Dirichlet Characters, Gauss Sums, and Inverse Z Transform

and Applied Analysis 3 The function χ is called a Dirichlet character modulo q. The principal character χ1 is that which has the properties χ1 n { 1 if ( n, q ) 1, 0 if ( n, q ) > 1. 2.6 If q ≥ 1, the Euler’s totient φ q is defined to be the number of positive integers not exceeding q that are relatively prime to q. There are φ q distinct Dirichlet characters modulo q, each of which is completely multiplicative and periodic with period q. That is, we have χ mn χ m χ n ∀m,n, 2.7 χ ( n q ) χ n ∀n. 2.8 Conversely, if χ is completely multiplicative and periodic with period q, and if χ n 0 if n, q > 1, then χ is one of the Dirichlet characters modulo q. Let f n be an arithmetical function. Series of the form ∑∞ n 1 f n /n s are called Dirichlet series with coefficients f n . If f n χ n , then the series are called Dirichlet Lfunctions. For any Dirichlet character χ mod q, the sum G ( n, χ ) q ∑ m 1 χ m e2πimn/q 2.9 is called the Gauss sums associated with χ. If χ χ1, then the Gauss sums reduce to Ramanujan’s sum G ( n, χ1 ) q ∑ m 1 m,q 1 e2πimn/q cq n . 2.10 See 10 . Let χ be a Dirichlet character modulo q. We have ∑ k|m χ k μ ( m k ) χ ( m k ) χ m ∑ k|m μ ( m k ) δ1m. 2.11 In this way, we have defined a generalized Möbius transform pair. Lemma 2.2. Let χ be a Dirichlet character modulo q; then transform pair sn ∞ ∑ k 1 χ k fkn, fn ∞ ∑ k 1 μ k χ k skn 2.12


Introduction
We consider a causal, linear, time-invariant system with an infinite impulse response {c j } ∞ j 1 .The system is assumed to be stable and the Z transform X z ∞ j 1 c j z −j is convergent for |z| > r, where r < 1.The frequency response of the system is obtained by evaluating the Z 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 M öbius transform and gave a general formulation.
In 8 , Schiff et al. applied Wintner's algorithm for the computation of the inverse Ztransform of an infinite causal sequence.Hsu et al. 9 applied two special M öbius inversion formulae to the inverse Z-transform.
The transform pairs play a central part in the arithmetic Fourier transform and inverse Z-transform.In this paper, based on Dirichlet characters, we presented a generalized M öbius 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 Z transform on the unit circle.The algorithm computes each term c j of the infinite impulse response from sampled values of the Z 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.

The Algorithm
According to the M öbius inversion formula for finite series 4 , if n is a positive integer and f n , g n are two number-theoretic functions, then where y denotes the integer part of real number y and μ n is the M öbius function: otherwise.

2.2
Knockaert 6 extended the M öbius inversion formula and proved the following proposition.
Proposition 2.1.Let f 1 , f 2 , . . .be a sequence of real numbers and α n , β n two arithmetical functions.For the transform pair to be valid for all sequences f n , it is necessary and sufficient that Let G be the group of reduced residue classes modulo q.Corresponding to each character f of G, we define an arithmetical function χ χ f as follows: where n {x : x ≡ n mod q } and a, b denotes the greatest common divisor of a and b.

Abstract and Applied Analysis 3
The function χ is called a Dirichlet character modulo q.The principal character χ 1 is that which has the properties If q ≥ 1, the Euler's totient φ q is defined to be the number of positive integers not exceeding q that are relatively prime to q.There are φ q distinct Dirichlet characters modulo q, each of which is completely multiplicative and periodic with period q.That is, we have χ n q χ n ∀n.

