A fast and low complexity operator for the computation of the arctangent of a complex number
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The document presents a novel method and hardware architecture for computing the arctangent of a complex number (atan2) efficiently and with low complexity. It outlines a two-stage approach that utilizes a coarse approximation followed by a lookup table to enhance accuracy, which leads to improvements in latency, speed, and resource use compared to traditional methods. This method addresses the trade-offs between computation speed, resource utilization, and output accuracy, making it suitable for various applications in hardware systems.
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Range Reduction Obtaining
| fr| requires the computation of y/x [see Fig. 2(a)], which involves the computation using
an LUT of 1/x. Since 1/x can
Fig. 3. Absolute error for one of our w = 12 atan2(a, b)/2π operators.
be extremely large for small values of x, a scaling operation is performed on |a| and |b|.
This block detects the leading-zeros in both |a| and |b|, scaling both by the same factor
(2s) so the MSB of x, the biggest one of both outputs, is always 1.Thanks to this block, x
is always in the [0.5, 1) range and the biggest possible value of 1/x is 2. C. Computation
of the Reciprocal For the computation of 1/x, two different strategies, both table based,
are used: direct tabulation for the w = 8 bit operator and bipartite tables for w = {12, 16}
bits. In both cases, all the bits from the input word are used. Therefore, for the direct
tabulation the only errors are those created by rounding the words stored in the table, and
for the bipartite tables, the maximum absolute value of the error can be estimated from
the second derivative of the stored function and also from the rounding errors. The value
stored in the first address of this table should be 2, but 2 − 2−n+1 is stored for a table
with n-bit words, so the MSB of all the stored words is the same, and therefore, it does
not need to be stored. The size of this table is 64 × 6 for w = 8, 128 × 14 + 128 × 7 for w
= 12, and 1k × 18 + 512 × 8 for w = 16 bit.](https://image.slidesharecdn.com/afastandlow-complexityoperatorforthecomputationofthearctangentofacomplexnumber-180918043636/85/A-fast-and-low-complexity-operator-for-the-computation-of-the-arctangent-of-a-complex-number-5-320.jpg)
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(SEMICONDUCTOR IP &PRODUCT DEVELOPMENT)
(ISO : 9001:2015Certified Company),
# 45, Vivekanandar Street, Dhevan kandappa Mudaliar nagar, Nainarmandapam,
Pondicherry– 605004, India.
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_________________________________________________________________
Advantages:
Latency is more
Speed and efficiency is more
References:
[1] J.-M. Muller, Elementary Functions: Algorithms and Implementation. Cambridge, MA, USA:
Birkhäuser, 1997.
[2] R. Gutierrez and J. Valls, “Low-power FPGA-implementation of atan(Y/X) using look-up table
methods for communication applications,” J. Signal Process. Syst., vol. 56, no. 1, pp. 25–33, 2009.
[3] F. de Dinechin and M. Istoan, “Hardware implementations of fixed-point atan2,” in Proc. 22nd
Symp. Comput. Arithmetic, Jun. 2015, pp. 34–40.
[4] R. Lyons, “Another contender in the arctangent race,” IEEE Signal Process. Mag., vol. 21, no. 1, pp.
109–110, Jan. 2004.
[5] S. Rajan, S. Wang, R. Inkol, and A. Joyal, “Efficient approximations for the arctangent function,”
IEEE Signal Process. Mag., vol. 23, no. 3, pp. 108–111, May 2006.
[6] X. Girones, C. Julia, and D. Puig, “Full quadrant approximations for the arctangent function,” IEEE
Signal Process. Mag., vol. 30, no. 1, pp. 130–135, Jan. 2013.
[7] J. M. Shima, “FM demodulation using a digital radio and digital signal processing,” M.S. thesis,
Univ. Florida, Gainesville, FL, USA, 1995.
[8] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions With Formulas, Graphs, and
Mathematical Tables, 10th ed. New York, NY, USA: Dover, 1964.
