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Time-Varying Delay Effects

In the category of delay effects, variable delay lines are used for (While reverberators need not be time varying, nowadays they typically are [104,58].)

In digital waveguide synthesis, variable delay lines are used for After an implementation note on time-varying delay lines, we will look at a number of time-varying delay effects requiring variable delay lines. The use of variable delay lines in digital waveguide models will be taken up Chapter 6.

Variable Delay Lines

Time varying delay lines are fundamental building blocks for delay effects, synthesis algorithms, and computational acoustic models of musical instruments. Let A denote an array of length $ N$. Then we can implement an $ M$-sample variable delay line in the C programming language as shown in Fig.5.1. We require, of course, $ M\leq N$.
Figure 5.1: The $ M$-sample variable delay line using separate read- and write-pointers.

   static double A[N];
   static double *rptr = A; // read ptr
   static double *wptr = A; // write ptr

   double setdelay(int M) {
       rptr = wptr - M;
       while (rptr < A) { rptr += N }

   double delayline(double x)
     double y;
     A[wptr++] = x;
     y = A[rptr++];
     if ((wptr-A) >= N) { wptr -= N }
     if ((rptr-A) >= N) { rptr -= N }
     return y;

The Synthesis Tool Kit, Version 4 [86] contains the C++ class ``Delay'' which implements this type of variable (but non-interpolating) delay line. There are additional subclasses which provide interpolating reads by various methods. In particular, the class DelayL implements continuously variable delay lengths using linear interpolation. The code listing in Fig.5.1 can be modified to use linear interpolation by replacing the line
  y = A[rptr++];
  long rpi = (long)floor(rptr);
  double a = rptr - (double)rpi;
  y = a * A[rpi] + (1-a) * A[rpi+1];
  rptr += 1;
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
above to
  rptr += 1 - g; // pointer update
When g is 0, we have a fixed delay line. 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}$. In §5.7.2, this will be applied to simulation of the Doppler effect.

Doubling and Slap-Back

The doubling effect is a studio recording technique often used to ``thicken'' vocals in which the same part is sung twice by the same person. In other words, doubling is a ``chorus of two'', where both parts are sung ``in unison'' by the same person. As an example, the Beatles used doubling very often, such as on the track ``Hard Day's Night''. A single variable delay line can simulate doubling very effectively. The related term slap back refers to the use of a single echo on a recorded track. The echo is often placed in a different spatial location in the stereo mix. Normally the echo delay is just large enough to be heard as a discrete echo on careful listening (e.g., on the order of tens of milliseconds). Compared to doubling, slap back uses a larger delay relative to the first voice, and the delay need not vary.


Flanging is a delay effect that has been available in recording studios since at least the 1960s. Surprisingly little literature exists, although there is some [32,429,59,17,104,242].6.1 The term ``flanging'' is said to have arisen from the way the effect was originally achieved by two tape machines set up to play the same tape in unison, with their outputs added together (mixed equally), as shown in Fig.5.2. To achieve the flanging effect, the flange of one of the supply reels is touched lightly to make it play a littler slower. This causes a delay to develop between the two tape machines. The flange is released, and the flange of the other supply reel is touched lightly to slow it down. This causes the delay to gradually disappear and then begin to grow again in the opposite direction. The delay is kept below the threshold of echo perception (e.g., only a few milliseconds in each direction). The process is repeated as desired, pressing the flange of each supply reel in alternation. The flanging effect has been described as a kind of ``whoosh'' passing subtly through the sound.6.2The effect is also compared to the sound of a jet passing overhead, in which the direct signal and ground reflection arrive at a varying relative delay [59]. If flanging is done rapidly enough, an audible Doppler shift is introduced which approximates the ``Leslie'' effect commonly used for organs (see §5.9).
Figure 5.2: Two tape machines configured to produce a flanging effect.
Flanging is modeled quite accurately as a feedforward comb filter, as discussed in §2.6.1, in which the delay $ M$ is varied over time. Figure 5.3 depicts such a model. The input-output relation for a basic flanger can be written as

