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 (see §4.4) or other techniques for implementation of fractional delay [269,386].
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. (5.6) to Eq. (5.2), 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. (5.2) and Eq. (5.6),
where the approximation is valid for . In Section 5.7.4, a simplified approach is proposed based on moving the delay input instead of its output.
The time-varying delay line was described in §5.1. As discussed there, to implement a continuously varying delay, we add a ``delay growth parameter'' g to the delayline function in Fig.5.1, and change the line
rptr += 1; // pointer updateto
rptr += 1 - g; // pointer updateWhen g is 0, we have a fixed delay line, corresponding to in Eq. (5.6). When , the delay grows samples per sample, which we may also interpret as seconds per second, i.e., . By Eq. (5.7), we see that we need
to simulate a listener traveling toward the source at speed .
Note that when the read- and write-pointers are driven directly from a model of physical propagation-path geometry, they are always separated by predictable minimum and maximum delay intervals. This implies it is unnecessary to worry about the read-pointer passing the write-pointers or vice versa. In generic frequency shifters , or in a Doppler simulator not driven by a changing geometry, a pointer cross-fade scheme may be necessary when the read- and write-pointers get too close to each other.