Time-Varying Delay-Line Reads

If $x(t)$ denotes the input to a time-varying delay, the output can be written as

$$y(t)=x(t-D_t).$$

where $D_t$ denotes the time-varying delay in seconds.

Frequency shift of a complex sinusoid:

$$x(t) = e^{j\omega_s t}$$

The output is now

$$y(t)= x(t-D_t) = e^{j\omega_s \cdot (t-D_t)}.$$

Instantaneous phase:

$$\theta(t)= \angle y(t) = \omega_s \cdot(t-D_t)$$

Differentiate to get instantaneous frequency:

$$\omega_l = \omega_s ( 1 - {\dot D_t}) \protect$$

where

• $\omega_l$ denotes the output frequency, and
• ${\dot D_t}\mathrel{\stackrel{\mathrm{\Delta}}{=}}\frac{d}{dt}D_t$ denotes the time derivative of $D_t$

Thus, the delay growth-rate, ${\dot D_t}$, equals the relative frequency downshift:

$${\dot D_t}= \frac{\omega_s -\omega_l }{\omega_s }.$$

Thus, time-varying delay most naturally simulates Doppler shift caused by a moving listener, with

$${\dot D_t}= -\frac{v_{ls}}{c}. \protect$$

That is, the delay growth-rate, ${\dot D_t}$, should be set to the speed of the listener away from the source, normalized by sound speed $c$.

For a moving source, use time-varying delay-line writes.