Two's Complement Wrap-Around

In this section, we give an example showing how temporary overflow in two's complement fixed-point causes no ill effects.

In 3-bit signed fixed-point arithmetic, the available numbers are as shown in Table 9.1.

Let's perform the sum , which gives a temporary overflow (, which wraps around to ), but a final result () which is in the allowed range :^10.3

Now let's do in three-bit two's complement:

In both examples, the intermediate result overflows, but the final result is correct. Another way to state what happened is that a positive wrap-around in the first addition is canceled by a negative wrap-around in the second addition.

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.