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Title	Pink Noise
Description	Laboratory Grade Pink Noise
Keywords	cooper, baker, cooperbaker, noise, pink, filter, filters, gain, coef, coeff, coefficients, pure, data, pd, object, algorithm, math, pinknoise, amazing, accurate, lab, laboratory, music, random, spectrum, audio, sound, signal, dsp, process, processing, digital, computer, music
Content	<?php dropIcon(); ?> Pink noise is very useful in computer music and audio engineering, and I have always wanted an accurate pink noise object in the Pure Data environment. Generating accurate pink noise is not as simple as it might seem, so I spent some time designing a so-called pinking filter, and generalized it for any audio sample rate. This page describes the filter I ended up with, shows how its parameters are calculated, and displays the Pd object source code at the bottom of the page. <br> <br> <?php ruler( 'Download' ); ?> <?php download( 'pink~.zip', 'pink~.zip' ); ?> — Pd object (os x), help patch, source code, and matlab parameter calculator <br> <br> <?php ruler( 'Algorithm' ); ?> The pinking filter creates pink noise by filtering white noise to create the desired pink energy spectrum. The filter's frequency response is -3dB per octave starting at approximately four Hertz and continuing to just below the Nyquist frequency. The pinking filter consists of several one-pole low-pass filters, where each low-pass is spaced two octaves from its neighbors and filter gains are attenuated in 6dB steps. This configuration results in a nearly linear -3dB per octave overall frequency response when the low-pass filters are summed. <br> <br> This algorithm is designed to work with any sample rate up to 192kHz. Filter coefficients are calculated by first determining the maximum cutoff frequency resulting in a filter coefficient less than one, described by the following formula where <i>f</i> is frequency and <i>sr</i> is sample rate. The <i>- 1</i> ensures that the coefficient resulting from <i>f</i> is just below 1 to prevent the first filter from exploding. <br> <center> <i><pre>f = sr / 2π - 1</i></pre> </center> Following this, a series of low-pass coefficents are calculated by successively dividing frequency by four while it remains larger than one. This operation is expressed by the following while loop, where <i>a</i> is an array of coefficients and <i>i</i> is the array index starting at zero. <br> <center> <table> <tr> <td> <i> <pre> while( f > 1 ) { a[i] = 2π * f / sr f = f / 4 i++ }</pre> </i> </td> </tr> </table> </center> The low-pass filter gains are calculated starting at 0dB for the lowest frequency filter, and continue downward in 6dB steps for each successive filter. An additional output gain factor is calculated so that the summed low-pass filters result in an overall frequency response that starts at 0dB-FS before proceeding downward. This process is described by the following while loop where <i>g</i> is an array of gain values, <i>dB</i> is the decibel gain value starting at zero, <i>s</i> is the sum of gain values starting at zero, </i><i>og</i> is the output gain factor, and <i>i</i> is the array index starting at the number of filters. <br> <center> <table> <tr> <td> <i> <pre> while( i-- > 0 ) { g[i] = pow( 10, dB / 20 ) s += g[i]; dB -= 6.0; } og = pow( 10, ( ( -20 * log10( s ) ) / 20 )</pre> </i> </td> </tr> </table> </center> Finally, pink noise is created by filtering white noise through the low-pass filters. The filters are implemented within a loop so that a variable number of filters may be used depending on sample rate, and this loop runs once for each sample of output. This implementation is expressed by the following statements where <i>w</i> is the white noise sample between -1 and 1, <i>p</i> is the pink noise sample initialized to zero, <i>i</i> is the array index starting at -1, <i>f</i> is the number of filters, <i>y</i> is the array of filtered samples, <i>a</i> is the array of filter coefficients, <i>g</i> is the array of filter gains, <i>o</i> is the output sample, and <i>og</i> is the output gain. <br> <center> <table> <tr> <td> <i> <pre> while( ++i < f ) { y[i] = a[i] * w + ( 1 - a[i] ) * y[i]; p += y[i] * g[i]; } o = og * p;</pre> </i> </td> </tr> </table> </center> The graph below shows the pinking filter's frequency response starting at 0dB, as well as the individual low-pass filters' frequency responses below that. Sample rate is 44.1kHz. <br> <br> <?php image( 'bode.png' ); ?> <br> <?php ruler( 'Cutoff Frequencies, Coefficients, and Gains' ); ?> Below are lists of filter pararameters for common audio sample rates. Parameters for other sample rates may be calculated with the pink.m matlab script in <?php download( 'pink~.zip', 'pink~.zip' ); ?