Complex FIR coefficients
in [DSP]
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From: Mikolaj on 7 Apr 2010 10:32 on 07-04-2010 o 13:03:04 Mikolaj <sterowanie_komputerowe(a)poczta.onet.pl> wrote: > It seems that sometimes, luckily > when you use complex (compressed, packed, combined, compact) > way of describing few dependent physical things the imaginary part of that complex representation (additional dimension used for compression) > can be human understandable > and could have interpretation. > > But you can always decompress complex matrix > to it's scalar version equations. -- Mikolaj
From: Michael Plante on 7 Apr 2010 14:17 glen wrote: >Mikolaj <sterowanie_komputerowe(a)poczta.onet.pl> wrote: >(snip) > >> I can just put 'j' or 'i' befor any physical value >> and than interpret it at will. >> I can fit j to anything I want. > >> But when we have description of something >> than partial imaginary result numbers >> can have no physical interpretation. > >I believe that in some cases complex physical quantities >that are in exponents have a physical interpretation. >As examples, the dielectric constant and its square root, >the index of refraction. Other than in exponents, >the use of complex numbers for physical quantities, >such as describing phase shifts, seems more of a >convenience, and not something with a physical >interpretation. Complex eigenvalues often have physical interpretation. A non-hermitian hamiltonian is sometimes used when particles leave the group of states being considered (for example, atoms that become ionized, and assume that you no longer care about those as part of your ensemble, so you don't consider those states). It is a bit of a hack, but the point is that the complex eigenvalues in that case then have the interpretation of loss over time (where, for other situations, "loss" might be of either sign).
From: glen herrmannsfeldt on 7 Apr 2010 21:29 Michael Plante <michael.plante(a)n_o_s_p_a_m.gmail.com> wrote: (snip, I wrote) >>I believe that in some cases complex physical quantities >>that are in exponents have a physical interpretation. (snip) > Complex eigenvalues often have physical interpretation. A non-hermitian > hamiltonian is sometimes used when particles leave the group of states > being considered (for example, atoms that become ionized, and assume that > you no longer care about those as part of your ensemble, so you don't > consider those states). It is a bit of a hack, but the point is that the > complex eigenvalues in that case then have the interpretation of loss over > time (where, for other situations, "loss" might be of either sign). Are these solutions of differential equations such that the complex value is in an exp()? If so, then the imaginary terms (multiplied by i) are the decay (or absorption) term. -- glen
From: Michael Plante on 8 Apr 2010 09:11 >Michael Plante <michael.plante(a)n_o_s_p_a_m.gmail.com> wrote: >(snip, I wrote) > >>>I believe that in some cases complex physical quantities >>>that are in exponents have a physical interpretation. >(snip) > >> Complex eigenvalues often have physical interpretation. A non-hermitian >> hamiltonian is sometimes used when particles leave the group of states >> being considered (for example, atoms that become ionized, and assume that >> you no longer care about those as part of your ensemble, so you don't >> consider those states). It is a bit of a hack, but the point is that the >> complex eigenvalues in that case then have the interpretation of loss over >> time (where, for other situations, "loss" might be of either sign). > >Are these solutions of differential equations such that the >complex value is in an exp()? If so, then the imaginary terms >(multiplied by i) are the decay (or absorption) term. It's similar to your example. One could find a basis where the time evolution operator U=exp(-i.H.t/hb) is diagonal. So it could be seen that way.
From: Michael Plante on 8 Apr 2010 09:48
Michael Plante wrote: >glen wrote: >>Michael Plante <michael.plante(a)n_o_s_p_a_m.gmail.com> wrote: >>(snip, I wrote) >> >>>>I believe that in some cases complex physical quantities >>>>that are in exponents have a physical interpretation. >>(snip) >> >>> Complex eigenvalues often have physical interpretation. A >non-hermitian >>> hamiltonian is sometimes used when particles leave the group of states >>> being considered (for example, atoms that become ionized, and assume >that >>> you no longer care about those as part of your ensemble, so you don't >>> consider those states). It is a bit of a hack, but the point is that >the >>> complex eigenvalues in that case then have the interpretation of loss >over >>> time (where, for other situations, "loss" might be of either sign). >> >>Are these solutions of differential equations such that the >>complex value is in an exp()? If so, then the imaginary terms >>(multiplied by i) are the decay (or absorption) term. > >It's similar to your example. One could find a basis where the time >evolution operator U=exp(-i.H.t/hb) is diagonal. So it could be seen that >way. While what I wrote was inspired by your post, my point was not so much about the ability to reduce it to that form, which depends on being able to find a clean solution, since this sort of perturbation is probably less necessary in simple cases. Rather, the value is in being able to directly interpret perturbations to the Hamiltonian without trying to find a solution. An interesting application is when a suitable "gain" mechanism is present (not explicitly included) to balance this loss, but the the gain introduces unpolarized atoms, whereas the "loss" removes from consideration whatever's available. Then the interpretation of this perturbation is depolarization of the ensemble over time. Michael |