Supriya Aggarwal

Area-Time Efficient Scaling-Free CORDIC Using Generalized Micro-Rotation Selection

This paper presents an area-time efficient CORDIC algorithm that completely eliminates the scale-factor. By suitable selection of the order of approximation of Taylor series the proposed CORDIC circuit meets the accuracy requirement, and attains the desired range of convergence. Besides we have proposed an algorithm to redefine the elementary angles for reducing the number of CORDIC iterations. A generalized micro-rotation selection technique based on high speed most-significant-1-detection obviates the complex search algorithms for identifying the micro-rotations. The proposed CORDIC processor provides the flexibility to manipulate the number of iterations depending on the accuracy, area and latency requirements. Compared to the existing recursive architectures the proposed one has 17% lower slice-delay product on Xilinx Spartan XC2S200E device.

Scale-Free Hyperbolic CORDIC Processor and Its Application to Waveform Generation

This paper presents a novel completely scaling-free CORDIC algorithm in rotation mode for hyperbolic trajectory. We use most-significant-1 bit detection technique for micro-rotation sequence generation to reduce the number of iterations. By storing the sinh/cosh hyperbolic values at octant boundaries in a ROM, we can extend the range of convergence to the entire coordinate space. Based on this, we propose a pipeline hyperbolic CORDIC processor to implement a direct digital synthesizer (DDS). The DDS is further used to derive an efficient arbitrary waveform generator (AWG), where a pseudo-random number generator modulates the linear increments of phase to produce random phase-modulated waveform. The proposed waveform generator requires only one DDS for generating variety of modulated waveforms, while existing designs require separate DDS units for different type of waveforms, and multiple DDS units are required to generate composite waveforms. Therefore, area complexity of existing designs gets multiplied with the number of different types waveforms they generate, while in case of proposed design that remains unchanged. The proposed AWG when mapped on Xilinx Spartan 2E device, consumes 1076 slices and 2016 4-input LUTs. The proposed AWG involves significantly less area and lower latency, with nearly the same throughput compared to the existing CORDIC-based designs.

Concept, Design, and Implementation of Reconfigurable CORDIC

This brief presents the key concept, design strategy, and implementation of reconfigurable coordinate rotation digital computer (CORDIC) architectures that can be configured to operate either for circular or for hyperbolic trajectories in rotation as well as vectoring-modes. It can, therefore, be used to perform all the functions of both circular and hyperbolic CORDIC. We propose three reconfigurable CORDIC designs: 1) a reconfigurable rotation-mode CORDIC that operates either for circular or for hyperbolic trajectory; 2) a reconfigurable vectoring-mode CORDIC for circular and hyperbolic trajectories; and 3) a generalized reconfigurable CORDIC that can operate in any of the modes for both circular and hyperbolic trajectories. The reconfigurable CORDIC can perform the computation of various trigonometric and exponential functions, logarithms, square-root, and so on of circular and hyperbolic CORDIC using either rotation-mode or vectoring-mode CORDIC in one single circuit. It can be used in digital synchronizers, graphics processors, scientific calculators, and so on. It offers substantial saving of area complexity over the conventional design for reconfigurable applications.

Reconfigurable CORDIC architectures for multi-mode and multi-trajectory operations

This paper presents reconfigurable CORDIC (Coordinate Rotation Digital Computer) architectures which can be configured to operate either for circular or hyperbolic trajectories in rotation as well as vectoring-modes. We propose three reconfigurable CORDIC designs: a reconfigurable rotation-mode CORDIC that operates either for circular or hyperbolic trajectory, a reconfigurable vectoring-mode CORDIC for circular and hyperbolic trajectories, and a generalized reconfigurable CORDIC that can operate in any of the modes for both circular as well as hyperbolic trajectories. The reconfigurable CORDIC can perform the computation of various trigonometric and exponential functions, logarithms, square-root, etc. of circular and hyperbolic CORDICs using either rotation-mode or vectoring-mode of operation in one single circuit. It can be used in digital synchronizers, graphics processors, scientific calculators and many other applications, with significant area saving over that of using two CORDICs for different trajectories.

Efficient Window-Architecture Design Using Completely Scaling-Free CORDIC Pipeline

Filtering being one of the most important modules in signal processing paradigm, this paper presents an FPGA implementation of various window-functions using CORDIC algorithm to minimize area-delay product. The existing window-architecture uses a linear CORDIC processor in series with circular CORDIC processor, that results in a long pipeline. Firstly, we replace the linear CORDIC with multiple optimized shift-add networks to reduce area and pipeline depth. Secondly, the conventional circular CORDIC processor is replaced by a completely scaling-free CORDIC processor to further improve the area-time efficiency of the existing design. As a result, the proposed window-architecture, on an average requires approximately 64.34% less pipeline stages and saves upto 48% area. Both the existing and the proposed window-architecture are capable of generating Hanning, Hamming and Blackman window families.