$\displaystyle y(n) = x(n) + g x[n-M(n)] \protect$ (6.1)

where $ x(n)$ is the input signal amplitude at time $ n=0,1,2,\dots$, $ y(n)$ is the output at time $ n$, $ g$ is the ``depth'' of the flanging effect, and $ M(n)$ is the length of the delay-line at time $ n$. The delay length $ M(n)$ is typically varied according to a triangular or sinusoidal waveform. We may say that the delay length is modulated by an ``LFO'' (Low-Frequency Oscillator). Since $ M(n)$ must vary smoothly over time, it is clearly necessary to use an interpolated delay line to provide non-integer values of $ M$ in a smooth fashion.
Figure 5.3: The basic flanger effect.
As shown in Fig.2.25, the frequency response of Eq.$ \,$(5.1) has a ``comb'' shaped structure. For $ g>0$, there are $ M$ peaks in the frequency response, centered about frequencies

$\displaystyle \omega^{(p)}_k = k \frac{2\pi}{M}, \quad k=0,1,2,\dots,M-1.

For $ g=1$, the peaks are maximally pronounced, with $ M$ notches6.3occurring between them at frequencies $ \omega^{(n)}_k = \omega^{(p)}_k
+ \pi/M$. As the delay length $ M$ is varied over time, these ``comb teeth'' squeeze in and out like the pleats of an accordion. As a result, the spectrum of any sound passing through the flanger is ``massaged'' by a variable comb filter. As is evident from Fig.2.25, at any given time there are $ M(n)$ notches in the flanger's amplitude response (counting positive- and negative-frequency notches separately). The notches are thus spaced at intervals of $ f_s/M$ Hz, where $ f_s$ denotes the sampling rate. In particular, the notch spacing is inversely proportional to delay-line length. The time variation of the delay-line length $ M(n)$ results in a ``sweeping'' of uniformly-spaced notches in the spectrum. The flanging effect is thus created by moving notches in the spectrum. Notch motion is essential for the flanging effect. Static notches provide some coloration to the sound, but an isolated notch may be inaudible [139]. Since the steady-state sound field inside an undamped acoustic tube has a similar set of uniformly spaced notches (except at the ends), a static row of notches tends to sound like being inside an acoustic tube.

Flanger Speed and Excursion

As mentioned above, the delay-line length $ M(n)$ in a digital flanger is typically modulated by a low-frequency oscillator (LFO). The oscillator waveform is usually triangular, sinusoidal, or exponential (triangular on a log-frequency scale). In the sinusoidal case, we have the following delay variation:

$\displaystyle M(n) = M_0 \cdot \left[1 + A\sin(2\pi f n T)\right]

where $ f$ is the ``speed'' (or ``rate'') of the flanger in cycles per second, $ A$ is the ``excursion'' or ``sweep'' (maximum delay swing) which is often not brought out as a user-controllable parameter, and $ M_0$ is the average delay length controlling the average notch density (also not normally brought out as a user-controllable parameter).

Flanger Depth Control

To obtain a maximum effect, the depth control, $ g$ in Fig.5.3, should be set to 1. A depth of $ g=0$ gives no effect.

Flanger Inverted Mode


Flanger Feedback Control

Many modern commercial flangers have a control knob labeled ``feedback'' or ``regen.'' This control sets the level of feedback from the output to the input of the delay line, thereby creating a feedback comb filter in addition to the feedforward comb filter, in the same manner as in the creation of a Schroeder allpass filter in §2.8.1 (see Fig.2.30).

