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Such a device consists of a sensor pad that is placed under a mattress:

sesnor pad placing under a mattress

What's the principle it uses to detect breathing?

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I had never heard of such a thing, and after a little digging, I can't find any hint of how it even claims to work, except by a combination of "technology" and warm fuzzy feelings. It's quite likely it doesn't actually do anything, and this is one more completely unnecessary product marketed at over-reactive parents who hear about child deaths but who don't understand statistics. – Chris White Feb 23 '13 at 22:20
@ChrisWhite I used it myself and I can confirm from my experience it works. I just don't know how. – Petr Pudlák Feb 24 '13 at 8:01
I'm guessing your question is: a pressure sensitive band around the baby's waist would obviously be able to pick up breathing motions. But how can a sensor pad underneath the mattress do that? I've been reading amazon reviews of one under-mattress-sensor-pad device. Reviewers describe for instance that even footsteps in the same room get picked up by the sensor pad as cot movement. One reviewer noticed that a close by air conditoning unit satisfied the sensor pad, so they moved the cot away from the air conditioning unit. – Cleonis Feb 24 '13 at 12:22
Is a better home for this question? – Qmechanic Feb 24 '13 at 19:20
@Qmechanic I'm not interested in parenting issues, but in the underlying scientific/engineering principle. – Petr Pudlák Feb 24 '13 at 19:37

Baby Alert... has a special microphone to be placed under a cot mattress, which sends out an alarm if the child's heart stops beating for four seconds, or if breathing fails for 15 seconds.According to one recent newspaper report, the alarm, has been shown to "wake the child and prompt breathing", and could dramatically cut the number of fatalities from sudden infant death syndrome, or "cot death". ... Some are attached to the baby's tummy; others, like the Baby Alert, use a special pressure pad or mattress, while one wall unit beams out ultrasound waves above the cot.

All these devices work by detecting changes in breathing movements in the chest and abdomen. When breathing stops for longer than a pre-set time (usually 20 seconds), an alarm will ring and a light start to flash. Some monitors also measure temperature and heartbeat.

quoted from The truth about baby breathing monitors (read more at the link)

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