I.
Detecti on -Trac ki n g
Performance with Combined
Waveforms
CONSTANTINO RAG0
Scientific Systems
PETER WILLETT
YAAKOV BAR-SHALOM, Fellow, IEEE
University of Connecticut
It is commonly understood that in radar or active sonar
detection systems constant-frequency (CF) pulscs correspond
to good Doppler but poor delay resolution capability; and that
linearly swept frequency (FM) pulses have the opposite behavior.
Many systems are capable of both types of operation, and
hence in this paper the fusion of such pulses is examined from
a system point of view (i.e., tracking performance) via hybrid
conditional averaging (HYCA), a new but increasingly accepted
technique for evaluating tracking performance in the presence
of missed detections. It is shown that tracking errors are highly
dependent on the waveform used, and in many situations tracking
performance using a good heterogeneous waveform is improved
by an order of magnitude when compared with a scheme using a
homogeneous pulse (CF or FM) with the same energy. Between
the two types of FM considered (upsweep and downsweep), it is
shown that the upsweep is always superior. Also investigated is
INTRODUCTION
Many active detection systems (radar and sonar)
are capable of waveform agility-particularly of
constant frequency (CF) and linearly swept frequency
(FM, or chirp)-in their transmitted pulses. It appears
that such systems presently use their abilities, but
do not fuse the results in an optimal way. The
purpose of this work is to examine the possible
benefits from such fusion. In a recent paper [l] a
similar problem was analyzed, but there the system
assumed perfect detection and the estimation of
the range and range-rate reach the corresponding
CRLB (Cramer-Rao lower bound). The importance
of waveform design from a tracking point of view is
discussed in [2].
We derive the relationship between the pulse shape
and the detection performance, and based on that
we compute the corresponding measurement noise
covariance. To be specific, we assume the following.
1) A Swerling I target and additive white Gaussian
noise. A target made up of many reflectors simplifies
the analysis, as will be seen.
2) A combined FM+-CF, FM--CF or FM+-FMc
transmitted pulse with the two waveforms abutted.
FM+stands for FM with a positive sweep rate
(upsweep), FM-for FM with a negative sweep rate
(downsweep). CF indicates a constant-frequency pulse
or subpulse. We do not consider phase-coded or other
broadband pulses.
3) A constant target RCS (radar cross section) for
the CF and F M subpulses. In the case of our “abutted”
pulses this makes sense, as one would expect little
change in the observed target scattering characteristics
for the duration of a pulse.
4) Known ambient noise power.
5 ) A single target, and no false alarms.
an alternating-pulse system, which, while suboptimal, appears to
The apportionment of energy between CF and FM or
offer robust performance.
FM+ and FM- subpulses is a parameter, and as such
Manuscript received December 16, 1996; revised April 1 , 1997
IEEE Log No. T-AESI34/2/03204.
This research was supported through Rome Laboratory under
AFOSR Contracts F30602-94-C-0060 and F49620-95-1-0229, and
by the Office of Naval Research Contracts N66604-92-C-1386 and
N00014-91-J-l950.
Authors’ addresses: C. Rago, Scientific Systems, 400 W. Cummings
Park, Ste. 3950, Woburn, MA 01801; P. Willett and Y. Bar-Shalom,
University of Connecticut, U-157, Storrs, CT 06269-2157.
0018-9251/98/$10.00 @ 1998 IEEE
612
we include full CF and full FM as special cases.
It is well known [3, 41 that a CF pulse has
good range-rate (i.e., Doppler frequency) resolution
characteristics, but is relatively poor at range
resolution; conversely, that an FM pulse can
have excellent range, but comparatively poor
range-rate resolution. The idea of this research is to
investigate whether a combined system (using some
apportionment between both kinds of pulses) can
perform better than either extreme.
Our goal, therefore, is to compare the
performances, with respect to tracking error, of a
number of schemes:
1) A pure CF waveform, of duration Tp.
2) A pure linear-FM waveform, of duration Tp;
sweep rate (denoted by 6 ) is a parameter. As will be
shown, the sign of the sweep rate has a significant
influence on the tracking performance.
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 34, NO. 2 APRIL 1998
~
3) Combined CF-FM- waveform, with the first
(1 - y>?:, seconds being of constant frequency, and the
remaining yT, of linearly-negatively-swept frequency.
As above, the sweep rate is a parameter.
4) A combined FM+-FM- waveform, with the
first yT, seconds being of linearly-positively-swept
frequency, and the remaining (1 - y)T, of linearlynegatively-swept frequency. As above, sweep rate is a
parameter.
5 ) A combined FM+-CF waveform, with the
first yT, seconds baing of linearly-positively-swept
frequency, and the remaining (1 - y)T, of constant
frequency. As above, the sweep rate is a parameter.
6) FM and CF waveforms, each of duration T,,
alternating on successive scans.
To be fair, the energy per pulse is independent of
the pulse shape employed. Comparison is on the
basis of tracking performance as specified by the
algebraic Riccati equation (ARE) modified by the
hybrid conditional iiveruging (HYCA) technique to
account for missed detections [5]-the technique is
analytical but approximate, and extensive justification
of its use is given in [6].
A major conclusion of this study is that, from
a system level (i.e.. tracking) point of view, FM'
(upsweep) is alwaq s superior to FM- (downsweep).
Overall, a 50%-50% division of energy between FM'
and FM- seems the best choice.
In Section I1 we give mathematical preliminaries
and signal models; naturally, we discuss ambiguity
functions and probability of detection. Section I11
details our analysis procedure, specifically relating to
our assumptions about measurement accuracy and the
use of the HYCA approach. Finally, Section IV gives
our results and recommendations.
II.
BACKGROUND
A.
