Bilateral Laplace Transforms on Time Scales: Convergence, Convolution, and the Characterization of Stationary Stochastic Time Series

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1 DOI /s Bilateral Laplace Transforms on Time Scales: Convergence, Convolution, the Characterization of Stationary Stochastic Time Series John M. Davis Ian A. Gravagne Robert J. Marks II Received: 30 May 2009 / Revised: 29 September 2009 Springer Science+Business Media, LLC 2010 Abstract The convergence of Laplace transforms on time scales is generalized to the bilateral case. The bilateral Laplace transform of a signal on a time scale subsumes the continuous time bilateral Laplace transform, the discrete time bilateral z-transform as special cases. As in the unilateral case, the regions of convergence ROCs) time scale Laplace transforms are determined by the time scale s graininess. ROCs for the bilateral Laplace transforms of double sided time scale exponentials are determined by two modified Hilger circles. The ROC is the intersection of points external to modified Hilger circle determined by behavior for positive time the points internal to the second modified Hilger circle determined by negative time. Since graininess lies between zero infinity, there can exist conservative ROCs applicable for all time scales. For continuous time R) bilateral transforms, the circle radii become infinite results in the familiar ROC between two lines parallel to the imaginary z axis. Likewise, on Z, the ROC is an annulus. For signals on time scales bounded by double sided exponentials, the ROCs are at least that of the double sided exponential. The Laplace transform is used to define the box minus shift through which time scale convolution can be defined. Generalizations of familiar This work was supported by National Science Foundation grant CMMI # J.M. Davis ) Department of Mathematics, Baylor University, One Bear Place #97328, Waco, TX , USA John_M_Davis@Baylor.edu I.A. Gravagne R.J. Marks II Department of Electrical Computer Engineering, Baylor University, One Bear Place #97356, Waco, TX , USA I.A. Gravagne Ian_Gravagne@Baylor.edu R.J. Marks II r.marks@ieee.org

2 properties of signals on R Z include identification of the identity convolution operator, the derivative theorem, characterizations of wide sense stationary stochastic processes for an arbitrary time scales including autocorrelation power spectral density expressions. Keywords Time scales Laplace transform z-transforms Region of convergence Hilger circle Stationarity Autocorrelation Power spectral density Hilger delta 1 Introduction A time scale, T, is any closed subset of the real line. Continuous time, R, discrete time Z, are special cases. The calculus of time scales was introduced by Hilger [7]. Time scales have found utility in describing the behavior of dynamic systems [1, 11] have been applied to control theory [2, 3, 6]. This is the second in a series of monographs outlining regions of convergence applications of Laplace transforms on time scales. The first paper was dedicated to the causal or one sided) Laplace transform on a time scale [5]. This paper extends these results to the bilateral Laplace transform on a time scale its use in defining convolution on an arbitrary time scale. Time scale convolution, in turn, allows modeling of wide sense stationary stochastic processes on time scales using autocorrelation power spectral density descriptors. For the convergence problem, there are three cases of bilateral time scales considered. 1. For time scales whose graininess is bounded from above below over the entire time scale or asymptotically. 2. For time scales whose asymptotic graininess approaches a constant. R Z are special cases. All time scales in this class are also asymptotically a member of the time scales in For all time scales. This can be considered a limiting special case of 1 since all time scales are bounded between zero infinity. 2 Time Scales Our introduction to time scales is limited to that needed to establish notation. A more detailed explanation is in our first paper [5] a complete rigorous treatment is in the text by Bohner Peterson [1]. 1. A time scale, T, is any collection of closed intervals on the real line. We will assume the origin is always a component of the time scale. 2. The graininess of a time scale at time t T is defined by ) μt) inf τ t. τ>t,τ T

3 3. The Hilger derivative of an image xt) at t T is x t) : xtσ ) xt) μt) where t σ : t + μt). When μt) 0, the Hilger derivative is interpreted in the limiting sense x t) d dt xt). 4. If yt) x t), then the definite time scale integral is b a yt) t xb) xa). 5. When x0) 1, the solution to the Hilger differential equation, x t) zxt), is xt) e z t) where the generalized exponential is t e z t) : exp As a consequence, for z 0, τ0 6. The circle minus operator is defined by ln1 + zμτ)) τ μτ) y z : y z 1 + zμt). The notation z in interpreted as y z with y The generalized exponential has the property that [1] e z t) 1 e z t). ). e 0 t) ) 3 Bilateral Laplace Let T denote a bilateral time scale let ft)be an image on T. Define the bilateral Laplace transform as Fz): ft)e z σ t) t 3.1) where e σ z t) : e zt σ ). The continuous time bilateral Laplace discrete time z-transforms are special cases.

