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XAPP552 (v1.0) June 1, 2012 Application Note: Spartan-6, Virtex-6, 7 Series, and Zynq-7000 Devices Parameterizable CORDIC-Based Floating-Point Library Operations Authors: Nikhil Dhume and Ramakrishnan Srinivasakannan Summary This application note presents a design methodology using the Xilinx System Generator for DSP tool to create a parameterizable floating-point library computation method for trigonometric, power, and logarithmic operations based on the coordinate rotational digital computer (CORDIC) algorithm [Ref 1]. The design methodology leverages the fixed-point CORDIC LogiCORE™ IP v5.0 block, along with floating-point building blocks such as adders, multipliers, comparators, ROM, and FIFOs to create a set of floating-point CORDIC functions that can be used as building blocks in applications. These functions are an essential requisite in a wide range of engineering applications such as image processing, manipulator kinematics, radar signal processing, robotics, and optimization processes in which a large number of trigonometric or power operations must be computed in an efficient manner. The library has been designed using System Generator for DSP, version 13.4, and supports single- and double-precision input as defined by the IEEE 754 floating-point standard [Ref 2]. Introduction The arithmetic unit is one of the important components of CPU design. For computation of complex arithmetic functions on hardware, the CORDIC algorithm is an attractive fixed-point algorithm that uses a sequence of simple “shift and add” operations to compute a wide variety of arithmetic functions. However, many applications are required to work not only with high precision but also a large dynamic range. Floating-point arithmetic is a feasible solution for such high-performance systems providing a dynamic range for representing real numbers and capabilities to retain resolution and accuracy. Floating-Point Solution for Xilinx FPGAs Xilinx FPGAs have long been used to implement fixed-point DSP and video algorithms in hardware. The flexibility of programmable logic allows fixed-point arithmetic to use custom bit widths that are not bound to the 8-, 16-, or 32-bit boundaries of a fixed-point processor. Fixed-point bit widths can grow as needed to accommodate applications that require large dynamic range. However, as the dynamic range needs to grow, a fixed-point implementation becomes increasingly expensive. Although floating-point solutions on FPGAs are inherently slower than contemporary processors, the inherent massive parallelism allows these solutions to be competitive to the software equivalent. For this reason, FPGAs are increasingly being used as floating-point accelerators. To benefit from the parallelism, there is a requirement to use hardware-efficient algorithms for FPGAs. More complex floating-point systems on FPGAs require good implementations of elementary functions such as logarithmic, power, and trigonometric. The System Generator for DSP tool meets this demand by supporting design and implementation of floating-point algorithms from within the Simulink modeling environment. System Generator for DSP also has the flexibility of optimizing an implementation that is bitand cycle-accurate to the original model. This library has been designed as an extension for customers familiar with the flow of the System Generator for DSP tool. © Copyright 2012 Xilinx, Inc. Xilinx, the Xilinx logo, Artix, ISE, Kintex, Spartan, Virtex, Zynq, and other designated brands included herein are trademarks of Xilinx in the United States and other countries. AMBA and ARM are registered trademarks of ARM in the EU and other countries. All other trademarks are the property of their respective owners. XAPP552 (v1.0) June 1, 2012 1 Fixed-Point CORDIC Algorithm Fixed-Point CORDIC Algorithm CORDIC algorithms are a class of iterative solutions for trigonometric and other transcendental functions that use only shifts and adds to perform. The trigonometric functions are implemented based on vector rotation. Incremental functions such as logarithm and power are performed with a simple extension to the hardware architecture and, while not CORDIC in the strictest