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Have been wondering about this always and finally, my curiosity overcame my laziness.

I thought the unit for a noise density should be (rad/s)/Hz as the noise density is simply divided by the sampling rate. But it is (rad/s)/sqrt(Hz).

Why there is a square root? What am I missing here?

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Answer from ChatGPT:

The noise density of a gyro sensor is represented as (rad/s)/sqrt(Hz) rather than (rad/s)/Hz because it accounts for the statistical nature of noise in the frequency domain. Let me explain why this representation is used:

Noise as a Random Process: Noise in sensors, including gyro sensors, is often considered a random process. It means that the noise at any given moment is not deterministic but follows a statistical distribution. The nature of this noise can be modeled as Gaussian (normal) noise.

Frequency Binning: When we analyze noise in the frequency domain (using techniques like Fourier analysis or power spectral density estimation), we often divide the signal into different frequency bins or intervals. We want to know how much noise power is present in each of these bins.

Units of Noise Power: Noise power is typically measured in (rad/s)^2/Hz, which represents the variance of angular velocity per unit frequency. So, (rad/s)^2/Hz tells you how much angular velocity variance exists in a specific frequency range (1 Hz, for example).

Noise Amplitude: To get the noise amplitude (i.e., the standard deviation or RMS value of the noise) at a particular frequency, you take the square root of the noise power in that frequency bin. This is where the (rad/s)/sqrt(Hz) unit comes into play. It represents the amplitude of the noise in radians per second per square root of the hertz.

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  • $\begingroup$ Seems like ChatGPT gave me a clear answer. Leaving the question just in case there is someone else wondering about this, $\endgroup$ Commented Sep 5, 2023 at 17:59

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Because the underlying phenomenon is a power spectral density that has units of variance per Hz, not standard deviation per Hz. This is not a choice about how to represent something, as ChatGPT seems to suggest - standard deviation per Hz is simply not a valid way to describe the density of noise as a function of sampling bandwidth.

You could report units of variance per Hz, that is (rad/sec)^2/Hz. To have the spec conveniently "look like" a standard deviation that is a little easier to think about, people take the square root of the quantity. Then you know that if you are going to sample it with 1000 Hz bandwidth, your noise will be 1000 times the spec.

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