Summary of Flanging

In view of the above, we may define a flanger in general as any filter which modulates the frequencies of a set of uniformly spaced notches and/or peaks in the frequency response. The main parameters are
  • Depth $ g\in[0,1]$ -- controlling notch depth
  • Speed $ f$ -- speed of notch movement
  • Phase -- switch to subtract instead of adding the direct signal with the delayed signal
Possible additional parameters include
  • Average Delay $ M_0$
  • Excursion or Sweep $ A$ -- amount by which the delay-line grows or shrinks
  • Feedback or Regeneration $ a_M \in(-1,1)$ -- feedback coefficient from output to input
Note that flanging provides only uniformly spaced notches. This can be considered non-ideal for several reasons. First, the ear processes sound over a frequency scale that is more nearly logarithmic than linear [459]. Therefore, exponentially spaced notches (uniformly spaced on a log frequency scale) should sound more uniform perceptually. Secondly, the uniform peaks and notches of the flanger can impose a discernible ``resonant pitch'' on the program material, giving the impression of being inside a resonant tube. Third, when $ g<0$ (inverted flanging), it is possible for a periodic tone to be completely annihilated by harmonically spaced notches if the harmonics of the tone are unlucky enough to land exactly on a subset of the harmonic notches. In practice, exact alignment is unlikely; however, the signal loudness can be modulated to a possibly undesirable degree as the notches move through alignment with the signal spectrum. For this reason, flangers are best used with noise-like or inharmonic sounds. For harmonic signals, it makes sense to consider methods for creating non-uniform moving notches. A Faust software implementation of flanging may be found in the file effect.lib within the Faust distribution [154,170]. The Faust programming example phaser_flanger.dsp may be run to hear the effect on a test signal and experiment with its parameters in real time.


The phaser, or phase shifter, is closely related to the flanger in that it also works by sweeping notches through the spectrum of the input signal. While the term phasing is sometimes used synonymously with flanging [32],6.4 typical commercial phase shifters have been observed to implement nonuniformly spaced notches.6.5 Furthermore, phasers such as the Univibe (1960s) were intended to simulate the Leslie rotating speaker effect (§5.9) as opposed to being a low-cost analog flanging approximation. In other words, the conceptual unification of phasing and flanging seems to be technical in nature; i.e., based on the fact that both effects operate by sweeping notches through the spectrum. Flangers are constrained to have an ``infinite series'' of harmonically spaced notches (§2.6.3), while phasers have a limited number of nonuniformly spaced notches. In both cases motion of the notches over time is essential to the effect, and this motion is classically periodic. We will therefore define a phaser as any linear filter which modulates the frequencies of a set of non-uniformly spaced notches, while a flanger will remain any device which modulates uniformly spaced notches. Digital implementations of phase shifters are discussed in §8.9. A Faust phase-shifter may be found in the file effect.lib within the Faust software distribution [154,170]. The Faust programming example phaser_flanger.dsp demonstrates the phasing effect.

Vibrato Simulation

The term vibrato refers to small, quasi-periodic variations in the pitch of a tone. On a violin, for example, vibrato is produced by wiggling the finger stopping the string on the fingerboard; a violin vibrato frequency can be very slow, or a bit faster than 6 Hz. A typical vibrato depth is on the order of 1 percent (a semitone is $ 2^{1/12}\approx 6$ percent). In the singing voice, vibrato is produced by modulating the tension of the vocal folds. Vibrato is typically accompanied by tremolo, which is amplitude modulation at the same frequency as the vibrato which causes it. For example, in the violin, the frequency-modulations of the string vibrations are translated into amplitude modulations by the complex variations in the frequency response of the violin body. To apply vibrato to a sound, it is necessary to apply a quasi-periodic frequency shift. This can be accomplish using a modulated delay line. This works because a time-varying delay line induces a simulated Doppler shift on the signal within it. The flanger in effect.lib (Faust distribution) has a vibrato mode in which it becomes a pure time-varying delay line. This mode can be accessed via a checkbox in the example phaser_flanger.dsp.

Doppler Effect

The Doppler effect causes the pitch of a sound source to appear to rise or fall due to motion of the source and/or listener relative to each other. You have probably heard the pitch of a horn drop lower as it passes by (e.g., from a moving train). As a pitched sound-source moves toward you, the pitch you hear is raised; as it moves away from you, the pitch is lowered. The Doppler effect has been used to enhance the realism of simulated moving sound sources for compositional purposes [80], and it is an important component of the ``Leslie effect'' (described in §5.9). As derived in elementary physics texts, the Doppler shift is given by

$\displaystyle \omega_l = \omega_s \frac{1+\frac{v_{ls}}{c}}{1-\frac{v_{sl}}{c}} \protect$ (6.2)

where $ \omega_s $ is the radian frequency emitted by the source at rest, $ \omega_l $ is the frequency received by the listener, $ v_{ls}$ denotes the speed of the listener relative to the propagation medium in the direction of the source, $ v_{sl}$ denotes the speed of the source relative to the propagation medium in the direction of the listener, and $ c$ denotes sound speed. Note that all quantities in this formula are scalars.

