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Applied Research Laboratory
Sonar Signal Processing
Course developed by:
Dr. Martin A. Mazur and Dr. R. Lee Culver
Presented by:
Dr. Martin A. Mazur
1
Applied Research Laboratory
Course Outline
Sonar Signal Processing
• Review of the Sonar Equation
• Some Mathematical Preliminaries
• Time/Frequency Signal Processing
• Space/Angle Signal Processing
• Decision Theory and Receiver Design
• Examples (Time Allowing)
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Applied Research Laboratory
What We Will Not Cover (In Any Great Depth)
• Digital signal processing concepts and techniques
• Adaptive signal processing or beamforming
• Post-detection signal processing (e.g. classification,
tracking)
• Random variable theory, stochastic processes
• Sonar implementation concepts (covered in a separate
course):
– Detailed transmitter/receiver block diagram
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Applied Research Laboratory
References • A. D. Waite, SONAR for Practising Engineers, Third Edition, Wiley (2002) (B)
• W. S. Burdic, Underwater Acoustic System Analysis, Prentice-Hall, (1991) Chapters 6-9, 11, 13, 15. (M)
• R. J. Urick, Principles of Underwater Sound, McGraw-Hill, (1983). (M)
• X. Lurton, An Introduction to Underwater Acoustics, Springer, (2002) (M)
• J. Minkoff, Signal Processing: Fundamentals and Applications for Communications and Sensing
Systems, Artech House, Boston, (2002). (M)
• Richard P. Hodges, Underwater Acoustics: Analysis, Design, and Performance of Sonar, Wiley, (2010). (M)
• Digital Signal Processing for Sonar, Knight, Pridham, and Kay, Proceedings of the IEEE, Vol. 69, No. 11,
November 1981. (M)
• M. A. Ainslie, Principles of Sonar Performance Modeling, Springer-Praxis, 2010. (A)
• H. L. Van Trees, Detection, Estimation and Modulation Theory, Part I, Wiley (1968) (A)
• J-P. Marage and Y. Mori, Sonar and Underwater Acoustics, Wiley, (2010) (A)
• Richard O. Nielsen, Sonar Signal Processing, Artech House, (1991) (A)
• B. D. Steinberg, Principles of Aperture and Array System Design, Wiley, New York (1976). (A)
(B) – Basic/Overview/Reference (M) – Mid-level/General/Reference (A) – Advanced/Special Topic
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Applied Research Laboratory
• The sonar equation, in its simplest form, is:
• In decibels, referred to the sonar system input:
• Referred to the receiver output, the sonar equation becomes
where
Threshold Detection Power Noise
Power Signal
Review of the Sonar Equation
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Applied Research Laboratory
Sonar Signal Processing Gain
Detectors
Receive
Beamformer
Receive
Time/Frequency
Processor
Downstream Processing
(e.g. classification,
tracking)
Processing Gain
PG = AG + G
M
Receiver
Channels
Receiver Gain
G
Array Gain
AG
Input SNR
Channel Output SNR
Thresholds:
DT = Input Detection Threshold
OT = Receiver Output Threshold
OT = DT + PG
Space-Angle
Signal Processing Time-Frequency
Signal Processing
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Applied Research Laboratory
Mathematical Tools
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Applied Research Laboratory
• In this section we touch briefly on some of the
mathematical concepts and tools used in signal
processing.
• Most of the section is definitions and a quick review of
the mathematical notation.
Outline
8
Applied Research Laboratory
Real and Complex Signals
• A real-valued function of time, f(t), or space, f(x), or both, f(x,t), is
often called a “real signal”.
• It is sometimes useful for purposes of analysis to represent a
signal as a complex valued function of space, time, or both:
• More often, such a function is written in polar form:
• The real-world signal f(t) represented by s(t) is just the real part of
s(t):
)()()( tvituts
(phase) )t(u
)t(varctan)t(
)(magnitude )t(v)t(u)t(R
where
e)t(R)t(s )t(i
22
))(cos()()()}({)( ttRtutstf e
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Applied Research Laboratory
Symbols and Identities
s(t) - a function of time, real or complex valued
s*(t) - the complex conjugate of s(t), i.e. if
then
|s(t)|2 - the squared magnitude of s(t), equal to s(t) ·s*(t)
S( ) - for deterministic signals, the frequency
or spectrum of s(t) (the Fourier transform of s(t))
S(f) - for random signals, the power spectrum of s(t)
(the Fourier transform of the autocorrelation
function of s(t)) Note: = 2 f
u(x) - a function of a spatial coordinate
U( ) - the angular spectrum of u(x) (the Fourier
transform of u(x))
Euler’s Identity: ei = cos + i ·sin
)()()()( )( tietRtvituts
)()()()( )(* tietRtvituts
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Name Symbol Graph Properties
Impulse or delta (t) Defined through its integral:
function
unit step u(t)
rectangular
function
Sinc function
T
trect
T
tsinc
)t(fdt)tt()t(f
dt)t(
00
1
T
t
)T
tsin(
T
tsinc
0
1
Special Functions
0
... 1
0
1
-T/2 T/2
0
-T/2 T/2
1
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Time/Frequency Signal Processing
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• In this section we touch briefly on some of the
concepts and tools used in spectral analysis, one of
the most important areas of signal processing.
• The section is a barebones outline, but because it is
just a sample of a very large topic, it is still a whirlwind
tour!
• Very little in the vast realm of digital signal processing
(DSP) can be covered in this small introduction.
Outline
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Spectral (Fourier) Analysis
• Any signal, real or complex, that varies with time can be “broken up into its spectrum” in a way similar to that in which light breaks up into its constituent colors by a prism
• The mathematical operation by which this is accomplished is the Fourier transform
• The Fourier transform of a time signal yields the “frequency content” of a signal
• Much signal processing is done in the “frequency domain” by means of mathematical operations (filters) on the Fourier transforms of the signals of interest
• The result of the processing can be converted back to a filtered time signal by means of the inverse Fourier transform 14
Applied Research Laboratory
• Periodic signals can be represented as a sum of
complex exponentials at discrete frequencies. For s(t) =
s(t + nT),
• The ck are complex numbers representing the spectrum
of s(t) and are called the Fourier coefficients of s(t)
T where , ec )t(s
k
tikk
20
0
k
k
T
tikk
)n(c)(S
dte)t(sT
c
0
01
Spectral Analysis of Periodic Signals
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• An arbitrary signal can be represented as an integral of
weighted complex exponentials over all frequencies:
• The function S( ) is the frequency spectrum of s(t) and
is called the Fourier transform of s(t).
