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This four-day course is designed for SONAR systems engineers, combat systems engineers, undersea warfare professionals, and managers who wish to enhance their understanding of passive and active SONAR or become familiar with the "big picture" if they work outside of either discipline. Each topic is presented by instructors with substantial experience at sea. Presentations are illustrated by worked numerical examples using simulated or experimental data describing actual undersea acoustic situations and geometries. Visualization of transmitted waveforms, target interactions, and detector responses is emphasized.
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2REVERBERATION
TARGET
TRANSMISSION
DELAY ANDATTENUATION
TIME-VARYINGMULTIPATH
DELAY ANDATTENUATION
AMBIENT NOISE
ACTIVE SONAR DETECTION MODEL
TIME-VARYING MULTIPATH
RECEPTION
PRE-DETECTIONFILTER
DETECTOR
POST-DETECTIONPROCESSOR
DECISION: H1/H0 ?
3
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.
n
n
•I
nn
Acoustic intensity, I
:
t);z,y(x,II
= Watts/(Meter)2
;I up
≡ where is the fluid velocity at (x,y,z;t). u
4
ACOUSTIC INTENSITY LEVELS EXPRESSED IN DECIBELS
)ref10 averagelog10 I|I| /(
=dB60
)ref10 averagelog I|I| /(
=6
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
5
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)(
,2
|I| ρrmsp
waveplane a for =
waterrefwaterref |I|I
≡
airrefairref |I|I
≡
.
6
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
7
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.
X
REPLICA CORRELATOR
x[k]
y[k-n]
g[n]∑+∞=
−∞=
k
kn]-y[kx[k]
o
oREPLICA CORRELATION (1 of 6)
INPUT
REPLICA
OUTPUT
8
1020
3040
INPUT (SIGNAL)
x[k],
y[k],REPLICA
k
k
12 3 4
12 3 4
y[k-n],INCREASES
k
SHIFTED REPLICA
12 3 4
y[k-k(1)],
k
SHIFTED REPLICA ALIGNED
WITH INPUT
0 1 2 3
k(1), FIXED BULK DELAY
k(1)
NON-ZERO VALUES OF x[k]
= 4k(2)
k(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)
k(2)
n
n
n
REPLICA CORRELATION (2 of 6)
0
0
0
9
1020
3040
12 3 4
110200
300
40
g[n],
k
k
n
200110 40
1 2 3 4
x[k],
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
0k
−∑+≥
=
= 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],
NON-ZERO VALUES OF x[k]
= 4k(2)
k(2)
k(1) k(1) + k(2)
n = k(1)
n = k(1)-k(2) n = k(1)+k(2)
k(1)
y[k-n] when n=k(1)
n + k(2)n
INPUT (SIGNAL)
OUTPUT OF REPLICA
CORRELATOR
REPLICA CORRELATION (3 of 6)
10
Instead of writing
]ny[kx[k]g[n](2)(1) kkk
0k
−∑+≥
=
=
we can write
∑∞+=
∞−=−=
k
k]ny[kx[k]g[n]
Since x[k] = 0 if k < 0 or k > k(1) + k(2).
1020
3040
k
x[k],
FIXEDk(1)
k(1) k(1) + k(2)
NON-ZERO VALUES OF x[k]
= 4k(2)
k(2)
INPUT (SIGNAL)
REPLICA CORRELATION (4 of 6)
11
XX
REPLICA CORRELATOR
x[k]
y[k-n]
g[n]∑+∞=
−∞=
k
kn]-y[kx[k]INPUT
REPLICA
OUTPUTSIGNAL
+NOISE
SIGNAL +
NOISE
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:
o
o
REPLICA CORRELATION (5 of 6)
2k
k
2]ny[kx[k]g[n] |||| ∑
∞+=
∞−=−=
SIGNAL+NOISEREPLICA
|g[n]|2
INSTANTANEOUS POWER OUTPUT OF REPLICA
CORRELATOR,SIGNAL + NOISE
n
BULK DELAY, k(1)
n = k(1)
THRESHOLD
12
XX
REPLICA CORRELATOR
x[k]
y[k-n]
g[n]∑+∞=
−∞=
k
kn]-y[kx[k]INPUT
REPLICA
OUTPUTSIGNAL
+NOISE
SIGNAL +
NOISE
REPLICA CORRELATION (6 of 6)
2k
k
2]ny[kx[k]g[n] |||| ∑
∞+=
∞−=−=
SIGNAL+NOISEREPLICA
|g[n]|2
INSTANTANEOUS POWER OUTPUT OF REPLICA
CORRELATOR,SIGNAL + NOISE
n
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.