Vector Formulation

Denote the sound-source velocity by $ \underline{v}_s(t)$ where $ t$ is time. Similarly, let $ \underline{v}_l(t)$ denote the velocity of the listener, if any. The position of source and listener are denoted $ \underline{x}_s(t)$ and $ \underline{x}_l(t)$, respectively, where $ \underline{x}\isdef (x_1,x_2,x_3)^T$ is 3D position. We have velocity related to position by

$\displaystyle \underline{v}_s= \frac{d}{dt}\underline{x}_s(t) \qquad \underline{v}_l= \frac{d}{dt}\underline{x}_l(t). \protect$ (6.3)

Consider a Fourier component of the source at frequency $ \omega_s $. We wish to know how this frequency is shifted to $ \omega_l $ at the listener due to the Doppler effect. The Doppler effect depends only on velocity components along the line connecting the source and listener [349, p. 453]. We may therefore orthogonally project the source and listener velocities onto the vector $ \underline{x}_{sl}=\underline{x}_l-\underline{x}_s$ pointing from the source to the listener. (See Fig.5.8 for a specific example.) The orthogonal projection of a vector $ \underline{x}$ onto a vector $ {\underline{y}}$ is given by [451]

$\displaystyle {\cal P}_{\underline{y}}(\underline{x}) = \frac{\left<\underline{...

Therefore, we can write the projected source velocity as

$\displaystyle \underline{v}_{sl}= {\cal P}_{\underline{x}_{sl}}(\underline{v}_s...
...line{x}_s\,\right\Vert^2}\left(\underline{x}_l-\underline{x}_s\right). \protect$ (6.4)

In the far field (listener far away), Eq.$ \,$(5.4) reduces to

$\displaystyle \underline{v}_{sl} \approx \frac{\left<\underline{v}_s,\underline...
...derline{x}_l\,\right\Vert\gg\left\Vert\,\underline{x}_s\,\right\Vert). \protect$ (6.5)

Doppler Simulation

It is well known that a time-varying delay line results in a frequency shift. Time-varying delay is often used, for example, to provide vibrato and chorus effects [17]. We therefore expect a time-varying delay-line to be capable of precise Doppler simulation. This section discusses simulating the Doppler effect using a variable delay line [468].6.6 Consider Doppler shift from a physical point of view. The air can be considered as analogous to a magnetic tape which moves from source to listener at speed $ c$ (see Fig.5.4). The source is analogous to the write-head of a tape recorder, and the listener corresponds to the read-head. When the source and listener are fixed, the listener receives what the source records. When either moves, a Doppler shift is observed by the listener, according to Eq.$ \,$(5.2).6.7
Figure 5.4: Magnetic tape analogy.

Doppler Simulation via Delay Lines

Figure 5.5: Delay line with variable read-pointers and write-pointers.
This analogy also works for a delay-line based computational model, as depicted in Fig.5.5. The magnetic tape is now the delay line, the tape read-head is the read-pointer of the delay line, and the write-head is the delay-line write-pointer. In this analogy, it is readily verified that modulating delay by changing the read-pointer increment from 1 to $ 1+v_{ls}/c$ (thereby requiring interpolated reads) corresponds to listener motion away from the source at speed $ v_{ls}$. It also follows that changing the write-pointer increment from $ 1$ to $ 1+v_{sl}/c$ corresponds source motion toward the listener at speed $ v_{sl}$. When this is done, we must use interpolating writes into the delay memory. Interpolating writes may be called de-interpolation [502], and they are formally the graph-theoretic transpose of interpolating reads (ordinary ``interpolation'') [333]. A review of time-varying, interpolating, delay-line reads and writes, together with a method using a single shared pointer, are given in [383].

Time-Varying Delay-Line Reads

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

$\displaystyle 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 [267,383]. Let's analyze the frequency shift caused by a time-varying delay by setting $ x(t)$ to a complex sinusoid at frequency $ \omega_s $:

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

The output is now

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

The instantaneous phase of this signal is

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

which can be differentiated to give the instantaneous frequency

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

$\displaystyle {\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

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

$\displaystyle {\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
  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

$\displaystyle \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 [275], 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.

