fundamentals of passive and active sonar technical training short course sampler

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2REVERBERATION

TARGET

TRANSMISSION

DELAY ANDATTENUATION

TIME-VARYINGMULTIPATH

AMBIENT NOISE

ACTIVE SONAR DETECTION MODEL

TIME-VARYING MULTIPATH

RECEPTION

PRE-DETECTIONFILTER

DETECTOR

POST-DETECTIONPROCESSOR

DECISION: H1/H0 ?

DEFINITIONS

Acoustic pressure, p: The difference between the total pressure, ptotal, and the hydrostatic (or undisturbed) pressure po.

ptotal = ptotal(x,y,z;t)

po = po(x,y,z)p = p (x,y,z;t) = ptotal - po

Pascals or Newtons/(meter)2

A vector whose component in the direction of any unit vector is the rate at which energy is being transported in the direction across a small plane element perpendicular to , per unit area of that plane element.

Acoustic intensity, I

t);z,y(x,II

= Watts/(Meter)2

≡ where is the fluid velocity at (x,y,z;t). u

ACOUSTIC INTENSITY LEVELS EXPRESSED IN DECIBELS

)ref10 averagelog10 I|I| /(

)ref10 averagelog I|I| /(

refx II 610+= [ Watts/(meter2) ]

Intensity level, I, usually refers to a normalized intensity magnitude, ( /Iref), expressed in decibels. For example, I = 60 dB means

PRESSURE1 Pascal = 1 Newton/(1 Meter)2

ENERGY1 Joule = 1 Newton x 1 Meter

POWER1 Watt = 1 Joule/(1Second)

INTENSITY (or ENERGY FLUX) Watts/(Meter)2

Joules/Second /(Meter)2

average|I|

refref

rmsp waveplane a For c

2)(I , ρ=

pressure square-mean-Root prms =

Density=ρvelocitySonic =c

)26 secondkgm/(meter 10x1.5water+= cρ

Pascals10 p 6-waterref ,rms =

Pascals10x20 p 6-airref ,rms =

)2secondkgm/(meter 430air =cρ

pressureacoustic of square-mean-Root prms =

Density=ρ

velocitySonic =c

)26 secondkgm/(meter 10x1.5 += cρ

218 (meter)/Watts10x0.67 −=water 2waterref [/ Pascals][10 6 ]I cρ−≡

)2secondkgm/(meter 430 =cρ

212 (meter)/Watts10−≈air Pascals]10 x 20 [ [ 2

air fre /6 ]I cρ−≡

For water:

For air:

ACOUSTIC INTENSITY REFERENCE VALUES: AIR AND WATER

magnitudeintensity acoustic Average c)(

|I| ρrmsp

waveplane a for =

waterrefwaterref |I|I

airrefairref |I|I

10log10( )= X watts/meter2

6.76 x 10-19 watts/meter2

X watts/meter2

I ref water watts/meter2 10log10( )=

10log10( )= Y watts/meter2

10-12 watts/meter2

100 dB air re 20 x 10-6 Pa 10log10( )=Y watts/meter2

I ref air watts/meter2

For water: X watts/meter2 = 6.76 x 10-9 watts/meter2

For air: Y watts/meter2 = 10-2 watts/meter2

10log10( )10-2 watts/meter2

6.76 x 10-9 watts/meter2 = 61.7 dB difference in intensity level.

100 dB water re 10-6 Pa

DECIBELS MEASURED IN AIR AND WATER

A cross-correlation sequence can be calculated for any pair of causal sequences x[k] and y[k], and there is no implied restriction on their relationship:

x[k]=y[k], i.e, auto-correlation,

x[k] may be the result of applying a Doppler shift to y[k],

x[k] is the result of applying a (fixed) shift n to y[k], i.e., x[k]=y[k-n], or

x[k] = A y[k-n], where A is a constant and the same for all k. When this is the case, it is helpful to think of x[k] and y[k] as structurally matched or simply matched.

The term ‘replica correlator’ or ‘replica correlation’, is applied when there is some relationship between the input x[k] and the replica y]k], for example:

g[n]n]-y[kx[k]n]y;xcorr[x,k

k≡= ∑

−∞=where g[n] is used to denote a cross-correlation sequence whose constituent sequences are understood but not explicitly stated.

REPLICA CORRELATOR

y[k-n]

g[n]∑+∞=

−∞=

kn]-y[kx[k]

oREPLICA CORRELATION (1 of 6)

REPLICA

OUTPUT

INPUT (SIGNAL)

y[k],REPLICA

12 3 4

y[k-n],INCREASES

SHIFTED REPLICA

12 3 4

y[k-k(1)],

SHIFTED REPLICA ALIGNED

WITH INPUT

0 1 2 3

k(1), FIXED BULK DELAY

NON-ZERO VALUES OF x[k]

= 4k(2)

k(1) + k(2)

n + k(2)

= k(1)

k(1) k(1) + k(2)

Each n > 0 produces a different (shifted)

y[k-n] sequence.

NON-ZERO VALUES OF y[k]

= 4k(2)

REPLICA CORRELATION (2 of 6)

12 3 4

110200

200110 40

1 2 3 4

SHIFTED REPLICA

The cross-correlation g[n] of x[k[ and y[k] is the output of the replica correlator, and

]ny[kx[k]g[n](2)(1) kkk

−∑+≥

= where n is the shift of y[k] with respect to x[k].

FIXEDk(1)

INCREASESn

k(1) k(1) + k(2)

y[k-n],

= 4k(2)

k(1) k(1) + k(2)

n = k(1)

n = k(1)-k(2) n = k(1)+k(2)

y[k-n] when n=k(1)

n + k(2)n

INPUT (SIGNAL)

OUTPUT OF REPLICA

CORRELATOR

Instead of writing

]ny[kx[k]g[n](2)(1) kkk

−∑+≥

we can write

∑∞+=

∞−=−=

k]ny[kx[k]g[n]

Since x[k] = 0 if k < 0 or k > k(1) + k(2).

FIXEDk(1)

k(1) k(1) + k(2)

= 4k(2)

INPUT (SIGNAL)

REPLICA CORRELATOR

y[k-n]

g[n]∑+∞=

−∞=

kn]-y[kx[k]INPUT

REPLICA

OUTPUTSIGNAL

+NOISE

SIGNAL +

The instantaneous power output of the replica correlator |g[n]|2 is calculated for each n and is the detection statistic, i.e., the statistic used to decide if a signal matching, or nearly matching, the replica is embedded in the noise.