• S( ) is calculated from s(t) as follows:
de)(S2
1 )t(s ti
dtetstsS ti)()}({)( F
Spectral Analysis of Continuous Signals
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Applied Research Laboratory
A Few Fourier Transform Relationships
filter passlow a of response Impulse
pulse time a of transform Fourier
function Delta
inverse its and
transform Fourier of defintion of Similarity
Versa- Vicend A
tionmultiplica Frequency nconvolutio Time
Modulation F(
shift Time
Scaling 1
nconjugatio Complex
Linearity
Definition
nDescriptio
-
-
)2
T(rectTA)T/tsinc(A
)2
Tsinc(TA)T/t(rectA
e)t(
)(f2F(t)
)(H)(F)t(hf(t)
)(H)(F)t(h)t(fdt)t(f(t)h
))t(fe
e)(F)t(f
Ft)f(
)(*F f*( t )
)(G)(F)t(g)t(f
dtf(t)e)t(f
)(F)t(f
i
0
ti
i
ti
0
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• It generally takes a month or two of hard homework
assignments for an engineering student to become
facile with the mathematics, concepts, and implications
of the previous table!
Note On The Last Table
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Frequency Domain Signal Processing
• Frequency domain signal processing, or “filtering” alters the frequency spectrum of a time signal to achieve a desired result.
• Examples of filters: band pass, band stop, low pass, high pass, “coloring”,
• Analog filters are electrical devices that work directly on the time signal and shape its spectrum electronically.
• Most filtering nowadays is done digitally. The spectrum of a sampled, digitized time signal is calculated using the Fast Fourier Transform (FFT). The FFT is a sampled version of the signal’s spectrum. Mathematical operations are performed on the sampled spectrum.
• Samples of the filtered signal are recovered using the inverse FFT. 19
Applied Research Laboratory
• A linear system is one described by an equation y(t) = L[x(t)] that
obeys the superposition property:
– If y1(t) is the response, or output, of the system to the input x1(t), and
y2(t) is the response to x2(t), and if a and b are arbitrary constants,
then
ay1(t) + by2(t) = L[ ax1(t) + bx2(t)]
• A system is time-invariant if when the input is shifted in time, the
output is correspondingly shifted.
• The response of a linear time-invariant system can be deduced by
its response to an impulse:
h(t) Impulse response function = L[ (t)]
• For any input x(t), the output is the convolution integral of x(t) with
the impulse response function h(t):
)t(h)t(xd)t(h)(x)]t(x[L)t(y
Response of a Linear Time-Invariant System
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Applied Research Laboratory
Suppose a linear system L has impulse response h(t).
• The Fourier transform of h(t), H( ), determines the “frequency response”
of L. How?
• The “convolution property” of the Fourier transform allows us to easily
calculate the spectrum of the output of the system for any input x(t):
convolution property: convolution in time domain multiplication in frequency domain
(AND VICE VERSA)
• The output time function y(t) can be obtained by calculating the inverse Fourier transform of Y( ):
deYYty ti- )(2
1)}({)( 1F
Frequency Domain Analysis of Linear Systems
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Applied Research Laboratory
Suppose we have two inputs to a receiver, only one of which is desirable.
Suppose the inputs are separated in frequency:
Example of Frequency Domain Processing:
Low Pass Filter
Frequency Domain
))](2())(2([2
))](2())(2([2
)(
22
11
B
AX
0 1 2 1 2
0 1 2 1 2 3 3
))](2())(2([2
)()()( is filter this ofoutput the Then 11
AXHY
And y(t) = A cos t
tcosBtcosA)t(x 21
Desired Unwanted
Time Domain
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• Sampling Theorem: A real, band limited, finite energy
signal can be reconstructed from samples taken at the
Nyquist rate.
– Assume that the spectrum of a signal s(t) is zero outside the
interval |f| < B.
– If samples sn are taken at the Nyquist rate of fs = 1/ts = 2B, then s(t) can
be reconstructed from its samples using the formula
– Sampling at less than the Nyquist rate can result in overlap of periodic
extensions in frequency domain (“aliasing”), resulting in distortion of
the reconstructed signal.
An Important Theorem In Spectral Analysis
n s
sn
)ntt(B2
)]ntt(B2sin[s)t(s
23
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Effects Of Sampling
• When a real signal is sampled, copies of its spectrum appear at multiples of the sampling frequency => aliasing if signal is not band-limited. But any time-limited signal is not band-limited!
• Effects of aliasing can be abated by using time-windowing, filtering, and overlapping of data. This process somewhat increases the bandwidth and complexity of a signal processor, but greatly diminishes aliasing distortion.
• Digital signal processors usually work with a finite set of samples of a signal and another finite set of samples of the signal’s spectrum called the Discrete Fourier Transform (DFT).
• The Fast Fourier Transform (FFT) is a computationally efficient method of computing the DFT. 24
Applied Research Laboratory
Effects Of Sampling (Illustration)
B -B
Band limited
signal spectrum
Sampled signal spectrum
Sampling @Nyquist rate
f
f B
B
-B
-B
Sampled signal spectrum
Sampling < Nyquist rate
Overlap of spectral copies = aliasing
fs = 2B
2fs fs < 2B 25
Applied Research Laboratory
• Let x(t) and y(t) be two random signals from processes that are
stationary (statistics do not vary with time) and ergodic (statistics
of the process can be estimated by taking time-averages of
representative signals).
• The cross-correlation of x and y is a measure of how “statistically
alike” x and y are at particular spacings in time:
• The autocorrelation of x, Rx( ), is a measure of how statistically
alike x is to itself at different times.
For stationary, ergodic signals
correlated over a long time T
T
xy dttytxT
tytxER0
)(*)(1
~)](*)([ )(
Random Signals: Correlation
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• The Fourier transforms of Rx and Rxy are the power
spectral density, Sx( ), of the process x, and the cross-
power spectral density of the processes x and y, Sxy( ),
respectively. They are measures of the frequency
spectral content of the process x, and the shared
spectral content of x and y, respectively.
Spectra of Random Signals
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• If a random process x with power spectral density Sx( )
is passed through a linear, time-invariant filter with
transfer function H(f), then the power spectral density
of the output process y is given by:
Spectra of Filtered Random Signals
)f(S)f(H)f(S xy
2
28
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Spatial Signal Processing
(Beamforming)
29
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Analogy Between Spatial Filtering (Beamforming)
and Time-Frequency Processing
Goals of Spatial Filtering:
1. Increase SNR for plane wave
signals in ambient ocean noise.
2. Resolve (distinguish between)
plane wave signals arriving
from different directions.
3. Measure the direction from
which plane wave signals are
arriving.
Goals of Time-Frequency Processing:
1. Increase SNR for narrowband
signals in broadband noise.
2. Resolve narrowband signals at
different frequencies.
3. Measure the frequency of
narrowband signals.
