[<<][math][>>][..]Tue Jan 4 11:09:54 EST 2011

Following up on the references from Tim Stilson's PhD[2] thesis. Already have the book "Finite Difference Equations"[1] from 1961 by Levy and Lessman. ( I found the original in a used book store without ISBN, [1] is the Dover reissue. ) The references Tim mentions are [3][4][5][6][7][8] (books and paywalled papers). I found one accessible paper on the subject[9]. For x[k] a real valued sequence |N - > |R, we define the operator z: z x[k] = x[k+1] Then we can define d = z - 1 or d x[k] = x[k+1] - x[k] These are normalized equations with time step equal to 1. Naively relating d to the s-plane requires s/f_s normalization. Before getting into cargo cult DSP math, one needs to keep in mind that: * The d and z transforms of a discrete filter contain the same information: they are merely described in terms of different primitives: the unit integrator (accumulator) vs. the unit delay. * In the limit case of infinite sampling rate filter coefficients of d-plane and s-plane coincide. However, naively converting between the discrete and continuous case for finite sample rates is still wrong! In fact, naive conversion is the forward difference transform wich maps s -> z - 1, and which has known stability issues for coefficients away from z = 1. More general, all the caveats concerning traditional filter discretizations still apply. One can derive reformulated versions of the well known transforms: forward/backward difference, bilinear, impulse response invariance (pole mapping) and pole-zero mapping. These then look like naive (f_s scaled) direct s -> d mappings + some correction terms O(s / f_s). The main use seems to be one of implementation: coefficient sensitivity will be less in d-plane formulation for low-frequency poles and zeros. Of course one can start by simply substituting z = d + 1 in a fully expanded z-plane formulation, but that is not very insightful. Instead we do the following: * Factor the z-plane formulation first as a product of first order terms (z-z_k), including complementary complex ones. Performing the substitution in this form yields terms (d + 1 - z_k). We name d_k = 1 - z_k. For low-frequency phenomena it happens that the z_k are all close to one meaning the d_k are small in magnitude. So essentially, a formulation in d-plane form eliminates a subtraction that is the cause of loss of precision in the z-plane formulation. * 1st order IIR sections in the form of b/(d-a) can be implemented directly using leaky integrators[11]. A state variable 2nd order section can be used to implement the complex conjugate 1st order sections. For IIR filter design (when everything is analytic) there is little practical difference between z-plane and d-plane approach: simply transform (analytically) when you're done. For FIR filter design which is mostly numerical in nature, the d-plane approach seems also useful for avoiding numerical instabilities. I.e. to formulate the optimization problem in such a way that the FIR filter appears in factored form, instead of multiplied out. ( From my own experience I've found that FIR filter design problems involving low frequencies are seriously plagued by convergence issues due to insufficient numerical precision. I recall that using a spectral factorization method to design an FFT window filter I would obtain results that where clearly wrong after visual inspection, i.e. one coefficient value "sticking out" while there should be overall symmetry. It would surprise me if this hasn't been realized long before. Look into ladder FIR filters. ) [1] isbn://0486672603 [2] https://ccrma.stanford.edu/~stilti/papers/Welcome.html [3] http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=1104162 [4] http://md1.csa.com/partners/viewrecord.php?requester=gs&collection=TRD&recid=2270973CI&q=&uid=790228084&setcookie=yes [5] isbn://9780817639341 [6] http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=0123294 [7] http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=73564 [8] http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=219661 [9] http://sites.google.com/site/jamesgibsonhomepage/projects/DeltaOperatorCaseStudy.pdf [11] entry://20110104-124734

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