Help/UGens/FFT/FFT Overview.html

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  35 <p class="p1"><b>FFT Overview</b></p>
  36 <p class="p2"><br></p>
  37 <p class="p3"><b>FFT and IFFT</b></p>
  38 <p class="p2"><br></p>
  39 <p class="p4">SuperCollider implements a number of UGens supporting FFT based processing. The most basic of these are <a href="FFT.html"><span class="s1">FFT</span></a> and <a href="IFFT.html"><span class="s1">IFFT</span></a><b> </b>which convert data between the time and frequency domains:</p>
  40 <p class="p2"><br></p>
  41 <p class="p4"><span class="Apple-tab-span">     </span><b>FFT(buffer, input)</b></p>
  42 <p class="p4"><b><span class="Apple-tab-span">  </span>IFFT(buffer)</b></p>
  43 <p class="p2"><br></p>
  44 <p class="p4">FFT stores spectral data in a local buffer (see <a href="../../ServerArchitecture/Buffer.html"><span class="s1">Buffer</span></a>) in the following order: DC, nyquist, real 1f, imag 1f, real 2f, imag 2f, ... real (N-1)f, imag (N-1)f, where f is the frequency corresponding to the window size, and N is the window size / 2.</p>
  45 <p class="p2"><br></p>
  46 <p class="p4">The buffer's size must correspond to a power of 2, and must also be a multiple of SC's block size. The window size is equivalent to the buffer size, and the window overlap defaults to 2. Both FFT and IFFT use a Sine window by default, the combination of which (i.e. raised sine, that is, sine squared) is a Hanning window.</p>
  47 <p class="p2"><br></p>
  48 <p class="p3"><b>Phase Vocoder UGens and Spectral Processing</b></p>
  49 <p class="p2"><br></p>
  50 <p class="p4">In between an FFT and an IFFT one can chain together a number of Phase Vocoder UGens (i.e. 'PV_...') to manipulate blocks of spectral data before reconversion. The process of buffering the appropriate amount of audio, windowing, conversion, overlap-add, etc. is handled for you automatically.</p>
  51 <p class="p2"><br></p>
  52 <p class="p5"><span class="Apple-tab-span">     </span></p>
  53 <p class="p6"><span class="Apple-tab-span">     </span>(</p>
  54 <p class="p6"><span class="Apple-tab-span">     </span>{ <span class="s2">var</span> in, chain;</p>
  55 <p class="p6"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span>in = <span class="s2">WhiteNoise</span>.ar(0.8);</p>
  56 <p class="p6"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span>chain = <span class="s2">FFT</span>(<span class="s2">LocalBuf</span>(2048), in);</p>
  57 <p class="p6"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span>chain = <span class="s2">PV_RandComb</span>(chain, 0.95, <span class="s2">Impulse</span>.kr(0.4));<span class="Apple-converted-space"> </span></p>
  58 <p class="p6"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span><span class="s2">IFFT</span>(chain);</p>
  59 <p class="p6"><span class="Apple-tab-span">     </span>}.play(s);</p>
  60 <p class="p6"><span class="Apple-tab-span">     </span>)</p>
  61 <p class="p2"><br></p>
  62 <p class="p4">PV Ugens write their output data <i>in place</i>, i.e. back into the same buffer from which they read. PV UGens which require two buffers write their data into the first buffer, usually called 'bufferA'.</p>
  63 <p class="p5"><br></p>
  64 <p class="p5"><span class="Apple-tab-span">     </span></p>
  65 <p class="p7"><span class="s3"><span class="Apple-tab-span">    </span>d = </span><span class="s2">Buffer</span><span class="s3">.read(s,</span>"sounds/a11wlk01.wav"<span class="s3">);</span></p>
  66 <p class="p5"><span class="Apple-tab-span">     </span></p>