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ψ1 ψ0 Spatial angle ψ
Ambient noise
angular density
Plane wave at ψ1 Plane wave at ψ0
Narrow spatial
filter at ψ0
f1 f0 Frequency
Broadband
noise spectrum
Sine wave at f1 Sine wave at f0
Narrowband
filter at f0
Time-Frequency Filtering and Beamforming
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SNR Calculation: Time-Frequency Filtering
response power Filter
(W/Hz) density spectral power Noise
(W/Hz) density spectral power Signal
at sinusoid Complex 2
2
0
2
00
)f(H
)f(N
)ff(
f)tfiexp(Define
Signal Power Is:
(watts) 2
0
22
0
2 )f(Hdf)f(H)ff(Ps
otherwise 022
1 02
,
ff,)f(H
If we assume
Idealized
rectangular
filter with
bandwidth β
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Applied Research Laboratory
SNR Calculation: Time-Frequency Filtering (Cont’d)
Then the noise power is:
(watts) 0
2Ndf)f(H)f(NPN
N0 is the
average
noise level
in band
And SNR is:
β
α
0
2
NP
PSNR
N
sNote that SNR can be
increased by narrowing
the bandpass filter.
f
P
N0
N(f)
f0
H(f)
33
Applied Research Laboratory
SNR Calculation: Spatial Filtering
response power angular filter Spatial )(
an)(W/steradi density angular power Noise )(
an)(W/steradi density angular power Signal )(
angleat arriving waveplane Acoustic)2exp(
2
0
2
00
G
N
xiDefine
Signal Power Is:
(watts) 2
0
2
4
2
0
2 )(Gd)(G)(Ps
If we assume
Idealized
“cookie cutter”
beam pattern
with width
34
Applied Research Laboratory
SNR Calculation: Spatial (Cont’d)
Then the noise power is:
K0 is the
average
noise intensity
in beam
And SNR is: Again, SNR can be
increased by
narrowing “filter”
P
K0
N( )
0
G( )
35
Applied Research Laboratory
Array Gain and Directivity Calculations
OH
Array
SNR
SNR Gain Array
Define
Then
4
2
4
2
4
2
0
2
d)(Nd)(G)(N
d)(G)(
P
PSNR
OH
OH
N
sOH
Assume
• plane wave signal
• arbitrary noise distribution
all for 12
)(G
For the omnidirectional hydrophone,
36
Applied Research Laboratory
Array Gain and Directivity Calculations (Cont’d)
For the array, assume it is steered in the direction of Ω0 and that
4
2
2
4
2
4
2
0
2
d)(G)(Nd)(G)(N
d)(G)(
P
PSNR
arrayarray
array
N
sarray
Putting these together yields
1Ω2
0
2 )(G
Then
4
2
4
d)(G)(N
d)(N
AG
array
37
Applied Research Laboratory
Array Gain and Directivity Calculations (Cont’d)
If the noise is isotropic (the same from every direction)
Then the Array Gain (AG) becomes the Directivity Index (DI), a performance index
For the array that is independent of the noise field.
K)(N
π4
2
ΩΩ
π4
d)(GDI
array
Array Gain and Directivity Index are usually expressed in decibels.
38
Applied Research Laboratory
Line Hydrophone Spatial Response
plane wave signal
wave fronts
L
L/2
-L/2
0
plane wave has wavelength
= c/f,
where f is the frequency
c is the speed of sound
x
x1 X1sin
39
Applied Research Laboratory
Line Hydrophone Spatial Response (Cont’d)
The received signal is
Let the hydrophone’s response or sensitivity at the point x be g(x).
Then, the total hydrophone response is
xc
sinxts
)t(s
point at
origin theat
dx)c
sinxt(s)x(g)t(s
/L
/L
out
2
2
ψ
40
Applied Research Laboratory
Line Hydrophone Spatial Response (Cont’d)
Using properties of the Fourier Transform:
And:
)f(S)c
sinfxiexp(dte)
c
sinxt(s fti ψπ2ψ π2
dte)t(s)f(S fti π2
df)f(S)ftic
sinfxiexp()
c
sinxt(s π2
ψπ2ψ
Or:
41
Applied Research Laboratory
Line Hydrophone Spatial Response (Cont’d)
Thus, the total hydrophone response can be written:
Where
dfdx)f(S)ftic
sinfxiexp()x(g)t(s
/L
/L
out π2ψπ2
2
2
We call g(x) the aperture function and G((fsin c) the pattern function.
They are a Fourier Transform pair.
dfedx)c
sinfxiexp()x(g)f(S fti
/L
/L
π2
2
2
ψπ2
dfe)c
sinf(G)f(S fti π2ψ
2
2
ψπ2
ψ/L
/L
dxx)c
sinf(iexp)x(g)
c
sinf(G
42
Applied Research Laboratory
Response To Plane Wave
An Example:
Unit Amplitude Plane Wave from direction
Note that the output is the input signal modulated by the value of the
pattern function at
tfie)t(s 02
dfe)c
sinf(G)ff()t(s fti
out
200
tfjec
sinfG 0π200 ψ
The Line Hydrophone Response is:
)ff()f(S 0δ
43
Applied Research Laboratory
Response To Plane Wave (Cont’d)
The pattern function is the same as the angular power
response defined earlier.
Sometimes we use electrical angle u:
λ
ψψ sin
c
sinfu
instead of physical angle
44
Applied Research Laboratory
Uniform Aperture Function
Consider a uniform aperture function
otherwise
,
Lx
L,
L)x(g
0
22
1
The pattern function is:
)Lu(
Lu
)Lusin(
Luj
e
Luj
ee
Luj
eedx
Ldxe)x(g)u(G
/L
/L
uxj/Luj/Luj
/Luj/Luj/L
/L
uxj
sinc
e uxj2
π
π
π2π2
π2
1
2
2
π22π22π2
2π22π22
2
ππ2
45
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Rectangular Aperture Function and Pattern Function
46
Applied Research Laboratory
Array Main Lobe Width (Beamwidth)
2
1
2
ψ2
3dBG
3 dB (Half-Power) Beamwidth
To find it, solve
for dB3ψ
47
Applied Research Laboratory
Beamwidth Example: Uniform Weighting
Lu
Lu
dBdB
dB
39.12sin)(sin
39.12
1
3
1
3
3
2
1
2
π
2
π
3
3
dB
dB
Lu
Lusin
. increasing ofeffect the Note For
deg 50
radians 8858851
L.L
L
L.
L.sin
48
Applied Research Laboratory
Discrete Elements: Line Array
)(1
n
N
n
n xxδag(x)
)()(
, spacing uniform and (Odd), 1 2M For
ndxaxg
dN
M
Mn
n
Aperture Function:
dxexgG(u) πuxj2)(
dxendxa uxj
n
M
Mn
2)(
Pattern Function: (N odd, uniform spacing)
nduj
n
M
Mn
ea π2
x
L = Nd
d
49
Applied Research Laboratory
Uniformly Weighted Discrete Line Array
πduj
n
er
n,N
a
2
:define yTemporaril
odd. spacing, uniform 1
Assume
2/12/1
2/2/11)(
then
rr
rr
Nar
NrG
NN
n
M
Mn
n
)sin(
)sin(1)(
or
ud
uNd
NuG
50
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Pattern Function For Uniform Discrete Line Array
51
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Notes On Pattern Function For Discrete Line Array
λ
1
λ
1
,
λ
1
λ
1u
d
•u Only has physical significance over the range
•The region may include more than
one main lobe if , which causes ambiguity
(called grating lobes).
