Trimming the Split Ends of Waveforms

In a recent post about making sounds in F# (using Pulse Code Modulation, PCM) I noted that my sound production method was creating popping sounds at the ends of each section of sound. I suggested this was due to sudden changes in phase, which I am persuaded is at least involved. Whether or not it is the fundamental cause, it may be a relevant observation about instrumental sound production.

One way to make the consecutive wave forms not have pops between them might well be to carry the phase from one note to the next as it would lessen the sudden change in the sound. Another way, probably simpler, and the method I pursue in this post is to "back off" from the tail—the "split end" of the wave form—and only output full cycles of waves with silence filling in the left over part of a requested duration. My experimentation with it suggests that this approach still results in a percussive sound at the end of notes when played on the speakers. 

The wave forms I was previously producing can be illustrated by Fig. 1. We can reduce the popping by avoiding partial oscillations (Fig. 2.). However, even on the final wave form of a sequence wave forms, there is an apparent percussive sound suggesting that this percussive effect is not (fully) attributable to the sudden start of the next sound. Eliminating this percussive effect would probably involve dampening.[1] Either the dampening would need to be a tail added to the sound beyond the requested duration or a dampening period would need to be built in to the duration requested.

Fig. 1. A partial oscillation is dropped and a new wave form starts at a phase of 0.0. The "jog" is audible as a "pop".

Fig. 2. "Silent partials" means we don't output any pulse signal for a partial oscillation. The feature as played on my speakers left the appearance of a slight percussive sound, though less annoying than the pop.

It's worth thinking a bit about how typical "physical" sound production has mechanisms in it which "naturally" prevent the popping that we have to carefully avoid in "artificial" sound production.

• In a wind instrument, you can't have a sudden enough change in air pressure or sudden enough change in the vibration of the reed or instrument body to create pops.
• In a stringed instrument, alteration of the frequency of the vibration of the string will maintain the phase of the vibration from one frequency to the next. 
• The wave form is not "interrupted" by anything you can do to the string. There are no truly "sudden" changes you can make to the vibration of the string. 
• Any dampening you can do is gradual in hearing terms. 
• A "hammer-on" a guitar string does not suddenly move the position of the string with respect to its vibration period—phase is maintained.
• In a piano, dampening does not create sufficiently sudden changes in the wave form to create a pop.

In short, (non-digital) instrumental sound production avoids "pops" by physical constraints that produce the effects of dampening and/or phase continuity. Digital sound production is not inherently constrained by mechanisms that enforce these effects.

I have altered the sinewave function to fill the last section of the sound with silence, following the pattern suggested by Fig. 2. This does leave an apparent percussive effect, but is a slight improvement in sound.

// duration is in seconds let sinewave volume freq_hertz sample_rate duration = let TWOPI = 2.0 * System.Math.PI // how long is a single step? let timestep = 1.0 / (float sample_rate) // how many total steps will fit? let steps = int(floor(duration / timestep)) // number of full cycles in duration let fullcycles = int(floor(duration * freq_hertz)) // time for one cycle let cycletime = 1.0 / freq_hertz // time for all of the full cycles let fullcyclestime = cycletime * (float fullcycles) let fullCycleSteps = int(floor(fullcyclestime / timestep)) let silentSteps = steps - fullCycleSteps // amplitude ranges between negative and positive (shifted down by half volume) let amp = volume / 2.0 let theta = freq_hertz * TWOPI * timestep seq { // full oscillations for i = 1 to fullCycleSteps do yield amp * sin(theta * float(i)) // fill remainder with silence for i = 1 to silentSteps do yield 0.0 }

Something this experiment does not deal with is whether we are hearing an effect of the wave form, per se, or whether we are hearing the behavior of the speakers that are outputting the sound. A sudden stop of something physical within the speaker might generate an actual percussive sound. Also still outstanding is whether the human ear can perceive phase discontinuity in the absence of an amplitude discontinuity.

I decided to pick this up again from the other direction and look at fast Fourier transforms done in Maxima here.

¹ Dampening is a (progressive) reduction in amplitude over several consecutive oscillations of a wave.