- Email: [email protected]
Comment on “Commutativity of block decimators and expanders with arbitrary rational sampling ratios and block lengths”
JID:YDSPR AID:1616 /DIS
[m5G; v 1.134; Prn:27/05/2014; 15:35] P.1 (1-2)
Digital Signal Processing ••• (••••) •••–•••
Contents lists available at ScienceDirect
Digital Signal Processing www.elsevier.com/locate/dsp
Comment on “Commutativity of block decimators and expanders with arbitrary rational sampling ratios and block lengths” Didier Pinchon Institute of Mathematics, University Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex, France
a r t i c l e
i n f o
a b s t r a c t
Article history: Available online xxxx
In this short comment paper, it is proved that the unique theorem of [1] is not valid. © 2014 Elsevier Inc. All rights reserved.
Keywords: Block sampling Decimation Expansion Commutativity
1. Introduction
However ↓(1, 2) and ↑(2, 52 ) do not commute as shown in (1).
In [1], the authors give a series of three conditions, that are claimed to be equivalent, to the commutativity of a block decimator ↓(q1 , p 1 ) and a block expander ↑(q2 , p 2 ). Up to this publication, a characterization of commuting pairs of block decimators and expanders was still an open problem. The notations of this comment paper are strictly the same as in [1] and the reader is thus referred to this paper. In Section 2 of this comment, a simple counter-example is given proving that Theorem 1 in [1] is wrong, while in Section 3 it is explained how to produce many other counter-examples simply using language C programs.
3. Producing other counter-examples
2. A counter-example As Theorem 1 in [1] may be considered as an algorithm for a boolean function with arguments q1 , p 1 , q2 , p 2 that returns a true value if and only if the three conditions are satisfied, it is straightforward to track, step by step, the values taken by the local variables. Using q1 = 1, p 1 = 2, q2 = 2, p 2 = 52 , we can notice in this way that the three conditions are satisfied.
x0 , x1⏐ , x2 , x3
⏐ ↓(1, 2) ⏐ x0⏐ , x2 ⏐ 5 ⏐ ↑(2, 2 ) x0 , x2 , 0, 0, 0
x0 , x1⏐ , x2 , x3
⏐ ↑(2, 52 ) ⏐
x0 , x1 , 0, 0, 0⏐ , x2 , x3 , 0, 0, 0
⏐ ↓(1, 2) ⏐ x0 , 0, 0, x3 , 0
DOI of original article: http://dx.doi.org/10.1016/j.dsp.2012.03.002. E-mail address: [email protected]. http://dx.doi.org/10.1016/j.dsp.2014.05.010 1051-2004/© 2014 Elsevier Inc. All rights reserved.
(1)
For general parameters q1 , p 1 , q2 , p 2 , the left part of (1) may be implemented as a partial function f 1 : i ∈ D ( f 1 ) ⊂ Z → f 1 (i ) ∈ Z such that i ∈ Z \ D ( f 1 ) means that the symbol xi is discarded and y f 1 (i ) = xi where yn is the resulting sequence by the composed function ↑(q2 , p 2 ) ◦ ↓(q1 , p 1 ). yn is set to 0 when n is not in the image of f 1 . In a similar way, we define a function f 2 to implement the composed function ↓(q1 , p 1 ) ◦ ↑(q2 , p 2 ) corresponding to the right part of (1). Because of their periodic behavior, it is then sufficient to check the identity of f 1 and f 2 , definition domains and values, for 0 ≤ i ≤ q1 p 1 q2 − 1, to prove the commutativity of ↓(q1 , p 1 ) and ↑(q2 , p 2 ). As it is a thankless task to execute all the calculation by hand, functions f 1 , f 2 and the function implementing Theorem 1 of [1] have been coded in the C programming language. Using these functions, a program theorem1 with one argument B detects all the values of the parameters which supply counter-examples to Theorem 1 for q1 p 1 ≤ B and q2 p 2 ≤ B. For B = 40, we get 4396 counter-examples. Source programs and their documentation are supplied and may be downloaded online alongside the electronic version of this comment paper. References ´ Commutativity of block decimators and expanders [1] B.-K. Ling, C. Ho, Z. Cvetkovic, with arbitrary rational sampling ratios and block lengths, Digit. Signal Process. 22 (2012) 677–680.
Didier Pinchon received his Ph.D. degree in mathematics, in 1979, from Université Pierre et Marie Curie, Paris for contributions in Ergodic Theory. From 1973 to 1986, he worked in Laboratoire de Probabilité, and
JID:YDSPR
2
AID:1616 /DIS
[m5G; v 1.134; Prn:27/05/2014; 15:35] P.2 (1-2)
D. Pinchon / Digital Signal Processing ••• (••••) •••–•••
then in Laboratoire d’Analyse Fonctionnnelle in this university as a researcher of National Center of Scientific Research (CNRS). From 1986 to 1990, he was hired by IBM France in IBM Paris Scientific Center and turned his interest to computer algebra and its applications to industrial
problems. Back in CNRS, in 1990, he joined the Institute of Mathematics of Toulouse, in 1994, and his current research area is still applications of computer algebra, mainly in computational quantum chemistry and signal processing.
