CONCLUSION

The full bass response [fig 16] begins at 2.5dB and follows the same low pass slope that we expect from [fig 7]. This response also correctly intersects the full treble response at 1kHz seen in [fig 19].

Taking a closer look at the mid frequency response we can see that indeed we do not obtain a flat frequency response [4]; in [fig 20] we observe a ballpark figure of the specified 7dB loss at the notch (ours is closer to 8dB loss).

Figure 20. Mid Frequency Response 1kHz Notch

CONCLUSION

In conclusion, this project has been successful in achieving results in the digital domain that are extremely close, if not identical to, the original analog tone control and its associated frequency response.

However, it’s important to discuss some of the trade-offs that were made to obtain these results. The main trade-off pertains to the potValue values (�). In order to honour the precise values for (�) specified during the analog filter design process, the argument potValue is rounded to the nearest one decimal place and then (somewhat experimentally) scaled to adhere to the desired range and intermediary values. The discrepancy here is that this rounding produces a non-continuous knob. Instead, this produces a dial capable of only representing a limited number of increments. This could be misleading to a user if they wanted to enter a specific value between 0−1 that exceeds one decimal place, as the function will allow the input but will immediately round it.

For whatever reason, via pre-warping, we didn’t seem to generate cut-off frequencies that were substantially closer to that of our ideal design. What is meant by this, is that if we observe our original analog frequency response [fig 7] we want the junction of the full bass and the full treble response and the notch of the mid frequency to sit at 1kHz.

In [fig 6] when we originally test the analog transfer function of the low and high pass in parallel in MATLAB the points we are expecting to see at 1kHz are sitting around 850Hz. We hypothesise that this discrepancy may be partly attributable to frequency warping and consequently proceed to apply the frequency pre-warping to the time constants (�#,�$) (see full working in sections [3.1] [3.2]). However, the pre-warped values we obtained didn’t take us much closer to our ideal frequency response. It’s unclear whether this is due to some miscalculation of the cut-off frequency or the fact that the non-linearity of analog and digital frequencies is less apparent the further the frequency is from Nyquist [7].

Either way, we attempt to correct this experimentally by testing values for (�#,�$) until a frequency response with the desired characteristics (junction of the full bass and the full treble response and the notch of the mid frequency response @ 1kHz) is achieved. These values are: -

th = 0.0000765; %high pass time constant tl = 0.00029; %low pass time constant