sense, are often included because of close similarity. A detailed study of the algorithm is given in Fixed-Point CORDIC Algorithm. Design Approach A floating-point library for CORDIC trigonometric functions has been developed using the fixed-point CORDIC block and other basic blocks. The approach chosen has been to use underlying trigonometric relations to extend the range of the fixed-point CORDIC algorithm. The input floating-point number is passed through a range reduction step and then processed with a fixed-point CORDIC block. The range reduction step reduces the input range to the one allowed by the fixed-point CORDIC algorithm. A post-processing step performs the inverse of the range reduction step after fixed-point computation. This approach is detailed in Figure 1. The library has been made parameterizable to ensure maximum flexibility. The floating-point CORDIC library presented in this application note has been implemented using the Xilinx IP portfolio. System Generator for DSP, version 13.4, was used to implement the flow. X-Ref Target - Figure 1 Input FloatingPoint Number Range Reduction Fixed-Point Algorithm Range Extension Output FloatingPoint Number Fixed-Point CORDIC Algorithm Exception Handling X552_01_042612 Figure 1: Approach Chosen for Floating-Point CORDIC Library The CORDIC algorithm was initially designed to perform vector rotation, where the vector (X,Y) is rotated through an angle θ yielding a new vector (X',Y'). The vector rotation equations are: X′= ( cos (θ) × X – sin(θ ) × Y ) Equation 1 Y′= ( cos (θ) × Y + sin (θ ) × X ) Equation 2 θ′ = 0 Equation 3 The CORDIC algorithm performs a vector rotation as a sequence of successively smaller rotations, each of angle atan(2-i), known as micro rotations. Equation 4 through Equation 6 show the expression for the ith iteration, where i is the iteration index from 0 to n. The expression for the ith micro rotation is: Xi + 1 = xi – αi × yi × 2–i yi + 1 = yi + αi × xi × 2–i θi + 1 = θi – αi × atanh (2–i) Equation 4 Equation 5 Equation 6 Where αi is the direction of rotation and can have a value of ±1. A detailed description of the CORDIC algorithm is given in Floating-Point Algorithms and Library Interface Specifications, page 10. XAPP552 (v1.0) June 1, 2012 2 Floating-Point Algorithms Floating-Point Algorithms Floating-Point CORDIC sin-cos The rotational mode of fixed-point CORDIC can be used to simultaneously compute the sine (sin) and cosine (cos) of the input angle. Setting the y component of the input vector to zero reduces the rotation mode to: Xn = An × x0 × cos(θ) Equation 7 Yn = An × x0 × sin (θ) Where An is a gain factor corresponding to the ith micro rotation. Equation 8 Range Enhancement The range reduction step for the fixed-point CORDIC algorithm can be achieved by performing an angle rotation on the input. The input is rotated to a value between –π and +π, which can then be fed to the fixed-point CORDIC as input. This step is required because the fixed-point CORDIC only converges for the range between –π and +π. The rotation of the angle can be given by: Rotated angle(x) = remainder(x, 2 × π ) – π Equation 9 Because n is subtracted from the remainder, there is an inherent reflection in this step which needs to be adjusted for post-processing. Algorithm The steps in the algorithm are: 1. Range reduction: The input is rotated between –π and +π. 2. The fixed-point CORDIC block is used for computation of the sine and cosine of the number, as detailed above. 3. Post-processing: A reflection operation is performed on the output of the fixed-point CORDIC block. Floating-Point CORDIC sinh-cosh The close relationship between trigonometric and hyperbolic functions suggests that the same architecture can be used to compute hyperbolic functions. The CORDIC equations for hyperbolic rotations are derived by setting the αi factor in Equation 4, Equation 5, and Equation 6 by the amount shown in Equation 10. αi = –1 if i < 0, +1 otherwise Equation 10 \ This reduces the CORDIC output in rotation mode to Equation 11 and Equation 12. xn = An × [x0 × cosh (θ0) + y0 × sinh (θ0)] Equation 11 yn = An × [x0 × cosh (θ0) + y0 × sinh (θ0)] Equation 12 An = Π 1 – 2–2i ≅ 0.80 Equation 13 The value of hyperbolic sine (sinh) and hyperbolic cosine (cosh) can be found by setting y0 to 0. Range Enhancement The range reduction for the hyperbolic functions can be found by splitting the input into fractional and integer portions. The integer portion can be processed by means of a stored look-up table (LUT). The fractional portion can be processed separately using a stored LUT. Algorithm The steps in the algorithm are: XAPP552 (v1.0) June 1, 2012 3 Floating-Point Algorithms 1. Range reduction: The input is reduced in range by first finding the absolute value of the number and then splitting the number into integer (int) and fractional (frac) portions. 