Optimization and Minimum-Energy Approaches for Private Message Security in the Wavelet Domain of Audio Signals

Abstract

In the field of embedding digital information or private message into an audio signal, signal-to-noise ratio (SNR) and bit error rate (BER) are commonly performance indexes in measuring the distortion and robustness. The larger the value of SNR, the better the audio quality of the embedded information, and the smaller the value of BER, the more resistant the embedded information is to malicious attacks. This study first considers the unknown coefficients in the discrete wavelet transform and the unknown weights of these coefficients, as well as the state-switching embedded equations in an optimization model to embed digital information or private message. Next, Karush–Kuhn–Tucker theorem and minimum-energy approach play two essential roles to determine the unknown coefficients and the unknown weights so as to complete the information embedding or private message communication. In addition, the extraction of private message is similar to the private message embedding reversely. Finally, this study obtains an essential optimization-based system with state switching for embedding digital information or private message into an audio signal. Experimental results verify that the embedded audio in the proposed method has high SNR and low BER, indicating strong robustness against various attacks, such as re-sampling, amplitude scaling, and mp3 compression.

Appendices

Appendix I

$$ \frac{{\partial J_{1} }}{{\partial {\hat{\mathbf{C}}}_{N} }} = {\mathbf{0}} \Rightarrow \frac{{2({\hat{\mathbf{C}}}_{N} - {\mathbf{C}}_{N} )}}{{{\mathbf{C}}_{N}^{T} {\mathbf{C}}_{N} }} + \lambda_{1} (1 - 2\alpha ){\mathbf{W}}^{T} = {\mathbf{0}} \, $$

(18a)

$$\frac{{\partial J_{1} }}{{\partial \lambda_{1} }} = 0{}_{{}} \Rightarrow (1 - 2\alpha ){\mathbf{W}\hat{C}}_{N} + \alpha \gamma_{1} + (\alpha - 1)\gamma_{0} = 0.$$

(18b)

Multiply (18a) by ${\mathbf{W}}$ and rewrite (18b), we have

$$ 2({\mathbf{W}\hat{C}}_{N} - {\mathbf{WC}}_{N} ) + \lambda_{1} (1 - 2\alpha ){\mathbf{C}}_{N}^{T} {\mathbf{C}}_{N} {\mathbf{WW}}^{T} = {\mathbf{0}} \, $$

(19a)

$$ {\mathbf{W}\hat{C}}_{N} = \frac{{ - \alpha \gamma_{1} + (1 - \alpha )\gamma_{0} }}{1 - 2\alpha }. $$

(19b)

Replacing (19b) to (19a), we have

$$ 2\left[ {\frac{{ - \alpha \gamma_{1} + (1 - \alpha )\gamma_{0} }}{1 - 2\alpha } - {\mathbf{WC}}_{N} } \right] + \lambda_{1} (1 - 2\alpha ){\mathbf{C}}_{N}^{T} {\mathbf{C}}_{N} {\mathbf{WW}}^{T} = {\mathbf{0}} \, $$

(20)

Hence, the optimal solution for the multiplier $\lambda_{1}$ is

$$ \lambda_{1}^{*} = \frac{{2\left[ {{\mathbf{WC}}_{N} - \frac{{ - \alpha \gamma_{1} + (1 - \alpha )\gamma_{0} }}{1 - 2\alpha }} \right]}}{{(1 - 2\alpha ){\mathbf{C}}_{N}^{T} {\mathbf{C}}_{N} {\mathbf{WW}}^{T} }} \, $$

(21)

Replacing (21) to (18a), the optimal DWT coefficients are

$$ {\hat{\mathbf{C}}}_{N}^{*} = {\mathbf{C}}_{N} - {\mathbf{W}}^{T} ({\mathbf{WW}}^{T} )^{ - 1} \left[ {{\mathbf{WC}}_{N} - \frac{{(1 - \alpha )\gamma_{0} - \alpha \gamma_{1} }}{1 - 2\alpha }} \right] $$

(22)

Appendix II

$$ {\text{Minimize}}\frac{{\left( {{\hat{\mathbf{C}}}_{N}^{*} - {\mathbf{C}}_{N} } \right)^{T} \left( {{\hat{\mathbf{C}}}_{N}^{*} - {\mathbf{C}}_{N} } \right)}}{{{\mathbf{C}}_{N}^{T} {\mathbf{C}}_{N} }} $$