Multiple Read Pointers

Using multiple read pointers, multiple listeners can be simulated. Furthermore, each read-pointer signal can be filtered to simulate propagation losses and radiation characteristics of the source in the direction of the listener. The read-pointers can move independently to simulate the different Doppler shifts associated with different listener motions and relative source directions.

Multiple Write Pointers

It is interesting to consider also what effects can be achieved using multiple interpolating write pointers. From the considerations in §5.7.1, we see that multiple write-pointers correspond to multiple write-heads on a magnetic tape recorder. If they are arranged at a fixed spacing, they are equivalent to multiple read pointers, providing a basic multipath simulation. If, however, the write pointers are moving independently, they induce independent Doppler shifts due to source motion. In particular, each write-pointer can lay down a signal from a separate source to a single listener with its own Doppler shift. Furthermore, each write-signal can be passed through its own filter. Such an individualized source filter can implement all filtering incurred along the propagation path from each source to the listener. When all write pointers have the same input signal, their filters can be implemented using a series chain in which the outputs of successive filters in the chain correspond to progressively longer propagation paths (progressively more filtering). Such an implementation can greatly reduce the filter order required for propagation paths longer than the shortest. The write-pointers may cross each other with no ill effects, since all but the first6.8 simply sum into the shared delay line. We have seen that a single delay line can be used to simulate any number of moving listeners (§5.7.3) or any number of moving sources. However, when simulating both multiple listeners and multiple sources, it is not possible to share a single delay line. This is because the different listeners do not see the same Doppler shift for each moving source, and while the listener's read-pointer motion can be adjusted to correct for the Doppler shift seen from any particular source, it cannot correct for more than one in general. Thus, in general, we need as many delay lines as there are sources or listeners, whichever is smaller. More precisely, if there are $ N$ moving sources and $ M$ moving listeners, simulation requires $ \min(N,M)$ delay lines.

Stereo Processing

As a special case, stereo processing of any number of sources can be accomplished using two delay lines, corresponding to left and right stereo channels. The stereo mix may contain a panned mixture of any number sources, each with its own stereo placement, path filtering, and Doppler shift. The two stereo outputs may correspond to ``left and right ears,'' or, more generally, to left- and right-channel microphones in a studio recording set-up.

System Block Diagram

Figure 5.6: Block diagram of a stereo simulator for any number of moving sound sources (from [468]).

Chorus Effect

frequency shift, and
amplitude modulation.

The Leslie

The Leslie, named after its inventor, Don Leslie,6.9 is a popular audio processor used with electronic organs and other instruments [59,189]. It employs a rotating horn and rotating speaker port to ``choralize'' the sound. Since the horn rotates within a cabinet, the listener hears multiple reflections at different Doppler shifts, giving a kind of chorus effect. Additionally, the Leslie amplifier distorts at high volumes, producing a pleasing ``growl'' highly prized by keyboard players. At the time of this writing, there is a nice Leslie Wikipedia page, including a stereo sound-example link under the first picture (that is best heard in headphones). Papers on computational audio models of the Leslie include [468,191]. The Leslie consists primarily of a rotating horn and a rotating speaker port inside a wooden cabinet enclosure [189].

Rotating Horn Simulation

The heart of the Leslie effect is a rotating horn loudspeaker. The rotating horn from a Model 600 Leslie can be seen mounted on a microphone stand in Fig.5.7. Two horns are apparent, but one is a dummy, serving mainly to cancel the centrifugal force of the other during rotation. The Model 44W horn is identical to that of the Model 600, and evidently standard across all Leslie models [189]. For a circularly rotating horn, the source position can be approximated as

$\displaystyle \underline{x}_s(t) = \left[\begin{array}{c} r_s\cos(\omega_m t) \\ [2pt] r_s\sin(\omega_m t) \end{array}\right] \protect$ (6.8)

where $ r_s$ is the circular radius and $ \omega_m $ is angular velocity. This expression ignores any directionality of the horn radiation, and approximates the horn as an omnidirectional radiator located at the same radius for all frequencies. In the Leslie, a conical diffuser is inserted into the end of the horn in order to make the radiation pattern closer to uniform [189], so the omnidirectional assumption is reasonably accurate.6.10
Figure 5.7: Rotating horn recording set up (from [468]).
By Eq.$ \,$(5.3), the source velocity for the circularly rotating horn is