  67 <p class="p5"><span class="Apple-tab-span">     </span></p>
  68 <p class="p6"><span class="Apple-tab-span">     </span>(</p>
  69 <p class="p6"><span class="Apple-tab-span">     </span>{ <span class="s2">var</span> inA, chainA, inB, chainB, chain;</p>
  70 <p class="p6"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span>inA = <span class="s2">LFSaw</span>.ar([100, 150], 0, 0.2);</p>
  71 <p class="p6"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span>inB = <span class="s2">PlayBuf</span>.ar(1, d, <span class="s2">BufRateScale</span>.kr(d), loop: 1);</p>
  72 <p class="p6"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span>chainA = <span class="s2">FFT</span>(<span class="s2">LocalBuf</span>(2048), inA);</p>
  73 <p class="p6"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span>chainB = <span class="s2">FFT</span>(<span class="s2">LocalBuf</span>(2048), inB);</p>
  74 <p class="p6"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span>chain = <span class="s2">PV_MagMul</span>(chainA, chainB); <span class="s4">// writes into bufferA</span></p>
  75 <p class="p6"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span>0.1 * <span class="s2">IFFT</span>(chain);</p>
  76 <p class="p6"><span class="Apple-tab-span">     </span>}.play(s);</p>
  77 <p class="p6"><span class="Apple-tab-span">     </span>)</p>
  78 <p class="p5"><span class="Apple-tab-span">     </span></p>
  79 <p class="p6"><span class="Apple-tab-span">     </span>d.free;</p>
  80 <p class="p2"><br></p>
  81 <p class="p4">Because each PV UGen overwrites the output of the previous one, it is necessary to copy the data to an additional buffer at the desired point in the chain in order to do parallel processing of input without using multiple FFT UGens. <a href="PV_Copy.html"><span class="s1">PV_Copy</span></a> allows for this.</p>
  82 <p class="p2"><br></p>
  83 <p class="p6"><span class="Apple-tab-span">     </span>(</p>
  84 <p class="p8"><span class="s3"><span class="Apple-tab-span">    </span>b = </span><span class="s2">Buffer</span><span class="s3">.alloc(s,2048,1); </span>// use global buffers for plotting the data</p>
  85 <p class="p6"><span class="Apple-tab-span">     </span>c = <span class="s2">Buffer</span>.alloc(s,2048,1);</p>
  86 <p class="p6"><span class="Apple-tab-span">     </span>)</p>
  87 <p class="p5"><span class="Apple-tab-span">     </span></p>
  88 <p class="p8"><span class="s3"><span class="Apple-tab-span">    </span></span>//// proof of concept</p>
  89 <p class="p6"><span class="Apple-tab-span">     </span>(</p>
  90 <p class="p6"><span class="Apple-tab-span">     </span>x = { <span class="s2">var</span> inA, chainA, chainB;</p>
  91 <p class="p6"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span>inA = <span class="s2">LFClipNoise</span>.ar(100);</p>
  92 <p class="p6"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span>chainA = <span class="s2">FFT</span>(b, inA);</p>
  93 <p class="p6"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span>chainB = <span class="s2">PV_Copy</span>(chainA, c); <span class="Apple-converted-space"> </span></p>
  94 <p class="p8"><span class="s3"><span class="Apple-tab-span">    </span><span class="Apple-tab-span">    </span></span><span class="s2">IFFT</span><span class="s3">(chainA) - </span><span class="s2">IFFT</span><span class="s3">(chainB); </span>// cancels to zero so silent!</p>
  95 <p class="p6"><span class="Apple-tab-span">     </span>}.play(s);</p>
  96 <p class="p6"><span class="Apple-tab-span">     </span>)</p>