Receiver Model
Receiver processing is assumed, without loss
of generality, to be: performed at "baseband"; that
is, subsequent to carrier-removal via mixing. If the
transmitted baseband signal is s(t), the matched filter
is the one with an impulse response h(t) = s*(-t),
i.e., the impulse response of the matched filter is
the conjugate time-reversed baseband signal. With
ri(t) the received waveform, and r(t) the baseband
representation of ri(t), we have, as an output process
of our matched fili;er,
Envelope
L]
Matched Filter
Fig. 1. Complex envelope matched filter operation.
where f, is the carrier frequency, and s(t) is a replica
of the transmitted baseband signal. This standard
scheme is pictured in Fig. 1.
If the return signal is expected to be Doppler
shifted (as in the radar case), then the matched filter
to a return signal with an expected frequency shift fa
has an impulse response h(t) = s*(-t)exp(j2..rrfot), and
its output is given by
The (baseband) received signal will be modeled as
a return from a Swerling I target [3]:
= As(t - -r)ej2"fd'Z(target) + v(t)
(3)
where s(t,-r,fd) = s(t - .)ejzafdt is a delayed ( 7 ) and
Doppler-shifted (fd) replica of the emitted baseband
complex envelope signal s(t); I(target) is a target
indicator. The complex numbers Ai represent the
amplitude and phase of each of the target reflectors;
for a Swerling I model the assumption is that there
are many such reflectors, each one with a random
amplitude and phase. Hence A = CiAiapproaches a
complex Gaussian random variable with zero mean
and variance 2a;. We assume v(t) is complex white
Gaussian noise independent of A, with zero mean
and variance 2N,. The signal s ( t ) can be any of those
described in the previous section.
B. Ambiguity Function
Of major interest in the sequel is the "ambiguity
function" [7], given by
(4)
As we see next, the ambiguity function specifies the
output of the matched filter in the absence of noise;
its shape is related to the implied resolution cell
(to be defined in Subsection IIIC), and is strongly
dependent on the waveform. The relationship between
the ambiguity function and resolution cell (which
does not appear to be universally well-understood)
is discussed in the Appendix. Briefly, the relationship
is that the time-reversed ambiguity function defines a
RAG0 ET AL.: DETECTION-TRACKING PERFORMANCE WITH COMBINED WAVEFORMS
613
~
resolution cell "primitive", which, when enclosed as
tightly as possible by some tesselating shape such as a
parallelogram, defines the resolution cell itself.
Examples showing the effect of different energy
apportionments between CF and FM pulses are given
in Figs. 5 and 8.
C.
This random variable is still zero mean, with variance
given by
x [As(a!- T)ej2"f,ln+ v(a!)]*}dXda!
Detection
Here we discuss the detection of a delayed and
Doppler-shifted return. The implied test will use the
output of the envelope (magnitude square of the real
and imaginary parts according to Fig. 1) matched
filter. At time t the magnitude square of the output
of a filter matched to a zero delay and a zero Doppler
shift is
2
lx(t)I2 = I L r ( X ) s * ( x - t)dAl .
(5)
Since the delay and Doppler shift are unknown, we
,
examine without loss of generality this at time T ~and
hence the test is given by
2 K
d hH l7
I x ( T ~ ) ~ ~ = ~ ~ T o r ( X ) s *-( X~ ~ ) 2
=
LT0
v(X)s*(X - To)dX.
-
(6)
with 7 a threshold. We characterize now the random
~ ) the noise-only hypothesis ( H ) and
variable ~ ( 7under
under the target-present hypothesis ( K ) .
Noise-Only Hypothesis: Here we have, according
to our formalism, T(target) = 0, and hence
X(To)
where we have used the definition of the ambiguity
function A given by (4)(with the normalization factor
to take into account the nonunity energy signals).
Recall that the magnitude square of a complex
Gaussian random variable x N(0,a?) is
exponentially distributed, with density given by
(7)
We consequently have that the probability of false
alarm (P,) is given by
and the probability of detection (Pd) by
The random variable X ( T ~ )is complex Gaussian, with
zero mean and variance given by
(13)
using (12) we get
= 2N0[.
The last integral E is the energy of the transmitted
pulse.
Target Present: In this case we have I(target) = 1,
and the matched filter output is thus
This relationship has been plotted for a particular
probability of false alarm, as a function of 7- and f d
and for different FM+-CF apportionments in Fig. 5
and for FM+-FM- in Fig. 8. Observe the pronounced
variation in the shape of the "resolution cells" as a
function of the apportionment.
Ill. COMPARISON OF SCHEMES
(9)
614
In this Section we give details relating to our mode
of comparison of our various pulse-fusion schemes.
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 34, NO. 2 APRIL 1998
delay and Doppler shift:
Of major interest are: the signal model, the target
model, the measurement model, and the analytical
framework. We devote a subsection to each.
A.
y(k) = Hx(k) + ~ ( k )
where the measurement covariance E { w ( k ) ( ~ ( k ) >=~ }
R is a function of the waveform shape employed. We
write
Signal Model
In our studies we use a baseband signal s(t).
This signal consists, in general, of two different
subpulses, concatenated in such a way that the phase
is continuous and the total signal duration is Tp. The
extreme cases (Le., when a duration of one of the
subpulses is zero) are also included.
In the first set of signals, called CF-FM-, each s(t)
comprises (1 - y)Tp seconds of CF (which at baseband
has constant frequency) followed by yTp seconds of
linearly-negatively-swept FM. That is, we have
0 1.t < (1 - y)Tp
(1 - y)Tp 5 t < Tp *
0
else
(15)
Results using other configurations, such as FMfollowed by CF, exhibit different ambiguity functions;
however, results are similar.