A signal is declared to be present if | g[n] |2 exceeds a threshold:

2]ny[kx[k]g[n] |||| ∑

∞−=−=

SIGNAL+NOISEREPLICA

|g[n]|2

INSTANTANEOUS POWER OUTPUT OF REPLICA

CORRELATOR,SIGNAL + NOISE

BULK DELAY, k(1)

n = k(1)

THRESHOLD

REPLICA CORRELATOR

y[k-n]

g[n]∑+∞=

−∞=

kn]-y[kx[k]INPUT

REPLICA

OUTPUTSIGNAL

+NOISE

SIGNAL +

2]ny[kx[k]g[n] |||| ∑

∞−=−=

SIGNAL+NOISEREPLICA

|g[n]|2

INSTANTANEOUS POWER OUTPUT OF REPLICA

CORRELATOR,SIGNAL + NOISE

BULK DELAY, k(1)

n = k(1)

THRESHOLD

If zero-mean, statistically independent Gaussian noise masks the input signal, cross-correlation of the received data with a replica matching the signal is the optimum receiver structure.

s(t-τRD)t

tt = 0

t)()( tnts +− BDτ

BDτ=t

τt (1)RD=

REPLICA CORRELATION FOR A CONTINUOUS CW WAVEFORM (1 of 4)

TRANSMISSION

AMBIENT NOISE

REPLICAS OF THE TRANSMISSION FOR

DIFFERENT REPLICA DELAYS

RECEIVED DATA

)( BDτ−ts

BULK DELAY

REPLICA DELAYS

τt (2)RD=s(t-τRD)(2)

tt = 0

tREPLICAS OF THE

TRANSMISSION FOR DIFFERENT REPLICA DELAYS

RECEIVED DATA

Shifted replicas, each with the same shape but different delays,

τRD, i=1,2, …, n, are under consideration,

Only one record of the received data is under consideration.

)()( tnts +− BDτ

s(t-τRD)REPLICA DELAYS

s(t-τRD)(2)

BDτ=t

τt (1)RD=

τt (2)RD=

Time delay, τd

Time, t

Replica correlator outputand its envelopefor a CW waveform

Output of the replica correlator is the product of the echo and the replica for time delays τd.

REPLICA OF THE TRANSMISSION FOR

SOME REPLICA DELAY

Note triangular shape of upper

peaks.Peak value is at τ d= 0

INCREASING REPLICA DELAY

τd = τBD - τRD

τd = 0

−τd

Time, t

Replica correlator outputfor a CW waveform plus ambient noise.

REPLICA OF THE TRANSMISSION FOR

INCREASING REPLICA DELAY

Time delay, τd

τd = 0

−τd

τRDSOME REPLICA DELAY

0 200 400 600 800 1000 1200-15

Independent normally distributed noise only.

Normally distributed noise plus sine wave.

Sine wave 240 samples long with zero-padding on each side.

Replica of sine wave(offset by -5).

-600 -400 -200 0 200 400 600-150

TIME DELAY τd IN DATA SAMPLES

REPLICA CORRELATOR OUTPUT FOR A CW WAVEFORM MASKED BY NOISE (1 of 3)

Sine wave amplitudeσNOISE = 4.0

Sine wave powerNoise power = 32

10 log( ) = -15 dB321

SAMPLED DATA POINTS

0 500 1000 1500 2000 2500-15

Independentnormally distributed noise only.

Replica of sine wave (offset -5).

Sine wave 480 samples long with zero-padding on each side.Normally distributed noise plus sine wave.

0 10005000-300

-500-1000

Sine wave powerNoise power

=> -15 dB

As on previous slide,

SAMPLED DATA POINTS

0 500 1000 1500 2000 2500-15

Independentnormally distributed noise only.

Replica of sine wave (offset -5).

Sine wave 480 samples long with zero-padding on each side.Normally distributed noise plus sine wave.

0 10005000-300

-500-1000

Sine wave powerNoise power

=> -15 dB

As on previous slide,

SAMPLED DATA POINTS

Note triangular

INDEPENDENT NORMALLY DISTRIBUTED NOISE, σ = 1

0 100 200 300 400 500 600 700 800 900 1000-4-3-2-10123

SAMPLED DATA POINTS

0 100 200 300 400 500 600 700 800 900 1000-50-40-30-20-10

RANDOM WALK (SUM OF NOISE VALUES UP TO N DATA POINTS)

NUMBER OF STEPS, N

N INCREASING N

RANDOM WALK SAMPLE AND A GENERAL RESULT (1 of 2)

Root-mean-square departure from starting position is . N

0 100 200 300 400 500 600 700 800 900 1000-4-3-2-10123

0 100 200 300 400 500 600 700 800 900 1000-1

1REPLICA

0 100 200 300 400 500 600 700 800 900 1000-2-1012

POINT-BY-POINT PRODUCTS OF REPLICA AND NOISE

SAMPLED DATA POINTS

NOISE SAMPLE MULTIPLIED BY A REPLICA

INDEPENDENT NORMALLY DISTRIBUTED NOISE, σ = 1

POINT-BY-POINT PRODUCTS OF REPLICA AND NOISE(FROM PREVIOUS SLIDE)

0 100 200 300 400 500 600 700 800 900 1000-35-30-25-20-15-10-505

N, NUMBER OF PRODUCTS SUMMED

SUM OF ABOVE PRODUCT VALUES OUT TO N POINTS

N INCREASING N

0 100 200 300 400 500 600 700 800 900 1000-2-1012

SAMPLED DATA POINTS

RANDOM WALK SAMPLE AND A GENERAL RESULT (2 of 2)

Root-mean-square departure from starting position (zero) is proportional to . N

LENGTH OF REPLICA CORRELATION IN SAMPLED DATA POINTS, N

Peak replica correlator output with matched inputs is (nearly) proportional to N, not .

REPLICAS OF INCREASING DURATION

0 100 200 300 400 500 600 700 800 900 1000-1

1 ECHOES OF INCREASING DURATION

SAMPLED DATA POINTS

0 100 200 300 400 500 600 700 800 900 10000100200300400500

0 100 200 300 400 500 600 700 800 900 1000-1

SAMPLED DATA POINTS

PEAK REPLICA CORRELATOR OUTPUT WHEN ECHO MATCHES REPLICA

If the number of ‘matched’ sampled data points in a received signal and replica is N, and if the noise masking the signal is uncorrelated from sample-to-sample, then:

2) The peak output of the replica correlator due to

1) The expectation of the root-mean-square noise

3) The ratio of the peak signal output to the root-mean-square

noise output is expected to increase as ,

4) The peak signal-to-noise instantaneous power output of

the replica correlator is expected to increase as .NNN

output of the replica correlator increases as ,

SUMMARY

the signal increases (nearly) linearly with N.

