Stochastic Controledited Part 9 pdf

Stochastic Control312 Equation (23) has three arbitrary parameters   , , n K   , which can be adjusted such that the approximate curve coincides with the actual curve at different points. The reason for presenting 4 th order approximation of the DPSD is that we can compute explicit expressions for the constants   , , n K   as functions of specific points on the data-graphs of the DPSD. In fact, if the approximate density ( )  S f coincides with the exact density ( )S f at 0 f  and max f f , then the arbitrary parameters   , , n K   are computed explicitly as 2 max 2 max 2 1 (0) 1 1 , , (0) 2 ( ) 1 2 n n f S K S S f                    (24) Figure 6 shows   S f and its approximation   S f  via a 4 th order even function. In the next section, the approximated DPSD is used to develop stochastic STF channel models. 4.3 Stochastic STF Channel Models A stochastic realization is used here to obtain a state space representation for the inphase and quadrature components [32]. The SDE, which corresponds to ( )H s in (23) is 2 2 ( ) 2 ( ) ( ) ( ), (0), (0) are g iven n n d x t dx t x t dt KdW t x x       (25) where   0 ( ) t dW t  is a white-noise process. Equation (25) can be rewritten in terms of inphase and quadrature components as 2 2 2 2 ( ) 2 ( ) ( ) ( ), (0), (0) are g iven ( ) 2 ( ) ( ) ( ), (0), (0) are g iven I n I n I I I I Q n Q n Q Q Q Q d x t dx t x t dt KdW t x x d x t dx t x t dt KdW t x x             (26) where   0 ( ) I t dW t  and   0 ( ) Q t dW t  are two independent and identically distributed (i.i.d.) white Gaussian noises. -6 -4 -2 0 2 4 6 -25 -20 -15 -10 -5 0 S D (f),  H(j  )  2 in dB [a] v = 5 km/h, f c = 910 MHz,  m = 10 0 ; Frequency Hz. S D (f)  H(j  )  2 -150 -100 -50 0 50 100 150 -50 -45 -40 -35 -30 [b] v = 120 km/h, f c = 910 MHz,  m = 10 0 ; Frequency Hz. S D (f),  H(j  )  2 in dB S D (f)  H(j  )  2 Fig. 6. DPSD, ( ) D S f , and its approximation 2 ( ) ( )S H j     via a 4 th order transfer function for mobile speed of (a) 5 km/hr and (b) 120 km/hr. Several stochastic realizations [32] can be used to obtain a state space representation for the inphase and quadrature components of STF channel models. For example, the stochastic observable canonical form (OCF) realization [33] can be used to realize (26) for the inphase and quadrature components for the jth path as                               1 1 1 , , , 2 2 2 2 , , , 1 , 2 , 1 1 , , 2 2 2 , , 0 0 1 0 , , 2 0 1 0 , 0 1 2 I j I j I j I j n n n I j I j I j I j I j j I j Q j Q j n n n Q j Q j dX t X t X dt dW t K dX t X t X X t I t f t X t dX t X t dX t X t                                                                                                        1 , 2 , 1 , 2 , 0 0 , , 0 1 0 Q j Q j Q j Q j Q j j Q j X dt dW t K X X t Q t f t X t                              (27) where       1 2 , , , [ ] T I j I j I j X t X t X t and       1 2 , , , [ ] T Q j Q j Q j X t X t X t are state vectors of the inphase and quadrature components.   j I t and   j Q t correspond to the inphase and quadrature components, respectively,     0 I j t W t  and     0 Q j t W t  are independent standard Brownian motions, which correspond to the inphase and quadrature components of the jth path respectively, the parameters   , , n K   are obtained from the approximation of the Wireless fading channel models: from classical to stochastic differential equations 313 Equation (23) has three arbitrary parameters   , , n K   , which can be adjusted such that the approximate curve coincides with the actual curve at different points. The reason for presenting 4 th order approximation of the DPSD is that we can compute explicit expressions for the constants   , , n K   as functions of specific points on the data-graphs of the DPSD. In fact, if the approximate density ( )  S f coincides with the exact density ( )S f at 0 f  and max f f , then the arbitrary parameters   , , n K   are computed explicitly as 2 max 2 max 2 1 (0) 1 1 , , (0) 2 ( ) 1 2 n n f S K S S f                    (24) Figure 6 shows   S f and its approximation   S f  via a 4 th order even function. In the next section, the approximated DPSD is used to develop stochastic STF channel models. 