The second set of signals, called FM+-FM-, is
described by
The last set of signals, called FM+-CF is specified
by
lo
(18)
else
B. Target Model
Our target model is kinematic with
rangehange-rate as state variables. That is, we have
x(X:+ 1) = Fx(k) + ~ ( k )
where the process noise is white and zero-mean with
covariance E { v ( k ) v T ( k ) } = Q ( k ) . Our observations are
F=
[:4‘1
r-At3
At21
-
where At represents the time between samples, co is
the wave’s propagation speed in the medium, f , is
the carrier frequency, and
is the power spectral
density of the continuous-time white process noise
discretized in (18) [8, p. 2621. Note that for clarity we
have chosen to work in one-dimension, range, since
it is in this domain that the differences between the
various waveform types become apparent. The rms
target acceleration corresponding to uq is a,/&
[8, p. 2631.
cri
C. Measurement Model
In the most generous world it would be possible
to determine an optimal (presumably maximum
likelihood) estimate of delay T and shift f, for each
detection. However, due to the complexity of such
an operation most systems resort to a sampled or
quantized approach, which immediately gives rise
to the concept of a resolution ceZZ. For us, an ideal
resolution cell (to be designated as a resolution cell
primitive) is defined as the region in rangehange-rate
space where each point, conditioned on the target
present hypothesis, has a probability of detection Pd or
greater. Mathematically, this resolution cell is defined
by the test given in (6) and (13). If the output of the
magnitude square matched filter exceeds the threshold
7 ,then the parameters ( 7 , f d ) of the return signal
are inside the resolution cell defined by the level P,.
If a detection is declared, then the corresponding
measurement as supplied to the tracking algorithm
is the centroid of that resolution cell (i.e., the matched
filter parameters T ~fob>,
, with measurement covariance
R derivable from the size and shape of the cell,
assuming a uniform distribution of the measurements
in the cell. Ideally, therefore, resolution cells should
be sufficiently regular in geometry that they are
mutually exclusive and exhaustive of space (Le.,
they “tessellate”); and that a target located in a
given cell will not effect a detection in any other.
Unfortunately this is overly idealized even in the
pure CF or FM cases, in which constant-P, contours
RAG0 ET AL.: DETECTION-TRACKING PERFORMANCE WITH COMBINED WAVEFORMS
615
1,51
1
I
I
-1.5‘
-0.08
-0.06
-0.04
-0.02
0
0.02
0 04
0.06
I
0.08
Fig. 2. Resolution cell (shaded area) and corresponding
parallelogram that contains it.
tend to be elliptical in shape, and it is certainly not
true when a combined waveform is used. To handle
this problem, we define as a practical resolution cell
the parallelogram that contains the resolution cell
primitive (see Fig. 2). The average probability of
detection is, therefore, the conditional expected value
given by
Naturally, the area inside the resolution cell but
outside the resolution cell primitive contributes to the
miss probability.
4) For such a resolution cell, define the
measurement covariance by assuming the location
of the detection-producing target to be uniformly
distributed within the resolution cell primitive, that
is, the shaded region in Fig. 2.
Our resolution cell primitives do not tessellate;
however, the parallelograms that contain each (the
practical resolution cells) do. Our model is that at
each time sample a “coin” is flipped: with probability
(1 - pd) we have a miss, and there is no measurement.
And with probability Pd a measurement is generated
distributed uniformly within the resolution cell
primitive.
D. Analytical Approach
The state estimation-error covariance matrix is
updated by the Kalman filter as follows ([8]):
P(k + 1 1 k ) = FP(k 1 k)F’ + Q
S(k + 1) = HP(k + 1 I k)H’ + R(k)
W(k + 1) = P(k + 1 I k)H’S(k + l)-’
(22)
P(k + 1 I k + 1) = P(k + 1 I k) - W(k + 1)
where f ( 7 , f d ) is the probability density function of
) the
the target being at the coordinates ( ~ , f d and
integration is denoted by as being done over the
shaded area. Using the previous uniform distribution
assumption, this density is equal to the inverse of the
area of the parallelogram (Ap):
-
Our analysis would match reality most closely if
we were to assign, at each time, a uniform distribution
within each resolution cell to our target, and thence
to establish a corresponding measurement error (the
difference between the uniformly distributed variate
and the resolution-cell center) and probability of
detection (via (14) and the ambiguity function).
However, in order to fit with our HYCA approach
we must quantize Pd, and in the current analysis we
quantize to two levels. Our steps are thus as follows.
1 ) Define a resolution cell primitive from the
corresponding ambiguity function (see the Appendix)
by selecting a “detection threshold” within which a
prespecified probability of detection is consistently
exceeded.
2) Form the corresponding (practical) resolution
cell as the parallelogram which most snugly
circumscribes the resolution cell primitive,
3) For such a resolution cell, define the probability
of detection to be its conditional average value over
the parallelogram that contains the cell (see (21)).
616
x S(k
+ l)W’(k + 1).
With R(k) = R this evolves to a steady state
which characterizes the performance of the filter, and
this solution can be computed from the associated
ARE. However, in the case that detections can be
missed, the measurement noise has a stochastic
covariance (R(k)), and there is no steady-state, but a
stationary matrix-valued random process. Therefore,
it is legitimate to seek the expected value of the error
covariance.
Let us assume that the measurement noise
covariance matrix can be modeled as
detection
R(k) =
where, as described in the previous subsection, R is
derived from the size and shape of the resolution cell
and the probabilities of detection and miss are the
corresponding averaged values.
The method here employed is the HYCA method
of [SI. (For justification of the approach, and
comparison of HYCA results to simulation, the reader
is directed to [6].) Under this scheme we embed
a binary random process { b ( k ) } into the Riccati
equation such that
b(k) =
P { b ( k ) = i} =
1
detection
2
miss
i=l
(24)
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 34, NO. 2 APRIL 1998
Tx
Pl(k
targets), the other applicable to the radar situation
(high frequency, short pulses, fast-moving targets).
Both cases use a Swerling I target and additive white
Gaussian noise. We expect that results using other
models would give rise to similar conclusions.
