Motorola DSP56000, 24-Bit Digital Signal Processor manual Integer/Fractional Number Comparison

DATA REPRESENTATION AND ROUNDING

A1 or B1(see Figure 3-8).

TWOS COMPLEMENT INTEGER

TWOS COMPLEMENT FRACTIONAL

FRACTIONAL = INTEGER EXCEPT FOR X AND ⎟

N BITS

S	•
S•
	N BITS

–2(N–1) TO [+2(N–1) –1]

–1 TO [+1–2–(N–1)]

Figure 3-8 Integer/Fractional Number Comparison

A comparison between integer and fractional number representation is shown in Figure 3-8. The number representation for integers is between ±2(N-1); whereas, the fractional representation is limited to numbers between ±1. To convert from an integer to a frac- tional number, the integer must be multiplied by a scaling factor so the result will always be between ±1. The representation of integer and fractional numbers is the same if the numbers are added or subtracted but is different if the numbers are multiplied or divided. An example of two numbers multiplied together is given in Figure 3-9.The key difference is that the extra bit in the integer multiplication is used as a duplicate sign bit and as the least significant bit (LSB) in the fractional multiplication. The advantages of fractional data representation are as follows:

•The MSP (left half) has the same format as the input data.

•The LSP (right half) can be rounded into the MSP without shifting or updating the exponent.

•A significant bit is not lost through sign extension.

•Conversion to floating-point representation is easier because the industry-standard floating-point formats use fractional mantissas.

•Coefficients for most digital filters are derived as fractions by the high-level language programs used in digital-filter design packages, which implies that the results can be used without the extensive data conversions that other formats require.

Should integer arithmetic be required in an application, shifting a one or zero, depending on the sign, into the MSB converts a fraction to an integer.

The Data ALU MAC performs rounding of the accumulator register to single precision if requested in the instruction (the A1 or B1 register is rounded according to the contents of the A0 or B0 register). The rounding method is called round-to-nearest (even) number, or convergent rounding. The usual rounding method rounds up any value above one-half

MOTOROLA DATA ARITHMETIC LOGIC UNIT 3 - 13