

Gray code at the pediatrician's office

Last week we took Katara to the pediatrician for a checkup, during which they weighed, measured, and inoculated her. The measuring device, which I later learned is called a stadiometer, had a bracket on a slider that went up and down on a post. Katara stood against the post and the nurse adjusted the bracket to exactly the top of her head. Then she read off Katara's height from an attached display. How did the bracket know exactly what height to report? This was done in a way I hadn't seen before. It had a photosensor looking at the post, which was printed with this pattern: (Click to view the other pictures I took of the post.) The pattern is binary numerals. Each numeral is a certain fraction of a centimeter high, say 1/4 centimeter. If the sensor reads the number 433, that means that the bracket is 433/4 = 108.25 cm off the ground, and so that Katara is 108.25 cm tall. The patterned strip in the left margin of this article is a straightforward translation of binary numerals to black and white boxes, with black representing 1 and white representing 0: 0000000000

0000000001

0000000010

0000000011

0000000100

0000000101

0000000101

...

1111101000

1111101001

...

1111111111

If you are paying attention, you will notice that although the strip at left is similar to the pattern in the doctor's office, it is not the same. That is because the numbers on the post are Gray-coded. Gray codes solve the following problem with raw binary numbers. Suppose Katara is close to 104 = 416/4 cm tall, so that the photosensor is in the following region of the post: ...

0110100001 (417)

0110100000 (416)

0110011111 (415)

0110011110 (414)

...

But suppose that the sensor (or the post) is slightly mis-aligned, so that instead of properly reading the (416) row, it reads the first half of the (416) row and last half of the (415) row. That makes 0110111111, which is 447 = 111.75 cm, an error of almost 7.5%. (That's three inches, for my American and Burmese readers.) Or the error could go the other way: if the sensor reads the first half of the (415) and the second half of the (416) row, it will see 0110000000 = 384 = 96 cm. Gray code is a method for encoding numbers in binary so that each numeral differs from the adjacent ones in only one position: 0000000000

0000000001

0000000011

0000000010

0000000110

0000000111

0000000101

0000000100

0000001100

...

1000011100

1000011101

...

1000000000

This is the pattern from the post, which you can also see at the right of this article. Now suppose that the mis-aligned sensor reads part of the (416) line and part of the (417) line. With ordinary binary coding, this could result in an error of up to 7.75 cm. (And worse errors for children of other heights.) But with Gray coding no error results from the misreading: ...

0101110000 (417)

0101010000 (416)

0101010001 (415)

0101010011 (414)

...

No matter what parts of 0101110000 and 0101110001 are stitched together, the result is always either 416 or 417. Converting from Gray code to standard binary is easy: take the binary expansion, and invert every bit that is immediately to the right of a 1 bit. For example, in 1 1 1 1 1 0 1 0 00, each red bit is to the right of a 1, and so is inverted to obtain the Gray code 1 0 0 0 0 1 1 1 00. Converting back is also easy: Flip any bit that is anywhere to the right of an odd number of 1 bits, and leave alone the bits that are to the right of an even number of 1 bits. For example, Gray code 1 00001 1 1 00 contains four 1's, and the bits to the right of the first ( 00001 ) and third ( 1 ), but not the second or fourth, get flipped to 11110 and 0 , to give 1 11110 1 0 00. [ Addendum 20110525: Every so often someone asks why the stadiometer is so sophisticated. Here is the answer. ]

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