## Re: [Audacity-nyquist] 1st derivative, and print statement

 Re: [Audacity-nyquist] 1st derivative, and print statement From: Roger Dannenberg - 2010-05-17 14:20:06 Continuity is not just a curiosity. The magnitude spectrum of an impulse is a constant. The spectrum of a step function (the integral of an impulse) falls off at 6dB per octave in the limit. A second integral eliminates the discontinuity and falls off at 12dB per octave in the limit. In general, the spectrum of a continuous signal falls off at 12dB per octave (or greater). If the first derivative is continuous, the signal falls off at 18dB per octave, and each additional continuous derivative gives another 6dB of rolloff. Since we are sensitive to higher frequencies, it's better not to introduce discontinuities. Linear cross fades and linear breakpoint envelopes result in continuous signals with discontinuities in the first derivative. The artifacts you get and what you can hear depend on other factors, so linear breakpoint functions are not necessarily bad (consider a good linear approximation to a smooth "S" curve), but it's easy to see that smoother envelopes are going to reduce artifacts at higher frequencies. -Roger