$\displaystyle \underline{v}_s(t) = \frac{d}{dt}\underline{x}_s(t) = \left[\begi...\omega_m t) \\ [2pt] r_s\omega_m\cos(\omega_m t) \end{array}\right] \protect$ (6.9)

Note that the source velocity vector is always orthogonal to the source position vector, as indicated in Fig.5.8.
Figure 5.8: Relevant geometry for a rotating horn (from [468]).
Since $ \underline{v}_s$ and $ \underline{x}_s$ are orthogonal, the projected source velocity Eq.$ \,$(5.4) simplifies to


Arbitrarily choosing $ \underline{x}_l=(r_l,0)$ (see Fig.5.8), and substituting Eq.$ \,$(5.8) and Eq.$ \,$(5.9) into Eq.$ \,$(5.10) yields

$\displaystyle \underline{v}_{sl}= \frac{-r_l r_s\omega_m\sin(\omega_m t)}{r_l^2...
...l-r_s\cos(\omega_m t) \\ [2pt] -r_s\sin(\omega_m)t \end{array}\right]. \protect$ (6.11)

In the far field, this reduces simply to

$\displaystyle \underline{v}_{sl}\approx -r_s\omega_m\sin(\omega_m t) \left[\begin{array}{c} 1 \\ [2pt] 0 \end{array}\right]. \protect$ (6.12)

Substituting into the Doppler expression Eq.$ \,$(5.2) with the listener velocity $ v_l$ set to zero yields

$\displaystyle \omega_l = \frac{\omega_s }{1+r_s\omega_m\sin(\omega_m t)/c} \approx \omega_s \left[1-\frac{r_s\omega_m}{c}\sin(\omega_m t)\right], \protect$ (6.13)

where the approximation is valid for small Doppler shifts. Thus, in the far field, a rotating horn causes an approximately sinusoidal multiplicative frequency shift, with the amplitude given by horn length $ r_s$ times horn angular velocity $ \omega_m $ divided by sound speed $ c$. Note that $ r_s\omega_m $ is the tangential speed of the assumed point of horn radiation.

Rotating Woofer-Port and Cabinet

It is straightforward to extend the above computational model to include the rotating woofer port (``baffle'') and wooden cabinet enclosure as follows:
  • In [189], it is mentioned that an AM ``throb'' is the main effect of the rotating woofer port. A modulated lowpass-filter cut-off frequency has been used for this purpose by others. Measured data can be used to construct angle-dependent filtering in a manner analogous to that of the rotating horn, and this ``woofer filter'' runs in parallel with the rotating horn model.
  • The Leslie cabinet multiply-reflects the sound emanating from the rotating horn. The first few early reflections are simply handled as additional sources in Fig.5.6.
  • To qualitatively simulate later, more reverberant reflections in the Leslie cabinet, we may feed a portion of the rotating-horn and speaker-port signals to separate states of an artificial reverberator (see Chapter 3). This reverberator may be configured as a ``very small room'' corresponding to the dimensions and scattering characteristics of the Leslie cabinet, and details of the response may be calibrated using measurements of the impulse response of the Leslie cabinet. Finally, in order to emulate the natural spatial diversity of a radiating Leslie cabinet in a room, ``virtual cabinet vent outputs'' can be extracted from the model and fed into separate states of a room reverberator. An alternative time-varying FIR filtering approach based on cabinet impulse-response measurements over a range of horn angles is described in [191].
In summary, we may use multiple interpolating write-pointers to individually simulate the early cabinet reflections, and a ``Leslie cabinet'' reverberator for handling later reflections more statistically.

Recent Research Modeling the Leslie

As mentioned above, modeling the Leslie via interpolating delay-line writes and cabinet image-sources was described in [468].6.11More recently, Leslie simulation via time-varying FIR filtering has been developed [191]. See these papers and their cited references for further details.
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