Signal Processing 93 (2013) Contents lists available at SciVerse ScienceDirect. Signal Processing

Transcription

1 Signal Processing 93 (03) Contents lists available at SciVerse ScienceDirect Signal Processing journal homepage: Lainiotis filter, golden section and Fibonacci sequence Nicholas Assimakis a, Maria Adam b,n, Chrissavgi Triantafillou c a Department of Electronics, Technological Educational Institute of Lamia, 3rd km Old National Road, Lamia-Athens, P.O. 300 Lamia, Greece b Department of Computer Science and Biomedical Informatics, University of Central Greece, -4 Papasiopoulou str., P.O. 300 Lamia, Greece c Department of Electronics, Technological Educational Institute of Lamia, Greece article info Article history: Received 6 May 0 Accepted 9 September 0 Available online 3 October 0 Keywords: Lainiotis filter Random walk Riccati equation Golden section Fibonacci sequence abstract The relation between the discrete time Lainiotis filter on the one side and the golden section and the Fibonacci sequence on the other is established. As far as the random walk system is concerned, the relation between the Lainiotis filter and the golden section is derived through the Riccati equation since the steady state estimation error covariance is related to the golden section. The relation between the closed form of the Lainiotis filter and the Fibonacci sequence is also derived. It is shown that the steady state Lainiotis filter computes the state estimate using a linear combination of the previous estimate and of the current measurement with coefficients related to the golden section. A Finite Impulse Response (FIR) implementation of the steady state Lainiotis filter is also proposed, where the filter computes the state estimate as a linear combination of a well-defined set of the last measurements with coefficients which are powers of the golden section. Finally, the scalar generic stochastic dynamic system is considered and the relation between its parameters and the golden section is investigated. & 0 Elsevier B.V. All rights reserved.. Introduction Although the connection between the golden section and Fibonacci numbers with nature, arts and architecture is known for centuries, there is presently a huge interest of modern sciences in these classical theories. Particularly, researchers in the computer science (measurement theory and communication systems [4]) and cryptography [] exhibit a substantial interest in these classical theories and use them in order to model phenomena in their field. The above are only a few applications of the golden section and the Fibonacci numbers that imply a new mathematical direction which is the creation of a fascinating and beautiful subject of the Mathematics of Harmony [3]. n Corresponding author. addresses: assimakis@teilam.gr (N. Assimakis), madam@ucg.gr (M. Adam), chrtriantafillou@teilam.gr (C. Triantafillou). Filtering plays an important role in many fields of science: applications to aerospace industry, chemical process, communication systems design, control, civil engineering, filtering noise from -dimensional images, pollution prediction and power systems are mentioned in []. In the field of signal processing, measurements are available containing the signal and the noise and the task is to produce an estimate of the signal through processing of the measurements by a filter. In this area the discrete time Kalman filter [,8] and Lainiotis filter [9,0] are wellknown algorithms that solve the filtering problem. Lainiotis filter uses the partitioning approach to estimation leading to robust, computationally effective and fast filtering algorithms [0]. The two filters are equivalent to each other [3] since they compute theoretically the same estimations. A key difference between the two filters is the fact that Kalman filter computes the estimation through prediction, while Lainiotis filter computes the estimation through smoothing. Another difference is that in the Kalman filter case the initial state has a /$ - see front matter & 0 Elsevier B.V. All rights reserved.