(23a)

$$ {\text{Subject to}} \sum\limits_{j = 1}^{N} {w_{j} } = M. $$

(23b)

Use the fact that ${\hat{\mathbf{C}}}_{N}^{*} = {\mathbf{C}}_{N} - {\mathbf{W}}^{T} ({\mathbf{WW}}^{T} )^{ - 1} \left[ {{\mathbf{WC}}_{N} - \frac{{(1 - \alpha )\gamma_{0} - \alpha \gamma_{1} }}{1 - 2\alpha }} \right]$, the term $\frac{{\left( {{\hat{\mathbf{C}}}_{N}^{*} - {\mathbf{C}}_{N} } \right)^{T} \left( {{\hat{\mathbf{C}}}_{N}^{*} - {\mathbf{C}}_{N} } \right)}}{{{\mathbf{C}}_{N}^{T} {\mathbf{C}}_{N} }}$can be simplified as $\frac{1}{{\left( {\sum\limits_{j = 1}^{N} {w_{j}^{2} } } \right)\left( {\sum\limits_{j = 1}^{N} {\left| {c_{j} } \right|^{2} } } \right)}}\left\{ {\sum\limits_{j = 1}^{N} {w_{j} } \left| {c_{j} } \right| - \frac{{(1 - \alpha )\gamma_{0} - \alpha \gamma_{1} }}{1 - 2\alpha }} \right\}^{2}$

