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 bandlimited interpolation  or other techniques for implementation of fractional delay [9,13].
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
Comparing Eq. (5) to Eq. (1), we find that the time-varying delay most naturally simulates Doppler shift caused by a moving listener, with
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 input instead of its output.