1
g–h AND g–h–k FILTERS
1.1 WHY TRACKING AND PREDICTION ARE NEEDED IN A
RADAR
Let us first start by indicating why tracking and prediction are needed in a radar.
Assume a fan-beam surveillance radar such as shown in Figure 1.1-1. For such
a radar the fan beam rotates continually through 360

, typically with a period of
10 sec. Such a radar provides two-dimensional information about a target. The
first dimension is the target range (i.e., the time it takes for a transmitted pulse
to go from the transmitter to the target and back); the second dimension is the
azimuth of the target, which is determined from the azimuth angle (see Figure
1.1-1) the fan beam is pointing at when the target is detected [1]. Figures 1.1-2
through 1.1-6 show examples of fan-beam radars.
Assume that at time t ¼ t
1
the radar is pointing at scan angle  and two
targets are detected at ranges R
1
and R
2
; see Figure 1.1-7. Assume that on the
next scan at time t ¼ t
1
þ T (i.e., t
1
þ 10 see), again two targets are detected;
see Figure 1.1-7. The question arises as to whether these two targets detected on
the second scan are the same two targets or two new targets. The answer to this
question is important for civilian air traffic control radars and for military

The chances of incorrect association could be greatly reduced if we could
accurately predict ahead of time where the echoes of targets 1 and 2 are to be
expected on the second scan. Such a prediction is easily made if we had an
estimate of the velocity and position of targets 1 and 2 at the time of the first
scan. Then we could predict the distance target 1 would move during the scan-
to-scan period and as a result have an estimate of the target’s future position.
Assume this prediction was done for target 1 and the position at which target 1
is expected at scan 2 is indicated by the vertical dashed line in Figure 1.1-7.
Because the exact velocity and position of the target are not known at the time
of the first scan, this prediction is not exact. If the inaccuracy of this prediction
is known, we can set up a Æ 3 (or Æ 2) window about the expected value,
where  is the root-mean-square (rms), or equivalently, the standard deviation
of the sum of the prediction plus the rms of the range measurement. This
window is defined by the pair of vertical solid lines straddling the expected
position. If an echo is detected in this window for target 1 on the second scan,
Figure 1.1-3 Fan-beam track-while-scan S-band and X-band radar antennas emplaced
on tower at Prince William Sound Alaska (S-band antenna on left). These radars are part
of the Valdez shore-based Vessel Traffic System (VTS). (Photo courtesy of Raytheon
Company.)
WHY TRACKING AND PREDICTION ARE NEEDED IN A RADAR
5
then with high probability it will be the echo from target 1. Similarly, a Æ 3
window is set for target 2 at the time of the second scan; see Figure 1.1-7.
For simplicity assume we have a one-dimensional world. In contrast to a
term you may have already heard, ‘‘flatland’’, this is called ‘‘linland’’.We
assume a target moving radially away or toward the radar, with x
n
representing
the slant range to the target at time n. In addition, for further simplicity we
assume the target’s velocity is constant; then the prediction of the target

6
g–h AND g–h–k FILTERS
Figure 1.1-5 L-band fan-beam track-while-scan Pulse Acquisition Radar of HAWK
system, which is used by 17 U.S. allied countries and was successfully used during
Desert Storm. Over 300 Hawk systems have been manufactured. (Photo courtesy of
Raytheon Company.)
Figure 1.1-6 New fan-beam track-while-scan L-band airport surveillance radar ASR-
23SS consisting of dual-beam cosecant squared antenna shown being enclosed inside
50-ft radome in Salahah, Oman. This primary radar uses a 25-kW peak-power solid-
state ‘‘bottle’’ transmitter. Mounted on top of primary radar antenna is open-array
rectangular antenna of colocated MSSR. This system is also being deployed in Hong
Kong, India, The People’s Republic of China, Brazil, Taiwan, and Australia.
WHY TRACKING AND PREDICTION ARE NEEDED IN A RADAR
7
that we can easily extend our results to the real, multidimensional world where
we have changing velocity targets.
The –, ––, and Kalman tracking algorithms described in this book are
used to obtain running estimates of x
n
and
_
x
n
, which in turn allows us to do the
association described above. In addition, the prediction capabilities of these
filters are used to prevent collisions in commercial and military air traffic
control applications. Such filter predictions also aid in intercepting targets in
defensive military situations.
The fan-beam ASR-11 Airport Surveillance Radar (ASR) in Figure 1.1-2 is
an example of a commercial air traffic control radar. The fan-beam marine radar

