Abstract
The chirp signal is a typical example of CAZAC (constant amplitude zero autocorrelation) sequence. Using the chirp signals, the chirp z transform and the chirp-Fourier transform were defined in order to calculate the discrete Fourier transform. We define a transform directly from the chirp signals for an even or odd number and the continuous version. We study the fundamental properties of the transform and how it can be applied to recursion problems and differential equations. Furthermore, when is not prime and , we define a transform skipped and develop the theory for it.
1. Introduction
Chirp signal has been widely used in technology, for example, the radar system [1], as the linear FM pulse signal, the spectrum analyser as the sweep signal, the communications as the chirp modulation signal [2], and CAZAC sequence. Since the CAZAC sequences have constant amplitude and zero autocorrelation properties, they are now widely used in the fields of channel estimation and time and frequency synchronizations for OFDM (orthogonal frequency division multiplexing) and CDMA (code division multiple access) techniques which are employed as the standard transmission techniques in the wireless communications systems [3–5]. For the discrete Fourier transform theory, attached the chirp signal and Fourier transform, two kinds of transforms were already defined, namely, chirp z transform and chirp-Fourier transform. The first transform is (i) Fourier transform, (ii) product with the chirp signal, the second one is (i) product with the chirp signal (ii) Fourier transform [6]. These transforms were investigated mainly for the calculation of discrete Fourier transform (DFT).
In this paper, we define another transform directly treating the chirp signal, which is equal to (i) the product of the chirp signal, (ii) Fourier transform, and (iii) the product of the chirp signal, and which has a new meaning. Our motivation is totally different with the ones for the chirp z transform and the chirp-Fourier transform. We name it as chirp signal transform and consider the transform for an even or odd number , continuous case, and skipped version in the case of . We calculate firstly some examples for the transform of discrete, continuous, and skipped versions. Secondly, we show the inverse transform for the transform elementary, not using usual Fourier transform theory. Thirdly we study the properties of the original chirp signal, which is fundamental for our theory. Finally, we apply the transform to the theory of recursion problems and ordinary differential and partial differential equations.
Chirp z transform and chirp-Fourier transform are based on the “orthogonality” of , ; on the other hand, our chirp signal transform is founded on the orthogonality of , . These three kinds of transforms are changed to each other by the unitary transform and ; however, the represented meanings are completely different. The two transforms’ is the meaning of the frequency and the remaining one’s is the meaning of the position. The chirp signal is useful for the radar, because it can be generated by simple liner FM pulse which can increase the frequency bandwidth of pulse and accordingly improve the accuracy of range measurement. Moreover, the chirp signals (LFM or NLFM) have satisfied properties of ambiguity function for radar application. The chirp signal transform shows that any -function is represented as a sum of such chirp signals with different central positions.
Furthermore, let and be and . Since , the set generates the Lie algebra . Additionally, and . It seems that the set has a good property for representation theory. It is very similar to the construction for the solution of harmonic oscillator in quantum mechanics by and . We expect to develop the algebraic theory for using the relationship of the differential operators , and the chirp transforms , in order to solve some differential equations.
2. Definitions, Notations, and Examples
In this section, we define chirp signal transform for discrete, continuous, and skipped versions.
For an even (resp., odd) and a constant number with , we define chirp signal transform of the function over the set . Consider When is odd especially, can be also defined by .
On the other hand, we also define chirp signal transform for continuous version, for , Since , the left hand side is equal to We denote to be and the Fourier transform as ; then, is also written as . It is also the same for discrete cases. The transforms and are called as chirp z transform and chirp-Fourier transform [6].
Let be even and divided as . Then, we write We define chirp signal transform as .
If is odd, both and are even or odd with , and , then we define
In the following, we calculate some examples for the above chirp signal transform .
(i) For , let be the step function or 0 .
Then, the chirp signal transform of is the following:
(ii) We also calculate the chirp signal transform for the Gaussian distribution . Consider Since is holomorphic and is constant up to , we denote it as .
The above is equal to . Therefore,
When is odd and , we calculate the skipped chirp transform of at with respect to , , and . Now, is odd, and is also odd. Consider
Finally, we explain briefly that the discrete chirp signal transform becomes the continuous one, when increases to the infinity.
