On 18/08/2015, Peter S <peter.schoffhau...@gmail.com> wrote: > > Similarly, even if frequency f=0.5 may be considered ill-specified > (because it's critical frequency), you can still approach it to > arbitrary precision, and the gain will approach -infinity. So > > f=0.4 > f=0.49 > f=0.499 > f=0.4999 > f=0.4999999999 > f=0.499999999999999999999999999999999999999999999999999 > etc. > > The more you approach f=0.5, the more the gain will approach > -infinity.
I made an actual test program to confirm this numerically: http://morpheus.spectralhead.com/txt/fracdelay.tcl.txt Here are the results, testing half-sample delay gain at varying frequencies: f = 0.4 => -10.2 dB f = 0.49 => -30 dB f = 0.499 => -58.3 dB f = 0.4999 => -98 dB f = 0.49999 => -138 dB f = 0.499999 => -178 dB f = 0.4999999 => -218 dB f = 0.49999999 => -258 dB f = 0.499999999 => -297.9 dB f = 0.4999999999 => -325.1 dB f = 0.49999999999 => gives -Inf dB, because it reaches limit of 64-bit precision At 44.1 kHz sampling rate, f=0.4999999999 equals 22049.99999559 Hz, that's the closest I can get to f=0.5 using 64 bit floating point precision. At that frequency, measured gain is -325.1 dB (~= 5.55e-17), which is practically zero. For comparison, 24 bit PCM signals' dynamic range is only 144 dB. To represent -325.1 dB in fixed point, you'd need at least 54 bits precision. Using arbitrary precision maths, you can approach f=0.5 arbitrary closely, and the gain will tend towards -Inf decibels. (But I doubt you'll ever want to reach 22050 Hz with less than 0.00001 Hz error...) Now I could make a fancy graph of this, but right now I won't bother. _______________________________________________ music-dsp mailing list music-dsp@music.columbia.edu https://lists.columbia.edu/mailman/listinfo/music-dsp
