Signal Processing - November 2017 - 165

ratio t m /T will be dimensionless and
i will be measured in radians. In the
first-stage processing we will estimate
t m and show how this will yield an estimate of angle i, for a given accuracy,
more efficiently than performing other
atan2(a,b) algorithms.
It can easily be seen that the sequence
p [n] = {b, a, - b, - a} is a sampled
version of p (t), as defined by (1) as

Im
a

c = b + ja
b

p(t )

b
tm = f

Case (1) -a
Re

c = b + ja a

p(t ) a

b

Case (2)
Re

-a
-b

Case (4) -a

b

b
-a

b
a

Re
a

tm

-b

c = b + ja

fr = -b
2a

b
t

f

Offset



fr = a
2b
t

c = b + ja
p(t )

b

-a

tm

a

b

fr = -b
2a
t

-a

Re

a

-a

-b

tm

b
p(t )

Case (3)

b

= c · $ cos ^-ih, cos ` r - i j,
2

cos ` 2r - i j, cos ` 3r - i j.
2
2
= c · cos ` nr - i j = p ^ t h t = nT
2
4
where n = 0, 1, 2, 3.
(3)

-a

b
t

-b

p [n] = {b, a, - b, - a} = c · {cos ^ i h,
sin ^ i h, - cos ^ i h, - sin ^ i h,

Our goal is to compute t m from the
p [n] samples.
To clarify our scenario here, Figure 2
shows the various p (t) waveforms that
result from various values of our complex
number input c. The time location of the
absolute maximum of the sinusoidal p (t)
waveform, t m, is proportional to the angle
of c.
The first step of the first-stage processing is to determine the time location of the
largest sample of four-sample sequence
p [n] (determined from the signs of a + b
and a - b), a parameter that we call offset. The second step of the first-stage processing computes the time location of the
maximum value of p (t) relative to offset,
a parameter that we call f.
Based on the previously given concepts and relationships, we conclude the
first step of the first-stage processing by
determining the value for offset, which
will be 0, 1, 2, or 3. In the second step of
the first-stage processing, we complete
the estimation of t m approximating the
value of time variable f. Specifically,
we approximate the p (t) signal by a
second-order Taylor series in the vicinity of the largest p [n] sample as detailed
in "Appendix," which gives us fr, an
approximation of the time location of
the maximum value of p (t) relative to
that sample

fr = a
2b

a

tm

FIGURE 2. The proposed atan2(a,b) algorithm illustrated for four different possible c = b + ja

values.

	

f . fr /

occurred prior to the largest sample in
p [n].)
Based on the values for offset and
fr from Table 1 and using (2), assuming T = 4, the result of the first-stage
processing is an approximation of
atan2(a,b), normalized to the range
[0, 1), as follows:

- pl (0)
p (1)
=
.(4)
pm (0)
2p (0)

In a general case, this computation
would require three samples: the biggest
of the four samples of the waveform, and
also the two samples adjacent to that sample, as depicted for case 1 in Figure 2. But
since those two adjacent samples have the
same absolute value and opposite sign,
only two samples are required in (4):
the largest sample p (0) and its following sample p (1). Using (4), we compile
our desired processing parameters in
Table 1. (Note that a negative value of
fr indicates that p (t) maximum value

atan2 (a, b)
= i = t m mod 1
2r
4
2r
 

offset + fr
	
.
mod 1,
4
(5)
where the mod operator is needed to
translate negative values to the desired

Table 1. The deduction of the expression for fr as a function of the signs of a + b and a - b.
Case

a+b 2 0

a -b 2 0

p(0)

p(1)

Offset

fr =

1

1

0

b

a

0

a
2b

2

1

1

a

-b

1

-b
2a

3

0

1

-b

-a

2

a
2b

4

0

0

-a

b

3

-b
2a

IEEE SIGNAL PROCESSING MAGAZINE

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November 2017

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p (1)
2p (0)

165
Table of Contents for the Digital Edition of Signal Processing - November 2017