Let be an even positive integer, and is denoted by . Then, we define for in , Now, If the final coincides to under , the above for the continuous case is the limit of the for the discrete case. When is odd, we can do the similar calculation. Furthermore, this correspondence can be represented as an infinitesimal lattice Fourier transform in nonstandard analysis [7–9].
3. The Inverse Chirp Signal Transform
In this section, we show the inverse transform of the chirp signal transform defined in the first section. Let be an integer, and let be the function space from to the complex number field . For , the inner product is defined by If , are orthogonal to each other, then an arbitrary function is represented as a linear combination of . For example, in the case of even , satisfies such property. Now, Since is for , or for , the above is equal to . Hence, , and .
Similarly, when is odd, satisfies the same property; in fact, .
Hence, .
We remark that the suffix of is changed from positive to negative as to .
Next, we prove that for the continuous version. We try to prove it directly.
Let be for . Then, the Fourier transform of is calculated as follows: and we put as Now, and is equal to for and for . We denote to be ; then, is equal to .
Theorem 1. for .
Proof. Consider
by Fubini’s theorem,
Finally, we show the inverse transform of the chirp signal transform for skipped version. When is even, we obtain the following.
Theorem 2. Functions , are orthogonal to each other.
Proof. For , if ; then, by the definition. If , then is written as . Consider
Hence, if , then ; otherwise, .
We assume that is odd. We obtain the following.
Theorem 3. Functions , are orthogonal to each other.
Proof. We denote to be . For , if , then by the definition. If , then is written as . Consider
since .
Hence, if , then ; otherwise, .
The above theorem leads to the same property as the even case. Hence, for both cases, we obtain
4. Properties of the Chirp Signal
In this section, we show some properties of the chirp signal which is fundamental for our chirp signal transform.
Since , .
Now, we write the differential operators , as , . The differential operators and satisfy the following properties:
Then, the chirp signal is the eigen function of with the eigen value . Let satisfy the equation . Then, . Similarly, ; that is, is the eigen function for with eigen value .
Let be an odd integer, and let , be integers. Then, we define as . We have the following.
Theorem 4. The function is periodic iff both and are odd or even.
Proof. Consider
iff is even. Since is odd and both and are odd or even, is always even.
Theorem 5. , are orthogonal to each other, iff .
Proof. Consider
Secondly, we calculate the Fourier transform for the chirp signal. Consider
Since is periodic,
we denote by
Now, since Fourier transform preserves the norm, .
Hence, the Fourier transform for is equal to .
Now, we assume is even, divides , and . Consider
We calculate the usual Fourier transform of . Consider
Next, we calculate the Fourier transform of . Consider
Let . Then, and . Hence, . It is a chirp signal with central position , which is a wave of and .
5. Application for Recursion Formulas and Ordinary Differential Equations
In this section, we develop some applications of the chirp signal transform.
(i) For a function from to , we calculate the chirp signal transform of with even , . Consider We apply this formula to the following equation for a constant and a given function : We transform this equation by the chirp signal transform using formula (30), We assume is not equal to for any . For example, the absolute value of is not equal to . Consider hence, .
Similarly, we calculate the chirp signal transform of . Consider Let be defined by . Then, is equal to , that is, .
We consider the following equation: We put the chirp signal transform of it. Consider By equality (34), If , for , then
Hence,
(ii) We put the assumption that . Consider by the assumption, Hence, and We write the differential operator as ; then, . We consider the following differential equation, for coefficients and a given function :
We apply the chirp signal transform: If for arbitrary , then .
Therefore, , formally.
For example, we consider the ordinary differential equation where is the delta function. Since is equal to , the solution is just , that is, . The integral is , , , in case , , respectively. Hence, for , respectively.
6. Application for Partial Recursion Formulas and Differential Equations
In this section, we consider the application for partial recursion formulas and differential equations.
(i) Let be two variable function on , and let Then
(ii) Let be two variables function on . We write .
Then, it is already shown as
Let be complex coefficients. Then, we consider the following partial differential equation: We put the chirp transform. Consider We divide the equation by , We put , We write , Hence,
7. Conclusions
In this paper, we define the chirp signal transform using the chirp signal. Furthermore, we develop the theory of the chirp signal transform and apply it to the recursion problem, the ordinary differential equation, and the partial differential equation.
Conflict of Interests
The authors declare that they have no conflict of interests regarding the publication of this paper.
Acknowledgment
The authors would like to thank professor K. Hidano for his valuable suggestion and encouragement.