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MICROCOPY RESOLUTION TEST CHART
NATIONAL BUREAU Of STANDARDS -1963 •
■MOST Project
^DETECTABILITY OF .LINEAR JM PULSES TRANSMITTED THROUGH A RANDOM MEDIUM
S. M. /Garber
Presented at NATO-Marina Italiana Advanced Study Institute on Stochastic
Problems in Underwater Sound Propagation, 18-23 September 1967, Lerici, Italy
D D C
reenact
OCT 85 19TT
General Electric Company / UlksEUUI
Heavy Military Electronics Department v 7^, 07 A-
Syracuse, New York 13201
■dStrtout^^
Approved lor pub.Uc release
Distribution Unlimited
ABSTRACT
The objective of the work to be described is to determine some of the implica-
tions of medium distortion on. waveform and processor design for active echo-ranging
sonar. In particular, the questions of bandwidth and duration of linear FM signals
are considered, as well as various combinations of coherent and incoherent pro-
cessing.
Sonar echos are modeled as a multiplicity of discrete arrivals, the arrival
time (range), doppler shift and amplitude being chosen at random from specified
probability densities. Noise is modeled as white with gaussian amplitude statistics,
and reverberation as having gaussian amplitude statistics but the power spectrum
of the transmitted signal.
With these models, computer simulation is employed to examine the detectability
of echos produced by the transmission of linear FM pulses of various bandwidths and
durations. The processor consists of a correlator followed by various amounts and
shapes of incoherent integration.
Monte Carlo techniques are used to generate ROC curves, showing the probability
of an echo being detected on a given ping vs the probability of a false detection
in the absence of an echo, with input signal-to-noise ratio or signal-to-reverbera-
tion ratio as a parameter. _
It is shown that, wherythe background is predominantly reverberation, in spite
of severe time spreading of echos significant gains in detection are still attain-
able through the use of^xtreme bandwidths . There is an accompanying loss in de-
tection, of the wi^er^Dand signals compared to the narrower band signals, with a
noise backgrgyndrT^ This loss is less, however, than the gain achieved when rever-
beration- limited.
\A
TThe conclusion reached from these results is that best detection performance
is obtained by use of a pulse with as much bandwidth as system constraints will
permit .
It^s also shown that post detection integration, following coherent processing,
sometimes improves detectability, and sometimes degrades detectability, depending
on the "denseness" of the multipath compared to the time resolution of the transmitted
signal. If the signal energy is divided between a few widely spaced arrivals, such
as might result from an extended target, it is better not to integrate. On the
other hand, if the spread is more or less continuous, such as could occur by reflec-
tion from the surface or bottom, it is better to integrate over the spread.
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1
The primary purpose of the work reported in this paper is to evaluate, using
computer simulation techniques, the performance of linear frequency modulated active
sonar signals and processors in a randomly fluctuating medium. This problem has
1 2
been treated by Price and Green and by this author in a previous paper . In both
cases ,. range-doppler spreading was assumed uniform and constant. In this paper,
range and doppler spreading are treated as random variables in order to more closely
represent the conditions that signals actually encounter in the underwater acoustic
operating environment. In general, spreading is introduced by the target, the medium
and the echo ranging platform. These are combined in the single spreading model
employed in this study.
Echo energy splitting is treated as a stationary random process whose parameters
can be varied to study their effect on detectability of LFM echos. It will be shown
that, at least for the energy splitting models assumed, the LFM pulse holds up ex-
ceedingly well, and that, in most sonar applications, system constraints rather than
the propagating medium or target, will limit the choice of transmitted pulse band-
width.
In order to carry out this study, the signals, the noise, and the processor
were simulated on a computer and Monte Carlo methods used to obtain the processor
output statistical quantities of probability of false alarm (PFA) and probability
of detection (PD). Comparisons are then made on the basis of required input signal-
to-interference ratio (SIR) to produce a specified value of PD when a threshold is
set for a specified PFA. Figure 1 shows a block diagram of the processes that are
simulated.
The discussion to follow is divided into four parts. First - a discussion of
the energy splitting model; second - a brief mention of the interference model;
third - a description of the processor; and fourth - the results that have been ob-
tained and the conclusions they have led to.
1. Price and Green, Signal Processing in Radar Astronomy - Communication via Fluct-
uating Multipath Media", Tech. Report No. 231 2*, Lincoln Laboratory, MIT, 6 Oct. I960.
