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Cited by 5Digital Filter Design and ImplementationMathematical Analysis and Transform MethodsAdvanced Adaptive Filtering Techniques

Solution and cost analysis of general multi-channel and multi-stage filtering circuits

Open access
Kutay, M.A., Ozaktas, H., Erden, M.F., Ozaktas, H.M., Arikan, O.·2002·Proceedings of the IEEE-SP International Symposium on Time-Frequency and Time-Scale Analysis (Cat. No.98TH8380)
DOI 10.1109/tfsa.1998.721466
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Abstract

The fractional Fourier domain multi-channel and multi-stage filtering configurations that have been previously proposed enable us to obtain either exact realizations or useful approximations of linear systems or matrix-vector products in many different applications. We discuss the solution and cost analysis for these configurations. It is shown that the problem can be reduced to a least squares problem which can be solved with fast iterative techniques.

Integrity signals

Registry

Crossref · 10.1109/tfsa.1998.721466

Cites

12 of 12 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

0 of 1 cited sources checked · 0 supported · 0 flagged

  • Findingnone given

    The formulation and cost analysis can be generalized to arbitrary rectangular matrices T of dimension Ng x Nf of arbitrary rank R.

    “In this work we provide a generalization of both the formulation and cost analysis to arbitrary rectangular matrices T of dimension N , x N f of arbitrary rank R.”
    Quote in the paperSubstantial claim
  • Findingown data

    In the image restoration example, multi-stage filtering with M=8 yields a mean-square estimation error of 4%.

    “The mean-square estimation error is 12% in the multi-stage case and 16% in the multi-channel case with M = 4, and 4% in the multistage case and 6% in the multi-channel case with M = 8.”
    Quote in the paperNot a substantial claim
  • Findingown data

    In the image restoration example, ordinary Fourier domain filtering yields a mean-square estimation error of 34%.

    “Ordinary Fourier domain filtering gives very poor results, resulting in an error of 34%.”
    Quote in the paperSubstantial claim
  • Findingown data

    The denoising result corresponds to an SNR improvement of 25.8 dB.

    “This corresponds to an SNR improvement of 25.8 dB.”
    Quote in the paperNot a substantial claim
  • Findingreasoning

    If the goal is to minimize the Frobenius norm difference between T and a specified matrix H, the problem can be exactly posed as a least-squares optimization problem leading to normal equations or solved with standard techniques.

    “For instance, if we wish to minimize IIT-HIIF, where H is a specified matrix, the problem can be exactly posed as a least-squares optimization problem leading to an associated set of normal equations or which can be solved with other standard techniques.”
    Quote in the paperSubstantial claim
  • May combine two claims. 4
  • Classified as probably not a substantial claim. 4

  • May combine two claims.

    Findingreasoning

    Choosing M larger results in both higher cost and higher flexibility.

    “If we choose M larger, cost and flexibility are both higher; as M approaches N I the number of degrees of freedom approaches that of a general linear system.”
    Quote in the paperSubstantial claim

  • Classified as probably not a substantial claim.

    Findingnone given

    The filter vector h that minimizes the squared error satisfies the normal equations associated with the least squares problem.

    “The filter vector h which minimizes l l €J -& [ I 2 is known to satisfy the so-called normal equations associated with the least squares problem:”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingnone given

    In the multi-stage case, the overall kernel T depends nonlinearly on the filter coefficients hk[j], so the optimization problem is much more difficult to solve.

    “In the multi-stage case, the overall kernel T depends nonlinearly on the filter coefficients hk [ j ] , so that solution of the optimization problem arising in this case is much more difficult.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning

    Choosing M small results in both low cost and low flexibility.

    “If we choose M to be small, cost and flexibility are both low; M = 1 corresponds to single-stage filtering.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Findingnone given

    M=1 corresponds to single-stage filtering.

    “If we choose M to be small, cost and flexibility are both low; M = 1 corresponds to single-stage filtering.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S4 (based).

    Premisenone given

    In many applications, filtering in fractional rather than ordinary Fourier domains achieves greater flexibility and performance at no additional cost.

    “In a variety of applications, greater flexibility and performance can be achieved at no additional cost, by filtering in fractional rather than ordinary Fourier domains”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationnone given

    These results represent significant improvements with respect to single domain filtering but are much cheaper to implement than general linear filtering.

    “These results represent significant improvements with respect to single domain filtering but are much cheaper to implement than general linear filtering.”
    Quote in the paperNot a substantial claim
    • Rests on an assumption: Most matrices are not wholly arbitrary and exhibit some kind of internal structure, although that structure may not be easy to identify or characterize.

  • Probably not a substantial claim: may not pass S4 (based).

    Premisenone given

    General linear systems have Ng Nf degrees of freedom.

    “The M-channel or hl-stage filtering configuration has about M N degrees of freedom, as opposed to genera!linear systems which have Ng N j degrees of freedom and shift-invariant systems which have about N degrees of freedom.”
    Quote in the paperNot a substantial claim
8 of 8

  • The paper relies on this without support.

    Most matrices are not wholly arbitrary and exhibit some kind of internal structure, although that structure may not be easy to identify or characterize.

    “In practice, most matrices are not wholly arbitrary and exhibit some kind of internal structure, although that structure may not be easy to identify or characterize.”

  • The paper relies on this without support.

    The number of channels or stages M=10 is sufficient to obtain useful approximations to the desired linear system matrix in a variety of applications.

    “The number of channels or stages M = 10 assumed in these plots has been shown to be sufficient to obtain useful approximations to the desired linear syst,em matrix in a variety of applications.”
2 of 2

Cites

Flagged first, then by citations

Research works printed but not linked: 1

  1. candan, 1998, The fractional discrete Fourier transform

Cited by

Sorted by citations

Related

Authors
Kutay, M. A., Ozaktas, H., Erden, M. F., Ozaktas, H. M., & Arikan, O.
Affiliations
  • Bilkent University (TR)
  • Tampere University of Technology (FI)
ORCID
Kutay, M.A.
Venue
Proceedings of the IEEE-SP International Symposium on Time-Frequency and Time-Scale Analysis (Cat. No.98TH8380)
Publisher
IEEE
Published
2002-11-27
Volume
35
Pages
481-484
DOI
10.1109/tfsa.1998.721466
OpenAlex
W1558237505
Cited by
5
References
12
FWCI
0.63
Type
conference paper
Language
en
Open access
Locations
  • IEEE-SP International Symposium on Time-Frequency and Time-Scale Analysis (publishedVersion)
  • Bilkent University Institutional Repository (Bilkent University) (submittedVersion)
Topics
  • Physical Sciences › Computer Science › Signal Processing › Digital Filter Design and Implementation
  • Physical Sciences › Mathematics › Applied Mathematics › Mathematical Analysis and Transform Methods
  • Physical Sciences › Engineering › Computational Mechanics › Advanced Adaptive Filtering Techniques
Keywords
fractional Fourier transform, multichannel filtering, multi-stage filtering, cost analysis, least squares, fast iterative solvers, matrix-vector multiplication

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