Computationally efficient linear-phase finite impulse response filter

Claim 1

A method of digital signal processing using a computational means for filtering an input signal to obtain an output signal in accordance with a desired filter response, the method comprising the steps of: selecting a first filter response vector and a second filter response vector, determining a third filter response vector from the first filter response vector and the second filter response vector by an algorithm including the synthetic division algorithm, and generating the output signal from the input signal, the first filter response vector, the second filter response vector, and the third filter response vector by executing on the computational means a set of filter operation instructions.

Claim 2

A method of digital signal processing using a computational means for filtering an input signal to obtain an output signal in accordance with a desired filter response, the method comprising the steps of: selecting a first filter response vector and a second filter response vector, determining a third filter response vector from the first filter response vector and the second filter response vector by an algorithm including the synthetic division algorithm, and generating the output signal from the input signal, the first filter response vector, the second filter response vector, and the third filter response vector by executing on the computational means a set of filter operation instructions.

Claim 8

A method of digital signal processing for filtering an input signal, represented by an input signal sequence, x[0], x[1], x[2], . . . , x[n], to obtain an output signal, represented by an output signal sequence, y[0], y[1], y[2], . . . , y[n], the method employing a computational means and comprising the steps of: choosing two filter response vectors, b'=(b' 0 , . . . , b' P ) and a=(a 1 , . . . , a P ), which specify a desired filter response, calculating a third filter response vector, b"=(b" 0 , . . . , b" P ), determined by a polynomial remainder B"(z)=b" 0 +b" 1 z -1 + . . . +b" P z -P resulting from the division of a polynomial B'(z)z N =(b' 0 +b' 1 z -1 + . . . +b' P z -P )z N by a polynomial A(z)=1+a 1 z -1 + . . . +a P z -P , encoding in the computational means a set of filter operation instructions to calculate the output signal sequence, y[0], y[1], y[2], . . . , y[n], from the input signal sequence, x[0], x[1], x[2], . . . , x[n], the operation instructions comprising a calculation of an output relation, y[n]=b'·x'-b"·x"-a·y, and generating with the computational means the output signal from the input signal by computing the output signal sequence from the input signal sequence in accordance with the filter operation instructions, whereby the input signal is efficiently filtered to produce the output signal in a manner determined by the desired filter response.

Claim 9

A method of digital signal processing for filtering an input signal, represented by an input signal sequence, x[0], x[1], x[2], . . . , x[n], to obtain an output signal, represented by an output signal sequence, y[0], y[1], y[2], . . . , y[n], the method employing a computational means and comprising the steps of: choosing two filter response vectors, b'=(b' 0 , . . . , b' P ) and a=(a 1 , . . . , a P ), which specify a desired filter response, calculating a third filter response vector, b"=(b" 0 , . . . , b" P ), determined by a polynomial remainder B"(z)=b" 0 +b" 1 z -1 + . . . +b" P z -P resulting from the division of a polynomial B'(z)z N =(b' 0 +b' 1 z -1 + . . . +b' P z -P )z N by a polynomial A(z)=1+a 1 z -1 + . . . +a P z -P , encoding in the computational means a set of filter operation instructions to calculate the output signal sequence, y[0], y[1], y[2], . . . , y[n], from the input signal sequence, x[0], x[1], x[2], . . . , x[n], the operation instructions comprising a calculation of an output relation, y[n]=b'·x'-b"·x"-a·y, and generating with the computational means the output signal from the input signal by computing the output signal sequence from the input signal sequence in accordance with the filter operation instructions, whereby the input signal is efficiently filtered to produce the output signal in a manner determined by the desired filter response.

Claim 13

A method of digital signal processing for filtering an input signal, represented by an input signal sequence, x[0], x[1], x[2], . . . , x[n], to obtain an output signal, represented by an output signal sequence, y[0], y[1], y[2], . . . , y[n], the method employing a computational means and comprising the steps of: choosing two filter response vectors, b'=(b' 0 , . . . , b' P ) and a=(a 1 , . . . , a P ), which specify a desired filter response, calculating a third filter response vector, b"=(b" 0 , . . . , b" P ), determined by a polynomial remainder B"(z)=b" 0 +b" 1 z -1 + . . . +b" P z -P resulting from the division of a polynomial B'(z)z N =(b' 0 +b' 1 z -1 + . . . +b' P z -P )z N by a polynomial A(z)=1+a 1 z -1 + . . . +a P z -P , encoding in the computational means a set of filter operation instructions to calculate the output signal sequence, y[0], y[1], y[2], . . . , y[n], from the input signal sequence, x[0], x[1], x[2], . . . , x[n], the operation instructions comprising: calculating an output relation, y 1 [n]=b'·x'-b"·x"-a·y 1 , for a primary filter, calculating an output relation, y[n]=b'·x'-b"·x"-a·y, for an auxiliary filter, calculating an output relation, y[n]=b'·x'-b"·x"-a·y, for a time-reversed filter defined by a time-reversed transfer function, H(z), combining the calculated output relations for the primary filter, the auxiliary filter and the time-reversed filter to obtain the output signal sequence y[0], y[1], y[2], . . . , y[n], copying periodically an output delay line of the auxiliary filter to an output delay line of the primary filter, clearing periodically a tail end of an input delay line of the primary filter, clearing periodically an input delay line of the auxiliary filter, and clearing periodically the output delay line of the auxiliary filter, and generating with the computational means the output signal from the input signal by computing the output signal sequence from the input signal sequence in accordance with the filter operation instructions, whereby the desired filter response has a linear-phase response.