2. Garber, "High Resolution Sonar Signals in a Multipath Environment", Supplement to
IEEE Transactions on Aerospace and Electronic Systems, Vol. AES-2, No. 6, Nov^1966.
2
TARGET ECHO MODEL
The choice of signal spreading model is difficult because insufficient experi-
mental data exists on which to base a good model. The model that has been adopted
for this study is a highly flexible one which was originally developed by the U. S.
Navy Underwater Sound Laboratory. It is intended to account for the spreading that
could occur upon reflection from the surface or bottom and to account for extended
targets. There are certain non-random multipath modes, such as the surface triplet
associated with a target below the surface or any fixed echo structure of the tar-
get that are not accounted for in this model.
The model assumes that the received signal energy is divided among a multipli-
city of replicas of the transmitted signal, each replica or arrival having associated
with it a particular range delay doppler and amplitude a^. For a given echo
the total received energy is equal to the sum of the energies in all of the individ-
ual arrivals. Figure 2 illustrates the idea. It shows the range delay, doppler
shift and amplitude of each of six arrivals in one echo. It is assumed that the
time variations of the target and path are slow enough that range, doppler and amp-
litude do not change during a pulse length, but fast enough that their values from
one ping to the next are completely independent. These assumptions seem to be borne
out by experience.
It is further assumed that the variables within a ping are independent of each
other. Hence, the amplitude of a path does not depend on its position in range or
doppler, nor does the doppler shift depend on time delay, etc. Thus, t, <J> and a are
independent, stationary random variables. Range delay is assumed normally distri-
buted about a mean x with a standard deviation of ot seconds. Similarly, doppler
shift is assumed normally distributed about zero mean with a standard deviation of
Hz. Finally, energy is assumed to be log-normally distributed with a standard
deviation of a db.
a
REVERBERATION AND NOISE MODEL
Reverberation is the summation of echos from a multiplicity of scatterers.
Therefore, the same model with different parameters could be used to represent rever-
beration as is used for target echos. However, in this study we have been primarily
concerned with the effects of target echo distortion, and hence, have used a simpler
reverberation model. It is assumed that the scatterers are closely spaced, and ran-
domly distributed in range and angle, and that the average scattering strength is
constant with range over a pulse length. The doppler spread of the scatterers is
assumed zero, since scatterer doppler spreading is typically much smaller than LFW
bandwidths and hence, would have negligible effect anyway. If the average spacing
of the scatterers is small compared to the size of a range resolution cell for any
bandwidths under consideration, then the reverberation appearing at the receiver
input can, with the postulated model, be approximated with stationary gaussian noise
having a power density spectrum equal to the energy density spectrum of the trans-
mitted signals.
One of the things of interest in this study will be the effect on signal-to-
noise and signal-to-reverberation performance of the variation of LFM pulse band-
width (W) and pulse length (T). For this purpose it is assumed that background
noise is white so that the noise pr-wer in the band of the signal is directly pro-
portional to signal bandwidth. at'iermore, it is assumed that reverberation power
at the input to the signal prc '.'or is directly proportional to pulse length. We
also assume that receiver input reverberation level does not depend on the bandwidth
of the transmitted signal.
DETECTION
The processor for the LFM waveform is a baseband I and Q correlator in which
the received signals are correlated with a reference signal consisting of a delayed
replica of the transmitted signal. See Figure 3. Correlations are computed at
closely spaced intervals of delay over all delays for which significant signal re-
turns are expected. In addition, an integrator is included to combine returns
that would normally appear at the correlator output at different resolvable times,
due to range and doppler spreading of the echo. Results are obtained both with
and without the use of the integrator.
Figure 4 shows the correlator output waveform for the case of three paths dis-
tributed as shown in range and doppler. This illustrates the well known range-
doppler ambiguity of LFM. The path that is doppler shifted appears at the correla-
tor output with a shift in time compared to where it would appear if its doppler
were zero. In fact it may be shown that if the doppler shift is much less than the
pulse bandwidth, a signal with an equivalent time delay and zero doppler will
produce the same correlator output waveform as a signal with an actual delay t and
doppler p, where
Furthermore , with our spreading model, in which range delay and doppler shift are in-
dependent gaussian random variables, a signal with multiple arrivals spread in time
only with standard deviation
v-/".2 * <VH)2
will produce the same results as a signal spread in both time and doppler having
standard deviations a and a, respectively. Thus, in this study, it was not consid-
T 9
ered necessary to treat time spreading and doppler spreading as separate effects.
Hence, doppler spread has been assumed zero for all results that will be shown.