  97 <p class="p6"><span class="Apple-tab-span">     </span>x.free;</p>
  98 <p class="p8"><span class="s3"><span class="Apple-tab-span">    </span></span>// IFFTed frames contain the same windowed output data</p>
  99 <p class="p6"><span class="Apple-tab-span">     </span>b.plot(<span class="s5">\b</span>, <span class="s2">Rect</span>(200, 430, 700, 300), <span class="s2">nil</span>, <span class="s2">nil</span>); c.plot(<span class="s5">\c</span>, <span class="s2">Rect</span>(200, 100, 700, 300), <span class="s2">nil</span>, <span class="s2">nil</span>);</p>
 100 <p class="p6"><span class="Apple-tab-span">     </span>[b, c].do(<span class="s2">_</span>.free);</p>
 101 <p class="p2"><br></p>
 102 <p class="p4">Note that PV UGens convert as needed between cartesian (complex) and polar representations, therefore when using multiple PV UGens it may be impossible to know in which form the values will be at any given time. FFT produces complex output (see above), so while the following produces a reliable magnitude plot:</p>
 103 <p class="p2"><br></p>
 104 <p class="p8"><span class="s3"><span class="Apple-tab-span">    </span>b = </span><span class="s2">Buffer</span><span class="s3">.alloc(s,1024); </span>// use global buffers for plotting the data</p>
 105 <p class="p6"><span class="Apple-tab-span">     </span>a = { <span class="s2">FFT</span>(b, <span class="s2">LFSaw</span>.ar(4000)); 0.0 }.play;</p>
 106 <p class="p6"><span class="Apple-tab-span">     </span>(</p>
 107 <p class="p6"><span class="Apple-tab-span">     </span>b.getn(0, 1024, { <span class="s2">arg</span> buf;</p>
 108 <p class="p6"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span><span class="s2">var</span> z, x;</p>
 109 <p class="p6"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span>z = buf.clump(2).flop;</p>
 110 <p class="p6"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span>z = [<span class="s2">Signal</span>.newFrom(z[0]), <span class="s2">Signal</span>.newFrom(z[1])];</p>
 111 <p class="p6"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span>x = <span class="s2">Complex</span>(z[0], z[1]);</p>
 112 <p class="p6"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span>{x.magnitude.plot}.defer</p>
 113 <p class="p6"><span class="Apple-tab-span">     </span>})</p>
 114 <p class="p6"><span class="Apple-tab-span">     </span>)</p>
 115 <p class="p6"><span class="Apple-tab-span">     </span>a.free; b.free;</p>
 116 <p class="p2"><br></p>
 117 <p class="p4">any Synth using PV UGens might not.</p>
 118 <p class="p2"><br></p>
 119 <p class="p4">It is possible to manipulate the FFT data directly within a synth graph (if there doesn't already exist a PV UGen which will do what you want), using the methods <a href="pvcalc.html"><span class="s6">pvcalc</span></a>, <a href="pvcalc2.html"><span class="s6">pvcalc2</span></a>, <a href="pvcollect.html"><span class="s6">pvcollect</span></a>. Here's an example which uses the <a href="../../Collections/SequenceableCollection.html"><span class="s6">SequenceableCollection</span></a> methods <i>clump</i> and <i>flop</i> to rearrange the order of the spectral bins:</p>
 120 <p class="p2"><br></p>
 121 <p class="p7"><span class="s3"><span class="Apple-tab-span">    </span>c = </span><span class="s2">Buffer</span><span class="s3">.read(s,</span>"sounds/a11wlk01.wav"<span class="s3">);</span></p>
 122 <p class="p5"><span class="Apple-tab-span">     </span></p>
 123 <p class="p6"><span class="Apple-tab-span">     </span>(</p>