2. The int portion is processed by means of a stored LUT while the frac portion is processed by means of a fixed-point CORDIC algorithm. 3. Post-processing: The outputs from the int and frac portions are combined using Equation 14 and Equation 15. cosh (int + frac) = cosh (int) × cosh (frac) + sinh (int) × sinh (frac) Equation 14 sinh (int + frac) = cosh (int) × sinh(frac) + cosh (frac) × sinh (int) Equation 15 4. Final stage: A reflection operation is performed when the input is negative. Floating-Point CORDIC Power The algorithm for exponential (ex) is computed by using the sinh and cosh computed from Floating-Point CORDIC sinh-cosh. In addition to exponential, powers of 10 and 2 are supported in the library. Algorithm 1. The exponential value of a number can be calculated from the sinh and cosh from Equation 16. ex = sinh (x) + cosh(x) Equation 16 2. The power value of 10 and 2 is computed by the core from Equation 17 and Equation 18. 10x = e(x ⋅ ln(10)) Equation 17 2x = e(x ⋅ ln(2)) Equation 18 Floating-Point CORDIC atan The floating-point CORDIC computes arctangent (atan (y/x)) directly using the vectoring mode of the CORDIC rotator if the angle is initialized with 0. The argument must be presented as a ratio of x/y. The angle accumulator output is given by Equation 19. Range Enhancement θn = θ0 + tan –1  xy----00- Equation 19 The range reduction step is performed by removing the sign portion of the input numbers and adjusting the input so that the imaginary part is always greater in magnitude than the real part. This adjustment is done to ensure that the output angle is always present in the first quadrant. Algorithm The steps in the algorithm are: 1. The absolute value of the input is found, and real and imaginary portions are adjusted so that the real portion is greater than the imaginary portion. 2. The fixed-point CORDIC atan is used to compute the output. 3. The output is rotated to the correct quadrant based on the input sign and whether the real portion of the number is greater than the imaginary portion. Floating-Point CORDIC log The CORDIC logarithm (ln) is implemented using the hyperbolic vectoring mode of CORDIC. The hyperbolic arctangent (atanh) can be used to compute log by using Equation 20. Logarithms to the base 10 and 2 are also supported. XAPP552 (v1.0) June 1, 2012 4 System Generator Implementation ln ( w ) = 2 × atanh   w-w-----+-–----11-- Range Enhancement Equation 20 There is an inherent range reduction involved in the previous step. Using w–1 and w+1 ensures that the real portion is always less than the imaginary portion and that the real portion is never equal to the imaginary portion. If the real portion equals the imaginary portion, the output of atanh goes to infinity (which cannot be represented by the fixed-point CORDIC block). The other range reduction algorithm used is separation of input into the mantissa and exponent portion, which are processed separately. The mantissa is processed by means of the fixed-point CORDIC, and the exponent is processed by means of a multiplier. Algorithm The steps in the algorithm are: 1. Split the input into the exponent and mantissa portion. The exponent portion can be added back to the processed mantissa at the end. 2. Process the mantissa by using the fixed-point CORDIC algorithm as mentioned above. 