General trade-offs in array design:
1)Want L large so that beamwidth is small and resolution is good
2)Want λd to avoid grating lobes.
3)Since L=Nd, or N=L/d, increasing L and decreasing d
Both cause N to increase, which costs more money
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Applied Research Laboratory
• Shading reduces sidelobe levels at the expense of
widening the main lobe.
• For other aperture functions:
First sidelobe
Aperture 3dB ,degrees level, dB
Rectangular 50 /L -13.3
Circular 58 /L -17.5
Parabolic 66 /L -22.0
Triangular 73 /L -26.5
Effects of Array Shading
(Non-Uniform Aperture Function)
53
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Beam Steering
g(x)e
dpepGe
dueuuGxg
dueuGg(x)
xπuj
πpxjxπuj
πuxj
πuxj
0
0
2
22
2
0
2
)(
)()(
)(
Want to shift the peak of the pattern function from u = 0 to u = u0.
What is the aperture function needed to accomplish this?
G(u- u0)
Beam steering is accomplished by multiplying the non-steered aperture
function by a unit amplitude complex exponential, which is just a delay
whose value depends on x.
54
Applied Research Laboratory
Beam Steering For Discrete Arrays
).u(sin 0
1
0 λψ
...)()()()0()()(...
)()(
00
0
22
2
1
dxedgxgdxedg
endxndgg(x)
ndujnduj
ndujN
n
The phase shift at element n is equivalent to a time shift:
The steered aperture function becomes
The steered physical angle is
.fc
sinndf
sinndndu nτπ2
ψπ2
λ
ψπ2π2 00
0
c
sinndn
0ψτ where
Is the time shift which must be applied to the nth element.
55
Applied Research Laboratory
Effect Of Beam Steering On Main Lobe Width
Beam steering produces a projected
aperture. Since reducing the aperture
increases the beam width, beam steering
causes the width of the (steered) main
lobe to increase. The lobe distorts
(fattens) more on the side of the beam
toward which the beam is being steered,
56
Applied Research Laboratory
Beam Steering: An Example
2
100
22
u-G( G(u )u)u
)](sin[
)](sin[1)(
0
00
uud
uuNd
NuuG
For a uniformly weighted, evenly spaced (d= /2), 8 element array, the
pattern function is
To find the beamwidth, set
1750ππ 00 .)uud)uud -( (
Which yields
is condition the and Using λψ2λ /sinu/d
11140ψψψψ 00 .ssinsin sin in-
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Applied Research Laboratory
Beam Steering: An Example (Cont’d)
As an example, take N=8, (d= /2).
0
0
3
00
0
5124
50λ50
λ50
812ψ46ψ46ψ
11140ψψ0ψ
.ndL
...
.sinsinsin
dB
:Note
:1 Case
58
Applied Research Laboratory
Beam Steering: An Example (Cont’d)
optimal is whyis This
at lobe grating a is therepoint at which
until lobe grating no is therethat Notice
:lsymmetricanot is beam But,
(wider)
:2 Case
2
λ
90ψ
90ψ
4863645
9945954
318ψ
636707011140ψ
954707011140ψ
7070ψ45ψ
0
0
0
000
000
0
3
01
01
0
0
0
d
..
..
.
.)..(sin
.)..(sin
.sin
dB
59
Applied Research Laboratory
Beam Pattern Effects Number of Elements (Array Size)
60
Applied Research Laboratory
Beam Pattern
1 Element Line Array – /2 spacing
61
Applied Research Laboratory
Beam Pattern
2 Element Line Array – /2 spacing
62
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Beam Pattern
3 Element Line Array – /2 spacing
63
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Beam Pattern
4 Element Line Array – /2 spacing
64
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Beam Pattern
5 Element Line Array – /2 spacing
65
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Beam Pattern
6 Element Line Array – /2 spacing
66
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Beam Pattern
7 Element Line Array – /2 spacing
67
Applied Research Laboratory
Beam Pattern
8 Element Line Array – /2 spacing
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Applied Research Laboratory
Beam Pattern
9 Element Line Array – /2 spacing
69
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Beam Pattern
10 Element Line Array – /2 spacing
70
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Beam Pattern Effects Element Spacing (Sampling)
71
Applied Research Laboratory
Beam Pattern
10 Element Line Array – /10 spacing
72
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Beam Pattern
10 Element Line Array – /4 spacing
73
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Beam Pattern
10 Element Line Array – /2 spacing
74
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Beam Pattern
10 Element Line Array – 3/4 spacing
75
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Beam Pattern
10 Element Line Array – spacing
76
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Beam Pattern
10 Element Line Array – 3/2 spacing
77
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Beam Pattern Effects Beam Pointing
78
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Beam Pattern
10 Element Line Array – /2 spacing – 0 deg pointing
79
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Beam Pattern
10 Element Line Array – /2 spacing – 5 deg pointing
80
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Beam Pattern
10 Element Line Array – /2 spacing – 10 deg pointing
81
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Beam Pattern
10 Element Line Array – /2 spacing – 20 deg pointing
82
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Beam Pattern
10 Element Line Array – /2 spacing – 30 deg pointing
83
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Beam Pattern
10 Element Line Array – /2 spacing – 45 deg pointing
84
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Beam Pattern
10 Element Line Array – /2 spacing – 60 deg pointing
85
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Beam Pattern
10 Element Line Array – /2 spacing – 90 deg pointing
86
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Where is the unit direction vector in polar (azimuth – elevation ) coordinates,
an is the element weight of the nth element, and is the wavenumber.
Three-Dimensional Arrays
A three dimensional array has M transducers placed at the vector locations .
The beam pattern is given by:
Beamsteering to an arbitrary direction vector is done as in the line array,
by simply substituting for in the beam pattern function.
πf/ck 2
v
0v)vv( 0 v
87
Applied Research Laboratory
Decision Theory and Receiver Design
88
Applied Research Laboratory
Signal
Processor Decide
Noise
Signal Detection and Performance Estimation
Signal is present
or
Signal is not present
Noise
Signal (?)
• Problem: How should received signals be
processed in order to detect signals in noise?
What kind of detection performance can be
expected?
• The approach to solution:
• Must be statistical, since noise is involved
• Implement hypothesis testing 89
Applied Research Laboratory
Hypothesis Testing
• Possible Hypotheses:
– H0 : Only noise is present
– H1 : Signal is present in addition to noise
• Steps in forming hypotheses:
– Process array output to obtain a detection statistic x.
– Calculate the a posteriori probabilities P(H0|x)
and P(H1 |x).
– Pick the hypothesis whose probability is the highest:
the maximum a posteriori, or MAP estimate.