+ l/k + I)
1 - I’d
pl(klk)
P2(k/L)
P?(k+l/b+l)
L
1 - P,i
Fig. 3. Illustration of procedure for finding steady-state
covariance for Kalman filter operating in stochastic environment
Y i a HYCA procedure.
Then, with Pij(k + 1 1 k + 1) defined as the estimation
error covariance given b(k) = i and b(k + 1) = j
(obtainable from (22) with R(k) according to (23)),
we get
L
Pj(k + 1 I k
+. 1) = x P i j ( k I k)P{b(k)
= i}.
i= 1
(25)
The state probabilities at time k , pi(k) = P { b ( k ) = i } ,
i = 1,2 are derived from the state probabilities at time
k - 1 and the Mark.ov transition matrix:
The matrices P,(k 1 k ) reflect the state error covariance
given that s(k) = i. The mean state covariance matrix
(i.e., what we want) is given by
2
~ ( I kk) = C P j ( k I k ) ~ j ( k ) .
(27)
.j=1
This can be evaluated when this matrix Markov
process reaches steady state. Fig. 3 shows a
flow-diagram that reflects the mechanics of the
iterations.
IV.
RESULTS
We have developed two sets of representative
plots: one with parameters suitable for sonar (Le.,
low carrier frequency, long pulses, slow-moving
A.
Sonar Case
For the (active) sonar case we assume a pulse
length of Tp = 1 s, a scan-to-scan time of At = 30 s,
and a carrier frequency of f , = 3.5 kHz. In each case
here we have SNR = 20 dB and K, = 50 Hz/s or )i =
-50 Hz; this corresponds to a maximum frequency
sweep of 50 Hz, found in case y = 1.
The first waveform analyzed was the combined
CF-FM- (a CF subpulse followed by downsweep (see
Fig. 4(a)). For various values of y we show in Fig. 5
representative constant-Pd contours for the CF-FMwaveform. As discussed earlier, the “resolution cell
primitive” is determined by the volume enclosed by
one of these contours, and for the CF and FM cases
these are as expected: in the former we have good
Doppler and comparatively poor range resolutions
(Fig. 5(a)); and in the latter the reverse (Fig. 5(c)).
Intermediate values of y give more complicated
Pd contours. Here the resolution cell primitive is
determined by the Pd = 0.80 contour. In Fig. 6 we
show the corresponding measurement errors; these are
the diagonal elements of the measurement covariance
matrix R (corresponding to the delay variance and the
Doppler-shift variance) and the correlation coefficient
between delay and Doppler-shift measurements.
(Note that a negative
correlation between delay and
Doppler-shift corresponds to a positive correlation
between range and range-rate measurements. Negative
measurement correlation is usually more favorable for
tracking .)
Fig. 6 deals with measurement covariances. As
discussed earlier, this corresponds to the covariance of
a uniformly distributed target within the resolution cell
primitive, for example the shaded region of Fig. 2.
It is worth noting here that an earlier contention,
that FM provides good delay information, does not
seem at first to be borne out by this figure. The
2-
-12
3
0 0 1 0 2 0 3 0 4 0 5 06 0 7 08 09 i
time
time
time
(a)
(b)
(C)
Fig. 4. Frequency variation for waveforms used in analyses. (a) CF-FM-. (b) FM+-FM-. (c) FM+-CF.
RAG0 ET AL.: DETECTION-TRACKING PERFORMANCE WITH COMBINED WAVEFORMS
617
0
-0.5
--
-0.5
:
-0.5
0
0
:
o
,
o
0.02
-1
0.5 -0.5
0.5 -0.2
0
I
0,061
1
10
0
20
30
40
50
60
70
80
90
100
0.2
0
(C)
I
10‘
I
1
0
lo-10
-1
-0.2
0
:Kl‘:\I
0
(e)
0.2 -10
-0.4 -0.2 0
(f)
20
30
40
50
60
70
80
90
100
(b)
0.2
0
(d)
-2
-0.2
10
0.2
20
-0 5
0
-1
-20
-0.5
1
1
10
10
20
30
40
50
60
70
BO
90
100
(4
0
FM- pulse
0.5
(9)
Fig. 5. Resolution cells (probability of detection contours) and
parallelograms that contain them in delaylDoppler space for
CF-FM- waveforms, ’P = 0.001 and SNR = 20 dB, frequency
sweep factor IC. = 50 H d s and probability of detection threshold of
0.80. (a) y = 0% (CF). (b) y = 20%. (c) y = 40%. (d) y = 50%.
(e) y = 60%. (f) y = 80%. (g) y = 100% (FM-). Horizontal and
vertical axes are in seconds and Hertz, respectively, a 1 s pulse
with 50 H d s is assumed.
reason is that in Fig. 6(a) we are plotting the delay
measurement error alone, and with reference to Fig. 5
this is as large as for the pure-CF case. What is not
plotted in Fig. 6(a) (but is available in Fig. 6(c)) is
that delay and Doppler measurements are strongly
correlated, meaning that if something is known about
the range-rate (for example, that it is small) then delay
can be inferred quite accurately.
In Fig. 7 we show steady-state tracking errors
for the CF-FiW waveform obtained via our HYCA
procedure, for various values of the target motion
uncertainty (quantified by the process noise power
spectral density a t ) . Note the difference between
Figs. 6 and 7: the former deals with measurements as
supplied to the tracker, and the latter to the posterior
tracking errors using these measurements.
There are a number of items of note as follows.
duration ( 7 . in
X of the total pulse duration)
Fig. 6. Measurement error variances for CF-FM- waveform as
function of y;same parameters as Fig. 5. (a) Delay variance.
(b) Doppler-shift variance. (c) Correlation coefficient between
delay and Doppler shift.
