Generalized Orthogonal Discrete W Transform and Its Fast Algorithm

Sun, Jichao; Zhang, Zhengping

doi:https://doi.org/10.1155/2021/2529754

Discrete Dynamics in Nature and Society

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Discrete Dynamics of Complex Interactions between Natural and Artificial Systems

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Research Article | Open Access

Volume 2021 | Article ID 2529754 | https://doi.org/10.1155/2021/2529754

Generalized Orthogonal Discrete W Transform and Its Fast Algorithm

Jichao Sun^1,2and Zhengping Zhang¹

Academic Editor: Stefania Tomasiello

Received14 Jul 2020

Revised19 Nov 2020

Accepted26 Nov 2020

Published11 Jan 2021

Abstract

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.

1. Introduction

The 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 [1–6]. Therefore, the method of constructing orthogonal transforms has become an important research topic. The 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.

The 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 [9–11].

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.

The kernel function of the DFT is , where x = 0, 1, …, N − 1, = 0, 1, …, N − 1. If and , where (n is a positive integer), is called an Nth-order primitive root of unity. It is easy to verify that is an Nth-order primitive root of unity in the field of complex numbers (N ≥ 2). In fact, the base of the kernel function can be further extended to any Nth-order primitive root of unity in the field of complex numbers, and can be extended to , where and can be any real numbers. In this paper, we prove that if 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 are orthogonal to each other, where l and x denote, respectively, the row and column indices of the transform matrix, while the and parameters can be any real numbers. The 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. Therefore, a real-valued transform was studied to replace the DFT. The 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. The DHT and the DFT are related by a simple conversion relationship; that is, the DHT kernel function = + , where x, l = 0, 1, …, N − 1 is the sum of the real and imaginary parts of the DFT kernel function = + , where i is the imaginary unit, x, l = 0, 1, …, N − 1, and denotes for a real number x. Therefore, 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. There 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 [17–20]. These 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. These transforms have been widely used in communication and signal processing [21–30]. Numerous approaches on fast algorithms for discrete Hartley-type transforms have been reported [31–34].

Thus, 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. The GODWT is a Hartley-type transform, and it has its fast algorithm. The 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. The GODWT is applied to digital hiding technology in the transform domain. The 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. The 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. The proposed fractional multiple frequency and phase basis functions provide a basis for simplifying some problems. The choice of the basis function is actually a subtle matter. For example, if the function 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. Thus, 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.

2. 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 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. Then, the product of q and n is not divisible by N, that is, N ∤ qn: when Hence, is the N^th-order primitive root of unity in the complex number field. Obviously, = is also an N^th-order primitive root of unity in the complex number field.(b)If = is an N^th-order primitive root of unity, then where is any integer: When = 0, , and becomes a first-order primitive root of unity. This root has no practical significance and will not be considered. When is a positive integer, we get , where and are relatively prime. Otherwise, if and have a common divisor s, s ≠ 1, then , , and ; then Therefore, is not an N^th-order primitive root of unity. This contradicts the assumptions. When is a negative integer, we similarly get , where and N are relatively prime. From (a) and (b), we realize that an N^th-order primitive root of unity in the complex number field can only be , where is a positive or negative integer and and N are relatively prime (N ≥ 2).(c)If is a real number and the conjugate of is denoted by , then Now, we prove the orthogonality of the rows of the transform matrix that is associated with the kernel function , where l and x denote the row and column indices of the transform matrix, respectively. The parameters and can be any real numbers; is N^th-order primitive root of unity. For any two row vectors and , where , ∈ {}, x = 0, 1, …, N − 1, denote their dot product by : when ≠ , ≠ 1, and ; therefore, Thus, the row vectors of the transform matrix associated with the kernel function are orthogonal to each other. When = , The modulus of each row vector is . If is a complex matrix of order N and , is called unitary matrix, where is the transposed conjugate matrix of , and is a unit matrix. Therefore, the N × N matrix generated by the kernel function = is a unitary matrix, where l, x = 0, 1, …, N − 1, and each of the parameters and can be any real number. The kernel function lies at the l-th row and the x-th column of the unitary matrix. When Therefore, only the kernel function constructed by the primitive root of unity should be considered. The transform with the kernel function is expressed as The transform in (11) is called a generalized discrete Fourier transform, and its inverse transform is

3. Construction of the GODWT

The DWT with two parameters, α and β, is defined as follows:where x = 0, 1, …, N − 1, = 0, 1, …, N − 1, and DWT is orthogonal for four (α, β) values, that is, (α, β) = {(0, 0), (, 0), (0, ), (, )}. The 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.

The 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.

Lemma 1. If is positive integer , , and both h and m are integers, then

Proof. If N ∤ m, , N ∤ pm, ≠ 1,From (16) and (17), we get . And,If m is divisible by N, that is, N|m, m = Nd,Q.E.D.

Lemma 2. A real or complex sequence , where , can be regarded as a function defined on the set . A finite Abelian group can be formed through modulo-N addition on . This group is called a residual-class additive group and is denoted by .