Storm for the intercept of SCUD missiles. Another example of such a radar is
the AEGIS wide-angle electronically scanned radar of Figure 1.1-10.
The Kalman tracking algorithms discussed in this book are used to
accurately predict where ballistic targets such as intercontinental ballistic
missiles (ICBMs) will impact and also for determining their launch sites (what
country and silo field). Examples of such radars are the upgraded wide-angle
electronically steered Ballistic Missile Early Warning System (BMEWS) and
the Cobra Dane radars of Figures 1.1-11 and 1.1-12 [1–3]. Another such wide-
angle electronically steered radar is the tactical ground based 25, 000-element
X-band solid state active array radar system called Theater High Altitude Area
Figure 1.1-10 Multifunction shipboard AEGIS electronically scanned phased-array
radar used to track many targets while also doing search on a time-shared basis. [1, 3].
Two hundred thirty-four array faces built each with about 4000 radiating elements and
phase shifters [137]. (Photo courtesy of Raytheon Company.)
10
g–h AND g–h–k FILTERS
Figure 1.1-11 Upgrade electronically steered phased-array BMEWS in Thule,
Greenland [1]. (Photo courtesy of Raytheon Company.)
Figure 1.1-12 Multifunction electronically steered Cobra Dane phased-array radar (in
Shemya, Alaska). Used to track many targets while doing search on a time-shared basis
[1, 3]. (Photo by Eli Brookner.)
WHY TRACKING AND PREDICTION ARE NEEDED IN A RADAR
11
Defense (THAAD; formerly called GBR) system used to detect, track, and
intercept, at longer ranges than the PATRIOT, missiles like the SCUD; see
Figure 1.1-13 [136, 137]. Still another is the Pave Paws radar used to track
satellites and to warn of an attack by submarine-launched ballistic missiles; see
Figure 1.1-14 [1–3].
Figure 1.1-13 A 25,000-element X-band MMIC (monolithic microwave integrated
circuit) array for Theater High Altitude Area Defense (THAAD; formerly GBR) [136,

position and velocity at time n À 1. (Later we shall show how we get our initial
estimates for the target position and velocity.)
Assume the target is estimated to have a velocity at time n À 1 of 200 ft=sec.
Let the scan-to-scan period T for the radar be 10 sec. Using (1.1-a) we estimate
the target to be (200 ft=sec) (10 sec) ¼ 2000 ft further away at time n than it was
at time n À 1. This is the position x
n
indicated in Figure 1.2-1. Here we are
assuming the aircraft target is flying away from the radar, corresponding to the
situation where perhaps enemy aircraft have attacked us and are now leaving
Figure 1.1-16 Short-range, limited-scan, electronically scanned (phase-frequency)
phased-array artillery locating Firefinder AN=TPQ-36 radar [1]. Two hundred forty-
three have been built. (Photo courtesy of Hughes Co.)
14
g–h AND g–h–k FILTERS
(a)
(b)
Figure 1.1-17 Very long range (over 1000 nmi) one-dimensional electronically
scanned (in azimuth direction) phased-array ROTHR: (a) transmit antenna; (b) receive
antenna [1]. (Photos courtesy of Raytheon Company.)
g–h FILTERS
15
Figure 1.1-18 AN=SPG-51 TARTAR dedicated shipboard tracking C-band radar
using offset parabolic reflector antenna [3]. Eighty-two have been manufactured. (Photo
courtesy of Raytheon Company.)
Figure 1.1-19 Hawk tracker-illuminator incorporating phase 3 product improvement
kit, which consisted of improved digital computer and the Low Altitude Simultaneous
HAWK Engagement (LASHE) antenna (small vertically oriented antenna to the left of
main transmit antenna, which in turn is to the left of main receive antenna). Plans are
underway to use this system with the AMRAAM missile. (Photo courtesy of Raytheon

not due to the target’s higher velocity. However, the target could really be going
faster than we anticipated, so we would like to allow for this possibility. We do
this by not giving the target the full benefit of the 6-ft=sec apparent increase in
velocity but instead a fraction of this increase. Let us use the fraction
1
10
th of the
Figure 1.1-20 NATO SEASPARROW shipborne dedicated tracker-illuminator
antenna [3]. One hundred twenty-three have been built. (Photo courtesy of Raytheon
Company.)
g–h FILTERS
17
6-ft=sec apparent velocity increase. (How we choose the fraction
1
10
will be
indicated later.) The updated velocity now becomes
Updated velocity ¼ 200 ft=sþ
1
10
60 ft
10 sec

¼ 200 ft=sec þ 0:60 ft=sec ¼ 200:6ft=sec
ð1:2-2Þ
In this way we do not increase the velocity of the target by the full amount. If
the target is actually going faster, then on successive observations the observed
position of the target will on the average tend to be biased further in range than
the predicted positions for the target. If on successive scans the target velocity is
increased by 0.6 ft=sec on average, then after 10 scans the target velocity will be

