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Abstract and Applied Analysis

VolumeÂ 2012Â (2012), Article IDÂ 821949, 9 pages

http://dx.doi.org/10.1155/2012/821949

## Dirichlet Characters, Gauss Sums, and Inverse Z Transform

^{1}Department of Mathematical Sciences, Xi'an Jiaotong University, Xi'an, Shaanxi, China^{2}Department of Mathematics, Northwest University, Xi'an, Shaanxi, China

Received 26 December 2011; Accepted 9 January 2012

Academic Editor: Karl JoachimÂ Wirths

Copyright Â© 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.

#### Abstract

A generalized MĂ¶bius 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.

#### 1. Introduction

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 [1] at the beginning of the last century. Similar algorithms were studied by Wintner [2] and Sadasiv [3] for the calculation of the Fourier coefficients of even periodic functions. This method was extended in [4] to calculate the Fourier coefficients of both the even and odd components of a periodic function. The Bruns approach was incorporated in [5] resulting in a more computationally balanced algorithm. In [6, 7], Knockaert presented the theory of the generalized MĂ¶bius transform and gave a general formulation.

In [8], Schiff et al. applied Wintner's algorithm for the computation of the inverse Z-transform of an infinite causal sequence. Hsu et al. [9] applied two special MĂ¶bius 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 MĂ¶bius 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 [10] play an important role in the paper.

#### 2. The Algorithm

According to the MĂ¶bius inversion formula for finite series [4], if is a positive integer and , are two number-theoretic functions, then where denotes the integer part of real number and is the MĂ¶bius function: Knockaert [6] extended the MĂ¶bius 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 Eulerâ€™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 [10].

Let be a Dirichlet character modulo . We have In this way, we have defined a generalized MĂ¶bius 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 MĂ¶bius transform. In fact,(a)let in Lemma 2.2; we have
which is Theoremâ€‰â€‰3 in [4] and Lemmaâ€‰â€‰1 in [8];(b)let and in Lemma 2.2, where is a positive integer; we have
which is Caseâ€‰â€‰1 of Lemmaâ€‰â€‰1 in [9];(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 Caseâ€‰â€‰2 of Lemmaâ€‰â€‰1 in [9];(d)let in Lemma 2.2; we have
which is transform pair I of Theoremâ€‰â€‰4 in [7];(e)let or , and in Lemma 2.2, where is an odd prime, , and is the Legendre's symbol defined as follows:
From [10], we know that admits a primitive root and . We have
which is transform pair II of Theoremâ€‰â€‰4 in [7].

From these facts, we claim that Lemma 2.2 is actually an important extension on the MĂ¶bius 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
*

*Proof. *On , let us write .

Define

Note that for a positive integer
we have
Let ; then
Note that ,â€‰â€‰so if and only if ; therefore,
By Lemma 2.2, we have
This completes the proof of Theorem 2.3.

*Remarks 2. *Let in Theorem 2.3; we have
which is the theorem in [8].

Let in Theorem 2.3 or and in Theorem 2.3; we easily have

Let and in Theorem 2.3; we have

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 Lemmaâ€‰â€‰3 of [8].

Taking as in Theorem 2.3, we maintain the following theorem.

Theorem 3.2. *The truncation error satisfies
**
where is the Lipschitz constant.*

*Proof. *Note that we have

Moreover, by the analyticity of . By Theorem 2.3 and Lemma 3.1, we have

This completes the proof of Theorem 3.2.

#### 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.

#### 5. Conclusion

A general algorithm offers a general way to compute the inverse transform. It is based on generalized MĂ¶bius 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.

#### Acknowledgments

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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