25τd = 0

HYPOTHETICAL TRANSMISSION, RECEIVED DATA, AND INSTANTANEOUS POWER OUTPUT OF A REPLICA CORRELATOR

OUTPUT OF REPLICA CORRELATOR

INSTANTANEOUS POWER OUTPUT OF REPLICA CORRELATOR AND ITS PEAK

POWER ENVELOPE

RECEIVED DATA,SIGNAL MATCHING REPLICA PLUS NOISE

TIME DELAY

TIME DE;AY

N SAMPLES

ROOT-MEAN SQUARE VALUE OF NOISE ALONE ~ N1/2

OUTPUT DUE TO NOISE ALONE

OUTPUT AT τd = 0DUE TO SIGNAL ~ N

OUTPUT AT τ=0 DUE TO SIGNAL ~ N2

MEAN VALUE OF NOISE POWER ALONE ~ N

(Too weak to be seen on this scale.)

REPLICA OF TRANSMISSION FOR TIME SHIFT τd

-τd +τd

INSTANTANEOUS POWER OUTPUT OF A REPLICA CORRELATOR FOR A CW WAVEFORM MASKED BY NOISE

TOO HIGH A THRESHOLD LEADS TO MISSED DETECTIONS TOO LOW A THRESHOLD

LEADS TO EXCESSIVE FALSE ALARMS

THRESHOLD SETTING DETERMINES (WITHIN LIMITS) THE RESULTING PROBABILITIES OF DETECTION AND FALSE ALARM

10005000-500-1000

SAMPLED DATA POINTS FORWARD AND BACKWARD IN TIME FROM EXACT OVERLAP OF CW WAVEFORM AND ITS REPLICA

TARGET ISPRESENT

TARGET ISABSENT

THRESHOLD IS CROSSED,

DECIDE TARGET PRESENT

CORRECTDETECTION

FALSE ALARM

MISSEDDETECTION

NO ACTION

TRUTHCOLUMNS

INSTANTANEOUS POWER OUTPUT OF

REPLICA CORRELATOR

For example, a detector might be designed to provide a 50% probability of detection while maintaining a probability of false alarm below 10-6.

PROBABILITIES OF FOUR POSSIBLE OUTCOMES

pd pfa

1 - pd 1 - pfa

pd is the probability a threshold crossing will occur when a target is actually present.

pfa is the probability a threshold crossing will occur when a target is not present.

THRESHOLD NOTCROSSED,

DECIDE TARGET ABSENT

SEPARATE AND DISTINCT ENSEMBLES

28REVERBERATION

TRANSMISSION

REPLICA CORRELATOR, DFT DETECTOR

DECISION: H1/H0 ?

TARGET

FOR CW TRANSMISSIONS

AMBIENT NOISE RECEPTION(Receive array)

(Beamformer)

PRE-DETECTIONFILTER

ACTIVE SONAR DETECTION MODEL(DISCUSSED SO FAR, ALONG WITH SONAR EQUATION)

29REVERBERATION

TRANSMISSION

AMBIENT NOISE

DECISION: H1/H0 ?

TARGET

FOR FREQUENY MODULATED

TRANSMISSIONS

ACTIVE SONAR DETECTION MODEL(NEXT STEPS)

PRE-DETECTIONFILTER

(Beamformer)

RECEPTION(Receive array)

REPLICA CORRELATOR, DFT DETECTOR

Time, t

τd is the time delay of the echo with respect to the replica established in the correlator.

Replica correlator outputand its envelope

for a CW waveform.+τd-τd

τd = 0

REPLICA CORRELATION FOR A CONTINUOUS CW WAVEFORM

Replica matches echo except for time delay.

Time, tE

EFFECT SHIFTING A CW WAVEFORM’S FREQUENCY BY φUSING THE ‘NARROWBAND’ APPROXIMATION

Replica correlator outputand its envelope

for a CW waveform

Frequency = fo + φ

Frequency = fo

+τd-τd

τd = 0

φ is the frequency difference (shift) of the echo with respect tothe replica established in the correlator. The greater | φ |, the narrower the envelope of the correlator’s output

Envelopenarrows

f6Received CW frequenciesg1, g2, g3, … , g6are each the result of the same narrowband Doppler shift with respect to equally spaced transmitted CW frequenciesf1, f2, f3, … , f6

TimeT RECEIVE

TRANSMITTED AND RECEIVED CW FREQUENCIES AFTER APPLYING A NARROWBAND DOPPLER SHIFTIn

TTRANSMIT

Received CW frequenciesg1, g2, g3, … , g6are each the result of a different narrowband Doppler shift with respect to equally spaced transmitted CW frequenciesf1, f2, f3, … , f6

TimeTRECEIVE

TRANSMITTED AND RECEIVED CW FREQUENCIES FOR AN ECHO PRODUCED BY A CLOSING TARGET,

A WIDEBAND DOPPLER TRANSFORMATION

TTRANSMIT

whereν = Constant range-rate of a reflector, positive closing, andc = Sonic velocity in the medium.

DOPPLER PARAMETER S AND THEWIDEBAND DOPPLER TRANSFORMATION

( ) 1c/ifc/21c)/(1/c)/(1 <<−=+−≡ νννν s

fkgki = = fk /sic/21 iν+ k = 1,2, …, N

νi siEach closing range-rate νi maps into a ‘stretch’ parameter si:

= fk si

si1 -gki fk_

Each transmitted CW pulse of frequency fk becomes a received CW pulse of frequency gki that depends upon si:

The difference between the received and transmitted CW frequencies depends on both si and fk:

1 to 1

Time, tE

φ is the frequency shift of the echo with respect to the replica established in the correlator. In the ‘wideband’ case the echo undergoes a Doppler transformation and not simply a Doppler shift.

Frequency = fo + φ

Frequency = fo

Envelope of replica correlator output

+τd-τd

τd = 0

EFFECT SHIFTING A CW WAVEFORM’S FREQUENCY BY φUSING THE WIDEBAND DOPPLER TRANSFORMATION

Time, t

Replica correlator output

and its envelope for a frequency-modulated waveform

+τd-τd

τd = 0

No frequency

REPLICA CORRELATOR OUTPUT FOR AFREQUENCY-MODULATED WAVEFORM (1 of 3)

Replica correlator output

and its envelope for a frequency-modulated waveform

+τd-τd

τd = 0

Time resolution of the envelope is ~W-1 where W is the bandwidth of the waveform.