3. The log10 and log2 is found in the CORDIC core using Equation 21. log b(w) = l-l-oo---g-g---(-(--wb----))- Equation 21 System Generator Implementation Setting Up the Library For usage of the library in the System Generator for DSP tool, a patch has been created in the TAR file contained in the reference design (see Reference Design, page 16). After the overlay is installed, the library should appear similar to Figure 2. The patch works with the System Generator for DSP tool, version 13.4 for nt, nt64, lin, and lin64 builds. To set up the library: 1. Clear the MATLAB and System Generator for DSP tool caches using the using xlCache ('clear all') command. 2. Extract the patch on top of the IDS build using WinZip or the tar –xvf command. 3. Open the System Generator for DSP tool to the corresponding IDS build. 4. The floating-point blocks are now visible as a part of the reference blockset as a “Floating Point” library. 5. These blocks are present as part of the library: Absolute, Conditional Negate, Floor-ceil, Split, Merge, Remainder, Cordic Sin-cos, Cordic Sinh-cosh, Cordic Atan, Cordic Log, and Cordic Power. 6. The library works for single and double floating-point data types. Library Usage The library is available in the floating-point section of the reference blockset, as shown in Figure 2. To use this library, the user can drag and drop any of the blocks to a new model file. XAPP552 (v1.0) June 1, 2012 5 System Generator Implementation X-Ref Target - Figure 2 X552_02_042612 Figure 2: Floating-Point Blocks of the Xilinx Reference Blockset Implementation Details This section describes some of the System Generator for DSP tool-specific implementation details and application of additional blocks present in the library. Parameterization of Blocks The individual blocks in the library are created as subsystems. The GUI is customized using the parameters tab from the block mask. Additionally, user preferences (such as single or double, and size of the LUT) are handled by means of an initialization pane from the edit block mask window. The block mask window with parameters for sin-cos is shown in Figure 3. The initialization section also handles tuning latencies. Detailed instructions on how to create block masks can be found in Trade-offs in the Current Approach, page 13. XAPP552 (v1.0) June 1, 2012 6 System Generator Implementation X-Ref Target - Figure 3 Figure 3: Mask Editor: CORDIC Sin-cos Parameters X552_03_021612 Latency Handling for AXI Blocks The CORDIC LogiCORE IP v5.0 block has an AXI stream interface, while the basic blocks do not. This necessitates some input parameters to be propagated to the output side to adjust for latencies. For this purpose, FIFOs are used in the design. An example of how FIFO blocks are used to adjust for latency of AXI blocks is shown in Figure 4. For example, the sin-cos block requires sign information to be made available at the output. The read enable of the FIFOs are driven by the output TVALID of the CORDIC block. XAPP552 (v1.0) June 1, 2012 7 System Generator Implementation X-Ref Target - Figure 4 phase_tvalid phase_tready 1 Out1 z-4cast Convert phase_tdata_phase dout_tvalid dout_tdata_imag phase_tlast dout_tdata_real 4 dout_tready In4 dout_tlast CORDIC_5_0 z-5 Delay2 [A] Goto z-4cast Convert1 [C] From2 z-4cast Convert2 2 Out2 in z-1 out sign float Conditional-negate 3 Out3 z-1 x(-1) Negate 4 Out4 z-5 Delay5 5 Out5 z-4cast din dout cast z-4 [C] Convert3 we empty Terminator1 Convert4 Delay3 Goto2 [A] re full From FIFO Terminator X552_04_021612 Figure 4: Usage of FIFOs in Sin-cos Block to Adjust for Latencies in CORDIC LogiCORE IP v5.0 Block Split and Merge For some blocks in the library, the exponent and mantissa at the input can be processed separately as fixed-point numbers (e.g., log, atan) and then recombined to form a meaningful floating-point output. The split and merge blocks are useful for this operation. The split block splits the input floating-point numbers into exponent and mantissa while providing information on whether the input was a not a number (NaN) signal. The merge block does the inverse operation, recombining normalized mantissa and exponent to form an output floating-point number. XAPP552 (v1.0) June 1, 2012 8 System Generator Implementation Remainder The range reduction step of some algorithms involves computing the modulo of a number (e.g., sin-cos and sinh-cosh). For this purpose, a remainder block is useful and is implemented by means of a floating-point divide and a floor operation. This