Choose
Choose
1
1
0
1
0
1
H
H
,
,
)x|H(P
)x|H(P
90
Applied Research Laboratory
Hypothesis Testing (Cont’d)
• Equivalently, we can use Bayes’ rule to write:
• P(H1) and P(H0) are called a priori probabilities
• Then the test can be written:
)H(P)H|x(P)x(P)x|H(P 111
Choose
Choose
1
1
0
1
00
11
0
1
H
H
,
,
)H(P)H|x(P
)H(P)H|x(P
)x|H(P
)x|H(P
)H(P)H|x(P)x(P)x|H(P 000
91
Applied Research Laboratory
Hypothesis Testing (Cont’d)
• An equivalent test is
• is called the likelihood ratio
0
1
0
1
1
0
0
1
Choose
Choose
H,)H(P
)H(P
H,)H(P
)H(P
)H|x(P
)H|x(P
)x()H|x(P
)H|x(P
0
1
92
Applied Research Laboratory
Aside: Bayes’ Rule and Notation
• Probability density functions are often used to describe
continuous random variables:
• Bayes’ Rule as written for probabilities also holds for probability
density functions (pdf).
• A compact notation is used in what follows:
• Likelihood ration test written in terms of probability density
functions
0
0
x
dx)x(p)x(P
)x(p)H|x(p 11 )x(p)H|x(p 00
0
1
0
1
1
0
0
1
Choose
Choose
H,)H(P
)H(P
H,)H(P
)H(P
)x(p
)x(p
93
Applied Research Laboratory
A First Example: Constant Signal
• The possible inputs are:
•
noise) plus (Signal
only) (Noise
1
0
)t(n)t(x:H
)t(n)t(x:H
:below shown as are (x)p) and (x)p)
then 0 and ddistribute Gaussian is If
0011 H|x(pH|x(p
,)t(n
2
22/12
1
2
22/12
0
2
)(exp)2()(
2exp)2()(
xxp
xxp
94
Applied Research Laboratory
• At time t, we receive a signal x(t). Knowing p0(x) and p1(x), we can calculate the likelihood ratio
and compare it to a threshold
and decide accordingly:
• Note that is the value of x at which (x) = 0 in the figure.
)x(p
)x(p)x(
0
1
00
10
Choose
Choose
H,
H,)x(
)H(P
)H(P
1
00
95
Applied Research Laboratory
Errors and Correct Decisions
• The possible errors are:
– False Alarm: We choose H1 when H0 is the right answer.
– False Dismissal: We choose H0 when H1 is the right answer.
• The possible correct decisions are:
– Detection: We choose H1 when it is the right answer.
– Correct Dismissal: We choose H0 when it is the right answer.
96
Applied Research Laboratory
Probabilities of Errors and Correct Decisions
dx)x(pP
dx)x(pP
FD
FA
1
0
dx)x(pP
dx)x(pP
CD
D
0
1
-
DFDFACD dx)x(pPPPP 1 because 1 :Note
Errors
Correct Decisions
97
Applied Research Laboratory
Neyman-Pearson Criterion
• Usually we don’t know P(H1) and P(H0) and thus cannot
calculate 0 from their ratio.
• Instead, we can specify a desired PFA, or false alarm rate, and
use it to obtain .
• Then we can calculate
or just compare x to directly.
yprobabilit alarm false specified 0 dx)x(pPFA
)(p
)(p
0
10
98
Applied Research Laboratory
Same Example: Multiple Samples
2
22/12
12
22/12
02
)(exp)2()( ,
2exp)2()( i
ii
i
xxp
xxp
For each sample xi=x(ti), the probabilities are:
99
Applied Research Laboratory
• If we have a set of M multiple, independent samples,
then their joint probability density functions under H1 and H0 are
and
,...x,xx 21
M
i
i/M xexp)()x(p
12
222
02
2
M
i
i/M xexp)(
12
222
22
M
i
i/M )x(exp)()x(p
12
222
12
2
Same Example: Multiple Samples (Cont’d)
100
Applied Research Laboratory
• The likelihood ratio becomes:
where is the mean value of the samples.
• Note that each xi is Gaussian with mean 0 under H0 or under
H1. Also, each xi has variance 2 under both H0 and H1.
• Then y is also Gaussian, with the same mean, but with variance
M
i
ixM
y1
1
2
2
21
2
22
0
1
22
My
Mexp
x)x(exp
)x(p
)x(p)x(
M
i
ii
M
2
Same Example: Multiple Samples (Cont’d)
101
Applied Research Laboratory
• y is a detection statistic (i.e. it is a sufficient statistic)
• Using the Neyman-Pearson criterion, the probability of a false
alarm
can be used to obtain a threshold for y.
• Note that using satisfies our intuition that the
receiver should counter the effects of noise by averaging the
samples.
M
i
ixM
y1
1
dy)y(pPFA 0
Same Example: Multiple Samples (Cont’d)
102
Applied Research Laboratory
Second Example: Arbitrary But Known Signal
• Possible receiver inputs are:
H0: x(t) = n(t) H1: x(t) = s(t) + n(t)
(Noise only) (Signal plus noise)
• If the signal is present, we know its shape exactly.
• Assume we have M samples in the interval (0,T). The
probabilities are:
)t(ss ii
M
i
i/M xexp)()x(pH
12
222
002
2 : Under
M
i
ii/M )sx(exp)()x(pH
12
222
112
2 : Under
103
Applied Research Laboratory
• The likelihood ratio is:
• The second term can be calculated before receiving the
samples.
• As we sample more finely in the interval (0,T), the summation
becomes the integral:
where Es is the energy in the signal s.
M
i
i
M
i
ii
M
i
iii ssxexpx)sx(
exp)x(p
)x(p)x(
1
2
21
21
2
22
0
1
2
11
2
T
s
M
i
i dt)t(ss0
2
1
2
104
Applied Research Laboratory
• The test statistic in this case is:
• Note that the received signal x(t) is being correlated with the signal
we are trying to detect s(t).
• Equivalently, we can filter x(t) using a filter with impulse response
function
h(t) = s(T - )
as can be seen from this equation:
• A filter whose impulse response function is matched to the signal in
this way is called a matched filter.
TM
i
ii dt)t(s)t(xsx)x(y01
ydt)t(x)t(sd)T(x)T(sd)T(x)(hT TT
0 00
105
Applied Research Laboratory
Test Statistic SNR
• We can define the SNR of y to be:
• The expected values of the test statistic y under H0 and H1 are
• The variance of y under H0 is (using the shorthand ):
s
T
T
dt)t(s))t(n)t(x(Ey)H|y(E
dt)t(s)t(xEy)H|y(E
0
11
0
00 0
T T
dtdntnstsEyyEy0 0
2
000 )()()()(])[()var(
00 H|yy
)var(
)]()([SNR
0
2
01
y
yEyEy
106
Applied Research Laboratory
• Let n(t) be Gaussian white noise with spectral level , i.e.:
• Then
• And so:
• As long as n(t) is white Gaussian noise (WGN), there is no other
receiver, i.e. no other test statistic y, which has a higher SNR. For
many other types of noise, the matched filter is optimal or near
optimal as well. This is why the matched filter is used.