Figure 5 shows correlator output waveforms for six successive pings all from
the same population, for the case
WT * 50
N = 6 paths
ot = 2/W
a * 3 db
a
o — 0 Hz
9
In each ping, the amplitudes of the individual arrivals are normalized such that the
total energy (sum of the square of the amplitudes) is constant. If all of the energy
were concentrated in a single arrival, the correlator output would peak at the level
marked "theoretical maximum". Notice that the peaks are somewhat reduced from the
theoretical maximum and there are usually several peaks.
When dealing with discrete numbers of arrivals, it is helpful to be aware of the
denseness with which these arrivals are "packed" relative to the time resolution of
the signal. For time delays distributed normally about a mean t, the densest packing
will occur, on the average over many pings, in the time resolution cell centered on
t. We will define a quantity, to be a measure of the denseness:
n s I ----- -- - - - - w
n Itheoretical maximum correlator output peak
This can be shown to be equal, to, for 9 = 0
peak correlator output for tq = t
3
n " /_„|A(t-To’0)|2 Pt(T-To)/aT] dT
where
E(X) = Expectation of X
A(t,9) = Ambiguity function of the transmitted waveform
P(x) « Zero mean gaussian probability density function with standard deviation
of unity
FIGURE 5. CORRELATOR OUTPUT WAVEFORMS
n/N is the fraction of signal energy, averaged over many pings, contributing to the
correlator output at the reference delay, = x, most likely to produce a correla-
tor peak. In other words, it is a measure of the signal energy contained within the
particular range resolution cell which is most likely to contain the greatest signal
energy. On the average, n is approximately equal to the number of arrivals that
appear in this resolution cell. If n<<l, since we are dealing with discrete arrivals,
the arrivals are distributed thinly among time resolution cells, and the correlator
output will appear "spiky". If n^l* corresponding to, on the average, many more
than one arrival in one resolution cell, the output waveform will tend to be smoother.
In the results to be shown, we will see that the effect of over-averaging, or
smoothing the correlator output is different depending on whether n is greater than
or less than unity.
In the Monte Carlo procedure employed in this study, a detection is counted if
the correlator output waveform exceeds the threshold at any time during the ping
and that threshold crossing would not have occurred in the absence of the echo.
The decision process is primarily sensitive to peaks of the output waveform. Thus,
for example, if there are N arrivals, and n<<l so that they are thinly distributed,
the largest peak of the correlator output waveform will be caused by the arrival
containing the most energy. If the energy were equally divided between all arrivals,
then there would be N equal output peaks with an amplitude of 1/*^N of the amplitude
which would result if all the energy were contained within a single arrival. How-
ever, if the energy is distributed unequally among the N arrivals, as with our model,
then, for any given ping, the arrival which contains the greatest energy will deter-
mine the detectability of that echo, all other arrivals contributing essentially
nothing. Furthermore, the fraction of the total energy contained in this maximum
arrival is always greater than 1/N. Hence, the detectability of signals consisting
of multiple arrivals unequally distributed in energy can be expected to be better
than the detectability of echos whose energy is equally divided among all arrivals.
As we will see in the results to follow, the presence of multiple arrivals has a
surprisingly small effect on detection, and the reason for this is in part the un-
equal division of energy between arrivals.
RESULTS
Results are shown in terms of the minimum detectable correlator input signal-
to- interference ratio (SIR) required to produce, at the processor output, a detection
probability of 0.5 when the output threshold is set for a false alarm probability of
0.01. Detection probability, as we have stated, is the probability that at least one
threshold crossing will occur, on a given ping, with an echo present that would not
have occurred with the echo absent. False alarm probability is the fraction of time,
over all pings, that the correlator output voltage is above the threshold in the
absence of signal. The input interference is measured in the band of the signal and
can be interpreted as either reverberation or noise.
Let us look first at the effect of the number of paths, N. For this, a is
held constant and N varied from 1 to 50. Figure 6 shows typical correlator output
waveforms for:
TW = 50
= 2/W sec
a = 6 db
a
a, = 0 Hz
<P
The waveform at the extreme left is for N = 1. With only a single arrival, the cor-
relator peak achieves its theoretical maximum*. For N=3, we see two clearly resolved
arrivals with reduced peaks. For N=6, we find three peaks and for N=12 four peaks.