2 7 N. Assimakis et al. / Signal Processing 93 (03) Gaussian probability density function (pdf), while in the Lainiotis filter case the pdf could be also non-gaussian []. Which time invariant filter is faster depends on the relation between the state and measurement dimensions: in fact KalmanfilterisfasterthanLainiotisfilterinmulti-state problems (the state dimension is enough greater than the measurement dimension), while Lainiotis filter is faster than Kalman filter in the multi-sensor problems (the measurement dimension is enough greater than the state dimension), as for instance the multi-sensor systems with many sensing devices are considered in the seismic processing, see [3]. Recently, the relation between the discrete time Kalman filter and the golden section is established [4,]. The novelty of the paper is to establish the relation between the discrete time Lainiotis filter on the one side and the golden section and the Fibonacci sequence on the other. In fact, the relation between the discrete time Lainiotis filter and the golden section is described for the scalar generic stochastic dynamic system; similar relation has been described for the Kalman filter assuming scalar systems with special output coefficient (equal to one). It is also shown that the prediction/estimation/ smoothing error covariances are related to the golden section. Finally, a FIR implementation of the steady state Lainiotis filter is proposed, where the filter coefficients are powers of the golden section. The paper is organized as follows: In Section and 3 a brief review of the golden section and the Fibonacci sequence, respectively, is presented while in Section 4 the discrete time Lainiotis filter is presented. In Section the random walk system is considered. The relation between the discrete time Lainiotis filter and the golden section is established through the Riccati equation. Also, the relation between the closed form of the Lainiotis filter and the Fibonacci sequence is derived. Moreover, the relation between the steady state Lainiotis filter and the golden section is described. In addition a Finite Impulse Response (FIR) implementation of the steady state Lainiotis filter is proposed, where the coefficients of the filter are related to the golden section. In Section 6 the relation between the Lainiotis filter, the Kalman filter and the golden section is described, for the random walk system. In Section 7 the scalar generic stochastic dynamic system is considered and the relation between the system parameters and the golden section is investigated. Finally, Section 8 summarizes the conclusions.. Golden section The Golden Ratio as a concept has a long history in mathematics. Although the term golden section appears in print for the first time by the German mathematician Martin Ohm, the younger brother of the well-known physicist George Simon Ohm, in a footnote in the 83 second edition of Die Reine Elementar-Mathematik [6], the mathematical concept of Golden Ratio traces back to the famous Greek mathematician Euclid from Alexandria. Particularly, Euclid s Elements [7] (Greek: StoiwEia) provide the first known written definition of what is now called the golden ratio in order to solve a geometrical problem concerning the division of a line segment in extreme and mean ratio. The definition is the following: A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less. We provide below the essence of this geometrical problem: A line segment AB must be divided with a point C into two parts so that the ratio between the shorter part AC and the longer one CB is equal to the ratio between the longer part CB and the whole line segment AB, i.e.: l AC CB CB AB Using the relationship AB ACþCB we take: l AC CB CB AB CB ACþCB CB=CB AC=CBþCB=CB lþ Hence, the equation to calculate the ratio l is l þl 0 ðþ with two roots: l, 7 p ffiffiffi The positive root of Eq. () is the so-called golden section: a þ p ffiffiffi 0:68 ðþ The golden section has the following property: a a ð3þ The reciprocal of the golden section is the golden ratio. So, the well-known number f (phi), the golden ratio, is given as follows: f þ p ffiffiffi :68 ð4þ The relations between the golden section and the golden ratio are derived by () and (4) and are given below: f a þa The golden section seems to appear in many of the proportions of famous ancient buildings, such as the Parthenon in Athens. Also the proportions of famous paintings seem to be designed according to the golden section, for example Botticelli s Venus in the painting La Primavera or Vergine delle Rocce created by Leonardo Da Vinci. However, there is no original documentary evidence that these buildings and paintings were deliberately designed using the golden section. The golden ratio has been of interest to mathematicians, philosophers, architects but in the last decades computer scientists and engineers have also concentrated their work on this ratio. Recently, concerning the signal processing research area, the relation between the discrete time Kalman filter and the golden section is established [4,]. 3. Fibonacci sequence Fibonacci, an Italian born mathematician, discovered his unique number sequence theory in around 00 AD; ðþ

