## 6 Practical Synthesis Usage
Here, we propose using the wavefolding structure shown in Figure 19 to expand the synthesis capabilities of the Lockhart wavefolder. This design, while not based on any existing circuit, is comparable to that of the Yusynth Metalizer which also utilizes four Lockhart circuits in series [55]. The following paragraphs describe the sections of this proposed topology. Its frequency-domain behavior is then examined and compared with that of the Serge middle VCM. The blocks labeled \"LWF\" in Figure 19 correspond to the proposed Lockhart wavefolder model. In order for this cascade of Lockhart wavefolders to behave as expected, we need to make sure that the individual folding stages satisfy two criteria. Firstly, the individual folders must provide approximately unity gain for small input values, i.e., below the folding point, and approximately negative unity gain  
Figure 17: (**a**) Transfer function of the proposed Serge middle VCM; and (**b**) its output when driven by a 100-Hz sinewave for \(G_{\rm S}=6\) and zero dc offset. Figure 18: Spectrogram of: (**a**) a 150-Hz sinewave with peak amplitude 1 V processed by the proposed Serge middle VCM with varying gain \(G_{\rm S}\) from 0–6; and (**b**) a 200-Hz sinewave processed with varying gain \(G_{\rm S}\) from 0–3 and dc offset from 0–3 V.  
beyond the folding point. Secondly, each stage should start folding at the same point with respect to its individual input. We can meet these criteria with the proposed Lockhart model by selecting an appropriate value for \(R_{\mathrm{L}}\) and adding static gain stages before and after the folding stages. These gain blocks will also help compensate for the attenuation introduced by the folding operation. First, we choose a value of \(R_{\mathrm{L}}\) for which the Lockhart wavefolder exhibits unity gain for small input values. Having found this resistance value, the pre- and post-gain stages can be determined by measuring the value of \(|V_{\mathrm{out}}|\) at exactly the folding point. The pre-gain is taken to be approximately this value, and the post-gain is taken to be its inverse. In Section 3.1, it was shown that for \(R_{\mathrm{L}}=7.5\,\mathrm{k}\Omega\) the Lockhart wavefolder exhibits approximately unity gain below the folding point. This value leads to pre- and post-gains of approximately 1/3 and 3, respectively. Figure 20a shows the transfer function of the proposed structure measured at the output of the post-gain block. We can observe how the folds introduced by this structure are evenly distributed, unlike those in Figure 17a. As with the Serge middle VCM, timbral control is achieved by modulating the value of \(G_{\mathrm{L}}\) and by adding a dc offset. The static gain blocks ensure the amplitude of the folded output is bounded between approximately \(\pm 1\,\mathrm{V}\) for values of \(G_{\mathrm{L}}\) between \(-10\) and \(10\) (assuming once more that \(V_{\mathrm{in}}\) has a peak amplitude of \(1\,\mathrm{V}\)). Figure 20b shows the time-domain result of processing a 100-Hz sinusoidal input with the proposed structure for \(G_{\mathrm{L}}=10\) and zero dc offset. In this particular design, additional timbral control can be achieved by modulating the value of \(R_{\mathrm{L}}\). Lastly, we add two optional blocks. The first is a \(\tanh()\) function after the post-gain block to model the behavior of an output buffering stage and to limit the range of the output waveform. This \(\tanh()\) block can also be antialiased using the antiderivative method described by Equation (45). The antiderivative of the \(\tanh()\) function is given by \(\log(\cosh())\)[48]. The second optional block is a static one-pole lowpass filter with a cutoff at \(f_{\mathrm{c}}=1.3\,\mathrm{kHz}\) whose purpose is to act as a simple tone control. A similar static filtering stage can be found at the output of the Buchla 259 wavefolder [17]. The s-domain transfer function of this filtering stage is defined as  
\[H(s)=\frac{w_{\mathrm{c}}}{s+w_{\mathrm{c}}}, \tag{51}\]  
where \(w_{\mathrm{c}}=2\pi f_{\mathrm{c}}\).