4 Here are some properties. 1. Integration property. F0) f t) t. This follows immediately from 3.1) 2.1). 2. The derivative theorem. When ft)e z t) goes to zero as t ± Fz) converges, f t) zf z). 3.2) Proof f t) f t)e z σ t) t. Using integration by parts [1] f t) ft)e z t) ft) [ e z t) ] t ft) z)e z t) t. Since z)e z t) ze z σ t), the result follows immediately. 3. Special cases. For continuous time, T R,wehave e z σ t) e zt 3.1) becomes the conventional bilateral Laplace transform Fz) For discrete time, T Z,wehave 3.1) becomes Fz) The bilateral z-transform is ft)e zt dt. e σ z t n) 1 + z) n+1) n F Z z) fn)1 + z) n+1). fn)z n. 3.3) n

5 Thus or, equivalently, Fz) F Zz + 1) z + 1 F Z z) zf z 1). Note, then, that the conventional z-transform is related to a shifted time scale Laplacetransformfor T Z. Forexample, theunitcircleforthe z-transform is centered about the origin. The time scale version is the same circle now shifted to be centered at z Variation. An alternate form of the bilateral transform which will prove useful in the characterization of stationary stochastic processes on a time scale is F z) : ft)ez σ t) t. 3.4) Although the analysis of convergence in this paper is for Fz), the convergence properties for F z) are similar follow immediately. We can break up the transform definition in 3.1) as Fz) F + z) + F z) 3.5) where We recognize that F + z) 0 ft)e z σ t) t 0 F z) ft)e z σ t) t. F + z) F u z) where F u z) is the notation for the causal Laplace transform on a time scale [5]. F u z) is alternately referred to as the unilateral or one sided Laplace transform. The regions of convergence for the causal Laplace transform has been established for causal functions with finite area for transcendental functions arising from solution of linear time invariant differential equations on time scales [5]. 3.1 Asymptotic Graininess Bounds Here are some graininess bounds useful in determining the convergence of bilateral transforms. All upper bounds are bounded by infinity all lower bounds must equal or exceed zero.

6 1) Constant Asymptotic Graininess. The graininess of some time scales asymptotically approach a constant at t ±. In such cases, we define the constant asymptotic graininesses as μ + lim t μt) μ lim t μt). More rigorously, μ + is the positive constant asymptotic graininess if lim μt) μ + 0. t Likewise, the negative constant asymptotic graininess lim μt) μ 0. t 2) Bounds. Graininess on a time scale is asymptotically be bounded from above below. Entire Bounds. The positive upper lower entire bounds for graininess are μ 0 + sup μt) t T,t 0 `μ 0 + inf μt). t T,t 0 The negative upper lower bounds for graininess are μ 0 sup μt) t T,t<0 `μ 0 inf μt). t T,t<0 Attained Bounds. The positive upper lower attained bounds for graininess are μ T + sup μt) t T,t T `μ T + inf μt). t T,t T The negative upper lower attained bounds for graininess are μ T sup μt) t T,t<T