13
s(t-τRD)t
tt = 0
t)()( tnts +− BDτ
BDτ=t
t
τt (1)RD=
s(t)
t)(tn
t
REPLICA CORRELATION FOR A CONTINUOUS CW WAVEFORM (1 of 4)
TRANSMISSION
ECHO
AMBIENT NOISE
REPLICAS OF THE TRANSMISSION FOR
DIFFERENT REPLICA DELAYS
RECEIVED DATA
)( BDτ−ts
BULK DELAY
REPLICA DELAYS
(1)
τt (2)RD=s(t-τRD)(2)
14
t
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
(1)
s(t-τRD)(2)
BDτ=t
τt (1)RD=
τt (2)RD=
(i)
REPLICA CORRELATION FOR A CONTINUOUS CW WAVEFORM (2 of 4)
ECHO
15
Sig
nal
Rep
lica
Time delay, τd
Time, t
τd
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
REPLICA CORRELATION FOR A CONTINUOUS CW WAVEFORM (3 of 4)
BDτ
τRD
τd = τBD - τRD
τd = 0
−τd
16
Rep
lica
Time, t
Rec
eive
d da
ta
Replica correlator outputfor a CW waveform plus ambient noise.
REPLICA OF THE TRANSMISSION FOR
INCREASING REPLICA DELAY
Time delay, τd
τd = 0
−τd
τd
τRDSOME REPLICA DELAY
REPLICA CORRELATION FOR A CONTINUOUS CW WAVEFORM (4 of 4)
17
0 200 400 600 800 1000 1200-15
-10
-5
0
5
10
15
0 200 400 600 800 1000 1200-15
-10
-5
0
5
10
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).
INST
ANTA
NEO
US
AMPL
ITU
DES
REP
LIC
A C
OR
REL
ATO
R
OU
TPU
T
-600 -400 -200 0 200 400 600-150
-100
-50
0
50
100
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
116
5.0 =
= 1.0
10 log( ) = -15 dB321
SAMPLED DATA POINTS
SAMPLED DATA POINTS
18
0 500 1000 1500 2000 2500-15
-10
-5
0
5
10
15
0 500 1000 1500 2000 2500-15
-10
-5
0
5
10
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.
INST
ANTA
NEO
US
AMPL
ITU
DES
REP
LIC
A C
OR
REL
ATO
R
OU
TPU
T
0 10005000-300
-200
-100
0
100
200
300
-500-1000
REPLICA CORRELATOR OUTPUT FOR A CW WAVEFORM MASKED BY NOISE (2 of 3)
Sine wave powerNoise power
=> -15 dB
As on previous slide,
SAMPLED DATA POINTS
SAMPLED DATA POINTS
TIME DELAY τd IN DATA SAMPLES
19
0 500 1000 1500 2000 2500-15
-10
-5
0
5
10
15
0 500 1000 1500 2000 2500-15
-10
-5
0
5
10
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.
