https://sourceforge.net/p/playpcmwin/wiki/QuantizationBitRate/Recent changes to QuantizationBitRateenFri, 08 May 2020 16:40:18 -0000https://sourceforge.net/p/playpcmwin/wiki/QuantizationBitRate/<div class="markdown_content"><pre>--- v5 +++ v6 @@ -1,6 +1,6 @@ -# PCM Quantization bit rate and cardinarity +# PCM Quantization bit rate and cardinality -Quantization bit rate (bit) | Cardinarity +Quantization bit rate (bit) | Cardinality --------------------------------- | --------------- 1 | 2 8 | 256 @@ -15,7 +15,7 @@ Therefore, if quantization bit rate is increased to countably infinite, any real number (rational numbers and irrational numbers) can be expressed exactly. -※……There is the following relation among with those two infinite cardinarities: 2^{ℵ0} ＝ ℵ1 +※……There is the following relation among with those two infinite cardinalities: 2^{ℵ0} ＝ ℵ1 </pre> </div>yamamoto2002Fri, 08 May 2020 16:40:18 -0000https://sourceforge.net3eaf8c6966e65aed7224613165a56fb2ca0933d6https://sourceforge.net/p/playpcmwin/wiki/QuantizationBitRate/<div class="markdown_content"><pre>--- v4 +++ v5 @@ -11,7 +11,6 @@ 64 | 18446744073709551616 128 | 340282366920938463463374607431768211456 256 | 115792089237316195423570985008687907853269984665640564039457584007913129639936 -1024 | 179769313486231590772930519078902473361797697894230657273430081157732675805500963132708477322407536021120113879871393357658789768814416622492847430639474124377767893424865485276302219601246094119453082952085005768838150682342462881473913110540827237163350510684586298239947245938479716304835356329624224137216 ℵ0 (countably infinite) | ℵ1 (cardinality of the continuum) ※ Therefore, if quantization bit rate is increased to countably infinite, any real number (rational numbers and irrational numbers) can be expressed exactly. </pre> </div>yamamoto2002Fri, 26 Jul 2019 11:14:39 -0000https://sourceforge.net9c5b5a237160d44f5d4f3d823e7b9c1dd1252697https://sourceforge.net/p/playpcmwin/wiki/QuantizationBitRate/<div class="markdown_content"><pre>--- v3 +++ v4 @@ -14,7 +14,7 @@ 1024 | 179769313486231590772930519078902473361797697894230657273430081157732675805500963132708477322407536021120113879871393357658789768814416622492847430639474124377767893424865485276302219601246094119453082952085005768838150682342462881473913110540827237163350510684586298239947245938479716304835356329624224137216 ℵ0 (countably infinite) | ℵ1 (cardinality of the continuum) ※ -Therefore, if quantization bit rate is increased to countably infinite, any real number can be expressed exactly. +Therefore, if quantization bit rate is increased to countably infinite, any real number (rational numbers and irrational numbers) can be expressed exactly. ※……There is the following relation among with those two infinite cardinarities: 2^{ℵ0} ＝ ℵ1 </pre> </div>yamamoto2002Fri, 26 Jul 2019 11:13:52 -0000https://sourceforge.net958b3167de3979220d640b1e80aeb09a33b41b20https://sourceforge.net/p/playpcmwin/wiki/QuantizationBitRate/<div class="markdown_content"><pre>--- v2 +++ v3 @@ -11,6 +11,7 @@ 64 | 18446744073709551616 128 | 340282366920938463463374607431768211456 256 | 115792089237316195423570985008687907853269984665640564039457584007913129639936 +1024 | 179769313486231590772930519078902473361797697894230657273430081157732675805500963132708477322407536021120113879871393357658789768814416622492847430639474124377767893424865485276302219601246094119453082952085005768838150682342462881473913110540827237163350510684586298239947245938479716304835356329624224137216 ℵ0 (countably infinite) | ℵ1 (cardinality of the continuum) ※ Therefore, if quantization bit rate is increased to countably infinite, any real number can be expressed exactly. </pre> </div>yamamoto2002Fri, 26 Jul 2019 11:04:49 -0000https://sourceforge.net6c93e86dcc945a81532811705ee56e6398721856https://sourceforge.net/p/playpcmwin/wiki/QuantizationBitRate/<div class="markdown_content"><pre>--- v1 +++ v2 @@ -13,7 +13,7 @@ 256 | 115792089237316195423570985008687907853269984665640564039457584007913129639936 ℵ0 (countably infinite) | ℵ1 (cardinality of the continuum) ※ -Therefore, if quantization bit rate is increased to countably infinite, any real value can be expressed exactly. +Therefore, if quantization bit rate is increased to countably infinite, any real number can be expressed exactly. ※……There is the following relation among with those two infinite cardinarities: 2^{ℵ0} ＝ ℵ1 </pre> </div>yamamoto2002Fri, 26 Jul 2019 11:03:34 -0000https://sourceforge.netcd43693e02995dceb557720ef24046c30f73ce9ehttps://sourceforge.net/p/playpcmwin/wiki/QuantizationBitRate/<div class="markdown_content"><h1 id="pcm-quantization-bit-rate-and-cardinarity">PCM Quantization bit rate and cardinarity</h1> <table> <thead> <tr> <th>Quantization bit rate (bit)</th> <th>Cardinarity</th> </tr> </thead> <tbody> <tr> <td>1</td> <td>2</td> </tr> <tr> <td>8</td> <td>256</td> </tr> <tr> <td>16</td> <td>65536</td> </tr> <tr> <td>24</td> <td>16777216</td> </tr> <tr> <td>32</td> <td>4294967296</td> </tr> <tr> <td>48</td> <td>281474976710656</td> </tr> <tr> <td>64</td> <td>18446744073709551616</td> </tr> <tr> <td>128</td> <td>340282366920938463463374607431768211456</td> </tr> <tr> <td>256</td> <td>115792089237316195423570985008687907853269984665640564039457584007913129639936</td> </tr> <tr> <td>ℵ0 (countably infinite)</td> <td>ℵ1 (cardinality of the continuum) ※</td> </tr> </tbody> </table> <p>Therefore, if quantization bit rate is increased to countably infinite, any real value can be expressed exactly. </p> <p>※……There is the following relation among with those two infinite cardinarities: 2^{ℵ0} ＝ ℵ1</p></div>yamamoto2002Fri, 26 Jul 2019 11:00:41 -0000https://sourceforge.neta93569b7e13f29dc7eb090761b5326d801b66c74