4.3 Stochastic STF Channel Models A stochastic realization is used here to obtain a state space representation for the inphase and quadrature components [32]. The SDE, which corresponds to ( )H s in (23) is 2 2 ( ) 2 ( ) ( ) ( ), (0), (0) are g iven n n d x t dx t x t dt KdW t x x       (25) where   0 ( ) t dW t  is a white-noise process. Equation (25) can be rewritten in terms of inphase and quadrature components as 2 2 2 2 ( ) 2 ( ) ( ) ( ), (0), (0) are g iven ( ) 2 ( ) ( ) ( ), (0), (0) are g iven I n I n I I I I Q n Q n Q Q Q Q d x t dx t x t dt KdW t x x d x t dx t x t dt KdW t x x             (26) where   0 ( ) I t dW t  and   0 ( ) Q t dW t  are two independent and identically distributed (i.i.d.) white Gaussian noises. -6 -4 -2 0 2 4 6 -25 -20 -15 -10 -5 0 S D (f),  H(j  )  2 in dB [a] v = 5 km/h, f c = 910 MHz,  m = 10 0 ; Frequency Hz. S D (f)  H(j  )  2 -150 -100 -50 0 50 100 150 -50 -45 -40 -35 -30 [b] v = 120 km/h, f c = 910 MHz,  m = 10 0 ; Frequency Hz. S D (f),  H(j  )  2 in dB S D (f)  H(j  )  2 Fig. 6. DPSD, ( ) D S f , and its approximation 2 ( ) ( )S H j     via a 4 th order transfer function for mobile speed of (a) 5 km/hr and (b) 120 km/hr. Several stochastic realizations [32] can be used to obtain a state space representation for the inphase and quadrature components of STF channel models. For example, the stochastic observable canonical form (OCF) realization [33] can be used to realize (26) for the inphase and quadrature components for the jth path as                               1 1 1 , , , 2 2 2 2 , , , 1 , 2 , 1 1 , , 2 2 2 , , 0 0 1 0 , , 2 0 1 0 , 0 1 2 I j I j I j I j n n n I j I j I j I j I j j I j Q j Q j n n n Q j Q j dX t X t X dt dW t K dX t X t X X t I t f t X t dX t X t dX t X t                                                                                                        1 , 2 , 1 , 2 , 0 0 , , 0 1 0 Q j Q j Q j Q j Q j j Q j X dt dW t K X X t Q t f t X t                              (27) where       1 2 , , , [ ] T I j I j I j X t X t X t and       1 2 , , , [ ] T Q j Q j Q j X t X t X t are state vectors of the inphase and quadrature components.   j I t and   j Q t correspond to the inphase and quadrature components, respectively,     0 I j t W t  and     0 Q j t W t  are independent standard Brownian motions, which correspond to the inphase and quadrature components of the jth path respectively, the parameters   , , n K   are obtained from the approximation of the Stochastic Control314 deterministic DPSD, and   I j f t and   Q j f t are arbitrary functions representing the LOS of the inphase and quadrature components respectively, characterizing further dynamic variations in the environment. Expression (27) for the jth path can be written in compact form as                         0 0 0 0 0 0 I I I I I Q Q Q Q Q I I I Q Q Q dX t X t dW t A B dt A B dX t X t dW t X t f t I t C C X t f tQ t                                                                       (28) where 2 0 1 0 , , 1 0 2 I Q I Q I Q n n n A A B B C C K                              (29)       0 , I Q t W t W t  are independent standard Brownian motions which are independent of the initial random variables   0 I X and   0 Q X , and       , ; 0 I Q f s f s s t  are random processes representing the inphase and quadrature LOS components, respectively. The band-pass representation of the received signal corresponding to the jth path is given as                 cos sin ( ) I I I c Q Q Q c l j y t C X t f t t C X t f t t s t v t              (30) where ( )v t is the measurement noise. As the DPSD varies from one instant to the next, the channel parameters   , , n K   also vary in time, and have to be estimated on-line from time domain measurements. Without