3 N. Assimakis et al. / Signal Processing 93 (03) the Fibonacci sequence, where the seed values are f and f starting from two seeds values the f 0 and f, is defined by recurrence by taking each subsequent number as the sum of the two previous ones: f n þ f n þ þf n, nz, and f 0, f ð6þ Thus, the sequence of Fibonacci numbers is f0,,,,3,,8,3,,34,,89,44,...,g. Itiswell-knownthatthe Fibonacci sequence satisfies the limit properties [4]: calculated off-line: A ½HQH T þrš K n QH T A K m F T H T A P n Q QH T AHQ F n F QH T AHF O n F T H T AHF ðþ f lim n f a and lim n þ f ð7þ n- f n þ n- f n þ which provide the relation between the Fibonacci sequence on the one side and the golden section and the golden ratio on the other side. Today, it is believed that Fibonacci himself did not even realize the connection of the Fibonacci sequence to the golden ratio. Fibonacci numbers seem to appear also in nature. For example, many types of flowers have a Fibonacci number of petals: daisies tend to have 34 or petals, sunflowers have 89 or 44. Similarly, the numbers of rings on the trunks of palm trees and the scales on the surface of a pineapple follow a sequence of Fibonacci numbers. Recently, concerning the signal processing research area, the relation between the discrete time Kalman filter and the Fibonacci sequence is described [4,]. 4. Lainiotis filter Consider the time invariant stochastic dynamic system described by the following state space equations: xðkþþfxðkþþwðkþ zðkþþhxðkþþþvðkþþ ð8þ for k 0,,..., where xðkþ is n state vector at time k, zðkþ is m measurement vector, F is n n system transition matrix, H is m n output matrix, fwðkþg,fvðkþg are independent Gaussian zero-mean white and uncorrelated random processes, Q is n n plant noise covariance matrix, R is m m measurement noise covariance matrix, and xð0þ is a Gaussian random process with mean x 0 and covariance P 0. The filtering problem is to produce an estimate at time L of the state vector using measurements till time L, i.e.the aim is to use the measurements set fzðþ, zðþ,..., zðlþg in order to calculate an estimate value xðl=lþ of the state vector xðlþ. The discrete time Lainiotis filter [9,0] is a wellknown algorithm that solve the filtering problem, by computing the estimation xðk=kþ at time k, and the corresponding estimation error covariance matrix Pðk=kÞ such that Pðkþ=kþÞP n þf n ½IþPðk=kÞO n Š Pðk=kÞF T n xðkþ=kþþf n ½IþPðk=kÞO n Š xðk=kþ þðk n þf n ½IþPðk=kÞO n Š Pðk=kÞK m ÞzðkþÞ ð0þ for k 0,,..., with initial conditions Pð0=0ÞP 0, and xð0=0þx 0, where the following constant matrices are ð9þ Recall that the covariance matrices Q,R and Pðk=kÞ are non-negative definite matrices; hereafter these matrices are considered to be positive definite. Then, the existence of the m m symmetric matrix A ½HQH T þrš is guaranteed, when R is a positive definite ðr40þ, which means that no measurement is exact. This is reasonable in physical problems. Moreover, the existence of ½IþPðk=kÞO n Š is guaranteed due to the presence of the identity matrix I and due to the facts that Pðk=kÞ40 and O n 40, since A40. Note that if the signal process system is asymptotically stable (i.e. all the eigenvalues of F lie inside the unit circle), then there exist a unique positive definite steady state value P e of the estimation error covariance matrix, i.e. the estimation error covariance Pðk=kÞ tends to the steady state estimation error covariance: P e lim Pðk=kÞ ðþ k- This steady state solution P e can be calculated by recursively implementing the Riccati equation emanating from Lainiotis filter (0) with the initial condition Pð0=0ÞP 0. The steady state or limiting solution of the Riccati equation is independent of the initial condition. The steady state estimation error covariance matrix satisfies the algebraic Riccati equation emanating from Lainiotis filter: P e P n þf n ½IþP e O n Š P e F T n ð3þ The steady state estimation error covariance matrix may exist even if the system is not asymptotically stable.. Lainiotis filter for the random walk system Consider the random walk system, namely the scalar (n and m ) stochastic dynamic system in (8) with the transition and output coefficients equal to one, F f and H h : xðkþþxðkþþwðkþ zðkþþxðkþþþvðkþþ ð4þ and the process and measurement noise sources having equal noise covariances Q q, R r : ðþ Then the Lainiotis filter parameters by () are A s, K n, K m s, P n s, F n, O n ð6þ

