CN107168928A - The eight point Winograd Fourier transformers without rearrangement - Google Patents

Abstract

The present invention relates to a kind of eight point Winograd Fourier transformers without rearrangement, it is characterised in that the processor is main by input matrix I, variable diagonal matrix A, output matrix O and complex multiplier M₁~M₃Four parts are constituted.Input matrix I passes through complex multiplier M₁It is multiplied with input vector v and obtains vectorial p, variable diagonal matrix A passes through complex multiplier M₂It is multiplied with vectorial p and obtains vectorial q, output matrix O passes through complex multiplier M₃It is multiplied with vectorial q and obtains output vector V.Invention removes the index of N points with the rearrangement operation that eight point Winograd Fourier transformations are related in sequence prime factor algorithm, control logic is simplified, arithmetic speed is improved, memory consumption has been saved, hardware cost is reduced.

Description

Eight-point Winograd Fourier transformer without reordering

Technical Field

The invention relates to the field of digital signal processing, in particular to a method for realizing a small-point Winograd fast Fourier Transform Algorithm (WFTA) without reordering.

Background

With the ever-increasing growth of wireless communication services, the available spectrum resources are increasingly strained. In order to improve the spectrum utilization rate and the communication quality, the modern wireless communication system widely adopts an Orthogonal Frequency Division Multiplexing (OFDM) technology with strong immunity to Frequency selective fading. The core of OFDM technology is the FFT. The number of points of the FFT is divided into two categories, power-of-2 and non-power-of-2. The FFT algorithm with the number of points being the power of 2 is relatively mature. In contrast, FFT with a non-power-of-2 point number is more flexible and has recently been applied to DRM, DTMB, and LTE systems. Therefore, the algorithm and implementation of the non-power-of-2 point FFT is worthy of intensive study.

At present, Prime Factor Algorithm (PFA) is the most effective non-power-2 FFT, and a nested multidimensional structure is adopted, so that the calculation complexity can be effectively reduced. For an N-point non-power-of-2 FFT, it is assumed that N can be decomposed into the product of s two-by-two prime factors, i.e., N equals N₁N₂…N_s. The basic principle of N-point PFA is to map a one-dimensional large-point FFT to an s-dimensional small-point FFT, and perform N/N on the ith (i ═ 1,2, …, s) -dimensional FFT_iSub N_iPoint-by-point FFT. The small point number FFT may be aided by Cooley-Tukey algorithms, WFTA, and other efficient algorithms.

In some cases, the PFA needs to be reordered. The reordering is divided into pre-scrambling and post-scrambling according to where it is located during the calculation. Irrespective of N_i(i-1, 2, …, s) point FFT, which is co-located if the ith dimension FFT does not need to be reordered; otherwise, it is indexed, and the reordering is at N_iWithin the sequence of points, pre-scrambling and post-scrambling being performed at N_iThe point FFT is performed before and after. Similarly, regardless of the internal mechanism of each-dimensional FFT, an N-point PFA is in-order if it does not need to be reordered as a whole; otherwise, it is permuted, the reordering is performed within a sequence of N points, and the pre-scrambling and post-scrambling are performed before the first-dimension FFT starts and after the last-dimension FFT ends, respectively. Thus, PFA can be theoretically divided into 4 types: indexing and sequencing, same-address and sequence, and same-address and sequence.

Currently, PFA is either indexed in order or indexed in order, inevitably introducing reordering operations. As is well known, reordering means that a level one buffer must be added, more memory resources are consumed, and hardware cost is increased. In addition, reordering reduces the computation speed and increases the complexity of control. Compared with the same-address and same-sequence PFA, the same-address and same-sequence PFA consumes less memory resources, and the total time delay of reordering is the same, N clock cycles, so the same-address and same-sequence PFA is more preferable.

Disclosure of Invention

In response to the technical disadvantage of the need for reordering in the prior art implementation of PFA, the present invention provides an eight-point Winograd fourier transformer that does not require reordering. When N-point non-2-power FFT is realized by adopting indexing same-order PFA, if a certain mutlipin factor N of N_i8 (i-1, 2, …, s), then the i-th dimension FFT need not be reordered using this patent.