loss of generality, we consider the case of flat fading, in which the mobile-to-mobile channel has purely multiplicative effect on the signal and the multipath components are not resolvable, and can be considered as a single path [2]. The frequency selective fading case can be handled by including multiple time-delayed echoes. In this case, the delay spread has to be estimated. A sounding device is usually dedicated to estimate the time delay of each discrete path such as Rake receiver [26]. Following the state space representation in (28) and the band pass representation of the received signal in (30), the fading channel can be represented using a general stochastic state space representation of the form [28]                     dX t A t X t dt B t dW t y t C t X t D t v t     (31) where                                           0 0 , , , 0 0 cos sin , cos sin , T I I I Q Q Q c I c Q c c T T I Q I Q A t B t X t X t X t A t B t A t B t C t t C t C D t t t v t v t v t dW t dW t dW t                                                 (32) In this case, ( )y t represents the received signal measurements,   X t is the state variable of the inphase and quadrature components, and   v t is the measurement noise. Time domain simulation of STF channels can be performed by passing two independent white noise processes through two identical filters,   H s  , obtained from the factorization of the deterministic DPSD, one for the inphase and the other for the quadrature component [4], and realized in their state-space form as described in (28) and (29). Example 3: Consider a flat fading wireless channel with the following parameters: 900MHz c f  , 80 km/hv  , o 10 m   , and     0 I Q j j f t f t   . Time domain simulation of the inphase and quadrature components, attenuation coefficient, phase angle, input signal, and received signal are shown in Figures 7-9. The inphase and quadrature components have been produced using (28) and (29), while the received signal is reproduced using (30). The simulation of the dynamical STF channel is performed using Simulink in Matlab [37]. 4.4 Solution to the Stochastic State Space Model The stochastic TV state space model described in (31) and (32) has a solution [32, 38]             0 1 0 0 0 , , t L L L L L L t X t t t X t u t B u dW u                 (33) where L = I or Q, and   0 , L t t is the fundamental matrix, which satisfies       0 0 , , L L L t t A t t t    and   0 0 , L t t is the identity matrix. Wireless fading channel models: from classical to stochastic differential equations 315 deterministic DPSD, and   I j f t and   Q j f t are arbitrary functions representing the LOS of the inphase and quadrature components respectively, characterizing further dynamic variations in the environment. Expression (27) for the jth path can be written in compact form as                         0 0 0 0 0 0 I I I I I Q Q Q Q Q I I I Q Q Q dX t X t dW t A B dt A B dX t X t dW t X t f t I t C C X t f tQ t                                                                       (28) where 2 0 1 0 , , 1 0 2 I Q I Q I Q n n n A A B B C C K                              (29)       0 , I Q t W t W t  are independent standard Brownian motions which are independent of the initial random variables   0 I X and   0 Q X , and       , ; 0 I Q f s f s s t   are random processes representing the inphase and quadrature LOS components, respectively. The band-pass representation of the received signal corresponding to the jth path is given as                 cos sin ( ) I I I c Q Q Q c l j y t C X t f t t C X t f t t s t v t              (30) where ( )v t is the measurement noise. As the DPSD varies from one instant to the next, the channel parameters   , , n K   also vary in time, and have to be estimated on-line from time domain measurements. Without loss of generality, we consider the case of flat fading, in which the mobile-to-mobile channel has purely multiplicative effect on the signal and the multipath components are not resolvable, and can be considered as a single path [2]. The frequency selective fading case can be handled by including multiple time-delayed echoes. In this case, the delay spread has to be estimated. A sounding device is usually dedicated to estimate the time delay of each discrete path such as Rake receiver [26]. Following the state space representation in (28) and the band pass representation of the received signal in (30), the fading channel can be represented using a general stochastic state space representation of the form [28]                     dX t A t X t dt B t dW t y t C t X t D t v t     (31) where                                           0 0 , , , 0 0 cos sin , cos sin , T I I I Q Q Q c I c Q c c T T I Q I Q A t B t X t X t X t A t B t A t B t C t t C t C D t t t v t v t v t dW t dW t dW t                                                 (32) In this case, ( )y t represents the received signal measurements,   X t is the state variable of the inphase and quadrature components, and   v t is the measurement noise. Time domain simulation of STF channels can be performed by passing two independent white noise processes through two identical filters,   H s  , obtained from the factorization of the deterministic DPSD, one for the inphase and the other for the quadrature component [4], and realized in their state-space form as described in (28) and (29). Example 3: Consider a flat fading wireless channel with the following parameters: 900MHz c f  , 80 km/hv  , o 10 m   , and     0 I Q j j f t f t  . Time domain simulation of the inphase and quadrature components, attenuation coefficient, phase angle, input signal, and received signal are shown in Figures 7-9. The inphase and quadrature components have been produced using (28) and (29), while the received signal is reproduced using (30). The simulation of the dynamical STF channel is performed using Simulink in Matlab [37]. 4.4 Solution to the Stochastic State Space Model The stochastic TV state space model described in (31) and (32) has a solution [32, 38]             0 1 0 0 0 , , t L L L L L L t X t t t X t u t B u dW u                 (33) where L = I or Q, and   0 , L t t is the fundamental matrix, which satisfies       0 0 , , L L L t t A t t t    and   0 0 , L t t is the identity matrix. Stochastic Control316 0 0.05 0.1 -2 -1 0 1 2 I(t) In-phase 0 0.05 0.1 -2 -1 0 1 2 Q(t) Quadrature 0 0.05 0.1 0 1 2 3 time [sec.] sqrt(I 2 (t)+Q 2 (t)) 0 0.05 0.1 -100 -50 0 50 100 time [sec.] tan -1 [Q(t)/I(t)] Fig. 7. Inphase and quadrature components, attenuation coefficient, and phase angle of the STF wireless channel in Example 3. 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0 1 2 3 r(t) = sqrt(I 2 (t)+Q 2 (t)) Channel Envelop; v = 80 km/h,  m = 10 o 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 -30 -20 -10 0 10 time [sec.] r(t) [dB s] Fig. 8. Attenuation coefficient in absolute units and in dB’s for the STF wireless channel in Example 3. 0 0.05 0.1 -1 -0.5 0 0.5 1 s l (t) Flat-Slow Fading 0 0.05 0.1 -1 0 1 y(t) 0 0.05 0.1 -1 -0.5 0 0.5 1 time [sec.] s l (t) Flat-Fast Fading 0 0.05 0.1 -1 0 1 time [sec.] y(t) Fig. 9. Input signal, ( ) l s t , and the corresponding received signal, ( ) y t , for flat slow fading (top) and flat fast fading conditions (bottom). Further computations show that the mean of   L X t is given by [32]       0 0 , L L L E X t t t E X t           (34) and the covariance matrix of   L X t is given by                   0 1 1 0 0 0 0 0 , , , , t T T T L L L L L L L L t t t t Var X t u t B u B u u t du t t                         (35) Differentiating (35) shows that   L t satisfies the Riccati equation               T T L L L t A t t t A t B t B t      (36) For the time-invariant case,   L L A t A and   , L L B t B equations (33)-(35) simplify to Wireless fading channel models: from classical to stochastic differential equations 317 0 0.05 0.1 -2 -1 0 1 2 I(t) In-phase 0 0.05 0.1 -2 -1 0 1 2 Q(t) Quadrature 0 0.05 0.1 0 1 2 3 time [sec.] sqrt(I 2 (t)+Q 2 (t)) 0 0.05 0.1 -100 -50 0 50 100 time [sec.] tan -1 [Q(t)/I(t)] Fig. 7. Inphase and quadrature components, attenuation coefficient, and phase angle of the STF wireless channel in Example 3. 