Time, tNo

frequency shift

Replica correlator outputand its envelope for a

frequency-modulated echoexperiencing a frequency shift.

+τd-τd

τd = 0

With echofrequency

When a frequency-modulated echo experiences a frequency shift with respect to the replica, the peak output of the correlator is

diminished and his peak output is no longer at τd = 0.

dτ̂+

dτ̂ is the time delay when the replica correlator’s output envelope is a maximum.

THE LINEAR FREQUENCY MODULATED (LFM) WAVEFORM OF UNIT AMPLITUDE

-T/2 +T/2

+ t- t

Instantaneous frequency, Hz

fc , Carrierfrequency

Bandwidth, >> W

Time, Seconds

Instantaneous frequency

TIME-FREQUENCY DIAGRAM REFERRED TO REPLICA WAVEFORM (1 of 2)

In the case of the narrowband assumption, φ is the uniform upward frequency shift of the echo with respect to the replica established in the correlator.

EchoReplica

~-t ~t=0~+t

, frequency shift

In the wideband case, the echo’s frequency shift is not uniformover its duration, and any frequency shift brings about a dilation or contraction in the duration of the waveform.

Replica

Contraction if frequency increases

Frequency shift is not uniform (closing target produces increased echo frequencies)

Time~-t ~t=0

TIME-FREQUENCY DIAGRAM REFERRED TO REPLICA WAVEFORM (2 of 2)

(no narrowband assumption)

(narrowband assumption)

Time scale referred to replica waveform

Echo data moves withrespect to replica

established in correlator

TIME DELAY DIAGRAMS

Increasing clock time

s(t-τRD ) tt = 0

BDτ=t

τt (1)RD=

)( BDτ−tsBULK DELAY

REPLICA DELAY

Clock time

~+t~-t

REPLICA

Replica

s(t-τRD ) tt = 0

BDτ=t

τt (1)RD=

REPLICA DELAY

Clock time

~+t~-t

REPLICA

TIME DELAY DIAGRAMS

Replica

s(t-τRD ) tt = 0

BDτ=t

τt (2)RD=

REPLICA DELAY

Clock time

~+t~-t

REPLICA

TIME DELAY DIAGRAMS

established in correlatorReplica

s(t-τRD ) tt = 0

BDτ=t

τt (3)RD=

REPLICA DELAY

Clock time

~+t~-t

REPLICA

TIME DELAY DIAGRAMS

Replica

s(t-τRD ) tt = 0

BDτ=t

τt (4)RD=

REPLICA DELAY

0ˆ >= dd ττ

Clock time

~+t~-t

REPLICA

dd ττ ˆ= when overlap is greatest

TIME DELAY DIAGRAMS

Replica

s(t-τRD ) tt = 0

BDτ=t

τt (5)RD=

REPLICA DELAY

Clock time

~+t~-t

REPLICA

TIME DELAY DIAGRAMS

Replica

s(t-τRD ) tt = 0

BDτ=t

τt (6)RD=

REPLICA DELAY

Clock time

~+t~-t

REPLICA

TIME DELAY DIAGRAMS

Replica

s(t-τRD ) tt = 0

BDτ=t

τt (6)RD=

REPLICA DELAY

(7) REPLICA

Clock time

~+t~-t

TIME DELAY DIAGRAMS

Replica

Echo data moves withrespect to replica data

τd τd

USING THE TIME-FREQUENCY DIAGRAM TO ESTIMATE THE MAXIMUM OUTPUT OF A REPLICA CORRELATOR WHEN THE ECHO

HAS A FREQUENCY SHIFT φ

The greater the overlap region, the larger the maximum replica correlator output. In this example an LFM waveform has experienced a narrowband Doppler shift.

τd 0 τd > τd

Replica data

Overlap region producing maximum correlator power output for frequency shift φ

TIME-FREQUENCY DIAGRAM REFERRED TO REPLICA WAVEFORM

In the wideband case, the echo’s frequency shift is not uniformover its duration, and any frequency shift brings about a dilation or contraction in the duration of the waveform.

Replica

Contraction if frequency increases

Frequency shift is not uniform (closing target produces increased echo frequencies at higher frequencies)

Time~-t ~t=0

Overlap region producing maximum correlator output

USING THE TIME-FREQUENCY DIAGRAM TO ESTIMATE THE MAXIMUM OUTPUT OF A REPLICA CORRELATOR

Here an LFM echo has experienced a wideband Dopper transformationrather than a narrowband Doppler shift. The overlap with the replica is

reduced, and the corresponding replica correlator output is reduced.

Time scale referred to replica waveform~t=0

~+t~-t

Replica

-T/2 +T/2+ t- t

Instantaneous frequency (hyperbolic)

-T/2 +T/2+ t- t

HYPERBOLIC FREQUENCY MODULATED (HFM) WAVEFORMS, THEIR TIME-PERIOD AND TIME-FREQUENCY DIAGRAMS

HFM waveforms and linear period modulated (LPM) waveforms are the same.

∆τ secondsper cycle

= (τ1 )−1

= (τ2 )−1

Instantaneous period τ (t), linear

Higher frequency echo data moves withrespect to replica

Overlap region producing maximum correlator output

Here an HFM echo has experienced a wideband Dopper transformationand the HFM’s time-frequency distribution adjusts itself to remain ‘more’

overlapped than in a similar case for an LFM waveform.

~+t~-t

USING THE TIME-FREQUENCY DIAGRAM TO ESTIMATE THE MAXIMUM OUTPUT OF A REPLICA CORRELATOR

Replica

If U(f) is the Fourier transform of waveform u(t), U(f) can be folded over as shown below to obtain Fourier transform Ψ(f).

The inverse Fourier transform of Ψ(f), ψ(t), is a complex waveform in the time domain and provides a convenient way to describe the envelope and phase of the real waveform u(t). ψ(t) is the analytic signal of real waveform u(t), and the Fourier transform of ψ(t) (i.e., Ψ(f)) contains no negative frequencies.

