y (k) is a stationary signal which may or may not be di erent from x(k). A detailed analysis on the properties of the quantized correlators can be found in 2]. It has been shown that very reliable estimates for the correlations can be obtained by using as low as 3 to 4 bits in the quantizers. Multiplications by the samples of the roughly quantized input signal can be realized by simple shifts and additions. For example, multiplication by a 3-bit quantized number requires at most 2 shifts and 2 additions. This gure can be further reduced by building simple dedicated circuitry. This implies that if we use xq (k) in the computation of k(n + 1) we can decrease the computational complexity associated with existing fast RLS algorithms by converting multiplications with xq (k) to shifts and adds. To derive di erent versions of fast RLS algorithms corresponding to the use of roughly quantized input signal, we merely replace input signal x(n) by xq (n) in the computation of adaptation gain. Denoting the adaptation gain obtained in this way as kq (n) and by using (5-6) it can be easily shown that
Abstract
1
Fast RLS Algorithms Running On Roughly Quantized Signals
Murat Belge and Orhan Ar kan Bilkent University, 06533 Ankara, TURKEY
1. Introduction
3. Fast RLS Adaptation Framework
2
The basic idea behind the proposed adaptation framework is to use a roughly quantized auxiliary input signal. By a roughly quantized signal, we mean a discrete time sequence which is quantized to at most 4 bits (16 levels). This signal has the special property that its correlation function is exactly the same as that of the original input signal, except at the zeroth lag. The proposed adaptive ltering con guration is schematically represented in Fig. 1 . Here, xq (k) is a roughly quantized signal which is obtained by the quantization of original input signal plus a zero mean i.i.d. random reference signal, (n), which is uniformly distributed within one quantization step. If no clipping occurs at the quantizer, the following equations are satis ed 2]
w(n + 1) = w(n) + k(n + 1) d(n + 1) ? wt (n)x(n + 1)
hiBiblioteka (4)where k(n) = R?1 (n)x(n) is commonly known as the adaptation gain 1]. Computation of optimal adaptive xx lter coe cients requires the computation of the adaptation gain at each time instant. FKA algorithm requires 8N operations to update k(n) while FAEST or FTF algorithms require 5N operations. In FAEST algorithm, an alternative de nition of the adaptation gain, k (n + 1) = R?1 (n)x(n + 1) is employed to further reduce the xx computational complexity.
Fast RLS Algorithms Running On Roughly Quantized Signals
Murat Belge and Orhan Ar kan** Electrical and Computer Engineering Department 409 Dana Building Northeastern University, Boston MA 02115 Phone: 617 - 373 8783 E-mail: belge@ **Department of Electrical and Electronics Engineering, Bilkent University, 06533 Ankara, TURKEY. Fax: + 90 - 312 - 2664127 Phone: + 90 - 312 - 2664307 E-mail: oarikan@.tr Edics: SPL.SP.2.6
A framework for obtaining fast RLS algorithms which use a roughly quantized auxiliary input signal for e ciently updating the required adaptation gain vector is presented. The new adaptation procedure is very similar to the conventional fast RLS adaptation. Analytically, it is shown that the lter weights are found by solving almost the same normal equations corresponding to the RLS case. Obtained results are in good agreement with those of the fast RLS algorithms with a signi cant reduction in the number of multiplications.
2. Fast Recursive Least Squares Adaptation
Jw (n) =
n X k=1 n?k
d(k) ? wt (n)x(k)
2
(1)
where d(k) is the desired signal at time k, w(n) is the vector of optimal lter coe cients at time n, x(k) = x(k) x(k ? 1) : : :x(k ? N +1)]t is the vector which contains N most recent samples of input signal and 0 < 1 is an exponential forgetting factor. Minimization of (1) yields the following least squares solution for w(n)