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0 1 2 3 r(t) = sqrt(I 2 (t)+Q 2 (t)) Channel Envelop; v = 80 km/h,  m = 10 o 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 -30 -20 -10 0 10 time [sec.] r(t) [dB s] Fig. 8. Attenuation coefficient in absolute units and in dB’s for the STF wireless channel in Example 3. 0 0.05 0.1 -1 -0.5 0 0.5 1 s l (t) Flat-Slow Fading 0 0.05 0.1 -1 0 1 y(t) 0 0.05 0.1 -1 -0.5 0 0.5 1 time [sec.] s l (t) Flat-Fast Fading 0 0.05 0.1 -1 0 1 time [sec.] y(t) Fig. 9. Input signal, ( ) l s t , and the corresponding received signal, ( ) y t , for flat slow fading (top) and flat fast fading conditions (bottom). Further computations show that the mean of   L X t is given by [32]       0 0 , L L L E X t t t E X t           (34) and the covariance matrix of   L X t is given by                   0 1 1 0 0 0 0 0 , , , , t T T T L L L L L L L L t t t t Var X t u t B u B u u t du t t                         (35) Differentiating (35) shows that   L t satisfies the Riccati equation               T T L L L t A t t t A t B t B t      (36) For the time-invariant case,   L L A t A and   , L L B t B equations (33)-(35) simplify to Stochastic Control318                             0 0 0 0 0 0 0 0 0 L L L T T L L L L t A t t A t u L L L L t A t t L L t A t t A t t A t u A t u T L L L L t X t e X t e B dW u E X t e E X t t e Var X t e e B B e du                            (37) It can be seen in (34) and (35) that the mean and variance of the inphase and quadrature components are functions of time. Note that the statistics of the inphase and quadrature components, and therefore the statistics of the STF channel, are time varying. Therefore, these stochastic state space models reflect the TV characteristics of the STF channel. Following the same procedure in developing the STF channel models, the stochastic TV ad hoc channel models are developed in the next section. 5. Stochastic Ad Hoc Channel Modeling 5.1 The Deterministic DPSD of Ad Hoc Channels Dependent on mobile speed, wavelength, and angle of incidence, the Doppler frequency shifts on the multipath rays give rise to a DPSD. The cellular DPSD for a received fading carrier of frequency f c is given in (20) and can be described by [25]   1 2 1 1 1 , 1 / 0 , otherwise c c f f f S f f f pG f f                      (38) where 1 f is the maximum Doppler frequency of the mobile , p is the average power received by an isotropic antenna, and G is the gain of the receiving antenna. For a mobile-to- mobile (or ad hoc) link, with 1 f and 2 f as the sender and receiver’s maximum Doppler frequencies, respectively, the degree of double mobility, denoted by  is defined by     1 2 1 2 min , /max , f f f f       , so 0 1    , where 1   corresponds to a full double mobility and 0   to a single mobility like cellular link, implying that cellular channels are a special case of mobile-to-mobile channels. The corresponding deterministic mobile-to- mobile DPSD is given by [39-41]         2 2 2 1 K 1 , 1 1 2 / 0 , otherwise c c m m m f f S f f f f f pG f                                        (39) where   K  is the complete elliptic integral of the first kind, and   1 2 max , m f f f . Figure 10 shows the deterministic mobile-to-mobile DPSDs for different values of α’s. Thus, a generalized DPSD has been found where the U-shaped spectrum of cellular channels is a special case. Here, we follow the same procedure in deriving the stochastic STF channel models in Section 4. The deterministic ad hoc DPSD is first factorized into an approximate nth order even transfer function, and then use a stochastic realization [32] to obtain a state space representation for inphase and quadrature components. The complex cepstrum algorithm [36] is used to approximate the ad hoc DPSD and is discussed next. -1 -0.5 0 0.5 1 0 2 4 6 8 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 0 1 2 3 4 -4 -2 0 2 4 0 1 2 3 4 5 -2 0 2 0 2 4 6 8 10 12 alpha = 0 alpha = 0.25, f2 = 4*f1 alpha = 0.5, f2 = 2*f1 alpha = 1 Fig. 10. Ad hoc deterministic DPSDs for different values of 's  , with parameters 1 0, 1, c f f  and pG   . 