block is also available to the user as part of the library. LUTs The sinh-cosh and power blocks use internal LUTs to implement some portions of the processing. The output range of the above blocks tends to infinity very quickly, and the whole input range might not be of interest to the user. Therefore, control has been provided to the user to select the LUT size based on the input range. For implementation of user-configurable LUTs, ROM blocks have been used, as shown in Figure 5. This option allows the user to optimize an instance of the block for resource utilization. X-Ref Target - Figure 5 X552_05_022912 Figure 5: ROM Block Configured to Support User-Defined Input Range Latency Tuning The latencies of individual blocks in the subsystem are tuned to give approximately 280 MHz for a single precision floating-point data type and 240 MHz for a double precision floating-point data type. XAPP552 (v1.0) June 1, 2012 9 Library Interface Specifications Library Interface Specifications This section describes the block interface terminology used for each of the different blocks. A sample user interface for sinh and cosh generated by MATLAB is shown Figure 6. X-Ref Target - Figure 6 X552_06_021612 Figure 6: Block Mask for CORDIC Sinh-cosh Block Table 1 describes parameter interfaces used across different blocks in the CORDIC library. Table 1: Parameter Interfaces in the CORDIC Library Parameter Name Description Iterations Number of add-sub iterations. When set to 0, the number of iterations performed is determined by the required accuracy of the output. The default value of iterations is 0. Precision Configures the internal precision of the fixed-point CORDIC. When precision is set to 0, the internal precision is automatically determined. Data type The data types supported for this library are Single and Double, as defined by the IEEE 754 standard. Custom data types or parameter inference are not supported. XAPP552 (v1.0) June 1, 2012 10 Results and Discussion Table 1: Parameter Interfaces in the CORDIC Library (Cont’d) Parameter Name Description Pipeline mode The supported pipeline modes are: • Maximum: The CORDIC core is implemented with a pipeline after every shift-add substage. • Optimal: The CORDIC core is implemented with as many stages of pipelining as possible without using additional LUTs. Architectural configuration The architectural configurations supported are: • Parallel: CORDIC core has single cycle data throughput and large silicon area. • Word-serial: CORDIC core is implemented with multi-cycle throughput and a smaller silicon area. Maximum input value This is present for sinh, cosh, and power to optimize the size of the LUT to conserve area. This is expected to be useful because power operations quickly tend toward infinity. Table 2 describes port interfaces used across various blocks in the CORDIC library. Table 2: Port Interfaces in the CORDIC Library Port name Direction Description Cartesian_tvalid Phase_tvalid I Input tvalid handshake signal for the AXI stream I Tdata_imag I Real portion of the input data Tdata_real I Imaginary portion of the input data Tdata_phase I Angle input used for sin-cos and sinh-cosh modes Cartesian_tlast I Used to specify the last data in a stream Dout_tready I Handshake signal for AXI stream Cartesian_tready O Output tready signal used as handshake for AXI stream Phase_tready O Dout_tvalid O Signal valid data at the output Dout_tlast O Signal last data at the output Results and Discussion To make comparisons at the architecture level, estimates have been made of the speed, area, latency, and throughput for the different blocks in the library. The results presented are based on the Virtex-7 devices. The speed information is presented for speed grades -1,-2, and -3. The device utilization figures are presented for -2 devices. The performance summary for the single data type is shown in Table 3. XAPP552 (v1.0) June 1, 2012 11 Results and Discussion Table 3: Block Performance Summary for Single Data Type Block name Mode -3 Speed Grade LUTs & Flip-Flops -2 Speed Grade Slice Registers Sin-cos Word Serial Parallel 302 6,514 302 8,605 281 4,725 281 6,905 Sinh-cosh Word Serial Parallel 302 10,020 302 12,453 281 7,306 281 9,804 Power Word