Figure 1.2-1 Target predicted and measured position, x
n
and y
n
, respectively, on nth
scan.
18
g–h AND g–h–k FILTERS
measurement at time n is the same as the symbol for the velocity estimate at
time n just before the measurement was made, both using the variable
_
x
n
.To
distinguish these two estimates, a second subscript is added. This second
subscript indicates the time at which the last measurement was made for use in
estimating the target velocity. Thus (1.2-3) becomes
_
x
Ã
n;n
¼
_
x
Ã
n;nÀ1
þ h
n
y
n

n, T ¼ 10 sec later, the target with a radial velocity of 200 ft=sec is at a range
2000 ft further out. As before, assume that at time n the target is actually
y
This is the notation of reference 5. Often, as shall be discussed shortly, in the literature [6, 7] a
caret over the variable is used to indicate an estimate.
Figure 1.2-2 Target predicted, filtered, and measured positions using new notation.
g–h FILTERS
19
observed to be 60 ft further downrange from where predicted; see Figure 1.2-2.
Again we ask where the target actually is. At x
Ã
n;nÀ1
,aty
n
, or somewhere in
between? As before initially assume a very accurate laser radar is being used for
the measurements at time n À 1 and n. It can then be concluded that the target is
at the range it is observed to be at time n by the laser radar, that is, 60 ft further
downrange than predicted. Thus
Updated position ¼ 10 nmi þ 2000 ft þ 60 ft ð1:2-5Þ
If, however, we assume that we have an ordinary microwave radar with a 1
of 50 ft, then the target could appear to be 60 ft further downrange than expected
just due to the measurement error of the radar. In this case we cannot reasonably
assume the target is actually at the measured range y
n
, at time n. On the other
hand, to assume the target is at the predicted position is equally unreasonable.
To allow for the possibility that the target could actually be a little downrange
from the predicted position, we put the target at a range further down than
predicted by a fraction of the 60 ft. Specifically, we will assume the target is

6
is represented by the parameter g
n
, which can be dependent
on n. Equation (1.2-7) represents the desired equation for updating the target
position.
Equations (1.2-4) and (1.2-7) together give us the equations for updating the
target velocity and position at time n after the measurement of the target range
y
n
has been made. It is convenient to write these equations together here as the
present position and velocity g–h track update (filtering) equations:
_
x
Ã
n;n
¼
_
x
Ã
n;nÀ1
þ h
n
y
n
À x
Ã
n;nÀ1
T
!

g–h AND g–h–k FILTERS
estimate is in contrast to the prediction estimate x
n;nÀ1
, which is an estimate of
x
n
based on past measurements. The term smoothed is used sometimes in place
of the term filtered [8]. ‘‘Smoothed’’ is also used [7] to indicate an estimate of
the position or velocity of the target at some past time between the first and last
measurement, for example, the estimate x
Ã
h;n
, where n
0
< h < n, n
0
being the
time of the first measurement and n the time of the last measurement. In this
book, we will use the latter definition for smoothed.
Often in the literature [6, 7] a caret is used over the variable x to indicate that
x is the predicted estimate x
Ã
n;nÀ1
while a bar over the x is used to indicate that
x is the filtered estimate x
Ã
n;n
. Then g–h track update equations of (1.2-8a) and
(1.2-8b) become respectively
"

À
^
x
n
Þð1:2-9bÞ
It is now possible by the use of (1.1-1) to predict what the target position and
velocity will be at time n þ 1 and to repeat the entire velocity and position
update process at time n þ 1 after the measurement y
nþ1
at time n þ 1 has been
made. For this purpose (1.1-1) is rewritten using the new notation as the g–h
transition equations or prediction equations:
_
x
Ã
nþ1;n
¼
_
x
Ã
n;n
ð1:2-10aÞ
x
Ã
nþ1;n
¼ x
Ã
n;n
þ T
_

Ã
nþ1;n
¼
_
x
Ã
n;nÀ1
þ
h
n
T
ðy
n
À x
Ã
n;nÀ1
Þð1:2-11aÞ
x
Ã
nþ1;n
¼ x
Ã
n;nÀ1
þ T
_
x
Ã
nþ1;n
þ g
n

n
. The estimate y
n
is actually the radar measurement at time n. The
estimate x
Ã
n;nÀ1
is based on the measurement made at time n À 1 and all
preceding times. What we want to do is somehow combine these two estimates
to obtain a new best estimate of the present target position. This is the filtering
problem. We have the estimates y
n
and x
Ã
n;nÀ1
and we would like to find a
combined estimate x
Ã
n;n
, as illustrated in Figure 1.2-3. The problem we face is
how to combine these two estimates to obtain the combined estimate x
Ã
n;n
.Ify
n
and x
Ã
n;nÀ1
were equally accurate, then we would place x
Ã