To remove the reordering operation of the ith dimension 8-point FFT, the conventional 8-point WFTA needs to be modified. The diagonal matrix of the conventional 8-point WFTA is fixed and invariable, and the invention expresses the angle parameter theta as 2 pi/8 x by each element on the diagonal of the diagonal matrix<N/8>₈The function of (a), wherein,<N/8>₈representing a modulo-8 operation on N/8. The diagonal matrices of the modified 8-point WFTA are different for different N.

For the ith-dimension FFT, N/8 times of 8-point WFTA which does not need to be reordered is needed, reordering is not needed, control logic is simplified, N/8 × 8 is saved in total, N clock cycles are saved, operation speed is improved, memory consumption is reduced by half, and hardware cost is reduced.

The advantages and spirit of the present invention can be further understood by the following detailed description and accompanying drawings.

Drawings

FIG. 1 is a functional block diagram of a conventional eight-point Winograd Fourier transformer;

FIG. 2 is a concrete constitution of an input matrix I;

fig. 3 is a specific configuration of the output matrix O;

FIG. 4 is a specific composition on the diagonal of the diagonal matrix D;

FIG. 5 is a schematic diagram of a structure of a pre-scrambled eight-point Winograd Fourier transformer;

FIG. 6 is a schematic structural diagram of an eight-point Winograd Fourier transformer for post-scrambling;

FIG. 7 is a functional block diagram of an eight-point Winograd Fourier transformer without reordering;

fig. 8 is a specific configuration on the diagonal of the variable diagonal matrix a.

Detailed Description

The invention is further described with reference to the following figures and specific examples, which are not intended to be limiting.

FFT of N-point sequence x (N) as

Wherein N, k is 0,1, …, N-1, W_N＝e-^j2π/N. The multiplication and addition operations for directly computing the N-point FFT are both proportional to the square of N. When N is large, the amount of calculation is large.

To reduce computational complexity, an N-point FFT can be implemented using nested multidimensional PFA when N is not a power of 2. Suppose N can be decomposed into s products of two-by-two interdependent factors, i.e., N equals N₁N₂…N_s. That is, any two factors N_iAnd N_jThe greatest common divisor of (i, j ≠ j) is 1,2, …, s, and i ≠ j. Note that N_iNot necessarily a prime number. The basic principle of the N-point PFA is that one-dimensional large-point FFT is mapped into s-dimensional FFT, and the ith-dimensional FFT is used for N/N_iSub N_iPoint-by-point FFT. The small point number FFT may be aided by Cooley-Tukey algorithms, WFTA, and other efficient algorithms.

In order to make PFA generally in the same order, when mapping one-dimensional FFT to s-dimensional FFT, according to the chinese remainder theorem, the input index n and the output index k adopt the same mapping manner as follows:

wherein, the symbol<>_NRepresenting modulo N operation, N_i,k_i＝0,1,…,N_i-1. The formula (2) and the formula (3) are substituted into the formula (1) and can be obtained by finishing the following steps:

wherein,

comparing equations (2) and (3) it is easy to find that the mapping manner of indices n and k is essentially identical. Therefore, as long as the index n of each-dimensional FFT in the equation (4)_iAnd k_iAll are naturally ordered, and N-point PFA is in-order.

In equation (4), the fourier transform factor of the i (i ═ 1,2, …, s) -th dimension FFT can be written as

Or

In the formula,

as is well known, conventional N_iThe input and output of the point FFT algorithm are in natural order. If the ith dimension FFT in equation (4) adopts the conventional N_iPoint FFT algorithm, then equation (5) is in terms of n_iNatural order input of, k_i' the natural sequence is output, and the formula (6) is according to n_i' Natural order input, k_iIs output in a natural order. However, as can be seen from equations (2) and (3), the i-th dimension FFT in equation (4) is required for the same order PFA in terms of n_iNatural order input of, k_iIs output in a natural order. It can be seen that if the ith dimension FFT in equation (4) employs the conventional N_iPoint FFT algorithm, then reordering is necessary. Specifically, equations (5) and (6) are post-scrambled and pre-scrambled according to the rules in equations (7) and (8), respectively. It can be seen that the i-th dimension FFT in equation (4) is indexed. Indexing is achieved by reordering. To get rid of this extra operation of reordering, we have to modify the regular N_iA point FFT algorithm, which takes the reordering operation into account.

