How to square a fixed point number?

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I guess they are signed and that the value is between -1.0 and 1.0

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Why can we expect that the Q2.14 range (-2.0..+1.999) isn't utilized? Saturation logic should be used in the general case.

Squaring is only a special case of fixed point multiply, by the way. It requires the same scaling operations.

The simplest way to do the result scaling and saturation is to use IEEE fixed point package.