Generalized Orthogonal Discrete W Transform and Its Fast Algorithm

Based on the generalized discrete Fourier transform, the generalized orthogonal discrete W transform and its fast algorithm are proposed and derived in this paper. The orthogonal discrete W transform proposed by Zhongde Wang has only four types. However, the generalized orthogonal discrete W transform proposed by us has inﬁnite types and subsumes a family of symmetric transforms. The generalized orthogonal discrete W transform is a real-valued orthogonal transform, and the real-valued orthogonal transform of a real sequence has the advantages of simple operation and facilitated transmission and storage. The generalized orthogonal discrete W transforms provide more basis functions with new frequencies and phases and hence lead to more powerful analysis and processing tools for communication, signal processing, and numerical computing.


Introduction
e orthogonal transform is a good mathematical tool and has been used in many applications of digital signal processing, such as harmonic analysis, numerical computation, image processing, data compression, and information hiding [1][2][3][4][5][6]. erefore, the method of constructing orthogonal transforms has become an important research topic. e orthogonal transforms include many transforms such as discrete Fourier transform (DFT), discrete Hartley transform, discrete W transform, discrete cosine transform, discrete Walsh transform, and discrete Haar transform. e DFT is the most widely used and influential transform. Since it was proposed by Fourier in 1822, the Fourier transform has been the most basic analysis method in the analysis of continuous signal and system [7]. In order to adapt to the spectrum analysis by computer, DFT was proposed, which is the approximation of spectrum analysis for continuous time signals. Frequency domain analysis was often superior to time-domain analysis. However, it was impractical to use DFT in the spectrum analysis due to high computational complexity before the advent of the fast algorithm of DFT (FFT). In 1965, Cooley and Tukey [8] published a famous paper on the FFT, which reduced the computation time of DFT by several orders of magnitude. As a result, FFT technology has been widely used in various scientific fields. It broadly promoted the effective combination of engineering and computer technologies. Also, it realized the rapid and effective analysis and processing of engineering problems by enabling use of computers. Fourier transform has a wide range of applications in discrete dynamics, acoustics, optics, and other physics, as well as number theory, combinatorial mathematics, probability, statistics, signal processing, cryptography, and many other fields [9][10][11].
In order to extend the application scope of DFT, the kernel function of DFT should be generalized to construct as many basis functions with new frequencies and phases as possible.
e kernel function of the DFT is [exp(i(2π/N))] lx , where x = 0, 1, . . ., N − 1, l = 0, 1, . . ., N − 1. If ω N � 1 and ω n ≠ 1, where n < N (n is a positive integer), ω is called an Nth-order primitive root of unity. It is easy to verify that exp(i(2π/N)) is an Nth-order primitive root of unity in the field of complex numbers (N ≥ 2). In fact, the base exp(i(2π/N)) of the kernel function can be further extended to any Nth-order primitive root of unity in the field of complex numbers, and (l, x) can be extended to (l + α, x + β), where α and β can be any real numbers. In this paper, we prove that if ω N is an Nth-order primitive root of unity in the field of complex numbers (N ≥ 2), then the row vectors of the transform matrix associated with the kernel function ω (l+α)(x+β) N are orthogonal to each other, where l and x denote, respectively, the row and column indices of the transform matrix, while the α and β parameters can be any real numbers. e generalized discrete Fourier transform is constructed using the normalized kernel function However, since the DFT is a complex-valued transform, a real sequence becomes a complex sequence after DFT. Complex sequences are not as easy to transmit and store as real sequence. erefore, a real-valued transform was studied to replace the DFT. e discrete Hartley transform (DHT) advanced by Bracewell [12] is a real-valued orthogonal transform. When the DHT is applied to real sequences, it avoids complex operations and speeds up calculations [13]. A transformed real sequence is still a real sequence, which is easy to transmit and store. e DHT and the DFT are related by a simple conversion relationship; that is, the DHT kernel function cas((2π/N)lx) � cos ((2π/N)lx) + sin((2π/N)lx), where x, l � 0, 1, . . ., N − 1 is the sum of the real and imaginary parts of the DFT kernel function exp(i(2π/N)lx) � cos((2π/N)lx) + i sin((2π/N) lx), where i is the imaginary unit, x, l � 0, 1, . . ., N − 1, and cas(x) denotes cos(x) + sin(x) for a real number x. erefore, the DHT can be used widely instead of DFT [14]. Wang and Hunt [15] developed the discrete W transform (DWT), which is a generalized DHT. ere are four types of orthogonal discrete W transform (ODWT), whose kernel function is still the sum of real and imaginary parts of Fourier transform kernel function. DWT and DHT have attracted many scholars' attention due to their advantage compared to DFT in processing real number sequences [16].
