Home / Problem Set 2 / Stroboscopy

Stroboscopy

The questions below are due on Monday March 01, 2021; 10:00:00 PM.
 
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Note that this exercise is intended to be solved by hand, without the use of computation.

Illuminating a moving object with a flashing light can change the apparent speed of the object's motion. This is called the stroboscopic principle, and it is widely used to slow the apparent speed of fast motions (such as the rotation speed of a motor) so as to make them easier for a human to observe.

For this problem, we will consider an object moving in 1 dimension with a periodic motion (with period T), such that x(t) = x(t+T) represents the object's position at time t.

Assume that we observe x(\cdot) using a flashing light that flashes every \Delta seconds. This is equivalent to sampling x(\cdot) to derive a discrete-time signal y[\cdot], given by y[n] = x(n\Delta).

Assume that we wish to find ten samples of x(t) that are uniformly spaced from t=0 to T, i.e.,

y[n] = x\left(\frac{nT}{10}\right)

However, our sampling rate is limited so that \Delta > T (that is, we cannot sample faster than once per period of the object's motion).

Find a value of \Delta such that \Delta > T and y[n] = x(nT/10). Explain your result and your process.


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Is it possible to choose \Delta so that time appears to run backward? If so, find \Delta such that

y[n] = x\left(-\frac{nT}{10}\right)

If not, prove why this is not possible.


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For what values of \Delta is y[n] periodic?

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Assume that x(t) is sinusoidal with radian frequency 2\pi,

x(t) = \cos(2\pi t)

Find all the values of \Delta such that y[n] can be written in the form

y[n] = cos(\Omega n)

where \Omega is in the range 0 \lt \Omega \lt \frac{\pi}{2}.


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