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. In discrete-time implementations, when $D_t$ is not an integer multiple of the sampling interval, $x(t-D_t)$ may be approximated to arbitrary accuracy (in a finite band) using bandlimited interpolation [17] or other techniques for implementation of fractional delay [9,13].

Let's analyze the frequency shift caused by a time-varying delay by setting $x(t)$ to a complex sinusoid at frequency $\omega_s$:

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

The output is now

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

The instantaneous phase of this signal is

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

which can be differentiated to give the instantaneous frequency

$$\omega_l = \omega_s ( 1 - {\dot D_t}) \protect$$ (5)

where $\omega_l$ denotes the output frequency, and ${\dot D_t}\isdef \frac{d}{dt}D_t$ denotes the time derivative of the delay $D_t$. Thus, the delay growth-rate, ${\dot D_t}$, equals the relative frequency downshift:

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

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

$${\dot D_t}= -\frac{v_{ls}}{c}. \protect$$ (6)

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 $c$.

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),

$${\dot D_t}= - \frac{\frac{v_{ls}}{c} + \frac{v_{sl}}{c}}{1-\frac{v_{sl}}{c}} \approx - \left(\frac{v_{ls}}{c} + \frac{v_{sl}}{c}\right)$$

where the approximation is valid for $v_{sl}\ll c$. In Section 3.5, a simplified approach is proposed based on moving the delay input instead of its output.