Digital audio signal watermarking using minimum‐energy scaling optimisation in the wavelet domain

IET Signal ProcessingVolume 14, Issue 10 p. 791-802 Research ArticleFree Access Digital audio signal watermarking using minimum-energy scaling optimisation in the wavelet domain Chih-Yu Hsu, Chih-Yu Hsu Fujian Provincial Key Laboratory of Big Data Mining and Applications, School of Computer Science and Mathematics, Fujian University of Technology, Fuzhou, People's Republic of ChinaSearch for more papers by this authorShu-Yi Tu, Shu-Yi Tu Department of Mathematics, University of Michigan, Flint, MI, 48502 USASearch for more papers by this authorChao-Tung Yang, Chao-Tung Yang Department of Computer Science, Tunghai University, Taichung, 40704 Taiwan Research Center for Smart Sustainable Circular Economy, Tunghai University, Taichung City, 407224 Taiwan Research Center for Nanotechnology, Tunghai University, Taichung City, 407224 TaiwanSearch for more papers by this authorChing-Lung Chang, Ching-Lung Chang Department of Computer Science and Information Engineering, National Yunlin University of Science and Technology, Yunlin, 64002 TaiwanSearch for more papers by this authorShuo-Tsung Chen, Corresponding Author Shuo-Tsung Chen shough34@yahoo.com.tw Department of Computer Science and Information Engineering, National Yunlin University of Science and Technology, Yunlin, 64002 TaiwanSearch for more papers by this author Chih-Yu Hsu, Chih-Yu Hsu Fujian Provincial Key Laboratory of Big Data Mining and Applications, School of Computer Science and Mathematics, Fujian University of Technology, Fuzhou, People's Republic of ChinaSearch for more papers by this authorShu-Yi Tu, Shu-Yi Tu Department of Mathematics, University of Michigan, Flint, MI, 48502 USASearch for more papers by this authorChao-Tung Yang, Chao-Tung Yang Department of Computer Science, Tunghai University, Taichung, 40704 Taiwan Research Center for Smart Sustainable Circular Economy, Tunghai University, Taichung City, 407224 Taiwan Research Center for Nanotechnology, Tunghai University, Taichung City, 407224 TaiwanSearch for more papers by this authorChing-Lung Chang, Ching-Lung Chang Department of Computer Science and Information Engineering, National Yunlin University of Science and Technology, Yunlin, 64002 TaiwanSearch for more papers by this authorShuo-Tsung Chen, Corresponding Author Shuo-Tsung Chen shough34@yahoo.com.tw Department of Computer Science and Information Engineering, National Yunlin University of Science and Technology, Yunlin, 64002 TaiwanSearch for more papers by this author First published: 02 March 2021 https://doi.org/10.1049/iet-spr.2020.0220AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract This work's contributions include three innovative concepts, an improved model, two-stage Lagrange principle, and minimum-energy scaling optimisation, for quantisation audio watermarking in the wavelet domain. First, discrete wavelet transform (DWT) multi-coefficients quantisation, composed of arbitrary scaling on the lowest DWT coefficients, and the group-based signal-to-noise ratio (SNR) of these coefficients is connected in a model. Then, the two-stage Lagrange principle and minimum-energy approach play two essential roles to obtain the optimal scaling factors. With the proposed scheme, the best fidelity and robustness of embedded audio can be attained and the perceptual evaluation of audio quality (PEAQ) test with an illustration of the relationship between SNR and PEAQ is also performed as well. Simulation results show that each watermarked audio by the proposed method attains a high SNR, good PEAQ, and a low bit error rate (BER). The SNR of most watermarked audios in their method is above 35 or even above 40 and the corresponding subjective difference grade of PEAQ is close to 0. In terms of comparing BER, most of their BER is as low as 2% or less indicating better robustness against many attacks, such as re-sampling, amplitude scaling, and mp3 compression. 