Introduction to Acoustics

166 Part A Propagation of Sound
Part A 5.3
Probability density function
1
0.8
0.6
0.4
0.2
0
Noise only
Mn
d
DT
Msn
Signal + Noise
Received signal level
Fig. 5.19 Probability density functions (PDF)ofnoiseand
signal+noise. Various probabilities as explained in the text
are related to specific regions under the curves. Thus, the
probability of detection (PD) is the light-shaded area under
the signal+noise PDF curve that is to the right of DT,
the detection threshold (thick vertical line). Similarly, the
probability of false alarm is the dark shaded area under the
noise PDF curve that is to the right of DT
statistics. We first note that the PD is the area under the
signal-plus-noise curve to the right of the DT and the
PFA is the area under the noise curve to the right of the
DT. Then, for example, if the mean of the signal-plusnoise
PDF was further to the right [higher signal-to-noise
ratio (SNR)], the detection index and PD would be larger.
We could then move the DT to the right, changing
the PD,butthePFA would be smaller. Receiver operating
characteristics (ROC) curves are plots of PD versus
PFA parameterized by d (Fig. 5.19). Figure 5.20 gives
a typical example of using this methodology and refer
the reader to the SONAR literature [5.16] for a more
complete treatment. A square-law detector is commonly
used for a passive system dealing with unknown signal.
In that case, it has been shown that the detection index
for a small-SNR, narrow-band signal in Gaussian noise
is given by
d = ωt
� �2 S
, (5.25)
N
where ω, t, S and N are the bandwidth (taken to be
larger than the width of the spectral line of the signal),
integration time, signal power and noise power in the
bandwidth, respectively. If 1 Hz is the noise bandwidth
reference, the detection index is d = ωt (S/N0), where
N0 is the noise in a 1 Hz band. This gives a relation
between the signal-to-noise ratio at the output of an
energy detector of a narrow-band signal in Gaussian
noise to the detection index:
� �
d
SNR = 5log ≡ DT , (5.26)
ωt
where the equivalence symbol is for a detection criteria
of a specified PD, PFA.
In reality, there are correction factors for the detection
threshold related to the length of observations used
to make a decision, human factors, and others that we
omit from this discussion. Since the criteria are specified
through the ROC curves, we can now estimate the
DT for specific cases. We simply go to the the ROC
curves for a selected PD versus PFA and read off the detection
index, from which we can compute the detection
threshold. For example, from Fig. 5.20,foraPD of 50%
and a PFA of 0.01%, the detection index is d = 16. Using
unit bandwidth and integration time, the detection
threshold is DT= 5 log 16 = 6 dB. This methodology is
an example of the meaning of the DT term in the SONAR
equation below, though the relationship between DT and
d is different depending on the type of receiver and
SONAR.
5.3.2 Passive SONAR Equation
A passive SONAR system uses the radiated sound from
a target to detect and locate the target. A radiating object
of source level (SL) (all units are in decibels) is
received at a hydrophone of a SONAR system at a lower
signal level S because of the transmission loss (TL) it
suffers (e.g., cylindrical spreading plus attenuation or
a TL computed from one of the propagation models of
Sect. 5.4),
S = SL − TL . (5.27)
The noise, N at a single hydrophone is subtracted from
(5.27) to obtain the signal-to-noise ratio at a single
hydrophone,
SNR = SL − TL − N . (5.28)
Typically a SONAR system consists of an array or
antenna of hydrophones which provides signal-tonoise
enhancement through a beam-forming process
(Sect. 5.6). This process is quantified in decibels by the
array gain (AG) (see Sect. 5.6.2) that is added to the
single-hydrophone SNR to give the SNR at the output
of the beam-former,
SNR BF = SL − TL − N + AG . (5.29)