DWT and DHT are both discrete Hartley-type transforms. Besides DFT, other complex-valued orthogonal transforms have been recently gaining attention. For each of these complex-valued orthogonal transforms, a real-valued orthogonal transform can be obtained by adding the real and imaginary parts of the kernel function of the complex-valued orthogonal transform together to form the kernel function of the real-valued orthogonal transform. Many of these constructed real-valued orthogonal transforms were found to resemble the DHT and hence are collectively called the discrete Hartley-type transforms [17][18][19][20]. ese include the Zhang-Hartley transform and the discrete W transform (DWT). Such transforms have been applied in spectral analysis, data compression, convolution, data security, and so on. ese transforms have been widely used in communication and signal processing [21][22][23][24][25][26][27][28][29][30]. Numerous approaches on fast algorithms for discrete Hartley-type transforms have been reported [31][32][33][34].
us, for the generalized discrete Fourier transform, its corresponding discrete Hartley-type transform should be considered. Accordingly, the generalized ODWT (GODWT) is proposed in this paper. We show that ODWT is a special case of GODWT. Unlike the ODWT that has only four types, the proposed GODWT has infinite types, subsuming a family of symmetric transforms.
GODWT can replace generalized DFT in wide applications by inheriting the advantages of DHT and DWT. e GODWT is a Hartley-type transform, and it has its fast algorithm. e transformed signal of the real sequence is still of real values, which makes the transmission and storage convenient. Compared with DHT and DWT, GODWT provides a large number of basis functions with new frequency, phase, and a large number of new transformations. It indicates that more transmission methods of information are available when adopting GODWT. In the digital hiding technology, the password can be embedded into transform coefficients.
e GODWT is applied to digital hiding technology in the transform domain. e transform space is expanded from ODWT space to GODWT space and the secret key space increases significantly so that the decoding gets more difficult.
GODWT not only provides integer multiples of frequency and phase but also provides various fractional multiples of frequency and phase. e signals in the objective world are various, and it is likely to be a linear combination of several fractional frequency and phase basis functions. If only integer frequency and phase basis function are used for orthogonal decomposition, the number of basis functions will be increased. e proposed fractional multiple frequency and phase basis functions provide a basis for simplifying some problems. e choice of the basis function is actually a subtle matter. For example, if the function y � x 2 is expanded by the Maclaurin series in a zero-centered finite interval, only one term arises in the expansion. However, there are an infinite number of terms if the same function is expanded by a trigonometric series. From this example, we can see the importance of the basis function choice for function approximation, spectral analysis, data compression, etc. A problem can be well solved only if it uses basis functions whose frequencies and phases are appropriate for or match the problem. Based on the generalized DFT and GODWT, we have at our disposal a large number of new basis functions with rich frequency and phase information.
In this paper, we use the primitive roots of unity to construct the generalized DFT. Based on the generalized DFT, we also propose and prove the GODWT. us, ODWT is expanded from the original four types to infinite types. Moreover, we propose the fast algorithms for computing the generalized DFT and the GODWT. At the end of the paper, the fast algorithm of GODWT and the application of new frequency and phase basis function in communication are illustrated and given as an example. In addition, the key space of GODWT used in digital hiding technology is analyzed. In a word, our generalization provides better mathematical tools for analysis and processing in engineering fields such as communication.

