If denotes the input to a time-varying delay, the output can be written as

where denotes the time-varying delay in seconds. In discrete-time implementations, when is not an integer multiple of the sampling interval, may be approximated to arbitrary accuracy (in a finite band) using

Let's analyze the frequency shift caused by a time-varying delay by setting to a complex sinusoid at frequency :

The output is now

The instantaneous phase of this signal is

which can be differentiated to give the instantaneous frequency

where denotes the output frequency, and denotes the time derivative of the delay . Thus,

Comparing Eq.
(5) to Eq.
(1), we find that the
time-varying delay most naturally simulates Doppler shift caused by a
*moving listener*, with

That is, the delay growth-rate, , should be set to the speed of the listener

Simulating *source* motion is also possible, but the relation
between delay change and desired frequency shift is more complex, viz.,
from Eq.
(1) and Eq.
(5),

where the approximation is valid for . In Section 3.5, a simplified approach is proposed based on moving the delay

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