F. Beaufays. Transform domain adaptive filters: An analytical approach. IEEE Trans. Signal Proc., 43(2):422--431, Feb. 1995.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....these schemes also lead to efficient adaptive filter structures. We shall further present in Sec. IX examples where they can lead to better performance than the DFT based scheme. We should mention that these new structures are distinct from the so called transform domain algorithms (as, e.g. in [9]) which process the data on a sample by sample basis. The frequency domain structures, on the other hand, perform block by block processing, which is essential for efficient frequency domain implementations. In this paper we also clarify the connection between the MDF structure and the more ....

F. Beaufays, "Transform-domain adaptive filters: An analytical approach, "IEEE Trans. Signal Processing, vol. 43, no. 2, pp. 422-431, Jan. 1995.

....of overcoming these limitations is to use projections of the input signal on an orthogonal basis. This desirable objective can be attained by means of transform domain adaptive algorithms, so called because the adaptation is performed in the frequency domain rather than the original time domain [11 16]. For a more refined method, we may use a multi rate adaptive filter which provides better trade offs between performance improvement, computational complexity and transmission delay [17] 3. Recursive Least Squares (RLS) Algorithm The LMS algorithm attains simplicity of implementation by using ....

Beaufays, F., 1995. "Transform-domain adaptive filters: an analytical approach", IEEE H 5 Transactions on Signal Processing, Vol. 43, pp. 422-431.

....of the new technique to SWC RLS adaptive filters shows that it stabilizes the marginal instability of this system at almost no additional computational cost. We also provide an extension of the technique to the sliding discrete cosine transform (DCT) useful for adaptive filtering tasks [17] [19]. 2. A NUMERICALLY STABLE RECURSIVE SLIDING WINDOW ESTIMATOR The proposed method for approximating (1) is a periodically time varying linear system. If numerical errors are not present, the system is equivalent to y(k) L Gamma1 X m=0 h l (k; m)x(k Gamma m) 5) h l (k; m) ae e ....

F. Beaufays, "Transform-domain adaptive filters: an analytical approach," IEEE Trans. Signal Processing, vol. 43, pp. 422431, Feb. 1995.

....compute the outputs s0;k and s1;k at time k of the two filters B0(z) and B1 (z) given their common input xk and then apply s j;k , j = i mod2 to the filter c i =A i (z) to obtain i;k . Other transforms have similar filter structures; for more information on this topic, the reader is referred to [5, 8]. Note that the DCT is extremely simple to implement in this form, requiring only two non trivial multiplies per frequency value per iteration to compute. The DCT of the vector e k can also be computed using a filter bank if the step size k = is fixed. In this case, the elements of e k are ....

F. Beaufays, "Transform-domain adaptive filters: an analytical approach," IEEE Trans. Signal Processing, vol. SP-43, no. 2, pp. 422-431, February 1995.

....can be obtained [21] 5.2.5 Transform Domain LMS (TDLMS) All of the above approaches to improve the behaviour of the LMS algorithm are operating in the time domain. An intriguing idea, however, is to use the frequency domain in order to decorrelate the input signal of the LMS algorithm [22] [23]. An intuitive approach is to apply a Fourier series decomposition to the input signal x k which yields the coefficients for each of the frequency components. As the Fourier basis functions form an orthogonal set the different frequencies should be totally uncorrelated. After weighting each ....

....are involved. Hence the Fourier analysis is approximated by the Discrete Fourier Transform (DFT) for which fast algorithms exist. The block diagram for this approach is depicted in fig. 15. Not only the DFT but any other orthogonal transform can be used to decorrelate the input signal [22] [23]. In real time (or nonblock) algorithms, the flow of input samples is continuously transformed by a fixed, data independent transform which, in case of speech signals, is often chosen to be the Discrete Cosine Transform (DCT) This way of continuously transforming is also called sliding ....

Beaufays, F., Transform-Domain Adaptive Filters: An Analytical Approach, IEEE Trans. Sign. Proc., Vol. 43, No. 2, Feb. 1995, pp. 422 - 431.

....comparison, the eigenvalue spread before any transformation tends to (1 ae) 2 = 1 Gamma ae) 2 . For highly correlated signals, i.e. for signals with correlation ae close to one, the DCT preprocessing brings a dramatic improvement over plain LMS 1 . 1 These results were first published in [2]. CHAPTER 4. EIGENVALUE SPREAD COMPUTATION 39 4.1 Introduction To determine how well a given transform decorrelates certain classes of input signals and how fast the corresponding transform domain algorithm converges, one must set the problem in a mathematical framework. Transforming a signal x ....

F. Beaufays. Transform domain adaptive filters: an analytical approach. IEEE Trans. on Signal Processing, 43(2):422--431, February 1995.

....is further complicated by the fact that only asymptotically do the eigenvalues stabilize to fixed magnitudes independent of n, and that power normalization is a nonlinear operation. Successive matrix manipulations and passages to the limit allowed us to prove the following asymptotic results (see Beaufays, 1993, 1994) for more details) Eigenvalue spread after DFT = 1 ae 1 Gamma ae ; Eigenvalue spread after DCT = 1 ae: Note that with the DCT, the asymptotic eigenvalue spread is never higher than 2 As a numerical example, let the correlation ae be equal to 0.95. The eigenvalue spread before ....

Beaufays, F. and Widrow, B. 1993. Transform domain adaptive filters: an analytical approach. Submitted to IEEE Trans. on Signal Proc.

....analysis is further complicated by the fact that only asymptotically do the eigenvalues stabilize to fixed magnitudes independent of n, and that power normalization is a nonlinear operation. Successive matrix manipulations and passages to the limit allowed us to prove the following results (see [1, 3] for a complete derivation) lim n 1 i Eigenvalue spread after DFT j = 1 ae 1 Gamma ae ; E.22) lim n 1 i Eigenvalue spread after DCT j = 1 ae: E.23) Note that with the DCT, the eigenvalue spread is never higher than 2 As a numerical example, let the correlation ae be equal to ....

F. Beaufays and B. Widrow. Transform Domain Adaptive Filters: An Analytical Approach. IEEE Trans. on Signal Processing, 43(2), February 1995.

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F. Beaufays. Transform domain adaptive filters: An analytical approach. IEEE Trans. Signal Proc., 43(2):422--431, Feb. 1995.

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F. Beaufays, "Transform Domain Adaptive Filters: An Analytical Approach", IEEE Trans. on Signal Proc., vol. 43(2), pp. 422-431, February 1995.

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