Conventional N_iPoint WFTA can be represented by a continuous multiplication of a vector and a matrix, i.e.

V＝O*D*I*v (9)

Wherein V and V are each independently of the other N_iThe vector formed by the input and output sequences of points, I and O are the input and output matrices, respectively, and D is the diagonal matrix. In general, the elements in both matrices I and O are only possible 0, ± 1 and ± j, and multiplication with a vector involves no substantial multiplication. For the diagonal matrix D, the elements at other positions are 0 except for the elements at the diagonal.

When N is present_iWhen the value is 8, the i-th dimension FFT of the N-point indexed homosequential PFA may use 8-point WFTA. Fig. 1 shows a functional block diagram of a conventional eight-point Winograd fourier transformer. The 8-point input sequence forms a vector V, the vector V is multiplied by a matrix I, the vector obtained by operation is multiplied by a diagonal matrix D, the vector obtained by operation is multiplied by a matrix O, and the vector V obtained by operation is the 8-point output sequence. Fig. 2 and 3 show specific configurations of the input matrix I and the output matrix O, respectively. FIG. 4 shows the specific composition of the diagonal matrix D, where the elements from the top left corner to the bottom right corner are D₀～d₇。

Indexing of the ith dimension FFT of the indexed, in-order PFA is accomplished by reordering. When N is present_iFig. 5 and 6 show schematic diagrams of the eight-point Winograd fourier transformer for pre-scrambling and post-scrambling, respectively, at 8. To remove the reordering operation in pre-scrambling or post-scrambling, we have to modify the regular N_iThe reordering operation is taken in 8 points WFTA.

Fig. 7 shows a functional block diagram of an eight-point Winograd fourier transformer without reordering, which mainly comprises four functional blocks, i.e., an input matrix I, a variable diagonal matrix a, an output matrix O, and a complex multiplier. The output vector and the input vector satisfy: v ═ O × a ═ I × V. Compared with the conventional 8-point WFTA, the matrixes I and O of the 8-point WFTA without reordering are kept unchanged, the diagonal matrix is not a constant any more, and each element on the diagonal is modified into an angle parameter theta of 2 pi/8<N/8>₈The function of (a), wherein,<N/8>₈representing a modulo-8 operation on N/8. FIG. 8 shows the specific structure of the diagonal of the variable diagonal matrix A, where the elements from the top left corner to the bottom right corner are a₀～a₇. The diagonal matrices of 8-point WFTAs that do not need to be reordered are different for different N.

The invention provides a method for removing 8-point WFTA reordering in indexing same-order PFA, when N is_iWhen the value is 8, the i-th dimension FFT of the N-point indexed same-order PFA can be implemented by N/8 times of 8-point WFTA without reordering, and the steps are as follows:

(1) determining an angle parameter theta 2 pi/8 x from N<N/8>₈Initializing the value of each element on the diagonal of the variable diagonal matrix A on the basis of the specific value of (1), so that A becomes a constant, and initializing a variable l to be 0(0 is less than or equal to l)<N/8)；

(2) From the input sequence x n]Of 8 data whose index is n ═ c<N/8*m+8*l>_N(0≤m<8) They constitute a vector v;

(3) by means of a complex multiplier M₁Multiplying the input matrix I by the vector v to obtain a vector p;

(4) by means of a complex multiplier M₂Multiplying the variable diagonal matrix A by the vector p to obtain a vector q;

(5) by means of a complex multiplier M₃Multiplying the output matrix O by the vector q to obtain a vector V;

(6) writing 8 data in vector V into output sequence X [ k ] in sequence]In (2), the written index is identical to the read index, and k is still equal to<N/8*m+8*l>_N；

(7) And (5) gradually changing the value of l by taking 1 as a step length, and repeating the steps (2) to (6) until N/8 times of 8-point WFTA without reordering are completed.