INST
ANTA
NEO
US
AMPL
ITU
DES
REP
LIC
A C
OR
REL
ATO
R
OU
TPU
T
0 10005000-300
-200
-100
0
100
200
300
-500-1000
REPLICA CORRELATOR OUTPUT FOR A CW WAVEFORM MASKED BY NOISE (3 of 3)
Sine wave powerNoise power
=> -15 dB
As on previous slide,
SAMPLED DATA POINTS
SAMPLED DATA POINTS
Note triangular
shape
TIME DELAY τd IN DATA SAMPLES
20
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
01020
RANDOM WALK (SUM OF NOISE VALUES UP TO N DATA POINTS)
STEP
S R
EMO
VED
FR
OM
ST
ARTI
NG
PO
SITI
ON
NUMBER OF STEPS, N
NO
ISE
VALU
ES
N INCREASING N
RANDOM WALK SAMPLE AND A GENERAL RESULT (1 of 2)
Root-mean-square departure from starting position is . N
21
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
0
1REPLICA
NO
ISE
VALU
ESR
EPLI
CA
VALU
ES
0 100 200 300 400 500 600 700 800 900 1000-2-1012
POINT-BY-POINT PRODUCTS OF REPLICA AND NOISE
SAMPLED DATA POINTS
PRO
DU
CT
VALU
ES
SAMPLED DATA POINTS
SAMPLED DATA POINTS
NOISE SAMPLE MULTIPLIED BY A REPLICA
INDEPENDENT NORMALLY DISTRIBUTED NOISE, σ = 1
22
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
10
N, NUMBER OF PRODUCTS SUMMED
SUM
OF
PRO
DU
CT
VALU
ES
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
PRO
DU
CT
VALU
ES
RANDOM WALK SAMPLE AND A GENERAL RESULT (2 of 2)
Root-mean-square departure from starting position (zero) is proportional to . N
23
LENGTH OF REPLICA CORRELATION IN SAMPLED DATA POINTS, N
PEAK
REP
LIC
AC
OR
REL
ATO
R O
UTP
UT
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
0
1 ECHOES OF INCREASING DURATION
SAMPLED DATA POINTS
N
0 100 200 300 400 500 600 700 800 900 10000100200300400500
N
0 100 200 300 400 500 600 700 800 900 1000-1
0
1
SAMPLED DATA POINTS
N
PEAK REPLICA CORRELATOR OUTPUT WHEN ECHO MATCHES REPLICA
24
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
21NN
21
N
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
2
21
=
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
τd
INSTANTANEOUS POWER OUTPUT OF REPLICA CORRELATOR AND ITS PEAK
POWER ENVELOPE
RECEIVED DATA,SIGNAL MATCHING REPLICA PLUS NOISE
TIME
TIME
TIME DELAY
TIME DE;AY
+τd
ARB
ITR
ARY
UN
ITS
ARB
ITR
ARY
UN
ITS
ARB
ITR
ARY
UN
ITS
ARB
ITR
ARY
UN
ITS
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 +τd
26
-200
-100
REP
LIC
A C
OR
REL
ATO
R
POW
ER O
UTP
UT,
AR
BIT
RAT
Y U
NIT
S
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
0
27
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
DELAY ANDATTENUATION
TIME-VARYINGMULTIPATH
DELAY ANDATTENUATION
TIME-VARYING MULTIPATH
REPLICA CORRELATOR, DFT DETECTOR
POST-DETECTIONPROCESSOR
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
DELAY ANDATTENUATION
TIME-VARYINGMULTIPATH
DELAY ANDATTENUATION
AMBIENT NOISE
TIME-VARYINGMULTIPATH
POST-DETECTIONPROCESSOR
DECISION: H1/H0 ?
TARGET
FOR FREQUENY MODULATED
TRANSMISSIONS
ACTIVE SONAR DETECTION MODEL(NEXT STEPS)
PRE-DETECTIONFILTER
(Beamformer)
RECEPTION(Receive array)
REPLICA CORRELATOR, DFT DETECTOR
30
PR
ES
SU
RE
Time, t
+τd
Ech
oR
eplic
a
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.
