Data processing method and apparatus operable on an irrational mathematical value

A data processing apparatus which operates on data containing an irrational mathematical value, the apparatus comprising:

• first memory means for storing a first group of data elements having a predetermined relation to an irrational number;
• second memory means for storing a second group of data elements having a predetermined relation to a real number, the first group of data elements and the second group of data elements in combination corresponding to the irrational mathematical value;
• a multiplier for multiplying the first group of data elements and a representation of the irrational number to produce an intermediate result;
• and
• a summer for summing the intermediate result and the second group of data elements to produce a result, the result approximating the irrational mathematical value having minimal error due to truncation and rounding.

A method for operating on irrational mathematical values, the method comprising the steps of:

• storing in a first memory portion a first group of data elements representing a coefficient of an irrational mathematical value;
• storing in a second memory portion a second group of data elements representing a real number;
• combining the first group of data elements with input data elements representing another coefficient of the irrational mathematical value to produce a combined coefficient;
• multiplying the combined coefficient with an approximation of the irrational mathematical value to produce an intermediate result;
• and
• summing the intermediate result and the second group of data elements to produce a result having minimal error due to truncation and rounding.

A data processing apparatus which improves the accuracy of resultant data in operations with irrational mathematical values, the data processing apparatus comprising:

• an input of the apparatus configured to receive input data, the input data including data corresponding to an input coefficient to be multiplied by square root of two (.sqroot.2) and an input addend;
• a first memory for storing a coefficient of the square root of two;
• a second memory for storing an addend;
• a summer which independently sums the input coefficient and the coefficient to produce a combined coefficient and sums the input addend and the addend to produce an addend sum;
• a multiplier which multiplies the combined coefficient and an approximation of the square root of two to produce an intermediate result;
• and
• a summer which sums the intermediate result and the addend sum to produce the resultant data having minimal error due to truncation and rounding.

A method for reducing rounding error and truncation error of irrational mathematical values in a data processing system, the method comprising the steps of:

• storing a multiple-bit approximation of an irrational number;
• storing an exact representation of a coefficient associated with the irrational number;
• storing an exact representation of an addend associated with a product of the coefficient and the irrational number;
• multiplying the coefficient and the irrational number to produce an intermediate result;
• and
• summing the intermediate result and the addend to provide a result.

A method for digitally filtering a signal representing an irrational mathematical value, the method comprising the steps of:

• receiving an input signal;
• forming a digital representation of the input signal;
• mathematically operating on the digital representation by:
• storing a multiple-bit approximation of an irrational number;
• storing in a first memory portion an exact representation of a coefficient associated with the irrational number;
• storing in a second memory portion an exact representation of an addend associated with a product of the coefficient and the irrational number;
• combining the coefficient with input data elements representing another coefficient of the irrational number to produce a combined coefficient;
• multiplying the coefficient and the number to produce an intermediate result;
• summing the intermediate result and the addend to provide a result having minimal truncation and rounding error;
• and
• producing an output signal in response to the result.

A digital signal processor configured to digitally filter an input signal representing an irrational value, the digital signal processor comprising:

• an input of the processor configured to receive input data related to the input signal, the input data including data corresponding to an input coefficient to be multiplied by square root of two (.sqroot.2) and an input addend;
• a first memory for storing a coefficient of the square root of two;
• a second memory for storing an addend;
• a summer which independently sums the input coefficient and the coefficient to produce a combined coefficient and sums the input addend and the addend to produce an addend sum;
• a multiplier which multiplies the combined coefficient and an approximation of the square root of two to produce an intermediate result;
• a summer which sums the intermediate result and the addend sum to produce output data having minimal error due to truncation and rounding;
• and
• an output configured to provide an output signal corresponding to the output data.