KR101755987B1 - Apparatus and method for Sliding Discrete Fourier Transform - Google Patents

Apparatus and method for Sliding Discrete Fourier Transform Download PDF

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KR101755987B1
KR101755987B1 KR1020160024639A KR20160024639A KR101755987B1 KR 101755987 B1 KR101755987 B1 KR 101755987B1 KR 1020160024639 A KR1020160024639 A KR 1020160024639A KR 20160024639 A KR20160024639 A KR 20160024639A KR 101755987 B1 KR101755987 B1 KR 101755987B1
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박천수
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세종대학교산학협력단
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Abstract

A sliding discrete Fourier transform method and apparatus are disclosed. The sliding discrete Fourier transform method according to an embodiment of the present invention includes a step of calculating a sliding updating vector transform (SUVT) using an intermediate result calculated in a previous window and a result of a sliding updating vector transform And calculating an SDFT output.

Description

[0001] The present invention relates to a sliding discrete Fourier transform method and apparatus,

The present invention relates to a sliding discrete Fourier transform method and apparatus.

Discrete orthogonal transformations such as Discrete Fourier transform (DFT), Discrete Hartley Transform (DHT), and Walsh Hadamard Transform (WHT) play an important role in the field of digital signal processing, filtering and communication. Recently, interest in the sliding conversion process is increasing. This sliding conversion process is such that the conversion window is shifted by one sample at a time and the conversion process is repeated.

The sliding DFT (SDFT) algorithm proposed by Jacobsen and Lyons ([1] E. Jacobsen and R. Lyons, "An update to the sliding DFT," IEEE Signal Processing Mag. ? 111, Jan. 2004.) is to calculate the DFT value using the iterative scheme. This SDFT algorithm can drastically reduce the amount of DFT computation, but it must tolerate latent instability. Therefore, the following algorithms have been proposed to guarantee the stability of SDFT.

[2] E. Jacobsen and R. Lyons, "The sliding DFT," IEEE Signal Processing Mag., Vol. 20, no. 2, pp. 74 ?? 80, Mar. 2003.

[3] K. Duda, "Accurate, guaranteed stable, sliding discrete Fourier transform," IEEE Signal Processing Mag., Vol. 27, no. 6, pp. 124 ?? 127, Nov. 2010.

[4] C. S. Park and S.J. Ko, "The hopping discrete Fourier transform," IEEE Signal Processing Mag., Vol. 31, no. 2, pp. 135-139, Mar. 2014.

These algorithms guarantee the stability of the sliding conversion process by increasing the computational complexity or decreasing the calculation accuracy.

The present invention is to provide a new sliding DFT method and apparatus that is fast, accurate, and stable.

According to an aspect of the present invention, a sliding discrete Fourier transform (SDFT) method with stability is disclosed.

The sliding discrete Fourier transform method according to an embodiment of the present invention includes the steps of calculating a sliding updating vector transform (SUVT) using an intermediate result calculated in a previous window and a result of the sliding updating vector transform And calculating the SDFT output using the SDFT output.

The SUVT is defined by the following equation.

Figure 112016019957822-pat00001

Here, D n L (k) is the kth value of the updating vector transform (UVT) of the L point, k is the index of the frequency domain having a range of 0? K <M, n is the index of the time index ego,

Figure 112016019957822-pat00002
ego,
Figure 112016019957822-pat00003
Which is a complex twiddle factor.

remind

Figure 112016019957822-pat00004
And
Figure 112016019957822-pat00005
Have the same relationship, and the relationship is expressed by the following equation.

Figure 112016019957822-pat00006

The SDFT is defined by the following equation.

Figure 112016019957822-pat00007

Where X n (k) is the kth value of the M point DFT at time index n.

L is any one of M / 4, M / 2 and 3M / 4.

When L is M / 4, the rotation factor W M Lk is expressed by the following equation.

Figure 112016019957822-pat00008

The above-mentioned X n (k) is simplified by the following equation using j k .

Figure 112016019957822-pat00009

When L is M / 4, X n (k) is generalized by the periodic characteristic of j k according to the following equation.