1 Introduction An audio watermarking system usually consists of the embedding and extraction processes and satisfies three minimum requirements of audio watermarking standards set by the International Federation of Phonographic Industry (IFPI) requirements. According to the IFPI recommendation, every audio watermarking algorithm is set to meet the following specifications [1-10]: (i) audio watermark should be imperceptible of the original signal; (ii) signal-to-noise ratio (SNR) needs to be >20 dB and the embedding capacity should be >20 bits per second (bps); and (iii) watermark should be capable of resisting common attacks and audio processing operations. Audio watermarking techniques are usually classified according to the domain where the watermark is embedded. The purpose of these procedures is to embed a robust watermark and to preserve the reliability of the original signal. Several methods proposed previously involved in inserting watermarks in the time domain [11-21] or frequency domain [3-10, 22-25]. In this study, we focus on the frequency domain especially the wavelet domain. In the frequency domain, the authors in [3, 6, 24] proposed an entropy-based audio watermarking scheme in the wavelet domain. They defined energy entropy and derive many useful properties of this energy entropy with respect to audio watermarking, especially the invariance against amplitude scaling attack. However, the robustness against other attacks and the SNR of embedded audios is poor. Chen et al. [4, 8, 10] and Zhao et al. [22] used optimisation quantisation to ensure better embedded-audio quality. Their experimental results showed robust hidden data against most common attacks, but the audio quality is worse, and the embedded watermarks are not robust to amplitude scaling attacks due to the fact that they used fixed-scaling on coefficients in discrete wavelet transform (DWT). Jeyhoon et al. [23] proposed a blind audio watermarking algorithm based on Discrete Cosine Transform (DCT), linear regression, and standard deviation. Jeyhoon et al. [23] proposed an amplitude modification scheme in the wavelet domain to obtain good robustness. However, the embedding capacity and SNR of embedded audios are both low. To improve the performance of the method proposed by Xiang [5], optimal modification on low-frequency amplitude in the wavelet domain is proposed successfully in [7, 9, 26]. Li et al. [25] proposed a norm ratio-based audio watermarking scheme in the DWT domain. It combines quantisation and patchwork by a norm ratio to have novel and robustness. However, the shortcomings of the two methods cannot be improved. This paper aims to propose an innovative method, minimum-energy scaling on DWT coefficients, for optimisation quantisation audio watermarking in the wavelet domain. With the proposed scheme, the best fidelity and robustness of embedded audio can be attained. As an initial step, arbitrary scaling factors of DWT coefficients of the original audio are employed