1 0’
0
10
20
30
40
50
80
70
80
90
100
(4
I=--+
[(,,se,)’]
lo-‘
‘no
10-2----=,---==
zII=:::::____-_-__-:
_..-_____---
_ _ _ -
1) Comparing Figs. 6 and 7 it is clear that
system-level (i.e,, tracking) performance cannot
be inferred directly from measurement-level (Le.,
resolution-cell shape) performance.
2) For low motion uncertainty, corresponding to
nearly constant velocity target motions (process noise
uq = lop2 ds’.’), FM-outperforms CF. However,
for more-maneuvering targets (oq= 0.3 m / ~ ‘ . ~ )
the opposite is true. This underscores the need
for system-level comparison. In the former case,
the tracking and measurement uncertainties are
complementary; with respect to Fig. 5(g) their overlap
618
(essentially a “stripe” across this resolution cell) is
small, and the performance is good. In the latter case
the tracking uncertainty and filter gains are large;
hence the controlled joint rangelrange-rate uncertainty
of Fig. 5(a) is preferable to the large (albeit strongly
correlated) joint uncertainties of Fig. 5(g).
3 ) In all cases shown here a value y = 50% (half
CF, half €34-) is close to optimal. The improvement
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 34, NO. 2 APRIL 1998
1
'Fi:Fi
20
0
-10
-20
-0.5
[set?]
0
0.5
o':mi
-0.2
-0.5
-0.1
0
0.2
-5
-0.1
0
0.1
IW
vi2'
0
'
10
'
20
'
30
'
50
'
'
I
60
70
80
90
100
60
70
80
90
100
(b)
0.1
0
'
40
/
OI
-1
0
10
20
30
40
50
(C)
-*
-0.1
0
(e)
0.1
-0.2
0
(f)
0.2
-0.5
0
0.5
(9)
Fig. 8. Resolution cells (probability of detection contours) and
parallelograms that contain them in delay/Doppler space for
FM+-FM- waveforms,'P = 0.001 and SNR = 20 dB, frequency
sweep factor IF. = 50 Hds and K, = -50 Hds, respectively, and
probability of detection threshold of 0.80. (a) y = 0% (FM-).
(b) y = 20%. (c) y = 40%. (d) y = 50%. (e) y = 60%. (0 y = 80%.
(8) y = 100% (FM'). Horizontal and vertical axes are in seconds
and Hertz, respectively, a 1 s pulse with 50 Hz/s is assumed.
FM' pulse duration (7. in o/r of the total pulse duration)
Fig. 9. Measurement error variances for FM+-FM- pulse as
function of y,same parameters as Fig. 8. (a) Delay variance.
(b) Doppler-shift variance. (c) Correlation coefficient between
delay and Doppler shift.
in range estimation, as compared with FM-only or
CF-only can be as high as an order of magnitude.
4) The alternating-pulse scheme, pictured as the
horizontal lines in Fig. 7, has excellent performance
for nearly constant velocity targets.
10
For the FM+-FW waveform, the pure FM' and the
50%-50% apportionment are the most appealing.
RAG0 ET AL.: DETECTION-TRACKING PERFORMANCE WITH
30
40
50
60
70
80
90
100
50
60
70
80
90
100
(a)
[(m/sec)']
The lack of smocithness in these figures is attributable
to the sometimes weird resolution-cell shapes
encountered.
The second waveform analyzed was FM+-FM(upsweep followed by downsweep, see Fig. 4(b)). For
various values of y we show in Fig. 8 representative
constant-P, contclurs for this case. It should be
noted that even when the measurement errors in
range and range rate are symmetric around the
50%-50% pulse (see Fig. 9), the steady-state tracking
errors are not (see Fig. 10). This is because as we
go from a pure EM- waveform to a pure FM+,
the resolution cell rotates, going from a positive
correlation between range and range-rate errors to a
negative correlation, as illustrated in Fig. 8, case (a)
and (c). At the tracking level, negatively correlated
measurements errors tend to compensate (as opposed
to the positively correlated ones), leading to smaller
estimation errors than the ones corresponding to the
positive correlated case (see [8, p. 1191). We can
summarize this as follows.
20
10"
l0-'O
10
20
30
40
(b)
FM'
pulse duration (y. in % of the total pulse duration)
Fig. 10. Steady-state estimation errors as function of y,same
parameters as Fig. 8 (FM+-FM-), carrier frequency f, = 3.5 kHz.
(a) Range variance. (b) Range-rate variance. Dashed lines:
uz = 0.1; solid lines: u2 = 0.01; dash-dot lines: u,' = 0,0001. For
each case horizontal lines refer to case of alternating FM+/FMpulses.
In view of the last conclusion, we revisit the
CF-FM case, but now with a positively swept FM
pulse (FM+-CF (see Fig. 4(c)). The results in this
case are shown in Fig. 11 and 12. As we can note
now, only for high uncertainty motion targets the 50%
combined FM+-CF waveform is optimal. For the other
cases, the pure positively swept FM pulse performs as
good as or better than the combined one, however, not
by much.
Our findings are summarized in Table I. The
target acceleration corresponding to highest value of
COMBINED WAVEFORMS
619
I
0.06,
0
10
20
40
30
50
60
70
80
90
100
TABLE I
Estimation Range Error Variances in m2, for Various Schemes in
Sonar Case
I
9 x lo3
6 x lo4
1 x 104
pure CF
pure FM-
I
I0
'
FM+
D -I I~
r
-P
-
it 50% CF. 50% FM-
H 50% CF. 50% FM+
10
i0
30
20
11
/I
I
I
6 x 10'
3 x 10'
2 x io1
3 x io*
3 x 101
4 x lo1
1
ti
1
1oo
to-zO
I1
2 x lo3
2 x lo4
1 x io3 1
!
1 104 I 1 103 I
1 104
1 x 103
6 x lo3
7 x 10'
2x10~ 4x10~
1 x io3
1 x.104
4 x 103
2 104
60
50
70
80
90
100
-
50% FM+. 50% FMalt CF. FMalt CF. FM+
alt FM+. FM-
L
1x10~
3 x 10'
7 x 10'
Note: In each column, the best is in boldface.