ANALYTIC SIGNAL OF A REAL WAVEFORM (1 of 2)ο

- f + f+1

Ψ(f) = 2 Q(f) U(f)

)(FT)( ft Ψ →←ψ

)()( FT fUtu →←

- f + f

+ fo2- fo2

|U(f)|

|Ψ(f)|

+ f- f

- f + f

+ fo2- fo2

|U(f)|

|Ψ(f)|

+ f- f

ENVELOPE

If u(t) is a narrowband waveform, the frequency content of ψ (t),(i.e., Ψ(f),) is negligible near zero frequency as well as null for all f < 0.

If U(f) is the Fourier transform of waveform u(t), U(f) can be folded over as show below to obtain Fourier transform Ψ(f).

The inverse Fourier transform of Ψ(f), ψ(t), is a complex waveform in the time domain and provides a convenient way to describe the envelope and phase of the real waveform u(t). ψ(t) is the analytic signal of real waveform u(t).

ENVELOPES |Ψ (f)|

- f + f

|Ψ (f)|

|U (f)|

+ fo- fo

- f + f

- f + f+1

Ψ(f) = 2 Q(f) U(f)

)()( FT fUtu →←

ANALYTIC SIGNAL OF A REAL WAVEFORM (2 of 2)

- f + f

+ fo2- fo2

|U(f)|

|Ψ(f)|

+ f- f

- f + f

+ fo2- fo2

|U(f)|

|Ψ(f)|

+ f- f

- f + f+1

Ψ(f) = 2 Q(f) U(f)

)()( FT fUtu →←

ττπτψ dt

ujtut ∫∞+

∞− −+= )()()()(

ττπτ dt

u∫∞+

∞− − )()( is the Hilbert transform of u(t)

and is often denoted by .)(ˆ tu

ENVELOPE

)( )(Re)( ttu ψ=

ANALYTIC SIGNAL WITHOUT APPROXIMATION (1 of 2)

ττπτψ dt

ujtutujtut ∫∞+

∞− −+=+= )()()()(ˆ)()(

)(tψ is the analytic signal of real waveform u(t), and is always given by

even if u(t) is not narrowband! However, it may be difficult to find

).(ˆ tu

Time, seconds

-1.5 -1 -0.5 0 0.5 1 1.5 2-1.5

-0.5 < t < +0.5, Secondsu(t) = - sin(2πfot)

= 3 Hertz; T = 1 Secondfo

)(tu)(ˆ tu

Envelopeu(t) = 0; t > |0.5| Seconds

)(ˆ)( 22 tutu +

The envelope of u(t)is given by:

ANALYTIC SIGNAL WITHOUT APPROXIMATION (2 of 2)

ττπτψ dt

ujtut ∫∞+

∞− −+= )()()()(

]2[exp)]()([)(ˆ)( tfjtbjtatt oπψψ +=≅

If u(t) is narrowband, ψ(t) can be approximated by

The analytic signal is always given by

ENVELOPE

+fo is a central frequency of U(f),where a(t) and b(t) are real,

and a(t) and b(t) vary slowly compared to 2π fot.

|U (f)|

+ fo- fo

- f + f

|Ψ (f)|

- f + f

|Ψ (f)|

- f + f

- f + f+1

Ψ(f) = 2 Q(f) U(f)

)()( FT fUtu →←

APPROXIMATING THE ANALYTICAL SIGNAL WITH A NARROWBAND COMPLEX WAVEFORM (1 of 6)

- f + f

ENVELOPE OF |U( f , fo )|

|),(| offU

fo-fo - f + f

|),(| offΨ

- f + f

|)(~| fΨ)(~FT)( ft Ψ →←υ

ENVELOPE OF |Ψ ( f , fo )|

ENVELOPE OF |Ψ ( f )|~

U(f,fo) IS FOURIER TRANSFORM OF u(t.fo)ψ(t,fo) IS THE INVERSE FOURIER

TRANSFORM OF Ψ (f,fo)

υ(t) IS THE INVERSE FOURIER TRANSFORM OF Ψ(f)~

DOWNSHIFT BY fo

ANALYTIC SIGNAL AND COMPLEX ENVELOPE OF REAL WAVEFORM u(t,fo)

υ(t) is the complex envelopeof real waveform u(t,fo).

ψ(t,fo) is the analytic signalof real waveform u(t.fo).

),(FT),( offoft Ψ →←ψ

exp(+2π j fo t)

exp(-2π j fo t)absent

unit circle

Rea(t)

= [a(t) + j b(t)] exp(+2π j fo t)

FACTORS OF THE COMPLEX NARROWBAND WAVEFORM

a(t) and b(t) are real and vary slowly with respect to 2π fot.

| | = [a(t)2 + b(t)2 ]

)( )(ˆ)( ttu ψRe=

)(ˆ tψ

[a(t) + j b(t)]complex envelope of .)(ˆ tψ

)(ˆ tψ

is the

Re[ψ(t)] = a(t)cos(2π fo t) – b(t)sin(2π fo t)

Im[ψ(t)] = a(t)sin(2π fo t) + b(t)cos(2π fo t)

COMPLEX PLANE FOR THE NARROWBAND WAVEFORMIm

ψ(t), rotates counter clockwise

Re[ψ(t)]

Im[ψ(t)]

ψ(t), rotates counter clockwise

Re[ψ(t)]

Im[ψ(t)]

)(ˆ tψ

ψ(t) = [a(t) + j b(t)] exp(+2π j fo t)

|ψ(t)| )(tA(t)b(t)a 22 =+=

‹‹

Arg(ψ(t)) = tan-1 [ Im[ψ(t)] / Re[ψ(t)] ] = Φ(t)

‹‹‹

])([)()(ˆ tjtAt e Φ

=ψ )(2)(, ttft o θπ +=Φ

= u(t)

])([)()(ˆ tjtAt e Φ

=ψ is a particularly convenient form of the

analytic signal because Φ(t) is the argument of and the )(ˆ tψ

instantaneous phase of u(t) in radians.

][ )(21)()( t

dtdtftf iousinstantane Φ≡≡ π

Given fi(t), the argument of and instantaneous phase of u(t) is given by

o)(2)( θπ +=Φ ∫ dttift

The instantaneous frequency of u(t) in Hertz is

][ o)(2 θπ +∫ tdtifje)()(ˆ tAt =ψ

And we can write as )(ˆ tψ

)(ˆ tψ

DEFINITION OF INSTANTANEOUS FREQUENCY

In the expression

Variations in fi(t) are frequency modulation or FM.