4 74 N. Assimakis et al. / Signal Processing 93 (03) In this section, we deal with the discrete time Lainiotis filter and show how is related to the golden section as well as to the Fibonacci sequence... Lainiotis filter and the golden ratio Substituting the values of parameters by (6) in (9) and (0) the recursive form of the Lainiotis filter is derived: Pðkþ=kþÞ s þpðk=kþ þpðk=kþ s ð7þ xðkþ=kþþ þpðk=kþ xðk=kþþ s þpðk=kþ þpðk=kþ zðkþþ ð8þ for k 0,,..., with initial conditions Pð0=0ÞP 0, and xð0=0þx 0. Moreover, the steady state value P e of the estimation error covariance can be calculated by solving the following algebraic Riccati equation, resulting from (3) by substituting the values of Lainiotis filter parameters by (6): P e þs P e s 4 0 ð9þ The unique positive solution of Eq. (9) is the steady state estimation error covariance P e and is related to the golden section; in fact by () it is equal to the golden section times the covariance : P e þ p ffiffiffi a ð0þ Moreover, we observe that the negative solution of (9) is related also to the golden ratio: p ffiffiffi þ p ffiffiffi f Thus, it has been shown that for the random walk system, the steady state estimation error covariance of the corresponding Lainiotis filter is related to the golden section... Lainiotis filter and the Fibonacci sequence The recursive form (open form) of Lainiotis filter consists of the recursive equations (7) and (8). We are able to derive the non-recursive form (closed form) of Lainiotis filter from these equations. We are going to show that for the random walk system, the coefficients of the closed form of the Lainiotis filter are related to the Fibonacci numbers. The recursive equations of the Lainiotis filter (7) and (8) can be written in the following closed form of the Lainiotis filter: Pðkþ=kþÞ f k þ 3 þf k þ Pð0=0Þ f k þ 4 þf k þ 3 Pð0=0Þ s ðþ xðkþ=kþþ f k þ 4 þf k þ 3 Pð0=0Þ " # xð0=0þþ Xk þ ðf i þ þf i Pð0=0ÞÞzðiÞ ðþ for k 0,,..., with initial conditions Pð0=0ÞP 0, and xð0=0þx 0. Proof. The proof of () is based on the induction method. Using the definition of the Fibonacci sequence by (6) and the recursive equation (7) for k 0 we can write Pð=Þ s þp 0 þp 0 f 3 þf P 0 f 4 þf 3 P 0 which satisfies (). Assume that the formula in () is true for k, then by (7) we have Pðkþ=kþÞ s þpðkþ=kþþ þpðkþ=kþþ s In the last equation, substituting the assumption of the induction and using the definition of Fibonacci sequence by (6) we derive Pðkþ=kþÞ þ f k þ 3 þf k þ P 0 f k þ 4 þf k þ 3 P 0 þ f k þ 3 þf k þ P s 0 s f k þ 4 þf k þ 3 P 0 ðf k þ 3 þf k þ 4 Þ þðf k þ þf k þ 3 ÞP 0 ðf k þ 4 þf k þ 3 Þ þðf k þ 3 þf k þ ÞP 0 f k þ þf k þ 4 P 0 ðf k þ 4 þðf k þ 4 þf k þ 3 ÞÞ þðf k þ 3 þðf k þ 3 þf k þ ÞÞP 0 f k þ þf k þ 4 P 0 f k þ 6 þf k þ P 0 f ðk þ Þþ3 þf ðk þ Þþ P 0 f ðk þ Þþ4 þf ðk þ Þþ3 P 0 i.e. () holds also for kþ, which completes the induction method for (). The proof of () is also based on the induction method. In fact, for k 0 the recursive equation (8) yields xð=þ þp 0 x 0 þ s þp 0 þp 0 zðþ þp 0 ½ x 0 þð þp 0 ÞzðÞŠ f 4 þf 3 P 0 ½ x 0 þðf 3 þf P 0 ÞzðÞŠ which satisfies (). Assume that the formula in () is true for k, then by (8) we have xðkþ=kþþ þpðkþ=kþþ xðkþ=kþþ þ s þpðkþ=kþþ þpðkþ=kþþ zðkþþ In the last equation, substituting the assumption of the induction and using () and the definition of Fibonacci

5 N. Assimakis et al. / Signal Processing 93 (03) sequence by (6) we derive f xðkþ=kþþ k þ 4 þf k þ 3 P 0 ðf k þ 4 þf k þ 3 Þ þðf k þ 3 þf k þ ÞP 0 ( x f k þ 4 0 þ þf k þ 3 P 0 f k þ 4 þf k þ 3 P 0 ) Xk þ ðf i þ þf i P 0 ÞzðiÞ þ f k þ þf k þ 4 P 0 zðkþþ f k þ 6 þf k þ P 0 x ðf k þ 4 þðf k þ 4 þf k þ 3 ÞÞ 0 þðf k þ 3 þðf k þ 3 þf k þ ÞÞP 0 þ ðf k þ 4 þðf k þ 4 þf k þ 3 ÞÞ þðf k þ 3 þðf k þ 3 þf k þ ÞÞP 0 Xk þ ðf i þ þf i P 0 ÞzðiÞþ f k þ þf k þ 4 P 0 zðkþþ f k þ 6 þf k þ P 0 x f k þ 6 0 þ þf k þ P 0 f k þ 6 þf k þ P 0 Xk þ ðf i þ þf i P 0 ÞzðiÞþ f k þ þf k þ 4 P 0 zðkþþ f k þ 6 þf k þ P 0 x f k þ 6 0 þ þf k þ P 0 f k þ 6 þf k þ P 0 " # k X þ ðf i þ þf i P 0 ÞzðiÞþðf k þ þf k þ 4 P 0 ÞzðkþÞ x f k þ 6 0 þ þf k þ P 0 f k þ 6 þf k þ P 0 Xk þ ðf i þ þf i P 0 ÞzðiÞ i.e. () holds also for kþ, which completes the induction method for (). & Thus, it is obvious that for the random walk system, the coefficients of the closed form of the Lainiotis filter are related to the Fibonacci numbers and the relation between the Lainiotis filter and the Fibonacci sequence has been established by () and (). Remark.. Notice that the estimation error covariance converges to the steady state estimation error covariance irrespective of the initial condition Pð0=0ÞP 0. For every kz0 from () we have Pðkþ=kþÞ f k þ 3 þf k þ P 0 f k þ 4 þf k þ 3 P 0 f k þ 3 f s þ k þ P 0 f k þ 3 f k þ 4 þ f k þ 3 f k þ 4 P 0 s It is obvious that by () and (7) the above equality yields P e lim Pðk=kÞa s þap 0 s k- a þap 0.3. Steady state Lainiotis filter and the golden section The steady state Lainiotis filter for the random walk system in (4) with the process and measurement noise sources have noise covariances as in () takes advantage of the a priori knowledge of the steady state estimation covariance P e by (0). Then, substituting the estimation covariance Pðk=kÞ by the steady state estimation covariance P e a in (8) we have xðkþ=kþþ þa s xðk=kþþ s þa þas zðkþþ þa xðk=kþþ þa þa zðkþþ ð3þ Using þa =a from (), we derive þa þa þa þþa =a þð=aþ =a ðaþþ=a aþ a and þa þþa þð=aþ a aþ a Substituting the above quantities in (3) we derive the recursive form of the steady state Lainiotis filter xðkþ=kþþa xðk=kþþazðkþþ ð4þ for k 0,,..., with initial condition xð0=0þx 0. Thus, it becomes evident that the recursive form of the steady state Lainiotis filter computes the state estimate using a linear combination of the previous estimate and of the current measurement with coefficients related to the golden section. Furthermore, for k 0,,..., and xð0=0þx 0, the closed form of the steady state Lainiotis filter can be derived: xðkþ=kþþa ðk þ Þ xð0=0þþ Xk þ a ðk iþþ3 zðiþ ðþ Proof. The proof of () is based on the induction method. In fact, for k 0 we are able to write the recursive form of the steady state Lainiotis filter in (4) as xð=þa xð0=0þþazðþ which satisfies (). Assume that the formula in () is true for k, then by (4) and the assumption of induction we derive xðkþ=kþþa xðkþ=kþþþazðkþþ " # a a ðk þ Þ xð0=0þþ Xk þ a ðk iþþ3 zðiþ þazðkþþ a ðk þ Þ xð0=0þþ Xk þ a ðk iþþ zðiþþazðkþþ a ðk þ Þ xð0=0þþ Xk þ a ðk iþþ zðiþ i.e. () holds also for kþ, and hence for all kz0. It becomes obvious that the steady state Lainiotis filter computes the state estimate as a linear combination of the initial state estimate and of all previous measurements with coefficients, which are powers of the golden section..4. FIR steady state Lainiotis filter and the golden section Using the ideas in [] for the scalar stochastic dynamic system in (4) with q r we are able to derive a FIR &

6 76 N. Assimakis et al. / Signal Processing 93 (03) form of the steady state Lainiotis filter as xðk=kþ XN a ðn iþþ zðiþk NÞ for k4n, where N such that a N oe and e a small positive number. ð6þ ð7þ Proof. Using the closed form of the steady state Lainiotis filter in () for k4n we derive xðk=kþa k xð0=0þþ Xk a ðk jþþ zðjþ j a k xð0=0þþ Xk N a ðk jþþ zðjþþ j j Xk j k N þ a ðk jþþ zðjþ a k xð0=0þþ Xk N a ðk jþþ zðjþþ XN a ðn iþþ zðiþk NÞ ð8þ Note that the golden section has the property lim a N 0 N- due to ao. Consequently, by (7) there exists an integer N such that a N oe, which means that we are able to assume that a j 0, for every jzn, while a j a0, jon. The last property allows us to confirm that in (8) the coefficient of xð0=0þ tends to zero, since k4n, and all the coefficients of zðjþ of the first sum for rjrk N tend to zero, since ðk jþþznþ4n: Thus, it is obvious that (8) yields (6). & The proposed FIR implementation of the steady state Lainiotis filter computes the state estimate as a linear combination of a known number of the last measurements, which are powers of the golden section a. Table summarizes the computational requirements required for the computation of the estimate value xðk=kþ of the state variable at time k of all the Lainiotis filter algorithms for the random walk system presented above. An essential advantage of the FIR form of the steady state Lainiotis filter is that the calculation burden does not depend on the estimation time, leading to the reduction of the computational time in comparison to the other algorithms. The coefficients of the FIR form are powers of the golden section and are computed off-line. Recall that only N last measurements are required. The calculation burden depends on the length N. In fact, a small value of N is Table Computational requirements of Lainiotis filter algorithms for the random walk system. Lainiotis filter (LF) Algorithm Equations Calculation burden (scalar operations) Order Recursive form LF (7) and (8) 8k 8k Closed form LF () and () 7kþ6 7k Recursive form (4) 3k 3k steady state LF Closed form steady () 3kþ 3k state LF FIR form steady state LF (6) N N enough to give reliable estimates. This is confirmed through the experiment where a random walk system has been considered. Fig. depicts the estimates xðk=kþ computed using the recursive form of the steady state Lainiotis filter with initial condition xð0=0þ0 and the FIR form of the steady state Lainiotis filter for N. This choice is rational (a 0 0:0083 is of the order of 0 3 ). It is noticeable that the estimates are close to each other (almost equivalent). 