Generalised Fractional Indexes Approximation with Application to Discrete-Time Generalised Weyl Symbol Computation

Dynamical, linear discrete-time system can be described by finite set of coefficients of difference equations or state space model. The set define dynamics of the linear time-invariant system for all times k (infinite time horizon). In contradistinction the description of discrete-time linear time-varying systems requires in general definition of an infinite number of coefficients. In order to describe dynamics of time-varying discrete-time systems one can use following state space description equations with time-dependent matrices [1, 2, 16, 18]:


Introduction
Dynamical, linear discrete-time system can be described by finite set of coefficients of difference equations or state space model.The set define dynamics of the linear time-invariant system for all times k  (infinite time horizon).In contradistinction the description of discrete-time linear time-varying systems requires in general definition of an infinite number of coefficients.
In order to describe dynamics of time-varying discrete-time systems one can use following state space description equations with time-dependent matrices [1,2,16,18]: where Above model can be converted into more general operators description [1,2,16,18] with transfer operator defined by set of impulse responses 0,0 Nevertheless analyzing or processing data with infinite dimensional size is impossible.Additional simplifying assumptions allow one to describe the timevarying system with finite set of coefficients.Linear timevarying systems can be classified with respect to the simplifying assumption.Generally following classes of time-varying systems can be distinguished [3]: general time-varying, periodic time-varying, almost periodic timevarying, almost time-invariant.Independently on the class of the system, but especially for time-varying systems in the general form analysis can be realized only on finite time horizon.It mean that accessible system data is limited by two constraints for indexes There are no assumptions about past and future system behaviour.
Time-frequency methods for continuous time systems are well known [4][5][6][7][8][9][10] as well as frequency methods for discrete-time systems [11][12][13][14].Many investigations has been made until now.Recently there are also known successful applications of time-varying approximations for nonlinear systems [17].The time-frequency transform is formulated as parameterized extension of Laplace transform.General form of the transform for continuous time systems can be defined by generalised Weyl symbol [10,15].
Discrete-time formula of the Generalised Weyl Symbol can be written using digital set of parameterised impulse responses (5) and the Discrete Fourier Transform (DFT) in following way where   R is arbitrary real number, usually bounded such that 0.5 Time-frequency transformation can be computed directly from eq. ( 6) only for =  0.5 (time-varying Zadeh transfer function [4] 0.5

 
).For 0   one can apply fractional indexes approximation introduced in [15].Main aim of the paper is to develop new generalised fractional indexescomputational method which allows to determine generalised Weyl symbol for arbitrary real 0.5,0.5 ,

 
not only for 0.5

 
(integer indexes method) and 0   (fractional indexes [15]).Parameter  allows to shape the set of parameterised impulse responses.
The selection of the parameter  in the generalised Weyl symbol enables selection of the best accuracy region for the time-frequency transform.

Generalised fractional indexes approximation
Definition.Generalised fractional index discrete time is defined as linear interpolation of h taken in following way Definition.Generalised fractional indexes discrete time response value of two variables

h p m h p m h p m h p m h p m
Taking account (5) we have: where floor denotes round toward minus infinity.
Generalized fractional impulse response can be written as follows

kn h h p m h p m h p m h p m
Thus generalised discrete-time Weyl symbol approximation can be defined by substituting (10) in (6), in the following form: where variables , , , , , a a b b  are defined above (9).

Application of the generalised fractional indexes for generalised Weyl symbol computation.
Time-invariant systems are always defined on infinite time horizon, thus all elements of the impulse response are always definite.Responses for time-varying systems do not need to be definite in general for all k  .The system is defined only on some bounded time horizon (4).The high accuracy are in the beginning-middle part of the time horizon.Accuracy for the end and beginning part of the time horizon is worse.Applying for computations generalised Weyl symbol with fractional indexes it is possible to choose precisely the part of the time horizon to compute with the high accuracy.

Conclusion
Time-frequency transformation is well known tool for systems and signals analysis.Accuracy of discrete-time, time-frequency diagrams depends mostly on the length of the time window.For systems defined on finite time horizon the length samples outside the time horizon are inaccessible.Generally there are two ways to analyse the system: use very short time-window, at least 2 times shorter then time horizon, or analyse the system on the full time horizon with incomplete data (without data outside the defined time horizon. Short time horizons results in boundary effects (boundary distortions) on the time-frequency diagramthe beginning and the end of time horizon.Using additional parameter  one can continuously choose the best accuracy region from the finite time horizon.Negative values of the transformation parameter close to 0.5

 
results in the best accuracy at the beginning of the time horizon, whereas positive values close to 0.5

 
gives the best accuracy at the end of the time horizon.Middle values of  close to zero ensures the best accuracy in the middle of the time horizon.

Fig. 3 . 5 
Fig. 3. 3D Magnitude-Time-Frequency diagram calculated for 0.5   using fractional indexes impulse responses for system defined on finite time horizon Applying parameterized impulse response (5) for the discrete-time low-pass filter mentioned above defined on finite horizon, three following 3D time-frequency diagrams are calculated and plotted in Fig. 2, Fig. 3 (integer indexes 0.5   ) and 4 (generalised fractional indexes

Fig. 4 .
Fig. 4. 3D Magnitude-Time-Frequency diagram calculated for 0.3   using fractional indexes impulse responses for system defined on finite time horizon Fig. 2 shows Kohn-Nirenberg symbol [9] with 0.5   and the high accuracy at the beginning part of the time horizon while in fig. 3 is plotted time-varying transfer function [4] with 0.5   and the high accuracy at the end part of the time horizon.Fig. 4 is 3D magnitude of Generalised Weyl Symbol with 0   calculated using fractional indexes approximation for impulse responses.
 and is system response at time k 1 for shifted by time k 0 Kronecker delta