Variations in A(t) are amplitude modulation or AM

,)()()( )( 22 tbtatA +=

ο Most active sonar waveforms are either frequency or amplitude

Phase modulation or PM is used in underwater communications; for example, 90o or 180o phase changes are embedded in θ (t)

and the analytic signal is expressed as

modulated; frequency modulation is more common.

)]([ )( )(2exp)()(ˆ ttftAt oj θπψ +=

i.e., θ(t) ‘switches’ or ‘flips’ the phase of the carrier frequency.

AMPLITUDE, FREQUENCY, AND PHASE MODULATION

][ o)(2 θπ +∫ tdtifje)()(ˆ tAt =ψ

In the expressionsο

both the amplitude and frequency modulation of u(t) and

[a(t) + j b(t)]are embedded in the complex envelope :ψ(t)

= a(t)cos(2π fo t) – b(t)sin(2π fo t)u(t) = Re( )ψ(t)

dttd ][ )(

21 θπFrequency modulation is )( )()()( tbjtat += Argθwhere

Amplitude modulation is )(tA(t)b(t)a 22 =+

ψ(t) = [a(t) + j b(t)] exp(+2π j fo t)

All the properties of the waveform (favorable and unfavorable!) independent of fo are embedded in the complex envelope.

This means analyses of most waveform properties (e.g., Doppler tolerance, range resolution) can be carried out by considering only the complex envelope without regard to a ‘carrier’ frequency.

IMPORTANCE OF THE COMPLEX ENVELOPE

General form of the analytic signal is])([

)()(tj

tAt e Φ=ψ

][ )(21)()( t

∫≡Φ dttift )(2)( π

,)()()( )( 22 tbtatA +=

A(t) is often explicitly stated, i.e., rect(t/T).

Start with fi(t) for the waveform you want.

Next integrate to find the phase.

OBTAINING THE ANALYTIC SIGNAL AND REAL WAVEFORM

Re[ψ(t)] gives the real waveform, u(t).o

Need not be narrowband.

ottoft θθπ ++=Φ )(2)(where d[θ(t)]/dt << 2π fo

If narrowband.Finally, insert Φ(t) in the

general form for ψ(t).

General form of the analytic signal is])([

)()(tj

tAt e Φ=ψ

][ )(21)()( t

∫≡Φ dttift )(2)( π

,)()()( )( 22 tbtatA +=

A(t) is often explicitly stated, i.e., rect(t/T).

Start with fi(t) for the waveform you want.

Then integrate to find the phase.

OBTAINING THE ANALYTIC SIGNAL AND REAL WAVEFORM

Re[ψ(t)] gives the real waveform, u(t).o

An indefinite integral with a constant of integration.

ottoft θθπ ++=Φ )(2)(where d[θ(t)]/dt << 2π fo

If narrowband.Finally, insert Φ(t) in the

general form for ψ(t).

THREE COMMON WAVEFORMS

CW (continuous wave)

LFM (linear frequency modulation)

HFM (hyperbolic frequency modulation)

The following slides give complex representations, i.e., for three common waveforms:)(tψ

(2[exp)/(rectπ

θπ otcfjTt +

where cycle1cfT >>

=)/(rect Tt1 if |t/T| < 1/2

0 if |t/T| > 1/2

of the CW waveform.

θο radians is a constant that determines the phase

=)(tψ

-10 -8 -6 -4 -2 0 2 4 6 8 10T TT T T T TT T TT

Tf)Tf(

T ππsin )(FT)/(rect)( fUTttu == →←

FREQUENCY

COMPLEX NARROWBAND CW WAVEFORMS OF

UNIT AMPLITUDE

Shift to frequency fc

)(FT)2(exp)( cffUtcfjtu − →←π

to get U(f - fc )

Cycles /Second

)( 22)( otcfdtcft θππ +=Φ ∫≡

Instantaneous phase (radians) )(2)( ttcft θπ +=Φ= , t < |T/2| and TW >> 1

and instantaneous frequency (Hertz)

])([21])(2[2

1 tdtd

cfttcfdtd θπθππ

tktdtd =])([

21 θπ

-T/2 +T/2

+ t- t

fcCarrier

frequency

Linear frequency modulation means

Therefore the LFM’s instantaneous phase )22(2 2π

π θotktcf ++= (radians)

otkt θπθ += 22

Bandwidth, >> W

i.e., the instantaneous frequency is a linear function of time, and k = W/T from the figure.

])([21)( tdt

dtif Φ≡= π

+ t- t

][][ )22(2exprect)( 2π

πψ θotktcfjTtt ++=

Wcf >>

])([)()(

tjtAt e Φ=ψ

)22(2)( 2π

π θotktcft ++=Φ∫=Φ dttift )(2)( π

tkcftif +=)(][rect)(

TttA =Given: and

Step 1

Step 2

-T/2 +T/2+ t- t

Instantaneous frequency (hyperbolic) = Finst(t) = 1/τ(t)

F2 = 1/τ2

Instantaneous period (linear) = τ(t)

-T/2 +T/2+ t- t

HYPERBOLIC FREQUECY MODULATED (HFM) WAVEFORMS, THEIR TIME-PERIOD AND TIME-FREQUENCY DIAGRAMS

HFM waveforms and linear period modulated (LPM) waveforms are the same.

∆ττ2

F1 = 1/τ1

Instantaneous period (linear),

-T/2 +T/2+ t- t

1(21)21(

FFo +=+= τττ2

121 FF −=−=∆ τττ

ot τττ ∆−=)(

ttinstF−+

=∆−

)21(21

1)(τττ

121211ττ −−= =FFW

2/2/ TtT +≤≤−

INSTANTANTEOUS FREQUENCY FOR AN HFM WAVEFORM OF UNIT AMPLITUDE

= 1/F1

= 1/F2

dttFt tins∫=Φ )(2)( π

])([)()( tjtAt e Φ=ψ

,INSTANTANEOUS FREQUECY INTEGRATION FOR THE HFM

ttinstF−+

=∆−

)21(21

1)(τττ

2/2/ TtT +≤≤−,

( )oFFW

Tθπ +

121 ln2

][rect)(TttA =and

( ) ][][ 2212

21 ln2exprect)( |

θπψ oFFWT

2/2/ TtT +≤≤−

Instantaneous period (linear),

-T/2 +T/2+ t- t

θο radians is a constant that determines the phase of the HFM waveform.

