The Leslie

The Leslie, named after its inventor, Don Leslie,¹⁰is a popular audio processor used with electronic organs and other instruments [#!Bode84!#,#!Henricksen81!#]. 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.

The Leslie consists primarily of a rotating horn and a rotating speaker port inside a wooden cabinet enclosure [#!Henricksen81!#]. We first consider the rotating horn.

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. . 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 [#!Henricksen81!#]. For a circularly rotating horn, the source position can be approximated as

$$\underline{x}_s(t) = \left[\begin{array}{c} r_s\cos(\omega_m t) \\ [2pt] r_s\sin(\omega_m t) \end{array}\right] \protect$$ (12)

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 diffuser is inserted into the end of the horn in order to make the radiation pattern closer to uniform [#!Henricksen81!#], so the omnidirectional assumption is reasonably accurate.

By Eq.$\,$(7), the source velocity for the circularly rotating horn is

$$\underline{v}_s(t) = \frac{d}{dt}\underline{x}_s(t) = \left[\begi... ...in(\omega_m t) \\ [2pt] r_s\omega_m\cos(\omega_m t) \end{array}\right] \protect$$ (13)

Note that the source velocity vector is always orthogonal to the source position vector, as indicated in Fig. 1.1.

Since $\underline{v}_s$ and $\underline{x}_s$ are orthogonal, the projected source velocity Eq.$\,$(8) simplifies to

$$\underline{v}_{sl}= {\cal P}_{\underline{x}_{sl}}(\underline{v}_s... ...line{x}_s\,\right\Vert^2}\left(\underline{x}_l-\underline{x}_s\right). \protect$$ (14)

Arbitrarily choosing $\underline{x}_l=(r_l,0)$ (see Fig. 1.1), and substituting Eq.$\,$(12) and Eq.$\,$(13) into Eq.$\,$(14) yields

$$\underline{v}_{sl}= \frac{-r_l r_s\omega_m\sin(\omega_m t)}{r_l^2... ...l-r_s\cos(\omega_m t) \\ [2pt] -r_s\sin(\omega_m)t \end{array}\right]. \protect$$ (15)

In the far field, this reduces simply to

$$\underline{v}_{sl}\approx -r_s\omega_m\sin(\omega_m t) \left[\begin{array}{c} 1 \\ [2pt] 0 \end{array}\right]. \protect$$ (16)

Substituting into the Doppler expression Eq.$\,$(6) with the listener velocity $v_l$ set to zero yields

$$\omega_l = \frac{\omega_s }{1+r_s\omega_m\sin(\omega_m t)/c} \approx \omega_s \left[1-\frac{r_s\omega_m}{c}\sin(\omega_m t)\right], \protect$$ (17)

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.

Leslie Free-Field Horn Measurements

The free-field radiation pattern of a Model 600 Leslie rotating horn was measured using the experimental set-up shown in Fig.  [#!SmithEtAlDAFx02!#]. A matched pair of Panasonic microphone elements (Crystal River Snapshot system) were used to measure the horn response both in the plane of rotation and along the axis of rotation (where no Doppler shift or radiation pattern variation is expected). The microphones were mounted on separate boom microphone stands, as shown in the figure. A close-up of the plane-of-rotation mic is shown in Fig. .

Microphone close-up (from [#!SmithEtAlDAFx02!#]).

The horn was set manually to fixed angles from -180 to 180 degrees in increments of 15 degrees, and at each angle the impulse response was measured using 2048-long Golay-code pairs [#!FosterGolay!#].

Figure shows the measured impulse responses and Fig.  shows the corresponding amplitude responses at the various angles. Note that the beginning of each impulse response contains a fixed portion which does not depend significantly on the angle. This is thought to be due to ``leakage'' from the base of the horn. It arrives first since the straight-line path from the enclosed speaker to the microphone is shorter than that traveling through the horn assembly.

Measured impulse-responses of the Leslie 600 rotating-horn at multiples of 15 degrees. The middle trace is recorded with the microphone along the axis of the horn (from [#!SmithEtAlDAFx02!#]).

Measured amplitude-responses of the Leslie 600 rotating-horn at multiples of 15 degrees (from [#!SmithEtAlDAFx02!#]).

Separating Horn Output from Base Leakage

Note that Fig.  indicates the existence of fixed and angle-dependent components in the measured impulse responses. An iterative algorithm was developed to model the two components separately [#!SmithEtAlDAFx02!#].

Let $M=256$ denote the number of impulse-response samples in each measured impulse response,and let $N=25$ denote the number of angles (-180:15:180) at which impulse-response measurements were taken. We denote the $M\times N$ impulse-response matrix by ${\mathbf{h}}$. Each column of ${\mathbf{h}}$ is an impulse response at some horn angle. (Figure can be interpreted as a plot of the transpose of ${\mathbf{h}}$.)

We model ${\mathbf{h}}$ as

$${\mathbf{h}}=$$   $$\mbox{${\bm \alpha}$}$$$$+$$   $$\mbox{${\bm \gamma}$}$$$$\cdot$$   diag$$(z^{-\tau_i}) + {\mathbf{e}}$$

where $\tau_i$ is the arrival-time delay, in samples, for the horn output in the $i$th row (the delays clearly visible in Fig.  as a function of angle). These arrival times are estimated as the location of the peak in the cross-correlation between the $i$th impulse response and the same impulse response after converting it to minimum phase [#!OppenheimAndSchafer!#]. The diagonal matrix diag$(z^{-\tau_i})$ denotes a shift operator which delays the $i$th column of $\mbox{${\bm \gamma}$}$ by $\tau_i$ samples. Thus, $\mbox{${\bm \gamma}$}$ contains the horn-output impulse response (without the base leakage) shifted to time zero (i.e., the angle-dependent delay is removed). Finally, the error matrix ${\mathbf{e}}$ is to be minimized in the least-squares sense.