7 `μ T inf μt). t T,t<T Asymptotic Bounds. The positive upper lower asymptotic bounds for graininess are μ + lim T μt + lim T sup t T,t T ) μt) ) `μ + lim T `μt + lim inf μt). T t T,t T The negative upper lower asymptotic bounds for graininess are ) μ lim T μt lim μt) T sup t T,t T `μ lim T `μt lim T ) inf μt). 3.6) t T,t T When the positive or negative asymptotic bounds are equal, a time scale has a constant positive /or negative constant asymptotic graininess. 3) Global Asymptotic Bounds. Note that, since `μ + `μ are nonnegative, μ + < μ <, we can always set global upper lower bounds for both the positive negative cases as Regions of Convergence on the z Plane Define a modified Hilger circle parameterized by a possibly complex) number α a graininess μ, as the locus of points for which z + 1 μ α + 1 μ. The circle is centered at 1/μ on the negative real axis on the z plane passes through the point α [5]. From this definition we offer the following definitions of regions on the z plane. 1) The region Hα, μ) contains all points outside of the modified Hilger circle with parameters α μ. Hα, μ) is the region inside of the same circle. 2) The regions Rα, `μ, μ) Lα, `μ, μ), illustrated in Fig. 1, are defined as the intersections Rα, `μ, μ) Hα, μ) Hα, `μ) 3.7)

8 Fig. 1 Illustration of the regions Rα, `μ, μ) Lα, `μ, μ). In each case, the leftmost bold circle is centered on the negative real axis at 1/ `μ the rightmost bold circle is centered on the negative real axis at 1/ μ. Both circles pass through the point α. The lightly shaded area, external to all of the circles, are Rα, `μ, μ). The darker shaded region, internal to the circles, are Lα, `μ, μ). The examples here illustrate the regions for different values of α. a) Reα<0. b) Reα>0. c) α is negative real. Here, the region L is empty. d) α real positive Lα, `μ, μ) Hα, μ) Hα, `μ). 3.8) These regions have the obvious properties Rα,μ,μ) Hα, μ) 3.9)

9 Lα,μ,μ) Hα, μ). 3.10) 3) The regions Dα) Eα) are the limiting cases of R L. They are illustrated in Fig. 2. Define Dα) Rα, 0, ). Dα) consists of the intersection of all points external to a circle of radius α centered on the z plane all of the points to the right of the line z Re α. Fig. 2 Illustration of the regions Dα) Eα). In each case, the bold circle is centered at the origin passes through α. The vertical line is defined by the line z Re α. The lightly shaded area are Dα). The darker shaded region are Eα). The examples here illustrate the regions for different values of α. a) Reα<0. b) Reα>0. c) α is negative real. Here, the region L is empty. d) α real positive

10 Likewise, let Eα) Lα, 0, ). 3.11) These regions have the property that [5] Dα) Rα, `μ, μ) Eα) Lα, `μ, μ). 4 A Bilateral Laplace Transform Clearly, if F + u) converges in a region Z + F u) converges in a region Z, then Fu)converges at least in the region Z Z + Z. 4.1) We can use this to find the region of convergence ROC) of the double exponential function defined by { eα t); t 0, e β:α t) : 4.2) e β t); t 0. We will find useful the following shorth notation: T μ + ) means T has a constant positive asymptotic graininess of μ +. T μ ) means T has a constant negative asymptotic graininess of μ. T `μ +, μ + ) means T has lower upper positive t graininess bounds of `μ + μ +. These can be `μ 0 +, μ0 + ), `μ T +, μt + ),or `μ +, μ + ). T `μ, μ ) means T has lower upper negative t graininess bounds of `μ μ. These can be `μ 0, μ0 ), `μ T, μt ),or `μ, μ ). T0 +, + ) means T has lower upper positive t graininess bounds of 0. T0, ) means T has lower upper negative t graininess bounds of 0. Theorem 4.1 The bilateral Laplace transform of the double exponential function, e β:α t), when it exists, is given by Fz) F + z) + F z); z Z where F + z) 1 z α ; z Z +, 4.3) F z) 1 z β ; z Z 4.4)

11 Z Z + Z. The component ROCs are Hα, μ + ) for T μ + ), Z + Rα, `μ +, μ + ) for T `μ +, μ + ), Dα) for T0 +, + ) Hβ, μ ) for T μ ), Z Lβ, `μ, μ ) for T `μ, μ ), Eβ) for T0, ). 4.5) 4.6) 4.1 Examples Zero Asymptotic Graininess Let μ + μ 0. Continuous time, T R, is a special case as is the log time scale L { t n t n sgnn) log n +1 ) ;<n< }. For positive t, the region of convergence is Z + Hα, 0). The corresponding Hilger circle has infinite radius Hα, 0) is recognized as the set of all points to the right of the line z Re α. Likewise, for negative time, Z Hβ, 0), contains all of the points to the left of the line z Re β. The resultant slab of convergence, Z Z + Z, is the familiar region of convergence in the bilateral Laplace transform. It is illustrated on the left of Fig Unit Asymptotic Graininess Let μ + μ 1. Discrete time, T Z, is a special case as is the time scale It follows that S { t n t n sgnn) n + n ) ;<n< }. Z + Hα, 1) Z Hβ, 1). The intersection, an annulus region of convergence illustrated on the right of Fig. 3,is familiar in bilateral z-transforms. Instead of being centered at the origin, however, the circles are centered at z 1. This anomaly is an artifact of inclusion of the bilateral

