Quadrature mirror filter banks and method

Claim 1

A pseudo-quadrature-mirror filter bank for near-perfect-reconstruction pseudo-quadrature-mirror filtering of an input signal, comprising: a plurality of analysis filters, each of said plurality of analysis filters including, a first delay chain, operatively coupled to the input signal, forming a set of 2M parallel paths for buffering the input signal; a first cascade of 2M polyphase components of an impulse transfer function H(z) of impulse response, h(n), operatively coupled to said first delay chain; and means, operatively coupled to said first cascade of 2M polyphase components, for generating a 2M-point Discrete Fourier Transform (DFT) to implement a 2M-point Discrete Cosine Transform (DCT) of the input signal, with each analysis filter having an impulse response, h k (n), of a k th analysis filter, where M is the number of subband signals, obtained by cosine-modulating an impulse response, h(n), of a prototype filter with linear phase, according to: ##EQU65## and N is the length of the impulse response, h(n), of the prototype filter; a plurality of synthesis filters, each of said plurality of synthesis filters including, a second delay chain, operatively coupled to the input signal, forming a set of 2M parallel paths for buffering the input signal; a second cascade of 2M polyphase components of an impulse transfer function H(z) of impulse response, h(n), operatively coupled to said second delay chain; and means, operatively coupled to said second cascade of 2M polyphase components, for generating a 2M-point Discrete Fourier Transform (DFT) to implement a 2-point Discrete Cosine Transform (DCT) of the input signal, with each of said plurality of synthesis filters operatively coupled to a respective one of said plurality of analysis filters, each synthesis filter having an impulse response, f k (n), of a k th synthesis filter, obtained by cosine-modulating the impulse response, h(n), of the prototype filter according to: ##EQU66## and N is the length of the impulse response, h(n), of the prototype filter; and wherein each impulse response, h(n), is found in accordance with: ##STR5## for even N, where n=2M(m-l)+2m 1 -1 and x is the greatest integer less than x, for x equal to (m+1)/2, and in accordance with: ##STR6## for odd N, where n=2M(m-f)+2m 1 -1 and x is the greatest integer less than x, for x equal to any of 1+m/2 and m/2, where J is an inverse identity matrix, matrix V is defined to be: ##EQU67## wherein each impulse response, h(n), is found to minimize the stopband error: ##EQU68## where P is a real, symmetric and positive definite matrix, with the elements, using a notation P k,l for denoting a (k,l) th element of matrix P, ##EQU69## where N is even, and where P is a real, symmetric and positive definite matrix, with the elements ##EQU70## where N is odd, and where K is the number of stopbands of H(e j ω), β i are their relative weights and ω i ,1 and ω i ,2 are the bandedges of these stopbands, and ##EQU71## and wherein the filter H k (z) is optimized by finding a least squares optimization h opt such that: ##EQU72##

Claim 10

A method, using a pseudo-quadrature-mirror-filter bank, for near-perfect-reconstruction pseudo-quadrature-mirror filtering an input signal, comprising the steps of: finding an impulse response, h(n), of a prototype filter in accordance with: ##STR7## for even N, where n=2M(m-l)+2m 1 -1 and x is the greatest integer less than x, for x equal to (m+1)/2, and in accordance with: ##STR8## for odd N, where n=2M(m-l)+2m 1 and x is the greatest integer less than x, for x equal to any of 1+m/2 and m/2, where J is an inverse identity matrix, matrix V is defined to be: and ##EQU84## finding the impulse response, h(n), minimizing the stopband error: ##EQU85## wherein P is a real, symmetric and positive definite matrix, with the elements, using a notation P k,l for denoting a (k,l) th element of matrix P, ##EQU86## where N is even, and wherein P is a real, symmetric and positive definite matrix, with the elements ##EQU87## where N is odd, with K is the number of stopbands of H(e j ω), β i are their relative weights and ω i ,1 and ω i ,2 are the bandedges of these stopbands, and optimizing, by least squares optimization, H k (z) such that: ##EQU88## generating a plurality of analysis filters from the impulse response, h(n), each analysis filters having an impulse response, h k (n), of a k th analysis filter generated by cosine-modulating the impulse response, h(n), with linear phase, according to: ##EQU89## and N is the length of the impulse response, h(n), of the prototype filter; generating a plurality of synthesis filters, from the impulse response, h(n), each synthesis filter having an impulse response, f k (n), of a k th synthesis filter, of the prototype filter according to: ##EQU90## and N is the length of the impulse response, h(n), of the prototype filter; and coupling the output of each k th analysis filter with each k th synthesis filter, respectively.