6. Lainiotis filter, Kalman filter and the golden section In this section the relation between the Lainiotis filter [3,9,0], the Kalman filter [,8] and the golden section is described for the random walk system in (4) with q r as in (). It is known that the Lainiotis filter is equivalent to the Kalman filter [3]. Thus, for the random walk system both filters provide the same estimates and the same estimation error covariances. Of course they provide the same steady state estimation error covariances as well. From the Lainiotis filter equations the recursive Riccati equation for the estimation error covariance, Pðk=kÞ, formulates in (7): Pðkþ=kþÞðð þpðk=kþþ=ð þ Pðk=kÞÞÞ. The relation between the smoothing error covariance, Pðk=kþÞ, and the estimation error covariance is obtained by the Lainiotis filter equation Pðk=kþÞ½I þ Pðk=kÞO n Š Pðk=kÞ, where substituting O n by (6) arises: Pðk=kþÞ Pðk=kÞ þpðk=kþ s Since Eq. (9) yields ð9þ Pðk=kÞ Pðk=kþÞ Pðk=kþÞ s ð30þ combining (7) and (30) we are able to derive the recursive Riccati equation for the smoothing error covariance from (9) as follows: Pðk=kþÞ s þpðk =k Þ þpðk =k Þ s þ s þpðk =k Þ þpðk =k Þ s s þpðk =k Þ þ3pðk =k Þ Pðk =kþ þ s Pðk =kþ s þ3 Pðk =kþ Pðk =kþ s s þpðk =kþ 0 þpðk =kþ Thus, the recursive Riccati equation for the smoothing error covariance is given by Pðk=kþÞ 4s þpðk =kþ 0 þpðk =kþ s ð3þ

7 N. Assimakis et al. / Signal Processing 93 (03) Steady state Lainiotis filter SSLF FIR SSLF estimate x(k/k) for SSLF and FIR SSLF time k Fig.. Estimates computed using the recursive form of the steady state Lainiotis filter and the FIR form of the steady state Lainiotis filter for N. The relation between the prediction error covariance Pðkþ=kÞ, and the estimation error covariance, is obtained by the equation Pðkþ=kÞFPðk=kÞF T þq of the Kalman filter [3], where substituting F f, Q q arises: Pðkþ=kÞPðk=kÞþ ð3þ Combining (7) and (3) we are able to derive the recursive Riccati equation for the prediction error covariance: Pðkþ=kÞ s þpðk=k Þ þpðk=k Þ s ð33þ By (9) and (3) it is easy to conclude that Pðk=kþÞoPðk=kÞoPðkþ=kÞ ð34þ The relations in (34) mean that the smoothed value of the state is better that the estimated one and that the estimated value of the state is better that the predicted one. The smoothing/estimation/prediction error covariances tend to the corresponding steady state smoothing/estimation/prediction error covariances P s =P e =P p, respectively. Of course, from the recursive Riccati equations in (7), (3) and (33) and using the relations in (3) and () we are able to derive the corresponding algebraic Riccati equations and their steady state solutions, which are summarized in Table : P s lim Pðk=kþÞa 3 k- P e lim Pðk=kÞa k- P p lim Pðkþ=kÞ k- a s Table Steady state error covariances and golden section. Error covariance Algebraic Riccati equation By the three above equalities it is easy to conclude that P s op e op p It is clear that in the special case q r, we have P s a 3 op e aop p a It is apparent that for the random walk system in (4) and (), the steady state smoothing/estimation/prediction error covariances emanating from Lainiotis and Kalman filters are related to the golden section, as depicted in Table. 7. Lainiotis filter for the scalar generic stochastic dynamic system Consider the scalar generic stochastic dynamic system in (8) with the transition coefficient F f and output H h xðkþþfxðkþþwðkþ zðkþþhxðkþþþvðkþþ Steady state error covariance Smoothing P s þ8s P s 4s 4 0 P s a 3 Estimation P e þs P e s 4 0 P e a Prediction P p s P p s 4 0 P p a s

8 78 N. Assimakis et al. / Signal Processing 93 (03) for k 0,,..., and the process and measurement noise sequences with covariances given by Q q and R r, respectively, with q40 and r 40. Then the Lainiotis filter parameters by () are A qh þr, K n qh qh þr, K m fh qh þr, P n qr qh þr, F n rf qh þr, O n f h ð3þ qh þr The aim of this section is to investigate for which values of the parameters f,h,q,r the steady state Lainiotis filter is related to the golden section a. Substituting the values of the parameters by (3) in (9) and (0), the recursive form of the Lainiotis filter is derived: Pðkþ=kþÞ qr qh þr þ r f ðqh þrþðqh þr þf h Pðk=kÞÞ Pðk=kÞ ð36þ rf xðkþ=kþþ qh þr þf h Pðk=kÞ xðk=kþ! qh þ qh þr þ rf hpðk=kþ zðkþþ ðqh þrþðqh þr þf h Pðk=kÞÞ ð37þ for k 0,,..., with initial conditions Pð0=0ÞP 0, and xð0=0þx 0. Note that there always exists a unique positive steady state value P e of the estimation error covariance, i.e. the estimation error covariance Pðk=kÞ tends to the steady state estimation error covariance P e, which can be calculated by recursively implementing the Riccati equation emanating from Lainiotis filter (36) with initial condition Pð0=0ÞP 0. Furthermore, the steady state value P e of the estimation error covariance satisfies the corresponding algebraic Riccati equation: f h P e þðqh þr rf ÞP e qr 0 ð38þ Notice that the discriminant D ðqh þr rf Þ þ4qrf h is a positive number, thus the real roots of (38) exist always and its unique positive root is given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P e ðqh þr rf Þþ ðqh þr rf Þ þ4qrf h ð39þ f h The steady state Lainiotis filter for the scalar generic stochastic dynamic