in the embedding criterion, and formulation of SNR is introduced. Next, a matrix function connecting the multi-coefficients quantisation embedding criterion, composed of arbitrary scaling on the lowest DWT coefficients, and the group SNR of these coefficients is proposed successfully. In the third step, the two-stage Lagrange principle, Hessian matrix, and minimum-energy approach enter to deal with the optimal condition. Then, the optimal scaling factors of the DWT coefficients are finally achieved. Perceptual evaluation of audio quality (PEAQ), established during 1994–1998 by the International Telecommunication Union is one standardised algorithm [27-29] for objectively measuring perceived audio quality. To judge the watermarked audio quality practically, an objective measurement by PQevalAudio is performed to compare the PEAQ of four types of audios [28]. The measurement of the watermarked audio quality in this study is based on SNR and PEAQ. Moreover, the relationship between the SNR and PEAQ is illustrated. Experimental results confirm that the proposed minimum-energy optimising scaling method preserves audio in good quality and maintains satisfactory robustness against common attacks. The rest of this paper is structured as follows. Section 2 gives an overview of the DWT and Lagrange principle. The proposed embedding process and the extraction process are presented in Section 3. Experimental results and conclusions are shown in Sections 4 and 5, respectively. 2 Preliminaries and background An introduction of the DWT and some mathematical concepts that will be carried out in this study will be reviewed in this section. 2.1 Discrete wavelet transform The DWT is a technique to transform image signals to wavelets. Such a transform maps a function in L 2 ( R ) onto a scale-space plane, and wavelets are attained by scales and translate a mother wavelet ψ ( x ) [30, 31]. The normalised scaling functions and wavelets are defined as φ i , n ( t ) = 2 i / 2 h i φ ( 2 i t − n ) ψ i , n ( t ) = 2 i / 2 g i ψ ( 2 i t − n ) , where i and n denote the scale and translation parameters and h i and g i denote the low-pass and high-pass filters, respectively. It is noted that wavelet functions are usually normalised to have unit energy and the factor 2 j / 2 in front is to maintain L 2 norm. The purpose of utilising orthogonal wavelet basis functions is to ease the work in calculating coefficient expansion and spanning L 2 ( R ) in signal processing. In this study, the host digital audio signal S ( n ), denotes the nth sample-timed original audio signal S ( t ), is slit into several segments where DWT is performed. This process can be accomplished by utilising an orthogonal basis to realise DWT by a filter bank. Fig. 1 shows how input signals S ( n ) are decomposed into eight non-overlapping multi-resolution sub-bands in DWT seven-level decomposition. The synchronisation codes and watermarks are both hided into the DWT low coefficients in the seventh level, denoted as A7 in Fig. 1. Fig. 1Open in figure viewerPowerPoint Discrete wavelet transformation with seven levels 2.2. Background This section reviews the background and related works. Definition 1.Suppose that C = [ c 1 c 2 ⋯ c n ] T is a n × 1 matrix and f ( C ) is a function of matrices. The gradient of f ( C ) is defined as ∇ f ( C ) = ∂ f ∂ C = ∂ f ∂ c 1 ∂ f ∂ c 2 ⋯ ∂ f ∂ c n T . In [8, 10, 24], Chen et al. balanced the conflicted relationship between the SNR and robustness in the wavelet domain by using the following two theorems [32]. Theorem 1.If A is a matrix with size n × n, and C ¯ and C are column vectors with size n × 1, then ∂ A C ¯ ∂ C ¯ = A T ∂ ( C ¯ − C ) T ( C ¯ − C ) ∂ C ¯ = 2 ( C ¯ − C ) . Theorem 2.