 124 <p class="p6"><span class="Apple-tab-span">     </span>x = {</p>
 125 <p class="p6"><span class="Apple-tab-span">     </span><span class="Apple-converted-space">  </span><span class="s2">var</span> in, numFrames=2048, chain, v;</p>
 126 <p class="p6"><span class="Apple-tab-span">     </span><span class="Apple-converted-space">  </span>in = <span class="s2">PlayBuf</span>.ar(1, c, loop: 1);</p>
 127 <p class="p6"><span class="Apple-tab-span">     </span><span class="Apple-converted-space">  </span>chain = <span class="s2">FFT</span>(<span class="s2">LocalBuf</span>(numFrames), in);</p>
 128 <p class="p5"><span class="Apple-tab-span">     </span><span class="Apple-converted-space"> <span class="Apple-tab-span">      </span></span></p>
 129 <p class="p6"><span class="Apple-tab-span">     </span><span class="Apple-converted-space">  </span>chain = chain.pvcalc(numFrames, {<span class="s2">|mags, phases|</span></p>
 130 <p class="p8"><span class="s3"><span class="Apple-tab-span">    </span><span class="Apple-converted-space">      </span></span>/* Play with the mags and phases, then return them */</p>
 131 <p class="p6"><span class="Apple-tab-span">     </span><span class="Apple-converted-space">      </span>[mags, phases].flop.clump(2).flop.flatten</p>
 132 <p class="p6"><span class="Apple-tab-span">     </span><span class="Apple-converted-space">  </span>}, tobin: 250);</p>
 133 <p class="p5"><span class="Apple-tab-span">     </span><span class="Apple-converted-space"> </span></p>
 134 <p class="p6"><span class="Apple-tab-span">     </span><span class="Apple-converted-space">  </span><span class="s2">Out</span>.ar(0, 0.5 * <span class="s2">IFFT</span>(chain).dup);</p>
 135 <p class="p6"><span class="Apple-tab-span">     </span>}.play(s);</p>
 136 <p class="p6"><span class="Apple-tab-span">     </span>)</p>
 137 <p class="p6"><span class="Apple-tab-span">     </span>x.free; c.free;</p>
 138 <p class="p2"><br></p>
 139 <p class="p2"><br></p>
 140 <p class="p3"><b>Multichannel Expansion with FFT UGens</b></p>
 141 <p class="p2"><br></p>
 142 <p class="p4">Care must be taken when using multichannel expansion with FFT UGens, as they require separate buffers. Code such as this can be deceptive:</p>
 143 <p class="p2"><br></p>
 144 <p class="p6">chain = <span class="s7">FFT</span>(bufnum, {<span class="s7">WhiteNoise</span>.ar(0.2)}.dup);</p>
 145 <p class="p2"><br></p>
 146 <p class="p4">The above may seem to work, but does not. It does result in two FFT UGens, but as they both write to the same buffer, the second simply overwrites the data from the first, thus wasting CPU and accomplishing nothing.</p>
 147 <p class="p2"><br></p>
 148 <p class="p4">When using multichannel expansion with FFT UGens it is necessary to ensure that each one writes to a different buffer. Here's an example of one way to do this:</p>
 149 <p class="p2"><br></p>
 150 <p class="p2"><br></p>
 151 <p class="p5"><br></p>
 152 <p class="p5"><br></p>
 153 <p class="p6">(</p>
 154 <p class="p7"><span class="s2">SynthDef</span><span class="s3">(</span>"help-multichannel FFT"<span class="s3">, { </span><span class="s2">arg</span><span class="s3"> out=0; </span><span class="s4">// bufnum is an array</span></p>
 155 <p class="p6"><span class="s8"><span class="Apple-tab-span">    </span></span><span class="s2">var</span> in, chain;</p>
 156 <p class="p6"><span class="s8"><span class="Apple-tab-span">    </span></span>in = [<span class="s2">SinOsc</span>.ar(200, 0, 0.2), <span class="s2">WhiteNoise</span>.ar(0.2)];</p>