$$\begin{aligned} & \frac{{\left( {\hat{\mathbf{C}}_{N}^{*} - {\mathbf{C}}_{N} } \right)^{T} \left( {\hat{\mathbf{C}}_{N}^{*} - {\mathbf{C}}_{N} } \right)}}{{{\mathbf{C}}_{N} ^{T} {\mathbf{C}}_{N} }} \hfill \\ & = \frac{{\begin{gathered}\left\{ {{\mathbf{C}}_{N} - {\mathbf{W}}^{T} ({\mathbf{WW}}^{T} )^{{ - 1}} \left[ {{\mathbf{WC}}_{N} - \frac{{(1 - \alpha )\gamma _{0} - \alpha \gamma _{1} }}{{1 - 2\alpha }}} \right] - {\mathbf{C}}_{N} } \right\}^{T}\hfill\\ \left\{ {{\mathbf{C}}_{N} - {\mathbf{W}}^{T} ({\mathbf{WW}}^{T} )^{{ - 1}} \left[ {{\mathbf{WC}}_{N} - \frac{{(1 - \alpha )\gamma _{0} - \alpha \gamma _{1} }}{{1 - 2\alpha }}} \right] - {\mathbf{C}}_{N} } \right\}\end{gathered}}} {{{\mathbf{C}}_{N} ^{T} {\mathbf{C}}_{N} }} \hfill \\ & = \frac{{\left\{ { - {\mathbf{W}}^{T} ({\mathbf{WW}}^{T} )^{{ - 1}} \left[ {{\mathbf{WC}}_{N} - \frac{{(1 - \alpha )\gamma _{0} - \alpha \gamma _{1} }}{{1 - 2\alpha }}} \right]} \right\}^{T} \left\{ { - {\mathbf{W}}^{T} ({\mathbf{WW}}^{T} )^{{ - 1}} \left[ {{\mathbf{WC}}_{N} - \frac{{(1 - \alpha )\gamma _{0} - \alpha \gamma _{1} }}{{1 - 2\alpha }}} \right]} \right\}}}{{{\mathbf{C}}_{N} ^{T} {\mathbf{C}}_{N} }} \hfill \\ &= \frac{{\left\{ {{\mathbf{W}}^{T} ({\mathbf{W}}^{T} )^{{ - 1}} {\mathbf{W}}^{{ - 1}} \left[ {{\mathbf{WC}}_{N} - \frac{{(1 - \alpha )\gamma _{0} - \alpha \gamma _{1} }}{{1 - 2\alpha }}} \right]} \right\}^{T} \left\{ {{\mathbf{W}}^{T} ({\mathbf{WW}}^{T} )^{{ - 1}} \left[ {{\mathbf{WC}}_{N} - \frac{{(1 - \alpha )\gamma _{0} - \alpha \gamma _{1} }}{{1 - 2\alpha }}} \right]} \right\}}}{{{\mathbf{C}}_{N} ^{T} {\mathbf{C}}_{N} }} \hfill \\ &= \frac{{\left\{ {{\mathbf{IW}}^{{ - 1}} \left[ {{\mathbf{WC}}_{N} - \frac{{(1 - \alpha )\gamma _{0} - \alpha \gamma _{1} }}{{1 - 2\alpha }}} \right]} \right\}^{T} \left\{ {{\mathbf{W}}^{T} ({\mathbf{WW}}^{T} )^{{ - 1}} \left[ {{\mathbf{WC}}_{N} - \frac{{(1 - \alpha )\gamma _{0} - \alpha \gamma _{1} }}{{1 - 2\alpha }}} \right]} \right\}}}{{{\mathbf{C}}_{N} ^{T} {\mathbf{C}}_{N} }} \hfill \\ &= \frac{{\left[ {{\mathbf{WC}}_{N} - \frac{{(1 - \alpha )\gamma _{0} - \alpha \gamma _{1} }}{{1 - 2\alpha }}} \right]^{T} \left( {{\mathbf{W}}^{{ - 1}} } \right)^{T} {\mathbf{W}}^{T} ({\mathbf{WW}}^{T} )^{{ - 1}} \left[ {{\mathbf{WC}}_{N} - \frac{{(1 - \alpha )\gamma _{0} - \alpha \gamma _{1} }}{{1 - 2\alpha }}} \right]}}{{{\mathbf{C}}_{N} ^{T} {\mathbf{C}}_{N} }} \hfill \\ &= \frac{{\left[ {{\mathbf{WC}}_{N} - \frac{{(1 - \alpha )\gamma _{0} - \alpha \gamma _{1} }}{{1 - 2\alpha }}} \right]^{T} \left( {{\mathbf{W}}^{T} } \right)^{{ - 1}} {\mathbf{W}}^{T} ({\mathbf{WW}}^{T} )^{{ - 1}} \left[ {{\mathbf{WC}}_{N} - \frac{{(1 - \alpha )\gamma _{0} - \alpha \gamma _{1} }}{{1 - 2\alpha }}} \right]}}{{{\mathbf{C}}_{N} ^{T} {\mathbf{C}}_{N} }} \hfill \\ &= \frac{{\left[ {{\mathbf{WC}}_{N} - \frac{{(1 - \alpha )\gamma _{0} - \alpha \gamma _{1} }}{{1 - 2\alpha }}} \right]^{T} ({\mathbf{WW}}^{T} )^{{ - 1}} \left[ {{\mathbf{WC}}_{N} - \frac{{(1 - \alpha )\gamma _{0} - \alpha \gamma _{1} }}{{1 - 2\alpha }}} \right]}}{{{\mathbf{C}}_{N} ^{T} {\mathbf{C}}_{N} }} \hfill \\ &= \frac{{({\mathbf{WW}}^{T} )^{{ - 1}} \left[ {{\mathbf{WC}}_{N} - \frac{{(1 - \alpha )\gamma _{0} - \alpha \gamma _{1} }}{{1 - 2\alpha }}} \right]^{T} \left[ {{\mathbf{WC}}_{N} - \frac{{(1 - \alpha )\gamma _{0} - \alpha \gamma _{1} }}{{1 - 2\alpha }}} \right]}}{{{\mathbf{C}}_{N} ^{T} {\mathbf{C}}_{N} }} \hfill \\ &= \frac{1}{{\left( {\sum\limits_{{j = 1}}^{N} {w_{j}^{2} } } \right)\left( {\sum\limits_{{j = 1}}^{N} {\left| {c_{j} } \right|^{2} } } \right)}}\left\{ {\left[ {\begin{array}{*{20}c} {w_{1} } & {w_{2} } & {...} & {w_{N} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\left| {c_{1} } \right|} \\ & {\left| {c_{2} } \right|} \\ & \vdots \\ & {\left| {c_{N} } \right|} \\ \end{array} } \right] - \frac{{(1 - \alpha )\gamma _{0} - \alpha \gamma _{1} }}{{1 - 2\alpha }}} \right\}^{T} \\ &\quad\times\left\{ {\left[ {\begin{array}{*{20}c} {w_{1} } & {w_{2} } & {...} & {w_{N} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\left| {c_{1} } \right|} \\ {\left| {c_{2} } \right|} \\ \vdots \\ {\left| {c_{N} } \right|} \\ \end{array} } \right] - \frac{{(1 - \alpha )\gamma _{0} - \alpha \gamma _{1} }}{{1 - 2\alpha }}} \right\} \hfill \\ &= \frac{1}{{\left( {\sum\limits_{{j = 1}}^{N} {w_{j}^{2} } } \right)\left( {\sum\limits_{{j = 1}}^{N} {\left| {c_{j} } \right|^{2} } } \right)}}\left\{ {w_{1} \left| {c_{1} } \right| + w_{2} \left| {c_{2} } \right| + ... + w_{N} \left| {c_{N} } \right| - \frac{{(1 - \alpha )\gamma _{0} - \alpha \gamma _{1} }}{{1 - 2\alpha }}} \right\}^{T}\hfill\\ &\quad\times \left\{ {w_{1} \left| {c_{1} } \right| + w_{2} \left| {c_{2} } \right| + ... + w_{N} \left| {c_{N} } \right| - \frac{{(1 - \alpha )\gamma _{0} - \alpha \gamma _{1} }}{{1 - 2\alpha }}} \right\} \hfill \\ &= \frac{1}{{\left( {\sum\limits_{{j = 1}}^{N} {w_{j}^{2} } } \right)\left( {\sum\limits_{{j = 1}}^{N} {\left| {c_{j} } \right|^{2} } } \right)}}\left\{ {\sum\limits_{{j = 1}}^{N} {w_{j} } \left| {c_{j} } \right| - \frac{{(1 - \alpha )\gamma _{0} - \alpha \gamma _{1} }}{{1 - 2\alpha }}} \right\}^{2}. \hfill \\\end{aligned} $$