(Lagrange principle): Assume that g is a continuously differentiable function of C on a subset of the domain of a function f. If C 0 minimises (or maximises) f ( C ) subject to the constraint g ( C ) = 0, then there exists a scalar multiplier λ such that ∇ f ( C 0 ) = λ ⋅ ∇ g ( C 0 ) , provided that ∇ g ( C 0 ) ≠ 0 . By defining an augmented matrix function H ( C , λ ) = f ( C ) + λ ⋅ g ( C ), the optimisation problem will then be treated as optimising H ( C , λ ) without any constraint. The necessary conditions for the existence of the extreme are ∂ H / ∂ λ = 0 and ∂ H / ∂ C = 0. Based on Theorems 1 and 2, they solve the optimisation-based quantisation problem minimise ( C ¯ N − C N ) T ( C ¯ N − C N ) C N T C N subject to A C ¯ N = γ 1 ( or A C ¯ N = γ 0 ) by setting the Lagrange multiplier λ to have a function without constraint H ( C ¯ N , λ ) = ( C ¯ N − C N ) T ( C ¯ N − C N ) C N T C N + λ T [ A C ¯ N − γ 1 ] and then the following necessary conditions derive the optimal solution ∂ H ∂ λ = 0 and ∂ H ∂ C ¯ N = 0 where C ¯ N = c ¯ 1 c ¯ 2 ⋯ c ¯ N T denotes the watermarked DWT-coefficient vector with respect to a given coefficient vector C N = c 1 c 2 ⋯ c N T ; A = a 1 a 2 ⋯ a N denotes a scaling vector, which has arbitrary scalar entries assigned by an encoder; γ 1 and γ 0 are two given objectives of the scaling of the coefficient vector C N . However, these concepts did not consider the optimisation of the scaling vector A. 3 Proposed embedding and extraction With the motivation in Section 2.2, we propose a better model for quantisation audio watermarking in the wavelet domain and combine the two-stage Lagrange principle with minimum-energy optimisation to optimise the scaling of DWT coefficients in watermark embedding. The embedding process and extraction process are shown in Figs. 2 and 3. The embedding process and extraction process are discussed in Sections 3.1–3.3 and Section 3.4 in detail. Fig. 2Open in figure viewerPowerPoint Watermark embedding process Fig. 3Open in figure viewerPowerPoint Watermark extraction process (a) Love song, (b) Symphony, (c) Dance, (d) Folklore 3.1 Embedding technique with amplitude scaling factors The watermarked audio may encounter various attacks of shifting or cropping a short segment inside the audio signal, synchronisation codes need to be put into audio with informative watermarks, so the generating watermark is capable of self-synchronisation. Before the embedding process, synchronisation codes and watermarks are generated in the form of a binary pseudorandom noise (PN) sequence { β i } , β i = 0 or 1. To achieve robustness performance, the synchronisation codes and the watermark are both placed into the DWT low-frequency coefficients. Every N consecutive absolute-valued DWT coefficients are collected and arranged as C N = c 1 c 2 ⋯ c N T . A detailed embedding { β i } process is illustrated as follows: consider a positive scalar matrix A = [ a 1 a 2 ⋯ a N ], where each entry is assigned arbitrarily by an encoder. All unknown values of watermarked lowest-frequency absolute DWT coefficients, C ¯ N = c ¯ 1 c ¯ 2 ⋯ c ¯ N T , needed to be assessed. These coefficients are related to the original DWT coefficients C N and ∑ j = 1 N a j c ¯ j = γ 1 , if β i = 1 γ 0 , if β i = 0 . where γ 1 and γ 0 denote the scales of C N . To bypass the possibility of any entries in A grows enormously, we assume all entries of A add up to a constant M, for example, N = 4, M = 4, and A = 0.9 1.2 1.2 0.7 . The SNR between the original audio S ( n ) and the watermarked audio S ¯ ( n ) is defined by SNR = − 10 ⋅ log ∑ n S ( n ) − S ¯ ( n ) 2 2 ∑ n S ( n ) 2 2 (1a) or SNR = 10 ⋅ log ∑ n S ( n ) 2 2 ∑ n S ( n ) − S ¯ ( n ) 2 2 (1b) One property the SNR holds is that a larger value corresponds to better robustness. Though the watermarked audio with a higher SNR is not guaranteed to possess good imperceptibility, higher fidelity can be achieved by maximising the SNR in the embedding process. Parseval theorem refers to the result that the sum of the square of S ( n ) is equal to the sum of the square of the DWT coefficients. Since all DWT coefficients are employed with orthogonal wavelet bases and those of high-frequency sub-band are not affected by the proposed algorithm but only the lowest DWT coefficients are adjusted. Hence the SNR in (1) is rewritten by SNR = − 10 ⋅ log ∑ C N − C ¯ N 2 2 ∑ C N 2 2 + ∑ ( high frequency sub − band DWT coefficients ) 2 ≤ − 10 ⋅ log ∑ C N − C ¯ N 2 2 ∑ C N 2 2 . (2a) To have as larger an SNR as possible, we can increase its upper bound. Whence we define the so-called group SNR, SNRL, in the wavelet domain by SN R L = − 10 ⋅ log C N − C ¯ N 2 2 C N 2 2 = − 10 ⋅ log ( C N − C ¯ N ) T ( C N − C ¯ N ) C N T C N (2b) From (2a) and (2b), it is easy to verify that the sum of all SNRL's in the wavelet domain is obviously an upper bound of the SNR. Therefore, the scaling factors for the lowest DWT wavelet coefficients will be adjusted to maximise the group SNR. To understand how variation of scaling factors affects behaviours of SNRL, we set a 1 + a 2 = 2 and show the relationship in Fig. 4. All plots indicate that SNRL attains its maximum as a 2 = 1, and declines as a 2 drops. Since logarithmic functions are invertible, SNRL in (2) is maximised when ( C ¯ N − C N ) T ( C ¯ N − C N ) / C N T C N is minimised. Fig. 4Open in figure viewerPowerPoint Variation of SNRL versus a 2 for four types of songs when N = 2 and M = 2 3.2 Optimal embedding technique Since C N T C N = ∑ j = 1 N c j 2 is a constant, to embed the binary bit β i = 1, the optimisation quantisation problem for the proposed embedding technique can be described as minimise ( C ¯ N − C N ) T ( C ¯ N − C N ) (3a) subject to A C ¯ N = γ 1 ( or A C ¯ N = γ 0 ) , (3b) ∑ j = 1 N a j = M . (3c) This optimisation problem can be solved in two steps: (i) solve the optimal problem in (3a) and (3b) and (ii) adjust the optimal solution obtained from (i) to simultaneously fulfil (3c). According to Theorem 2, there exists a scalar multiplier λ 1 so that the optimiser (3a) and the constraint (3b) can be combined into a constraint-free matrix function H 1 ( C ¯ N , λ 1 ) = ( C ¯ N − C N ) T ( C ¯ N − C N ) + λ 1 ( A C ¯ N − γ 1 ) . The necessary conditions for the existence of the minimum of H 1 ( C ¯ N , λ ) are ∂ H 1 ∂ C ¯ N = 0 ⇒ 2 ( C ¯ N − C N ) + A T λ 1 = 0 (4a) ∂ H 1 ∂ λ 1 = 0 ⇒ A C ¯ N − γ 1 = 0 . (4b) Multiply (4a) by the amplitude scaling matrix A and apply (4b) to obtain 2 ( γ 1 − A C N ) + A A T λ 1 = 0 which leads the optimal multiplier λ 1 ∗ and the optimal DWT coefficients C ¯ N ∗ to be λ 1 ∗ = 2 ( A A T ) − 1 ( A C N − γ 1 ) C ¯ N ∗ = C N − 1 2 A T λ 1 ∗ = C N − A T ( A A T ) − 1 ( A C N − γ 1 ) . By replacing γ 1 with γ 0 , the optimal DWT coefficients for embedding β i = 0 can be derived similarly. 