Figure 1: Circuit diagram for the Moog VCF from original patent [1]

Figure 2: Block diagram of Moog VCF in continuous time [2]

The low-pass filter block seen in Figure 1 is actually four stages of low-pass filters. Each transistor pair forms a low-pass filter. However, all stages are tuned to a single cutoff frequency, seen in Figure 2 as G₁(s). Part of the “voltage controlled” aspect of the filter is provided by a feedback path. In the circuit this can be seen as a potentiometer in Figure 1,  and can been seen in the block diagram as a gain with a variable value of k in Figure 2. Depending on the value of the potentiometer, you could adjust the resonance at the cutoff frequency set for the filter. When the gain in the feedback path is at its lowest setting, the VCF acts like a typical low-pass filter. As the value increases, resonance in the form of “corner peaking” occurs [2].

Figure 3: Frequency response of Moog VCF for different feedback path gains [2]

Tim Stilson and Julius Smith performed a thorough analysis of the filter in continuous time in their paper “Analyzing the Moog VCF with considerations for digital implementation” before attempting to discretize the filter. One particularly desirable property they found in their analysis was that if they set the feedback gain (k) to a constant value and swept the cutoff frequency control, the filter exhibited constant quality factor (Q). As well, the control for feedback gain and the control for cutoff frequency were uncoupled, so changing one would not change the other. They sought to preserve these good properties that the filter had in continuous time when they attempted to discretize the filter [2].

Stilson and Smith model

Stilson and Smith attempted to discretize the transfer function for each low-pass filter section using the Bilinear transform and the Backwards Difference transform. They set out three design goals to meet: that the controls were uncoupled, that the controls changed useful parameters (such as frequency and Q), and that the controls were efficiently implemented without too much computational overhead. The results from the two transformations did not meet their design criteria, mainly due to the addition of a delay term in the feedback loop to help computability. However, when they compared the results of the two filters, they noticed something interesting. For the bilinear transformed filter, Q increased as cutoff frequency increased. For the backwards difference transformed filter, Q fell as cutoff frequency increased. The only fundamental difference between the two was the location of the zeros. The bilinear transformed filter had a zero at z = -1  and the backwards difference transformed filter had a zero at z = 0. They attempted to find a “compromise” between the two which would yield constant Q. They found one that was very close to meeting this requirement which had a zero at z=-0.3 [2]. The following is the transfer function for one low-pass filter section:

Where the following is the overall transfer function of the filter with the feedback loop:

This “compromise” filter met their design requirements to a certain extent. It was then used as the basis for further research by Välimäki and Huovilainen to improve upon the filter.

Välimäki and Huovilainen model

Välimäki and Huovilainen had analyzed the filter further and had developed some adjustments to Stilson and Smith’s compromise filter. This allowed them to more accurately model the Moog VCF. They proposed the following block diagram for each low-pass filter section:

Figure 4: Block diagram for proposed change to each low-pass filter section [3]

Where the value of g is defined in the paper [3] as:

As well they proposed the following block diagram as an improved way of modeling the Moog VCF:

Figure 5: Block diagram for proposed model for Moog VCF [3]

Where G_res is defined in the paper [3] as:

And C_res and G_comp can be values between 0 and 1. The values A, B, C, D, and E can be used to obtain different filters, such as band-pass and high-pass filters, but in the case of the 4 pole low-pass filter, the values of A, B, C, and D are set to 0 and the value of E is 1. As well, to accurately model the non-linearities in the original Moog VCF, a waveshaper block is added before the input to the first low-pass filter section. This waveshaper can be any “smoothly saturating function”; Välimäki and Huovilainen used a hyperbolic tangent function [3].

Using this difference equation, the transfer function was derived, and it was determined that it did in fact maintain the zero at z = -0.3 from the Stilson and Smith compromise filter:

for i = 1:N
    %Input to first LP filter stage (after waveshaping)
    input = tanh(x(i) - 4*Gres*(ym1(4) - Gcomp * x(i)));
    % Send input through 4 stages of LP filtering
    for n = 1:4
        %Output of each stage
        output = ((glp1*input + glp2*xm1(n) - ym1(n))*g) + ym1(n);
        xm1(n) = input;     %store x[n-1] for each stage
        ym1(n) = output;    %store y[n-1] for each stage
        input = output;     %input to next stage is this stage's output
    end
    
    y(i) = output;
end

Figure 6: Frequency response for multiple G_comp values. f_c = 500 Hz C_res = 0.9

From this plot it was clear there was some effect, but it was not very much. The pass-band region was determined to have been effected the greatest by a variation in G_comp and was magnified in the plot below:

Figure 7: Zoomed in view of pass-band for frequency response in Fig. 6

As can be seen, a higher value of G_comp reduces the amount of attenuation in the pass-band and keeps the magnitude more or less constant, but seems to have diminishing returns after G_comp=0.5. The paper recommends to keep the value of G_comp at 0.5 because it results in “a 6-dB pass-band gain decrease at the maximum resonance” [3], so for the rest of the scripts it will remain at a value of 0.5.

Then, a script was written to plot the frequency response for the filter at five different values of C_res between 0 and 1 to see its effect on the filter. The value of G_comp was set to 0.5 and the cutoff frequency value was set to 500 Hz. The following plot was obtained in MATLAB with the CresEffectPlot.m script:

Figure 8: Frequency response for multiple values of C_res. f_c = 500 Hz  G_comp= 0.5

As can be seen from the above figure, the value of C_res controls the amount of resonance that occurs at the cutoff frequency. A value of 0 provides a typical low-pass response without any corner peaking at the cutoff frequency. As C_res increases, the peak becomes more pronounced. However, a value of C_res = 1.0 does not produce a typical attenuation of the frequencies above the cutoff. It was determined that a value of 0.9 would produce a good amount of resonance, while still keeping the typical attenuation of high frequencies.

Then, a script was written to plot the frequency response for the filter at five different values of cutoff frequencies between 250 and 1500 Hz to see if it would move the cutoff frequencies correctly. The value of G_comp was set to 0.5 and the value of C_res was set to 0.9, as established by the previous analyses. The following plot was obtained in MATLAB with the fcEffectPlot.m script:

Figure 9: Frequency response for multiple values of f_c C_res = 0.9  G_comp= 0.5

From this graph, it can be seen that it did indeed move the cutoff frequency as assumed. Peaks in Fig. 9 were found with the Data Cursor tool in MATLAB at: 242.6 Hz, 485.2 Hz, 974.8 Hz, 1182 Hz, and 1473 Hz. This means the maximum amount of resonance is provided at slightly below the specified cutoff frequency, but they are within an acceptable limit.

After analyzing the effects of the frequency response graphically, the filter was tested on audio. Thanks to Pekonen et al., it was possible to pass recordings of the sawtooth waveform from an original Moog oscillator [4]. To download the recordings, see the companion website for the paper and download the reference recordings they have.

Firstly, a recording of the waveform at a fundamental frequency of 495.44 Hz was passed through the discretized Moog lowpass VCF model at several different cutoff frequencies. The following audio file was generated using the MATLAB script fcSweep.m. The audio contains the original waveform, then the filtered waveforms at 1500 Hz, 1200 Hz, 1000 Hz, 500 Hz and finally 250 Hz.