# A Low-Latency RDP-CORDIC Algorithm for Real-Time Signal Processing of Edge Computing Devices in Smart Grid Cyber-Physical Systems

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

- We proposed a rotation direction prediction method of the CORDIC algorithm, which completed the calculation of all the micro-rotation directions by inputting the angle and direction prediction constants, providing the basis for the subsequent merge iteration;
- A constant compensation algorithm for direction prediction was proposed to achieve higher accuracy of direction prediction, being able to solve the problem of large memory consumption under the condition of high accuracy;
- The single-stage iterative structure of the CORDIC algorithm was replaced by a three-stage and multi-stage iterative structure. Based on this structure, the CORDIC algorithm design with high accuracy, low latency, and low power consumption was achieved.

## 2. Related Work

## 3. Conventional CORDIC Algorithm

_{1}is rotated by θ to obtain vector v

_{2}, and the coordinates of v

_{1}and v

_{2}are (x

_{0}, y

_{0}), (x

_{t}, y

_{t}) respectively, then, the equation of change of vector coordinates can be expressed by Equation (1).

_{i}= tan

^{−1}(2

^{−i}), each rotation can be expressed by the iterative Equation (2).

_{i+}

_{1}indicates the remaining angle and d

_{i}is the direction of rotation. (x

_{i+}

_{1,}y

_{i}

_{+1}) indicates the coordinates of (x

_{i}, y

_{i}) after the next rotation. In the rotation mode, the remaining angle z

_{i}

_{+1}value was used as a direction reference, and after n iterations, the z

_{i}

_{+1}value tended to zero and the vector v

_{i}almost tended to the vector, thus realizing the successive approximation calculation.

_{i}in Equation (2) involved multiplication in the iterative calculation process, it can be proposed not to participate in the iterative operation. Let $K={{\displaystyle \prod}}_{i=0}^{n}\mathrm{cos}\theta i\text{}$=1/[(1 + 2

^{−2i})

^{0.5}], 1/K is the scaling factor mentioned above, then the iteration equations of the radix-2 CORDIC algorithm in rotation mode at the (i + 1)th step are as follows:

_{0}= $\theta $, the coordinates of Equation (3) are (x

_{n}, y

_{n}) after n iterations of calculation.

_{0}= K and y

_{0}= 0, after n iterations of calculation, x

_{n}and y

_{n}will be equal to the values of cos θ and sin θ, respectively. Therefore, the calculation of sine and cosine functions based on the CORDIC algorithm was implemented.

## 4. RDP-CORDIC Algorithm

_{2}

^{3}]/3 iterations, thus reducing the number of iterations by about 1/2. The BBR is impressed by decomposing the input angle θ into a combination of a larger angle and several 2

^{−i}radians so that the direction of rotation is determined by the binary bit value of θ each rotation. Note that, the BBR method requires a ROM to store all the computation results after N/3 − 1 iterations, and the ROM consumption increases as precision gets higher, e.g., 16-bit precision requires a ROM of 26 × 16 × 2 (bit) size.

#### 4.1. Rotation Direction Prediction

^{−i}, resulting in large consumption of ROM resources. Therefore, the micro-rotation angle chosen for the RDP-CORDIC algorithm was tan

^{−1}(2

^{−i}), and a new rotation direction prediction method needs to be sought.

_{1}, d

_{2}, d

_{3}, d

_{4}and d

_{5}in various combinations for 16-bit precision. In order to determine the rules for the value of λ, the cumulative value of the rotation angle corresponding to λ was viewed as the angle reference, which was denoted as θ

_{cp}. It should be noted that in the calculation, the values for d

_{6}~d

_{16}were 0. When calculating θ

_{cp}, not only the sum of the angles of d

_{1}~d

_{5}rotation may be covered, but also the micro-selected rotation angles of d

_{6}~d

_{16}should be accumulated. Equation (9) is the calculation of the reference angle value θ

_{cp(m)}.

_{θ}, it is necessary to make the accuracy of λ higher than that of d

_{θ}. For the 16-bit precision of d

_{θ}, λ needs 17-bit size precision. Since the integer bits of both λ and θ are 0, each prediction constant in ROM only needs 16 bits in size. Therefore, to implement the RDP-CORDIC algorithm with N bit precision, the size of the prediction constants λ and θ in ROM is also the same as N bit.