system takes advantage of the a priori knowledge of the steady state estimation covariance P e by (39). Then, substituting the estimation covariance Pðk=kÞ by the steady state estimation covariance P e in Eq. (37), the recursive form of the steady state Lainiotis filter is derived: rf xðkþ=kþþ xðk=kþ qh þr þf h P e! qh þ qh þr þ rf hp e zðkþþ ð40þ ðqh þrþðqh þr þf h P e Þ for k 0,,..., with initial condition xð0=0þx 0. The relation between the steady state estimation error covariance P e, the parameters f,h,q,r and the golden section a is presented in the following supposing that holds: 0oa q r h o In fact, the following relations are equivalent: (i) P e r h a (ii) f r aqh ra ð4þ ð4þ ð43þ Proof. (i) ) (ii) Equating the forms of P e by (39) and (4) we take qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðqh þr rf Þ þ4qrf h arf þqh þr rf ð44þ The right part of (44) is positive, since it is written ða Þrf þqh þr a 3 rf þqh þr due to a a a að aþa 3. Hence, the form of f in (43) is derived directly by (44). It is obvious that f 40 due to inequality in (4). (ii) ) (i) Substituting f by (43) in (39) the estimation error covariance P e in (4) directly arises. & Thus, it has been shown that for the scalar generic stochastic dynamic system, the steady state estimation error covariance P e is related to the golden section, under the assumption that the parameters f,h,q,r are related to the golden section a with the relation in (43). In the special case where the inequality in (4) holds, substituting the quantities P e,f by (4) and (43) in (40) and using (), the special recursive form of the steady state Lainiotis filter is derived: xðkþ=kþþa fxðk=kþþ a h zðkþþ ð4þ for k 0,,..., with initial condition xð0=0þx 0. Thus, it has been shown that for the scalar generic stochastic dynamic system, the corresponding steady state Lainiotis filter is related to the golden section, under the assumption that the parameters f,h,q,r are related to the golden section a with the relation in (4). Remark 7.. It is clear that the Lainiotis filter as well as the steady state Lainiotis filter for the scalar stochastic dynamic system presented in Section is verified. In fact, for the values of parameters f h and q r the recursive form of the Lainiotis filter in (36) and (37) takes the form in (7) and (8). Obviously, the choice of parameters h and q r satisfies the inequality in (4) and gives P e a,asin (0); in this case, from (43) arises f and choosing f the special recursive form of the steady state Lainiotis filter in (4) takes the form in (4). Furthermore, in the special case where the inequality in (4) and the equivalent relations in (4) and (43) hold, the special closed form of the steady state Lainiotis filter is

9 N. Assimakis et al. / Signal Processing 93 (03) derived for k 0,,... xðkþ=kþþða f Þ k þ xð0=0þþ a h Xk þ ða f Þ k i þ zðiþ ð46þ Proof. It is easy to see that for k 0 the special recursive form in (4) is written as xð=þa fxð0=0þþ a h zðþ, which satisfies (46). By induction, supposing that Eq. (46) holds for k, then by (4) we derive xðkþ=kþþða f Þxðkþ=kþÞþ a h zðkþþ " ða f Þ ða f Þ k þ xð0=0þþ a # Xk þ ða f Þ k i þ zðiþ þ a h h zðkþþ ða f Þ k þ xð0=0þþ a h Xk þ ða f Þ ðk þ Þ i þ zðiþ i.e. (46) holds also for kþ, and hence for all kz0. It becomes obvious that, when the inequality in (4) and one of two equivalent relations in (4) and (43) hold, the steady state Lainiotis filter computes the state estimate as a linear combination of the initial state estimate and of all previous measurements with coefficients, which are related to the golden section a. Finally, following the methodology used for the derivation of the FIR steady state Lainiotis filter in Section, in the special case where the inequality in (4) and the equivalent relations in (4) and (43) hold, we are able to derive the special FIR form of the steady state Lainiotis filter for the scalar generic stochastic dynamic system as xðk=kþ a h X N ða f Þ N i zðk N þiþ for k4n, where N such that ða f Þ N oe and e a small positive number. & ð47þ ð48þ Proof. Using the special closed form of the steady state Lainiotis filter in (46) for k4n we derive: xðk=kþða f Þ k xð0=0þþ a h ða f Þ k xð0=0þþ a h X k N j X k j ða f Þ k j zðjþ ða f Þ k j zðjþþ a h X k j k N þ ða f Þ k j zðjþ quantity a f has the property lim N- ða f Þ N 0 due to sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9a f 9 a r aqh a aqh o: ra r Consequently, by (48) there exists an integer N such that ða f Þ N oe, which means that we are able to assume that ða f Þ j 0, for every jzn, while ða f Þ j a0, jon: The last property allows us to confirm that in (49) the coefficient of xð0=0þ tends to zero, since k4n, and all the coefficients of zðjþ of the first sum for