Transformation between time domain and frequency domain based on nearly orthogonal filter banks

A filter bank for signal composition, comprising:

• a plurality of filter units having one output and two inputs, wherein the two inputs comprise two paths that have complementary transfer functions, and wherein the plurality of filter units are connected to one another to form a tree structure,
• wherein the filter bank composes signals having Nc sub-carrier signals, the filter bank having Ns stages, stage s having 2s levels, wherein Ns=log2Nc, s is a stage number, and sε[0, 1 . . . Ns−1],
• wherein an nth order impulse response coefficient of an sth stage, qth level filter unit, hs,q(n), is calculated by multiplying an nth order impulse response coefficient of sth stage, pth level filter unit, hs,p(n), and a rotation factor, where the rotation factor is a complex exponential factor, and
• wherein the rotation factor is
• e
• j
• ⁢
• 2
• ⁢
• ⁢
• π
• N
• c
• ⁢
• (
• p
• ~
• -
• q
• ~
• )
• ,
• wherein {tilde over (p)} is a first value of a first bit reversed version of a first Ns−1 bits binary encode of p, and {tilde over (q)} is a second value of a second bit reversed version of a second Ns−1 bits binary encode of q.

A signal composing method, comprising:

• feeding Nc sub-carrier signals into Nc inputs of a tree-structured filter bank, respectively, wherein the filter bank has a plurality of filter units, each filter unit having one output and two inputs, wherein the two inputs comprise two paths that have complementary transfer functions;
• and
• obtaining a composed signal containing the Nc sub-carrier signals from an output of the filter bank,
• wherein the filter bank composes signals having Nc sub-carrier signals, the filter bank having Ns stages, stage s having 2s levels, wherein Ns=log2Nc, s is a stage number, and sε[0, 1 . . . Ns−1],
• wherein an nth order impulse response coefficient of an sth stage, qth level filter unit, hs,q(n), is calculated by multiplying an nth order impulse response coefficient of sth stage, pth level filter unit, hs,p(n), and a rotation factor, where the rotation factor is a complex exponential factor, and
• wherein the rotation factor is
• e
• j
• ⁢
• 2
• ⁢
• ⁢
• π
• N
• c
• ⁢
• (
• p
• ~
• -
• q
• ~
• )
• ,
• wherein {tilde over (p)} is a first value of a first bit reversed version of a first Ns−1 bits binary encode of p, and {tilde over (q)} is a second value of a second bit reversed version of a second Ns−1 bits binary encode of q.

A signal decomposing method comprising:

• feeding a signal containing Nc sub-carrier signals into a tree structured filter bank having one input and Nc outputs, where the filter bank has a plurality of filter units having one input and two outputs, where the two outputs comprise two paths that have complementary transfer functions;
• and
• obtaining the Nc sub-carrier signals from the Nc outputs of the filter bank, respectively,
• wherein the filter bank decomposes signals having Nc sub-carrier signals, the filter bank having Ns stages, stage s having 2s levels, wherein Ns=log2Nc, s is a stage number, and sε[0, 1 . . . Ns−1],
• wherein an nth order impulse response coefficient of an sth stage, qth level filter unit, hs,q(n), is calculated by multiplying an nth order impulse response coefficient of sth stage, pth level filter unit, hs,p(n), and a rotation factor, wherein the rotation factor is a complex exponential factor, and
• wherein the rotation factor is
• e
• j
• ⁢
• 2
• ⁢
• ⁢
• π
• N
• c
• ⁢
• (
• p
• ~
• -
• q
• ~
• )
• ,
• wherein {tilde over (p)} is a first value of a first bit reversed version of a first Ns−1 bits binary encode of p, and {tilde over (q)} is a second value of a second bit reversed version of a second Ns−1 bits binary encode of q.

A communication method, comprising:

• composing Nc sub-carrier signals using a first tree structured filter bank having Nc channels to obtain a composed signal containing the Nc sub-carrier signals;
• and
• decomposing the composed signal using a second tree structured filter bank having Nc channels to obtain the Nc sub-carrier signals,
• where vector form transfer function {right arrow over (H)}r,p of channel p of the second filter bank is orthogonal to vector form transfer function {right arrow over (H)}t,q of channel q of the first filter bank.