Figure 112016019957822-pat00010

When k = 4i, the relationship between Xn (4i) and Xn- M / 4 (4i) is expressed by the following equation.

Figure 112016019957822-pat00011

Here, the expressions of Re () and Im () represent the real part and the imaginary part of a complex number, respectively.

According to another aspect of the present invention, a sliding discrete Fourier transform apparatus for performing a sliding discrete Fourier transform (SDFT) method with stability is disclosed.

A sliding discrete Fourier transform apparatus according to an embodiment of the present invention includes a memory for storing an instruction and a processor for executing the instruction, the instruction including a sliding updating vector transformation (SUVT) using an intermediate result calculated in a previous window, : calculating a sliding updating vector transform; and calculating the SDFT output using the result of the sliding updating vector transform.

The sliding discrete Fourier transform method and apparatus according to an embodiment of the present invention can provide fast, accurate, and stable stability. Among the SDFT algorithms that are guaranteed to be stable, the smallest amount of computation is required, and DFT Mathematically equivalent output values can be provided.

BRIEF DESCRIPTION OF THE DRAWINGS FIG. 1 is a diagram illustrating an example of a sliding updating vector conversion method according to an embodiment of the present invention; FIG.
2 is a flowchart illustrating a sliding discrete Fourier transform method according to an embodiment of the present invention.
3 is a diagram illustrating calculation complexity according to an M value for each algorithm;
4 is a diagram illustrating a simulation result of a sliding discrete Fourier transform method according to an embodiment of the present invention.
5 is a diagram schematically illustrating a configuration of a sliding discrete Fourier transform apparatus according to an embodiment of the present invention.

As used herein, the singular forms "a", "an" and "the" include plural referents unless the context clearly dictates otherwise. In this specification, the terms "comprising ", or" comprising "and the like should not be construed as necessarily including the various elements or steps described in the specification, Or may be further comprised of additional components or steps. Also, the terms "part," &quot; module, "and the like described in the specification mean units for processing at least one function or operation, which may be implemented in hardware or software or a combination of hardware and software .

First, prior to describing the sliding discrete Fourier transform method according to the embodiment of the present invention, the SDFT algorithm which is guaranteed to be stable will be described in order to facilitate understanding and explanation of the present invention.

In the SDFT algorithm, a transformation value is calculated in a fixed-length window. The complex input signal (x (n), n = 0, 1, 2, ...) is divided into M overlapping windows. Assuming that k is an index of the frequency domain having a range of 0? K < M, the kth value of the M point DFT at the time index n can be calculated by the following equation.

Figure 112016019957822-pat00012

here,

Figure 112016019957822-pat00013
ego,
Figure 112016019957822-pat00014
to be.

Therefore, the formula of SDFT [1] by the circular shift characteristic can be expressed by the following equation.

Figure 112016019957822-pat00015

Here, the periodic characteristic of a complex twiddle factor is used (

Figure 112016019957822-pat00016
).

SDFT can have a slightly stable transfer function because the pole is placed in the unit circle of the z domain. In practice, the complex rotation factor in equation (2) can be expressed in a floating point format with finite precision. Numerical rounding of the complex rotation factor can move the poles out of the unit circle and cause instability of the system. In order to solve this problem, SDFT algorithm with stability is proposed using a simple iterative updating scheme [2]. This algorithm is realized using a periodic time-varying system, so that numerical errors caused by limited precision calculations can be reduced exponentially to zero on the time axis.

An algorithm called rSDFT [3] has been proposed as an SDFT algorithm with other stability. This algorithm can ensure stability by placing the poles in the r radius inside the unit circle using the damping factor r. The DFT approximation using the damping index r can be expressed by the following equation.

Figure 112016019957822-pat00017

Where 0 &quot; r &lt; 1, and the tilde indicates the approximation of the DFT. Similarly, the SDFT repetition equation of Equation (2) can be expressed by the following equation.

Figure 112016019957822-pat00018

The output value of rSDFT is different from the DFT of Equation (1). Furthermore, an error can be accumulated in the output value.