2.8
Conversely, if χ is completely multiplicative and periodic with period q, and if χ n 0 if n, q > 1, then χ is one of the Dirichlet characters modulo q.
Let f n be an arithmetical function.Series of the form ∞ n 1 f n /n s are called Dirichlet series with coefficients f n .If f n χ n , then the series are called Dirichlet Lfunctions.For any Dirichlet character χ mod q, the sum 2.9 is called the Gauss sums associated with χ.If χ χ 1 , then the Gauss sums reduce to Ramanujan's sum G n, χ 1 q m 1 m,q 1 e 2πimn/q c q n .2.10 See 10 .
Let χ be a Dirichlet character modulo q.We have

2.11
In this way, we have defined a generalized M öbius transform pair.
Lemma 2.2.Let χ be a Dirichlet character modulo q; then transform pair is valid for all q.
Remarks 1.The transform pairs play a central part in the arithmetic Fourier transform and inverse Z-transform.It is not hard to show that all the transform pairs used in the mentioned papers are the special cases of our generalized M öbius transform.In fact, a let q 1 in Lemma 2.2; we have c let q 2 α , α ≥ 2, and in Lemma 2.2, then χ 2 is one of the Dirichlet characters modulo q since χ 2 k is completely multiplicative and periodic with period q.We have which is Case 2 of Lemma 1 in 9 ; d let χ χ 1 in Lemma 2.2; we have which is transform pair I of Theorem 4 in 7 ; e let q 4, p α or 2p α , and χ 3 k k/q in Lemma 2.2, where p is an odd prime, α ≥ 1, and k/q is the Legendre's symbol defined as follows: 1 and n is a quadratic residue mod q, −1 if k, q 1 and n is not a quadratic residue mod q, 0 if k, q > 1.

2.18
Abstract and Applied Analysis 5 From 10 , we know that q admits a primitive root and k/q −1 ind k .We have which is transform pair II of Theorem 4 in 7 .
From these facts, we claim that Lemma 2.2 is actually an important extension on the M öbius 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 X z ∞ j 1 c j z −j be convergent for |z| > r, where r < 1.For any fixed q ≥ 1 and Dirichlet character χ modulo q, the coefficients are given by X e 2πi/kn l r/q .2.20 Proof.On |z| 1, let us write X θ X e iθ ∞ j 1 c j e −ijθ .Define X e 2πi/n l r/q .

2.21
Note that for a positive integer k we have c ln e −2πilr/q .

Abstract and Applied Analysis
Note that 1 ≤ m, s ≤ q, so q | m − s if and only if m s; therefore,

2.25
By Lemma 2.2, we have X e 2πi/kn l r/q .

2.26
This completes the proof of Theorem 2.3.
Remarks 2. Let q 1 in Theorem 2.3; we have X e 2πi/kn l 1 , 2.27 which is the theorem in 8 .Let q 2 in Theorem 2.3 or q 4 and χ χ 1 in Theorem 2.3; we easily have X e 2πi/kn l 1/2 .

2.29
In practice, a large number of coefficients c n may be calculated.We suppose that a truncation is employed.Next we estimate the error due to the truncation of the series.

Error Estimate
In order to estimate the error due to truncation of the series representation of the coefficients c n , we require the following lemma.
uniformly in θ, where C is the Lipschitz constant.

Abstract and Applied Analysis 7
Proof.This is Lemma 3 of 8 .
Taking X z as in Theorem 2.3, we maintain the following theorem.
Theorem 3.2.The truncation error satisfies where C is the Lipschitz constant.
Proof.Note that we have X e iϕ dϕ.

3.3
Moreover, X ∈ Lip 1 0, 2π by the analyticity of X.By Theorem 2.3 and Lemma 3.1, we have Cφ q n 2 N .

3.4
This completes the proof of Theorem 3.2.

An Example
Consider the function The few first coefficients are c 1 2, c 2 1, and c 3 5/12.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 q and the Dirichlet character, we will greatly improve the algorithm.

Conclusion
A general algorithm offers a general way to compute the inverse Z transform.It is based on generalized M öbius transform, Dirichlet characters, and Gauss sums.The algorithm computes each term c j of the infinite impulse response from sampled values of the Z 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 q 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 Z transform.