Synthesis and Realization of 3-D Orthogonal FIR Filters Using Pipeline Structures

Circuits, Systems, and Signal Processing, Aug 2017

The authors present a novel design algorithm for 3-D orthogonal filters. Both separable and non-separable cases are discussed. In the separable case, the synthesis leads to a cascade connection of 1-D systems. In the latter case, one obtains 2-D systems followed by a 1-D one. Realization techniques for these systems are presented which utilize Givens rotations and delay elements. The results are illustrated by examples of separable and non-separable 3-D system designs, i.e., Gaussian and Laplacian filters.

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Synthesis and Realization of 3-D Orthogonal FIR Filters Using Pipeline Structures

Circuits, Systems, and Signal Processing April 2018, Volume 37, Issue 4, pp 1669–1691 | Cite as Synthesis and Realization of 3-D Orthogonal FIR Filters Using Pipeline Structures AuthorsAuthors and affiliations P. PoczekajłoR. T. Wirski Open Access Article First Online: 04 August 2017 470 Downloads Abstract The authors present a novel design algorithm for 3-D orthogonal filters. Both separable and non-separable cases are discussed. In the separable case, the synthesis leads to a cascade connection of 1-D systems. In the latter case, one obtains 2-D systems followed by a 1-D one. Realization techniques for these systems are presented which utilize Givens rotations and delay elements. The results are illustrated by examples of separable and non-separable 3-D system designs, i.e., Gaussian and Laplacian filters. Keywords3-D Orthogonal FIR filter Synthesis Pipeline structure Givens rotation  1 Introduction Since the first pulse-code modulation transmission of digitally quantized speech, in World War II, digital signal processing (DSP) began to proliferate to all areas of human life. A classic DSP is based on linear systems described by impulse response functions and transfer functions implemented by structures built from adders, multipliers, and unit delays. Another approach was initiated by [31], known as the state space approach. It was also extended to the 2-D case by Roesser [18], as well as to three-dimensional (3-D) [9]. The steady increase in computational power encourages applying DSP techniques to multidimensional processing. However, the n-dimensional (n-D) DSP development has encountered difficulties caused by n-D polynomials [5]. Namely, there is no straightforward generalization of the fundamental theorem of algebra to higher dimensions. Classical digital systems are known to possess poor parameters under finite-precision arithmetic, like the sensitivity of the frequency response to changes in the structural parameters, noise, intrinsic oscillations, and limit cycles. These effects have led to the invention of wave filters [6] and orthogonal filters [2, 3]. The most common approach to orthogonal filter synthesis is a transfer function decomposition and the state space approach. When it comes to multidimensional DSP, the former technique is of a limited use due to the n-D polynomials. In contrast, the latter provides an opportunity to extend 1-D state space techniques to higher dimensions, thanks to the 2-D, 3-D, and possibly n-D state space equations. The state space approach to lossless systems was initiated by the famous paper [30], where paraunitary matrix synthesis techniques were developed for the 2-D transfer function of a continuous system. The state space approach was also used to develop 2-D orthogonal filter synthesis [16] and simplified to cover a class of separable-denominator orthogonal filters [28] which found to be useful in real-time processing [21]. Nowadays, one can observe that a processed data becomes n-D like video, multichannel audio, machine vision, to name a few. The 3-D processing is especially important in medicine [1] and image/video processing [5, 10] but also finds applications in other areas like material structures [12]. DSP synthesis is based on difference equations usually transformed by the \(\mathcal {Z}\) transform. For the 3-D function \(f(x_1, x_2 ,x_3)\), this is given by $$\begin{aligned} \begin{aligned} \mathcal {Z}_3 \left\{ f(x_1, x_2 ,x_3) \right\} = \sum _{x_1=-\infty }^\infty \sum _{x_2=-\infty }^\infty \sum _{x_3=-\infty }^\infty f(x_1, x_2 ,x_3) z_1^{-x_1} z_2^{-x_2} z_3^{-x_3}, \end{aligned} \end{aligned}$$ (1) where \(z_1\), \(z_2\), and \(z_3\) are complex numbers. Linear time invariant filters are usually classified into recursive and non-recursive. The latter, called finite impulse response (FIR) filters, are very popular due to their simplicity and natural stability. Typically, they are described by a transfer function which is the \(\mathcal {Z}\) transform of its impulse response. For the 1-D case, it is given by $$\begin{aligned} T(z)=a_{0}+a_{1}z^{-1}+ \cdots +a_nz^{-n}=\sum _{i=0}^na_{i}z^{-i}, \end{aligned}$$ (2) where \(a_0,\ldots ,a_n\) are real constant coefficients. In the 3-D case, the transfer function of an FIR filter extends to $$\begin{aligned} T(z_{h},z_{v},z_{d})=\sum _{i=0}^{n} \sum _{j=0}^{m} \sum _{k=0}^{l} a_{ijk}z_{h}^{-i}z_{v}^{-j}z_{d}^{-k}. \end{aligned}$$ (3) In this paper, we deal with a class of orthogonal filters. Introducing the energy of a 3-D real vector function \(f(x_1, x_2 ,x_3)\), in the form $$\begin{aligned} \mathcal {E}\{f(x_1, x_2 ,x_3)\}=\sum _{x_1=-\infty }^\infty \sum _{x_2=-\infty }^\infty \sum _{x_3=-\infty }^\infty f^\mathrm{T}(x_1, x_2 ,x_3)f(x_1, x_2 ,x_3), \end{aligned}$$ (4) we define an orthogonal filter to be a system which preserves the energy, i.e., the input energy equals the output energy. Technically, we are about to find a net of Givens rotati (...truncated)


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P. Poczekajło, R. T. Wirski. Synthesis and Realization of 3-D Orthogonal FIR Filters Using Pipeline Structures, Circuits, Systems, and Signal Processing, 2017, pp. 1669-1691, Volume 37, Issue 4, DOI: 10.1007/s00034-017-0618-2