Sonar Signal Processing I - Pennsylvania State · PDF file• Richard O. Nielsen, Sonar...

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


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)

2


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

3


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

4


• 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

5


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

6


Mathematical Tools

7


• 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


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

9


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

10


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

11


Time/Frequency Signal Processing

12


• 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

13


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


• 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

15


• 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

16


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

17


• 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

18


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


• 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

20


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

21


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

22


• 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


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


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


• 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

26


• 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

27


• 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


Spatial Signal Processing

(Beamforming)

29


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.

30


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

31


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

32


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


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


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


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


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


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


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


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



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



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


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


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


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


Rectangular Aperture Function and Pattern Function

46


Array Main Lobe Width (Beamwidth)

2

1

2

ψ2

3dBG

3 dB (Half-Power) Beamwidth

To find it, solve

for dB3ψ

47


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


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


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


Pattern Function For Uniform Discrete Line Array

51


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

52


• 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


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


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


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


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-

57


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


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


Beam Pattern Effects Number of Elements (Array Size)

60


Beam Pattern

1 Element Line Array – /2 spacing

61


Beam Pattern


62


Beam Pattern


63


Beam Pattern


64


Beam Pattern


65


Beam Pattern


66


Beam Pattern


67


Beam Pattern


68


Beam Pattern


69


Beam Pattern


70


Beam Pattern Effects Element Spacing (Sampling)

71


Beam Pattern


72


Beam Pattern


73


Beam Pattern


74


Beam Pattern

10 Element Line Array – 3/4 spacing

75


Beam Pattern

10 Element Line Array – spacing

76


Beam Pattern

10 Element Line Array – 3/2 spacing

77


Beam Pattern Effects Beam Pointing

78


Beam Pattern

10 Element Line Array – /2 spacing – 0 deg pointing

79


Beam Pattern


80


Beam Pattern


81


Beam Pattern


82


Beam Pattern


83


Beam Pattern


84


Beam Pattern


85


Beam Pattern


86


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


Decision Theory and Receiver Design

88


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


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


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


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


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


• 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


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


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


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


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


• 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


• 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


101


• 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


102


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


• 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


• 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


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


• 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


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


• 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


• Assume s(t) is a tone burst:

• The autocorrelation function is:

110


• 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


• 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


• The probability density functions are plotted below

• Can use the Neyman-Pearson criterion to get , then calculate

PD

dz)z(pPFA 0

113


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


Passive Broadband Detection

Want to detect targets with broadband signatures:

Assume we know the ambient noise power spectrum

115


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


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



• 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



• 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


Passive Narrowband Detection

• Want to detect targets that emit pure tone signatures:

• Receiver is shown below (essentially a spectrum analyzer)

120


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


• 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


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


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


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


• 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


• 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


Additional Material

128


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


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


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


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


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


Array Gain: Discrete Line Array

).L),sin(KL

,sinK)(N B

B

10( 090

900

:Model Noise

00

00

134


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


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


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



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



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


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


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


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



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

sinf(G)t(s

)ff()f(S

dfe)c

sinf(G)f(S

dx)c

sinxt(s)x(g)t(s

L

tfj

RR

ftj

R

RR

π

2ππ2ψ

2ππ2ψ

ψ

δ

ψ

ψ

ψ

000

000

π200

0

π2

0

143



)}()(Re{

)}()(Im{tansin

for solve And

)sin

tan()}()(Re{

)}()(Im{

compute , angle the find To

)sin

sin()sin

cos()sin

(

)sin

exp()sin

()()(

:together signals two the multiply ,difference phase theget To . angle arrival the estimate to

arraysright andleft the ofout coming signals the between difference phase the use We

11

0

:0

0

0

00

2

00

0

2

00

0

tsts

tsts

L

L

tsts

tsts

Lj

L

c

fG

Lj

c

fGtsts

RL

RL

RL

RL

RL

144


)2/2cos()( and )2/2cos()( are signalsinput The

signals. two the between

difference phase the is wheresin angle arrival an estimate to

to outputs array aperture-split processing for ionmechanizat the isBelow

1

0

fttsftts

L

LR

Split Aperture Mechanization

145


Examples

146


Example : Passive Broadband

• Given T = 1000

• Find: The required SNRy such that PD = 0.9 and PFA = 10 -4

147


• 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)

148


System Performance Analysis

Narrowband Passive Detection

• A narrowband passive receiver looks for target radiated tones. It

is essentially a spectrum analyzer

149


• 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

150


• 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

151


• 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

152


• 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)

'

f

cdlog

LlogDI 10

500

5000 13

10

100210

210

153


• 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)

154


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.

155


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


• The target echo level (in dB) is

• Where TS is the target strength:

TSrTLSLEL )(2

PowerIncident

Power Scattered StrengthTarget TS

Active Sonar: Echo Level

157


• 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

158


• 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

159


• 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

160


• Given:

• Find:

The detection range for PD = .5 and PFA = 10 -4

Active Sonar: Example

161


• 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)

162


• 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.


163


• 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


164

Sonar Signal Processing I - Pennsylvania State · PDF file• Richard O. Nielsen, Sonar...

Documents

Transcript of Sonar Signal Processing I - Pennsylvania State · PDF file• Richard O. Nielsen, Sonar...