Improved (?) javadoc (MATH-677).
git-svn-id: https://svn.apache.org/repos/asf/commons/proper/math/trunk@1208986 13f79535-47bb-0310-9956-ffa450edef68
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Sebastien Brisard
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@ -75,18 +75,17 @@ public class FastHadamardTransformer implements RealTransformer {
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}
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/**
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* The FHT (Fast Hadamard Transformation) which uses only subtraction and addition.
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* <br>
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* Requires <b>Nlog2N = n2</b><sup>n</sup> additions.
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* <br>
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* <br>
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* <b><u>Short Table of manual calculation for N=8:</u></b>
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* The FHT (Fast Hadamard Transformation) which uses only subtraction and
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* addition. Requires {@code N * log2(N) = n * 2^n} additions.
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*
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* <h3>Short Table of manual calculation for N=8</h3>
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* <ol>
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* <li><b>x</b> is the input vector we want to transform</li>
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* <li><b>y</b> is the output vector which is our desired result</li>
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* <li>a and b are just helper rows</li>
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* <li><b>x</b> is the input vector to be transformed,</li>
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* <li><b>y</b> is the output vector (Fast Hadamard transform of <b>x</b>),</li>
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* <li>a and b are helper rows.</li>
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* </ol>
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* <table border="1" align="center">
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* <table align="center" border="1" cellpadding="3">
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* <tbody align="center">
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* <tr>
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* <th>x</th>
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* <th>a</th>
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@ -94,109 +93,127 @@ public class FastHadamardTransformer implements RealTransformer {
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* <th>y</th>
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* </tr>
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* <tr>
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* <td>x<sub>0</sub></td>
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* <th>x<sub>0</sub></th>
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* <td>a<sub>0</sub> = x<sub>0</sub> + x<sub>1</sub></td>
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* <td>b<sub>0</sub> = a<sub>0</sub> + a<sub>1</sub></td>
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* <td>y<sub>0</sub> = b<sub>0</sub >+ b<sub>1</sub></td>
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* </tr>
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* <tr>
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* <td>x<sub>1</sub></td>
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* <th>x<sub>1</sub></th>
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* <td>a<sub>1</sub> = x<sub>2</sub> + x<sub>3</sub></td>
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* <td>b<sub>0</sub> = a<sub>2</sub> + a<sub>3</sub></td>
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* <td>y<sub>0</sub> = b<sub>2</sub> + b<sub>3</sub></td>
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* </tr>
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* <tr>
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* <td>x<sub>2</sub></td>
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* <th>x<sub>2</sub></th>
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* <td>a<sub>2</sub> = x<sub>4</sub> + x<sub>5</sub></td>
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* <td>b<sub>0</sub> = a<sub>4</sub> + a<sub>5</sub></td>
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* <td>y<sub>0</sub> = b<sub>4</sub> + b<sub>5</sub></td>
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* </tr>
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* <tr>
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* <td>x<sub>3</sub></td>
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* <th>x<sub>3</sub></th>
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* <td>a<sub>3</sub> = x<sub>6</sub> + x<sub>7</sub></td>
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* <td>b<sub>0</sub> = a<sub>6</sub> + a<sub>7</sub></td>
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* <td>y<sub>0</sub> = b<sub>6</sub> + b<sub>7</sub></td>
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* </tr>
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* <tr>
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* <td>x<sub>4</sub></td>
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* <th>x<sub>4</sub></th>
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* <td>a<sub>0</sub> = x<sub>0</sub> - x<sub>1</sub></td>
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* <td>b<sub>0</sub> = a<sub>0</sub> - a<sub>1</sub></td>
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* <td>y<sub>0</sub> = b<sub>0</sub> - b<sub>1</sub></td>
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* </tr>
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* <tr>
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* <td>x<sub>5</sub></td>
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* <th>x<sub>5</sub></th>
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* <td>a<sub>1</sub> = x<sub>2</sub> - x<sub>3</sub></td>
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* <td>b<sub>0</sub> = a<sub>2</sub> - a<sub>3</sub></td>
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* <td>y<sub>0</sub> = b<sub>2</sub> - b<sub>3</sub></td>
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* </tr>
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* <tr>
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* <td>x<sub>6</sub></td>
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* <th>x<sub>6</sub></th>
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* <td>a<sub>2</sub> = x<sub>4</sub> - x<sub>5</sub></td>
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* <td>b<sub>0</sub> = a<sub>4</sub> - a<sub>5</sub></td>
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* <td>y<sub>0</sub> = b<sub>4</sub> - b<sub>5</sub></td>
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* </tr>
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* <tr>
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* <td>x<sub>7</sub></td>
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* <th>x<sub>7</sub></th>
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* <td>a<sub>3</sub> = x<sub>6</sub> - x<sub>7</sub></td>
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* <td>b<sub>0</sub> = a<sub>6</sub> - a<sub>7</sub></td>
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* <td>y<sub>0</sub> = b<sub>6</sub> - b<sub>7</sub></td>
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* </tr>
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* </tbody>
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* </table>
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*
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* <b><u>How it works</u></b>
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* <h3>How it works</h3>
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* <ol>
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* <li>Construct a matrix with N rows and n+1 columns<br> <b>hadm[n+1][N]</b>
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* <br><i>(If I use [x][y] it always means [row-offset][column-offset] of a Matrix with n rows and m columns. Its entries go from M[0][0] to M[n][m])</i></li>
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* <li>Place the input vector <b>x[N]</b> in the first column of the matrix <b>hadm</b></li>
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* <li>The entries of the submatrix D<sub>top</sub> are calculated as follows.
