~ × & Σ …
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~ × & Σ …
Let's start with a simple sinusoid …
As you all know (or should by now) a sinusoid can be expressed as
y = A · sin (ω·t)
where A is a constant, ω is its frequency expressed in radians/sec and t is time expressed in secs. This is all well and good.
If we define φ,
φ ≡ ω/ωn
as some driving frequency ratio, where ωn is defined as a resonant frequency of the system in which we are interested (and constant for purposes of this discussion), then, substituting this into the sinusoid expression, one obtains
y = A · sin (φ·τ)
where τ, tau, is defined as dimensionless time with units of [1]. It can be calculated using the expression,
τ ≡ ωn · t
As you all know (or should by now) a sinusoid can be expressed as
y = A · sin (ω·t)
where A is a constant, ω is its frequency expressed in radians/sec and t is time expressed in secs. This is all well and good.
If we define φ,
φ ≡ ω/ωn
as some driving frequency ratio, where ωn is defined as a resonant frequency of the system in which we are interested (and constant for purposes of this discussion), then, substituting this into the sinusoid expression, one obtains
y = A · sin (φ·τ)
where τ, tau, is defined as dimensionless time with units of [1]. It can be calculated using the expression,
τ ≡ ωn · t

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Re: ~ × & Σ …
Here we have the simple sine curve.
y = A ‧ sin(x)
The abscissa, x, is given in radians not degrees, the amplitude is 1.0 and the period is 2Π or 6.283185308…
The reader should realize, that in the previous post, τ which is given by ωn · t, is also expressed in radians. This can be easily shown by looking at the units of each parameter.
ωn is given in radians/sec, and t is in sec, so multiplying them together we obtain radians/sec × sec ⟿ radians. A radian is a "dimensionless" parameter.
y = A ‧ sin(x)
The abscissa, x, is given in radians not degrees, the amplitude is 1.0 and the period is 2Π or 6.283185308…
The reader should realize, that in the previous post, τ which is given by ωn · t, is also expressed in radians. This can be easily shown by looking at the units of each parameter.
ωn is given in radians/sec, and t is in sec, so multiplying them together we obtain radians/sec × sec ⟿ radians. A radian is a "dimensionless" parameter.

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Re: ~ × & Σ …
Got it so far...
Mike
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Re: ~ × & Σ …
happy to read that …
Here is a plot of sin(x) and cos(x). Note that cos(x) = sin(x+Π/2) where Π/2 is the phase angle. The waveform, cos(x) leads sin(x) in time (dimensionless or not).
Here is a plot of sin(x) and cos(x). Note that cos(x) = sin(x+Π/2) where Π/2 is the phase angle. The waveform, cos(x) leads sin(x) in time (dimensionless or not).

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Re: ~ × & Σ …
A further peculiarity of sinusoids.
If a particle were to move, having its position (in 2D) vary harmonically with time, given by
The movement is circular. Any point on the curve represents the position of the particle at some time, t. If the movement goes forward in time, the particle position on the curve proceeds in a counterclockwise manner; starting at the coordinates, (1,0).
This representation is known as a parametric plot.
If a particle were to move, having its position (in 2D) vary harmonically with time, given by
 x(t)=cos(t) and
 y(t)=sin(t)
The movement is circular. Any point on the curve represents the position of the particle at some time, t. If the movement goes forward in time, the particle position on the curve proceeds in a counterclockwise manner; starting at the coordinates, (1,0).
This representation is known as a parametric plot.

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Re: ~ × & Σ …
If we let x(t)=cos(t), then, the derivative with respect to time is dx(t)/dt=sin(t). This means, if x(t) denotes the position of a particle, as it oscillates to and fro, dx/dt is its velocity.
If we plot velocity versus postion, we obtain again a circular plot.
However, at t=0 the motion now starts at coordinates (1,0) and as time progresses, say Π/2, the coordinates have become (0,1). IOW, the direction of the path followed is now CW.
If we plot velocity versus postion, we obtain again a circular plot.
However, at t=0 the motion now starts at coordinates (1,0) and as time progresses, say Π/2, the coordinates have become (0,1). IOW, the direction of the path followed is now CW.

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Re: ~ × & Σ …
I am going to return to this thread, but first some lateral thinking …

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Re: ~ × & Σ …
changing ω from 3 to 1,

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Re: ~ × & Σ …
To summarize,

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Re: ~ × & Σ …
some lateral drifting …

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Re: ~ × & Σ …
One sinusoid in a random (noisy) environment can be extracted …

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Re: ~ × & Σ …
Interestingly, I witnessed a demonstration on National Public Radio of the ability of the human ear/brain to perform this exact kind of extraction. A highpitched sine wave below the volume of normal perception was delivered with white noise superimposed on it. With the noise level low, there was no perception of the tone. Amazingly, as the sound level of the superimposed noise increased, without increasing the amplitude of the tone, the tone became audible. The higher the noise level, the more apparent the tone became (at least, up to a point)! This phenomenon was referred to as 'stochastic resonance'.WebPilot wrote:One sinusoid in a random (noisy) environment can be extracted …
L Cottrill

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Re: ~ × & Σ …
Interesting, but strictly speaking, stochastic resonance occurs in bistable systems, …
… continuing ⟿
and here is the analysis:
REM A little clipping and reduction in amplitude is observed, but the end result is still here; a 100 Hz signal is found embedded in the noise.
… continuing ⟿
 I numerically created the above waveform (noise and sine);
 I then exported it to a .wav format (binary) file;
 I then reloaded this waveform and analyzed it.
and here is the analysis:
REM A little clipping and reduction in amplitude is observed, but the end result is still here; a 100 Hz signal is found embedded in the noise.

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Re: ~ × & Σ …
I may need this capabilility in the not too distant future …
Vector Field
Vector Field

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Re: ~ × & Σ …
Thanks for this. Very fun to watch, Forrest.
Mike
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