Ground rules: You are free to discuss approaches to the problems with your fellow students, and talk over issues when looking at schematics, but your solutions should be your own. In particular, you should never be looking at another student's solutions at the moment you are putting pen to paper on your own solution. That's called "copying," and it is lame. Unpleasantness may result from such behavior.
Late policy: If you show up really late, suggesting that you were doing the homework during class, then I'll take off up to 10 points based on my mood. If you turn it in later that day, say before 6:00 PM or so (that's when my 2025 recitation ends - you can find me in Van Leer 361 from 3 to 6). After that, I'll take off 20 per day. This isn't to be mean - it's to encourage you to get it turned in and get on with whatever other work you have to do in other classes, even if it's not perfect - but also to encourage you to go ahead and do the work and turn it in and learn some stuff and get some points, even if you're past the deadline.
Read Waveshaping using Chebychev Polynomials, by Miller Puckette, to review the material needed to do this problem.
According to some slides from ECE2025, you can synthesize a rough periodic vowel sound by setting the 2nd, 4th, 5th harmonics of a wave equal to 12, 29, and 49. We will let the remaining harmonics be zero. (Notice the fundamental and third harmonics are missing).
(a) Find a fifth order polynomial, represented as a function of f, f(x), such that f(cos(omega t)) has the spectrum described above, with fundamental frequency omega. (I would recommend using the provided functions chebpolycoefs.m and chebpolysum.m. Include a plot of your polynomial f(x) over the range -1 to 1.
(b) Synthesize a tone f(a(t) cos(omega t)), where a(t) is a decaying function of time that starts at $a(0) = 1$. You may choose the function a(t) (a simple linear function is fine, or you could try more complicated things if you wanted). the duration of the tone, and the frequency omega. Choose a sampling frequency that is a little bit higher than what you would need to safely represent the 5th harmonic (according to the Nyquist criterion). Experiment until you think you have an interesting example. Listen to the tone, and display its spectrogram. Turn in a listing of your program and a printout of your spectrogram.
You may look to sawchebdemo.m for inspiration.
Make the following modifications to syncdemo.m:
Note: There are a lot of different definitions of soft-sync; I just made up the one above. Different pieces of hardware use the term "soft-sync" to refer to very different features.
Skim An Introduction To FM, by Bill Schottstaedt. (There's Lisp code; don't let that scare you. We won't need it.)
(a) Use MATLAB to synthesize a tone via Frequency Modulation (although it's actually Phase Modulation, technically speaking). Generate the wave according to the formula in the document following the sentence "Given our formula for FM, let's assume, for starters, that f(t) is a sinusoid." Let the modulation index B be your shoe size, and let the carrier frequency equal 200 + the last two digits of your phone number, in Hertz. Let the modulation frequency equal the carrier frequency.
I have provided the script fmsynth.m to get you started. (Notice that I've added a phase variable theta; if you play with changing the theta variable over time, say by uncommenting the theta = linspace(0,2*pi,length(tt)) line, you'll hear that the phase relationship between the carrier and the modulator does matter. The effect is extremely subtle, but it is audible. In what follows, we will leave the phase constant throughout the tone.
This particular choice of modulator and carrier frequencies will produce a harmonic spectrum. Estimate the amplitude of the fourth harmonic from the magnitude of the FFT as plotted by the code provided by eyeballing the height of the fourth harmonic. Include your FFT plot. (I have multiplied the FFT by 2, and divided by the length of the FFT, so that a cosine with an amplitude of 1 will appear as a spike with height 1. Don't worry if you haven't had ECE4270; if you have, you'll follow why I need to do those calibration steps, but if you haven't, you can take it on faith.)
Then, compute the exact value of the amplitude of the fourth harmonic via the formula in the document that appears right after the phrase "or in slightly more compact form." Write one line of MATLAB code that will produce this value. Recall it will consist of the sum of two Bessel functions; one will correspond to a sideband with a negative frequency that folds over.
(b) Now let's complicate things by repeating the steps of (a), except now use the slightly more general expression that appears in the left hand side of the equation right after the phrase "This is Chowning's version of the expansion. In general:" You can let phi be zero, since it won't change the sound at all, but let theta = pi/7 (note the case in (a) corresponded to theta = 0). Estimate the amplitude of the fourth harmonic from the height of the appropriate spike on the FFT plot (be sure to include the plot), and then compute the exact value using the right hand side of the equations right after the phrase "This is Chowning's version of the expansion. In general:" Write one line of MATLAB code that will produce this value. Notice that you will be adding two cosines with the same frequency but different phases, so you will need to remember how to do "phasor addition" to do this problem!