2
2
0
0 0
00
s
T T
N
dtd)t()(s)t(sN
)yvar(
)(N
)(Rnn2
0
0
2
NSNR s
y
20N
107
Applied Research Laboratory
Third Example: Signal Known
Except Amplitude and Start Time
• This is the most common case, in which we are
– Looking for a target echo
– Listening for a radiated signal
• Exact arrival time and signal amplitude are
unknown. The hypotheses are:
• As before, T is the duration of s(t)
unknown) (
only) (Noise
001
0
a, t)t(n)tt(sa)t(x:H
)t(n)t(x:H
108
Applied Research Laboratory
• We apply the signal to a matched filter. under H1, the output
is
• The first term is the autocorrelation function as s at a lag of
t –T -t0. It is maximum when t = T+ t0, the time
corresponding to the end of the pulse arrival
• The second term is random due to the noise.
T
T
T
dtnTstTt
dtnttsaTs
dtxhty
00s
00
0
)()()(R a
)]()()[(
)()()(
109
Applied Research Laboratory
• Assume s(t) is a tone burst:
• The autocorrelation function is:
110
Applied Research Laboratory
• Autocorrelation function is written:
• Can get the envelope of Rs(p) by squaring and low-pass
filtering
• This is maximum when p = (t-T-t0) = 0 or t=T+t0.
• Thus the peak in [y2(t)]lpf occurs at t= T+t0 , and since we
know T, can get t0
otherwise 0
,22
0
,
Tp-T )tfcos()pT(aA
)p(Rs
222
2 )(8
)]([ pTaA
pR lpfs
111
Applied Research Laboratory
• Therefore, we define a new test statistic Z(t):
• The probability density functions of Z(t) under H0 and H1 are
shown by Burdic to be:
• Where and is the zero-order modified
Bessel function.
lpftytZ )]([ )( 2
21
20221
02
220
-S- 1
2 -
1
yyy
sy
yy
zSzI
zexp)z(p
N,
zexp)z(p
yS SNR )(I 0
112
Applied Research Laboratory
• The probability density functions are plotted below
• Can use the Neyman-Pearson criterion to get , then calculate
PD
dz)z(pPFA 0
113
Applied Research Laboratory
Fourth Example: Possible Doppler Shift
• Non-zero radial motion between a transmitter (or reflector)
and receiver causes the frequency of the received signal to
be shifted relative to the transmitted signal. This is called
Doppler Shift.
• This complication is usually met by implementing a parallel
bank of filters (or FFT), each matched to a different
frequency.
y1
yl
yL
114
Applied Research Laboratory
Passive Broadband Detection
Want to detect targets with broadband signatures:
Assume we know the ambient noise power spectrum
115
Applied Research Laboratory
Passive Broadband Detection (Cont’d)
• Use the receiver shown below, where h1(t) and h2 (t) are
filters whose impulse functions need to be determined.
116
Applied Research Laboratory
Passive Broadband Detection (cont.)
• It has been shown that the Eckart Filter is optimal for h1(t):
• Note: when the noise is white, H1(f) looks like s(f).
Otherwise, H1(f) is minimized when n (f) is large
• The power spectrum of y under H0 and H1 is then:
FilterEckart 2
2
1)f(
)f()f(H
n
s
dff
f
dff
f
dff
dfff
and
f
f
f
ffHfff
f
ffHff
n
s
n
s
y
yy
y
n
s
n
snsy
n
sny
)(
)(
)(
)(
)(
)]()([
SNR
)(
)(
)(
)()())()(()(
)(
)()()()(
2
2
2
22
1
2
1
0
01
1
0
117
Applied Research Laboratory
Passive Broadband Detection (cont.)
• Burdic shows that the SNR of the output of the envelope detector
is
• The commonly-used post detection filter is an averager whose
duration is as long as possible,
• The product of is typically large, where is the effective
noise bandwidth at the output of the pre-detection filter h1( ), i.e.
is the width of a rectangular filter which admits the same noise
power. The frequency domain expression for is derived by
Burdic in section 8-4 to be
2SNRSNR 2 yy
otherwise 0
22
1
2
,
TT,
T)t(h
2
4
1
2
2
1
)()(
])()([
dffHf
dffHf
n
n
118
Applied Research Laboratory
Passive Broadband Detection (cont.)
• Using the Eckert Filter
• Given large T , Burdic shows that the SNR at the averager
output is
• Using the expressions for SNRy and
• Note the effect on SNRz of increasing T.
2
2
2
df)f(
)f(
df)f(
)f(
n
s
n
s
2SNRSNRSNR 2 yyz TT
df)f(
)f(T
df)f(
)f(
df)f(
)f(
df)f(
)f(
df)f(
)f(
Tn
s
n
s
n
s
n
s
n
s
z 2
2
2
2
2
22
2
2SNR
119
Applied Research Laboratory
Passive Narrowband Detection
• Want to detect targets that emit pure tone signatures:
• Receiver is shown below (essentially a spectrum analyzer)
120
Applied Research Laboratory
Passive Narrowband Detection (Cont’d)
• Typically implemented by Fournier transforming the input
signal. Second filter is an integrator (“averager”). Long
averages are usually employed, so that T >>1.
aroundConstant :Spectrum Noise
otherwise 0
22- 1
(f)H :Filter
:Spectrum Signal
0n
0
2
1
0
2
f)f(
,
ff,
)ff(a)f(
:If
s
121
Applied Research Laboratory
• Then
• As before, the SNR of the test statistic Z is
• Putting these together
)f(
a
n
y
0
2
SNR
2SNR SNR yz T
2
0
2
SNR)f(
aT
n
z
Passive Narrowband Detection (Cont’d)
122
Applied Research Laboratory
Receiver Operating Characteristic Curves
• ROC curves graphically display the relationship between test
statistic SNR, PFA , and PD . Each receiver has its own ROC curve.
The one below is for large time-bandwidth products.
123
Applied Research Laboratory
Signal Parameter Estimation
• Now consider the case where we want to estimate
- arrival time
- signal frequency, f
- signal arrival angle,
of a signal s with energy Es
• We want unbiased estimators. For example, if 0 is the true
arrival time, and is the estimate, then is unbiased if
• The other desirable feature of an estimator is that it have
variance that decreases asymptotically with increasing
SNR. Low variance means small error.
ˆ
0)ˆ(E
ˆ
124
Applied Research Laboratory
Minimum Bounds on Variances
• It can be shown that the minimum possible (Cramer-Rao)
variance of the time delay estimate is
• Where is the mean-square bandwidth of the signal
• Note that the variance of the time delay estimate is reduced
(which is good) by increasing signal bandwidth and input
SNR
2
0
2
1
ss
N
)ˆvar(
2
s
125
Applied Research Laboratory
• Similarly, the Cramer-Rao lower bound for frequency estimates is
Where is the mean-square signal duration.
• The variance of the frequency estimate is reduced by increasing
signal duration and input SNR.
