Chirp Signal Transform and Its Properties

The chirp signal exp(iπ(x−y)2) 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 N 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 N is not prime and N = ML, we define a transform skipped L and develop the theory for it.


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][4][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 exp(2),  ∈ R; on the other hand, our chirp signal transform is founded on the orthogonality of exp(( − ) 2 ),  ∈ R.These three kinds of transforms are changed to each other by the unitary transform exp( 2 ) and exp( 2 ); 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 exp(( − V) 2 ) 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  2 -function is represented as a sum of such chirp signals with different central positions.
Furthermore, let  and  be (/) + 2 and (/) − 2.Since [, ] = 4, the set {, } generates the Lie algebra (2, C).Additionally,  = 2 and  = −2.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.

Definitions, Notations, and Examples
In this section, we define chirp signal transform for discrete, continuous, and skipped versions.
(i) For  < , let  [,] be the step function Then, the chirp signal transform of  [,] is the following: (ii) We also calculate the chirp signal transform for the Gaussian distribution exp(− 2 ).Consider Since exp(−(1 The above is equal to exp(−( When  is odd and  = , we calculate the skipped chirp transform of () = exp(−2(/)) at  = 0 with respect to exp((( + )/)),  = , 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 1/ is denoted by .Then, we define   for  in  2 (R), Now, If the final lim  → ∞ lim  → ∞ coincides to lim  → ∞ 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][8][9].
Similarly, when  is odd, () = exp((( + )/)) satisfies the same property; in fact, Hence, We remark that the suffix of  is changed from positive to negative as  to  −1 .
Next, we prove that  =  for the continuous version.We try to prove it directly.
2 )  () ) ; (16) 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.
We assume that  is odd.We obtain the following.The above theorem leads to the same property as the even case.Hence, for both cases, we obtain , ( − )  () . (20)

Properties of the Chirp Signal
In this section, we show some properties of the chirp signal exp(( − ) 2 ) which is fundamental for our chirp signal transform. Since Now, we write the differential operators (/) + 2, (/)−2 as , .The differential operators  and  satisfy the following properties: Let  be an odd integer, and let ,  be integers.Then, we define  ,, () as exp(−(( + )/)).We have the following.
iff  +  is even.Since  is odd and both  and  are odd or even,  +  is always even.