mate of x
Ã
n;n
based on measurement y
n
and prediction x
Ã
n;nÀ1
.
22
g–h AND g–h–k FILTERS
The above equation gives the updated estimate as a weighted sum of the two
estimates. The selection of the fraction g
n
determines whether we put the
combined estimate closer to y
n
or to x
Ã
n;nÀ1
. For example, if y
n
and x
Ã
n;nÀ1
are
equally accurate, then we will set g
n
equal to
1

closer to x
Ã
n;nÀ1
. How we select g
n
shall be shown later.
1.2.2 a–b Filter
Now that we have developed the g–h filter, we are in a position to develop the
– filter. To obtain the – filter, we just take (1.2-11) and replace g with 
and h with —we now have the – filter. Now you know twice as much,
knowing the – filter as well as the g–h filter.
1.2.3 Other Special Types of Filters
In this section we will increase our knowledge 22-fold because we will cover 11
new tracking filters. Table 1.2-1 gives a list of 11 new tracking filters. The
equations for all of these filters are given are given by (1.2-11). Consequently,
all 11 are g–h filters. Hence all 11 are – filters. Thus we have increased our
tracking-filter knowledge 22 fold! You are a fast learner! How do these filters
differ? They differ in the selection of the weighting coefficients g and h as shall
be seen later. (Some actually are identical). For some of these filters g and h
depend on n. This is the case for the Kalman filter. It is worthwhile emphasizing
that (1.2-11a) and (1.2-11b) are indeed the Kalman filter prediction equations,
albeit for the special case where only the target velocity and position are being
tracked in one dimension. Later we will give the Kalman filter for the multi-
TABLE 1.2-1. Special Types of Filters
1. Wiener filter
2. Fading-memory polynomial filter
3. Expanding-memory (or growing-memory) polynomial filter
4. Kalman filter
5. Bayes filter
6. Least-squares filter

n
ÞþÁt
_
xðt
n
Þþ
ðÁtÞ
2
2!

xðt
n
Þ
þ
Át
3
ðÞ
3!
_xðt
n
ÞþÁÁÁ ð1:2-13Þ
For
ðÁtÞ
2
2!

xðt
n
Þ
Figure 1.2-4 The g–h filter predicts position of constant-velocity target perfectly in

b
Ã
 b
Ã
nþ1;n
¼À
xT
2
h
ð1:2-15Þ
Figure 1.2-5 illustrates the constant lag error prediction resulting when the
tracking equations of (1.2-11) are used for a constant accelerating target.
Figure 1.2-5 Constant lag error b
Ã
that results in steady state when tracking a target
having a constant acceleration with a constant g–h filter.
g–h FILTERS
25
Equation (1.2-15) is not surprising. The acceleration error is proportional to
1
2
(

xT
2
) and we see that correspondingly b
Ã
is proportional to
1
2

Alternatively, the smaller is h, the more sluggish is the filter. Thus quite
reasonably the lag error for the filter is inversely proportional to h.
When tracking a constant-accelerating target with a g–h filter, there will also
be in steady state constant lag errors for the filtered target position x
Ã
n;n
and the
velocity
_
x
Ã
n;n
given respectively by [12]
b
Ã
n;n
¼À

xT
2
1 À g
h
ð1:2-16aÞ
_
b
Ã
n;n
¼À

xT

truncation error, since the error results from the truncation of the acceleration
term in the Taylor expansion given by (1.2-13).
1.2.4.4 Tracking Errors due to Random Range Measurement Error
The radar range measurement y
n
can be expressed as
y
n
¼ x
n
þ 
n
ð1:2-17Þ
where x
n
without the asterisk is the true target position and 
n
is the range
measurement error for the nth observation. Assume that 
n
is a random zero
mean variable with an rms of 

that is the same for all n. Because 

represents the rms of the range x measurement error, we shall replace it by 
x
from here on. The variance of the prediction x
Ã
nþ1;n

2
x
. This
normalized variance is called the variance reduction factor (VRF). Using
the tracking prediction equations of (1.2-11), in steady state the VRF for
x
Ã
nþ1;n
for a constant g–h filter is given by [12; see also problems 1.2.4.4-1 and
1.2.6-2]
VRFðx
Ã
nþ1;n
Þ¼
VARðx
Ã
nþ1;n
Þ

2
x
¼
2g
2
þ 2h þ gh
gð4 À 2g À hÞ
ð1:2-19Þ
The corresponding VRFs for x
Ã
n;n

_
x
Ã
nþ1;n
Þ

2
x
¼
1
T
2
2h
2
gð4 À 2g À hÞ
ð1:2-21Þ
Thus the steady-state normalized prediction error is given simply in terms of g
and h. Other names for the VRF are given in Table 1.2-3.
1.2.4.5 Balancing of Lag Error and rms Prediction Error
Equation (1.2-19) allows us to specify the filter prediction error VARðx
nþ1;n
Þ in
terms of g and h. The non random lag prediction error b
Ã
is given in turn by
g–h FILTERS
27