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 (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 $x(t)$ 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

$$\omega_l = \omega_s ( 1 - {\dot D_t}) \protect$$ (6.6)

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.(5.6) to Eq.(5.2), we find that the time-varying delay most naturally simulates Doppler shift caused by a moving listener, with

$${\dot D_t}= -\frac{v_{ls}}{c}. \protect$$ (6.7)

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.(5.2) and Eq.(5.6),

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

where the approximation is valid for $v_{sl}\ll c$ . In Section 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 update

to

  rptr += 1 - g; // pointer update

When g is 0, we have a fixed delay line, corresponding to ${\dot D_t}=0$ in Eq.(5.6). When $\texttt{g}>0$ , the delay grows $\texttt{g}$ samples per sample, which we may also interpret as seconds per second, i.e., ${\dot D_t}=\texttt{g}$ . By Eq.(5.7), we see that we need

$$\texttt{g} = -\frac{v_{ls}}{c}$$

to simulate a listener traveling toward the source at speed $v_{ls}$ .

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 [277], 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.