31
PR
ES
SU
RE
Time, tE
cho
Rep
lica
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
32
f1
f2
f3
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
stan
tane
ous
frequ
ency
f5
f4
TTRANSMIT
g1
g2
g3
g6
g5
g4
33
•
f1
f2
f3
f6
g 1
g2
g3
g6
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
Inst
anta
neou
s fre
quen
cy
f5
f4
g5
g4
TTRANSMIT
34
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:
o
o
o
o
1 to 1
35
PR
ES
SU
RE
Time, tE
cho
Rep
lica
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
CLC
CLC
+τd-τd
τd = 0
-τd
EFFECT SHIFTING A CW WAVEFORM’S FREQUENCY BY φUSING THE WIDEBAND DOPPLER TRANSFORMATION
36
PR
ES
SU
RE
Time, t
Replica correlator output
Ech
oR
eplic
a
and its envelope for a frequency-modulated waveform
-τd
+τd-τd
τd = 0
τd is the time delay of the echo with respect to the replica established in the correlator.
No frequency
shift
REPLICA CORRELATOR OUTPUT FOR AFREQUENCY-MODULATED WAVEFORM (1 of 3)
37
PR
ES
SU
RE
Replica correlator output
Ech
oR
eplic
a
and its envelope for a frequency-modulated waveform
-τd
+τd-τd
τd = 0
τd is the time delay of the echo with respect to the replica established in the correlator.
Time resolution of the envelope is ~W-1 where W is the bandwidth of the waveform.
Time, tNo
frequency shift
REPLICA CORRELATOR OUTPUT FOR AFREQUENCY-MODULATED WAVEFORM (2 of 3)
38
CLC
PR
ES
SU
RE
Ech
oR
eplic
a
Replica correlator outputand its envelope for a
frequency-modulated echoexperiencing a frequency shift.
+τd-τd
τd = 0
With echofrequency
shift
CLC
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+
dτ̂ is the time delay when the replica correlator’s output envelope is a maximum.
REPLICA CORRELATOR OUTPUT FOR AFREQUENCY-MODULATED WAVEFORM (3 of 3)
39
THE LINEAR FREQUENCY MODULATED (LFM) WAVEFORM OF UNIT AMPLITUDE
W
-T/2 +T/2
+ t- t
Instantaneous frequency, Hz
fc , Carrierfrequency
Bandwidth, >> W
Time, Seconds
40
Instantaneous frequency
Time
TIME-FREQUENCY DIAGRAM REFERRED TO REPLICA WAVEFORM (1 of 2)
PR
ES
SU
RE
Ech
oR
eplic
a
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.
φ
T
EchoReplica
~-t ~t=0~+t
~+t
~+t
, frequency shift
41
PR
ES
SU
RE
Ech
oR
eplic
a
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.
T
Echo
Replica
Contraction if frequency increases
Frequency shift is not uniform (closing target produces increased echo frequencies)
Time~-t ~t=0
~+t
~+t
~+t
TIME-FREQUENCY DIAGRAM REFERRED TO REPLICA WAVEFORM (2 of 2)
Instantaneous frequency
(no narrowband assumption)
42
Instantaneous frequency
(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=
tECHO
)( BDτ−tsBULK DELAY
REPLICA DELAY
(1)
0>dτ
~t=0
Clock time
~+t~-t
REPLICA
dτ
Replica
43
Time scale referred to replica waveform
Increasing clock time
s(t-τRD ) tt = 0
BDτ=t
τt (1)RD=
tECHO
)( BDτ−tsBULK DELAY
REPLICA DELAY
(1)
~t=0
Clock time
~+t~-t
REPLICA
Instantaneous frequency
(narrowband assumption)
TIME DELAY DIAGRAMS
0>dτ
Replica
Echo data moves withrespect to replica
established in correlator
44
Time scale referred to replica waveform
Increasing clock time
s(t-τRD ) tt = 0
BDτ=t
τt (2)RD=
tECHO
)( BDτ−tsBULK DELAY
REPLICA DELAY
(2)
~t=0