[9] M. Arnold, T. Bailey, and J. Cowles, “Error analysis of the kmetz/maenner algorithm,” J. VLSI
Signal Process. Syst. Signal, Image Video Technol., vol. 33, no. 1, pp. 37–53, 2003. [Online]. Available:
http://dx.doi.org/10.1023/A:1021189701352](https://image.slidesharecdn.com/afastandlow-complexityoperatorforthecomputationofthearctangentofacomplexnumber-180918043636/85/A-fast-and-low-complexity-operator-for-the-computation-of-the-arctangent-of-a-complex-number-6-320.jpg)
![NXFEE INNOVATION
(SEMICONDUCTOR IP &PRODUCT DEVELOPMENT)
(ISO : 9001:2015Certified Company),
# 45, Vivekanandar Street, Dhevan kandappa Mudaliar nagar, Nainarmandapam,
Pondicherry– 605004, India.
Buy Project on Online :www.nxfee.com | contact : +91 9789443203 |
email : nxfee.innovation@gmail.com
_________________________________________________________________
[10] D. D. Sarma and D. W. Matula, “Faithful bipartite ROM reciprocal tables,” in Proc. 12th Symp.
Comput. Arithmetic, Jul. 1995, pp. 17–28.
[11] M. J. Schulte and J. E. Stine, “Symmetric bipartite tables for accurate function approximation,” in
Proc. 13th IEEE Symp. Comput. Arithmetic, Jul. 1997, pp. 175–183.
[12] M. J. Schulte and J. E. Stine, “Approximating elementary functions with symmetric bipartite
tables,” IEEE Trans. Comput., vol. 48, no. 8, pp. 842–847, Aug. 1999.](https://image.slidesharecdn.com/afastandlow-complexityoperatorforthecomputationofthearctangentofacomplexnumber-180918043636/85/A-fast-and-low-complexity-operator-for-the-computation-of-the-arctangent-of-a-complex-number-7-320.jpg)
A fast and low complexity operator for the computation of the arctangent of a complex number
- 1. NXFEE INNOVATION (SEMICONDUCTOR IP &PRODUCT DEVELOPMENT) (ISO : 9001:2015Certified Company), # 45, Vivekanandar Street, Dhevan kandappa Mudaliar nagar, Nainarmandapam, Pondicherry– 605004, India. Buy Project on Online :www.nxfee.com | contact : +91 9789443203 | email : [email protected] _________________________________________________________________ A Fast and Low-Complexity Operator for the Computation of the Arctangent of a Complex Number Abstract: The computation of the arctangent of a complex number, i.e., the atan2 function, is frequently needed in hardware systems that could profit from an optimized operator. In this brief, we present a novel method to compute the atan2 function and a hardware architecture for its implementation. The method is based on a first stage that performs a coarse approximation of the atan2 function and a second stage that improves the output accuracy by means of a lookup table. We present results for fixed-point implementations in a field-programmable gate array device, all of them guaranteeing last-bit accuracy, which provide an advantage in latency, speed, and use of resources, when compared with well-established fixed-point options. Software Implementation: Modelsim Xilinx 14.2 Existing System: The computation of the arctangent function atan2(a, b) (see Fig. 1), i.e., obtaining the angle of a complex number c = b + ja, has been the subject of extensive study, because this computation is required in many applications. In hardware approximations for atan2(a, b), there is often a tradeoff between the use of resources and the computation speed and/or latency. For example, the fastest option for the computation of any function may always be the direct implementation with a lookup table (LUT), but, since the atan2(a, b) is a function of two input variables, in such a case, if the precision of the input data increases by one bit, the amount of memory needed increases by a factor of four. On
- 2. NXFEE INNOVATION (SEMICONDUCTOR IP &PRODUCT DEVELOPMENT) (ISO : 9001:2015Certified Company), # 45, Vivekanandar Street, Dhevan kandappa Mudaliar nagar, Nainarmandapam, Pondicherry– 605004, India. Buy Project on Online :www.nxfee.com | contact : +91 9789443203 | email : [email protected] _________________________________________________________________ the other hand, iterative algorithms, such as the Coordinated Rotation Digital Computer (CORDIC), can be implemented with a minimal use of resources, but at the cost of a low processing speed. If more parallelism is introduced in the implementation, much higher throughputs can be achieved, but the use of resources and the computational delay are increased. Fig. 1. 3-D plot of atan2(a, b)/2π for the first octant Fig. 1(b). Simplified scheme of the proposed approximation for atan2(a, b)/2π Smaller LUTs can be achieved by using the recip-mult-atan method (RMAM), first, z = a/b is calculated by computing 1/b with an LUT (avoiding thus the large LUT needed for a two-variable function) and multiplying the result by a, and finally, the one-variable function atan(z) is computed using another LUT. Another option is to use high-order algebraic polynomials, like Chebyshev polynomials or Taylor series. These methods offer