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* <br>D<sub>top</sub> goes from entry [0][1] to [N/2-1][n+1].
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* <br>The columns of D<sub>top</sub> are the pairwise mutually exclusive sums of the previous column
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* <li>Construct a matrix with {@code N} rows and {@code n + 1} columns,
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* {@code hadm[n+1][N]}.<br/>
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* <em>(If I use [x][y] it always means [row-offset][column-offset] of a
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* Matrix with n rows and m columns. Its entries go from M[0][0]
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* to M[n][N])</em></li>
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* <li>Place the input vector {@code x[N]} in the first column of the
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* matrix {@code hadm}.</li>
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* <li>The entries of the submatrix {@code D_top} are calculated as follows
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* <ul>
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* <li>{@code D_top} goes from entry {@code [0][1]} to
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* {@code [N / 2 - 1][n + 1]},</li>
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* <li>the columns of {@code D_top} are the pairwise mutually
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* exclusive sums of the previous column.</li>
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* </ul>
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* </li>
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* <li>The entries of the submatrix D<sub>bottom</sub> are calculated as follows.
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* <br>D<sub>bottom</sub> goes from entry [N/2][1] to [N][n+1].
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* <br>The columns of D<sub>bottom</sub> are the pairwise differences of the previous column
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* <li>The entries of the submatrix {@code D_bottom} are calculated as
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* follows
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* <ul>
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* <li>{@code D_bottom} goes from entry {@code [N / 2][1]} to
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* {@code [N][n + 1]},</li>
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* <li>the columns of {@code D_bottom} are the pairwise differences
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* of the previous column.</li>
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* </ul>
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* </li>
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* <li>How D<sub>top</sub> and D<sub>bottom</sub> you can understand best with the example for N=8 above.
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* <li>The output vector y is now in the last column of <b>hadm</b></li>
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* <li><i>Algorithm from: http://www.archive.chipcenter.com/dsp/DSP000517F1.html</i></li>
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* <li>The consputation of {@code D_top} and {@code D_bottom} are best
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* understood with the above example (for {@code N = 8}).
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* <li>The output vector {@code y} is now in the last column of
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* {@code hadm}.</li>
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* <li><em>Algorithm from <a href="http://www.archive.chipcenter.com/dsp/DSP000517F1.html">chipcenter</a>.</em></li>
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* </ol>
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* <br>
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* <b><u>Visually</u></b>
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* <table border="1" align="center">
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* <h3>Visually</h3>
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* <table border="1" align="center" cellpadding="3">
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* <tbody align="center">
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* <tr>
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* <td>0</td>
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* <td>1</td>
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* <td>2</td>
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* <td>3</td>
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* <td>...</td>
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* <td>n + 1</td>
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* <td></td><th>0</th><th>1</th><th>2</th><th>3</th>
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* <th>…</th>
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* <th>n + 1</th>
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* </tr>
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* <tr>
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* <td>0</td>
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* <th>0</th>
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* <td>x<sub>0</sub></td>
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* <td colspan="5">↑</td>
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* <td colspan="5" rowspan="5" align="center" valign="middle">
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* ↑<br/>
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* ← D<sub>top</sub> →<br/>
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* ↓
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* </td>
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* </tr>
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* <tr><th>1</th><td>x<sub>1</sub></td></tr>
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* <tr><th>2</th><td>x<sub>2</sub></td></tr>
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* <tr><th>…</th><td>…</td></tr>
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* <tr><th>N / 2 - 1</th><td>x<sub>N/2-1</sub></td></tr>
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* <tr>
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* <th>N / 2</th>
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* <td>x<sub>N/2</sub></td>
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* <td colspan="5" rowspan="5" align="center" valign="middle">
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* ↑<br/>
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* ← D<sub>bottom</sub> →<br/>
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* ↓
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* </td>
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* </tr>
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* <tr><th>N / 2 + 1</th><td>x<sub>N/2+1</sub></td></tr>
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* <tr><th>N / 2 + 2</th><td>x<sub>N/2+2</sub></td></tr>
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* <tr><th>…</th><td>…</td></tr>
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* <tr><th>N</th><td>x<sub>N</sub></td></tr>
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* </tbody>
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* </table>
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* <pre>
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* +--------+---+---+---+-----+---+
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* | 0 | 1 | 2 | 3 | ... |n+1|
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* +------+--------+---+---+---+-----+---+
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* |0 | x<sub>0</sub> | /\ |
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* |1 | x<sub>1</sub> | || |
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* |2 | x<sub>2</sub> | <= D<sub>top</sub> => |
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* |... | ... | || |
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* |N/2-1 | x<sub>N/2-1</sub> | \/ |
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* +------+--------+---+---+---+-----+---+
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* |N/2 | x<sub>N/2</sub> | /\ |
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* |N/2+1 | x<sub>N/2+1</sub> | || |
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* |N/2+2 | x<sub>N/2+2</sub> | <= D<sub>bottom</sub> => | which is in the last column of the matrix
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* |... | ... | || |
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* |N | x<sub>N/2</sub> | \/ |
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* +------+--------+---+---+---+-----+---+
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* </pre>
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*
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* @param x input vector
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* @return y output vector
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* @param x the input vector
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* @return the output vector, {@code y}
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* @exception IllegalArgumentException if input array is not a power of 2
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*/
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protected double[] fht(double[] x) throws IllegalArgumentException {
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