FM'
pulse duration (7.in
% of the total pulse duration)
Fig. 11. Measurement error variances for FM+-CF waveform as
function of y,same parameters as Fig. 5 but with positive FM
swept. (a) Delay variance. (b) Doppler-shift variance.
(c) Correlation coefficient between delay and Doppler shift.
1
0
10
20
30
40
50
60
70
80
90
100
10'1
0
10
'
'
20
30
40
50
60
70
60
90
100
1
(b)
FMt pulse duration ( 7 . in 9% of the total pulse duration)
10-31
0
10
3
20
i
30
40
50
60
70
80
90
Fig. 13. Steady-state estimation errors as function of y
(FM+-FM-) for radar case described in Section IVB, same
parameters as Fig. 8), carrier frequency f, = 40 GHz, pulse length
T = 100 ,us. (a) Range variance. (b) Range-rate variance. Dashed
lines: o2 = 10; solid lines: 'e = 100; dash-dot lines: e: = 1000.
For zach case horizontal fines refer to case of alternating
FM*/FM- pulses.
100
(4
FM' pulse duration (7:in % of the total pulse duration)
Fig. 12. Steady-state estimation errors as function of y
(FM+-CF), carrier frequency ,& = 3.5 kHz. (a) Range variance.
(b) Range-rate variance. Dashed lines: e: = 0.1; solid lines:
e,"= 0.01; dash-dot lines: e2 = 0,0001. For each case horizontal
lines refer to case of alternating CF/FM+ pulses.
oq considered is 0.06 d s 2 , typical of the turn of a
submarine.
8.
Radar Case
In this case we have assumed a relatively long
radar pulse, Tp = 100 p s and a carrier frequency
off, = 40 GHz. This pulse combined with the
high carrier frequency gives a frequency resolution
well below the expected Doppler shift. Given the
620
previous results for the sonar case, we investigate
the FM+-FM- and FM+-CF cases only. The FM
modulation used corresponds in this case to IC. =
+500 kHz/s. The scan-to-scan time is At = 1 s; the
SNR is 20 dB.
The first waveform we analyzed was the
FM'-FM-. The steady-state tracking errors for three
different process noise levels are shown in Fig. 13,
and as we can see, the pure FM' pulse performs
the best. Next we analyzed the FM+-CF waveform,
for the same radar conditions. As in the previous
example, the pure FM' waveform performs the best
in terms of steady-state estimation errors, but in this
case the 50%-50% pulse performs nearly optimally
(see Fig. 14).
These results are strongly dependent on the pulse
duration (or the range and range-rate resolution). An
interesting point to note here is that as we increase
the pulse length, the pure FM' is no longer optimal;
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 34, NO. 2 APRIL 1998
"[I
1'
0
1
I (m/sec) *]
1
0
0
,
10
k
20
30
50
40
60
70
I
1W
90
80
(b)
FMt pulse duration (7,in % of the total pulse duration)
Fig. 14. Steady-state estimation errors as function of y for radar
case described in Section IVB (FM+-CF). (a) Range variance.
(b) Range-rate variance. Dashed lines: g
: = 1000; solid lines:
g,' = 100; dash-dot lines: 0: = 10. For each case horizontal lines
refer to case of alternating CFIFM+ pulses.
,*
,
.
.
30
do
TABLE I1
Estimation Range Error Variances in m2, for Various Process
Noise Intensities and Waveform Schemes, in Radar Case (Pulse
Length 100 p s )
-----
f
'Q1o
Fig. 16. Steady-state estimation errors as function of y for same
case as Fig. 14 (FM+-CF) but with pulse length of Tp = 200 p s .
(a) Range variance. (b) Range-rate variance. Dashed lines:
0," = 1000; solid lines: 0,'= 100; dash-dot lines: of = 10. For each
case horizontal lines refer to case of alternating CFlFM' pulses.
A
10
50
$0
A
80
90
pure
i o
FM-
(a)
alt CF.
FM+
_.
H nlt FM+. FM10
20
30
SO
40
60
70
80
90
100
(b)
FM' pulse duration
(y
in
I
I
I
2 x 104 I 5 x 103
Nore: In each column, the best is in boldface.
,--_
"'0
~
I1
7% of the total pulse duratlon)
Fig. 15. Steady-state estimation errors as function of for same
case as Fig. 13 (FM+-FM-) but with pulse length of Tp = 200 p s .
(a) Range variance. (b) Range-rate variance. Dashed lines:
o2 = 1000; solid lines: ut = 100; dash-dot lines: u,' = 10. For each
case horizontal lines refer to case of alternating FM+/FM- pulses.
the 50%-50% waveform replaces it. Also worth
mentioning here is the fact that the alternating scheme
can perform slightly better than the 50%-50% pulse
for low motion uncertainty targets. The steady-state
tracking errors for a pulse length Tp = 200 ps (the
SNR and P
' are assumed the same as with the 100 ps
waveform, i s . , the RCS is smaller) are shown in
Figs. 15 (FM+-FM-) and 16 (FW-CF). Our findings
are summarized in Tables I1 and 111. The target
acceleration corresponding to the highest value of crq
is 32 d s 2 .
II
2.5 x 103
i
TABLE I11
Estimation Range Error Variances in m2, for Various Process
Noise Intensities and Waveform Schemes, in Radar Case (Pulse
Length 200 p s )
1) 0,"= 1000 I
0II
e.mire CF
-~
purc FMpure FM+
50% CF, 50% FM+
50%
_.FMt. 50% FMalt CF. FM+
alt FM+, FM-
11 8 x 10'
I1
4.5 x 106
G x lo4
2.5 x 104
1 x lo4
II 2 5 x 104
II
104
Note: In each column. the best is in
H
H
It
I
u,"= 100
8 x lo5
I
3 x 106
3.5 x lo4
7 x 103
4 x lo3
I
I 4.5 x 103
I 5.5 103
boldface.