5.2 Approximating the Deterministic Ad Hoc DPSD Since the ad hoc DPSD is more complicated than the cellular one, we propose to use a more complex and accurate approximation method: The complex cepstrum algorithm [36]. It uses several measured points of the DPSD instead of just three points as in the simple method (described in Section 4.2). It can be explained briefly as follows: On a log-log scale, the magnitude data is interpolated linearly, with a very fine discretization. Then, using the complex cepstrum algorithm [36], the phase, associated with a stable, minimum phase, real, rational transfer function with the same magnitude as the magnitude data is generated. With the new phase data and the input magnitude data, a real rational transfer function can be found by using the Gauss-Newton method for iterative search [35], which is used to Wireless fading channel models: from classical to stochastic differential equations 319                             0 0 0 0 0 0 0 0 0 L L L T T L L L L t A t t A t u L L L L t A t t L L t A t t A t t A t u A t u T L L L L t X t e X t e B dW u E X t e E X t t e Var X t e e B B e du                            (37) It can be seen in (34) and (35) that the mean and variance of the inphase and quadrature components are functions of time. Note that the statistics of the inphase and quadrature components, and therefore the statistics of the STF channel, are time varying. Therefore, these stochastic state space models reflect the TV characteristics of the STF channel. Following the same procedure in developing the STF channel models, the stochastic TV ad hoc channel models are developed in the next section. 5. Stochastic Ad Hoc Channel Modeling 5.1 The Deterministic DPSD of Ad Hoc Channels Dependent on mobile speed, wavelength, and angle of incidence, the Doppler frequency shifts on the multipath rays give rise to a DPSD. The cellular DPSD for a received fading carrier of frequency f c is given in (20) and can be described by [25]   1 2 1 1 1 , 1 / 0 , otherwise c c f f f S f f f pG f f                      (38) where 1 f is the maximum Doppler frequency of the mobile , p is the average power received by an isotropic antenna, and G is the gain of the receiving antenna. For a mobile-to- mobile (or ad hoc) link, with 1 f and 2 f as the sender and receiver’s maximum Doppler frequencies, respectively, the degree of double mobility, denoted by  is defined by     1 2 1 2 min , /max , f f f f       , so 0 1    , where 1   corresponds to a full double mobility and 0   to a single mobility like cellular link, implying that cellular channels are a special case of mobile-to-mobile channels. The corresponding deterministic mobile-to- mobile DPSD is given by [39-41]         2 2 2 1 K 1 , 1 1 2 / 0 , otherwise c c m m m f f S f f f f f pG f                                        (39) where   K  is the complete elliptic integral of the first kind, and   1 2 max , m f f f . Figure 10 shows the deterministic mobile-to-mobile DPSDs for different values of α’s. Thus, a generalized DPSD has been found where the U-shaped spectrum of cellular channels is a special case. Here, we follow the same procedure in deriving the stochastic STF channel models in Section 4. The deterministic ad hoc DPSD is first factorized into an approximate nth order even transfer function, and then use a stochastic realization [32] to obtain a state space representation for inphase and quadrature components. The complex cepstrum algorithm [36] is used to approximate the ad hoc DPSD and is discussed next. -1 -0.5 0 0.5 1 0 2 4 6 8 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 0 1 2 3 4 -4 -2 0 2 4 0 1 2 3 4 5 -2 0 2 0 2 4 6 8 10 12 alpha = 0 alpha = 0.25, f2 = 4*f1 alpha = 0.5, f2 = 2*f1 alpha = 1 Fig. 10. Ad hoc deterministic DPSDs for different values of 's  , with parameters 1 0, 1, c f f  and pG   . 