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 , that is,be a normalized orthogonal function system, where and are the real and imaginary parts of , respectively. Then, the condition for the real-valued function set,to be a normalized orthogonal function system iswhere , ∈ {0, 1, …, N − 1}.

Proof. is a normalized orthogonal system. Hence, for any , ∈ {0, 1, …, N − 1},which can be rewritten asHence, we find thatFrom (25) and (26), we can getFrom (27), we find that, for any , ∈ {0, 1, …, N − }, the condition is the necessary and sufficient condition for to be a real normalized orthogonal function system.

3.1. The Generalized Orthogonal Discrete W Transform

The unitary matrix with the kernel function is constructed as discussed above. The N row vectors of an N × N unitary matrix form a normalized orthogonal system. The sum of the real and imaginary parts of the kernel function is given asand 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 = , both and are positive integers, (, N) = 1, (, N) = 1; let = and , where and h can be any integers; by substitution in (31), we get = = is an integer and is also an integer.

According to Lemma 1, we get

Then, when and , equation (29) holds, and the N functions,form a normalized orthogonal basis, where the independent variable x = 0, 1, …, N − 1.

The 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. The transformis GODWT, and its inverse transform is

If is the factor of 2 and is the factor of 2 , we get 2 = and 2 = . So, parameter of (28) and (29) becomes = = . Similarly, parameter of (28) and (29) becomes = = . Then, we set = and = , where and can be any integers. Finally, we obtain = and = . Thus, the N row vectors of the real matrix with the kernel functions form a normalized orthogonal basis. If you change and into and h, or any other two letters, the new letters express the same settings as long as they are any integers.

Let = and = , where and h can be any integers; if = = 1, the factors of 2 and of 2 can only be taken as 1 or 2. When (α, β) ∈ {(0, 0), (, 0), (0), ( )}, the four transforms are given by the kernel function , and it is obvious that they are DWT-I, DWT-II, DWT-III, and DWT-IV, respectively, that is, ODWT.

For the GODWT, when q = 1, it reduces to the ODWT, and when q ≠ 1, a new family of orthogonal transforms arises. This family greatly expands the ODWT and also subsumes a family of symmetric transforms.

If q contains a factor , we can construct a transform matrix such that and , where f can be any integer. This transform matrix includes two special cases, namely, (, ) and (, ). In these cases, the real-valued orthogonal matrix is a symmetric matrix, and the forward and inverse transforms have the same form.

4. Fast Algorithms for the Generalized Discrete Fourier and GODWT

4.1. The Fast Algorithm for the Generalized Discrete Fourier Transform

Let be a sequence, where , and let be an N^th-order primitive root of unity in the complex number field. Then, the transform of by the kernel function (the generalized DFT) is expressed as

From (37), we can identify two steps for the transform of with the kernel function :(1)When N is a composite number (nonprime, N ≥ 2), is transformed by the kernel function . This is adopted for the generalized fast Fourier transform (FFT) [35].(2)The result from the first step is multiplied by .

So, when N is a composite number, there is a fast algorithm for computing the transform of with the kernel function .

If is an N^th-order primitive root of unity in the complex field, then 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

4.2. The Fast Algorithm for the GODWT

Let be a unitary matrix: = + i, where and are real matrices and = + forms an orthogonal matrix. Then, the inverse transform matrix of is its conjugate transposed matrix = − i. The inverse transform matrix of is its transposed matrix = + .

If is a real-valued column vector, then the actions of the matrices on this vector are as follows:

From (39) and (40), we see that is the sum of the real and imaginary parts of . Moreover, from (41) and (42), we see that is the real part minus the imaginary part of .

For forward and inverse transforms of the GODWT, the transformed sequences are all real value sequences. Using the above relations of and as well as and , 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. This is one of the problems in the anticipated future work on the GODWT.

Example 1. Let , , , , , and . Therefore, = = , and = . The transform isThe fast algorithm of the transform in (43) is given as follows.
Due to the fact that the kernel function cas[] is the sum of the real and imaginary parts of the generalized DFT kernel function , where = , we firstly calculate the transform of with the kernel function :In order to get , first, we calculate , , where x = 0, 1, …, 7. Second, is transformed with the kernel function into ; then we calculate , , where l = 0, 1, …, 7. We get after the real and the imaginary parts of are added.
The matrix is formed with the kernel function , x = 0, 1, …, 7, l = 0, 1, …, 7. It can be decomposed into the product of a sparse matrix according to the method in [35], so it has a fast algorithm = ,where .By using the sparse matrix, it is easy to draw a signal flow diagram that a column vector is transformed by the matrix . The FFT flow diagram is depicted in Figure 1.
We calculate the inverse transform of . First, we calculate the inverse transform of the transform associated with the kernel as follows:where is the conjugate of .
In order to get , first, is computed as . Then, the FFT of is computed using a kernel function . The transform matrix formed by kernel function is the conjugate transpose matrix of . is the symmetric matrix. So . In the product of sparse matrix of , that is, ∗∗ and its corresponding signal flow diagram, is replaced with , the product of the sparse matrix decomposed by and its corresponding signal flow diagram can be obtained. The input and output of the signal flow diagram are and , respectively. Then, given = , is obtained using the real part minus the imaginary part of .
Another method of computing is as follows:
= , so, in order to calculate the FFT of with kernel function , we can firstly calculate the FFT of with kernel function and then take the conjugate. The product of sparse matrix and signal flow graph of can be directly used for FFT with kernel function .
In this example, the transform = is symmetric transform; thus, the inverse transform is the same as the positive transform. The inverse transform isTherefore, we can use the same fast algorithm as the positive transform.

