In this letter, we have extended the existing results on the admissible set of periodic symbolic sequences of a second-order digital filter with marginally stable system matrix to the unstable case. Based on this result, the initial conditions can be computed using the symbolic sequences. The truncation error of the representation of an initial condition due to the use of a finite number of symbols is studied

Ho, Yuk-Fan

Ling, Wing-Kuen

Tam, Peter Kwong-Shun

English

University of Lincoln Institutional Repository

 1 Admissibility of Unstable Second-order Digital Filters with Two’s Complement Arithmetic  Bingo Wing-Kuen Ling, Charlotte Yuk-Fan Ho and Peter Kwong-Shun Tam Department of Electronic and Information Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China  SUMMARY In this letter, we have extended the existing results on the admissible set of periodic symbolic sequences of a second-order digital filter with marginally stable system matrix to the unstable case. Based on this result, the initial conditions can be computed using the symbolic sequences. The truncation error of the representation of an initial condition due to the use of a finite number of symbols is studied.  KEY WORDS: Admissibility, two’s complement arithmetic, symbolic sequences.  1. INTRODUCTION It is well known that the autonomous response of a second-order digital filter with marginally stable system matrix implemented using two’s complement arithmetic may exhibit chaotic behaviors, dependent on the initial conditions [1]-[4]. In order to analyze these complex behaviors, symbolic sequences are introduced. The symbolic sequences depend on the initial conditions. It is found that the map from the set of initial conditions to the set of symbolic sequences is neither injective nor surjective. Some researchers have worked out the admissible set of periodic sequences [2]-[4]. In this letter, we extend the results to the case with unstable system matrix and some  2 interesting phenomenon was found. The organization of this letter is as follows: The system is described in section 2. The admissible set of symbolic sequences of an unstable second-order digital filter with two’s complement arithmetic is discussed in section 3. Finally, a conclusion is summarized in section 4.  2. SYSTEM DESCRIPTION Assume a second-order digital filter with two’s complement arithmetic is realized in direct form. The state space model of the feedback system can be represented as follows:          kxakxbfkxkxkxk21221111x , for 0k , (1) where a  and b  are the filter parameters,  kx1  and  kx2  are the state variables, and f  is the nonlinearity due to the use of two’s complement arithmetic. The nonlinearity f  can be modeled as:   nf  2  (2) such that 1212  nn   and Zn . (3) Hence, the state vector is confined in a square defined as follows:        11,11: 2121221 kxkxkxkxIkxkx, for 0k . (4)  By introducing the symbolic sequences, the state space model of the digital filter can further be represented as:         kskxakxbkxk21122x  (5)  3    ksk 20xA , for 0k , (6) where ab10A , (7) and    mmks ,,1,0,1,,   , for 0k , (8) in which m  is the minimum integers satisfying:     1212 21  mkxakxbm , for 0k . (9) The admissible set of periodic symbolic sequences with period M  [2] is given by      1sin2sin12cos,mod1:10MjMMjissMj , for 1,,1,0  Mi  , (10) where  2cos 1 a , (11)  qp,mod  is the reminder of qp , and     ,1,0 sss  . (12)  3. ADMISSIBILITY AND INVERTIBILITY OF A SECOND-ORDER DIGITAL FILTER WITH TWO’S COMPLEMENT ARITHMETIC Let 1  and 2  be the eignevalues of A . In this section, we assume that: 11   (13) and 12  . (14) Define  4     mmkss ,,1,0,1,,:   , (15) and 2: IS . (16) Obviously, S  is not surjective and the set   is not admissible. Lemma 1: Define    ,1,0for  2: 212111221nIjssnjjnjnjnjnb, then the set b  is admissible and bb IS 2:  is surjective. Proof: Since bs  , we have:   2121112212 Ijsnjjjjj. (17) Let    njjjjjjs1211122120x . (18) Then      jskkjjkk  101200 AxAx  (19)       kjjkjkjkjkkjjkjkjkjk jsjsjs1211121210 1211121222, for 0k . (20) Since   2121112122 Ijskjjkjkjkjk, for 0k , (21) 11  , (22) 12  , (23)  5 and   2Ik x , for 0k , (24) we have    jsjs  , for 1,,1,0  kj   and 0k . (25) This implies ss   (26) and    bsS 0x . (27) Hence, the set b  is admissible and bb IS 2:  is surjective. This completes the proof.   