Claim 11

A method, using a pseudo-quadrature-mirror-filter bank, for near-perfect-reconstruction pseudo-quadrature-mirror filtering an input signal, comprising the steps of: finding an impulse response, h(n), of a prototype filter in accordance with: ##STR7## for even N, where n=2M(m-l)+2m 1 -1 and x is the greatest integer less than x, for x equal to (m+1)/2, and in accordance with: ##STR8## for odd N, where n=2M(m-l)+2m 1 and x is the greatest integer less than x, for x equal to any of 1+m/2 and m/2, where J is an inverse identity matrix, matrix V is defined to be: and ##EQU84## finding the impulse response, h(n), minimizing the stopband error: ##EQU85## wherein P is a real, symmetric and positive definite matrix, with the elements, using a notation P k,l for denoting a (k,l) th element of matrix P, ##EQU86## where N is even, and wherein P is a real, symmetric and positive definite matrix, with the elements ##EQU87## where N is odd, with K is the number of stopbands of H(e j ω), β i are their relative weights and ω i ,1 and ω i ,2 are the bandedges of these stopbands, and optimizing, by least squares optimization, H k (z) such that: ##EQU88## generating a plurality of analysis filters from the impulse response, h(n), each analysis filters having an impulse response, h k (n), of a k th analysis filter generated by cosine-modulating the impulse response, h(n), with linear phase, according to: ##EQU89## and N is the length of the impulse response, h(n), of the prototype filter; generating a plurality of synthesis filters, from the impulse response, h(n), each synthesis filter having an impulse response, f k (n), of a k th synthesis filter, of the prototype filter according to: ##EQU90## and N is the length of the impulse response, h(n), of the prototype filter; and coupling the output of each k th analysis filter with each k th synthesis filter, respectively.

Claim 15

A method, using a pseudo-quadrature-mirror-filter bank, for near-perfect-reconstruction pseudo-quadrature-mirror filtering an input signal, comprising the steps of: finding an impulse response, h(n), of a prototype filter minimizing the stopband error: ##EQU92## optimizing the impulse response, h(n), of the prototype filter; generating a plurality of analysis filters from the impulse response, h(n), each analysis filters having an impulse response, h k (n), of a k th analysis filter generated by cosine-modulating the impulse response, h(n), with linear phase, according to: ##EQU93## and N is the length of the impulse response, h(n), of the prototype filter; generating a plurality of synthesis filters, from the impulse response, h(n), each synthesis filter having an impulse response, f k (n), of a k th synthesis filter, of the prototype filter according to: ##EQU94## and N is the length of the impulse response, h(n), of the prototype filter; and coupling the output of each k th analysis filter with each k th synthesis filter, respectively.

Claim 16

A method, using a pseudo-quadrature-mirror-filter bank, for near-perfect-reconstruction pseudo-quadrature-mirror filtering an input signal, comprising the steps of: finding an impulse response, h(n), of a prototype filter minimizing the stopband error: ##EQU92## optimizing the impulse response, h(n), of the prototype filter; generating a plurality of analysis filters from the impulse response, h(n), each analysis filters having an impulse response, h k (n), of a k th analysis filter generated by cosine-modulating the impulse response, h(n), with linear phase, according to: ##EQU93## and N is the length of the impulse response, h(n), of the prototype filter; generating a plurality of synthesis filters, from the impulse response, h(n), each synthesis filter having an impulse response, f k (n), of a k th synthesis filter, of the prototype filter according to: ##EQU94## and N is the length of the impulse response, h(n), of the prototype filter; and coupling the output of each k th analysis filter with each k th synthesis filter, respectively.