Tone-Stack
The tone stack is the three knobs, labeled low, mid, and treble, on the top of the amplifier. It acts as equalization which is just applying filters to a signal to change the amount of certain frequencies in the signal. As you can see if Figure 1, the three knobs have levels from 1-10.
Figure 1.
We know that this type of filtering is generally linear time-invariant, so in order to analyze the effect of these knobs, we ran a white noise signal (full frequency at a constant magnitude) signal through the amplifier and took measurements with each knob at levels 1-10. We recorded in the Audio Studio in the Duderstadt and used several different microphones and placements since these are both factors that could skew our data. Figure 2 shows our recording process and microphone choices and placements.
Figure 2.
We have taken Fast Fourier Transforms in Matlab of the many audio segments we have recorded to understand their frequency content since this is the most important aspect being affected in this part of the signal chain. Figures 3-5 show the effects of each of the tone knobs at varying levels. The spectrogram images each have ten resulting pulses (seen separated by the darker lines) of impulse through the amp at varying gain levels of the tone knob. the first of which is gain euals one and the last of which is gain equals 10. One can note the increase of intensity (with color intensity) at each increase of gain and the increase of more high frequencies with each increase of the tone knob.
Figure 3. Treble Knob, 1-10.
Figure 4. Mid Knob, 1-10.
Figure 5. Bass Knob, 1-10.
Next, we used what we know about our input, unaffected signal x(t), and the signal outputs that we calculated y(t) to find a transform h(t) that models the behavior of each of the knobs. This relatively accurately affected the range of frequencies that the knob specifically focuses on and amplify them by the correct amount given a knob input gain.
Figure 6. The Freq Gain of Each Knob on 10 w/ Smoothing Function
Gain adjustments were calculated for each knob at each 1-10 parameter. We then applied a smoothing function to remove the peaks and outlying values that can be seen in the "FFT Spectrum Difference" images in Figures 3-5. The resulting smoothed amplitude adjustments, for a gain of 10 of each knob, can be seen in figure 6. When the user inputs which knob parameters they wish to use, the filters for the specific knob at those parameters are selected and added together with the other parameters as shown in Figure 7 for Treble 8, Mids 6, and Bass 10.
Figure 7. Treble 8, Mid 6, and Bass 10 Filters added together
To apply this gain adjustment to theaudio input, we switched the input audio into the freq domain and padded the gain adjustment filter to match the audio input. From here we multiplied the two singles together and brought them back into the time domain. The resulting audio then had the effects of the tone knobs applied to it. A before and after gain for the time domain can be seen in figure 8.
Figure 8. Audio Wav Before and After Tone Knobs (Treb 8 | Mid 6 | Bass 10)
After all of this analysis and work, we finally got the tone knobs working, however they sounded very much unlike the actual knobs in that even with a setting of 1 on the bass, it was extremely overpowering on the entirety of the signal. This could be due to the equalization settings coming directly from what we recorded out and this signal was also being effected by other factors in the amp which were greatly effecting the apparent frequency response. For the demo, we ended up using what we learned from this analysis to design peak filters using Matlab's iirpeak function for three different center frequencies that were estimated using the above plots. The resulting Matlab file is included below. It also has the bright switch code included since the process for that filtering was similar.
Figure 9. Matlab For Tone Related Effects.
Below are examples of the filter using the real data measurements, our estimates using these data measurements, and the actual recordings with these same settings. (Names formatted treble (T), mid (M), bass (B), bright (G))
T:10, M:1, B:1, G:off
Recorded
Calculated
Estimated
T:1, M:10, B:1, G:off
Recorded
Calculated
Estimated
T:1, M:1, B:10, G:off
Recorded
Calculated
Estimated
The estimations ended up being more useful for making the final product sound good and were still informed by the responses that we analyzed in the frequency domain. To get a more accurate response, it would make more sense we believe to analyze the circuits to create the transfer functions being used in the amp for this part of the response and rather focus on smaller scale parts of the hardware that are more effected by being hardware such as the introduction of heat. This would mean instead creating a theoretical model of the amp using knowledge from the circuitry and comparing the output of that to recordings and analyzing more on the component level maybe instead, like how heat introduced to transistors might make them behave different.