An Algebraical Superposition Technic for Transformation From Z Domain to Time Domain

An algebraical superposition technic for trasformation from z domain to time domain is presented. The establishing model process is: starting the inverse z transforms integral formula, and in its region of convergence based on thelsquocomplex function integralrsquothe inverse z transform integral is represented by 2k-1 term series. When the transform function on iterm series along integral circle are conjugated complex number distribution,the bidirectional series sum on k[-K,K] term series can be expressed by a monomial trigonomial function series sum on k[0,K],in which the members are easy calculation and sum. In the paper the solution process and main points are presented.nbsp The application examples are shown,the resules are supported to the algebraical superposition technic.The technic can be used to solve the problem which are difficult to be solved by presented method#65288such as Partial Fraction Exparation method,etc#65289.


I. INTRODUCTION
In the fields of electronics, dynamics, controls and other fields of science and technology, the acquirement of system parameters is very important. Some system parameters vary with time, while some vary with frequency.The former is called time-domain parameter, and the latter is called frequency-domain parameter. We know one from another by the transformation between time domain and frequency domain.The signal variation with time often has two states: continuous-time variation and discrete-time variation, the later is often from sampling a continuous-time variation signal. With regard to time domain-frequency domain transforrm, Laplace trasform is often used for continuous-time variation signal. In discrete-time signal system z transform is a strong tool to analysis linear time-invariant system .The z transform pay an important role in solving difference equation,which is simillar to that Laplace trasform pay an important role in solving differential equation [1][2] [3].
For Laplace trasform and z transform, to perform transform from time domain to frequency domain, some results can be obtained from trasform-pair Table or from the numerical integral [1] [2]. For a inverse transform form frequency domain\、z domain to time domain it is often necessary to use one kind of inverse transform formular ,but its time domain solution may appear data including 0 t  , that is disagreement to ‚no meaning or no necessary for 0 t  ‛. May be, it is a puzzle.
In the paper we reserch to use algebraical superposition technic to inverse z transform for extending the domain of inverse z transform Ben Qing GAO, School of Information and Electronics ， Beijing Institute of Technology，Beijing 100081，China. and finding easily out solution data only for 0 t  .The mathmatical mode for algebraical superposition technic and the main points for executive calculation will be shown in next two sections. Calculation examles will be exhibited to prove algebraical superposition technic in another section .It should point out that known method for inverse z transform are direct division,partial-fraction expansion,contour integation [1] [2]. But those require the function formula to be rational function. The algebraical superposition technic in the paper is suit for both rational functions and irrational functions.
In the paper we reserch to use algebraical superposition technic to inverse z transform for extending the domain of inverse z transform and finding easily out solution data only for 0 t  .The mathmatical mode for algebraical superposition technic and the main points for executive calculation will be shown in next two sections. Calculation examles will be exhibited to prove algebraical superposition technic in another section .It should point out that known method for inverse z transform are direct division, partial-fraction expansion,contour integation [1] [2]. But those require the function formula to be rational function. The algebraical superposition technic in the paper is suit for both rational functions and irrational functions.

II. PRINCIPLE MODEL
We now enter into a principle to perform inverse z transform by algebraical superposition technic.Setting up process as follows: For a causal signal to find out time domain sequences based on inverse z transform [3] is Where 0 n  is the symbol of causal signal, ( ) 0 xn  for 0 n  . The above integral path c is a closed circle with radius r on the convergence region of z transform. The region of convergence for transform in complex z-plane is located at the region where a rr  , a r is convergence radius. Otherwise, the z transforms formula correspponding to equation ( 1 ) is in the following To get the integral calculation for(1), ‚complex functions integral‛ [4] is used. For the integral calculation, the integral path c can be choosed to consist of 2k small arc elements and corresponding equally divided points are 0  (5) Another equivalent formula to(4)is 21 1 0.5 0.5 1 0 Practical calculation show: the two sequence values respectively due to (4)and (6)is much closed. For example the real parts of two sets of data have agreement in 12 digits.

III. EXECUTIVE MAIN POINTS
(1)The first step to find out the time domain sequence is to determine the region of convergence(ROC)of z thransform in the complex z-plane. The ROC has often characters as follows: (a) The ROC is often a cirque form, where its center is the coordinate origin point. (d) If () Xzgiven is over presented materials, finding out the polar points by observing and testing, in the ROC any polar points should not be appeared.
(2)It is necessary to calculate and analyse () Ikvariation with k by (4)

IV. APPLICATION EXAMPLES
Two examples will be exhibited. First one is to know its time domain solution, so it is easy using the time domain solution to validate the algorithm in the paper. Second one exhibits that the algorithm in the paper is also to fit for the transform of an irrational function in z region.
So the theoretical value of example1 is from(9). It is first needed to determine b that is due to minimum radius of convergence region,