> . <br> <br> <center> <table> <tr> <td> <pre> <b>Sample Rate : 8000</b> Output Gain : 0.506845730 Freq Coeff Gain ------------------------------------------- 1.242421430 0.000975796 1.000000000 4.969685722 0.003903182 0.501187234 19.878742886 0.015612728 0.251188643 79.514971546 0.062450913 0.125892541 318.059886184 0.249803650 0.063095734 1272.239544735 0.999214602 0.031622777 </pre> </td> <td> </td> <td> <pre> <b>Sample Rate : 11025</b> Output Gain : 0.506845730 Freq Coeff Gain ------------------------------------------- 1.712581296 0.000976006 1.000000000 6.850325186 0.003904024 0.501187234 27.401300744 0.015616095 0.251188643 109.605202974 0.062464381 0.125892541 438.420811897 0.249857524 0.063095734 1753.683247588 0.999430097 0.031622777 </pre> </td> </tr> <tr> <td> <pre> <b>Sample Rate : 16000</b> Output Gain : 0.506845730 Freq Coeff Gain ------------------------------------------- 2.485819423 0.000976179 1.000000000 9.943277693 0.003904716 0.501187234 39.773110773 0.015618864 0.251188643 159.092443092 0.062475456 0.125892541 636.369772368 0.249901825 0.063095734 2545.479089470 0.999607301 0.031622777 </pre> </td> <td> </td> <td> <pre> <b>Sample Rate : 22050</b> Output Gain : 0.506845730 Freq Coeff Gain ------------------------------------------- 3.426139155 0.000976284 1.000000000 13.704556622 0.003905137 0.501187234 54.818226487 0.015620548 0.251188643 219.272905949 0.062482191 0.125892541 877.091623794 0.249928762 0.063095734 3508.366495176 0.999715048 0.031622777 </pre> </td> </tr> <tr> <td> <pre> <b>Sample Rate : 32000</b> Output Gain : 0.502806702 Freq Coeff Gain ------------------------------------------- 1.243153852 0.000244093 1.000000000 4.972615409 0.000976371 0.501187234 19.890461636 0.003905483 0.251188643 79.561846546 0.015621932 0.125892541 318.247386184 0.062487728 0.063095734 1272.989544735 0.249950913 0.031622777 5091.958178941 0.999803650 0.015848932 </pre> </td> <td> </td> <td> <pre> <b>Sample Rate : 44100</b> Output Gain : 0.502806702 Freq Coeff Gain ------------------------------------------- 1.713313718 0.000244106 1.000000000 6.853254873 0.000976423 0.501187234 27.413019494 0.003905693 0.251188643 109.652077974 0.015622774 0.125892541 438.608311897 0.062491095 0.063095734 1754.433247588 0.249964381 0.031622777 7017.732990353 0.999857524 0.015848932 </pre> </td> </tr> <tr> <td> <pre> <b>Sample Rate : 48000</b> Output Gain : 0.502806702 Freq Coeff Gain ------------------------------------------- 1.864852849 0.000244109 1.000000000 7.459411395 0.000976435 0.501187234 29.837645580 0.003905739 0.251188643 119.350582319 0.015622955 0.125892541 477.402329276 0.062491819 0.063095734 1909.609317103 0.249967275 0.031622777 7638.437268411 0.999869100 0.015848932 </pre> </td> <td> </td> <td> <pre> <b>Sample Rate : 64000</b> Output Gain : 0.502806702 Freq Coeff Gain ------------------------------------------- 2.486551845 0.000244117 1.000000000 9.946207381 0.000976467 0.501187234 39.784829523 0.003905867 0.251188643 159.139318092 0.015623466 0.125892541 636.557272368 0.062493864 0.063095734 2546.229089470 0.249975456 0.031622777 10184.916357881 0.999901825 0.015848932 </pre> </td> </tr> <tr> <td> <pre> <b>Sample Rate : 88200</b> Output Gain : 0.502806702 Freq Coeff Gain ------------------------------------------- 3.426871577 0.000244123 1.000000000 13.707486309 0.000976493 0.501187234 54.829945237 0.003905972 0.251188643 219.319780949 0.015623887 0.125892541 877.279123794 0.062495548 0.063095734 3509.116495176 0.249982191 0.031622777 14036.465980705 0.999928762 0.015848932 </pre> </td> <td> </td> <td> <pre> <b>Sample Rate : 96000</b> Output Gain : 0.502806702 Freq Coeff Gain ------------------------------------------- 3.729949838 0.000244125 1.000000000 14.919799352 0.000976499 0.501187234 59.679197409 0.003905994 0.251188643 238.716789638 0.015623977 0.125892541 954.867158551 0.062495909 0.063095734 3819.468634205 0.249983638 0.031622777 15277.874536822 0.999934550 0.015848932 </pre> </td> </tr> <tr> <td> <pre> <b>Sample Rate : 176400</b> Output Gain : 0.500806513 Freq Coeff Gain ------------------------------------------- 1.713496824 0.000061033 1.000000000 6.853987295 0.000244132 0.501187234 27.415949181 0.000976528 0.251188643 109.663796724 0.003906111 0.125892541 438.655186897 0.015624443 0.063095734 1754.620747588 0.062497774 0.031622777 7018.482990353 0.249991095 0.015848932 28073.931961410 0.999964381 0.007943282 </pre> </td> <td> </td> <td> <pre> <b>Sample Rate : 192000</b> Output Gain : 0.500806513 Freq Coeff Gain ------------------------------------------- 1.865035954 0.000061033 1.000000000 7.460143817 0.000244133 0.501187234 29.840575267 0.000976531 0.251188643 119.362301069 0.003906122 0.125892541 477.449204276 0.015624489 0.063095734 1909.796817103 0.062497955 0.031622777 7639.187268411 0.249991819 0.015848932 30556.749073644 0.999967275 0.007943282 </pre> </td> </tr> </table> </center> <br> <?php ruler( 'References' ); ?> The technique described above was inspired by Paul Kellet's filter coefficients, located here: <br> · <?php href( 'http://www.musicdsp.org/files/pink.txt', 'Filter to make pink noise from white'); ?> <br> <br> Larry Trammell's method is described here: <br> · <?php href( 'http://home.earthlink.net/~ltrammell/tech/pinkalg.htm', 'Pink Noise Generator'); ?> <br> · <?php href( 'http://musicdsp.org/archive.php?classid=1#220', 'Trammell Pink Noise (C++ class)'); ?> <br> <br> An excellent reference page about pink noise techniques is located here: <br> · <?php href( 'http://www.firstpr.com.au/dsp/pink-noise/', 'DSP generation of Pink (1/f) Noise'); ?> <br> <br> My exploratory Pd object source code for the Kellet and Trammel techniques is available here: <br> · <?php download( 'pink~.ref.zip', 'pink~.ref.zip' ); ?> <br> <br> <?php ruler( 'pink~.c' ); ?> <?php code( 'pink~.c' ); ?>
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