[m5G; v 1.134; Prn:27/05/2014; 15:35] P.1 (1-2)
Digital Signal Processing ••• (••••) •••–•••
Contents lists available at ScienceDirect
Digital Signal Processing www.elsevier.com/locate/dsp
Comment on “Commutativity of block decimators and expanders with arbitrary rational sampling ratios and block lengths” Didier Pinchon Institute of Mathematics, University Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex, France
a r t i c l e
i n f o
a b s t r a c t
Article history: Available online xxxx
In this short comment paper, it is proved that the unique theorem of [1] is not valid. © 2014 Elsevier Inc. All rights reserved.
Keywords: Block sampling Decimation Expansion Commutativity
1. Introduction
However ↓(1, 2) and ↑(2, 52 ) do not commute as shown in (1).
In [1], the authors give a series of three conditions, that are claimed to be equivalent, to the commutativity of a block decimator ↓(q1 , p 1 ) and a block expander ↑(q2 , p 2 ). Up to this publication, a characterization of commuting pairs of block decimators and expanders was still an open problem. The notations of this comment paper are strictly the same as in [1] and the reader is thus referred to this paper. In Section 2 of this comment, a simple counter-example is given proving that Theorem 1 in [1] is wrong, while in Section 3 it is explained how to produce many other counter-examples simply using language C programs.
3. Producing other counter-examples
2. A counter-example As Theorem 1 in [1] may be considered as an algorithm for a boolean function with arguments q1 , p 1 , q2 , p 2 that returns a true value if and only if the three conditions are satisfied, it is straightforward to track, step by step, the values taken by the local variables. Using q1 = 1, p 1 = 2, q2 = 2, p 2 = 52 , we can notice in this way that the three conditions are satisfied.
x0 , x1⏐ , x2 , x3
⏐ ↓(1, 2) ⏐ x0⏐ , x2 ⏐ 5 ⏐ ↑(2, 2 ) x0 , x2 , 0, 0, 0
x0 , x1⏐ , x2 , x3
⏐ ↑(2, 52 ) ⏐
x0 , x1 , 0, 0, 0⏐ , x2 , x3 , 0, 0, 0
⏐ ↓(1, 2) ⏐ x0 , 0, 0, x3 , 0
DOI of original article: http://dx.doi.org/10.1016/j.dsp.2012.03.002. E-mail address: [email protected]. http://dx.doi.org/10.1016/j.dsp.2014.05.010 1051-2004/© 2014 Elsevier Inc. All rights reserved.
(1)
For general parameters q1 , p 1 , q2 , p 2 , the left part of (1) may be implemented as a partial function f 1 : i ∈ D ( f 1 ) ⊂ Z → f 1 (i ) ∈ Z such that i ∈ Z \ D ( f 1 ) means that the symbol xi is discarded and y f 1 (i ) = xi where yn is the resulting sequence by the composed function ↑(q2 , p 2 ) ◦ ↓(q1 , p 1 ). yn is set to 0 when n is not in the image of f 1 . In a similar way, we define a function f 2 to implement the composed function ↓(q1 , p 1 ) ◦ ↑(q2 , p 2 ) corresponding to the right part of (1). Because of their periodic behavior, it is then sufficient to check the identity of f 1 and f 2 , definition domains and values, for 0 ≤ i ≤ q1 p 1 q2 − 1, to prove the commutativity of ↓(q1 , p 1 ) and ↑(q2 , p 2 ). As it is a thankless task to execute all the calculation by hand, functions f 1 , f 2 and the function implementing Theorem 1 of [1] have been coded in the C programming language. Using these functions, a program theorem1 with one argument B detects all the values of the parameters which supply counter-examples to Theorem 1 for q1 p 1 ≤ B and q2 p 2 ≤ B. For B = 40, we get 4396 counter-examples. Source programs and their documentation are supplied and may be downloaded online alongside the electronic version of this comment paper. References ´ Commutativity of block decimators and expanders [1] B.-K. Ling, C. Ho, Z. Cvetkovic, with arbitrary rational sampling ratios and block lengths, Digit. Signal Process. 22 (2012) 677–680.
Didier Pinchon received his Ph.D. degree in mathematics, in 1979, from Université Pierre et Marie Curie, Paris for contributions in Ergodic Theory. From 1973 to 1986, he worked in Laboratoire de Probabilité, and
JID:YDSPR
2
AID:1616 /DIS
[m5G; v 1.134; Prn:27/05/2014; 15:35] P.2 (1-2)
D. Pinchon / Digital Signal Processing ••• (••••) •••–•••
then in Laboratoire d’Analyse Fonctionnnelle in this university as a researcher of National Center of Scientific Research (CNRS). From 1986 to 1990, he was hired by IBM France in IBM Paris Scientific Center and turned his interest to computer algebra and its applications to industrial
problems. Back in CNRS, in 1990, he joined the Institute of Mathematics of Toulouse, in 1994, and his current research area is still applications of computer algebra, mainly in computational quantum chemistry and signal processing.