= 1/F1

= 1/F2

W = F2 – F1 , (F2 > F1), TW >> 1, and F1 >> W

=)/(rect Tt 1 if |t/T| < 1/2

0 if |t/T| > 1/2

THE HYPERBOLIC-FREQUENCY-MODULATED (HFM) WAVEFORM OF UNIT AMPLITUDE

( ) ][][ 2212

21 ln2exprect)( |

θπψ oFFWT

2/2/ TtT +≤≤−

( ) ][][ 222

21 ln2exp21rect)( |

−=π

θπψ oFFWT

+T/2-T/2

][rectTt

+t-t+1

][ 21rect −

CAUTION

CANNOT be shifted in time like the rect function, i.e.,HFM

t |)(ψ

,Tt0 ≤≤If the analytic signal for an HFM is

( ) ][][ 2212

21 ln2exprect)( |

θπψ oFFWT

2/2/ TtT +≤≤−

and the above is not a simple time shift of

77REVERBERATION

TARGET

TRANSMISSION

AMBIENT NOISE

ACTIVE SONAR DETECTION MODEL

RECEPTION

PRE-DETECTIONFILTER

REPLICA CORROR DFT

DECISION: H1/H0 ?

(NEXT STEPS)

PULSED WAVEFORMS

• In general pulses can be: 'SHAPED' or UNIFORM, OVERLAPPED, or

NOT OVERLAPPED,

• Individual pulses are identified by their carrier frequencies,

• However, the frequency content of all pulses is ‘spread’ around the carrier frequency.

Note: ∆B is the uniform separation distance of thecarrier frequencies, not a frequency spread.

f23456

WWWB∆

f23456

WWWWB∆B∆

f23456

The power spectral density of any pulse is ‘spread’ about it's carrier frequency:

∆F is a measure of this spread.

∆ B ∆ F

f6f5f4f3f2f1

SPECTRUM OF PULSED WAVEFORMS

Impulsivesource R

Rectangular∆T pulse

∆T ∆T

OpeningDoppler

∆T (∆Ts)

ClosingDoppler

∆T (∆Ts)

s < 1s > 1

TRANSMITTED AND RECEIVED DURATION TIMES

BEAMFORMED PULSED

WAVEFORM DATA

S1S2.Si.SM

FILTEROPERATIONSFOR EACH

COHERENT AND SEMI-COHERENT DETECTOR STRUCTURES

SEMI-COHERENT PROCESSING: The output of each Doppler channel Si is the result of applying a separate filter operation, i.e., a separate CW replica, to each pulse g1i, g2i, … , gNi and adding the results.

For each beamformed channel in, there are M Doppler channels, each corresponding to a discrete, a priori, wideband Doppler hypothesis Si.

COHERENT PROCESSING: The output for each Doppler channel Si is the result of applying the replica matching g1i, g2i, … , gNi to the entirereceived signal.

PEAK PICK

DOPPLER ESTIMATE

DETECTIONSTATISTIC

Detector is a bank of narrowband filters or replicas each centered on different receive CW frequencies g1i, g2i, … , gNi determined by si.

Assume target’s Doppler motion produces an echo with stretch factor si.

Transmit frequency-hop pulses at frequencies f1, f2, … , fNo

SCHEMATIC DIAGRAM FOR A SEMI-COHERENT PROCESSOR MATCHED TO STRETCH FACTOR Si

Output for g1i

Filter No. NFilter No. N Delay τNDelay τN

Filter No. 1Filter No. 1

• • • • • • • • • • • • • • •ΣΣ

Output

Output for gNi

Filter No. 2Filter No. 2 Delay τ2Delay τ2

Output for g2i

Delays differ by ∆Tsi

andDetection Statistic

1 2 3 4 5 6 7 8 9 10

NUMBER OF PULSES, N

1 2 3 4 5 6 7 8 9 10

NUMBER OF PULSES, N

EXPECTED PROCESSING GAIN FOR FULLY COHERENT PROCESSING OVER N PULSES

Lower frequency

Higher frequency

84REVERBERATION

TRANSMISSION

DECISION: H1/H0 ?

TARGET

AMBIENT NOISE RECEPTION(Receive array)

(Beamformer)

PRE-DETECTIONFILTER

ACTIVE SONAR DETECTION MODEL(NEXT STEPS)

REPLICA CORRELATOR

DETECTOR1) COMPARISON WITH

A ‘MATCHED FILTER’

2) AMBIGUITYFUNCTIONS

The term ‘matched filter’ is applied when the filter’s response function is proportional to the time-reversed input sequence f[n-k] for some value of n = m; i.e., when for some shift m of f[-k], A h[k] = f[m-k] ; then the output at n = m is given by

For a linear, time invariant filter:

THE ‘MATCHED FILTER’

~ ∑∑∑+∞=

−∞=

−∞==−=−=

2}h[k]{Ak][mfh[k]k][fk]h[mm]g[

If f[n] is complex, the ‘matching’ condition is Ah[k] = f [m-k]. *o When the matching condition holds, the filter is essentially cross-

correlating the input data f[n] with complex conjugate of the input data.

Linear,time invariant

filter, h[n]

Input signal, noise-free,

f[n]Output,

∑+∞=

−∞=−

kk]h[n[k]f

∑+∞=

−∞=−

kk]h[k][nf

=n]g[~

Some authors reserve the term ‘matched filter’ for an analog linear time-invariant filter whose response function is exactly matched to the filter’s time-reversed input function. Common usage relaxes this requirement for ‘exact’ matching – as in the case of replica correlation.

THE ‘MATCHED FILTER’

From now on we will examine the output of a replica correlator with the understanding that its output is equivalent to a matched filter with a response function obtained by time-reversing the replica established in the correlator.

For example, when perturbations in time delay and frequency shift affect f[k], a fixed h[k] and the same ‘matched filter’ is under consideration even though the perturbed f[k] is no longer an exact ‘match’ to h[k].

Other detection methods, e.g., the DFT (discrete Fourier transform) should not be confused with the term ‘matched filter’. Neither a replica nor an impulse response is embedded in the DFT.

USING THE TIME-FREQUENCY DIAGRAM TO ESTIMATE THE MAXIMUM OUTPUT OF A REPLICA CORRELATOR WHEN THE ECHO

HAS A FREQUENCY SHIFT φ

The greater the overlap region, the larger the maximum replica correlator output. In this example an LFM waveform has experienced a narrowband Doppler shift.

