An important issue in multi-resolution analysis is that of optimal basis selection. An optimal P-band perfect reconstruction filter bank (PRFB) is derived in this seminar, which minimizes that approximation error (in the mean square) between the original signal and its low resolution principal components with deceasing variance. Optimality issues are further analyzed in the special case of stationary and cyclosationary processes. By exploiting the connection between discrete-time filter bank and continuous wavelets, an optimal multi-resolution decomposition of L2(R) is obtained. Analogies results are also derived for deterministic signals. Some illustrative examples and simulations are presented.
The representation of signals in many resolution or scales has drawn attention in several diverse fields and has shown potential in many applications. In computer vision, the pyramid representation has been used for motion estimation and stereo vision problems. In discrete-time signal processing, filter bank analysis has been employed in sub band coding & multi-resolution techniques have found applications and medical imaging. Furthermore, an elegant mathematical theory has been developed (Wavelet analysis), which unifies the discrete and continuous time representations.
[...] V p (ω ) P ω 2π ( p S xx P [ ] [ ] Based on and proposition 1 one is tempted to associated the vectors Vk(ω) with the optimal analysis filter bank and the diagonal elements Sxx(ω-2πk/P) with the spectral density of the resulting principal components. However, two subtle issues should be carefully considered in this interpretation so that the derivation is meaningful. First, Notice that the LHS of is periodic with period 2π, while each matrix involved in the RHS is periodic with period 2πP. [...]
[...] Independent work from, which came to our attention after the submission of this paper, reaches similar conclusions and confirms our analysis for the case of two-band PEFB's and stationery inputs REFERENCES 1995. M. K. Tsatsanis and G. B. Giannakis, “Principal component filter banks for optimal analysis,” IEEE Trans. Signal Processing, vol pp. 1766–1777, Aug. A. N. Akansu, P. Duhamel, X. Lin, and M. de. Courville, “Orthogonal transmultiplexers in communications: A review,” IEEE Trans. Signal Processing, vol pp. 979–995, Apr S.Akkarakaran and P. P. Vaidyanathan, optimization of filter banks with denoising applications,” in Proc. IEEE ISCAS'99, Orlando, FL, June 1999. J. [...]
[...] In the next subsection, we focus on the specific form of the principal component PRFB for stationary 4. PCFB FOR STATIONARY CASE Let x be a zero-mean process with absolutely summable autocorrelation r γχ = E{x(n + τ) } and spectral density Sxx (ω). We wish to compute the eigenvectors and eigenvalues of the spectral matrix Sxx(ω) analytically. If we use the block version of then the entry of the autocorrelation matrix Rxx(τ) = E{x(n + τ)xT(n)} = [Rxx(m,l) T ] is Rxx(m,l) = E { x(nP + τP + x (nP-l } = rxx (τP + l Fourier transforming and using the under sampling formula (e.g we obtain the entry of Sxx as follows S xx ( m ) (ω ) = rxx (τP + l m jωτ τ j 1 P = S xx (ω 2πk ) / P e P k S xx ( m ) ω 2πk p m ) From we can write the spectral matrix as 1 jφ (ω ) p ω 2πk e 1 S xx (ω ) = S xx p k P : j ( p (ω ) e jφ (ω ) . [...]
[...] A. C. Bingham, “Multicarrier modulation for data transmission: An idea whose time has IEEE Commun. Mag., pp. May 1990. I. Djokovic and P. P. Vaidyanathan, optimal analysis/synthesis filters for coding gain optimization,” IEEE Trans. Signal Processing, vol pp. 1276–1279, May 1996. N. S. Jayant and P. Noll, Digital Coding of Waveforms. Englewood Cliffs, NJ: Prentice-Hall I. Kalet, “Multitone modulation,” in Subband and Wavelet Transforms, A. N. Akansu and M. J. Smith, [...]
[...] It does not make sense, for example, when S zz = constant; in this case one can find a number of different filter banks that attain the optimum performance. Proposition 4 is rather surprising in its simplicity. One might be intrigued in seeing a constant response H = being analyzed in polyphase components as in and not in the more familiar h k k = , hk(i) = l which is implied by decimation in time. However, we should keep in mind that the result of holds for a given and H k is not constant for all ω but rather piecewise constant. [...]
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