3.3 Embedding technique with minimum energy amplitude scaling factors In this section, we will explain how the amplitude scaling matrix A can be optimized. Using the optimal DWT coefficients C ¯ N ∗ , optimisation of A can be described as follows: minimise C ¯ N ∗ − C N T C ¯ N ∗ − C N (5a) subject to ∑ j = 1 N a j = M . (5b) Use the fact that C ¯ N ∗ = C N − A T ( A A T ) − 1 ( A C N − γ 1 ), the term C ¯ N ∗ − C N T C ¯ N ∗ − C N can be simplified as C ¯ N ∗ − C N T C ¯ N ∗ − C N = C N − A T ( A A T ) − 1 ( A C N − γ 1 ) − C N T C N − A T ( A A T ) − 1 ( A C N − γ 1 ) − C N = − A T ( A A T ) − 1 ( A C N − γ 1 ) T − A T ( A A T ) − 1 ( A C N − γ 1 ) = A T ( A T ) − 1 A − 1 ( A C N − γ 1 ) T A T ( A A T ) − 1 ( A C N − γ 1 ) = ( A C N − γ 1 ) T ( A A T ) − 1 ( A C N − γ 1 ) = ( A A T ) − 1 ( A C N − γ 1 ) T ( A C N − γ 1 ) = 1 ∑ j = 1 N a j 2 a 1 a 2 … a N c 1 c 2 ⋮ c N − γ 1 T a 1 a 2 … a N c 1 c 2 ⋮ c N − γ 1 = 1 ∑ j = 1 N a j 2 a 1 c 1 + a 2 c 2 + ⋯ + a N c N − γ 1 T a 1 c 1 + a 2 c 2 + ⋯ + a N c N − γ 1 = 1 ∑ j = 1 N a j 2 ∑ j = 1 N a j c j − γ 1 2 . To minimise C ¯ N ∗ − C N T C ¯ N ∗ − C N subject to ∑ j = 1 N a j = M , the necessary condition ∂ H 2 / ∂ a k = 0 , k = 1 , 2 , … , N, where H 2 ( a j , λ 2 ) = 1 / ∑ j = 1 N a j 2 ∑ j = 1 N a j c j − γ 1 2 + λ 2 ∑ j = 1 N a j − M needs to be satisfied. More specifically ∂ H 2 ∂ a k = 2 ∑ j = 1 N a j 2 c k ∑ j = 1 N a j c j − γ 1 − 2 a k ∑ j = 1 N a j 2 2 ∑ j = 1 N a j c j − γ 1 2 + λ 2 = 0 leads to the optimal solution λ 2 ∗ = − 2 ∑ i = 1 N a j c j − γ 1 ∑ j = 1 N a j 2 c k − a k ∑ j = 1 N a j c j − γ 1 ∑ j = 1 N a j 2 , forall k . If c k − a k ∑ j = 1 N a j c j − γ 1 ∑ i = 1 N a j 2 = ρ for all k, where ρ is a constant, then | c 1 | = | c 2 | = ⋯ = | c N |. However, it is not always true in this case. Therefore, we consider the other situation where ∑ j = 1 N a j | c j | − γ 1 = 0 with ∑ j = 1 N a j = M . (6) Since the associated Hessian matrix ∂ H ∂ a 1 2 ⋯ ∂ H ∂ a 1 ∂ a N ⋮ ⋱ ⋮ ∂ H ∂ a N ∂ a 1 ⋯ ∂ H ∂ a N 2 = 2 ∑ j = 1 N a j 2 ρ 1 2 ⋯ ρ 1 ρ N ⋮ ⋱ ⋮ ρ N ρ 1 ⋯ ρ N 2 = 2 ∑ j = 1 N a j 2 ρ 1 ⋮ ρ N ρ 1 ⋯ ρ N , where ρ k = c k − 2 a k / ∑ j = 1 N a j 2 ∑ i = 1 N a j c j − γ 1 is non-negative-definite; the amplitude scaling factors a 1 , a 2 , … , a N obtained from (6) are sufficient to obtain the minimum solution of (5a) and (5b). Use a N = M − ∑ j = 1 N − 1 a j to rewrite (6) as a 1 c 1 + a 2 c 2 + ⋯ + M − ∑ j = 1 N − 1 a j c N − γ 1 = 0 . i.e. c 1 − c N a 1 + c 2 − c N a 2 + ⋯ + c N − 1 − c N a N − 1 = γ 1 − M c N (7) Since N − 1 unknowns are contained in one equation, infinitely many numbers of solutions are there to (7) and such equation is frequently resolved using the method of minimum-energy solution. Set P = c 1 − c N c 2 − c N ⋯ c N − 1 − c N , x = a 1 a 2 ⋯ a N − 1 T , b = γ 1 − M c N , then (7) can be expressed as P x = b. The minimum-energy solution is given by x + = P T v, where v satisfies P P T v = b. Through algebraic operations, we have v = γ 1 − M c N ∑ j = 1 N − 1 c j − c N 2 and x + = c 1 − c N c 2 − c N ⋮ c N − 1 − c N γ 1 − M c N ∑ j = 1 N − 1 c j − c N 2 . Thus, the amplitude scaling factors corresponding to the optimal solution are a j = c j − c N ∑ k = 1 N − 1 c k − c N 2 γ 1 − M c N , j = 1 , 2 , … , N − 1 , a N = M − ∑ j = 1 N − 1 a j = ∑ j = 1 N − 1 c j − c N γ 1 − M c j ∑ k = 1 N − 1 c k − c N 2 . (8) Suppose that all a j ′ s in (8) are positive, the existence of an optimal solution to the system of linear equations in (6) is guaranteed; if some a j ′ s are either negative or zero, an optimal solution does not exist and a near optimal solution will be searched instead. In the case of a k ≤ 0 for some k, which occurs when the signs of c k − c N and γ 1 − M c N in (8) are opposite, we have γ 1 − M c N = γ 1 − M c k + M c k − c N indicating either γ 1 / M < c N < c k or c k < c N < γ 1 / M holds. Assume that sign c j − c N = sign γ 1 − M c N , for 1 ≤ j ≠ k ≤ N, then c k is either the largest or smallest absolute DWT coefficients C N depends on whether γ 1 / M < c N or c N < γ 1 / M. In order not to discredit the SNRL, the DWT coefficient term c k needs to be un-scaled, namely, a k = 1, and we turn to resolve (7) for the optimal solution with other N − 1 scaling factors. If more than one scaling factors are non-negative, we seek for the other factors to maximise the SNRL and the optimisation process will be performed iteratively. This optimisation process is summarised as follows: Step 1: To minimise ( C ¯ N − C N ) T ( C ¯ N − C N ) subject to A C ¯ N = γ 1 which give us C ¯ N ∗ = C N − A T ( A A T ) − 1 ( A C N − γ 1 ). Step 2: To minimise ( C ¯ N ∗ − C N ) T ( C ¯ N ∗ − C N ) subject to ∑ j = 1 N a j = M , which give us the scaling factors as described in (8). Step 3: Check the positivity of the scaling factors in (8). Set a k = 1 for negative a k and repeat Step 2 with N being N−1. To embed the binary bit ' β i = 0', we replace γ 1 with γ 0 and all a j ′ s can be obtained similarly. γ 1 , γ 0 are calculated using the quantisation rules To embed β i = 1 in C N , ∑ j = 1 N c j is adjusted to γ 1 = ∑ j = 1 N c j / Q Q + 3 4 Q To embed β i = 0 in C N , ∑ j = 1 N c j is adjusted to γ 0 = ∑ j = 1 N c j / Q Q + 1 4 Q where c j represents DWT lowest-frequency sub-band coefficients of the original signal, Q denotes the quantisation parameter, and is the greatest integer function. 3.4 Extraction techniques Watermark extraction is reverse of embedding. DWT and segmentation are applied to extract the test audio signal. Let C ^ N = [ | c ^ 1 | | c ^ 2 | ⋯ | c ^ N | ] T be the consecutive low-frequency sub-band DWT coefficients in the seventh level of the segmented test signal. Depending on the two scenarios, all binary bits { β ^ i } in the PN sequence can be extracted from C ^ N and a j ∗ , the optimal amplitude scaling factor If ∑ j = 1 N a j ∗ c ^ j − ∑ j = 1 N a j ∗ c ^ j / Q Q ≥ Q 2 , set β ^ i = 1. If ∑ j = 1 N a j ∗ c ^ j − ∑ j = 1 N a j ∗ c ^ j / Q Q < Q 2 , set β ^ i = 0. Once the corresponding orthonormal PN sequence is formulated, synchronisation codes are determined and the embedded watermark is extracted. 3.5 Almost invariance feature of the scaling factors In the wavelet domain, any audio signal S ( t ) ∈ L 2 ( R ) can be expressed as a series expansion of orthogonal scaling functions and wavelets S ( t ) = ∑ ℓ c j 0 ( ℓ ) φ j 0 , ℓ ( t ) + ∑ k ∑ j = j 0 ∞ d j ( k ) ψ j , k ( t ) , where c j ( ℓ ) = ∫ R S ( t ) φ j , ℓ ( t ) d t and d j ( k ) = ∫ R S ( t ) ψ j , k ( t ) d t denote the sequences of low-pass and high-pass coefficients, respectively; j 0 be the integer to define an interval on which S ( t ) is a piecewise constant. Let S ¯ ( t ) and S ~ ( t ) denote the signals prior to and after amplitude modification manipulation. Since they form a scale multiple of each other, there exists an amplitude modification factor τ such that S ~ ( t ) = τ ⋅ S ¯ ( t ) and we have S ~ ( t ) = τ ⋅ S ¯ ( t ) = τ ∑ k c ¯ j 0 ( k ) φ j 0 , k ( t ) + τ ∑ k ∑ j = j 0 ∞