2
0
0
2
1
TN
)f̂var(s
2
0T
Minimum Bounds on Variances (Cont’d)
126
Applied Research Laboratory
• Finally , the Cramer-Rao lower bound for arrival angle estimates
for a line array of length L is
where is the wavelength of the signal: =c/f, where c is the
speed of sound and f is the frequency.
• The variance of the arrival angle estimates is reduced by
increasing the aperture size and by higher input SNR.
3
32
0 2
2
1
L
N
)ˆvar(s
Minimum Bounds on Variances (Cont’d)
127
Applied Research Laboratory
Additional Material
128
Applied Research Laboratory
Spectra Of Real Signals
• In general, the Fourier transform of any signal is a
complex function of a real variable, the frequency.
• The Fourier transform of a real signal is conjugate
symmetric: S*(f) = S(-f)
• Because of this, spectra of real signals have negative
frequency components that are equal in magnitude to the
positive frequency components, but opposite in phase.
• Thus, all spectral information about a real signal is
contained in the positive frequencies. Negative
frequency components are redundant!
• To fully take advantage of this in signal processing, need
the concept of analytic signals. 129
Applied Research Laboratory
Analytic Signals
• Real signals are often represented, for analysis, as the real part of a
complex signal:
• Often, the variation of the signal near its center frequency is of interest.
Can rewrite the complex signal as:
• This representation conveniently encapsulates information about the
center frequency, amplitude and phase of a signal. This representation is
a special case of an analytic signal.
• If the spectrum of the complex envelope is confined to a narrow band of
frequencies, then the spectrum of f(t) is confined to positive frequencies.
)]}(exp[{)cos()( 00 tiAtAtx e
tii eAetiAtf 0 )](exp[)( 0
Complex Envelope
Center frequency
0
Spectrum of
analytic time signal
0
Basebanded spectrum
of envelope
130
Applied Research Laboratory
Properties of Analytic Signals
• Whereas real signals have both positive and negative frequency components, the analytic signal representation only has the positive set.
• Analytic signals are convenient for use in modern complex array processors and are well suited to Fourier analysis.
• Drawback: real and imaginary, or “in-phase” and “quadrature” samples must be taken.
• This is more than made up for in simplicity of processing and bandwidth savings in other parts of processing stream.
131
Applied Research Laboratory
Processing of Analytic Signals
• To create an analytic signal representation sa(t) of a signal s(t), it is necessary to create a fictitious complex signal.
• The real part of the analytic signal is the real signal s(t). The imaginary part of the analytic signal is called the Hilbert transform of s(t).
• For narrowband signals, samples of the Hilbert transform of s(t) can be obtained by sampling the signal again 90° out-of-phase of the “real” samples.
• The analytic signal can be conveniently digitally shifted in frequency and subsampled to further decrease bandwidth requirements.
• Only the information-containing part of the signal need be processed, not the negative frequencies or the empty space down to DC. 132
Applied Research Laboratory
Sampling with Analytic Signal Output
Sample Pair
Outputs
Real + Imaginary
Real Signal
Input Band Pass
Filter
Sample
Clock
fs>BW
A/D
period
delay
I R I R I R
time
delay => 90° at fc
period = 1/fs
Analytic sampling can also be accomplished with phase shifters 133
Applied Research Laboratory
Array Gain: Discrete Line Array
).L),sin(KL
,sinK)(N B
B
10( 090
900
:Model Noise
00
00
134
Applied Research Laboratory
Array Gain: Discrete Line Array (Cont’d)
Nine-element vertical line array gain versus elevation steering
angle in a surface-generated ambient noise field 135
Applied Research Laboratory
Array Performance Analysis
)r(T
d),f(G),f(
log)f(S
)r(T:r
),f(log),f(SL
:f
),f(
ref
arrays
array
ref
s
s
array
4
2
10
:array the by received level Signal
rangeat loss onTransmissi
10
direction , frequencyat level Source
:source theat density angular / spectrum power Signal
effects. bandwidth filter and signal
include and equation sonar the of form the in SNR writeto Want
136
Applied Research Laboratory
Array Performance Analysis (Cont’d)
.r,r
)r(Tlog)r(TL
log)r(TL)f(SL
)r(T
)f(logS
,f)f(G
)r(T
df)f(G)f(
log)f(S
)r(T
)f(G)f(log)f(S
ref
ref
c
ref
csarray
carray
ref
arrays
array
ref
arrays
array
1 10
where
10
10
:Then . width
withat centered and rrectangula is Assume
10
:is power signal received total The
10
Then source. the of
direction the in steered array the withsignal, waveplane a Assume
2
2
2
137
Applied Research Laboratory
Array Performance Analysis (Cont’d)
ref
array
array
ref
array
array
ref
i
dfd)f,(G)f,(N
logNL
d)f,(G)f,(N
log)f(NL
:f
d)f,(N
log)f(NL
f
)f,(N
f
4
2
4
2
4
10
:array the by received power noiseambient Total
10
frequencyat array by received level spectral noise Ambient
10
:array the ofinput theat frequencyat level spectral noise Ambient
: direction , frequencyat density ngularspectrum/a power noise Ambient
138
Applied Research Laboratory
Array Performance Analysis (Cont’d)
SNR. increases gain array higherthat Note
Then
10
writecan we, the Using
1010
is gain array thethat Recall
10
:Then . width with,at
centered filter, pass) (band rrectangula a is that assumeNow
4
2
4
4
2
2
)f(AG)f(NL)r(TL)f(SL
NLSSNR
log)f(AG)f(NLNL
AG
)f(NLlog)f(NLd)f,(G)f,(N
d)f,(N
log)f(AG
d)(G)f,(N
logNL
f
)f,(G
ccic
arrayarray
cciarray
carrayci
carrayc
c
c
ref
c
array
c
array
139
Applied Research Laboratory
Directivity Index For A Discrete Line Array
DI for a discrete line array, broadside and endfire with N=9.