On both (a) and (b), note that your FFT plots need not (and in fact shouldn't) include the entire FFT; just plot the interesting parts showing the main harmonics.
Check out Ray Wilson's Voltage Controlled Low Pass Filter (Four Pole 24db/Oct): The input and feedback resistors are 100K; it looks like the divider is made with a 1K to ground. (I find it interesting that he chooses to use TL084 op amps as buffers instead of the buffers built in to the LM13700. Maybe this is to avoid having to deal with the weird 1.4 V drop you get from the LM13700 buffers? The TL084 also are probably better quality than just the simple Darlington pair in the LM13700.)
In parts (a) through (g), we will consider the gain of just one of the filter stages, either the second, third, or fourth (they are all the same; I'm not including the first one so we can avoid the effect of resisor coupling in the resonant feedback loop while working (a) and (b)).
a) Find the voltage at the input terminal of the OTA in terms of the voltage at the output of the buffer and voltage at the input of the filter block. Don't make any approximations concerning the resistors (i.e., if you use superposition, note that you must compute the value of the little resistor in parallel with the big resistor to solve this.)
b) In class, I attempted to use vigorous handwaving to attempt to convince you that part (a) could be approximated as
v_at_ota = (v_input + v_output) * (little_resistor / (little_resistor + big_resistor))
Comment on how close this approximation is to what you found in (a).
c) In class, I used even more vigorous handwaving to attempt to convince you that part (a) could be further approximated as
v_at_ota = (v_input + v_output) * (little_resistor / big_resistor)
Comment on how close this approximation is to what you found in (a) and (b).
d) Assume that the transductance gain of the OTA is 19.2*I_con, where I_con is the current flowing into the control pin of the OTA. What is the cutoff frequency of the filter block in terms of (I_con) in Hertz, using the approximation in part (c)? (Remember that the transconductance gain just takes the place of 1/R in the usual single-pole cutoff freuqency calculation, and for convenience we include the scaling of the resistive divider as part of the transconducance gain.)
e) Given the result in (d), what value I_con would be needed for the cutoff frequency of one stage to be 3000 Hz?
f) What single-stage cutoff frequency would you compute if you used the I_con you computed in (e), but you used the no-approximation technique of part (a)?
g) What single-stage cutoff frequency would you compute if you used the I_con you computed in (e), but you used the approximation in part (b)? Comment on how close the cutoffs computed in (f) and (g) are to 3000 Hz.
h) Now let's consider the full four-pole cascade with feedback level denoted as K, as in lecture. Let the cutoff frequency of a single stage be 3000 Hz. On the same plot, show the magnitude of the frequency response (with the horizontal axis in Hertz), from DC to some value that you think best shows off the curves, for four cases: K=0, K just big enough so that you can just barely see a resonance "bump" in the curve, K close to 4 (but not so big that it swamps your other curves), and a K somewhere between the last two cases that you think is interesting. Make sure the value at DC corresponds with the results computer by the simple formula derived in lecture.
In this problem we'll look at the Oberheim OB-Mx. Strangely, Tom Oberheim had nothing to do with this synth; Gibson had bought the rights to the Oberheim name. Don Buchla was called in to try to save the project, but it eventually wound up released before it was really ready against Buchla's wishes.
If you don't see a specific unit on a capacitor, there's usually an implied "microfarads."
a) The Moog transistor ladder VCF contains a cascade of four one-pole lowpass filter sections. Find the cutoff frequency of one of those sections in the OB-MX's transistor ladder as a function of the control current being pulled from the tied emitters of the transistor pair that feeds the ladder. Two things to note: 1) Notice that when analyzing the Moog VCF, we don't include a resistive divider in the gain as we've done in other OTA-C cutoff computations; there is a resistive divider right at the first input, but it's not important for our frequency analysis. 2) I initially wrote expressions on the board for a one-sided ladder; for a real Moog two-sided ladder, the control current gets split between the two halves of the ladder, so you get a transconductance gain from each transistor pair that's like that of an OTA, and the formula for the cutoff is basically the same as for the OTA-C filters we looked at earlier (except we leave out the resistive divider).
b) Let's do some DC analysis. At DC, the caps are open circuits. Supposing that the transistors draw negligible current through the bases, what are the voltages at the bases of the four stages of the ladder? (Number the stages 1 through 4, from bottom to top).