Each column of the matrix $\mbox{${\bm \alpha}$}$ contains a copy of the estimated horn-base leakage impulse-response:

   $$\mbox{${\bm \alpha}$}$$$$= \underline{a}\cdot\mathbf{1}^T$$

where $\mathbf{1}^T = [1,1,\dots,1]$.

The estimated angle-dependent impulse-responses in $\mbox{${\bm \gamma}$}$ are modeled as linear combinations of $K=5$ fixed impulse responses, viewed (loosely) as principal components:

   $$\mbox{${\bm \gamma}$}$$$$= {\mathbf{g}}\cdot {\mathbf{w}}$$

where ${\mathbf{g}}$ is the $M\times K$ orthonormal matrix of fixed filters (principal components), and ${\mathbf{w}}$ is a $K\times N$ matrix of weights, found in the usual way by a truncated singular value decomposition (SVD) [#!Golub!#].

Algorithm

To start the separation algorithm, $\mbox{${\bm \gamma}$}$$_0$ is initialized to the zero-shifted impulse response data ${\mathbf{h}}\cdot$diag$(z^{\tau_i})$, ignoring the tails of the base-leakage they may contain. Then $\mbox{${\bm \alpha}$}$$_0$ is estimated as the mean of ${\mathbf{h}}-$$\mbox{${\bm \gamma}$}$$_0$diag$(z^{-\tau_i})$. This mean is then subtracted from ${\mathbf{h}}$ to produce ${\mathbf{b}}_1 = ({\mathbf{h}}-$   $\mbox{${\bm \alpha}$}$$_0)$diag$(z^{-\tau_i})$ which is then then converted to $\mbox{${\bm \gamma}$}$$_1 = {\mathbf{g}}_1 \cdot {\mathbf{w}}_1$ by a truncated SVD. A revised base-leakage estimate $\mbox{${\bm \alpha}$}$$_1$ is then formed as ${\mathbf{h}}-$$\mbox{${\bm \gamma}$}$$_1$diag$(z^{-\tau_i})$, and so on, until convergence is achieved.

Results

Figure plots the $K=5$ weighted principal components identified for the angle-dependent component of the horn radiativity. Each component is weighted by its corresponding singular value, thus visually indicating its importance. Also plotted using the same line type are the zero-lines for each principal component. Note in particular that the first (largest) principal component is entirely positive.

First 5 principal components weighted by their corresponding singular values. Each angle-dependent impulse response is modeled as a linear combination of these angle-independent impulse-response components (from [#!SmithEtAlDAFx02!#]).

Figure shows the complete horn impulse-response model ( $\mbox{${\bm \alpha}$}$$+$   $\mbox{${\bm \gamma}$}$$\cdot$   diag$(z^{-\tau_i})$), overlaid with the original raw data ${\mathbf{h}}$. We see that both the fixed base-leakage and the angle-dependent horn-output response are closely followed by the fitted model.

Overlay of measured (solid) and modeled (dotted) impulse-responses at multiples of 15 degrees (from [#!SmithEtAlDAFx02!#]).

Figure shows the estimated impulse response of the base-leakage component $\underline{a}(n)$, and Fig.  shows the modeled angle-dependent horn-output components $\mbox{${\bm \gamma}$}$ delayed out to their natural arrival times.

Modeled base-leakage impulse-response (angle-independent) (from [#!SmithEtAlDAFx02!#]).

Modeled horn-output impulse-responses at multiples of 15 degrees (from [#!SmithEtAlDAFx02!#]).

Figure shows the average power response of the horn outputs. Also overlaid in that figure is the average response smoothed according to Bark frequency resolution [#!SmithAndAbel99!#]. This equalizer then becomes $H_0(z)$ in Fig. 1.1. The filters $H_{0L}(z)$ and $H_{0R}(z)$ in Fig. 1.1 are obtained by dividing the Bark-smoothed frequency-response at each angle by $H_0(z)$ and designing a low-order recursive filter to provide that equalization dynamically as a function of horn angle. The impulse-response arrival times $\tau_i$ determine where in the delay lines the filter-outputs are to be summed in Fig. 1.1.

Average angle-dependent amplitude response overlaid with Bark-smoothed response to be used as a fixed equalization applied to the source (from [#!SmithEtAlDAFx02!#]).

Figure shows a spectrogram view of the angle-dependent amplitude responses of the horn with $H_0(z)$ (Bark-smoothed curve in Fig. ) divided out. This angle-dependent, differential equalization is used to design the filters $H_{0L}(z)$ and $H_{0R}(z)$ in Fig. 1.1. Note that below 12 Barks or so, the angle-dependence is primarily to decrease amplitude as the horn points away from the listener, with high frequencies decreasing somewhat faster with angle than low frequencies.

Angle-dependent amplitude response divided by Bark-smoothed average response to be used as the basis for design of time-varying, angle-dependent equalization to be applied after $H_0(z)$ (from [#!SmithEtAlDAFx02!#]).