Recently, a modulated SDFT (mSDFT: modulated SDFT) algorithm [4] has been introduced. The mSDFT first performs a modulation sequence on the input signal (

Figure 112016019957822-pat00019
), Thereby calculating a modulated sequence. Next, using the modulated sequence, the mSDFT can calculate the iteration formula of DFT when k = 0 by the following equation.

Figure 112016019957822-pat00020

here,

Figure 112016019957822-pat00021
to be. As a result, X n (k) can be calculated by the following equation.

Figure 112016019957822-pat00022

That is, by excluding the complex rotation factor from the feedback of the resonator, the mSDFT can have a pole located precisely at the unit circle, and the stability can be guaranteed without any condition. However, this mSDFT has more than twice the computational complexity of other SDFT algorithms.

Hereinafter, a sliding discrete Fourier transform method according to an embodiment of the present invention will be described.

As described above, the stability guaranteed SDFT filter can be realized by excluding an incorrect rotation factor from the feedback loop. Thus, it is necessary to examine the peculiar relationship between the DFT output values that can be expressed without using the rotation factor.

Basically, SDFT can calculate the DFT value by taking advantage of the iterative relationship between successive DFT values. That is, by extending the relationship between successive values, a general formula between DFT values with a distance of L hops can be derived.

Figure 112016019957822-pat00023
, The following equation can be obtained by repeatedly replacing X n (k) by L times using X n-1 (k) in Equation (2).

Figure 112016019957822-pat00024

D n L (k) may be defined as the kth value of the updating vector transform (UVT) of the L point, and can be expressed by the following equation.

Figure 112016019957822-pat00025

Here, 0? K <M. Using this notation, Equation (7) can be simplified and expressed by the following equation.

Figure 112016019957822-pat00026

This allows the DFT output value at the time index n to be calculated directly from the DFT output value at the time index (nL) by using D n L (k). The sliding updating vector conversion method according to an embodiment of the present invention will be described later in detail with reference to FIG.

In Equation (9), the rotation factor W M Lk is multiplied by the delayed output value, and the value of the rotation factor changes in accordance with the time hop L. Hereinafter, it is assumed that L = M / 4. However, the present invention is not limited thereto. For example, L = M / 2 and L = 3M / 4.

That is, when L is M / 4, the rotation factor W M Lk can be expressed by the following equation.

Figure 112016019957822-pat00027

Therefore, Equation (9) can be simplified to the following equation.

Figure 112016019957822-pat00028

Then, according to the periodicity of j, the following equation can be obtained.

Figure 112016019957822-pat00029

Here, i = 0, 1, ... , M / 4-1. This can form the basis of an efficient scheme for calculating the DFT output value of the shifted window. Given the DFT output value of the previous window in the time index (nM / 4), the DFT output value at the time index n can be calculated directly without multiplying with the rotation factor. For example, when k = 4i, the relationship between Xn (4i) and XnM / 4 (4i) can be expressed by the following equation.

Figure 112016019957822-pat00030

Here, the expressions of Re () and Im () represent a real part and an imaginary part of a complex number, respectively. Then, each part of X n (4i) can be obtained by the following equation.

Figure 112016019957822-pat00031

k = 4i + 1, 4i + 2, 4i + 3. This result shows that only two real parts are needed to compute X n (k) using X nM / 4 (k) and D n M / 4 (k). Moreover, since the multiplication by an incorrect rotation factor is excluded from the iterative calculation, no numerical error is accumulated. Therefore, the sliding discrete Fourier transform method according to the embodiment of the present invention can guarantee the stability (gSDFT: guaranteed-stable SDFT).

1 is a diagram illustrating an example of a sliding updating vector conversion method according to an embodiment of the present invention.

The quick calculation of UVT in Equation (8) is an important part because it determines the calculation cost of the sliding discrete Fourier transform method according to the embodiment of the present invention. Since the calculation of the UVT value is similar to the DFT, a conventional fast Fourier transform (FFT) algorithm can be used. A radix-2 decimation-in-time (DIT) algorithm is used in the sliding updating vector transform (SUVT) method according to an embodiment of the present invention. This algorithm divides the L size UVT into two L / 2 sized interleaved UVTs at each iteration stage. On the basis of the time-division algorithm, D n L (k) using the decimated sequence can be expressed by the following equation.

