## 3 Results
### The Moog Ladder Filter  
As an example of a musically interesting application of ODES and RK4 in audio synthesis that is not based on harmonic motion, we return to the 'ladder filter' invented by [PERSON] in 1965 [30]. This filter has been subject to numerous studies since [1]. Most digitizations use the bilinear transform [31] or some similar procedure [32] to discretize the linear portion (small signal response) of the analog transfer function. As mentioned above, since there is feedback in the circuit, digitizations of this filter often include unit sample feedback (\(z^{-1}\)). Again, this is problematic because the delay couples the gain parameter \(g\) (which controls the cutoff frequency of the filter) to the resonance parameter \(r\) (which controls the bandwidth of the filter). This causes additional phase shifting so that effect of the resonance parameter varies with frequency and vice versa. Many adjustments have been suggested to ameliorate this artifact of digitization [33] or to avoid the addition of delay to the feedback part of the filter without resorting to iteration. Examples of this include solution through the use of Volterra series [34; 35] and linear compensation filters [36].  
The Moog ladder filter is a four-pole network, wherein the voltage across each stage is defined:  
\[\dot{y}_{i}=\omega(\tanh(y_{i-1})-\tanh(y_{i})) \tag{27}\]  
where \(r\) is the gain that governs the cutoff frequency of the filter, \(\dot{y}_{i}\) is the change in voltage across stage \(i\), \(y_{i}\) is the present voltage across stage \(i\), and \(y_{i-1}\) is the present voltage across the previous stage for \(i=1,2,3\). Ideally, the cutoff frequency has a ratio of \(f_{c}=\omega/2\pi\) Hz for all frequencies.  
Since each stage of the filter shifts the input signal phase by \(\pi/4\) radians, the output of the filter is inverted and fed back into the input to create resonance. So, for \(i=0\) we have:  
\[\dot{y}_{0}=\omega(\tanh(V_{in}-rV_{3})-\tanh(V_{0})) \tag{28}\]  
Figure 8: RK4 solving Equations (25) using (**a**) the continuous friction model in Equation (24) and (**b**) Equation (26). (**a**) Differentiable friction model; (**b**) Discontinuous friction model.  
when \(r=4\), the filter saturates and will oscillate on its own. In the results presented here, we simply plug Equations (27) and (28) into RK4 and grind out the samples. As indicated by the results, no further analysis, linearization, or digitization than this is necessary to achieve good results.  
Figure 9a shows a comparison of measured frequency and amplitude for impulses fed to the filter tuned to various values for the parameter \(f_{c}\). This compare favorably with the results presented in [32] where compensation for the zero introduced by unit delay in the feedback path results in unbounded amplitude growth at higher values of \(f_{c}\). The relationship between \(f_{c}\) and the measured frequency peaks in Figure 9b compares favorably with a similar analysis of another delay-free implementation given in [36]. Here there is noticeable flattening that results from lowering the value of \(r\). This suggests that some coupling between this parameter and \(f_{c}\) persists.