Signal Processing - November 2017 - 168

is used, the approximation of the
atan2(a,b), has a maximum error of
! 4.07c, which corresponds to 6.5 exact
bits [i.e., the number of most significant
bits that are zero in the binary representation of the maximum absolute value
of the error; this value can be obtained
as - log 2 (max (abs (error))), assuming
a [0, 1) normalization of the values].
Another coarse approximation was
presented in [6, eq. (12)], achieving the
same accuracy with higher computational cost.
As shown in Table 2, by using a small
error LUT of 32 values our proposal
achieves similar accuracy to Lyons' [2,
eq. (2)] and Rajan et al.'s [3, eq. (9)], but
requiring three and two fewer multiplications, respectively. If larger LUT sizes
are used, similar accuracy to [4]-(18)
and (16) can be achieved with fewer
arithmetic resources by our method.
Finally, another option considered
for comparison purposes is the method described in [7], which we call the
Ratio+LUT method: first, the ratio
z = a/b is calculated with a division
operation (to avoid a large LUT storing
a two-variable function), second, the
one-variable function atan(z) is computed using a LUT. Note that, to extend the
computation to a full-quadrant atan2,
four additions would be needed. As seen
in Table 2, when using the Ratio+LUT,
the size of this LUT would be four times
larger than in our proposal, for the same
final accuracy. Moreover, in fixed-point
implementations of the algorithm, the
largest value stored in the atan2 LUT
would be 3.5 bits larger than in our case,
making the quantization noise worse for
the same LUT size and word length.

Table 2. The computational cost comparison of our proposed algorithm with various
previously published arctangent algorithms.
Method

/

+

LUT Size (Words)

Maximum Error (Degrees)

[5, eq. (4.13)]

1

0

3

-

! 4.075

[6, eq. (12)]

2

0

4

-

! 4.075

Ours

1

0

5

32

! 0.249

[2, eq. (2)]

1

3

5

-

! 0.276

Ratio+LUT

1

0

4

128

! 0.224

Ours

1

0

5

64

! 0.126

[4, eq. (18)]

1

4

4

-

! 0.162

[3, eq. (9)]

1

3

5

-

! 0.086

Ratio+LUT

1

0

4

256

! 0.112

Ours

1

0

5

1K

! 0.008

[4, eq. (16)]

1

7

6

-

! 0.008

Ratio+LUT

1

0

4

4K

! 0.007

5

[5]
[6]

Our Proposal
[2, eq. (2)]+LUT
Ratio+LUT
[4, eq. (18)]+LUT
[6, eq. (12)]+LUT

1
[2]
0.1

[4]

10

15

0.01

# atan2 Exact Bits

Maximum atan2 Error (°)

10

*

0.001
20
0 2

8

32 128 512 2K
LUT Size (Words)

8K 32K

FIGURE 5. The maximum arctan error and number of exact bits versus LUT size.

an algorithm is proposed for a two-octant
one-variable atan(x), three extra addition/
subtractions are included in this table as
the cost of extending the approximation
to all the quadrants. In Table 2, we only
consider divisions, multiplications, additions/subtractions, and storage requirements to evaluate the computational cost
of the algorithms. Other required operations have a computational cost that can
be highly platform dependent. Operators
like binary shifts, mod(), floor(), bit string
concatenations, etc. have no cost at all in
fixed-point application-specific integrated
168

circuit (ASIC) or field-programmable
gate array (FPGA) implementations but
may result in additional computational
time in pure software implementations.
For example, in a fixed-point FPGA
implementation, the computation of
offset = 2*not(b0)+not(xor(b1,b0)) doesn't
involve multiplications nor additions, just
bit concatenations and simple logic operators. In such a case, the hardware architecture could be implemented following the
data flow illustrated in Figure 3.
When only the first stage of the
algorithm, i.e., without the error LUT,
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November 2017

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Using a different first stage
As we have shown (see Table 2), the first
stage of our algorithm requires the least
arithmetic resources. Nevertheless, in this
section, we explore the idea of adding a
second stage based on an error LUT to
other atan2 algorithms. Adding a second
stage could be an easy way of improving
the accuracy of an existing implementation with minimum design cost.
We have used [6, eq. (12)], [2, eq.
(2)], and [4, eq. (18)] as first stages in
a two-stage algorithm. Their error was
Table of Contents for the Digital Edition of Signal Processing - November 2017
