[sdiy] Frequency follower idea. Question about FM.
cheater cheater
cheater00 at gmail.com
Sat Sep 11 07:46:42 CEST 2010
thanks =) i'm wondering about how someone could go about implementing this...
D.
On Thu, Sep 9, 2010 at 18:37, Giorgio <dancemachine at gmail.com> wrote:
> good stuff. you might want to look at the harvestman hertz donut if you haven't already. it has some very interesting and musical oscillator sync/tracking behaviors
>
> sent from phone
>
> On Sep 9, 2010, at 12:12 PM, cheater cheater <cheater00 at gmail.com> wrote:
>
>> Hi guys,
>> I was wondering recently about syncing VCOs to other oscillators, and
>> I came up with a framework idea of how this could be done in a new and
>> I think interesting way. I wonder what you guys think about it.
>>
>> One of the problems is: you want an analogue LFO that is tempo-synced,
>> but you want e.g. a sinewave, or a triangle wave, and very importantly
>> it must reach the amplitude maximum. If you just coarsely set the
>> frequency of the VCO and then sync it to sequencer pulses, then you
>> lose the amplitude (the typical DCO problem). If on the other hand you
>> don't want to lose the amplitude (because, for example, the LFO
>> modulates the pitch and it's important that it reaches a certain
>> interval), then the only other way for the VCO to catch up is for its
>> pitch to be raised temporarily. It's similar to how beatmatching works
>> with vinyl. Of course, this can work well for audio-rate VCOs as well.
>>
>> So here's the idea with how vinyl works vs how VCOs work. We have two
>> VCOs vs two adapters with vinyl records on them.
>>
>> 1. unsynced VCOs - two records are playing with separate tempo. There
>> is no sync.
>>
>> 2. Hardsynced VCOs: every time VCO1 reaches its sync point, the VCO2
>> is also reset towards its own sync point. This can be seen as speeding
>> up VCO2 to a very high speed of oscillation until it reaches its sync
>> point. The analogy here would be that when vinyl 1 reaches a kick,
>> vinyl 2 is sped up to nearly-infinite speed until it reaches the kick.
>> Since the speed is infinite, the operation takes an infinitesimal
>> amount of time, and in the next measure vinyl 1 and vinyl 2 start out
>> perfectly in phase. Because they are probably set to different tempos
>> (the probability of having the same tempos is 0), they will quickly
>> drift out of sync.
>>
>> 3. Beatmatching: this is how records are synced: vinyl 1 is playing at
>> a certain rate, vinyl 2 is playing at another rate. First the two are
>> synced together. Then the dj checks if vinyl 2 is slower or faster and
>> adjusts the rate of deck 2 to be higher or lower. At the same time he
>> either pushes vinyl 2 forward or taps it with the finger to slow it
>> down.
>>
>> This could look like this with VCOs: You sync the two VCOs together.
>> You have access to the phases of both VCOs and check how quickly they
>> fall apart. The derivative of their difference is the difference in
>> frequency that needs to be added to VCO2. Then, the oscillators are in
>> frequency sync (let's call this part we have just accomplished
>> "frequency calibration"). However, after this happens, the oscillator
>> needs to be 'pushed forward'. We can do this by injecting some phase
>> to the VCO. However, so that there is no click*, we can limit the rate
>> at which the VCO is being "pushed forward". We can do this by
>> frequency-modulating the VCO one time with a windowing function to
>> speed it up, or inverted windowing function to slow it down (let's
>> call that part "phase calibration"). During the phase calibration, the
>> frequency calibration should be somehow disabled. However, in a
>> running process, it would be undesirable to have a state-machine, so
>> maybe some form of continuous process could be invented from this.
>>
>> I have below included the mathematics behind this technique. One
>> question is how to realize a fairly smooth windowing function.
>>
>> *I believe if we have a carrier at frequency f_1 and it is modulated
>> by something that has frequency content (much) lower than f_1, then
>> you should not get any new audio-rate harmonics that are not "part of
>> f_1", should you? How does this work exactly? Is there a special rule
>> that says that your modulator needs to be band-limited to a certain
>> frequency dependent on f_1? I understand that this question is not
>> stated 100% properly, but any hints in this region are good. Basically
>> I have an audio-rate carrier, and I want to frequency-modulate it so
>> that no clicks/added harmonics are perceived in the output.
>>
>> Also on a similar note, the MF-107 FreqBox: Does anyone know how this
>> thing functions? Does it have a PLL inside? I assume it has a PLL and
>> a VCO, and the PLL can sync the VCO, while the original signal can FM
>> the VCO. An env follower can FM and AM the VCO.
>>
>> Here are the maths for the calibration:
>>
>> p_1, p_2
>> phases of oscillators 1 and 2
>>
>> r = p_1 - p_2
>> Phase difference between oscillators 1 and 2.
>>
>> dr/dt
>> the rate of change of the phase difference.
