Optimization of Number Representations - Signal Processing Systems - page 1281

Fig. 1 Integer and fractional data formats

Fig. 2 Generalized fixed-point data format

because all of the intermediate variables or constants should be represented with
only integers or fractions whose represented values are usually much different from
those of the corresponding floating-point data. The difference of the representation
format also hinders incremental conversion from a floating-point design to a fixed-
point one. For seamless floating-point to integer or fixed-point conversion, the
semantic gap between the floating-point and fixed-point data formats needs to be
eliminated.
To solve these problems, a generalized fixed-point data-type that contains both
integer and fractional parts can be used [ 17 ] . This fixed-point format contains the
attributes specified as follows.

< wordlength , integer wordlength , sign over f low quantization mode > (1)

The word-length (WL) is the total number of bits for representing a fixed-
point data. The integer word-length (IWL) is the number of bits to the left of
the (hypothetical) binary-point. The fractional word-length (FWL) is the number
of bits to the right of the (hypothetical) binary point. The sign can be either
unsigned(“u”) or two's complement(“t”). Thus, the word-length (WL) corresponds
to “IWL FWL” for unsigned data. If

the fractional word-length is 0, the data with this format can be represented with
integers. At the same way, it becomes the fractional format when the IWL is 0. Note
that the IWL or FWL can be even larger than the WL; in this case, the other part
has a minus word-length. Figure 2 shows an interpretation of an 8-bit binary data
employing the fixed-point format with the IWL of 3.
The overflow and quantization modes are needed for arithmetic or quantization
operations. The overflow mode specifies whether no treatment (“o”) or saturation

+ FWL + 1” for signed data, and is “IWL +