1
o,' = 10
1
I 8 x lo5 flU
I
7 x lo5
8 x lo3
I
I
I
5 x 103
4 x lo3
2 . 5 x 103
2.5 103
nH
C. Discussion of Results
The previous two sections have dealt with three
scenarios: sonar, shorter pulse radar, and longer pulse
radar. We cannot hope to be exhaustive in our study;
but there is some variety here, and we can claim some
insight.
RAG0 ET AL.: DETECTION-TRACKING PERFORMANCE WITH COMBINED WAVEFORMS
62 1
1) Downsweep FM (m-),
in its pure form or in
combination with CF7is almost never a good idea.
Intuitively, the problem is that range and range-rate
measurement errors are positively correlated, and this
is undesirable given a kinematic target motion model.
2) Pure upsweep FM (EM’) performs well in
almost all cases studied. This is due to the negative
correlation of the range and range-rate measurement
uncertainties, which yields a partial cancellation
of the errors [8]. It is occasionally outperformed
(for example, in the longer pulse radar case) by
combination CF/FM+ and FM+/F’Iv- waveforms.
3) The “alternating” schemes, in which a pulse
of one type is followed by a pulse of another type,
appear to offer a very good suboptimal performance.
It is our judgment that FM+/FM- is the most robust
performer, followed closely by CF/FM+.
V.
SUMMARY
In this paper we have investigated the performance
of combined constant- and swept-frequency waveform
fusion systems. Our comparison is in terms of
steady-state tracking performance, via the ARE
as modified to account for missed detections.
Fundamental to our analysis is the resolution cell
primitive (defined for our purposes as the volume
of rangekange-rate space wherein the probability
of detection exceeds a given threshold) and its
implied measurement error. Our results indicate
that the overall detection-tracking performance is
strongly dependent on the waveform used, and that
the use of the optimal waveform can lead to dramatic
improvement in tracking error (as much as an order of
magnitude range-estimation error variance reduction).
A major conclusion of this study is that, from
a system level (i.e., tracking) point of view, EM+
(upsweep) is always superior to FM- (downsweep).
Overall, a 50%-50% division of energy between EM+
and FM- seems the best choice. However, a scheme
using alternating CF and FM+ waveforms, available
through only minor modifications to conventional
signal processing, works well also.
4
Y(‘
1
f0)
fd
I I2
X
(s(i
- T o ) e > 2 ~ f o [ ~ - T o ) )*
Fig. 17. Magnitude square matched filter operation.
We begin by repeating the definition (4)
(28)
of the normalized ambiguity function, and to
introduce the magnitude-square matched-filter
operation of Fig. 17. With reference to this latter,
the signal y(7,fd; 7-o,fo) is that which would appear as
output for a filter matched to a signal having delay T~
and Doppler shift f o , when the actual signal has delay
T and Doppler shift fd.
Our goal is to relate y to the ambiguity function,
and this turns out to be straightforward:
Y(T>>,f3,.;df
IL
11:”
+a3
As(t - 7 ) e l z T f d ( t - T ) s * ( t - ,o)e-J2nfO(L-TO)
=
As(X)s*(X
=
=4 7 0 -
7-3
fd
- fo)
+
-
&
),lza(k-f0)X,-12nf0(T~T0)
I’
dX
)2
(29)
in which the effect of noise has been ignored.
We next turn to the concept of the resolution cell.
It is assumed that ( 7 ,fd)-space must be (uniquely
and exhaustively) tesselated into regions, usually
parallelograms,’ whose target occupancies will
be indicated by a threshold-exceedance at a filter
matched to the centroid ( 7 ,fd) value.2 Without loss
of generality we focus attention on the resolution
cell matched to zero delay and Doppler shift. Ideally
,
is
this cell ought to match a region where y ( ~fd;O,O)
large; however, since we can write
APPENDIX A. MATCHED FILTERS, AMBIGUITY
FUNCTIONS, AND RESOLUTION CELLS
we have that this cell ought to be a parallelogram into
which can be inscribed a time-reflected ambiguity
function contour, where the level provides some
minimum Pd value of interest (for much of this work
we have chosen this as Pd = 0.80). Thus we take
this element A(-7,,fd)to be, for want of a more
widely accepted term, a “resolution-cell primitive,”
from which a tesselating resolution cell grid can be
determined.
As there appears to be some confusion both in
common reference and in the relevant literature, we
offer this brief primer on the relationship between the
above concepts.
‘We are here assuming that only range and range-rate are of
interest, an inherently one-dimensional case; more generally we
have parallelopipeds.
’This amounts to sampling in (r,,fd)-space.
ACKNOWLEDGMENTS
Thanks to E E. Daum for the suggestion to
investigate the FM+-FM- cases; and to W. Cao for
his comments on the manuscript.
622
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 34. NO. 2 APRIL 1998
As a penultimate note, we recall that while
range is directly proportional to delay T ,
range-rate is proportional to the negative Doppler
frequency shift -jd. Thus, due to symmetry of the
ambiguity function, the resolution cell primitive in
rangehange-rate space has the same shape (although
different scale) and orientation as does the ambiguity
function in delay/Doppler space.
Since it appears that d(-7,fd)rather than d ( T , f d )
is the quantity of interest, it is reasonable to question
the motivation for defining the ambiguity function in
the way it is defined. As our final note, we respond
to this. The answer is that many current systems do
not attempt to tesselate delay/Doppler space with
resolution cells, but use a single filter matched to
f d = 0 for continuous (or finely sampled) values of
7 . The quantity of interest is in this case
~ ( 7 , f d ; ~ o , O=) 4 7 0 - 7 , f d )
(31)
meaning that a target with Doppler frequency shift f d
produces a matched-filter trace whose shape is as of a
horizontal “slice” of the ambiguity function of (4) at
frequency fd. This idea and its effects on tracking are
well discussed in [2].
