FIR Filter Design via Spectral Factorization and Convex Optimization

Wu, Shao-Po; Boyd, Stephen; Vandenberghe, Lieven

doi:10.1007/978-1-4612-0571-5_5

Cited by 103 publications

(144 citation statements)

References 33 publications

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“…The constraint (4c) requires that the response in Θ SL (or Θ N ) is at most U 2 (θ). The constraint (4d) is sufficient to ensure that the complex weights w i can be extracted (though not uniquely) from the obtained r w by spectral factorization [8]. Here, minimum phase spectral factor is used † .…”

Section: The Proposed Methodsmentioning

confidence: 99%

“…It is a common practice to approximate the semi-infinite constraints (4b)-(4d) by discretizing θ, like in [8,9]. This does not affect the convexity of the resulting constraints.…”

Section: Reformulation Of the Proposed Methods Via Lmi Constraintsmentioning

confidence: 99%

“…In contrast, the use of the LMI constraint in the proposed method represents the semi-infinite magnitude response constraint on the beampattern in a finite and convex manner. Due to this finite and precise representation of the semi-infinite constraint, there is no need to discretize the beampattern region, which are typically required by the methods of [7][8][9] so as to approximate the semi-infinite constraint. As such, the synthesized beampattern by the proposed method conforms to the prescribed specifications of an arbitrary beampattern strictly, unlike [7].…”

Section: Introductionmentioning

confidence: 99%

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Beampattern Synthesis With Linear Matrix Inequalities Using Minimal Array Sensors

Nai¹,

Ser²,

Yu³

et al. 2009

PIER M

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Abstract-A new beampattern synthesis formulation is proposed to compute the minimum number of array sensors required. In order to satisfy all the prescribed specifications of the beampattern, the proposed method imposes linear matrix inequality (LMI) constraints on the beampattern as developed by Davidson et al., which remove the need to discretize the beampattern region. As the proposed formulation is quasi-convex, an iterative procedure is used to decompose it into a systematic sequence of convex feasibility problems, in order to find the minimum number of sensors. The proposed method guarantees convergence if the globally optimum solution lies in the search interval, which is easy to ensure at the start of the search.Corresponding author: S. E. Nai (NAIS0001@ntu.edu.sg). 166Nai et al.

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Section: The Proposed Methodsmentioning

confidence: 99%

“…It is a common practice to approximate the semi-infinite constraints (4b)-(4d) by discretizing θ, like in [8,9]. This does not affect the convexity of the resulting constraints.…”

Section: Reformulation Of the Proposed Methods Via Lmi Constraintsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Beampattern Synthesis With Linear Matrix Inequalities Using Minimal Array Sensors

Nai¹,

Ser²,

Yu³

et al. 2009

PIER M

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show abstract

“…Examples include antenna array synthesis (see, e.g., [12,13,16]), FIR filter design (see, e.g., [4,23,24]), and coronagraph design (see, e.g., [8,19,11,10,17,20,7,9,14]). If the design function f can be constrained to vanish outside a compact interval C = (−a, a) of the real line centered at the origin, then we can write the Fourier transform as f (ξ) = c(x)f (x)dx subject to −ε ≤ f (ξ) ≤ ε,…”

Section: Fourier Transforms In Engineeringmentioning

confidence: 99%

Fast Fourier optimization

Vanderbei

2012

Math. Prog. Comp.

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Many interesting and fundamentally practical optimization problems, ranging from optics, to signal processing, to radar and acoustics, involve constraints on the Fourier transform of a function. It is well-known that the fast Fourier transform (fft) is a recursive algorithm that can dramatically improve the efficiency for computing the discrete Fourier transform. However, because it is recursive, it is difficult to embed into a linear optimization problem. In this paper, we explain the main idea behind the fast Fourier transform and show how to adapt it in such a manner as to make it encodable as constraints in an optimization problem. We demonstrate a real-world problem from the field of high-contrast imaging. On this problem, dramatic improvements are translated to an ability to solve problems with a much finer grid of discretized points. As we shall show, in general, the "fast Fourier" version of the optimization constraints produces a larger but sparser constraint matrix and therefore one can think of the fast Fourier transform as a method of sparsifying the constraints in an optimization problem, which is usually a good thing.

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“…Therefore, it can be readily solved using interior-point methods; see, e.g., [2], [5], [6]. Semidefinite programming (SDP) techniques have also been used in the FIR filter design; see, e.g., [7].…”

Section: A Chebyshev Fir Equalizationmentioning

confidence: 99%

Robust Chebyshev FIR Equalization

Mutapcic

Kim

Boyd

2007

IEEE GLOBECOM 2007-2007 IEEE Global Telecommunications Conference

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Abstract-In Chebyshev finite-impulse response (FIR) equalization, we design an FIR filter that minimizes the Chebyshev equalization error, i.e., the maximum absolute deviation between the equalized and the desired frequency response functions, assuming the unequalized response function is known exactly. In robust Chebyshev FIR equalization, we take into account uncertainty in the unequalized response function, described as a set of possible values for the unequalized response at each frequency, by designing an FIR filter that minimizes worst-case Chebyshev equalization error over all possible unequalized response functions. When the uncertainty in unequalized response function is described by a complex uncertainty ellipsoid, at each frequency, we show that the robust Chebyshev FIR equalization design problem can be formulated as a semidefinite program (SDP), and therefore efficiently (and globally) solved. When the uncertainty is given by a complex disk, the design problem can be formulated as a second-order cone program (SOCP), which can be solved almost as fast as the nominal Chebyshev equalization problem (ignoring uncertainty). The robust equalizer design method is demonstrated with a numerical example.

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FIR Filter Design via Spectral Factorization and Convex Optimization

Cited by 103 publications

References 33 publications

Beampattern Synthesis With Linear Matrix Inequalities Using Minimal Array Sensors

Beampattern Synthesis With Linear Matrix Inequalities Using Minimal Array Sensors

Fast Fourier optimization

Robust Chebyshev FIR Equalization

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