## Re-creating the "breathing" LED

### October 31, 2011

Perhaps it’s not surprising, but Apple has a patent on the iconic "breathing" pattern used for the sleep indicator LED on all their computers.

Recently — for a personal project — I wanted to see if I could replicate this effect. It turns out that I wasn’t alone. Ladyada tried to reverse engineer the pattern a few years ago. Unfortunately, she stopped short of providing anything — like code — that the lazy web surfing Arduino hacker might use to recreate the effect. That’s the purpose of this short tutorial.

## Digging Into the Math

The Apple patent claims that the breathing pattern is a simple sinusoid, but observation of one’s own (heavy) breathing will show that the pattern is a little more complicated than that. Maybe a sine wave works for Apple, but it doesn’t look quite right to me. In my own breathing, I tend to ease in to a fast inhale, and stop suddenly before easing out to a fast exhale. Also, the period between inhalation and exhalation happens to be shorter than the period between exhalation and the next inhalation. This is not a simple sinusoid, as the oscilloscope plot from Ladyada’s own investigations will attest.

A commenter on Ladyada’s blog suggested that the pattern is probably more accurately modeled by f(x) = e^{sin(x)}. Plotting this equation gives the following:

Compare the curvature characteristics of this plot with the simple sinusoid below:

The plot of f(x) = e^{sin(x)} has wider "troughs" (periods between inhale and exhale) and narrower "peaks" (periods between exhale and inhale), more accurately matching natural breathing patterns. As a simple experiment, try breathing "sinusoidally" and you’ll see how unnatural it feels.

## Turning it Into Code

For my own experimentation, I used the ubiquitous Arduino. The Arduino supports analog output using pulse-width modulation (PWM) mapped to integer values from 0 to 255. To recreate the breathing LED, this means manipulating the original equation f(x) = e^{sin(x)} such that the amplitude fits within the PWM range.

I took enough math in school to know that the minima and maxima of any equation occur at critical points in the equation, where the derivative of that equation is either 0 or its not differentiable. Beyond that, I left it to Wolfram Alpha to do the hard work. It turns out that the minimum of the wave is 1/e, and the maximum is e. Using this information to adjust the amplitude of the equation such that it fits within the 0 to 255 range gives the following:

Swap *x* for the number of seconds that have elapsed, and map the above equation to PWM output on any supported Arduino pin, and you have the beginnings of a breathing pattern. The problem is that the frequency may be too high or low (depending on your preference), and so the breathing will appear fast or slow. Easy enough: Multiply *x* by any value to adjust the frequency. I like π/2.

Finally, 1 - e, and 255/(e - 1/e) are constants, and can be pre-calcuated to reduce overhead. The final Arduino sketch is as follows (with the LED connected to pin 11, a suitable resistor in series, yadda, yadda …):

```
#include <math.h>
void setup()
{
pinMode(11, OUTPUT);
}
void loop()
{
float val = (exp(sin(millis()/2000.0*PI)) - 0.36787944)*108.0;
analogWrite(11, val);
}
```

## Gratuitous Video

And, in case you don’t have an Arduino handy, here’s a short video of the final effect:

## Conclusion

You might be asking: Is it really that big of a difference? Wouldn’t a simple sinusoid suffice? To answer the latter question: yes. To answer the former: the difference is noticeable, but only slightly. Steve Jobs was a notorious perfectionist. I like to think that he would’ve cared about such things.