12 Fig. 3 Left: ROCs for time scales with zero asymptotic graininess for both positive negative time. Continuous time, R,is a special case.right: ROCs for time scales with unit asymptotic graininess for both positive negative time. Discrete time, Z, is a special case z-transform as a special case of the bilateral Laplace transform on time scales. The conventional z-transform can be generated from F b u) using 3.3) Periodic Graininess A discrete time scale, D, issaidtohaveperiodic graininess 1 is there is a period N where μt n ) μt n+n ) for all values of n. a) The exponential for time scales with periodic graininess. In general, for a discrete time scale [5] n 1 k0 1 + μt k)z); n>0, e z t n ) 1; n 0, 4.7) 1 kn 1 + μt k)z) 1 ; n<0. We can break the products into periods. For t>0, assume there are n P N replications of the graininess. Then n 1 k0 N 1 k0 2N 1 kn p+1)n 1 kpn 3N 1 k2n PN 1 kp 1)N n 1 kpn 1 Such time scales arise from recurrent nonuniform signal sampling, also called interlaced or bunched sampling [8 10, 14 17].

13 [ P 1 p+1)n 1 p0 kpn [ P 1 N 1 p0 q0 )] )] n 1 kpn n PN 1 where, in the second product we let q k Np in the third product q k NP. Imposing the periodicity of the graininess, we conclude that, for t>0, e z t n ) [ P 1 N 1 p0 q0 [ P 1 N 1 p0 k0 ) )] 1 + μq + pn) ) )] 1 + μq) q0 n PN 1 q0 n PN 1 q0 N 1 n PN 1 ) P ) 1 + μq) 1 + μkq) q0 n PN 1 q0 k0 N 1 ) P μq) qn PN 1 + μkq) ) 1 + μkq + PN) ) 1 + μq) ) P. 4.8) b) ROCs for bilateral Laplace transforms of signals with periodic time scales. For periodic time scales, μ 0 + μt + μ + μ0 μt μ. We will collectively refer to all of these upper bounds as Likewise, the lower bound will be referred to collectively as μ N 1 max m0 μt n). `μ 0 + `μt + `μ + `μ0 `μt `μ, `μ N 1 min n0 μt n). The ROC for the two sided exponential, e β:α t), follows as Z Z + Z Rα, `μ, μ) Lβ, `μ, μ). Examples of this ROC are shown in Figs In each of these figures:

14 Fig. 4 Illustration of the regions of convergence for time scales with periodic graininess for the double sided exponential e β:α t) in 4.2)whenα β are real β<α. The regions of convergence are shown blackened in a) b). As value of β starting from b) increases with all other values fixed, the smaller circle passing through β eventually becomes subsumed in the leftmost circle passing through α. When this happens, the Z ROCisempty.Thisisshowninc) where,sinceβ 1/ `μ, the smaller circle passing through β has shrunk to zero. As β α move to the right with the circle centers fixed, the ROC Z remains empty Fig. 5 This is the same case treated in Fig. 4 except β>α. Each illustration is the mirror image of that in Fig. 4 The locations of α β are labeled depicted by dots white in the middle fading to black at the dot s edges. The center of the modified Hilger circles are shown with a hollow dot,, corresponding to the value 1/ `μ on the negative real axis of the z plane, a solid dot,,isat 1/ μ is also on the negative real axis of the z plane. 2 2 Since `μ< μ the solid dot,, is always to the right of the hollow dot,.