In other words, the optimization model above can be rewritten as:

$$ {\text{minimize}}\frac{1}{{\left( {\sum\limits_{j = 1}^{N} {w_{j}^{2} } } \right)\left( {\sum\limits_{j = 1}^{N} {\left| {c_{j} } \right|^{2} } } \right)}}\left\{ {\sum\limits_{j = 1}^{N} {w_{j} } \left| {c_{j} } \right| - \frac{{(1 - \alpha )\gamma_{0} - \alpha \gamma_{1} }}{1 - 2\alpha }} \right\}^{2} $$

(24a)

$$ {\text{subject to}}\sum\limits_{j = 1}^{N} {w_{j} } = M. $$

(24b)

In order to minimize $\frac{1}{{\left( {\sum\limits_{j = 1}^{N} {w_{j}^{2} } } \right)\left( {\sum\limits_{j = 1}^{N} {\left| {c_{j} } \right|^{2} } } \right)}}\left\{ {\sum\limits_{j = 1}^{N} {w_{j} } \left| {c_{j} } \right| - \frac{{(1 - \alpha )\gamma_{0} - \alpha \gamma_{1} }}{1 - 2\alpha }} \right\}^{2}$ subject to $\sum\limits_{j = 1}^{N} {w_{j} } = M,$ we set

$$ H_{2} (w_{j} ,\lambda_{2} ) = \frac{1}{{\left( {\sum\limits_{j = 1}^{N} {w_{j}^{2} } } \right)\left( {\sum\limits_{j = 1}^{N} {\left| {c_{j} } \right|^{2} } } \right)}}\left\{ {\sum\limits_{j = 1}^{N} {w_{j} } \left| {c_{j} } \right| - \frac{{(1 - \alpha )\gamma_{0} - \alpha \gamma_{1} }}{1 - 2\alpha }} \right\}^{2} + \lambda_{2} \left( {\sum\limits_{j = 1}^{N} {w_{j} } - M} \right). $$

(25)

