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This work examines the use of the time-varying delay line (TVDL) to implement loopback frequency modulation (LBFM), an oscillator that loops back to modulate its own frequency. Digital delay lines are used regularly in sound synthesis/processing to model the pure delay associated with one-dimensional acoustic propagation. When the delay is made time varying, the TVDL time warps the input according to a delay function, altering the input's instantaneous frequency and phase. As a result, the TVDL is well suited for delay-based effects and, in particular, those involving frequency/phase modulation for which the TVDL delay function is oscillatory and thus bounded by a maximum and minimum delay. Sustaining a constant change in sounding frequency however, corresponds to a delay function having a term that is linear in time, making it limited only by the length of the input signal. While TVDLs may still be used when there is a pitch shift, limiting the delay by simple wrapping of the delay function and/or cross fading between multiple TVDLs may not be adequate to avoid audible artifacts. In LBFM, the resulting phase has both linear and oscillating terms and the resulting signal undergoes a sustained shift in the fundamental frequency that makes a TVDL implementation more challenging. An alternate closed-form representation of the LBFM oscillator, however, provides the information necessary for accurately wrapping the TVDL delay function and ensuring it is suitably bounded so that the produced sound is free of phase distortion and audible artifacts.

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