.resolution ldirectiona poorer hence beamwidth, broaderbut DI, higher has
beam fire End octave. per 3dBabout at drops When
ambiguity. cause
lobes grating However, 10about oscillates When
10 1 When
2 at which frequency the is
weightingUniform
0
0
0
0
0
DI,ff
.NlogDI,ff
NlogDI),f
f(ff
/df
140
Applied Research Laboratory
024
204
022
2
xL/),/Lx(g(x)g
L/x),/Lx(g(x)g
otherwise,
Lx
L,
L)x(g
L
R
:are functions aperture The
Effect of translation of
the beam pattern
Target Angle Estimation Using Split Apertures
141
Applied Research Laboratory
Target Angle Estimation Using Split Apertures (Cont’d)
)u(G)/Lujexp()u(G
)u(G)/Lujexp(
dye)y(g)/Lujexp(
dxe)/Lx(g)u(G
)/Lu(csinLu/
)Lu/(sin
dxe)x(g)u(G
L
uyj/L
/L
uxj/L
R
uxj/L
/L
2π
2π
2π
4
22π
2π
π24
4
π22
0
π24
4
:similarly And
:And
:are functions Pattern
142
Applied Research Laboratory
Target Angle Estimation Using Split Apertures (Cont’d)
signals. two the between difference phase the is
:similarly And
:f frequency and angleat waveplane a For
:be to shown previously werearrays the ofout signals The
00
Luj
)/Lujtfjexp()c
sinf(G)u(s
)/Lujtfjexp()c
sinf(G
e)c
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Target Angle Estimation Using Split Apertures (Cont’d)
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Applied Research Laboratory
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Split Aperture Mechanization
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Examples
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Example : Passive Broadband
• Given T = 1000
• Find: The required SNRy such that PD = 0.9 and PFA = 10 -4
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Applied Research Laboratory
• Solution: from the ROC curve PD and PFA cross at SNRz = 25, using
the relationship between SNRy and SNRz derived earlier for passive
broadband detection
Or
Or
158.01000
25SNR
21
y
22 SNR1000SNR SNR yyz T
dBy 0.8)SNRlog(10
Example : Passive Broadband (Cont’d)
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System Performance Analysis
Narrowband Passive Detection
• A narrowband passive receiver looks for target radiated tones. It
is essentially a spectrum analyzer
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• At the output of the array, we saw that the noise and signal
levels were
• There fore, at the output of the filter centered at f0, the SNR
in dB is
• For the narrowband passive receiver, we saw that
Or in dB
2
1
ZSNRSNR
Ty
)at tone pure a (for ),()()(
log10)()()(
0000array
array
ffrTLfSLfSL
fAGfNLfNL s
log10)()(),()()SNRlog(10 0000 fAGfNLfrTLfSL sy
log5log5)SNRlog(5)SNRlog(10 TZy
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• Putting these together yields
This equation relates the test statistic z SNR, the averaging time
T, the filter bandwidth , the source level SL, the transmission
loss TL, the ambient noise level NLs, and the array gain AG.
• We are usually given an allowable false alarm time FA. Since
there are N channels, and 1 output each T seconds, there are
N/T false alarms opportunities per second. The number of false
alarms per second is then
This equation can be solved for PFA. Then, using PD, we can get
required SNRz.
FA
FAT
NP
1
)()(),()(log5log5)SNRlog(5 0000 fAGfNLfrTLfSLT sZ
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• Given:
Target Signal: Pure sinusoid at f1 = 500 Hz with source level
SL = 125 dB// Pa@1m.
Ambient noise level: NLs = +65 dB// PA in 1 Hz band at 500 Hz.
(Corresponds approximately to Sea State 3.)
Spatial Filter: Horizontal line array with L = 100 ft.
Predetection processor: Bank of contiguous narrowband filters
with = 0.5 Hz covering a 500 Hz band centered at 500 Hz.
Postdetection processor: Linear integrators with T = 100 sec.
Desired system : 1000 sec.
• Find: The detection range for PD = 50%
Example : Passive Narrowband
FA
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• Solution:
From the previous slide,
• Since T = 100·0.5 = 50, use the ROC curve for large T and
PD = 50% to get d = 14 = SNR
• The noise is isotropic, so AG=DI and DI is found by Burdic to
be
41010001000
100
FA
FAN
TP
Example : Passive Narrowband (Cont’d)
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f
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LlogDI 10
500
5000 13
10
100210
210
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Applied Research Laboratory
• The use the equation derived earlier, solving for TL
• Assuming spherical spreading with no absorption, the
detection range is 8,700 meters.
dB
TDIfNLfSLfrTL zS
8.78
5.
100log5)14log(101365125
log5SNR5)()(),( 000
Example : Passive Narrowband (Cont’d)
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Aside: Processing Gain Against Reverberation
• The spectrum of reverberation is very similar to that of the
transmitted signal spectrum.
• For windowed tones, the spectrum is heavily concentrated
around the transmit frequency (zero doppler shift). This is
called the “reverb ridge”.
• So is the spectrum of the echo from a non-moving target. And
target echo strength falls off faster than reverb strength
=> Echoes of windowed tones from low-doppler targets are
easily drowned out by reverberation.
• When the target is moving, the spectrum is shifted away from
the concentrated reverb spectrum at zero doppler.
• It is easier to detect high doppler targets using tone pulses.
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Applied Research Laboratory
Active Sonar
0 Doppler (Reverberation Limited) Target
• Assume that sound backscattered by the ocean surface (called
reverberation) causes the noise or interference, against which
we are trying to detect a target echo.
• The receiver for this example is simply one channel of the
system shown in the figure on Slide 114. 156
Applied Research Laboratory
• The target echo level (in dB) is
• Where TS is the target strength:
TSrTLSLEL )(2
PowerIncident
Power Scattered StrengthTarget TS
Active Sonar: Echo Level
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Applied Research Laboratory
• The reverberation level is
• Here B is the effective, two-way azimuthal
beamwidth, and
is the area of the surface which is insonified,
called the scattering patch.
2log10))(()(2)( B
gs
rcrSrTLSLrRL
Brc
2
Active Sonar: Reverberation Level
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Applied Research Laboratory
• The ocean surface scatters sound back toward the receiver. The
amount scattered depends upon the grazing angle g, the surface
condition, and the size of the insonified area. We have models for
the scattered energy per unit surface area:
area surfaceunit
power scattered strength scattering surfacesS
Active Sonar: Surface Scattering Strength
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Applied Research Laboratory
• Using the equations developed for EL and RL, the
target echo (desired signal) to reverberation
(unwanted interference) level is called the signal
to interference (SIR) level:
• Note that source level and transmission loss do
not matter
2log10))(( B
gs
rcrSTSRLELSIR
Active Sonar: Signal to Interference Ratio
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Applied Research Laboratory
• Given:
• Find:
The detection range for PD = .5 and PFA = 10 -4
Active Sonar: Example
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Applied Research Laboratory
• From Urick, Table 8.1 we get
From which B = 0.156 radians.
• The target echo and reverberation both fill the 0 Doppler
frequency bin. The time-bandwidth product is 1. We get no
advantage from a post-detection filter. Therefore,
where y is the filter output and z is the integrator output.
69.10 or ,9.6log10)log(10 DD
BB
2
Z SNRSNR y
Active Sonar: Example (Cont’d)
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Applied Research Laboratory
• For PD = 0.5 and PFA = 10-4, the graph below gives the
required SNRz versus T .
• Given T = 1, we get that SNRz needs to be about 75.
Active Sonar: Example (Cont’d)
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Applied Research Laboratory
• Detection is achieved (with PD = 50% and PFA = 10 -4) when
• Using the equation derived for SIR and solving for r yields
• Performance will be improved by decreasing or B, but they
must be large enough for the target to be insonified
)SNRlog(10SIR yRLEL
.1000
or
9.294.92
)156.0()1.0(1500log10)40(10
)SNRlog(102
log10)log(10
mr
dB
cSTSr y
Bs
Active Sonar: Example (Cont’d)
164