>>
>> If for example r is negative, and keeps on falling (we are
>> "unwrapping" r, instead of counting it modulo 2pi) then raising the
>> pitch of vco2 a bit will make the difference r drop a little less.
>>
>> Therefore, we need to heighten the frequency by -h*dr/dt, where h is
>> some constant:
>>
>> f_2 |-> f_2 + (-h) * dr/dt
>>
>> After this happens, dr/dt ~= 0 (is nearly equal to zero). Therefore r
>> is constant. We need to inject the amount r of voltage into oscillator
>> 2's accumulator:
>>
>> r = p_1 - p_2
>> p_2 = p_1 - r
>>
>> if we want p_2 to be equal to p_1, we need to add something to it:
>>
>> p_2 + x = p_1
>> p_1 - r + x = p_1
>> x = r
>>
>> Therefore we need to add r to p_2 :)
>>
>> To add r to p_2 we can do a frequency modulation by a window function w(t).
>>
>> The (unwrapped) phase of vco2 called p_2 is the integral of its frequency f_2:
>>
>> p_2(t_now) = b Int_0^{t_now} f_2(t) dt + p_2(0)
>>
>> That is, the phase at time t=t_now is the integral from time t=0 to
>> t=t_now of the frequency control voltage f_2(t) over the time-variable
>> "t", multiplied by an amplitude "b" (which depends on the design of
>> the VCO), and all that added to the phase it had at t=0.
>>
>> similarly, for vco1:
>>
>> p_1(t_now) = b Int_0^{t_now} f_1(t) dt + p_1(0).
>>
>> We assume that f_1=f_2=const starting at some point we'll start
>> calling t=1. The number 1 is chosen arbitrarily. It might mean t=1
>> second. At t=1, the accumulators had phases p_1(1) and p_2(1) and the
>> VCOs were at the same frequencies, meaning the phase rises at the same
>> speed. We rebase our integral and with those constraints we can start
>> counting it from t=1.
>>
>> This means that
>> p_1(t_now) = b Int_1^{t_now} f_1(t) dt + p_1(1)
>> and
>> p_2(t_now) = b Int_1^{t_now} f_1(t) dt + p_2(1)
>>
>> therefore for t_now > 1, p_1(t_now) - p_2(t_now = p_1(1) - p_2(1).
>> This difference is our, now constant, value r.
>>
>> We need to add r to p_2 somehow. This means that starting with t_now >
>> T, we want this to be the case:
>>
>> p_2(t_now) = b Int_1^{t_now} f_2(t) dt + r + p_2(1)
>>
>> The only parameter we can manipulate is f_2(t). If we do not
>> manipulate it, after t=1 it will stay equal to f_1(t) and therefore
>> constant.
>>
>> So we can do this:
>>
>> p_2(t_now) = b Int_1^{t_now} f_2(t) + k*w(t) dt + p_2(1)
>>
>> = b Int_1^{t_now} f_2(t) dt + b Int_1^{t_now} k*w(t) dt + p_2(1)
>>
>> But what we want is p_2(t_now) = b Int_1^{t_now} f_2(t) dt + r + p_2(1)
>>
>> Therefore, we need to choose w(t) such that:
>> b Int_1^{t_now} k*w(t) dt = r
>>
>> We know b (it comes from the design of our oscillator), we know r, we
>> need to find k and w(t).
>>
>> We need our function w to be such that its integral is 1 when counted
>> from point t_0>1 where we begin emitting it, until T where we will
>> have stopped emitting it. We also want it to not add any phase
>> afterwards, that means, for t>T w(t)=0. And we don't want it to add
>> any frequency before we start emitting it, that means for t<t_0,
>> w(t)=0. This means that:
>>
>> Int_{t_0}^T w(t) dt = 1
>>
>> for u < t_0 < T < v: Int_u^v w(t) dt = 1
>>
>> So we have chosen our window function, therefore we have w(t).
>>
>> Our equation looks like this:
>>
>> b Int_1^{t_now} k*w(t) dt = r
>>
>> b*k Int_1^{t_now} w(t) dt = r
>>
>> Because 1 < t_0 and t_now > T, we can say:
>>
>> b*k * 1 = r
>>
>> Therefore k = r/b.
>>
>> This means we need to inject our windowing function once, while its
>> amplitude is multiplied by the number r/b.
>>
>> I believe it is possible to extend this technique to a continuous time
>> control system. The technique above will only work for when VCO1 has a
>> constant frequency. A continuous control system could work for the
>> case when VCO1 is of changing frequency.
>>
>> This technique could sound fairly differently depending on what
>> windowing function is chosen.
>>
>> A continuous technique could work like this:
>>
>> We define two areas in which p_2(t) can be:
>> (A) it's either in in the area p_2(t) \in (p_1(t)-pi, p_1(t) ) <mod 2pi>
>>
>> (B) or it's in the area p_2(t) \in (p_1(t), p_1(t) + pi) <mod 2pi>
>>
>> The cases p_2(t) = p_1(t) and p_2(t) = p_1(t)+pi <mod 2pi> are
>> excluded for clarity, since the probability they will happen is 0.