Serial Parallel 302 11,049 302 13,489 281 8,091 281 10,589 302 Word Serial 2,460 Log 302 Parallel 4,944 279 1,625 281 4,132 Atan Word Serial Parallel 302 2,682 302 4,935 279 1,834 281 4,152 -1 Speed Grade Slice LUTs 223 4,898 223 6,937 223 7,560 223 9,940 223 8,198 223 10,578 223 1,856 223 4,290 223 2,025 223 4,257 Latency Block RAM 130 1 130 1 138 3 138 3 171 3 171 3 64 1 64 1 67 2 67 2 The performance summary for the double data type is shown in Table 4. Table 4: Block Performance Summary for Double Data Type Block name Mode Sin-cos Word Serial Parallel -3 Speed Grade LUTs & Flip-Flops 282 16,598 297 23,437 -2 Speed Grade Slice Registers 246 12,504 258 19,542 -1 Speed Grade Slice LUTs 215 12,060 222 18,811 Latency Block RAM 171 1 171 1 285 251 217 179 Word Serial 24,183 17,898 18,219 5 Sinh-cosh 297 258 222 179 Parallel 31,607 25,498 25,553 5 Throughput DSP48E1 Slices 24 9 1 9 28 27 1 27 28 40 1 40 28 8 1 8 26 5 1 5 Throughput DSP48E1 Slices 45 20 1 20 49 73 1 73 XAPP552 (v1.0) June 1, 2012 12 Trade-offs in the Current Approach Table 4: Block Performance Summary for Double Data Type (Cont’d) Block name Mode -3 Speed Grade LUTs & Flip-Flops -2 Speed Grade Slice Registers -1 Speed Grade Slice LUTs Power Word Serial Parallel 285 26,257 297 33,681 229 19,438 229 27,038 217 19,663 222 27,997 283 Word Serial 3,075 Log 297 Parallel 12,045 248 2,787 258 10,463 216 3,575 222 11,020 Atan Word Serial Parallel 283 5,140 297 12,399 247 3,319 258 10,636 216 4,100 222 11,271 Latency Block RAM 212 5 212 5 85 1 85 1 88 3 88 3 Throughput DSP48E1 Slices 49 101 1 101 49 25 1 25 47 14 1 14 Trade-offs in the Current Approach In the current approach simple range-enhancement algorithms are used along with a fixed-point CORDIC algorithm. The resulting output is expected to be hardware-efficient and to deliver performance comparable to that of the fixed-point CORDIC block. For example, the underlying CORDIC block in 32-bit mode for fixed-point sin and cos is expected to run at 345 MHz for parallel mode and 222 MHz for word-serial architecture. This compares with 280 MHz achieved for the floating-point CORDIC algorithm in single mode. The percentage of error tends to increase at a lower input range, whereas absolute error tends to increase at a higher range. Detailed error profiles for different blocks are shown in Figure 7 to Figure 9. The error profiles are presented in terms of units in last place (ULP). The error profiles for sin-cos and sinh-cosh in single mode are given in Figure 7. The error profiles were obtained by providing a ramp signal as the input of the floating-point reference block and comparing it with the Simulink output. It can be seen that the error is maximum when the input value is closest to 0. XAPP552 (v1.0) June 1, 2012 13 Trade-offs in the Current Approach X-Ref Target - Figure 7 Error in ULP Error in ULP sin-cos Single 12 10 8 6 4 2 0 -5 -4 -3 -2 -1 0 1 2 Input sinh-cosh Single 6 5 4 3 2 1 0 -5 -4 -3 -2 -1 0 1 2 Input Figure 7: Error Profile for sin-cos and sinh-cosh sin cos 3 4 sinh cosh 3 4 X552_07_042712 XAPP552 (v1.0) June 1, 2012 14 Trade-offs in the Current Approach X-Ref Target - Figure 8 Error in ULP The error profiles for exponential and log are shown in Figure 8. Log Single 1 Log 0.8 0.6 0.4 0.2 0 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Input Power Single 4 Power 3 Error in ULP 2 1 0 0 1 2 3 4 5 6 7 8 9 Input X552_08_042712 Figure 8: Error Profile for Log and Power XAPP552 (v1.0) June 1, 2012 15 Reference Design X-Ref Target - Figure 9 Error in ULP 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 0 Reference Design The error profile for atan produced by providing a ramp for input 1 and inverse ramp for input 2 is shown in Figure 9. atan Single atan 20 40 60 80 100 120 140 Input X552_08_042712 Figure 9: Error Profile for atan The reference design for this application note can be downloaded from: The reference design matrix is shown in Table 5. Table 5: Reference Design Matrix Parameter General Developer name Target devices (stepping level, ES, production, speed grades) Source code provided Source code format Description Nikhil Dhume and Ramakrishnan Srinivasakannan Spartan-6, Virtex-6, 7 Series, and Zynq-7000 devices Yes System Generator MDL library