Then, the necessary condition $\frac{{\partial H_{2} }}{{\partial w_{k} }} = 0,\quad k = 1,2, \ldots ,N$ implies

$$ \frac{{\partial {\text{H}}_{2} }}{{\partial w_{k} }}{ = }\frac{{2\left| {c_{k} } \right|}}{{\left( {\sum\limits_{j = 1}^{N} {\left| {c_{j} } \right|^{2} } } \right)\left( {\sum\limits_{j = 1}^{N} {w_{j}^{2} } } \right)}}\left\{ {\sum\limits_{j = 1}^{N} {w_{j} \left| {c_{j} } \right|} - \frac{{(1 - \alpha )\gamma_{0} - \alpha \gamma_{1} }}{1 - 2\alpha }} \right\} - \frac{{2w_{k} }}{{\left( {\sum\limits_{j = 1}^{N} {\left| {c_{j} } \right|^{2} } } \right)\left( {\sum\limits_{j = 1}^{N} {w_{j}^{2} } } \right)^{2} }}\left\{ {\sum\limits_{j = 1}^{N} {w_{j} \left| {c_{j} } \right|} - \frac{{(1 - \alpha )\gamma_{0} - \alpha \gamma_{1} }}{1 - 2\alpha }} \right\}^{2} { + }\lambda_{2} = 0. $$

That is, the optimal solution for $\lambda$ is

$$ \lambda_{2}^{*} = \frac{{2w_{k} }}{{\left( {\sum\limits_{j = 1}^{N} {\left| {c_{j} } \right|^{2} } } \right)\left( {\sum\limits_{j = 1}^{N} {w_{j}^{2} } } \right)^{2} }}\left\{ {\sum\limits_{j = 1}^{N} {w_{j} \left| {c_{j} } \right|} - \frac{{(1 - \alpha )\gamma_{0} - \alpha \gamma_{1} }}{1 - 2\alpha }} \right\}^{2} - \frac{{2\left| {c_{k} } \right|}}{{\left( {\sum\limits_{j = 1}^{N} {\left| {c_{j} } \right|^{2} } } \right)\left( {\sum\limits_{j = 1}^{N} {w_{j}^{2} } } \right)}}\left\{ {\sum\limits_{j = 1}^{N} {w_{j} \left| {c_{j} } \right|} - \frac{{(1 - \alpha )\gamma_{0} - \alpha \gamma_{1} }}{1 - 2\alpha }} \right\} $$

or

$$ \lambda_{2}^{*} { = }\frac{{2\left\{ {\sum\limits_{i = 1}^{N} {w_{j} \left| {c_{j} } \right|} - \frac{{(1 - \alpha )\gamma_{0} - \alpha \gamma_{1} }}{1 - 2\alpha }} \right\}}}{{\left( {\sum\limits_{j = 1}^{N} {\left| {c_{j} } \right|^{2} } } \right)\left( {\sum\limits_{j = 1}^{N} {w_{j}^{2} } } \right)}}\left\{ {w_{k} \frac{{\sum\limits_{j = 1}^{N} {w_{j} \left| {c_{j} } \right|} - \frac{{(1 - \alpha )\gamma_{0} - \alpha \gamma_{1} }}{1 - 2\alpha }}}{{\sum\limits_{j = 1}^{N} {w_{j}^{2} } }} - \left| {c_{k} } \right|} \right\},{\text{ for all }}k. $$

${\text{If }}w_{k} \frac{{\sum\limits_{j = 1}^{N} {w_{j} \left| {c_{j} } \right|} - \frac{{(1 - \alpha )\gamma_{0} - \alpha \gamma_{1} }}{1 - 2\alpha }}}{{\sum\limits_{j = 1}^{N} {w_{j}^{2} } }} - \left| {c_{k} } \right|$ is a constant for all $k$; then, $|c_{1} | = |c_{2} | = \cdots = |c_{N} |$. However, the case $|c_{1} | = |c_{2} | = \cdots = |c_{N} |$ is not always true in DWT, respectively. Hence, we consider the other case

$$ \sum\limits_{j = 1}^{N} {w_{j} |c_{j} |} - \frac{{(1 - \alpha )\gamma_{0} - \alpha \gamma_{1} }}{1 - 2\alpha } = 0{\text{with}}\sum\limits_{j = 1}^{N} {w_{j} } = M. $$