Figure 112016019957822-pat00032

here,

Figure 112016019957822-pat00033
And
Figure 112016019957822-pat00034
Can be expressed by the following equation.

Figure 112016019957822-pat00035

Equation (15) shows that D n L (k) is obtained using two UVT values of the decimated sequence. Here, the decimated sequence is {d (n), d (n - 2), ... , d (n- L + 2)} and {d (n - 1), d (n - 3), ... , d (n - L + 1)}. The decimation process is repeated until the resulting sequence is reduced to a one-point sequence.

In the SDFT algorithm, UVT needs to be iteratively calculated at each time index. Thus, if the UVT output value of the current window can be obtained using the intermediate calculation of the preceding window, the amount of calculation of UVT can be further reduced. From this, it can be easily deduced that the UVT calculation in the continuous time index has a relationship represented by the following equation.

Figure 112016019957822-pat00036

Equation (17)

Figure 112016019957822-pat00037
This is exactly
Figure 112016019957822-pat00038
. This is important because the intermediate computation of the preceding window relative to the current window can be reused without lack of computational accuracy. Based on this result, a sliding updating vector conversion method according to an embodiment of the present invention can be derived.

At time index n, the SUVT algorithm according to an embodiment of the present invention first uses a time division (DIT) algorithm

Figure 112016019957822-pat00039
. Then, according to equation (15)
Figure 112016019957822-pat00040
Wow
Figure 112016019957822-pat00041
The UVT output value D n L (k) is obtained. here,
Figure 112016019957822-pat00042
Has already been acquired in the previous window. All that is needed is an additional memory to store the intermediate computation results of the previous window.

FIG. 1 shows an SUVT calculation process when M = 16 and L = 4, according to an embodiment of the present invention. In the example of FIG. 1, the normal line and the dotted line indicate addition and subtraction, respectively, and only the portion indicated by the black dot is used for the calculation. Since the rotation factor used in the SUVT is the same as that used in the conventional butterfly-based FFT algorithm, the accuracy of the gSDFT according to the embodiment of the present invention is the same as that of the conventional FFT.

2 is a flowchart illustrating a sliding discrete Fourier transform method according to an embodiment of the present invention.

The sliding discrete Fourier transform method according to the embodiment of the present invention repeatedly calculates a DFT output value by shifting a fixed-length window one sample at a time. Hereinafter, for the convenience of understanding and explanation of the invention, only L = M / 4 will be described. It is assumed that the time index is n and the window size is M.

In step S210, the sliding discrete Fourier transform device calculates d (n) using the input samples. At this stage, only one complex input sample is needed.

In step S220, the sliding discrete Fourier transform apparatus calculates {d (n), d (n - 2), ... , d (n- M / 4)}

Figure 112016019957822-pat00043
. Here, {d (n), d (n - 2), ... , d (n- M / 4)} are already obtained through the calculation process in the previous window. Decimation
Figure 112016019957822-pat00044
Times,
Figure 112016019957822-pat00045
In the calculation,
Figure 112016019957822-pat00046
Times complex multiplication and
Figure 112016019957822-pat00047
Complex input samples are needed.

In step S230, the sliding discrete Fourier transform device

Figure 112016019957822-pat00048
And
Figure 112016019957822-pat00049
To calculate the L-order UVT D n M / 4 (k). here,
Figure 112016019957822-pat00050
Is calculated in the calculation of the preceding window. The calculated D n M / 4 (k) becomes the input value in the next step. In this step, M / 2 complex multiplication and M complex input samples are required, as shown in equation (15). Calculated from the previous window
Figure 112016019957822-pat00051
An additional memory of M / 2 size may be required to be stored.

In step S240, the sliding discrete Fourier transform apparatus calculates X n (k) using X nM / 4 (k) and D n M / 4 (k) according to equation (12). At this stage, only 2M real numbers are needed.

FIG. 3 is a diagram illustrating a calculation complexity according to an M value for each algorithm. 3, R M and R A represent the number of real multiplication and the number of real part, respectively.