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Discrete Dynamics in Nature and Society

Construction of the Generalized DFT Using Primitive Roots of Unity
We show in three stages the construction of the generalized discrete Fourier transform using primitive roots of unity in the complex number field: (a) If q is a positive integer, q and N are relatively prime; that is, (q, N) = 1, and we can know that exp(i(2π/N)q) is the N th -order primitive root of unity in the complex number field (N ≥ 2). In fact, when n = 1, 2, . . ., N − 1, n is not divisible by N, that is, N ∤ n, and (q, N) = 1. en, the product of q and n is not divisible by N, that is, N ∤ qn: when n �N, Hence, exp[i(2π/N)q] is the N th -order primitive root of unity in the complex number field.
also an N th -order primitive root of unity in the complex number field. (b) If ω N = r · exp(iθ) is an N th -order primitive root of unity, then where k is any integer: When k = 0, ω N � 1, and ω N becomes a first-order primitive root of unity. is root has no practical significance and will not be considered. When k is a positive integer, we get ω N � exp (i(k2π/N)), where k and N are relatively prime. Otherwise, if k and N have a common divisor s, s ≠ 1, then k � sk 1 , N � sN 1 , and N 1 < N; then erefore, ω N is not an N th -order primitive root of unity. is contradicts the assumptions. When k is a negative integer, we similarly get ω N � exp(i(k2π/N)), where − k and N are relatively prime. From (a) and (b), we realize that an N th -order primitive root of unity ω N in the complex number field can only be exp(i(k2π/N)), where k is a positive or negative integer and |k| and N are relatively prime (N ≥ 2). (c) If τ is a real number and the conjugate of ω τ N is denoted by ω τ N , then Now, we prove the orthogonality of the rows of the transform matrix that is associated with the kernel function ω , where l and x denote the row and column indices of the transform matrix, respectively. e parameters α and β can be any real numbers; ω N is N th -order primitive root of unity. For any two row vectors ω Discrete Dynamics in Nature and Society us, the row vectors of the transform matrix associated with the kernel function ω (l+α)(x+β) N are orthogonal to each other. When l 1 = l 2 , e modulus of each row vector is is a unitary matrix, where l, x = 0, 1, . . ., N − 1, and each of the parameters α and β can be any real number. e kernel function (1 lies at the l-th row and the x-th column of the unitary matrix. When erefore, only the kernel function e transform in (11) is called a generalized discrete Fourier transform, and its inverse transform is

Construction of the GODWT
e DWT with two parameters, α and β, is defined as follows: where }. e orthogonal discrete W transform has only the four types. We denote the four transforms associated with these parameter value pairs by DWT-I, DWT-II, DWT-III, and DWT-IV, respectively, and we call them ODWT. e parameter q can be added to the expression in (13) to obtain At this point, many meaningful orthogonal transforms can be constructed. Two lemmas are proved as follows.

and both h and m are integers, then
From (16) and (17), we get N− 1 x�0 exp(i(2π/N)pmx) � 0. And, is group is called a residual-class additive group and is denoted by Z N .
Next, we derive below the conditions for constructing a real-valued orthogonal system by adding the real and imaginary parts of a complex-valued orthogonal system together.
Let the complex-valued function set on Z N , that is, be a normalized orthogonal function system, where R k (x) and N k (x) are the real and imaginary parts of Φ k (x), respectively. en, the condition for the real-valued function set, to be a normalized orthogonal function system is where Proof. Φ k (x) is a normalized orthogonal system. Hence, for any k 1 , which can be rewritten as Hence, we find that From (25) and (26), we can get From (27), we find that, for any k 1 , k 2 ∈ {0, 1, . . ., N − }, the condition N− 1 x�0 R k 1 (x)N k 2 (x) � 0 is the necessary and sufficient condition for and this sum is used as the kernel function of a new real matrix. According to Lemma 2, we know that the necessary and sufficient conditions for the orthonormality of the N row vectors of the real matrix can be formulated as Let Using the product-to-sum formula (i.e., the Prosthaphaeresis formula), from (30), we can get If q � q 1 q 2 , both q 1 and q 2 are positive integers, ∵(q, N) � 1, ∵(q 2 , N) � 1; let α � (g/2q 1 ) and β � (h/2q 2 ), where g and h can be any integers; by substitution in (31), we get ∴ q 1 (l 1 + l 2 + 2α) � q 1 (l 1 + l 2 + 2(g/2q 1 )) � q 1 (l 1 + l 2 ) + g is an integer and q 1 (l 1 − l 2 ) is also an integer.
e real-valued orthogonal matrix is obtained with the kernel functions in (34), where l and x are the row and column indices of the matrix. e transform is GODWT, and its inverse transform is If s 1 is the factor of 2 q 1 and s 2 is the factor of 2 q 2 , we get 2q 1 � t 1 s 1 and 2q 2 � t 2 s 2 . So, parameter α of (28) and (29) becomes α � (g/2q 1 ) � (g/t 1 s 1 ). Similarly, parameter β of (28) and (29) becomes β � (h/2q 2 ) � (h/t 2 s 2 ). en, we set g � t 1 g 1 and h � t 2 h 2 , where g 1 and h 2 can be any integers. Finally, we obtain α � (g 1 /s 1 ) and β � (h 2 /s 2 ). us, the N row vectors of the real matrix with the kernel functions (1/ �� N √ )cas[(2π/N)q(l + (g 1 /s 1 ))(x + (h 2 /s 2 ))] form a normalized orthogonal basis. If you change g 1 and h 2 into g and h, or any other two letters, the new letters express the same settings as long as they are any integers.
For the GODWT, when q � 1, it reduces to the ODWT, and when q ≠ 1, a new family of orthogonal transforms arises.
is family greatly expands the ODWT and also subsumes a family of symmetric transforms.
If q contains a factor s 2 , we can construct a transform matrix such that α � (f/2s) and β � (f/2s), where f can be any integer.