>>
>> There is a comparator that outputs 1 for case (A) and -1 for case (B).
>> Let's call it L(t).
>>
>> This means, for example, if VCO2 is just a tiny amount behind VCO1,
>> L(t) will be +1, and if it's just in front of VCO1, L(t) will be -1.
>>
>> Assume we have case (A). Every now and then, while L(t) is positive,
>> we feed a small amount of charge, say K, into the phase accumulator
>> p_2(t) to "catch up" and perform phase calibration. As soon as we
>> overtake VCO1, L(t) will become negative, and K will also become
>> negative, therefore "slowing down" p_2(t). This means that our p_2(t)
>> will be jumping back and forth from being just under to just above
>> p_1(t).
>>
>> Of course a continuous version of summation is an integration. So a
>> way to do the above is to have an current following L(t) and add that
>> current, call it M(t), to p_2(t). (I am, of course, not good enough
>> with electronics to work out the specifics of that).
>>
>> However, we still need to make sure that the change only happens
>> gradually: starts up slow, speeds up, and ends slow. Therefore before
>> we inject M(t) into p_2(t), we pull it through a sine shaper. The
>> shaper should somehow auto-adjust so that the electric charge
>> injection becomes slowest as p_2 approaches p_1. I am not sure what
>> the specifics could be of this. Does anyone have any ideas?
>>
>> To prevent oscillation of the system when p_2(t) ~= p_1(t) and when
>> p_2(t) ~= p_1(t) - pi <mod 2pi>, we put a low-pass filter of some
>> frequency F on M(t) so that it does not "zipper" along with L(t). In
>> the latter case the system should be knocked out of place into either
>> state domain (A) or state domain (B) by the fact that frequencies are
>> different.
>>
>> Of course, of f_1 > f_2, then we would need to keep on emitting M(t)
>> indefinitely, because p_2(t) would keep on lagging behind.
>>
>> However, the closer we get to the state p_1(t) ~= p_2(t), the easier
>> it will be for us to start marking up the frequency. This is because
>> the rate of change of r will be lower and lower. This means that we
>> have a system that is now under control and we can slowly start doing
>> other things with it. We look at L(t) statistically. If f_2 < f_1,
>> then L(t) will be mostly positive. If f_2 > f_1, then L(t) will be
>> mostly negative. The closer f_2 is to f_1, the more even the
>> distribution of L(t) will be across the values +1 and -1. We can take
>> the low-pass-filtered M(t) and inject it into some form of
>> sample-holding capacitor. The charge of that capacitor will be
>> mirrored into some form of positive voltage which raises or lowers the
>> frequency of VCO2. As f_2 -> f_1, M(t) will tend to 0, and the
>> frequency of f_2 will stabilize.
>>
>> This technique could work better than the previous one for the
>> situation where f_1 changes. However the faster f_1 changes, the
>> higher the frequency of the lowpass filter on M(t) should be, so that
>> f_2 has enough resource to catch up. Therefore: F(t) ~ f_1'(t) (note
>> the derivative sign)
>>
>> This fourth approach can be analogous to the DJ catching up with two
>> hands on the record adapter 2, trying to run his hands so quickly that
>> adapter 2 is in sync with adapter 1; then, a second person starts
>> correcting the pitch, during which the dj slows down his hands and
>> finally lets go of the record what so ever.
>>
>> Of course both techniques I described above can also be enhanced to
>> allow keeping the two oscillators at a certain desired difference of
>> phases r = r_0. This could prove sonically interesting.
>>
>> You can assume that the second technique will not work *ideally*, that
>> is, there will be some sway in the value of r, even if f_1 is
>> relatively constant. This will probably create a somewhat random
>> phasing sound which is very pleasing to the ear.
>>
>> If the low pass filtered M(t) is completely disconnected from the core
>> of VCO2, L(t) can be used as a measure of how close together f_1 and
>> f_2 are. You could do it this way: you check how symmetric L(t) is
>> (over a short period of time) and the longer it is symmetric, the
>> higher voltage you output from an integrator. On the other hand if
>> L(t) stops being symmetric, the charge in the integrator is reduced.
>> The integrator should have a low rise rate and a high fall rate. It
>> can be used to amplitude-modulate VCO2, making VCO2 output only when
>> VCO1 is near its frequency. This can be used for a resonator. Because
>> of the relatively low parts count for a minimum implementation using a
>> fixed frequency LPF for M(t) and a fixed frequency oscillator for
>> VCO2, you could easily imagine banks of such oscillators, working as a
>> very interesting resonator bank. This could be very musically
>> pleasing, giving a sweeping tone from VCO1 a nice arpeggiated
>> characteristic. You could even use some sort of fixed AD envelope to
>> give the resonances a plucked sound.
>>
>> The electronic implementation is left as an excercise for the reader.
>>
>> Cheers,
>> D.
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