Referring to FIG. 3, in particular, when M = 16, the multiplication number of the gSDFT algorithm according to the embodiment of the present invention is reduced to 50% and 70% for the rSDFT and mSDFT algorithms, respectively. When M <2 13 , the number of multiplications of the gSDFT algorithm is smaller than that of the existing mSDFT algorithm. Therefore, not to be the error accumulation value, and to the application is gopset number of times the most important consideration, and M is the number gSDFT algorithm to be used when less than 213, and in other cases may be used mSDFT algorithm.

The main features of the gSDFT algorithm according to the embodiment of the present invention are as follows.

1) When M is less than 2 13 , the multiplication times of the gSDFT algorithm are smaller than those of the conventional methods.

2) The iterative equation of Equation (12) is unconditionally stable and no numerical error is accumulated.

3) The gSDFT algorithm produces an output signal mathematically equivalent to the DFT output of Equation (1) at all time indices.

4) The SUVT is calculated independently of the iteration of the SDFT. This separation feature is a great advantage for hardware implementations.

4 is a diagram illustrating a simulation result of a sliding discrete Fourier transform method according to an embodiment of the present invention.

The efficiency of the sliding discrete Fourier transform method according to an embodiment of the present invention is investigated using a zero-mean Gaussian noise with a standard deviation of 0 as a test signal. The simulation was performed with 64-bit double-precision arithmetic, and M was set to 16. In the simulation, numerical errors of rSDFT, mSDFT and gSDFT were generated by iteratively calculating Equations 4, 5 and 12, respectively. This iterative calculation was repeated 10 6 times to scale the numerical error and a numerical error was calculated at each time index. All algorithms were executed using highly efficient ANSI-C code, and performance was evaluated on Intel i5 3.4 GHz CPU and 8 GB RAM.

Figure 4 shows the numerical error measurement results of all algorithms. Value error E n at time index n can be calculated by the following equation.

Figure 112016019957822-pat00052

Here, X n DFT (k) represents the kth value of the standard DFT of Equation (1), and X n Algorithm (k) represents the kth value of each algorithm.

As shown in FIG. 4, the mSDFT and gSDFT algorithms significantly reduce the numerical error compared to the rSDFT algorithm. Here, the damping index r of the rSDFT algorithm is set to 0.99999. 4, the average numerical error during the 64-hour index after the error scaling process

Figure 112016019957822-pat00053
Are shown. As shown in Fig. 4, rSDFT, mSDFT, and gSDFT
Figure 112016019957822-pat00054
Are 9.72 x 10 -3 , 7.41 x 10 -11 , and 7.17 x 10 -11 , respectively. This shows that the numerical error of gSDFT is smaller than that of mSDFT. Therefore, the sliding discrete Fourier transform method according to the embodiment of the present invention consistently outperforms other algorithms.

5 is a diagram schematically illustrating a configuration of a sliding discrete Fourier transform apparatus according to an embodiment of the present invention.

Referring to FIG. 5, a sliding discrete Fourier transform apparatus according to an embodiment of the present invention includes a processor 510, a memory 520, an input unit 530, and a display unit 540.

Processor 510 may be a CPU or a semiconductor device that executes processing instructions stored in memory 520.

The memory 520 may include various types of volatile or non-volatile storage media. For example, memory 520 may include ROM, RAM, and the like.

For example, the memory 520 may store instructions for performing the sliding discrete Fourier transform method according to an embodiment of the present invention.

The input unit 530 is a means for receiving various commands, information, and the like from the user to control the sliding discrete Fourier transform apparatus or to execute an application installed in the sliding discrete Fourier transform apparatus. The input unit 530 may include at least one key button, and may include a touch input module formed integrally with the display unit 540.

The display unit 540 is means for displaying various data inputted to the sliding discrete Fourier transform apparatus or stored in the sliding discrete Fourier transform apparatus in the form of time information. For example, the display unit 540 may be a liquid crystal display (LCD).

On the other hand, the components of the above-described embodiment can be easily grasped from a process viewpoint. That is, each component can be identified as a respective process. Further, the process of the above-described embodiment can be easily grasped from the viewpoint of the components of the apparatus.