(26)

Since $w_{N} = M - \sum\limits_{j = 1}^{N - 1} {w_{j} }$, we rewrite (26) as

$$ w_{1} \left| {c_{1} } \right| + w_{2} \left| {c_{2} } \right| + ... + \left( {M - \sum\limits_{j = 1}^{N - 1} {w_{j} } } \right)\left| {c_{N} } \right| - \frac{{(1 - \alpha )\gamma_{0} - \alpha \gamma_{1} }}{1 - 2\alpha } = 0, $$

(27)

i.e.,

$$ (\left| {c_{1} } \right| - \left| {c_{N} } \right|)w_{1} + (\left| {c_{2} } \right| - \left| {c_{N} } \right|)w_{2} + ... + (\left| {c_{N - 1} } \right| - \left| {c_{N} } \right|)w_{N - 1} = \frac{{(1 - \alpha )\gamma_{0} - \alpha \gamma_{1} }}{1 - 2\alpha } - M\left| {c_{N} } \right|. $$

(28)

Since $N - 1$ unknowns are contained in one equation, infinitely many numbers of solutions are there to (28) and such equation should be resolved using the minimum-energy method. Assume

$$ \begin{gathered} {\mathbf{P}} = \left[ {\left| {c_{1} } \right| - \left| {c_{N} } \right|{\mkern 1mu} \quad \left| {c_{2} } \right| - \left| {c_{N} } \right|\quad \cdots \quad \left| {c_{N - 1} } \right| - \left| {c_{N} } \right|} \right], \hfill \\ {\mathbf{x}} = \left[ {w_{1} \quad w_{2} \quad \cdots \quad w_{N - 1} } \right]^{T} ,\qquad \hfill \\ b = \frac{{(1 - \alpha )\gamma_{0} - \alpha \gamma_{1} }}{1 - 2\alpha } - M\left| {c_{N} } \right|, \hfill \\ \end{gathered} $$

then Eq. (28) can be expressed as ${\mathbf{Px}} = b$. The minimum-energy solution is given by ${\mathbf{x}}_{ + } = {\mathbf{P}}^{T} v$ where $v$ satisfies ${\mathbf{PP}}^{T} v = b$. Through algebraic operations, we have

$$ v = \frac{{\frac{{(1 - \alpha )\gamma_{0} - \alpha \gamma_{1} }}{1 - 2\alpha } - M\left| {c_{N} } \right|}}{{\sum\limits_{j = 1}^{N - 1} {\left( {\left| {c_{j} } \right| - \left| {c_{N} } \right|} \right)^{2} } }}{\text{and}}{\mathbf{x}}_{ + } = \left[ {\begin{array}{*{20}c} {\left| {c_{1} } \right| - \left| {c_{N} } \right|} \\ {\left| {c_{2} } \right| - \left| {c_{N} } \right|} \\ \vdots \\ {\left| {c_{N - 1} } \right| - \left| {c_{N} } \right|} \\ \end{array} } \right]\frac{{\frac{{(1 - \alpha )\gamma_{0} - \alpha \gamma_{1} }}{1 - 2\alpha } - M\left| {c_{N} } \right|}}{{\sum\limits_{j = 1}^{N - 1} {\left( {\left| {c_{j} } \right| - \left| {c_{N} } \right|} \right)^{2} } }}. $$

Thus, the optimal solutions of weights are

$$ w_{j} = \frac{{\left| {c_{j} } \right| - \left| {c_{N} } \right|}}{{\sum\limits_{k = 1}^{N - 1} {\left( {\left| {c_{k} } \right| - \left| {c_{N} } \right|} \right)^{2} } }}\left[ {\frac{{(1 - \alpha )\gamma_{0} - \alpha \gamma_{1} }}{1 - 2\alpha } - M\left| {c_{N} } \right|} \right],\quad j = 1,2, \ldots ,N - 1, \, $$