Claim 14

A method of digital signal processing for filtering an input signal, represented by an input signal sequence, x[0], x[1], x[2], . . . , x[n], to obtain an output signal, represented by an output signal sequence, y[0], y[1], y[2], . . . , y[n], the method employing a computational means and comprising the steps of: choosing two filter response vectors, b'=(b' 0 , . . . , b' P ) and a=(a 1 , . . . , a P ), which specify a desired filter response, calculating a third filter response vector, b"=(b" 0 , . . . , b" P ), determined by a polynomial remainder B"(z)=b" 0 +b" 1 z -1 + . . . +b" P z -P resulting from the division of a polynomial B'(z)z N =(b' 0 +b' 1 z -1 + . . . +b' P z -P )z N by a polynomial A(z)=1+a 1 z -1 + . . . +a P z -P , encoding in the computational means a set of filter operation instructions to calculate the output signal sequence, y[0], y[1], y[2], . . . , y[n], from the input signal sequence, x[0], x[1], x[2], . . . , x[n], the operation instructions comprising: calculating an output relation, y 1 [n]=b'·x'-b"·x"-a·y 1 , for a primary filter, calculating an output relation, y[n]=b'·x'-b"·x"-a·y, for an auxiliary filter, calculating an output relation, y[n]=b'·x'-b"·x"-a·y, for a time-reversed filter defined by a time-reversed transfer function, H(z), combining the calculated output relations for the primary filter, the auxiliary filter and the time-reversed filter to obtain the output signal sequence y[0], y[1], y[2], . . . , y[n], copying periodically an output delay line of the auxiliary filter to an output delay line of the primary filter, clearing periodically a tail end of an input delay line of the primary filter, clearing periodically an input delay line of the auxiliary filter, and clearing periodically the output delay line of the auxiliary filter, and generating with the computational means the output signal from the input signal by computing the output signal sequence from the input signal sequence in accordance with the filter operation instructions, whereby the desired filter response has a linear-phase response.

Claim 15

A method of digital signal processing for filtering an input signal, represented by an input signal sequence, x[0], x[1], x[2], . . . , x[n], to obtain an output signal, represented by an output signal sequence, y[0], y[1], y[2], . . . , y[n], the method employing a computational means and comprising the steps of: choosing two filter response vectors, b'=(b' 0 , . . . , b' P ) and a=(a 1 , . . . , a P ), which specify a desired filter response, calculating a third filter response vector, b"=(b" 0 , . . . , b" P ), determined by a polynomial remainder B"(z)=b" 0 +b" 1 z -1 + . . . +b" P z -P resulting from the division of a polynomial B'(z)z N =(b' 0 +b' 1 z -1 + . . . +b' P z -P )z N by a polynomial A(z)=1+a 1 z -1 + . . . +a P z -P , encoding in the computational means a set of filter operation instructions to calculate the output signal sequence, y[0], y[1], y[2], . . . , y[n], from the input signal sequence, x[0], x[1], x[2], . . . , x[n], the operation instructions comprising: calculating an output relation, y[n]=b'·x'-b"·x"-a·y, for a primary filter, attenuating the output signal y[n] by a factor of α[n] to obtain an attenuated output signal, where the factor of α[n] ranges between 0 and 1, shifting the attenuated output signal by an integral number of steps, calculating an output relation, y[n]=b'·x'-b"·x"-a·y, for an auxiliary filter, copying periodically an output delay line of the auxiliary filter to an output delay line of the primary filter, clearing periodically a tail end of an input delay line of the primary filter, clearing periodically an input delay line of the auxiliary filter, and clearing periodically the output delay line of the auxiliary filter, and generating with the computational means the output signal from the input signal by computing the output signal sequence from the input signal sequence in accordance with the filter operation instructions, whereby the output signal may be cross-faded with a signal from a second filter.

Claim 16

A method of digital signal processing for filtering an input signal, represented by an input signal sequence, x[0], x[1], x[2], . . . , x[n], to obtain an output signal, represented by an output signal sequence, y[0], y[1], y[2], . . . , y[n], the method employing a computational means and comprising the steps of: choosing two filter response vectors, b'=(b' 0 , . . . , b' P ) and a=(a 1 , . . . , a P ), which specify a desired filter response, calculating a third filter response vector, b"=(b" 0 , . . . , b" P ), determined by a polynomial remainder B"(z)=b" 0 +b" 1 z -1 + . . . +b" P z -P resulting from the division of a polynomial B'(z)z N =(b' 0 +b' 1 z -1 + . . . +b' P z -P )z N by a polynomial A(z)=1+a 1 z -1 + . . . +a P z -P , encoding in the computational means a set of filter operation instructions to calculate the output signal sequence, y[0], y[1], y[2], . . . , y[n], from the input signal sequence, x[0], x[1], x[2], . . . , x[n], the operation instructions comprising: calculating an output relation, y[n]=b'·x'-b"·x"-a·y, for a primary filter, attenuating the output signal y[n] by a factor of α[n] to obtain an attenuated output signal, where the factor of α[n] ranges between 0 and 1, shifting the attenuated output signal by an integral number of steps, calculating an output relation, y[n]=b'·x'-b"·x"-a·y, for an auxiliary filter, copying periodically an output delay line of the auxiliary filter to an output delay line of the primary filter, clearing periodically a tail end of an input delay line of the primary filter, clearing periodically an input delay line of the auxiliary filter, and clearing periodically the output delay line of the auxiliary filter, and generating with the computational means the output signal from the input signal by computing the output signal sequence from the input signal sequence in accordance with the filter operation instructions, whereby the output signal may be cross-faded with a signal from a second filter.