- 3. NXFEE INNOVATION (SEMICONDUCTOR IP &PRODUCT DEVELOPMENT) (ISO : 9001:2015Certified Company), # 45, Vivekanandar Street, Dhevan kandappa Mudaliar nagar, Nainarmandapam, Pondicherry– 605004, India. Buy Project on Online :www.nxfee.com | contact : +91 9789443203 | email : [email protected] _________________________________________________________________ good precision, but, since the arctangent is highly nonlinear, they lead to long polynomials and intensive computations. In other cases, approximations based on rational functions are used, as they may provide good enough results with a few elementary operations. As a general rule, in this kind of approximations, the division operation is the main contributor to their computational cost, but in addition to that division, they usually require one or more multiplication operations The architecture we propose is essentially a two-stage method, as shown in Fig. 1 (b). The First Stage uses a low-complexity coarse approximation for the two-input atan2(a, b)/2π. The Second Stage improves the accuracy by means of a small LUT that stores precomputed error values, as a function of the output of the First Stage. This table does not depend on the two inputs a and b of the atan2 operator and is, therefore, comparatively much smaller. As will be shown, the resulting operator is small and can compute the arctangent faster than other popular options, for the same output accuracy. Disadvantages: Less Latency Less speed and efficiency Proposed System: Hardware Architecture Fig. 2(a) shows the scheme of the proposed hardware architecture. It works with the absolute values of the inputs a and b, and their signs are used later in the scheme. The “sel” signal selects the operands for the division operation according to the signs of a + b and a − b. The division required for the computation of | fr| is implemented with a table that stores 1/x and a multiplier that completes the division. A scaling stage is added in order to reduce the size of the reciprocal table. The most relevant implementation details
- 4. NXFEE INNOVATION (SEMICONDUCTOR IP &PRODUCT DEVELOPMENT) (ISO : 9001:2015Certified Company), # 45, Vivekanandar Street, Dhevan kandappa Mudaliar nagar, Nainarmandapam, Pondicherry– 605004, India. Buy Project on Online :www.nxfee.com | contact : +91 9789443203 | email : [email protected] _________________________________________________________________ are commented upon in Section IV-A–E. Specifically, we give details for three accuracies: w = {8, 12, 16} bit. Fig. 2 (a) Implementation scheme for the proposed approximation for the atan2(a, b)/2π function. (b) Simplified model for error analysis. Datapath Dimensioning Fig. 2(a) shows the binary formats used in different buses of the system, where s(q, t) and u(q, t) denote signed and unsigned fixed point formats, respectively. 2q is the weight of the most significant bit (MSB) and 2t is the weight of the LSB Note that truncating the output of the multiplier does not introduce an additional error unless the truncated bits are used in the following processing steps. In a more realistic scenario, LBA can still be achieved with smaller LUTs that either do not use all the available bits as their address word or that are implemented as bipartite tables. Although in these cases the tables introduce bigger errors (as explained in the following), LBA could still be achieved, since and represent an upper bound that could not be reached. For this reason, we performed exhaustive tests for different LUT sizes looking for optimized implementations. Fig. 3 is an example of the error pattern obtained in one of the w = 12 operators.