(29)

and

$$ w_{N} = M - \sum\limits_{j = 1}^{N - 1} {w_{j} } = \frac{{\sum\limits_{j = 1}^{N - 1} {\left( {\left| {c_{j} } \right| - \left| {c_{N} } \right|} \right)} \left[ {\frac{{(1 - \alpha )\gamma_{0} - \alpha \gamma_{1} }}{1 - 2\alpha } - M\left| {c_{j} } \right|} \right]}}{{\sum\limits_{k = 1}^{N - 1} {\left( {\left| {c_{k} } \right| - \left| {c_{N} } \right|} \right)^{2} } }}. $$

(30)

Suppose that all $w_{j}^{\prime } s$ in (29) and (30) are positive, existence of an optimal solution to the system of linear equations in (26) is guaranteed; if some $w_{j}^{\prime } s\left[ {0,1} \right]$ are either negative or zero, an optimal solution does not exist and a near optimal solution will be searched instead. In the case of $w_{k} \le 0{\text{ for some }}k$, which occurs when the signs of $\left| {c_{k} } \right| - \left| {c_{N} } \right|$ and $\frac{{(1 - \alpha )\gamma_{0} - \alpha \gamma_{1} }}{1 - 2\alpha } - M\left| {c_{N} } \right|$ in (29) and (30) are opposite, we have

$$ \frac{{(1 - \alpha )\gamma_{0} - \alpha \gamma_{1} }}{1 - 2\alpha } - M\left| {c_{N} } \right| = \frac{{(1 - \alpha )\gamma_{0} - \alpha \gamma_{1} }}{1 - 2\alpha } - M\left| {c_{k} } \right| + M(\left| {c_{k} } \right| - \left| {c_{N} } \right|) $$

indicating either ${\raise0.7ex\hbox{${\frac{{(1 - \alpha )\gamma_{0} - \alpha \gamma_{1} }}{1 - 2\alpha }}$} \!\mathord{\left/ {\vphantom {{\frac{{(1 - \alpha )\gamma_{0} - \alpha \gamma_{1} }}{1 - 2\alpha }} M}}\right.\kern-0pt} \!\lower0.7ex\hbox{$M$}} < \left| {c_{N} } \right| < \left| {c_{k} } \right|or\left| {c_{k} } \right| < \left| {c_{N} } \right| < {\raise0.7ex\hbox{${\frac{{(1 - \alpha )\gamma_{0} - \alpha \gamma_{1} }}{1 - 2\alpha }}$} \!\mathord{\left/ {\vphantom {{\frac{{(1 - \alpha )\gamma_{0} - \alpha \gamma_{1} }}{1 - 2\alpha }} M}}\right.\kern-0pt} \!\lower0.7ex\hbox{$M$}}$ or holds.

Assume that $sign(\left| {c_{j} } \right| - \left| {c_{N} } \right|) = sign(\frac{{(1 - \alpha )\gamma_{0} - \alpha \gamma_{1} }}{1 - 2\alpha } - M\left| {c_{N} } \right|),{\text{ for }}1 \le j\not = k \le N$, then $\left| {c_{k} } \right|$ is either the largest or smallest absolute DWT coefficients ${\mathbf{C}}_{N}$ depending on whether ${\raise0.7ex\hbox{${\frac{{(1 - \alpha )\gamma_{0} - \alpha \gamma_{1} }}{1 - 2\alpha }}$} \!\mathord{\left/ {\vphantom {{\frac{{(1 - \alpha )\gamma_{0} - \alpha \gamma_{1} }}{1 - 2\alpha }} M}}\right.\kern-0pt} \!\lower0.7ex\hbox{$M$}} < \left| {c_{N} } \right|{\text{or}}\left| {c_{N} } \right| < {\raise0.7ex\hbox{${\frac{{(1 - \alpha )\gamma_{0} - \alpha \gamma_{1} }}{1 - 2\alpha }}$} \!\mathord{\left/ {\vphantom {{\frac{{(1 - \alpha )\gamma_{0} - \alpha \gamma_{1} }}{1 - 2\alpha }} M}}\right.\kern-0pt} \!\lower0.7ex\hbox{$M$}}$. In order not to discredit the SNR, the DWT coefficient term $c_{k}$ needs to be un-scaled, namely $w_{k} = 1$, and we turn to resolve (28) for the optimal solution with other $N - 1$ weights. If more than one weights are nonnegative, we seek for the other weights to maximize the SNR and the optimization process will be performed iteratively.