 157 <p class="p8"><span class="s9"><span class="Apple-tab-span">    </span></span><span class="s3">chain = </span><span class="s2">FFT</span><span class="s3">(</span><span class="s2">LocalBuf</span><span class="s3">([2048, 2048]), in); </span>// each FFT has a different buffer</p>
 158 <p class="p8"><span class="s3"><span class="Apple-tab-span">    </span></span>// now we can multichannel expand as normal</p>
 159 <p class="p6"><span class="s8"><span class="Apple-tab-span">    </span></span>chain = <span class="s2">PV_BrickWall</span>(chain, <span class="s2">SinOsc</span>.kr(-0.1));</p>
 160 <p class="p6"><span class="s8"><span class="Apple-tab-span">    </span></span><span class="s2">Out</span>.ar(out, <span class="s2">IFFT</span>(chain));</p>
 161 <p class="p6">}).play;</p>
 162 <p class="p6">)</p>
 163 <p class="p2"><br></p>
 164 <p class="p8">// or using global buffers</p>
 165 <p class="p2"><br></p>
 166 <p class="p6">b = {<span class="s2">Buffer</span>.alloc(s,2048,1)}.dup;</p>
 167 <p class="p5"><br></p>
 168 <p class="p6">(</p>
 169 <p class="p6"><span class="s2">SynthDef</span>(<span class="s10">"help-multichannel FFT"</span>, { <span class="s2">arg</span> out=0, bufnum= #[0, 1]; <span class="s4">// bufnum is an array</span></p>
 170 <p class="p6"><span class="s8"><span class="Apple-tab-span">    </span></span><span class="s2">var</span> in, chain;</p>
 171 <p class="p6"><span class="s8"><span class="Apple-tab-span">    </span></span>in = [<span class="s2">SinOsc</span>.ar(200, 0, 0.2), <span class="s2">WhiteNoise</span>.ar(0.2)];</p>
 172 <p class="p8"><span class="s9"><span class="Apple-tab-span">    </span></span><span class="s3">chain = </span><span class="s2">FFT</span><span class="s3">(bufnum, in); </span>// each FFT has a different buffer</p>
 173 <p class="p8"><span class="s3"><span class="Apple-tab-span">    </span></span>// now we can multichannel expand as normal</p>
 174 <p class="p6"><span class="s8"><span class="Apple-tab-span">    </span></span>chain = <span class="s2">PV_BrickWall</span>(chain, <span class="s2">SinOsc</span>.kr(-0.1));</p>
 175 <p class="p6"><span class="s8"><span class="Apple-tab-span">    </span></span><span class="s2">Out</span>.ar(out, <span class="s2">IFFT</span>(chain));</p>
 176 <p class="p6">}).play(s,[<span class="s5">\bufnum</span>, b]);</p>
 177 <p class="p6">)</p>
 178 <p class="p5"><br></p>
 179 <p class="p2"><br></p>
 180 <p class="p2"><br></p>
 181 <p class="p4">Note that dup on a UGen just makes a reference to that UGen, because UGen defines -copy to simply return the receiver. (See <a href="../UGen.html"><span class="s1">UGen</span></a> for more detail.)</p>
 182 <p class="p2"><br></p>
 183 <p class="p6">a = SinOsc.ar;</p>
 184 <p class="p6">a.dup[1] === a</p>
 185 <p class="p5"><br></p>
 186 <p class="p6">true</p>
 187 <p class="p2"><br></p>
 188 <p class="p4">Code like <span class="s11">IFFT</span><span class="s12">(chain).dup</span> is found throughout the PV help files , and is just a convenient way to copy a mono signal to stereo, without further computation.</p>
 189 <p class="p2"><br></p>
 190 <p class="p9"><span class="s3">See also <a href="../../Other Topics/MultiChannel.html"><span class="s8">MultiChannel</span></a>.</span></p>
 191 <p class="p2"><br></p>
 192 <p class="p3"><b>PV and FFT UGens in the Standard Library</b></p>
 193 <p class="p2"><br></p>
 194 <p class="p4">The following PV UGens are included in the standard SC distribution:</p>
 195 <p class="p2"><br></p>
 196 <p class="p4"><span class="Apple-tab-span">     </span><a href="FFT.html"><span class="s1">FFT</span></a><span class="Apple-tab-span">  </span>Fast Fourier Transform</p>