Note, though, that as N increases, the largest peak does not decrease substantially
beyond N=6. This is due to (a) the effect previously discussed regarding the unequal
splitting of energy among arrivals and that the greater N, the more likely the
occurrence of relatively large peaks; and (b) the fact that the average energy in a
resolution cell tends to approach a constant value as n becomes much greater than
unity. Note also that the "spikiness" of the output waveform diminishes as N in-
creases. From observing these waveforms , we would expect that detectability would
be best for N=l, and diminish gradually as N increases. This is borne out in Figure
7 which shows minimum detectable input SIR (called SIRm^n) vs N (solid line).
SIRmin is -10.6 db for a single arrival and increases to about -6 db for N=50, a
loss in detection of k.6 db. The dashed curve shows the effect of smoothing the
correlator output with an integrating filter whose impulse response is a rectangle
of duration 2 aT (= k/W). We see that, although detectability is degraded somewhat
by the smoothing for small N, where the arrivals are thinly spread, it is signifi-
cantly improved for large N. In fact, the effect of the integrator is to render
detectability almost constant at about -9 db for all N.
*The theoretical maximum is defined as the maximum correlator output voltage when
there is no interference and when all signal energy is contained within a single
resolution cell.
12
f
4 «
FIGURE 7. MINIMUM
* 50
* 0.04 T SEC
* 2/W
* 0 Hz
* 3dB
INPUT SIR vs NUMBER OF PATHS
f
Now let us hold N constant at 50 and see the effect of varying the amount of
spreading, a , again for WT=50. Typical correlator output waveforms are shown in
T %
Figure 8. For the first output waveform shown, a T ■ .01/W which implies essentially
no spreading at all and the correlator treats the echo as a single arrival. The
second trace is for * 1/W and some evidence of time spreading of the correlator
output is evident, as well as some reduction in the peak. This trend continues as
0^ increases. Note also the increase in "spikiness". We would expect detectability
to diminish as increases, and this is shown in Figure 9. The solid curve is a
rough smoothing of the experimental points (X's). Detectability degrades (SIRm^n
increases) up to a T * 5/W, but then reverses, unexpectedly, for o = 10/W. The
cause of this reversal is not well understood. It is a consequence of our choice
for detection probability of 0.5. If a higher detection probability of say 0.9 were
chosen, this reversal would not have occurred. Nevertheless, the greatest loss in
detectability for the range of <j from 0 to 10/W is only about 5 db. Furthermore,
with over-averaging of the . correlator by an amount 20^, the greatest loss is only
about 3 db. This comparison is a little bit unfair, because, in practice, we cannot
predict the amount of spreading and hence, cannot match the amount of over-averaging
to the spreading. However, detectability is a slowly varying function of averaging
time, as is suggested in Figure 10 which shows SIR^n vs averaging time for ot = 5/W
and N=1 and 30. For 30 arrivals, the best detection is obtained with an integration
time approximately equal to 2a . However, detection is only, at the most, 2 db
worse for no integration. The curve for N=1 illustrates what can happen if the
integration time is too long. For N=l, of course, the best detection is obtained
with no integration. However, if we over-average by as much as 10 range resolution
cells, the loss in detectability is less than 2 db. Beyond this point, however,
the losses begin to increase rather rapidly.
One of the most important questions that we are attempting to answer in this
study is: How should echo energy splitting effect the choice of the transmitted
signal bandwidth, W, and the transmitted pulse length, T? Were it not for energy
splitting, to obtain maximum detection performance, it would be best to use all
the bandwidth available when reverberation limited and all. the pulse length (trans-
mission time) available when noise limited. There we other constraints on these
parameters, however. Available bandwidth is limited by the efficient bandwidth of
the projector and the need for multiple frequency channels. Transmission time is
limited by the need to cover a wide search and by the non-stationarity of reverber-
ation during a ping cycle. Our concern, then, is whether the environment will place
15
40 dB
FIGURE 8. CORRELATOR OUTPUT WAVEFORMS vs TIME SPREADING
MINIMUM DETECTABLE INPUT SIR
. <dB)
e
0.5
12 5 10
AVERAGING TIME
20
WT
50
0.1 T » 5/W
0.03W1.5/T
3dB
Tav/T
WT
av
FIGURE 10. MINIMUM DETECTABLE INPUT SIR vs AVERAGING TIME
constraints on the choice of W and T that are more stringent than these other con-
straints. The following results provide some insight into this question.