5.2 Approximating the Deterministic Ad Hoc DPSD Since the ad hoc DPSD is more complicated than the cellular one, we propose to use a more complex and accurate approximation method: The complex cepstrum algorithm [36]. It uses several measured points of the DPSD instead of just three points as in the simple method (described in Section 4.2). It can be explained briefly as follows: On a log-log scale, the magnitude data is interpolated linearly, with a very fine discretization. Then, using the complex cepstrum algorithm [36], the phase, associated with a stable, minimum phase, real, rational transfer function with the same magnitude as the magnitude data is generated. With the new phase data and the input magnitude data, a real rational transfer function can be found by using the Gauss-Newton method for iterative search [35], which is used to Stochastic Control320 generate a stable, minimum phase, real rational transfer function, denoted by   H s  , to identify the best model from the data of   H f as       2 1 , min l k k k k a b wt f H f H f     (40) where   1 1 1 0 1 1 1 0 m m m m m b s b s b H s s a s a s a              (41)   1 0 , , m b b b   ,   1 0 , , m a a a   ,   wt f is the weight function, and l is the number of frequency points. Several variants have been suggested in the literature, where the weighting function gives less attention to high frequencies [35]. This algorithm is based on Levi [42]. Figure 11 shows the DPSD, ( )S f , and its approximation ( )S f  via different orders using complex cepstrum algorithm. The higher the order of ( )S f  the better the approximation obtained. It can be seen that approximation with a 4 th order transfer function gives a very good approximation. -50 -40 -30 -20 -10 0 10 20 30 40 50 1.5 2 2.5 3 3.5 4 4.5 Original S(f) Appr. with order = 2 Appr. with order = 4 Appr. with order = 6 S(f) Frequency (Hz) alpha = 0.25 Fig. 11. DPSD, ( )S f , and its approximations, ( )S f  , using complex cepstrum algorithm for different orders of ( )S f  . Figure 12(a) and 12(b) show the DPSD, ( )S f , and its approximation ( )S f  using the complex cepstrum and simple approximation methods, respectively, for different values of 's  via 4 th order even function. It can be noticed that the former gives better approximation than the latter; since it employs all measured points of the DPSD instead of just three points in the simple method. -60 -40 -20 0 20 40 60 0 0.05 0.1 0.15 0.2 0.25 Magnitude (W) Frequency (Hz) S(w) Appr. S(w) alpha = 0.5 alpha = 0.33 alpha = 0.25 alpha = 0.2 (a) -60 -40 -20 0 20 40 60 0 0.05 0.1 0.15 0.2 0.25 Frequency (Hz) Magnitude (W) S(w) Appr. S(w) alpha = 0.5 alpha = 0.33 alpha = 0.25 alpha = 0.2 (b) Fig. 12. DPSD,   S f , and its approximation,   S f  , via 4 th order function for different α’s using (a) the complex cepstrum, and (b) the simple approximation methods. [...]... mobile communications, IEEE Inc., NY, 197 4 [26] B Sklar, Digital communications: Fundamentals and applications Prentice Hall, 2nd Edition, 2001 [27] C.D Charalambous and N Menemenlis, Stochastic models for long-term multipath fading channels,” Proc 38th IEEE Conf Decision Control, pp 494 7- 495 2, Phoenix, AZ, Dec 199 9 [28] M.M Olama, S.M Djouadi, and C.D Charalambous, Stochastic differential equations for... systems,” Electronics Letters, vol 33, no 15, pp.1287-1288, 199 7 R.H Clarke, “A statistical theory of mobile radio reception,” Bell Systems Technical Journal, 47, 95 7-1000, 196 8 326 Stochastic Control [10] J.I Smith, “A computer generated multipath fading simulation for mobile radio,” IEEE Trans on Vehicular Technology, vol 24, no 3, pp 39- 40, Aug 197 5 [11] T