 197 <p class="p4"><span class="Apple-tab-span">     </span><a href="IFFT.html"><span class="s1">IFFT</span></a><span class="Apple-tab-span">        </span>Inverse Fast Fourier Transform</p>
 198 <p class="p4"><span class="Apple-tab-span">     </span><a href="PV_Add.html"><span class="s1">PV_Add</span></a><span class="Apple-tab-span">    </span>complex addition</p>
 199 <p class="p4"><span class="Apple-tab-span">     </span><a href="PV_BinScramble.html"><span class="s1">PV_BinScramble</span></a><span class="Apple-tab-span">    </span>scramble bins</p>
 200 <p class="p4"><span class="Apple-tab-span">     </span><a href="PV_BinShift.html"><span class="s1">PV_BinShift</span></a><span class="Apple-tab-span">  </span>shift and stretch bin position</p>
 201 <p class="p4"><span class="Apple-tab-span">     </span><a href="PV_BinWipe.html"><span class="s1">PV_BinWipe</span></a><span class="Apple-tab-span">    </span>combine low and high bins from two inputs</p>
 202 <p class="p9"><span class="s3"><span class="Apple-tab-span">    </span><a href="PV_BrickWall.html"><span class="s8">PV_BrickWall</span></a><span class="Apple-tab-span">        </span>zero bins</span></p>
 203 <p class="p4"><span class="Apple-tab-span">     </span><a href="PV_ConformalMap.html"><span class="s1">PV_ConformalMap</span></a><span class="Apple-tab-span">  </span>complex plane attack<span class="Apple-converted-space"> </span></p>
 204 <p class="p4"><span class="Apple-tab-span">     </span><a href="PV_Copy.html"><span class="s1">PV_Copy</span></a><span class="Apple-tab-span">  </span>copy an FFT buffer</p>
 205 <p class="p4"><span class="Apple-tab-span">     </span><a href="PV_CopyPhase.html"><span class="s1">PV_CopyPhase</span></a><span class="Apple-tab-span">        </span>copy magnitudes and phases</p>
 206 <p class="p4"><span class="Apple-tab-span">     </span><a href="PV_Diffuser.html"><span class="s1">PV_Diffuser</span></a><span class="Apple-tab-span">  </span>random phase shifting</p>
 207 <p class="p9"><span class="s3"><span class="Apple-tab-span">    </span><a href="PV_HainsworthFoote.html"><span class="s8">PV_HainsworthFoote</span></a></span></p>
 208 <p class="p9"><span class="s3"><span class="Apple-tab-span">    </span><a href="PV_JensenAndersen.html"><span class="s8">PV_JensenAndersen</span></a></span></p>
 209 <p class="p4"><span class="Apple-tab-span">     </span><a href="PV_LocalMax.html"><span class="s1">PV_LocalMax</span></a><span class="Apple-tab-span">  </span>pass bins which are a local maximum</p>
 210 <p class="p4"><span class="Apple-tab-span">     </span><a href="PV_MagAbove.html"><span class="s1">PV_MagAbove</span></a><span class="Apple-tab-span">  </span>pass bins above a threshold</p>
 211 <p class="p4"><span class="Apple-tab-span">     </span><a href="PV_MagBelow.html"><span class="s1">PV_MagBelow</span></a><span class="Apple-tab-span">  </span>pass bins below a threshold</p>
 212 <p class="p4"><span class="Apple-tab-span">     </span><a href="PV_MagClip.html"><span class="s1">PV_MagClip</span></a><span class="Apple-tab-span">    </span>clip bins to a threshold</p>
 213 <p class="p4"><span class="Apple-tab-span">     </span><a href="PV_MagFreeze.html"><span class="s1">PV_MagFreeze</span></a><span class="Apple-tab-span">        </span>freeze magnitudes</p>
 214 <p class="p4"><span class="Apple-tab-span">     </span><a href="PV_MagMul.html"><span class="s1">PV_MagMul</span></a><span class="Apple-tab-span">      </span>multiply magnitudes</p>