- 5. NXFEE INNOVATION (SEMICONDUCTOR IP &PRODUCT DEVELOPMENT) (ISO : 9001:2015Certified Company), # 45, Vivekanandar Street, Dhevan kandappa Mudaliar nagar, Nainarmandapam, Pondicherry– 605004, India. Buy Project on Online :www.nxfee.com | contact : +91 9789443203 | email : [email protected] _________________________________________________________________ Range Reduction Obtaining | fr| requires the computation of y/x [see Fig. 2(a)], which involves the computation using an LUT of 1/x. Since 1/x can Fig. 3. Absolute error for one of our w = 12 atan2(a, b)/2π operators. be extremely large for small values of x, a scaling operation is performed on |a| and |b|. This block detects the leading-zeros in both |a| and |b|, scaling both by the same factor (2s) so the MSB of x, the biggest one of both outputs, is always 1.Thanks to this block, x is always in the [0.5, 1) range and the biggest possible value of 1/x is 2. C. Computation of the Reciprocal For the computation of 1/x, two different strategies, both table based, are used: direct tabulation for the w = 8 bit operator and bipartite tables for w = {12, 16} bits. In both cases, all the bits from the input word are used. Therefore, for the direct tabulation the only errors are those created by rounding the words stored in the table, and for the bipartite tables, the maximum absolute value of the error can be estimated from the second derivative of the stored function and also from the rounding errors. The value stored in the first address of this table should be 2, but 2 − 2−n+1 is stored for a table with n-bit words, so the MSB of all the stored words is the same, and therefore, it does not need to be stored. The size of this table is 64 × 6 for w = 8, 128 × 14 + 128 × 7 for w = 12, and 1k × 18 + 512 × 8 for w = 16 bit.
- 6. NXFEE INNOVATION (SEMICONDUCTOR IP &PRODUCT DEVELOPMENT) (ISO : 9001:2015Certified Company), # 45, Vivekanandar Street, Dhevan kandappa Mudaliar nagar, Nainarmandapam, Pondicherry– 605004, India. Buy Project on Online :www.nxfee.com | contact : +91 9789443203 | email : [email protected] _________________________________________________________________ Advantages: Latency is more Speed and efficiency is more References: [1] J.-M. Muller, Elementary Functions: Algorithms and Implementation. Cambridge, MA, USA: Birkhäuser, 1997. [2] R. Gutierrez and J. Valls, “Low-power FPGA-implementation of atan(Y/X) using look-up table methods for communication applications,” J. Signal Process. Syst., vol. 56, no. 1, pp. 25–33, 2009. [3] F. de Dinechin and M. Istoan, “Hardware implementations of fixed-point atan2,” in Proc. 22nd Symp. Comput. Arithmetic, Jun. 2015, pp. 34–40. [4] R. Lyons, “Another contender in the arctangent race,” IEEE Signal Process. Mag., vol. 21, no. 1, pp. 109–110, Jan. 2004. [5] S. Rajan, S. Wang, R. Inkol, and A. Joyal, “Efficient approximations for the arctangent function,” IEEE Signal Process. Mag., vol. 23, no. 3, pp. 108–111, May 2006. [6] X. Girones, C. Julia, and D. Puig, “Full quadrant approximations for the arctangent function,” IEEE Signal Process. Mag., vol. 30, no. 1, pp. 130–135, Jan. 2013. [7] J. M. Shima, “FM demodulation using a digital radio and digital signal processing,” M.S. thesis, Univ. Florida, Gainesville, FL, USA, 1995. [8] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables, 10th ed. New York, NY, USA: Dover, 1964. [9] M. Arnold, T. Bailey, and J. Cowles, “Error analysis of the kmetz/maenner algorithm,” J. VLSI Signal Process. Syst. Signal, Image Video Technol., vol. 33, no. 1, pp. 37–53, 2003. [Online]. Available: http://dx.doi.org/10.1023/A:1021189701352
- 7. NXFEE INNOVATION (SEMICONDUCTOR IP &PRODUCT DEVELOPMENT) (ISO : 9001:2015Certified Company), # 45, Vivekanandar Street, Dhevan kandappa Mudaliar nagar, Nainarmandapam, Pondicherry– 605004, India. Buy Project on Online :www.nxfee.com | contact : +91 9789443203 | email : [email protected] _________________________________________________________________ [10] D. D. Sarma and D. W. Matula, “Faithful bipartite ROM reciprocal tables,” in Proc. 12th Symp. Comput. Arithmetic, Jul. 1995, pp. 17–28. [11] M. J. Schulte and J. E. Stine, “Symmetric bipartite tables for accurate function approximation,” in Proc. 13th IEEE Symp. Comput. Arithmetic, Jul. 1997, pp. 175–183. [12] M. J. Schulte and J. E. Stine, “Approximating elementary functions with symmetric bipartite tables,” IEEE Trans. Comput., vol. 48, no. 8, pp. 842–847, Aug. 1999.