CORDIC-based window implementation to minimise area and pipeline depth

Filtering is one of the most important modules in signal processing paradigm. This study presents a field-programmable gate array implementation of various window functions using coordinate rotation digital computer (CORDIC) algorithm to minimise area-delay product. First, the authors modify the Taylor series approximation order used in the scaling-free CORDIC, to completely eliminate the scale-factor and, yet, preserve the range of convergence spanning across the entire coordinate space. Secondly, the authors propose a new generalised technique for micro-rotation sequence identification to reduce the number of iterations required by the pipelined CORDIC processor. Then, this circular CORDIC processor is used to realise window functions. The existing window architecture uses a linear CORDIC processor in series with circular CORDIC processor, resulting in long pipeline. The authors replace the linear CORDIC with multiple optimised shift-add networks to reduce area and pipeline depth. As a result, the proposed window architecture, on an average requires approximately 64.34% less pipeline stages and saves up to 48% area. The authors have designed the processor to implement Hanning, Hamming and Blackman window families. The implementation of the proposed architecture is detailed in this study.

Hardware Efficient Architecture for Generating Sine/Cosine Waves

This paper presents a hardware efficient architecture for generating sine and cosine waves based on the CORDIC (Coordinate Rotation Digital Computer) algorithm. In its original form the CORDIC suffers from major drawbacks like scale-factor calculation, latency and optimal selection of micro-rotations. The proposed algorithm overcomes all these drawbacks. We use leading-one bit detection technique to identify the micro-rotations. The scale-free design of the proposed algorithm is based on Taylor series expansion of the sine and cosine waves. The 16-bit iterative architecture achieves approximately 4.5% and 6.7% lower slice-delay product as compared to the other existing designs. The algorithm design and its VLSI implementation are detailed.

Leading One Detection Hyperbolic CORDIC with Enhanced Range of Convergence

This paper focuses on developing an area efficient hyperbolic Coordinate Rotation Digital Computer (CORDIC) algorithm with performance improvement. The algorithm eliminates the need of scale factor calculation in the Range of Convergence (ROC). At the same time the range of convergence offered is higher than the conventional CORDIC ROC in the hyperbolic rotation mode. Being the only kind of algorithm in hyperbolic rotation with sign sequence μ = 1 always, one complete operation requires just 5 iterations. Thus the pipelined implementation has 5 stages which provides a 50% increase in throughput in comparison to conventional CORDIC. As far as the area improvement is considered, 16-bit processor can be realized using 56% less number of full adders required by Flat-CORDIC. The x and y datapath are based on series expansion of hyperbolic functions. The complete algorithm design along with pipelined architecture implementation is detailed.

Redesigned-scale-free CORDIC algorithm based FPGA implementation of window functions to minimize area and latency

One of the most important steps in spectral analysis is filtering, where window functions are generally used to design filters. In this paper, we modify the existing architecture for realizing the window functions using CORDIC processor. Firstly, we modify the conventional CORDIC algorithm to reduce its latency and area. The proposed CORDIC algorithm is completely scale-free for the range of convergence that spans the entire coordinate space. Secondly, we realize the window functions using a single CORDIC processor as against two serially connected CORDIC processors in existing technique, thus optimizing it for area and latency. The linear CORDIC processor is replaced by a shift-add network which drastically reduces the number of pipelining stages required in the existing design. The proposed design on an average requires approximately 64% less pipeline stages and saves up to 44.2% area. Currently, the processor is designed to implement Blackman windowing architecture, which with slight modifications can be extended to other widow functions as well. The details of the proposed architecture are discussed in the paper.

Low complexity VLSI implementation of CORDIC-based exponent calculation for neural networks

This article presents a low hardware complexity for exponent calculations based on CORDIC. The proposed CORDIC algorithm is designed to overcome major drawbacks (scale-factor compensation, low range of convergence and optimal selection of micro-rotations) of the conventional CORDIC in hyperbolic mode of operation. The micro-rotations are identified using leading-one bit detection with uni-direction rotations to eliminate redundant iterations and improve throughput. The efficiency and performance of the processor are independent of the probability of rotation angles being known prior to implementation. The eight-staged pipelined architecture implementation requires an 8 × N ROM in the pre-processing unit for storing the initial coordinate values; it no longer requires the ROM for storing the elementary angles. It provides an area-time efficient design for VLSI implementation for calculating exponents in activation functions and Gaussain Potential Functions (GPF) in neural networks. The proposed CORDIC processor requires 32.68% less adders and 72.23% less registers compared to that of the conventional design. The proposed design when implemented on Virtex 2P (2vp50ff1148-6) device, dissipates 55.58% less power and has 45.09% less total gate count and 16.91% less delay as compared to Xilinx CORDIC Core. The detailed algorithm design along with FPGA implementation and area and time complexities is presented.