τd 0 τd > τo

Replica data

τd = τd^

Time scale referred to replica waveformt~+0t =~t~−

φ = 0

Time scale referred to replica waveformt~+0t =~t~−

Complete overlap produces maximum correlator power output over all frequency

shifts and time delays

τd τd= = 0

If the echo and replica frequencies achieve partial overlap, the power output of the replica correlator is a local maximum for a fixed φ. The power output is a global maximum when there is complete overlap.

PARTIAL AND COMPLETE OVERLAP

τd = τd^

Increasing clock timeφτd t~+

0t =~t~−

We would like to know the normalized instantaneous power output of a replica correlator (or matched filter) as a function of a waveform’s frequency shift φ and time delay τd when:

A replica of the waveform is established in the correlator, and

by only a uniform frequency shift φ and time delay τd.The input data (the echo) differs from the replica (the waveform)

●●

NEED FOR A NARROWBAND AMBIGUITY FUNCTION

2|),(| φτχ

NARROWBAND AMBIGUITY FUNCTION FOR A TYPICAL LFM WAVEFORM

NARROWBAND AMBIGUITY FUNCTION FOR AN LFM WAVEFORM

T=1 sec W=10 Hz Volume=0.99dB

Impulsivesource R

Rectangular∆T pulse

∆T ∆T

OpeningDoppler

∆T (∆Ts)

ClosingDoppler

∆T (∆Ts)

s < 1s > 1

TRANSMITTED AND RECEIVED DURATION TIMES

-60 -40 -20 0 20 40 60

Stretch Parameter, S

TIME DELAY IN SAMPLING INTERVALS

WIDEBAND AMBIGUITY FUNCTION FOR A TYPICAL HFM WAVEFORM

CW WAVEFORM

DURATION = 0.25 SECONDS

CW FREQUENCY = 3500 Hz

HFM WAVEFORM

DURATION = 0.25 SECONDS

START FREQUENCY = 3450 Hz

END FREQUENCY = 3550 Hz

WIDEBAND HFM AMBIGUITY FUNCTION

WIDEBAND CW AMBIGUITY FUNCTION

TIME DELAY IN SECONDS WITH RESPECT TO WAVEFORM CENTERS

TIME DELAY AND FREQUENCY RESOLUTION FOR LFM AND CW WAVEFORMS

Time-delay spread and frequency spread degrade the above resolutions when T and W increase beyond limits imposed by these effects!

LFM ambiguity function

Time delay resolution is W -1

Frequency resolution is T -1

CW ambiguity function

Time delay resolution is T

Frequency resolution is T -1

POINT TARGETτοφο

φ = 0 corresponds to zero frequency shift.

τ = 0 corresponds to a point target’s bulk delay time.

The correlator output can’t distinguish between a time delay το and frequency shift φο and zero time delay and zerofrequency shift.

THE RANGE-DOPPLER COUPLING EFFECT APPLICABLE TO LFM AND HFM WAVEFORMS

Correlator outputs

RIDGE LINEφ

For το and φο

Source transmission of long CW with stable frequency

CLOCK TIME

POINT TARGET WITH CLOSING RANGE

MEDIUM FREQUENCY DISPERSION OF A LONG CW WAVEFORM (1 of 3)

Amplitude and Doppler shifted frequency of received echo vary (within limits) at random

due to non-stationary medium.

FREQUENCY

∆fDOPPLER

Center frequency of long CW transmission

Envelope of transmitted

CW waveformEnvelope of echo from closing target is ‘smeared’

on average over a frequency spread of B Hz.

A similar frequency spread occurs in the absence of a Doppler shift.( )

Fading of a received signal produced by medium dispersion on a long CW transmission.

• • •• • •

The amplitude and phase vary (within limits) at random.

The average duration of a reinforcement or fade is (1 cycle)/(B Hz) seconds.

••

1 cycleB Hz

CONVOLUTION OF CW TRANSMISSION AND

FREQUENCY SPREAD

f = fcarrier

∆f = 0

f = fcarrier

CONVOLUTION OF A RECTANGULAR CW’S SINC FUNCTION AND A MEDIUM’S FREQUENCY SPREAD

SINC FUNCTION OF CW TRANSMISSION

OF DURATION T.

MEDIUM’S FREQUENCY SPREAD WITH MEAN BB

f = fcarrier

∆f = 0

FREQUENCY SPREAD

SINC FUNCTION OF CW TRANSMISSION

OF DURATION T.

MEDIUM’S FREQUENCY SPREAD WITH MEAN B

CLOCKTIME

EXTENDED TARGET AT CONSTANT RANGE

TRANSMISSION TIME

Reception interval is ‘smeared’ on averageover a time-delay spread of L seconds.

Slope = c, Sonic velocityin the water

‘Smearing’ occurs due to extended target; or

‘unresolved’ multipath.

Bulk time delay(or just ‘time delay’)

MEDIUM TIME DISPERSION OF TRANSMITTED GAUSSIAN PULSE

FREQUENCY DIFFERENCE,(f1 – f2) , Hz

Fading of two fixed-frequency received CW echoes in the same

acoustic channel at time to

Probability received CW echoes at frequencies f1 and f2 will experience local fades or maxima within interval ∆T

Proportional to L-1

CW echo f2 experiences a local maximum but echo f1 does not.

TIME-DELAY SPREAD AFFECTS SIGNAL FADING

EFFECT OF TIME-DELAY & FREQUENCY SPREAD ON AN LFM WAVEFORM, TO = 1 Sec, W = 400 Hz

L = 5 ms, B = 2 Hz

- 0.01 - 0.005 0 0.005 0.01

Time Delay, τ Seconds

quency S

hift, H

quency S

hift, H

Ambiguity Function, B = 0 and L = 0

0 2 4 6 8 10

B, Frequency Spread, Hz

PEAK RESPONSE LOSS FOR LFM WAVEFORM SUBJECTED TO REPLICA CORRELATION

To = 1 Second, W = 400 Hz

Recall the results for the ratio of signal output power to interference output power for a replica correlator:

OUTPUT

SoT2(1 cycle) oΝ

OUTPUT

SW2(1 cycle) oR

when working against ambient noise, and

when working against reverberation.

BTo < 1 cycle

LW < 1 cycle

MEDIUM EFFECTS LIMIT REPLICA CORRELATOR PERFORMANCE

These results are for coherent processing only, and can be expected to apply within 1 or 2 dB only if:

fundamentals of passive and active sonar technical training short course sampler

k2 ykn

direction n

water x wattsmeter2

x iref wattsmeter2

k2 k1 k1

shift of yk

air wattsmeter2 y wattsmeter2

fixed shift n

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