Computing multidimensional DFTs in FPGA
First Claim
1. A method for computing multidimensional discrete Fourier transforms (DFTs) using a field programmable gate array (FPGA) comprising the steps of:
- a. ordering a set of polynomial data in a two dimensional matrix;
b. multiplying each element of the two dimensional matrix by ω
-n.sbsp.2 (where ω
=e-jπ
/N, e is a constant (ln(e)=1, where In indicates the natural logarithm), j=√
-1, n2 =the column index number in the matrix, and N=the transform length) to produce a premultiplication product;
c. performing a polynomial transform (PT) on the premultiplication product to produce a polynomial transform result;
d. performing N reduced DFTs of N terms on the polynomial transform result to produce a permuted output; and
e. reordering the permuted output to a natural order.
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Abstract
An FPGA configured for computation of an N×N discrete Fourier transform (DFT) using polynomial transforms defined in modified rings of transforms, comprising a first buffer for ordering a set of polynomial data in a two dimensional matrix, a multiplier for multiplying each element of the two dimensional matrix by ω-n.sbsp.2 (where ω=e-jπ/N, e is a constant (ln(e)=1),
j=√-1, n2 =the column index number in the matrix, and N=the transform length) to produce a premultiplication product, a polynomial transform circuit for performing a polynomial transform (PT) modulo (zN +1), size N, root z2 on the premultiplication product to produce a polynomial transform result, where z represents the unit delay operator, a reduced DFT calculator for performing N reduced DFTs of N terms on the polynomial transform result to produce a permuted output, and an address generator for reordering the permuted output to a natural order.
119 Citations
10 Claims
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1. A method for computing multidimensional discrete Fourier transforms (DFTs) using a field programmable gate array (FPGA) comprising the steps of:
-
a. ordering a set of polynomial data in a two dimensional matrix; b. multiplying each element of the two dimensional matrix by ω
-n.sbsp.2 (where ω
=e-jπ
/N, e is a constant (ln(e)=1, where In indicates the natural logarithm),j=√
-1, n2 =the column index number in the matrix, andN=the transform length) to produce a premultiplication product; c. performing a polynomial transform (PT) on the premultiplication product to produce a polynomial transform result; d. performing N reduced DFTs of N terms on the polynomial transform result to produce a permuted output; and e. reordering the permuted output to a natural order. - View Dependent Claims (2, 3)
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4. An apparatus for computation of an N×
- N discrete Fourier transform (DFT) using polynomial transforms defined in modified rings of transforms, comprising;
a. first buffer means for ordering a set of polynomial data in a two dimensional matrix; b. first multiplier means for multiplying each element of the two dimensional matrix by ω
-n.sbsp.2 (where ω
=e-jπ
/N, e is a constant (ln(e)=1, where ln indicates the natural logarithm), j=√
-1, n2 =the column index number in the matrix, and N=the transform length) to produce a premultiplication product;c. PT means for performing a polynomial transform (PT) modulo (zN +1), size N, root z2 on the premultiplication product to produce a polynomial transform result, where z represents the unit delay operator; d. reduced DFT means for performing N reduced DFTs of N terms on the polynomial transform result to produce a permuted output; and e. reordering means for reordering the permuted output to a natural order; wherein the first buffer means, the first multiplier means, the PT means, the reduced DFT means, and the reordering means are embodied in a field programmable gate array (FPGA). - View Dependent Claims (5, 6, 7)
- N discrete Fourier transform (DFT) using polynomial transforms defined in modified rings of transforms, comprising;
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8. An apparatus for computing multidimensional discrete Fourier transforms (DFTs) using a field programmable gate array (FPGA) comprising:
-
a. means for ordering a set of polynomial data in a multi-dimensional matrix; b. first multiplier means for multiplying each element of the multi-dimensional matrix by ω
-n.sbsp.2 (where ω
=e-jπ
/N, e is a constant (ln(e)=1, where In indicates the natural logarithm), j=√
-1, n2 =the column index number in the matrix, and N=the transform length) to produce a premultiplication product;c. PT means for performing a polynomial transform (PT) on the premultiplication product to produce a fast polynomial transform (FPT) result; d. reduced DFT means for performing N reduced DFTs of N terms on the FPT result to produce a permuted output; e. reordering means for reordering the permuted output to a natural order. - View Dependent Claims (9, 10)
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Specification