Clock time
~+t~-t
REPLICA
0>dτ
Instantaneous frequency
(narrowband assumption)
TIME DELAY DIAGRAMS
Echo data moves withrespect to replica
established in correlatorReplica
45
Time scale referred to replica waveform
Increasing clock time
s(t-τRD ) tt = 0
BDτ=t
τt (3)RD=
tECHO
)( BDτ−tsBULK DELAY
REPLICA DELAY
(3)
~t=0
Clock time
~+t~-t
REPLICA
0>dτ
Instantaneous frequency
(narrowband assumption)
TIME DELAY DIAGRAMS
Replica
Echo data moves withrespect to replica
established in correlator
46
Time scale referred to replica waveform
Increasing clock time
s(t-τRD ) tt = 0
BDτ=t
τt (4)RD=
tECHO
)( BDτ−tsBULK DELAY
REPLICA DELAY
(4)
0ˆ >= dd ττ
~t=0
Clock time
~+t~-t
REPLICA
Instantaneous frequency
(narrowband assumption)
dd ττ ˆ= when overlap is greatest
TIME DELAY DIAGRAMS
Echo data moves withrespect to replica
established in correlator
Replica
47
Time scale referred to replica waveform
Increasing clock time
s(t-τRD ) tt = 0
BDτ=t
τt (5)RD=
tECHO
)( BDτ−tsBULK DELAY
REPLICA DELAY
(5)
~t=0
Clock time
~+t~-t
REPLICA
0>dτ
Instantaneous frequency
(narrowband assumption)
TIME DELAY DIAGRAMS
Echo data moves withrespect to replica
established in correlator
Replica
48
Time scale referred to replica waveform
Increasing clock time
s(t-τRD ) tt = 0
BDτ=t
τt (6)RD=
tECHO
)( BDτ−tsBULK DELAY
REPLICA DELAY
(6)
~t=0
Clock time
~+t~-t
REPLICA
0=dτ
Instantaneous frequency
(narrowband assumption)
TIME DELAY DIAGRAMS
Replica
Echo data moves withrespect to replica
established in correlator
49
Time scale referred to replica waveform
Increasing clock time
s(t-τRD ) tt = 0
BDτ=t
τt (6)RD=
tECHO
)( BDτ−tsBULK DELAY
REPLICA DELAY
(7) REPLICA
~t=0
Clock time
~+t~-t
0<dτ
Instantaneous frequency
(narrowband assumption)
TIME DELAY DIAGRAMS
Echo data moves withrespect to replica
established in correlator
Replica
50
Instantaneous frequency
Time scale referred to replica waveform
Echo data moves withrespect to replica data
established in correlator
τd τd
φ
Instantaneous frequency
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.
Time scale referred to replica waveform
τd 0 τd > τd
Replica data
=
=
t~+
t~+
0t =~
0t =~
t~−
t~−
Overlap region producing maximum correlator power output for frequency shift φ
^
^
51
TIME-FREQUENCY DIAGRAM REFERRED TO REPLICA WAVEFORM
PR
ES
SU
RE
Ech
oR
eplic
a
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.
T
Echo
Replica
Contraction if frequency increases
Frequency shift is not uniform (closing target produces increased echo frequencies at higher frequencies)
Time~-t ~t=0
~+t
~+t
~+t
Instantaneous frequency
(no narrowband assumption)
52
Echo data moves withrespect to replica
Overlap region producing maximum correlator output
Instantaneous frequency
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.
τd
Time scale referred to replica waveform~t=0
~+t~-t
Time scale referred to replica waveform~t=0
~+t~-t
Instantaneous frequency
(no narrowband assumption)
Replica
53
W
-T/2 +T/2+ t- t
Instantaneous frequency (hyperbolic)
F2
F1
0
-T/2 +T/2+ t- t
το
τ1
τ2
0
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
54
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.