In addition, the above-described technical features may be implemented in the form of program instructions that can be executed through various computer means and recorded in a computer-readable medium. The computer-readable medium may include program instructions, data files, data structures, and the like, alone or in combination. The program instructions recorded on the medium may be those specially designed and constructed for the embodiments or may be available to those skilled in the art of computer software. Examples of computer-readable media include magnetic media such as hard disks, floppy disks and magnetic tape; optical media such as CD-ROMs and DVDs; magnetic media such as floppy disks; Magneto-optical media, and hardware devices specifically configured to store and execute program instructions such as ROM, RAM, flash memory, and the like. Examples of program instructions include machine language code such as those produced by a compiler, as well as high-level language code that can be executed by a computer using an interpreter or the like. The hardware device may be configured to operate as one or more software modules to perform the operations of the embodiments, and vice versa.

It will be apparent to those skilled in the art that various modifications, additions and substitutions are possible, without departing from the spirit and scope of the invention as defined by the appended claims. Should be regarded as belonging to the following claims.

510: Processor
520: Memory
530:
540:

Claims (10)

In a sliding discrete Fourier transform (SDFT) method with guaranteed stability,
Calculating a sliding updating vector transform (SUVT) using the intermediate result calculated in the previous window; And
Calculating the SDFT output using the result of the sliding updating vector transformation,
The SUVT comprises:
Using a decimation-in-time (DIT) algorithm, L-sized UVTs at each iteration stage are calculated as two UVTs in a decimated sequence divided by two L / 2 interleaved UVTs ,
Wherein one of the two UVTs is an intermediate result calculated in the previous window and the calculated SUVT is an input value of an SUVT calculation in the next window.
The method according to claim 1,
Wherein the SUVT is defined by the following equation.
Figure 112016019957822-pat00055

Here, D n L (k) is the kth value of the updating vector transform (UVT) of the L point, k is the index of the frequency domain having a range of 0? K <M, n is the index of the time index ego,
Figure 112016019957822-pat00056
ego,
Figure 112016019957822-pat00057
Which is a complex twiddle factor.
3. The method of claim 2,
remind
Figure 112016019957822-pat00058
The
Figure 112016019957822-pat00059
And the relation is expressed by the following equation. &Lt; EMI ID = 1.0 &gt;
Figure 112016019957822-pat00060

 3. The method of claim 2,
Wherein the SDFT is defined by the following equation.
Figure 112016019957822-pat00061

Where X n (k) is the kth value of the M point DFT at time index n.
5. The method of claim 4,
Wherein the L is any one of M / 4, M / 2, and 3M / 4.
6. The method of claim 5,
Wherein when the L is M / 4, the rotation factor W M Lk is expressed by the following equation.
Figure 112016019957822-pat00062

 The method according to claim 6,
Wherein X n (k) is applied with j k and is simplified by the following equation.
Figure 112016019957822-pat00063

 8. The method of claim 7,
Wherein, when L is M / 4, the X n (k) is generalized according to the periodic characteristic of j k by the following equation.
Figure 112016019957822-pat00064

 9. The method of claim 8,
Wherein the relationship between Xn (4i) and XnM / 4 (4i) when k = 4i is expressed by the following equation.
Figure 112016019957822-pat00065

Here, the expressions of Re () and Im () represent the real part and the imaginary part of a complex number, respectively.
1. A sliding discrete Fourier transform apparatus for performing a sliding discrete Fourier transform (SDFT)
A memory for storing instructions; And
And a processor for executing the instruction,
Wherein the command comprises:
Calculating a sliding updating vector transform (SUVT) using the intermediate result calculated in the previous window; And
And performing the sliding discrete Fourier transform on the SDFT output using the result of the sliding updating vector transformation,
The SUVT comprises:
Using a decimation-in-time (DIT) algorithm, L-sized UVTs at each iteration stage are calculated as two UVTs in a decimated sequence divided by two L / 2 interleaved UVTs ,
Wherein one of the two UVTs is an intermediate result calculated in the previous window and the calculated SUVT is an input value of the SUVT calculation in the next window.


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