XAPP552 (v1.0) June 1, 2012 16 Conclusion Conclusion References Table 5: Reference Design Matrix (Cont’d) Parameter Description Design uses code and IP from existing Xilinx Yes application note and reference designs, CORE Generator software, or third party Simulation Functional simulation performed Yes Timing simulation performed Yes Test bench used for functional and timing Yes simulations Test bench format System Generator MDL file Simulator software/version used System Generator for DSP, version 13.4 ISE Design Suite 13.4 SPICE/IBIS simulations No Implementation Synthesis software tools/version used Xilinx Synthesis Technology (XST) 13.4 Implementation software tools/versions used System Generator for DSP, version 13.4 ISE Design Suite 13.4 Static timing analysis performed Yes Hardware Verification Hardware verified No Hardware platform used for verification N/A A floating-point library for computation of trigonometric, power, and logarithmic operations has been designed by applying range extension algorithms to underlying fixed-point blocks. This approach has been demonstrated to give comparable performance to the underlying fixed-point block. The device utilization, latency, and maximum operating frequency have been documented for all the blocks in the library. This application note uses the following references: 1. Volder, Jack E., The CORDIC trigonometric computing technique. IRE Transactions on Electronic Computers, vol. EC-8, September 1959, pp. 330-334 2. IEEE Std. 754-2008, IEEE Standard for Floating-Point Arithmetic, August 2008. 3. WP409, High-Level Implementation of Bit- and Cycle-Accurate Floating-Point DSP Algorithms with Xilinx FPGAs 4. Andraka, Ray. A survey of CORDIC algorithms for FPGA based computers, Proceedings of the 1998 ACM/SIGDA sixth international symposium on Field programmable gate arrays, Feb 22–24, 1998. pp. 191–200 5. Lang, T. and E. Antelo, High-throughput CORDIC-based geometry operations for 3D computer graphics, IEEE Transactions on Computers, vol. 54, no. 3, March 2005, pp. 347–361 XAPP552 (v1.0) June 1, 2012 17 Revision History Revision History Notice of Disclaimer 6. DS858, LogiCORE IP CORDIC Product Specification 7. UG638, System Generator for DSP Reference Guide 8. Creating a Block Mask (in MATLAB) The following table shows the revision history for this document. Date 06/01/2012 Version 1.0 Description of Revisions Initial Xilinx release. The information disclosed to you hereunder (the “Materials”) is provided solely for the selection and use of Xilinx products. To the maximum extent permitted by applicable law: (1) Materials are made available “AS IS” and with all faults, Xilinx hereby DISCLAIMS ALL WARRANTIES AND CONDITIONS, EXPRESS, IMPLIED, OR STATUTORY, INCLUDING BUT NOT LIMITED TO WARRANTIES OF MERCHANTABILITY, NON-INFRINGEMENT, OR FITNESS FOR ANY PARTICULAR PURPOSE; and (2) Xilinx shall not be liable (whether in contract or tort, including negligence, or under any other theory of liability) for any loss or damage of any kind or nature related to, arising under, or in connection with, the Materials (including your use of the Materials), including for any direct, indirect, special, incidental, or consequential loss or damage (including loss of data, profits, goodwill, or any type of loss or damage suffered as a result of any action brought by a third party) even if such damage or loss was reasonably foreseeable or Xilinx had been advised of the possibility of the same. Xilinx assumes no obligation to correct any errors contained in the Materials or to notify you of updates to the Materials or to product specifications. You may not reproduce, modify, distribute, or publicly display the Materials without prior written consent. Certain products are subject to the terms and conditions of the Limited Warranties which can be viewed at; IP cores may be subject to warranty and support terms contained in a license issued to you by Xilinx. Xilinx products are not designed or intended to be fail-safe or for use in any application requiring fail-safe performance; you assume sole risk and liability for use of Xilinx products in Critical Applications: XAPP552 (v1.0) June 1, 2012 18




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