## 5 Results
### Frequency-Domain Behavior  
The spectrogram in Figure 11 shows the effect of increasing the value of \(R_{\text{L}}\) in the Lockhart wavefolder model for a constant 150-Hz sinusoidal input. As expected, the level of harmonic distortion introduced by the circuit is proportional to the value of this resistance. Therefore, this parameter can be used for additional timbral control. It should be noted that due to the antisymmetric nature of the folding function, the system introduces odd harmonics only for input signals centered around zero. Since the level of harmonics introduced by the [PERSON] wavefolder depends on the choice of \(R_{\text{L}}\), we consider the highest recommended case \(R_{\text{L}}=50\,\text{k}\Omega\) as a worst-case scenario in terms of aliasing distortion. Figure 12 shows the spectrograms of a linear sweep from 20 Hz-5 kHz with peak amplitude of 1 V processed by the proposed [PERSON] and [PERSON] wavefolder models. This frequency range was chosen as it covers all fundamental frequencies of musical interest. A sample rate \(F_{\mathrm{s}}=1\) MHz was used to generate these figures in order to simulate an ideal alias-free continuous-time behavior. These results will be used as a reference when evaluating the performance of the proposed antialiased forms. From these spectrograms we can observe how the Lockhart wavefolder is capable of generating brighter sounds. This perceptual attribute can be varied by changing the value of \(R_{\mathrm{L}}\). Figure 13a shows the result of processing the same linear sweep using the trivial (i.e., non-antialiased) Lockhart model at a standard audio rate of \(F_{\mathrm{s}}=44.1\) kHz. When compared to Figure 12a, we can clearly observe the high levels of aliasing distortion introduced by the model. This is somewhat ameliorated in Figure 13b, where the sweep has been processed at a sample rate of \(F_{\mathrm{s}}=88.2\) kHz (i.e., two-times oversampling). Figure 13c,d shows the result of processing the sweep using the proposed antialiasing method at audio rate and with two-times oversampling, respectively. As shown in these spectrograms, there is a significant reduction in aliasing, particularly below the fundamental frequency. This behavior is advantageous in music applications because at low frequencies the audibility of aliasing distortion is only limited by the hearing threshold. On the other end of the spectrum, the masking effects of harmonics will help suppress the audible effects of high-frequency aliases [65]. The spectrograms in Figure 14a,b show the outcome of processing the 1 V linear sweep with the proposed [PERSON] wavefolder model at audio rate and with two-times oversampling, respectively. When compared with Figure 13a,b it is evident that the Serge wavefolder model is less susceptible to aliasing distortion. This can be attributed to the fact that its transfer function is not as sharp as that of the Lockhart, particularly when \(R_{\mathrm{L}}=50\) k\(\Omega\). Figure 14c,d shows the result of processing the linear sweep using the antiderivative method at audio rate and with oversampling by two. In this case, operating at audio rate yields very effective results as there are very few visible aliases left below the fundamental. When combined with oversampling by two the antiderivative method produces a nearly alias-free spectrum for the measured frequency range. Figure 11: Spectrogram of the Lockhart wavefolder under 150-Hz sinusoidal input for values of \(R_{\mathrm{L}}\) between 1 and 50 k\(\Omega\). Figure 12: Spectogram for a linear sweep from 20 Hz to 5 kHz processed using: (**a**) the proposed Lockhart wavefolder model (\(R_{\mathrm{L}}=50\) k\(\Omega\)); and (**b**) the proposed [PERSON] wavefolder model. A sample rate \(F_{\mathrm{s}}=1\) MHz was used to simulate analog behavior. The performance of the proposed antialiasing method was further evaluated by computing the A-weighted noise-to-mask ratio (ANMR) for a set of sinusoidal input signals processed by both wavefolder models.