Daum, F. (1992)
A system approach to multiple target tracking.
In Y. Bar-Shalom (Ed.), Multitarget-Multisenor Tracking:
Applications and Advances, Vol. 11.
Artech House, 1992, 149-181.
[3] Levanon, N. (1988)
Radar Principles.
New York: Wiley, 1988.
[4] Skolnik, M. L. (1990)
Radar Handbook.
New York: McGraw-Hill, 1990.
[SI Li, X.-R., and Bar-Shalom, Y. (1995)
Stability evaluation and track life of the PDAF for
tracking in clutter.
IEEE Transactions on Automatic Control, 36, 5 (May
1991), 588-602.
[6] Li, X.-R., and Bar-Shalom, Y. (1995)
Performance prediction of hybrid state algorithms: The
scenario conditional averaging approach.
In C. T. Leondes (Ed.), International Series on Advances
in Control and Dynamic Systems, Vol. 72.
New York: Academic Press, 1995.
[7] Stein, S . (1981)
Algorithms for ambiguity function processing.
IEEE Transactions on Acoustic, Speech and Signal
Processing, ASSP-29, 3 (June 1981), 588-599.
[8] Bar-Shalom, Y., and Li, X.-R. (1993)
Estimation and Tracking: Principles, Techniques and
Software.
Boston: Artech House, 1993.
[2]
REFERENCES
[l]
Kershaw, D. J., and Evans, R. J. (1994)
Optimal waveform selection for tracking systems.
IEEE Transaction7 on Information Theory, 40, 5 (Sept.
1994), 1536-1550.
Constantino Rag0 received his Bachelor Degree from the University of La Plata,
Argentina, in 1986, and the M.Sc. and Ph.D. degrees from the University of
Connecticut, Storrs, in 1994 and 1995, respectively, all in electrical engineering.
From 1986 to 1991 he was affiliated with the “Laboratorio de Electrbnica
Industrial, Control e Instrumentacibn (LEICI)” at the Department of Electrical
Engineering, University of La Plata, working in the areas of digital signal
processing and spectral estimation. From 1991-1995 he was a graduate
researchheaching assistant in the Department of Electrical and Systems
Engineering, University of Connecticut. Since 1995 he has been with Scientific
Systems Company, Inc., Wobum, MA, as a research engineer. His research
interests include detection and estimation theory, failure detection and
identification, and data fusion and tracking.
Peter Willett received his B.A.Sc. in 1982 from the University of Toronto,
Toronto, Canada, and his Ph.D. in 1986 from Princeton University, Princeton, NJ.
Since 1986 he has been with the University of Connecticut, Storrs, where
he is an Associate Professor. His interests are generally in the areas of detection
theory and signal processing.
RAG0 ET AL.: DETECTION-TRACKING PERFORMANCE WITH COMBINED WAVEFORMS
623
Yaakov Bar-Shalom (S’63-M’66-SM’80-F’84)
was born on May 11, 1941.
He received the B.S. and M.S. degrees from the Technion, Israel Institute of
Technology, in 1963 and 1967 and the Ph.D. degree from Princeton University
in 1970, all in electrical engineering.
From 1970 to 1976 he was with Systems Control, Inc., Palo Alto, CA.
Currently he is Professor of Electrical and Systems Engineering and Director of
the ESP Lab (Estimation and Signal Processing) at the University of Connecticut.
His research interests are in estimation theory and stochastic adaptive control and
has published over 200 papers in these areas. In view of the causality principle
between the given name of a person (in this case, “(he) will track”, in the modern
version of the original language of the Bible) and the profession of this person,
his interests have focused on tracking. His other interests are stochastic control of
vertical airfoils and of pairs of inclined foot supports on crystals. He coauthored
the monograph Tracking and Data Association (Academic Press, 1988), the
graduate text Estimation and Tracking: Principles, Techniques and Sojhvare (Artech
House, 1993), the text Multitarget-Multisensor Tracking: Principles and Techniques
(YBS Publishing, 1995), and editcd the books Multitarget-Multisensor Tracking:
Applications and Advances (Artech House, Vol. I, 1990; Vol. 11, 1992). He has
been elected Fellow of IEEE for “contributions to the theory of stochastic systems
and of multitarget tracking”. He has been consulting to numerous companies, and
originated the series of Multitarget-Multisensor Tracking short courses offered via
UCLA Extension, at Government Laboratories, private companies and overseas.
He has also developed the commercially available interactive software packages
MULTIDATTMfor automatic track formation and tracking of maneuvering
or splitting targets in clutter, PASSDATm for data association from multiple
passive sensors, BEARDATTMfor target localization from bearing and frequency
measurements in clutter, IMDATTMfor image segmentation and target centroid
tracking and FUSEDATTMfor fusion of possibly heterogeneous multisensor data
for tracking.
During 1976 and 1977 he served as Associate Editor of the IEEE Transactions
on Automatic Control and from 1978 to 1981 as Associate Editor of Automatica.
He was Program Chairman of the 1982 American Control Conference, General
Chairman of the 1985 ACC, and Co-Chairman of the 1989 IEEE International
Conference on Control and Applications. During 1983-1987 he served as
Chairman of the Conference Activities Board of the IEEE Control Systems
Society and during 1987-1989 was a member of the Board of Governors of the
IEEE CSS. In 1987 he received the IEEE CSS Distinguished Member Award.
Since 1995 he is a Distinguished Lecturer of the IEEE AESS. He is co-recipient
of the M. Barry Carlton Award for the best paper in the IEEE Transactions on
Aerospace and Electronic Systems in 1995.
624
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 34, NO. 2 APRIL 1998