15 Fig. 6 An example of the ROC, Z, for the double exponential, e β:α t),forreβ<re α Im β>im α>0. As the leftmost α circle becomes larger than the leftmost β circle, there is no intersection Z is empty. This is illustrated in c). Note that if α β are interchanged in a), b), or c), the region Z will be empty The ROC Z, if not empty, is blackened. The region Z + Rα, `μ, μ) not in Z is shown lightly shaded. The region Z Lβ, `μ, μ) not in Z is shown more darkly shaded. Figures 4 5 illustrate scenarios where α β are real. ROCs for complex α β are shown in Figs In all cases, the blackened area, Z, is equal to the intersection of 1) Z + corresponding to the area outside the union of the circles passing through α with 2) Z which is the area inside the intersection of the β circles Zero Lower Bound No Upper Bound An example is shown in Fig. 8 using a time scale with no upper graininess bound a lower graininess bounds of zero. 4.2 Proof of Theorem 4.1 Using the decomposition in 3.5), F + z) 0 e α t)e z σ t) t. This integral is equivalent to the causal time scale Laplace transform of e α t) has been derived elsewhere [5].

16 Fig. 7 The ROCs here are the mirror images of those in Fig. 6. The ROCs, Z, are for the double exponential e β:α t). Like Fig. 6, Im β>im α>0. However, in the examples shown here, Re β<re α.forc), we see that Z is empty. Note that if α β are interchanged in a), b), or c), the region Z will be empty Fig. 8 For the values of α β shown, the double sided exponential, e β:α t), in4.2) converges in the ROC, Z, shown shaded black. The region Z + is shaded lightly Z is more darkly shaded. Their intersection Z given by 4.1)is shown shaded black The other component of the Laplace transform is similar. 0 F z) 0 0 e β t)e z σ t) t μt)z e βt)e z t) t μt)z e β zt) t. 4.9)

17 Motivated by the operator definition, we continue F z) 1 0 β z 1 β z 0 1 z β e β zt) The step between 4.10) 4.11) is valid if For discrete time scales when t<0[5], β z 1 + μt)z e β zt) t β z)e β z t) t ) 1 β z. 4.11) e β z t n ) e β z ) ) 1 kn 1 + μt k )z 1 + μt k )β. 4.13) T `μ 0, μ0 ). This product approaches zero if all terms do not exceed one. This is true if 1 + μtk )z < 1 + μtk )β or, equivalently z + 1 μt k ) < β + 1 μt k ). This is the definition of the region Hβ, μt k )).Fort<0, the graininess varies over the range `μ 0 μt k) μ 0, the overall region of convergence is the intersection of all of the H s in this interval. However, since 3 H β,μt k ) ) Hβ, `μ) Hβ, μ), `μ μt k ) μ we conclude from 3.8) that 4.12) istrueintheroc Z L α, `μ 0, ) μ ) 3 This is graphically evident of in Fig. 1 where a sequence of modified Hilger circles are drawn over a range of graininesses. The points internal to all of the modified Hilger circles is equal to the points inside both the leftmost rightmost circles.

18 T `μ T, μt ).From4.13), e β z ) T k μt k )z 1 + μt k )β kσt) 1 + μt k )z 1 + μt k )β. 4.15) For 4.12) to be true, only the first product needs to be zero. We can thus deal with graininesses over the interval <t T the ROC increases from 4.14)to Z L α, `μ T, μt ). 4.16) T `μ, μ ). In the limiting case, the ROC is Z L α, `μ, ) μ. 4.17) When the lower upper asymptotic bounds are equal, `μ μ μ, using 3.10) applied to 4.17), we have the region of convergence for T μ ) Z Hα, μ ). 4.18) Lastly, for T0, ), we apply 3.11) to4.17) obtain Z Eα). 5 Convergence of Signals Bounded by the Double Sided Exponential In this section, after establishing a sufficient condition for the bilateral Laplace transform of e β:α t) to converge at z see Lemma 5.1), we show under the same condition, the Laplace transform of ft)will converge at z see Corollary 5.1) when ft) eβ:α t). 5.1) The result is a special case of a more general theorem Theorem 5.1) that only requires the bound in 5.1)tobetrueforT t>t + for any finite T + T. Lemma 5.1 A sufficient condition for the bilateral Laplace transform of e β:α t) to converge at z is e β:α t)e z σ t) t <. 5.2) Proof We begin with the magnitude of the Laplace transform if of e β:α t) write e β:α t)e z σ t) t e β:α t)e z σ t) t <. Theorem 5.1 If there exists a bounded function ft)on regressive time scale, T