4.3. 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. The code division multiplexing and frequency division multiplexing can be combined by using GODWT. We still take the primitive unit root = , and three kernel functions of GODWT, that is, cas[], cas [], and cas[] can be obtained. For the three transmitted code sequences, that is, , , and , x = 0, 1, …, 7, the transforms of them are calculated, respectively, to get , , and , where l = 0, 1, …, 7. ; the equation is written in matrix forms as Let the carrier of code division multiplexing be : The carriers , …, are periodic and repetitive. The amounts of information carried by each carrier wave over a period are constant, which are , , respectively. The matrix equation above can be written as In order to transmit = , l = 0, 1, …, 7, the angular frequency of carriers is from to . It is called code division multiplexing where we have transmitting , l = 0, 1, …, 7 instead of transmitting , x = 0, 1, …, 7. Both the input and output are eight real numbers. In contrast, if we use the DFT, the output is eight complex numbers. The transmission of eight complex numbers requires the transmission of sixteen real numbers. Accordingly, the transmission efficiency of GODWT is two times that of DFT. Transmitting and instead of transmitting and is also code division multiplexing, but they all have different carriers. = , l = 0, 1, …, 7; when transmitting , the angular frequency of carriers is from to . = , l = 0, 1, …, 7; when transmitting , the angular frequency of carriers is from to . When they are transmitted together, it is frequency division multiplexing. The receiver would receive their sum: These three components are separated with a filter, and then , were restored using the inverse GODWT. Time-division multiplexing (TDM) is also a method to transmit multiway signals. Time-division multiplexing is used primarily for digital signals but may be applied in analog multiplexing in which two or more signals or bit streams are transferred appearing simultaneously as subchannels in one communication channel but are physically taking turns on the channel. Each signal appears on the line only a fraction of time in an alternating pattern. The time domain is divided into several recurrent time slots of fixed length. Transmitting , , and with different time slots, they can be separated at the receiving end, and then , , and were restored using the inverse GODWT. This 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.(2)In the digital hiding technology, the password can be embedded into transform coefficients. The secret key space of ODWT is (N, , ), where (α, β) = {(0, 0), (, 0), (0, ), (, )}, and N is a positive integer. Actually, N cannot be big; if N is bigger, it is not easy to calculate. The secret key space of GODWT is (N, q, , ). It includes the secret key space of ODWT. Because the value of q is not limited in principle and the parameter value pairs (α, β) changed from 4 pairs of ODWT to infinity pairs of GODWT: (α, β) = where = , and h can be any integers, the secret key space of GODWT has been greatly generalized.

5. Conclusions

The kernel function of the generalized DFT is , where (, N) = 1, and parameters and can be any real numbers. When l = 0, 1, …, N-1, these N functions form a complex-valued normalized orthogonal system.

For the kernel function of the generalized DFT, the real and imaginary parts are added together leading to the functions:

cas [] = , (l = 0, 1, …, N-1). If = , both and are positive integers; let = , , where and h are any integers. These N functions form a real normalized orthogonal system, where the independent variable x = 0, 1, …, N-1. The transform with the kernel function cas [] is GODWT. For setting the values of and , if the factor of 2 replaces 2and the factor of 2 replaces 2, the conclusion is still valid.

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. The 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 compression. Hence, it can be used as a more powerful analysis and processing tool for communication, signal processing, and numerical computing.

Abbreviations

N, , q, , , , s, t:	Positive integers
d, f, , , h, , , m:	Integers
, , :	Real numbers
:	Complex numbers
, , , , , , :	The matrices
:	Real-valued column vector
:	Nth-order primitive root of unity in the complex number field
:	Conjugate of
:	Conjugate matrix of the matrix
:	Transposed matrix of the matrix
:	Conjugate transposed matrix of the matrix .

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was supported by the projects of the Chinese Natural Science Foundation (Grant no. 61865002), Anhui Province Higher Education Promotion Plan (Grant no. TSKJ2017B30), the Natural Science Fund of Anhui Province (Grant no. 1908085ME128), the Natural Science Foundation of Education Department of Anhui Province (Grant no. KJ2018A0123), and Shanghai Key Laboratory of Materials Laser Processing and Modification (Grant no. MLPM2017-3).

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Copyright © 2021 Jichao Sun and Zhengping Zhang. 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.

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