Digital signal processing circuits, systems, and method implementing approximations for logarithm and inverse logarithm

A digital signal system for determining an approximate antilog x from a value of f(x), wherein x has a base b, wherein the value of f(x) has an integer portion and a decimal portion, comprising:

• an input for receiving a signal;
• circuitry for measuring an attribute of the signal, wherein the attribute relates at least in part to the value of fix);
• circuitry for identifying from the integer portion of f(x) a largest power of b less than the absolute value of x;
• and
• circuitry for determining from the decimal portion of f(x) a value r representative of a difference between x and b to the largest power.

A digital signal system for determining an approximate antilog x from a value of f(x), wherein x has a base b, wherein the value of f(x) has an integer portion and a decimal portion, comprising:

• an input for receiving a signal;
• circuitry for measuring an attribute of the signal, wherein the attribute relates at least in part to the value of f(x);
• circuitry for identifying from the integer portion of f(x) a largest power of b less than the absolute value of x;
• and
• circuitry for forming a product between the decimal portion of f(x) raised to the largest power and a value equal to the integer portion of f(x) plus one, wherein the product equals the approximate antilog x.

A digital signal system for determining an approximate antilog of a negative number x from a value of f(x), wherein x has a base b, wherein the value of f(x) has an integer portion and a decimal portion, comprising:

• an input for receiving a signal;
• circuitry for measuring an attribute of the signal, wherein the attribute relates at least in part to the value of f(x);
• circuitry for identifying from the integer portion of f(x) a largest power of b less than the absolute value of x, wherein the circuitry for identifying identifies that the largest power of b less than the absolute value of x equals the integer portion of f(x) minus one;
• and
• circuitry for determining from the decimal portion of f(x) a value r representative of a difference between x and b to the largest power, wherein the circuitry for determining the value r comprises:
• circuitry for increasing the decimal portion of f(x) to form a sum;
• and
• circuitry for forming a product between the sum and the base b raised to the largest power.

A digital signal system for determining an approximate antilog of a negative number x from a value of f(x), wherein x has a base b, wherein the value of f(x) has an integer portion and a decimal portion, comprising:

• an input for receiving a signal;
• circuitry for measuring an attribute of the signal, wherein the attribute relates at least in part to the value of f(x);
• circuitry for determining x, comprising:
• circuitry for increasing the integer portion of f(x) by one to form a sum;
• and
• circuitry for forming a product between the sum and the base b raised to a power of the integer portion of f(x) minus one.

A digital signal system for determining an approximate antilog of a negative number x from a value of f(x), wherein x has a base b, wherein the value of f(x) has an integer portion and a decimal portion, comprising:

• an input for receiving a signal;
• circuitry for measuring an attribute of the signal, wherein the attribute relates at least in part to the value of f(x);
• circuitry for determining x, comprising:
• circuitry for increasing the decimal portion of f(x) by one to form a first sum;
• circuitry for forming a first product between the first sum and a constant;
• circuitry for forming a second product between the first product and the base b raised to a power of the integer portion of f(x) minus two;
• and
• circuitry for forming a second sum by combining the second product with the base b raised to a power of the integer portion of f(x) minus one, wherein the second sum is the approximate antilog of the negative number x.