e Fast Algorithm for the Generalized Discrete Fourier
Transform. Let f(x) be a sequence, where x � 0, 1, . . . , N − 1, and let ω N be an N th -order primitive root of unity in the complex number field. en, the transform of f(x) by the kernel function From (37) )ω lx N . is is adopted for the generalized fast Fourier transform (FFT) [35].
(2) e result from the first step is multiplied by ω (l+α)β N . So, when N is a composite number, there is a fast algorithm for computing the transform of f(x) with the kernel function If ω N is an N th -order primitive root of unity in the complex field, then ω − 1 N is also an N th -order primitive root of unity in the complex field. Similarly, when N is a composite number, a fast algorithm exists for computing the inverse transform is a real-valued column vector, then the actions of the matrices on this vector are as follows: , we can obtain fast algorithms for computing the GODWT and its inverse transform. However, the advantages of real-valued operations can be brought into play only by developing direct fast algorithms of real-valued orthogonal transforms on real sequences. is is one of the problems in the anticipated future work on the GODWT.
By using the sparse matrix, it is easy to draw a signal flow diagram that a column vector is transformed by the matrix [Φ]. e FFT flow diagram is depicted in Figure 1.
Another method of computing 7 l�0 [ [ F(l)ω (1/6)l ]ω lx is as follows: )l ]ω lx , so, in order to calculate the FFT of F(l)ω (1/6)l with kernel function ω lx , we can firstly calculate the FFT of F(l)ω (1/6)l with kernel function ω lx and then take the conjugate. e product of sparse matrix and signal flow graph of Φ can be directly used for FFT with kernel function ω lx .

Examples of the GODWT Application
(1) In communication field, the communication between different users using different frequency bands is frequency division multiplexing, and the communication between different users relying on different address codes is code division multiplexing. e code division multiplexing and frequency division multiplexing can be combined by using GODWT. We still take the primitive unit root ω � exp[i(2π × 9/8)], and three kernel functions of GODWT, that is, ( ; the equation is written in matrix forms as [a(0), a(1), . . . , a (7) Let the carrier of code division multiplexing be Z 10 , Z 11 , . . . , Z 17 : a(0)Z 10 + a(1)Z 11 + · · · + a(7)Z 17 .
is realizes the combination of code-division multiplexing and time-division multiplexing. It shows that the proposed GODWT provides more means for the transmission in communication, compared with DHT and ODWT.
When N is a composite number, based on the fast algorithms of the generalized DFT and its inverse transform, we get the fast algorithm of the GODWT and its inverse transform.
GODWT can replace the generalized DFT in wide applications. e transformed signal of the real sequence is still of real values, which makes the transmission and storage convenient. Also, GODWT provides a large number of basis functions with a new frequency, phase, and a large number of new transforms and subsumes a family of symmetric transforms; their forward and reverse transformations can be implemented by the same computer programs or hardware.
It can be seen that GODWT can provide more transmission means for communication, better security for digital hiding technology, and many new methods for data Discrete Dynamics in Nature and Society compression. Hence, it can be used as a more powerful analysis and processing tool for communication, signal processing, and numerical computing.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.