_{cp}

_{5}for different values of λ is given in Table 3. The process of rotation direction prediction can be summarized as follows:

- Compare the input angle with θ
_{cp}in the direction prediction constant, and select the value of λ corresponding to a value close to and less than or equal to θ_{cp}; - The binary value d
_{θ}representing the micro-rotation direction was calculated based on λ. Finally, the prediction of the micro-rotation direction in the non-iterative case was performed.

#### 4.2. ROM Resource Optimization

^{[(N−log}

_{2}

^{3)/3]}× N bit size, and the ROM consumption increased sharply with the increase of accuracy. The reason for the sharp increase of ROM consumption was that the high accuracy direction prediction asked for more λ values to be selected, leading to an increase in the table of direction prediction constants. It may be useful to analyze the λ expansion and let m = [(N − log

_{2}

^{3})/3)], then λ

_{m}and λ

_{m+}

_{1}are as in Equations (10) and (11), respectively.

_{2}(3/20) − 3)/5], Equation (11) can be reduced to Equation (12)

_{m}

_{+1}and λ

_{m}, and similarly, the value of λ

_{m}

_{+i}can be calculated from λ

_{m}. Thus, an accuracy compensation algorithm for λ is proposed, where λ is composed of a fixed λ

_{s}and an accuracy compensation λ

_{c}.

_{2}(3/20) − 3)/5], the accuracy compensation λ

_{c}is calculated as Equation (14).

_{2}3 > m, d

_{i}in Equation (14) can be calculated based on λ

_{s}. To sum up, the ROM consumption of the direction prediction constant was reduced from ${2}^{\left[N-{\mathrm{log}}_{2}3\right]/3}\ast N$ bit to ${2}^{\left[N+{\mathrm{log}}_{2}\left(3/20\right)-3\right]/5}\ast N$ bit by using the direction prediction constant accuracy compensation method, which achieved high accuracy and reduced the ROM consumption.

#### 4.3. Iterative Merging

_{i}2

^{−(3i+3)}and y

_{i}2

^{−(3i+3)}dropped to machine zero, Equation (15) removed ${y}_{i}\cdot \frac{{d}_{i}\cdot {d}_{i+1}\cdot {d}_{i+2}}{{2}^{(3i+3)}}$ and ${x}_{i}\cdot \frac{{d}_{i}\cdot {d}_{i+1}\cdot {d}_{i+2}}{{2}^{(3i+3)}}$ two terms, the three-level combined iteration formula became Equation (16).

_{i}2

^{−(2i+1)}and y

_{i}2

^{−(2i+1)}are machine zeros, and the subsequent iterative process can be represented by the multilevel merge iteration of Equation (17).

- For the number of iterations i ≤ [(N − 3)/3], the three-stage merge iteration Formula (15) was used;
- When [(N − 3)/3] < i ≤ [(N − 1)/2], the three-stage merge iteration simplified Formula (16) was used;
- Finally when i > [(N − 1)/2], Formula (16) for multi-stage merge iteration calculation was used.

## 5. Hardware Design of RDP-CORDIC Algorithm

#### 5.1. RDP-CORDIC Algorithm Structure Design

_{n}, and the result needs to be iteratively calculated in a [(N − 1)/2] stage pipeline method. The structure of RDP-CORDIC algorithm consists of two parts: the direction prediction part on the left side and the rotation iteration on the right side. The direction prediction module calculates all micro-rotation directions in advance, while the rotation iteration part transforms the single-stage iterative structure into a three-stage and multi-stage iterative structure since all micro-rotation directions are known. The new structure cut off part of hardware overhead and latency compared to the conventional structure.

Algorithm 1 RDP-CORDIC workflow |

1. Directional rough prediction (1) Pre-store the direction prediction constants θ _{cp}, λ_{s}, and μ_{i} in a ROM of size 2[(N − (log2(3/20) − 3)/5] bits;(2) Use the MSB of the input angle θ as the lookup address of the ROM for reading out θ _{cp};(3) Send the sign bit of the value of the input angle θ minus θ _{cp} to the selection input port of the multiplexer, and the multiplexer outputs the corresponding value of λ_{s};(4) Add up λ _{s}, θ, and 0.5 − 0.5ε to get the rough rotation direction prediction value d_{ap}. |

