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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 821949, 9 pages
Dirichlet Characters, Gauss Sums, and Inverse Z Transform
1Department of Mathematical Sciences, Xi'an Jiaotong University, Xi'an, Shaanxi, China
2Department of Mathematics, Northwest University, Xi'an, Shaanxi, China
Received 26 December 2011; Accepted 9 January 2012
Academic Editor: Karl Joachim Wirths
Copyright © 2012 Jing Gao and Huaning Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
A generalized Möbius transform is presented. It is based on Dirichlet characters. A general algorithm is developed to compute the inverse transform on the unit circle, and an error estimate is given for the truncated series representation.
We consider a causal, linear, time-invariant system with an infinite impulse response . The system is assumed to be stable and the transform is convergent for , where . The frequency response of the system is obtained by evaluating the transform on the unit circle.
The arithmetic Fourier transform (AFT) offers a convenient method, based on the construction of weighted averages, to calculate the Fourier coefficients of a periodic function. It was discovered by Bruns  at the beginning of the last century. Similar algorithms were studied by Wintner  and Sadasiv  for the calculation of the Fourier coefficients of even periodic functions. This method was extended in  to calculate the Fourier coefficients of both the even and odd components of a periodic function. The Bruns approach was incorporated in  resulting in a more computationally balanced algorithm. In [6, 7], Knockaert presented the theory of the generalized Möbius transform and gave a general formulation.
In , Schiff et al. applied Wintner's algorithm for the computation of the inverse Z-transform of an infinite causal sequence. Hsu et al.  applied two special Möbius inversion formulae to the inverse -transform.
The transform pairs play a central part in the arithmetic Fourier transform and inverse -transform. In this paper, based on Dirichlet characters, we presented a generalized Möbius transform of which all the transform pairs used in the mentioned papers are the special cases. A general algorithm was developed in Section 2 to compute the inverse transform on the unit circle. The algorithm computes each term of the infinite impulse response from sampled values of the transform taken at a countable set of points on the unit circle. An error estimate is given in Section 3 for the truncated series representation. A numerical example is given in Section 4. Number theory and Dirichlet characters  play an important role in the paper.
2. The Algorithm
According to the Möbius inversion formula for finite series , if is a positive integer and , are two number-theoretic functions, then where denotes the integer part of real number and is the Möbius function: Knockaert  extended the Möbius inversion formula and proved the following proposition.
Proposition 2.1. Let be a sequence of real numbers and two arithmetical functions. For the transform pair to be valid for all sequences , it is necessary and sufficient that
Let be the group of reduced residue classes modulo . Corresponding to each character of , we define an arithmetical function as follows: where and denotes the greatest common divisor of and .
The function is called a Dirichlet character modulo . The principal character is that which has the properties
If , the Euler’s totient is defined to be the number of positive integers not exceeding that are relatively prime to . There are distinct Dirichlet characters modulo , each of which is completely multiplicative and periodic with period . That is, we have Conversely, if is completely multiplicative and periodic with period , and if if , then is one of the Dirichlet characters modulo .
Let be an arithmetical function. Series of the form are called Dirichlet series with coefficients . If , then the series are called Dirichlet L-functions. For any Dirichlet character , the sum is called the Gauss sums associated with . If , then the Gauss sums reduce to Ramanujan's sum See .
Let be a Dirichlet character modulo . We have In this way, we have defined a generalized Möbius transform pair.
Lemma 2.2. Let be a Dirichlet character modulo ; then transform pair is valid for all q.
Remarks 1. The transform pairs play a central part in the arithmetic Fourier transform and inverse -transform. It is not hard to show that all the transform pairs used in the mentioned papers are the special cases of our generalized Möbius transform. In fact,(a)let in Lemma 2.2; we have
which is Theorem 3 in  and Lemma 1 in ;(b)let and in Lemma 2.2, where is a positive integer; we have
which is Case 1 of Lemma 1 in ;(c)let , , and
in Lemma 2.2, then is one of the Dirichlet characters modulo since is completely multiplicative and periodic with period . We have
which is Case 2 of Lemma 1 in ;(d)let in Lemma 2.2; we have
which is transform pair I of Theorem 4 in ;(e)let or , and in Lemma 2.2, where is an odd prime, , and is the Legendre's symbol defined as follows:
From , we know that admits a primitive root and . We have
which is transform pair II of Theorem 4 in .
From these facts, we claim that Lemma 2.2 is actually an important extension on the Möbius inversion formula. In practice, we can choose the best possible transform pair.
We do not discuss the convergence of the transform pair since in practice it is used only on a truncated series. Next we establish our main theorem.
Theorem 2.3. Let be convergent for , where . For any fixed and Dirichlet character modulo , the coefficients are given by
In practice, a large number of coefficients may be calculated. We suppose that a truncation is employed. Next we estimate the error due to the truncation of the series.
3. Error Estimate
In order to estimate the error due to truncation of the series representation of the coefficients , we require the following lemma.
Lemma 3.1. If is a function of period , with , then uniformly in , where C is the Lipschitz constant.
Proof. This is Lemma 3 of .
Taking as in Theorem 2.3, we maintain the following theorem.
Theorem 3.2. The truncation error satisfies where is the Lipschitz constant.
4. An Example
Consider the function The few first coefficients are , , and . Employing formulae (2.27), (2.28), and (2.29), we obtain the results given in Tables 1, 2, and 3. The results show that formulae (2.28) and (2.29) is quite more accurate than formula (2.27). Choosing carefully the modulo and the Dirichlet character, we will greatly improve the algorithm.
A general algorithm offers a general way to compute the inverse transform. It is based on generalized Möbius transform, Dirichlet characters, and Gauss sums. The algorithm computes each term of the infinite impulse response from sampled values of the transform taken at a countable set of points on the unit circle. An error estimate and a numerical example are given for the truncated series representation. Choosing carefully the modulo and the Dirichlet character we will greatly improve the algorithm. But this is not exhaustive. Dirichlet characters and Gauss sums play an important role in number theory, and there are so many methods and results associated with them. Any development on the Dirichlet character and Gauss sums may be applied to the inverse transform.
This paper is supported by the National Natural Science Foundation of China under Grant no. 10901128, the Specialized Research Fund for the Doctoral Program of Higher Education of China under Grant no. 20090201120061, the Natural Science Foundation of the Education Department of Shaanxi Province of China under Grant no. 09JK762, and the Fundamental Research Funds for the Central University.
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