rjrk N tend to zero, since k jzn: Thus, it is obvious that (49) yields (47). & In the case where the inequality in (4) and the equivalent relations in (4) and (43) hold, the special FIR implementation of the steady state Lainiotis filter computes the state estimate as a linear combination of a known number of the last measurements with coefficients, which are powers of the golden section a. 8. Conclusions The relation between the discrete time Lainiotis filter on the one side and the golden section and the Fibonacci sequence on the other side is established. Consider the random walk system, i.e. the scalar stochastic dynamic system with the transition and output coefficients equal to one. It is shown that the Lainiotis filter computes the state estimate using a linear combination of the previous estimate and of the current measurement with coefficients related to the Fibonacci numbers. Furthermore, it is pointed out that the steady state estimation error covariance is related to the golden section. It is also shown that the recursive form of the steady state Lainiotis filter computes the state estimate using a linear combination of the previous estimate and the current measurement with coefficients related to the golden section, while the non-recursive form of the steady state Lainiotis filter computes the state estimate as a linear combination of the initial state estimate and of all previous measurements with powers of the golden section as coefficients. A FIR implementation of the steady state Lainiotis filter is also proposed, where the filter computes the state estimate as a linear combination of a well-defined set of the last measurements with coefficients which are powers of the golden section. Table 3 summarizes the relationship between the Lainiotis filter, the golden section and the Fibonacci numbers for the scalar stochastic dynamic system. Thus, it becomes evident that for the scalar stochastic dynamic Table 3 Relationship between the Lainiotis filter, the golden section and the Fibonacci numbers. ða f Þ k xð0=0þþ a h X k N j ða f Þ k j zðjþþ a h X N ða f Þ N i zðk N þiþ ð49þ Using the assumption 0oaðq=rÞh o by (4) and the relation f ðr aqh Þ=ra, it becomes clear that the Lainiotis filter algorithm Closed form Lainiotis filter Recursive form Lainiotis filter Recursive form steady state Lainiotis filter Closed form steady state Lainiotis filter FIR steady state Lainiotis filter Relation to Fibonacci numbers Golden section Golden section Golden section Golden section

10 730 N. Assimakis et al. / Signal Processing 93 (03) system the Lainiotis filter is fully governed by the golden section and the Fibonacci sequence. Concerning to the scalar generic stochastic dynamic system the relation between the parameters and the golden section was investigated. Results analogous to those for the scalar stochastic dynamic system were derived under the assumption that the parameters are related to the golden section with a specified relation. References [] B.D.O. Anderson, J.B. Moore, Optimal Filtering, Dover Publications, New York, 00. [] N. Assimakis, M. Adam, FIR implementation of the steady state Kalman filter, International Journal of Signal and Imaging Systems Engineering (3/4) (008) [3] N. Assimakis, M. Adam, Discrete time Kalman and Lainiotis filters comparison, International Journal of Mathematical Analysis (007) [4] A. Benavoli, L. Chisci, A. Farina, Fibonacci sequence, golden section, Kalman filter and optimal control, Signal Processing 89 (009) [] A. Capponi, A. Farina, C. Pilotto, Expressing stochastic filters via number sequences, Signal Processing 90 (00) 4 3. [6] R.H. Fischler, A Mathematical History of the Golden Number, Dover Publications, 998. [7] T.L. Heath, Euclid s Elements, Green Lion Press, Santa Fe, 00. [8] R.E. Kalman, A new approach to linear filtering and prediction problems, Transactions of the ASME, Journal of Basic Engineering 8 (960) [9] D.G. Lainiotis, Discrete Ricatti equation solutions: partitioned algorithms, IEEE Transactions on Automatic Control AC-0 (97) 6. [0] D.G. Lainiotis, Partitioned linear estimation algorithms: discrete case, IEEE Transactions on Automatic Control AC-0 (97) 7. [] D.G. Lainiotis, S.K. Park, R. Krishnaiah, Optimal state-vector estimation for non-gaussian initial state vector, IEEE Transactions on Automatic Control AC-6 (97) [] A.P. Stakhov, The golden matrices and a new kind of cryptography, Chaos Solitons and Fractals 3 (007) [3] A.P. Stakhov, The Generalized Principle of the Golden Section and its applications in mathematics, science, and engineering, Chaos, Solitons and Fractal6 () (00) [4] A.P. Stakhov, The golden section in the measurement theory, Computers & Mathematics with Applications 7 (4 6) (989)