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A frequency averaging framework for the solution of complex dynamic systems.

Lecomte C - Proc. Math. Phys. Eng. Sci. (2014)

Bottom Line: A frequency averaging framework is proposed for the solution of complex linear dynamic systems.It is remarkable that, while the mid-frequency region is usually very challenging, a smooth transition from low- through mid- and high-frequency ranges is possible and all ranges can now be considered in a single framework.An interpretation of the frequency averaging in the time domain is presented and it is explained that the average may be evaluated very efficiently in terms of system solutions.

View Article: PubMed Central - PubMed

Affiliation: Associate Member, Southampton Statistical Sciences Research Institute , University of Southampton , Southampton, UK.

ABSTRACT
A frequency averaging framework is proposed for the solution of complex linear dynamic systems. It is remarkable that, while the mid-frequency region is usually very challenging, a smooth transition from low- through mid- and high-frequency ranges is possible and all ranges can now be considered in a single framework. An interpretation of the frequency averaging in the time domain is presented and it is explained that the average may be evaluated very efficiently in terms of system solutions.

No MeSH data available.


Related in: MedlinePlus

Illustration of the two-dimensional inverse Fourier transform of the second frequency moments as described in (g). Three averaging width, a=1, 0.1 and 0.01 rad Hz, are considered as well as two kinds of second frequency moment terms: plots (a), (b) and (c) are the maps of non-zero values of the inverse Fourier transforms of the frequency averaged products, csq(g)[g1(ωA),g1(ωB)], whereas plots (d), (e) and (f) are those of the power term E[g1(ωA+Δω),g1(ωB+ Δω)H], both for a real Gaussian average. They are respectively labelled ‘’ and ‘’. While the specific g1 transfer function of §4 is used here for illustration, the same Gaussian ridge scaling functions, r(tA,tB) presented in (h), would affect the product of the exact impulse responses of any general system similarly averaged.
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RSPA20130743F7: Illustration of the two-dimensional inverse Fourier transform of the second frequency moments as described in (g). Three averaging width, a=1, 0.1 and 0.01 rad Hz, are considered as well as two kinds of second frequency moment terms: plots (a), (b) and (c) are the maps of non-zero values of the inverse Fourier transforms of the frequency averaged products, csq(g)[g1(ωA),g1(ωB)], whereas plots (d), (e) and (f) are those of the power term E[g1(ωA+Δω),g1(ωB+ Δω)H], both for a real Gaussian average. They are respectively labelled ‘’ and ‘’. While the specific g1 transfer function of §4 is used here for illustration, the same Gaussian ridge scaling functions, r(tA,tB) presented in (h), would affect the product of the exact impulse responses of any general system similarly averaged.

Mentions: Each of the two expressions (7.4) and (7.6) is a two-dimensional Fourier transform in the plane (tA,tB) of the product of an outer product of and by the weighting function, . While similar expressions would exist for other averaging distributions, the weighting function in the real Gaussian case, is remarkably also a Gaussian. Furthermore, since both impulse responses and are zero for negative value of their arguments, in the case of a causal system, and since has a negative argument in the expression of the second average, these inverse Fourier transforms of the average functions result in functions that are each non-zero only in a single quadrant. In both cases, the outer product of the impulse responses is scaled by the Gaussian ridge function, r(tA,tB), that becomes tighter and tighter for increasing values of a. The maps of the non-zero components of these functions are illustrated in figure 7 for the transfer function g1, which was introduced in §4.Figure 7.


A frequency averaging framework for the solution of complex dynamic systems.

Lecomte C - Proc. Math. Phys. Eng. Sci. (2014)

Illustration of the two-dimensional inverse Fourier transform of the second frequency moments as described in (g). Three averaging width, a=1, 0.1 and 0.01 rad Hz, are considered as well as two kinds of second frequency moment terms: plots (a), (b) and (c) are the maps of non-zero values of the inverse Fourier transforms of the frequency averaged products, csq(g)[g1(ωA),g1(ωB)], whereas plots (d), (e) and (f) are those of the power term E[g1(ωA+Δω),g1(ωB+ Δω)H], both for a real Gaussian average. They are respectively labelled ‘’ and ‘’. While the specific g1 transfer function of §4 is used here for illustration, the same Gaussian ridge scaling functions, r(tA,tB) presented in (h), would affect the product of the exact impulse responses of any general system similarly averaged.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
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getmorefigures.php?uid=PMC4042715&req=5

RSPA20130743F7: Illustration of the two-dimensional inverse Fourier transform of the second frequency moments as described in (g). Three averaging width, a=1, 0.1 and 0.01 rad Hz, are considered as well as two kinds of second frequency moment terms: plots (a), (b) and (c) are the maps of non-zero values of the inverse Fourier transforms of the frequency averaged products, csq(g)[g1(ωA),g1(ωB)], whereas plots (d), (e) and (f) are those of the power term E[g1(ωA+Δω),g1(ωB+ Δω)H], both for a real Gaussian average. They are respectively labelled ‘’ and ‘’. While the specific g1 transfer function of §4 is used here for illustration, the same Gaussian ridge scaling functions, r(tA,tB) presented in (h), would affect the product of the exact impulse responses of any general system similarly averaged.
Mentions: Each of the two expressions (7.4) and (7.6) is a two-dimensional Fourier transform in the plane (tA,tB) of the product of an outer product of and by the weighting function, . While similar expressions would exist for other averaging distributions, the weighting function in the real Gaussian case, is remarkably also a Gaussian. Furthermore, since both impulse responses and are zero for negative value of their arguments, in the case of a causal system, and since has a negative argument in the expression of the second average, these inverse Fourier transforms of the average functions result in functions that are each non-zero only in a single quadrant. In both cases, the outer product of the impulse responses is scaled by the Gaussian ridge function, r(tA,tB), that becomes tighter and tighter for increasing values of a. The maps of the non-zero components of these functions are illustrated in figure 7 for the transfer function g1, which was introduced in §4.Figure 7.

Bottom Line: A frequency averaging framework is proposed for the solution of complex linear dynamic systems.It is remarkable that, while the mid-frequency region is usually very challenging, a smooth transition from low- through mid- and high-frequency ranges is possible and all ranges can now be considered in a single framework.An interpretation of the frequency averaging in the time domain is presented and it is explained that the average may be evaluated very efficiently in terms of system solutions.

View Article: PubMed Central - PubMed

Affiliation: Associate Member, Southampton Statistical Sciences Research Institute , University of Southampton , Southampton, UK.

ABSTRACT
A frequency averaging framework is proposed for the solution of complex linear dynamic systems. It is remarkable that, while the mid-frequency region is usually very challenging, a smooth transition from low- through mid- and high-frequency ranges is possible and all ranges can now be considered in a single framework. An interpretation of the frequency averaging in the time domain is presented and it is explained that the average may be evaluated very efficiently in terms of system solutions.

No MeSH data available.


Related in: MedlinePlus