 215 <p class="p4"><span class="Apple-tab-span">     </span><a href="PV_MagDiv.html"><span class="s1">PV_MagDiv</span></a><span class="Apple-tab-span">      </span>division of magnitudes</p>
 216 <p class="p4"><span class="Apple-tab-span">     </span><a href="PV_MagNoise.html"><span class="s1">PV_MagNoise</span></a><span class="Apple-tab-span">  </span>multiply magnitudes by noise</p>
 217 <p class="p4"><span class="Apple-tab-span">     </span><a href="PV_MagShift.html"><span class="s1">PV_MagShift</span></a><span class="Apple-tab-span">  </span>shift and stretch magnitude bin position</p>
 218 <p class="p4"><span class="Apple-tab-span">     </span><a href="PV_MagSmear.html"><span class="s1">PV_MagSmear</span></a><span class="Apple-tab-span">  </span>average magnitudes across bins</p>
 219 <p class="p4"><span class="Apple-tab-span">     </span><a href="PV_MagSquared.html"><span class="s1">PV_MagSquared</span></a><span class="Apple-tab-span">      </span>square magnitudes</p>
 220 <p class="p4"><span class="Apple-tab-span">     </span><a href="PV_Max.html"><span class="s1">PV_Max</span></a><span class="Apple-tab-span">    </span>maximum magnitude</p>
 221 <p class="p4"><span class="Apple-tab-span">     </span><a href="PV_Min.html"><span class="s1">PV_Min</span></a><span class="Apple-tab-span">    </span>minimum magnitude</p>
 222 <p class="p4"><span class="Apple-tab-span">     </span><a href="PV_Mul.html"><span class="s1">PV_Mul</span></a><span class="Apple-tab-span">    </span>complex multiply</p>
 223 <p class="p4"><span class="Apple-tab-span">     </span><a href="PV_PhaseShift.html"><span class="s1">PV_PhaseShift</span></a><span class="Apple-tab-span">      </span>shift phase of all bins</p>
 224 <p class="p4"><span class="Apple-tab-span">     </span><a href="PV_PhaseShift270.html"><span class="s1">PV_PhaseShift270</span></a><span class="Apple-tab-span">        </span>shift phase by 270 degrees</p>
 225 <p class="p4"><span class="Apple-tab-span">     </span><a href="PV_PhaseShift90.html"><span class="s1">PV_PhaseShift90</span></a><span class="Apple-tab-span">  </span>shift phase by 90 degrees</p>
 226 <p class="p4"><span class="Apple-tab-span">     </span><a href="PV_RandComb.html"><span class="s1">PV_RandComb</span></a><span class="Apple-tab-span">  </span>pass random bins</p>
 227 <p class="p4"><span class="Apple-tab-span">     </span><a href="PV_RandWipe.html"><span class="s1">PV_RandWipe</span></a><span class="Apple-tab-span">  </span>crossfade in random bin order</p>
 228 <p class="p4"><span class="Apple-tab-span">     </span><a href="PV_RectComb.html"><span class="s1">PV_RectComb</span></a><span class="Apple-tab-span">  </span>make gaps in spectrum</p>
 229 <p class="p4"><span class="Apple-tab-span">     </span><a href="PV_RectComb2.html"><span class="s1">PV_RectComb2</span></a><span class="Apple-tab-span">        </span>make gaps in spectrum</p>
 230 <p class="p4"><span class="Apple-tab-span">     </span><a href="UnpackFFT.html"><span class="s6">UnpackFFT</span></a>, <a href="PackFFT.html"><span class="s6">PackFFT</span></a>, <a href="Unpack1FFT.html"><span class="s6">Unpack1FFT</span></a><span class="Apple-tab-span">        </span><span class="Apple-tab-span">    </span>"unpacking" components used in <a href="pvcalc.html"><span class="s6">pvcalc</span></a>, <a href="pvcalc2.html"><span class="s6">pvcalc2</span></a>, <a href="pvcollect.html"><span class="s6">pvcollect</span></a> (can also be used on their own)</p>
 231 <p class="p2"><br></p>
 232 </body>
 233 </html>