<p>Permutation of the natural ordered Hadamard matrix into the sequency ordered Hadamard matrix of order 16
</p><p>The former is usually referred to as <a href="https://en.wikipedia.org/wiki/Walsh_matrix" class="extiw" title="w:Walsh matrix">Walsh matrix</a>, but sometimes also the latter.
</p><p>The number of sign changes per row in the natural ordered matrix is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle wp(15,7,3,1)~=~(0,15,~~7,8,~~3,12,~~4,11,~~1,14,~~6,9,~~2,13,~~5,10)}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mi>w</mi><mi>p</mi><mo stretchy="false">(</mo><mn>15</mn><mo>,</mo><mn>7</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mtext> </mtext><mo>=</mo><mtext> </mtext><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>15</mn><mo>,</mo><mtext> </mtext><mtext> </mtext><mn>7</mn><mo>,</mo><mn>8</mn><mo>,</mo><mtext> </mtext><mtext> </mtext><mn>3</mn><mo>,</mo><mn>12</mn><mo>,</mo><mtext> </mtext><mtext> </mtext><mn>4</mn><mo>,</mo><mn>11</mn><mo>,</mo><mtext> </mtext><mtext> </mtext><mn>1</mn><mo>,</mo><mn>14</mn><mo>,</mo><mtext> </mtext><mtext> </mtext><mn>6</mn><mo>,</mo><mn>9</mn><mo>,</mo><mtext> </mtext><mtext> </mtext><mn>2</mn><mo>,</mo><mn>13</mn><mo>,</mo><mtext> </mtext><mtext> </mtext><mn>5</mn><mo>,</mo><mn>10</mn><mo stretchy="false">)</mo></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle wp(15,7,3,1)~=~(0,15,~~7,8,~~3,12,~~4,11,~~1,14,~~6,9,~~2,13,~~5,10)}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b72fac9948da6814eef38bf3d76ccc8d2f210ca" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:68.838ex; height:2.843ex;" alt="{\displaystyle wp(15,7,3,1)~=~(0,15,~~7,8,~~3,12,~~4,11,~~1,14,~~6,9,~~2,13,~~5,10)}"></span>,<br>
in the sequency ordered matrix the number of sign changes is consequtive.
</p><p><br></p><p><br></p><p>The permutation is the product of the <a href="https://en.wikipedia.org/wiki/Gray_code" class="extiw" title="w:Gray code">Gray code</a> permutation and the <a href="https://en.wikipedia.org/wiki/Bit-reversal_permutation" class="extiw" title="w:Bit-reversal permutation">Bit-reversal permutation</a>:
</p>
<a href="//commons.wikimedia.org/wiki/File:Gray_code_*_bit_reversal_16.svg" class="image"><img alt="Gray code * bit reversal 16.svg" src="https://upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Gray_code_%2A_bit_reversal_16.svg/500px-Gray_code_%2A_bit_reversal_16.svg.png" decoding="async" width="500" height="396" srcset="https://upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Gray_code_%2A_bit_reversal_16.svg/750px-Gray_code_%2A_bit_reversal_16.svg.png 1.5x, https://upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Gray_code_%2A_bit_reversal_16.svg/1000px-Gray_code_%2A_bit_reversal_16.svg.png 2x" data-file-width="850" data-file-height="673"></a>
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