View Fixed-Point Number Circles

Unsigned Integer Number Circle

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, define a 3-bit unsigned integer quantizer and enumerate its range.

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

u = 8×1

     0
     1
     2
     3
     4
     5
     6
     7

Display those values in binary.

b = num2bin(q,u)

b = 8x3 char array
    '000'
    '001'
    '010'
    '011'
    '100'
    '101'
    '110'
    '111'

Array them around a clock face with their corresponding binary and decimal values.

numbercircle(q);

Unsigned Fixed-Point Number Circle

Unsigned fixed-point values are unsigned integers that are scaled by a power of two. The negative exponent of the power of two is called the fraction length.

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

uf = u*2^-f

For example, define a 3-bit unsigned fixed-point quantizer with a fraction length of 1 and enumerate its range.

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

uf = 8×1

         0
    0.5000
    1.0000
    1.5000
    2.0000
    2.5000
    3.0000
    3.5000

Display those values in binary.

b = num2bin(q,uf)

b = 8x3 char array
    '000'
    '001'
    '010'
    '011'
    '100'
    '101'
    '110'
    '111'

Array them around a clock face with their corresponding binary and decimal values.

numbercircle(q);

Unsigned Fractional Fixed-Point Number Circle

Unsigned fractional fixed-point numbers are fixed-point numbers whose fraction length 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. Therefore, multiplication does not overflow.

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, define a 3-bit unsigned fractional fixed-point quantizer, which implies a fraction length of 3.

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

uf = 8×1

         0
    0.1250
    0.2500
    0.3750
    0.5000
    0.6250
    0.7500
    0.8750

Display those values in binary.

b = num2bin(q,uf)

b = 8x3 char array
    '000'
    '001'
    '010'
    '011'
    '100'
    '101'
    '110'
    '111'

Array them around a clock face with their corresponding binary and decimal values.

numbercircle(q);

Signed Two's-Complement Integer Number Circle

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, 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)'

s = 8×1

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

Display those values in binary.

b = num2bin(q,s)

b = 8x3 char array
    '100'
    '101'
    '110'
    '111'
    '000'
    '001'
    '010'
    '011'

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

Array them around a clock face with their corresponding binary and decimal values.

numbercircle(q);

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.

Signed Fixed-Point Number Circle

Signed fixed-point values are signed integers that are scaled by a power of two. The negative exponent of the power of two is called the fractionlength.

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

sf = s*2^-f

For example, define a 3-bit signed fixed-point quantizer with a fraction length of 1 and enumerate its range.

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

sf = 8×1

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

Display those values in binary.

b = num2bin(q,sf)

b = 8x3 char array
    '100'
    '101'
    '110'
    '111'
    '000'
    '001'
    '010'
    '011'

Array them around a clock face with their corresponding binary and decimal values.

numbercircle(q);

Signed Fractional Fixed-Point Number Circle

Signed fractional fixed-point numbers are fixed-point numbers whose fraction length 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.

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 word length in bits.

sf = s*2^-f

For example, define a 3-bit signed fractional fixed-point quantizer, which implies a fraction length of 2.

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

sf = 8×1

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

Display those values in binary.

b = num2bin(q,sf)

b = 8x3 char array
    '100'
    '101'
    '110'
    '111'
    '000'
    '001'
    '010'
    '011'

Array them around a clock face with their corresponding binary and decimal values.

numbercircle(q);

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