If the i-th dimension FFT of the N-point indexed co-ordered PFA uses a regular 8-point WFTA, then 8 clock cycles are required for each reordering, which means that N/8 × 8 cycles of the regular 8-point WFTA are required for N/8 × 8 reordering. Therefore, if the ith dimension FFT of the N-point index same-sequence PFA adopts the invention, the reordering is not needed, thereby simplifying the control logic, saving N clock cycles, improving the operation speed, reducing the memory requirement by half and reducing the hardware cost.

While the present invention has been described in detail and by way of examples and embodiments, it will be apparent to those skilled in the art that various changes and modifications can be made without departing from the spirit and scope of the invention.

Claims (4)

1. An eight-point Winograd Fourier transformer without reordering is nested in an s-dimensional N-point indexing same-order prime factor algorithm, wherein N is N₁N₂…N_sAny two different factors N_iAnd N_jReciprocity, i-1, 2, …, s, j-1, 2, …, s, when a certain factor N_iWhen the value is 8, the processor may be configured to remove reordering operations of an ith-dimension FFT of an N-point indexed prime factor algorithm, and the processor may include:

complex multiplier M₁～M₃They perform a momentMultiplication of the array and the vector;

an input matrix I which is passed through a complex multiplier M₁Multiplying the input vector v to obtain a vector p;

a variable diagonal matrix A which is passed through a complex multiplier M₂Multiplying the vector p to obtain a vector q;

an output matrix O which passes through a complex multiplier M₃And multiplying the vector q to obtain an output vector V.

2. The eight-point Winograd fourier transformer of claim 1, wherein the input matrix I and the output matrix O are the same as a conventional eight-point Winograd fourier transformer, and the diagonal matrix is modified from a conventional constant matrix to a variable matrix a.

3. The eight-point Winograd fourier transformer of claim 1, wherein each element on a diagonal of the variable diagonal matrix a is an angular parameter θ 2 pi/8 ·<N/8>₈The function of (a), wherein,<N/8>₈representing a modulo-8 operation on N/8.

4. An eight-point Winograd Fourier transform method for removing ith-dimension FFT reordering operation of an N-point index same-sequence prime factor algorithm, wherein N is N₁N₂…N_sAny two different factors N_iAnd N_jMutilins, i-1, 2, …, s, j-1, 2, …, s, N_iThe processing method is characterized by comprising the following steps:

(1) determining an angle parameter theta 2 pi/8 x from N<N/8>₈Initializing the value of each element on the diagonal of the variable diagonal matrix A on the basis of the specific value of (1), so that A becomes a constant, and initializing a variable l to 0, wherein l is not less than 0<N/8，<N/8>₈Represents the operation of taking the module 8 of the N/8;

(2) from the input sequence x n]Of 8 data whose index is n ═ c<N/8*m+8*l>_NThey form a vector v, where 0 ≦ m<8，<N/8*m+8*l>_NShows taking the modulus N for N/8 m +8 lOperating;

(3) by means of a complex multiplier M₁Multiplying the input matrix I by the vector v to obtain a vector p;

(4) by means of a complex multiplier M₂Multiplying the variable diagonal matrix A by the vector p to obtain a vector q;

(5) by means of a complex multiplier M₃Multiplying the output matrix O by the vector q to obtain a vector V;

(6) writing 8 data in vector V into output sequence X [ k ] in sequence]In (2), the written index is identical to the read index, and k is still equal to<N/8*m+8*l>_N；

(7) And (5) gradually changing the value of l by taking 1 as a step length, and repeating the steps (2) to (6) until N/8 times of 8-point WFTA without reordering are completed.