τd
Time scale referred to replica waveform~t=0
~+t~-t
USING THE TIME-FREQUENCY DIAGRAM TO ESTIMATE THE MAXIMUM OUTPUT OF A REPLICA CORRELATOR
Instantaneous frequency
(no narrowband assumption)
Replica
55
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
ο
Q(f)
- f + f+1
Ψ(f) = 2 Q(f) U(f)
)(FT)( ft Ψ →←ψ
)()( FT fUtu →←
- f + f
+ fo2- fo2
|U(f)|
|Ψ(f)|
+ fo2
+ f- f
- f + f
+ fo2- fo2
|U(f)|
|Ψ(f)|
+ fo2
+ f- f
ENVELOPE
ENVELOPE
56
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).
ο
ο
ο
ENVELOPES |Ψ (f)|
+ fo
- f + f
|Ψ (f)|
+ fo
- f
|U (f)|
+ fo- fo
- f + f
Q(f)
- f + f+1
Ψ(f) = 2 Q(f) U(f)
)(FT)( ft Ψ →←ψ
)()( FT fUtu →←
ANALYTIC SIGNAL OF A REAL WAVEFORM (2 of 2)
57
- f + f
+ fo2- fo2
|U(f)|
|Ψ(f)|
+ fo2
+ f- f
- f + f
+ fo2- fo2
|U(f)|
|Ψ(f)|
+ fo2
+ f- f
Q(f)
- f + f+1
Ψ(f) = 2 Q(f) U(f)
)(FT)( ft Ψ →←ψ
)()( FT fUtu →←
ττπτψ dt
ujtut ∫∞+
∞− −+= )()()()(
ττπτ dt
u∫∞+
∞− − )()( is the Hilbert transform of u(t)
and is often denoted by .)(ˆ tu
ENVELOPE
ENVELOPE
)( )(Re)( ttu ψ=
ANALYTIC SIGNAL WITHOUT APPROXIMATION (1 of 2)
58
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
ο
).(ˆ tu
Time, seconds
Inst
anta
neou
s Am
plitu
de
-1.5 -1 -0.5 0 0.5 1 1.5 2-1.5
-1
-0.5
0
0.5
1
1.5
-1.5 -1 -0.5 0 0.5 1 1.5 2-1.5
-1
-0.5
0
0.5
1
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)
59
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
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)|
+ fo
- f + f
|Ψ (f)|
+ fo
- f + f
Q(f)
- f + f+1
Ψ(f) = 2 Q(f) U(f)
)(FT)( ft Ψ →←ψ
)()( FT fUtu →←
APPROXIMATING THE ANALYTICAL SIGNAL WITH A NARROWBAND COMPLEX WAVEFORM (1 of 6)
60
- f + f
ENVELOPE OF |U( f , fo )|
|),(| offU
fo-fo - f + f
|),(| offΨ
fo
- 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
0
0
0
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
61
exp(+2π j fo t)
exp(-2π j fo t)absent
Re
Im
unit circle
Im
Rea(t)
b(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 ]
1/2
)( )(ˆ)( ttu ψRe=
)(ˆ tψ
)(ˆ tψ
[a(t) + j b(t)]complex envelope of .)(ˆ tψ
)(ˆ tψ
is the
62
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
Re
ψ(t), rotates counter clockwise
Re[ψ(t)]
Im[ψ(t)]
Φ(t)
Im
Re
ψ(t), rotates counter clockwise
Re[ψ(t)]
Im[ψ(t)]
Φ(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)
63
])([)()(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
64
In the expression
Variations in fi(t) are frequency modulation or FM.
Variations in A(t) are amplitude modulation or AM
,)()()( )( 22 tbtatA +=
ο
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.
ο
)]([ )( )(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
65
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)
‹
, and
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.
ο
ο
, and
IMPORTANCE OF THE COMPLEX ENVELOPE
66
General form of the analytic signal is])([
)()(tj
tAt e Φ=ψ
][ )(21)()( t
dtdtftf iousinstantane Φ≡≡ π
∫≡Φ dttift )(2)( π
Variations in fi(t) are frequency modulation or FM.