19 finite values T < 0 T + > 0 such that ft) e β:α t) for t<t t>t +, the sufficient condition in 5.2) is true, then the bilateral Laplace transform in ft)converges at z. Proof Let Fz)be the Laplace transform of ft). Then Fz) ft)e z σ t) t ft)e σ z t) t [ T + T+ + T T + ] ft)e σ z t) t. If z is regressive ft) is bounded, then the middle integral is finite. For the remaining integrals we impose 5.2) write [ T + T + ] ft)e σ z t) t [ T <. + T + ] e β:αt)e σ z t) t Therefore, Fz) <. The following corollary follows immediately as a special case. Corollary 5.1 Suppose z is regressive 5.1) 5.2) hold. Then the Laplace transform of ft)converges at z. 6 The Box Minus Shift The box minus shift ) on a time scale can be defined using the bilateral Laplace transform. The box minus shift, ft τ), reduces to the conventional shift operator, ft τ) on R Z. Definition 6.1 We define the box minus operation,, through ft τ) Fz)e z τ). 6.1)

20 Note that, as a consequence, t ) ln1 + μξ)z) e z t τ) exp ξ ξτ μξ) 6.2), interpreting f τ) f0 τ), from the semigroup property, Lemma 6.1 As a consequence, from 6.3), Proof Using 6.2), 0 e z t) exp e z t τ) e z t)e z τ). 6.3) e z t) e z t). e z t τ) e z t)e z τ). exp ξt t [ t exp [ e z t) ] 1 ln1 + μξ)z) μξ) ξ0 ξ0 Definition 6.2 Define the Hilger delta as [4] ln1 + μξ)z) μξ) ln1 + μξ)z) μξ) ) ξ ) ξ )] 1 ξ e z t). δ H t τ σ ) { δ[t τ]/μt); μt) > 0, : δt τ); otherwise, 6.4) where the Kronecker delta, δ[t], is one for t 0 is otherwise zero, δt) is the Dirac delta [10]. For R Z, the Hilger delta in 6.4) becomes δ[t τ] δt τ) respectively. 7 Time Scale Convolution The box minus operation allows definition of convolution of two signals on the same time scale.

21 Definition 7.1 Convolution on a time scale is defined as ft) ht) : fξ)h t ξ σ ) ξ. 7.1) ξ T This definition is consistent with the convolution of transcendental functions defined in Bohner Peterson [1] its generalization [4]. It differs, however, from the time scale convolutions defined using the Fourier transform on a time scale [10, 12] which is defined only over a special class of time scales. 4 On R Z,7.1) becomes conventional convolution that describes the response, gt) ft) ht) of a linear time invariant system LTI) system with impulse response, ht), to a stimulus of ft) [10]. The Laplace transform of the impulse response is the system function or the transfer function, Hz), which contains the amplitude phase changes imposed by the system on the stimulus. This property is generalized to an arbitrary time scale by the following theorem. Theorem 7.1 System function) Convolving a function ht) with a time scale exponential function yields, as a result, the same exponential weighted by the Laplace transform of the impulse response Proof e w t) ht) e w t)h w). 7.2) e w t) ht) hτ)e w t τ σ ) τ e w t) e w t) hτ)e w τ σ ) τ hτ)e w σ τ) τ from which 7.2) follows. Theorem 7.2 Convolution on a time scale corresponds to multiplication in the Laplace domain Proof gt) ft) ht) Gz) F z)hz). gt) ft) ht) fξ)h t ξ σ ) ξ ξ T 4 Specifically, additively idempotent time scales [10, 12].

