Fixed-Point Number Definitions

This example illustrates the definitions of unsigned and signed-two's-complement integer and fixed-point numbers.

Unsigned Integers.

Unsigned integers are represented in the binary number system in the following way. Let

b = [b(n) b(n-1) ... b(2) b(1)]

be the binary digits of an n-bit unsigned integer, where each b(i) is either one or zero. Then the value of b is

u = b(n)*2^(n-1) + b(n-1)*2^(n-2) + ... + b(2)*2^(1) + b(1)*2^(0)

For example, let's define a 3-bit unsigned integer quantizer, and enumerate its range.

originalFormat = get(0, 'format'); format

q = quantizer('ufixed',[3 0]);
[a,b] = range(q);
u = (a:eps(q):b)'

% Now, let's display those values in binary.
b = num2bin(q,u)

u =

     0
     1
     2
     3
     4
     5
     6
     7


b =

  8x3 char array

    '000'
    '001'
    '010'
    '011'
    '100'
    '101'
    '110'
    '111'

Unsigned Integer Number Circle.

Let's array them around a clock face with their corresponding binary and decimal values.

fidemo.numbercircle(q);

Unsigned Fixed-Point.

Unsigned fixed-point values are unsigned integers that are scaled by a power of two. We call the negative exponent of the power of two the "fractionlength".

If the unsigned integer u is defined as before, and the fractionlength is f, then the value of the unsigned fixed-point number is

  uf = u*2^-f

For example, let's define a 3-bit unsigned fixed-point quantizer with a fractionlength of 1, and enumerate its range.

q = quantizer('ufixed',[3 1]);
[a,b] = range(q);
uf = (a:eps(q):b)'

% Now, let's display those values in binary.
b = num2bin(q,uf)

uf =

         0
    0.5000
    1.0000
    1.5000
    2.0000
    2.5000
    3.0000
    3.5000


b =

  8x3 char array

    '000'
    '001'
    '010'
    '011'
    '100'
    '101'
    '110'
    '111'

Unsigned Fixed-Point Number Circle.

Let's array them around a clock face with their corresponding binary and decimal values.

fidemo.numbercircle(q);

Unsigned Fractional Fixed-Point.

Unsigned fractional fixed-point numbers are fixed-point numbers whos fractionlength f is equal to the wordlength n, which produces a scaling such that the range of numbers is between 0 and 1-2^-f, inclusive. This is the most common form of fixed-point numbers because it has the nice property that all of the numbers are less than one, and the product of two numbers less than one is a number less than one, and so multiplication does not overflow.

Thus, the definition of unsigned fractional fixed-point is the same as unsigned fixed-point, with the restriction that f=n, where n is the wordlength in bits.

  uf = u*2^-f

For example, let's define a 3-bit unsigned fractional fixed-point quantizer, which implies a fractionlength of 3.

q = quantizer('ufixed',[3 3]);
[a,b] = range(q);
uf = (a:eps(q):b)'

% Now, let's display those values in binary.
b = num2bin(q,uf)

uf =

         0
    0.1250
    0.2500
    0.3750
    0.5000
    0.6250
    0.7500
    0.8750


b =

  8x3 char array

    '000'
    '001'
    '010'
    '011'
    '100'
    '101'
    '110'
    '111'

Unsigned Fractional Fixed-Point Number Circle.

Let's array them around a clock face with their corresponding binary and decimal values.

fidemo.numbercircle(q);

Signed Two's-Complement Integers.

Signed integers are represented in two's-complement in the binary number system in the following way. Let

b = [b(n) b(n-1) ... b(2) b(1)]

be the binary digits of an n-bit signed integer, where each b(i) is either one or zero. Then the value of b is

s = -b(n)*2^(n-1) + b(n-1)*2^(n-2) + ... + b(2)*2^(1) + b(1)*2^(0)

Note that the difference between this and the unsigned number is the negative weight on the most-significant-bit (MSB).

For example, let's define a 3-bit signed integer quantizer, and enumerate its range.

q = quantizer('fixed',[3 0]);
[a,b] = range(q);
s = (a:eps(q):b)'

% Now, let's display those values in binary.
b = num2bin(q,s)

% Note that the most-significant-bit of negative numbers is 1, and positive
% numbers is 0.

s =

    -4
    -3
    -2
    -1
     0
     1
     2
     3


b =

  8x3 char array

    '100'
    '101'
    '110'
    '111'
    '000'
    '001'
    '010'
    '011'

Signed Two's-Complement Integer Number Circle.

Let's array them around a clock face with their corresponding binary and decimal values.

The reason for this ungainly looking definition of negative numbers is that addition of all numbers, both positive and negative, is carried out as if they were all positive, and then the n+1 carry bit is discarded. The result will be correct if there is no overflow.

fidemo.numbercircle(q);

Signed Fixed-Point.

Signed fixed-point values are signed integers that are scaled by a power of two. We call the negative exponent of the power of two the "fractionlength".

If the signed integer s is defined as before, and the fractionlength is f, then the value of the signed fixed-point number is

  sf = s*2^-f

For example, let's define a 3-bit signed fixed-point quantizer with a fractionlength of 1, and enumerate its range.

q = quantizer('fixed',[3 1]);
[a,b] = range(q);
sf = (a:eps(q):b)'

% Now, let's display those values in binary.
b = num2bin(q,sf)

sf =

   -2.0000
   -1.5000
   -1.0000
   -0.5000
         0
    0.5000
    1.0000
    1.5000


b =

  8x3 char array

    '100'
    '101'
    '110'
    '111'
    '000'
    '001'
    '010'
    '011'

Signed Fixed-Point Number Circle.

Let's array them around a clock face with their corresponding binary and decimal values.

fidemo.numbercircle(q);

Signed Fractional Fixed-Point.

Signed fractional fixed-point numbers are fixed-point numbers whos fractionlength f is one less than the wordlength n, which produces a scaling such that the range of numbers is between -1 and 1-2^-f, inclusive. This is the most common form of fixed-point numbers because it has the nice property that the product of two numbers less than one is a number less than one, and so multiplication does not overflow. The only exception is the case when we are multiplying -1 by -1, because +1 is not an element of this number system. Some processors have a special multiplication instruction for this situation, and some add an extra bit in the product to guard against this overflow.

Thus, the definition of signed fractional fixed-point is the same as signed fixed-point, with the restriction that f=n-1, where n is the wordlength in bits.

  sf = s*2^-f

For example, let's define a 3-bit signed fractional fixed-point quantizer, which implies a fractionlength of 2.

q = quantizer('fixed',[3 2]);
[a,b] = range(q);
sf = (a:eps(q):b)'

% Now, let's display those values in binary.
b = num2bin(q,sf)

sf =

   -1.0000
   -0.7500
   -0.5000
   -0.2500
         0
    0.2500
    0.5000
    0.7500


b =

  8x3 char array

    '100'
    '101'
    '110'
    '111'
    '000'
    '001'
    '010'
    '011'

Signed Fractional Fixed-Point Number Circle.

Let's array them around a clock face with their corresponding binary and decimal values.

fidemo.numbercircle(q);

set(0, 'format', originalFormat);
%#ok<*NOPTS,*NASGU>