Figure 11 shows correlator output waveforms for four different transmitted sig-
nal bandwidths. The number of arrivals, N, is 50 and they are spread in time delay
by ot * .OUT. The first waveform is for W ® 10/T (ot W = .4) and we see that all of
the 50 paths fall within a single range resolution cell and the correlator output
achieves its theoretical maximum value. As bandwidth is increased, the width of a
range resolution cell becomes smaller and more evidence of energy splitting is
observed. Hence, we would expect that the performance at the wider bandwidths
would be degraded from theoretical performance more than it would at the narrower
bandwidths. This is seen in Figure 12, which shows SIRmin vs signal bandwidth.
The lower curve, labeled o * 0, is the "theoretical" curve for no energy splitting.
This curve has the expected slope of 10 db per decade reflecting the well known
relationship between the input and output signal-to-noise ratio of a correlator:
output SNR = K*WT* input SNR
where K is a constant and the ratios are power ratios. The upper curve of Figure 12
is for the spreading conditions described above. We see that the loss due to spread-
ing (the vertical distance between curves) varies between 2 and 5 db, and does not
appear to increase very much with increasing bandwidth. As a matter of fact it
decreases slightly at the extreme bandwidths. This is the same effect as noted
above with large values of a^. The largest TW in Figure 12 is 250, for which W
= 10.
What does this say about how much bandwidth to use? The important parameter
in assessing the performance of signals and processors is not minimum detectable
input signal-to-interference ratio, as we have been using, but rather minimum
detectable input signal level. For discussions thus far the two are equivalent
since no parameters have been varied that effect input interference level. When
bandwidth is a parameter, one must account for the fact that, under the assumption
of white noise, the interference power level, when noise limited, is directly pro-
portional to bandwidth.
Thus, from Figure 12, two curves are derived showing minimum detectable input
signal vs W for noise limited and for reverberation limited conditions. These are
shown in Figure 13. Now we can Judge the effects on detection of employing wide
bandwidth. When reverberation limited, the wider bandwidths are clearly better
litfiilMiaiallflttatii
ififiWiI***..* limit -n- mi
INPUT SIGNAL ENERGY - CONSTANT
FIGURE 11. CORRELATOR OUTPUT WAVEFORMS vs PULSE BANDWIDTH
MINIMUM DETECTABLE INPUT SIR
(dB)
Pd * 0 5
Pf • 0.01
* 0.04 T
■ 0 Hz
or ■ 3 dB
0
N ■ 50
crT * 0
10 15 20 50 100 200 500 1000
Pd - 0.5
Pf • 0.01
•V • 004 T
#d • 0
r0 - 3dB
N • 50
_J 1 1 1 1 1 L
05 1 £ 5 10 20 ■*-
FIGURE 13. MINIMUM DETECTABLE INPUT SIGNAL LEVEL vs BANDWIDTH
« «* *»
than the narrower bandwidths. For WT = 100, reverberation limited performance is
about 5 db better than for WT = 20. When noise limited however, some loss in per-
formance is incurred at the wider bandwidths. For WT = 100, noise limited perform-
ance is about 2 db poorer than for WT = 20.
These results by themselves are not conclusive. We have used only one value
of signal spreading (o = and, for that matter, only one spreading model. In
addition, we have selected only one output criterion: pd = .5, pf = .01. However,
these results are encouraging. They appear to be telling us that even in the pre-
sence of signal spreading, wide pulse bandwidth can buy significant reverberation
limited performance at the cost of only a small degradation in noise limited per-
formance .
In reference 2, similar results are obtained for a spreading model in which
signal energy is spread uniformly in range and doppler and the correlator output
is integrated uniformly over the spread. For an LFM waveform with time spreading
of 5 range resolution cells (corresponding approximately to Wo^ = 10 for the random
spreading model), reverberation limited performance is only about 2 db worse than
theoretical — a strong Justification for the use of wide bandwidths.
CONCLUSIONS
1. Echo detectability is a function, not only of the number of resolution cells
over which signal energy is spread but also of the number of individual discrete
arrivals within a ping. Detectability also depends on the expected spread in amp-
litudes of the individual arrivals, being better with more amplitude spread.
2. For choices of bandwidths usually available in long range active sonar systems,
echo energy splitting does not appear to cause a serious limitation and, to attain
the best possible echo detectability under reverberation limited conditions, one
should use as much bandwidth as possible.
3. Over-averaging the correlator output provides modest gains in detection per-
formance when signal energy spreading is dense (n>l). However, when signal energy
is thinly spread (n<l), it is better not to over-average the correlator output.
23
As we will see in the results to follow, the presence of multiple arrivals has a
surprisingly small effect on detection, and the reason for this is in part the un-
equal division of energy between arrivals.
11
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