Re: rules of fixed-point arithmetic

On Mon, 08 Sep 2008 18:24:16 -0700, Sam Green wrote:

Hello,

I have come across an unpublished paper outlining some rules of fixed-
point arithmetic. These rules include things such as: "The scale of the
sum of two binary numbers scaled X(e,f) is X(e +1,f), i.e., the sum of
two M-bit numbers requires M +1 bits." Can anyone give me the name of a
book that has similar mathematical content dealing with fixed-point
arithmetic? I am working on an academic paper and need solid references.

Fixed point arithmetic is just integer arithmetic. Interpretation
usually involves a scale factor of a power of two, but that often doesn't
show up as actual manipulation (although some chips do have a shift-left
by one after a multiply, so that the usual binary points of source
operands and accumulators line up.)

Since it takes two bits to hold all of the results of adding two one bit
integers together, so it also takes an extra bit in fixed point.

Where this gets curly is that many processors that have special
arithmetic support for fixed point arithmetic work around the need for
extra precision on addition by "saturating" the result, or by having a
set of extra "guard bits" for holding these overflows for short-ish runs
of accumulation. An example of the former are the ARMv5e DSP
instructions, and of the latter are the Motorola DSP56000 series (and
most other fixed point DSP processors.)

Any text on digital signal processing will discuss this, as will the
architecture reference manuals for all "fixed point" processor chips.

Cheers,

--
Andrew
.

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