2. Accurate direction prediction |

(1) Shift and sum up the rough rotation direction prediction d_{s+1}~d_{m} with μs according to Equation (14) to calculate the compensation value λ_{c}.(2) Calculate the exact direction prediction value d _{θ} by re-summing λ, θ and 0.5 − 0.5ε.3. Iteration calculation (1) In the iterative calculation part uses multiple three-level merge iteration modules and one multi-level merge iteration module; (2) Set the input values of the iterative calculation module as x _{1} = K and y_{1} = 0;(3) The rotation directions d _{1}~d_{3s−1} are determined by the rough direction value dap, and the rotation directions d_{3s}~d_{n} are determined by the accurate direction value d_{θ}. |

#### 5.2. Calculation of Sine and Cosine Function Based on RDP-CORDIC Algorithm

^{−1}(2

^{−i}), where i was an integer greater than 0, the calculated angle range was limited to [0, $\mathsf{\pi}$/4]. The symmetry of trigonometric function were combined with the trigonometric change to expand the input angle range, and the final change relationship is shown in Table 5.

#### 5.3. More Applications of the RDP-CORDIC Algorithm

_{0}= x

_{0}+ j × y

_{0}, X

_{k}= x

_{k}+ j × y

_{k}, bringing ${W}_{N}^{nk}={e}^{-j\frac{2\pi}{N}nk}$ into Equation (18), the imaginary and real parts of X

_{k}after being simplified are as in Equation (19).

_{0}rotated by θ = −2nkπ/N, Then, using the CORDIC algorithm idea, we can transform Equation (19) into Equation (1). So the complex multiplication of FFT can then be implemented by the RDP-CORDIC algorithm.

_{R}and θ

_{L}are the left and right rotation angles, calculated by Equation (21). The values of ${\delta}_{1}$ and ${\delta}_{2}$ are the singular values of the matrix G. In the above operation, both the arc tangent funcion and the sine/cosine function can be implemented by the CORDIC algorithm. The structure of the 2 × 2 SVD module based on the RDP-CORDIC algorithm is shown in Figure 7.

## 6. Performance Testing and Analysis

#### 6.1. ROM Optimization Results of the RDP-CORDIC Algorithm

#### 6.2. Performance Comparison of CORDIC Algorithms

#### 6.3. Test of Calculation Error and Calculation Time of Variousfunctions

^{−5}. The reason for the different errors for each input angle is that the error is 0 only when the accumulated value of the angle of directional rotation is equal to the input angle. However, the direction of rotation is not certain for different input angles, which results in the difference between the totalized rotation value and the input angle value. For other functions, the input test data is limited to a different range due to the characteristic limitations of the CORDIC algorithm. As in Figure 9c–f, the input angles are limited to [0.2, 9.5], [0.03, 2], [−1.12, 1.12] and [−1.12, 1.12], respectively. The test results show that the RDP-CORDIC algorithm performs well on a variety of functions, with maximum absolute errors less than 7.7 × 10

^{−4}. Because the computation time of each function is the same through the CORDIC algorithm, the sine and cosine functions are used for the test computation time. The time of the single computation of the sine and cosinefunctions for different CORDIC algorithms are compared in Table 7, and it can be found that the RDP-CORDIC algorithm takes only 60 ns at a system clock of 100 MHz.

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 4.**Structure of the classical optimized CORDIC algorithm and the RDP_CORDIC algorithm: (

**a**) Classical optimized CORDIC algorithm pipeline structure; (

**b**) Structure of RDP-CORDIC algorithm.

**Figure 9.**The various functions calculation error base on RDP-CORDIC algorithm: (

**a**) Absolute error of sine signal value; (

**b**) Absolute error of cosine signal value; (

**c**) Absolute error of ln(x) value; (

**d**) Absolute error of sqrt value; (

**e**) Absolute error of sinh value; (

**f**) Absolute error of cosh value.

**Figure 10.**Maximum absolute error of sine and cosine signals calculated by various CORDIC algorithms at different bit widths.

Application Category | Functions Implemented |
---|---|

basic arithmetic | multiplication |

division | |

trigonometric function | sin x |

cos x | |

tan x | |

inverse trigonometric function | arcsin x |

arcsin^{−1} x | |

arctan^{−1} x | |

hyperbolic function | cosh x |

sinh x | |

tanh x | |

tanh^{−1} x | |

other common functions | $\sqrt{x}$ |

In(x) | |

e^{x} | |

other applications | Fast Fourier transform |

Matrix eigenvalue estimation | |

Singular value decomposition | |

Digital frequency synthesis |

CORDIC Algorithm | Radix | Rotation Direction Prediction | Fixed Scaling Factor |
---|---|---|---|