22 ξ T ξ T t T Hz) [ fξ)h t ξ σ ) ] ξ e z σ t) t ξ T [ h t ξ σ ) ] e z σ t) t ξ t T fξ) fξ) [ Hz)e σ z ξ)] ξ ξ T F z)hz) fξ)e z σ ξ) ξ Circuits Syst Signal Process Gz). The following results follow immediately. Convolution on a time scale is commutative, associative distributive over addition. The Sifting Property of the Hilger delta. If we define ft) δ H t) : fτ)δ H t τ σ ) τ t it follows that the Hilger delta is the identity operator for convolution on a time scale. ft) δ H t) ft). The sifting properties of the Dirac delta Kronecker delta on R Z follow as special cases. The Shift Property. If gt) ft) ht), then ft ξ) ht) ft) ht ξ) gt ξ). The Derivative Property. From the derivative theorem in 3.2), it follows immediately that g t) ft) h t) f t) ht). 8 Wide Sense Stationarity of a Stochastic Process on a Time Scale Let xt) be a real stochastic process on a time scale T. Its autocorrelation [10] is R x t, τ) E [ xt)xτ) ]. 8.1)

23 Definition 8.1 A stochastic process, xt),onatimescalet is wide sense stationary WSS) 5 [10, 13] if 6 R x t, τ) R x t τ). 8.2) As a consequence, the autocorrelation of a wide sense stationary WSS) stochastic process on a time scale can be represented by a single one-dimensional function, R x t). Notes 1. For R Z,8.2) takes on the familiar form 2. From 8.1), R x t, τ) R x τ, t); thus R x t, τ) R x t τ). R x t τ) R x τ t). Definition 8.2 The Laplace transform of the autocorrelation is the power spectral density, S x z) Theorem 8.1 On time scale T, let Then R x t) S x z). yt) xt) ht). 8.3) S y z) Hz)H z)s x z). 8.4) Proof Multiply both sides of 8.3) byxτ) expectate to give R xy τ, t) R x t τ) t ht) 8.5) where t denotes convolution with respect to the variable t. Rewrite 8.3) asyτ) xτ) τ hτ), multiply both sides by yt), expectate to give Substitute 8.3) gives R y t, τ) R xy τ, t) τ hτ). R y t, τ) ht) t R x t τ) τ hτ). 5 An further requirement of a constant first moment accompanies the classic definition of WSS stochastic processes [10, 13, 14]. For the treatment in this paper, however, such an assumption is not needed. 6 We use here a common abuse of notation [10, 13, 14]. The function R x cannot simultaneously be a two-dimensional function, R x t, τ), a one-dimensional function, R x t).

24 Laplace transform both sides with respect to t gives t [ t R y t, τ) ht) Rx t τ) τ hτ) ] e z σ t) t t T ξ T t T Circuits Syst Signal Process [ R x ξ τ)h t ξ σ ) ) ] τ ξ hτ) e z σ t) t ξ T [ R x ξ τ) h t ξ σ ) ] τ e z σ t) t hτ) ξ t T R x ξ τ) [ τ Hz)e z ξ) σ ξ)] hτ) ξ T [ ] τ Hz) R x ξ τ)e z σ ξ) hτ) ξ T Hz)S x z) [ e z τ) τ hτ) ] Hz)S x z) hη)e z τ η σ ) η Hz)S x z) η T η T Hz)S x z)e z τ) Hz)S x z)e z τ) hη)e z τ)e z η σ ) η η T η T hη)e z η σ ) η hη)e σ z η) η Hz)S x z)h z)e z τ). 8.6) The e z τ) term reveals R y t, τ) is of the form R y t τ). Therefore 8.4) follows immediately. 9 Conclusion We have established generalizations of the ROCs from unilateral to bilateral Laplace transforms. For double sided exponentials, the ROC, when it exists, are the intersection of points outside of a modified Hilger circle defined by behavior for positive time inside another modified Hilger circle determined by behavior for negative time. The ROCs revert to the familiar horizontal slab ROC for continuous time annulus for discrete time. Since graininess lies between zero infinity, there are conservative ROCs applicable for all time scales. Signals bounded by double sided exponentials were shown to converge in at least the ROC of the double sided exponential. The Laplace transform on a time scale is used to define a box minus operator that, in turn, allows definition of time scale convolution. Time scale convolution allows characterization of wide sense stationary stochastic processes on a time scale via its autocorrelation power spectral density.

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