Lemma 2: bb IS 2:  is injective. Proof: Let     221 0,0 Ixx . (28) Assume    00 21 xx   (29) and       bbb sSS  00 21 xx . (30) Since       jskkjjkk  10111200 AxAx , for 0k , (31) and      jskkjjkk  10122200 AxAx , for 0k , (32) we have  6         00 2121 xxAxx  kkk , for 0k . (33) Since 11  , (34) 12  , (35) and     221 , Ikk xx , for 0k , (36) we have    00 21 xx  . (37) This contradicts equation (29). Hence, bS  is injective, and completing the proof.  Remark 1: According to Lemma 1 and 2, bS  is bijective. Lemma 3: Define  2: IT bb  , then bT  is bijective and    0 121112212jjjjjbjssT. Proof: By applying similar methods in Lemma 1 and 2, we can easily prove that bT  is bijective. To show    0 121112212jjjjjbjssT, (38) since   20 Ix , we have      jskkjjkk  101200 AxAx , for 0k . (39) This implies that        jskkjjk  1011200 AxAx , for 0k . (40)  7        jskkjjkkk 101120limlim0 AxAx  (41)  0 121112212jjjjjjs. (42) This completes the proof.  Remark 2:    njjnjnjnjnjsn121112212x , for 0n . Proof: Since      jskkjjkk  101200 AxAx , for 0k , (43) and      0 1211122120jjjjjbjssTx , (44) we have    njjnjnjnjnjsn121112212x , for 0n . (45)   Remark 3: For a one-dimensional case, any number  1,1x  can be represented as an M -ary number with each bit    1,,1,0,1,,1  MMjb  , that is:    01jjMjbx . (46) Define     ,1,0 bbb  , (47) and  8  bone  . (48) Since 1101 jjMM , (49)   1101  jjMjb . (50) Hence, the mapping   oneoneS  1,1:  (51) is surjective. It is well known that oneS  is injective, so oneS  is bijective. However, this is not true for the two-dimensional case. Since:     111122210 12111221 mmjjjjj (52) and 12  mba , (53) we have    111221m . (54) Hence, S  is not surjective and the set   is not admissible. However, if we confine the set   by its subset b , then we guarantee that there exists   20 Ix . Hence, bS  is surjective and the set b  is admissible. Although an infinite number of bits is required to represent x  with infinite precision, we may truncate the representation by a finite number of bits and the quantization error is bounded by the magnitude represented by the last bit. That is 111 kkjj MMM . (55) However, for the two-dimensional case, the truncation error is  9     kkkkkjjjjjmkeke111212211211122121 22. (56) Since   0lim keik  (57) and   0kei  (58) for 2,1i ,  Zk0  such that  kei  for 2,1i , are monotonically decreasing with respect to k  for 0kk  . Hence, we still can truncate the representation of  0x  using a finite number of symbols. This property suggests that an information can be coded using the successive approximation technique. Compared to the existing successive approximation coding technique, the traditional one is to code the information directly, while this coding technique is to code the symbolic sequences. The security is improved.  4. CONCLUSION In this letter, we have extended the results on the admissible set of symbolic sequences of a marginally stable second-order system to an unstable system. Based on this result, the initial conditions can be computed by the symbolic sequences directly. Moreover, the truncation error of the representation of an initial condition due to the use of finite number of symbols is studied.  ACKNOWLEDGEMENT The work described in this letter was substantially supported by The Hong Kong Polytechnic University, partly form a studentship and partly by a research grant with account number G-YD26.  10 REFERENCES [1] L. O. Chua and T. Lin, “Chaos in digital filters,” IEEE Transactions on Circuits and Systems, vol. 35, no. 6, pp. 648-658, June 1988. [2] L. O. Chua and T. Lin, “Fractal pattern of second-order non-linear digital filters: a new symbolic analysis,” International Journal of Circuit Theory and Applications, vol. 18, pp. 541-550, 1990. [3] Z. Galias and M. J. Ogorzalek, “On symbolic dynamics of a chaotic second-order digital filter,” International Journal of Circuit Theory and Applications, vol. 20, pp. 401-409, 1992. [4] C. W. Wu and L. O. Chua, “Properties of admissible symbolic sequences in a second-order digital filter with overflow non-linearity,” International Journal of Circuit Theory and Applications, vol. 21, pp. 299-307, 1993. 

Chaos in digital filters,”

Fractal pattern of second-order non-linear digital filters: a new symbolic analysis,”

On symbolic dynamics of a chaotic second-order digital filter,”

Properties of admissible symbolic sequences in a second-order digital filter with overflow non-linearity,”

Admissibility of unstable second-order digital filter with two’s complement arithmetic

Admissibility of unstable second-order digital filter with two’s complement arithmetic

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