R-2 CORDIC [26] | R-2 | × | √ |

R-4 CORDIC [24] | R-4 | × | × |

R-8 CORDIC [28] | R-8 | × | × |

scaling-free CORDIC [31] | MIX-R | × | √ |

Mixed-R-scaling-free CORDIC [29] | MIX-R | × | √ |

BBR-CORDIC [34] | R-2 | √ | × |

CORDIC II [38] | R-2 | × | × |

RDP-CORDIC [proposed] | R-2 | √ | √ |

{d_{1},d_{2},d_{3},d_{4},d_{5}} | λ | θ_{cp5} |
---|---|---|

01111 | 0.03635239 | −0.0305780 |

10000 | 0.03636256 | 0.03190169 |

10001 | 0.03643358 | 0.09425964 |

10010 | 0.03644375 | 0.15673931 |

10011 | 0.03699740 | 0.21813201 |

10100 | 0.03700756 | 0.28061168 |

10101 | 0.03707859 | 0.34296963 |

10110 | 0.03708875 | 0.40544930 |

10111 | 0.04137373 | 0.45937935 |

11000 | 0.04138389 | 0.52185901 |

11001 | 0.04145492 | 0.58421697 |

11010 | 0.04146508 | 0.64669663 |

11011 | 0.04201873 | 0.70808934 |

11100 | 0.04202890 | 0.77056900 |

i | μ_{i} |
---|---|

1 | 0.0363523910 |

2 | 0.0050213369 |

3 | 0.0006450055 |

4 | 8.1190004043 × 10^{−5} |

5 | 1.0166569732 × 10^{−5} |

6 | 1.2713795232 × 10^{−6} |

7 | 1.5893989889 × 10^{−7} |

8 | 1.9868033028 × 10^{−8} |

9 | 2.4835211812 × 10^{−9} |

10 | 3.1044068054 × 10^{−10} |

Input Angle Range θ_{e} | Angle after Conversion θ | cos θ_{e} | sin θ_{e} |
---|---|---|---|

[0$,\text{}\mathsf{\pi}$/4) | θ_{e} | cos θ | sin θ |

[$\mathsf{\pi}$/4$,\text{}\mathsf{\pi}$/2) | $\mathsf{\pi}$/2 − θ_{e} | sin θ | cos θ |

[$\mathsf{\pi}$/2$,3\mathsf{\pi}$/4) | θ_{e} − $\mathsf{\pi}$/2 | −sin θ | cos θ |

[$3\mathsf{\pi}/4,\text{}\mathsf{\pi}$) | $\mathsf{\pi}$ − θ_{e} | −cos θ | sin θ |

[$\mathsf{\pi},5\mathsf{\pi}/4$) | θ_{e} − $\mathsf{\pi}$ | −cos θ | −sin θ |

[$5\mathsf{\pi}/4,\text{}3\mathsf{\pi}/2$) | $3\mathsf{\pi}$/2 − θ_{e} | −sin θ | −cos θ |

[$3\mathsf{\pi}/2,\text{}7\mathsf{\pi}/4$) | θ_{e} − $3\mathsf{\pi}$/2 | sin θ | −cos θ |

[$7\mathsf{\pi}/4,\text{}2\mathsf{\pi}$] | 2$\mathsf{\pi}$ − θ_{e} | cos θ | −sin θ |

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**MDPI and ACS Style**

Qin, M.; Liu, T.; Hou, B.; Gao, Y.; Yao, Y.; Sun, H. A Low-Latency RDP-CORDIC Algorithm for Real-Time Signal Processing of Edge Computing Devices in Smart Grid Cyber-Physical Systems. *Sensors* **2022**, *22*, 7489.
https://doi.org/10.3390/s22197489

**AMA Style**

Qin M, Liu T, Hou B, Gao Y, Yao Y, Sun H. A Low-Latency RDP-CORDIC Algorithm for Real-Time Signal Processing of Edge Computing Devices in Smart Grid Cyber-Physical Systems. *Sensors*. 2022; 22(19):7489.
https://doi.org/10.3390/s22197489

**Chicago/Turabian Style**

Qin, Mingwei, Tong Liu, Baolin Hou, Yongxiang Gao, Yuancheng Yao, and Haifeng Sun. 2022. "A Low-Latency RDP-CORDIC Algorithm for Real-Time Signal Processing of Edge Computing Devices in Smart Grid Cyber-Physical Systems" *Sensors* 22, no. 19: 7489.
https://doi.org/10.3390/s22197489