Aulin, “A modified model for fading signal at... vol 2006, Article ID 898 64, 13 pages, 2006 [31] C.D Charalambous and N Menemenlis, “General non-stationary models for shortterm and long-term fading channels,” EUROCOMM 2000, pp 142-1 49, April 2000 [32] B Oksendal, Stochastic differential equations: An introduction with applications, Springer, Berlin, Germany, 199 8 [33] W.J Rugh, Linear system theory, Prentice-Hall, 2nd Edition, 199 6 [34] R.E.A.C Paley... Math Soc Coll., Am Math., vol 9, 193 4 [35] J.E Dennis Jr., and R.B Schnabel, Numerical methods for unconstrained optimization and nonlinear equations, NJ: Prentice-Hall, 198 3 [36] A.V Oppenheim and R.W Schaffer, Digital signal processing, Prentice Hall, New Jersey, 197 5, pp 513 [37] http://www.mathworks.com/ [38] P.E Caines, Linear stochastic systems, New-York Wiley, 198 8 [ 39] R Wang and D Cox, “Channel... Abarbanel, 199 6), economics, material science, to name several In control theory, it helps to understand the information structure and hence characterize the cause-effect notion of causality in nonsequential stochastic control systems (e.g., Andersland & Teneketzis, 199 2) Given the well-known importance, it has been an active arena of research for several decades (e.g.,Kaneko, 198 6; Vastano & Swinney, 198 8;... Swinney, 198 8; Rosenblum et al., 199 6; Arnhold et al., 199 9; Schreiber, 2000; Kaiser & Schreiber, 2002) However, it was not until recently that the concept is formalized, on a rigorous mathematical and physical footing In this chapter we will introduce the rigorous formalism initialized in Liang & Kleeman (2005) and established henceforth; we will particularly focus on the part of the studies by Liang (2008)... reflections, theoretical and experimental waveform power spectra,” Bell Systems Technical Journal, 43, 293 5 297 1, 196 4 F Graziosi, M Pratesi, M Ruggieri, and F Santucci, “A multicell model of handover initiation in mobile cellular networks,” IEEE Transactions on Vehicular Technology, vol 48, no 3, pp 802-814, 199 9 F Graziosi and F Santucci, “A general correlation model for shadow fading in mobile systems,”... 306-3 09, 2002 [41] C.S Patel, G.L Stuber, and T.G Pratt, “Simulation of Rayleigh-faded mobile-to-mobile communication channels,” IEEE Trans on Comm., vol 53, no 11, pp 1876-1884, 2005 [42] E.C Levi, “Complex curve fitting,” IRE Trans on Automatic Control, vol AC-4, pp 3744, 195 9 328 Stochastic Control Information flow and causality quantification in discrete and continuous stochastic systems 3 29 17... 2, pp 353-357, May 199 6 [17] C.C Tan and N.C Beaulieu, “On first-order Markov modeling for the Rayleigh fading channel,” IEEE Transactions on Communications, vol 48, No 12, pp December 2000 [18] H.S Wang and N Moayeri, “Finite-state Markov channel: A useful model for radio communication channels,” IEEE Transactions on Vehicular Technology, Vol 44, No 1 , pp 163-171, February 199 5 [ 19] I Chlamtac, M Conti,... exchanged (cf Cove & Thomas, 199 1), but does not tell anything about the directionality of the exchange This is the major thrust that motivates many studies in this field, among which are Vastano & Swinney ( 198 8) and Schreiber (2000) Another thrust, which is also related to the above, is the concern over causality Traditionally, causality, such as the Granger causality (Granger, 196 9), is just a qualitative . pp.1287-1288, 199 7. [9] R.H. Clarke, “A statistical theory of mobile radio reception,” Bell Systems Technical Journal , 47, 95 7-1000, 196 8. Wireless fading channel models: from classical to stochastic. vol. 33, no. 15, pp.1287-1288, 199 7. [9] R.H. Clarke, “A statistical theory of mobile radio reception,” Bell Systems Technical Journal , 47, 95 7-1000, 196 8. Stochastic Control326 [10] J.I channels,” Proc . 38 th IEEE Conf. Decision Control, pp. 494 7- 495 2, Phoenix, AZ, Dec. 199 9. [28] M.M. Olama, S.M. Djouadi, and C.D. Charalambous, Stochastic differential equations for modeling, estimation