modeling vertical vibration noise
I was curious how the Metrigear Vector accelerometer-based cadence extraction. might be affected by noise. Noise is basically any component of a data which is unpredictable and uninteresting. But to consider this, I'd need to consider the characteristics of the noise as sensed by the accelerometers.
This would be simple to determine: go out on a ride, unclip, and coast, then download the data. Simple, sure, except I don't have a Metrigear Vector, or for that matter any other accelerometers. It's more fun to estimate stuff theoretically than measure it, anyway. Reality tends to get ugly.
To model noise, I break noise down into sinusoidal components. For every frequency, the noise can then be described by two coefficients: an amplitude and a phase. The phase is easy: it is randomly chosen from the set of angles from 0 to 2π (or 0 to 360° if you're afraid of irrational numbers).
That leaves amplitude. Everyone's favorite probability distribution is the normal one: the Central Limit Theorem justifies its popularity. Using this, the probability of a given frequency component having an amplitude between Aω and Aω + dA, where I assume only positive values of Aω are allowed, is:
P dA = sqrt[2 /πAω02] exp[‒Aω2 / 2 Aω02] dA
Fine: now what's needed is Aω. I'll assume the following form for Aω, again using the Gaussian function:
Aω0 = a0 sqrt[2 /π ω02] exp[‒ω2 / 2 ω02]
What this says if for relatively low frequencies, noise is relatively "white" (independent of frequency) while for high frequencies, the magnitude of noise falls off rapidly. While not necessarily applicable for simple systems like a pendulum, the Gaussian spectral distribution may nicely describe complex, well-damped systems, such as a bicycle rolling over a rough, undulating road.
This leaves two more parameters: a0 and ω0. The latter describes at what frequency the noise amplitudes begin to rapidly diminish. Once this is set, it's an easy matter to choose a0 to give the desired amount of noise.
Okay -- enough theory. What about a bicycle? Suppose I'm rolling a bicycle on inflated tires over a rough road. What should I choose for a0 and ω0?
I'll start with the easy one: a0. It's hard to imagine that the vertical accelerations when riding over bumps would much exceed gravity's acceleration (g). If they did, the bike would lose contact with the ground as it moved. Sure, this may happen during off-road downhills, but is clearly not typical during periods where accurately measuring transmitted power is a concern. So I'll recommend picking a0 such that the amplitude of the noise signal is typically confined to the range from -g to g.
Then there's ω0. A number of mechanisms provide for an attenuation of high-frequency noise transmitted to the bike. The most obvious is the inflated tires. There's a good reason Dunlap's invention proved so popular: inflated tires reduce the transmission of high-frequency acceleration.
There's two approaches one can take here: one is to calculate what the resonant frequency of the tire-bike system is, above which attenuation occurs, below which transmission occurs. The other is to guess. This last option is what I tried... hmm... 10 Hz (63 radians / second). Why 10 Hz? The threshold of human audible detection is 20 Hz. Bike frames don't generally make too much noise when you ride them, so I figure there's not a lot of transmission in the audible spectrum. 10 Hz will drop the amplitude of noise down to 13.5% of its peak by 20 Hz.
But such a choice isn't very satisfying. Next time I'll try actually using some physics.
But anyway, here's a plot showing a typical result. As you can see, I picked the amplitude so the signal generally bounces between -g and g.
This would be simple to determine: go out on a ride, unclip, and coast, then download the data. Simple, sure, except I don't have a Metrigear Vector, or for that matter any other accelerometers. It's more fun to estimate stuff theoretically than measure it, anyway. Reality tends to get ugly.
To model noise, I break noise down into sinusoidal components. For every frequency, the noise can then be described by two coefficients: an amplitude and a phase. The phase is easy: it is randomly chosen from the set of angles from 0 to 2π (or 0 to 360° if you're afraid of irrational numbers).
That leaves amplitude. Everyone's favorite probability distribution is the normal one: the Central Limit Theorem justifies its popularity. Using this, the probability of a given frequency component having an amplitude between Aω and Aω + dA, where I assume only positive values of Aω are allowed, is:
P dA = sqrt[2 /πAω02] exp[‒Aω2 / 2 Aω02] dA
Fine: now what's needed is Aω. I'll assume the following form for Aω, again using the Gaussian function:
Aω0 = a0 sqrt[2 /π ω02] exp[‒ω2 / 2 ω02]
What this says if for relatively low frequencies, noise is relatively "white" (independent of frequency) while for high frequencies, the magnitude of noise falls off rapidly. While not necessarily applicable for simple systems like a pendulum, the Gaussian spectral distribution may nicely describe complex, well-damped systems, such as a bicycle rolling over a rough, undulating road.
This leaves two more parameters: a0 and ω0. The latter describes at what frequency the noise amplitudes begin to rapidly diminish. Once this is set, it's an easy matter to choose a0 to give the desired amount of noise.
Okay -- enough theory. What about a bicycle? Suppose I'm rolling a bicycle on inflated tires over a rough road. What should I choose for a0 and ω0?
I'll start with the easy one: a0. It's hard to imagine that the vertical accelerations when riding over bumps would much exceed gravity's acceleration (g). If they did, the bike would lose contact with the ground as it moved. Sure, this may happen during off-road downhills, but is clearly not typical during periods where accurately measuring transmitted power is a concern. So I'll recommend picking a0 such that the amplitude of the noise signal is typically confined to the range from -g to g.
Then there's ω0. A number of mechanisms provide for an attenuation of high-frequency noise transmitted to the bike. The most obvious is the inflated tires. There's a good reason Dunlap's invention proved so popular: inflated tires reduce the transmission of high-frequency acceleration.
There's two approaches one can take here: one is to calculate what the resonant frequency of the tire-bike system is, above which attenuation occurs, below which transmission occurs. The other is to guess. This last option is what I tried... hmm... 10 Hz (63 radians / second). Why 10 Hz? The threshold of human audible detection is 20 Hz. Bike frames don't generally make too much noise when you ride them, so I figure there's not a lot of transmission in the audible spectrum. 10 Hz will drop the amplitude of noise down to 13.5% of its peak by 20 Hz.
But such a choice isn't very satisfying. Next time I'll try actually using some physics.
But anyway, here's a plot showing a typical result. As you can see, I picked the amplitude so the signal generally bounces between -g and g.
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