Variations in A(t) are amplitude modulation or AM
,)()()( )( 22 tbtatA +=
A(t) is often explicitly stated, i.e., rect(t/T).
o
o
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).
67
General form of the analytic signal is])([
)()(tj
tAt e Φ=ψ
][ )(21)()( t
dtdtftf iousinstantane Φ≡≡ π
∫≡Φ dttift )(2)( π
Variations in fi(t) are frequency modulation or FM.
Variations in A(t) are amplitude modulation or AM
,)()()( )( 22 tbtatA +=
A(t) is often explicitly stated, i.e., rect(t/T).
o
o
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).
68
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
o
o
o
69
])2
(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
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
T
T
T
T
T
T
T
T
Tf)Tf(
T ππsin )(FT)/(rect)( fUTttu == →←
U(f)
FREQUENCY
COMPLEX NARROWBAND CW WAVEFORMS OF
UNIT AMPLITUDE
Shift to frequency fc
)(FT)2(exp)( cffUtcfjtu − →←π
U(f)
to get U(f - fc )
Cycles /Second
)( 22)( otcfdtcft θππ +=Φ ∫≡
70
THE LINEAR FREQUENCY MODULATED (LFM) WAVEFORM OF UNIT AMPLITUDE
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
W
-T/2 +T/2
+ t- t
Instantaneous frequency
fcCarrier
frequency
Linear frequency modulation means
Therefore the LFM’s instantaneous phase )22(2 2π
π θotktcf ++= (radians)
otkt θπθ += 22
2)(
Bandwidth, >> W
i.e., the instantaneous frequency is a linear function of time, and k = W/T from the figure.
so
])([21)( tdt
dtif Φ≡= π
71
W
+T/2
+ t- t
Instantaneous frequency
][][ )22(2exprect)( 2π
πψ θotktcfjTtt ++=
Wcf >>
THE LINEAR FREQUENCY MODULATED (LFM) WAVEFORM OF UNIT AMPLITUDE
-T/2
])([)()(
tjtAt e Φ=ψ
)22(2)( 2π
π θotktcft ++=Φ∫=Φ dttift )(2)( π
tkcftif +=)(][rect)(
TttA =Given: and
Step 1
Step 2
72
W
-T/2 +T/2+ t- t
Instantaneous frequency (hyperbolic) = Finst(t) = 1/τ(t)
F2 = 1/τ2
0
Instantaneous period (linear) = τ(t)
-T/2 +T/2+ t- t
το
τ1
0
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
73
Instantaneous period (linear),
-T/2 +T/2+ t- t
το
τ1
τ2
0
∆τ
)2
11
1(21)21(
21
FFo +=+= τττ2
11
121 FF −=−=∆ τττ
Tt
ot τττ ∆−=)(
)(tτ
tFFWT
FFWT
toTT
ttinstF−+
=∆−
==
)21(21
21)(
1)(τττ
121211ττ −−= =FFW
2/2/ TtT +≤≤−
INSTANTANTEOUS FREQUENCY FOR AN HFM WAVEFORM OF UNIT AMPLITUDE
,
= 1/F1
= 1/F2
74
dttFt tins∫=Φ )(2)( π
])([)()( tjtAt e Φ=ψ
,INSTANTANEOUS FREQUECY INTEGRATION FOR THE HFM
tFFWT
FFWT
toTT
ttinstF−+
=∆−
==
)21(21
21)(
1)(τττ
2/2/ TtT +≤≤−,
( )oFFW
TtFFW
Tθπ +
−+=
2121
121 ln2
][rect)(TttA =and
( ) ][][ 2212
11
21 ln2exprect)( |
+−+
=π
θπψ oFFWT
tFFWT
jTtt
HFM
2/2/ TtT +≤≤−
75
Instantaneous period (linear),
-T/2 +T/2+ t- t
το
τ1
τ2
0
∆τ
)(tτ
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
11
21 ln2exprect)( |
+−+
=π
θπψ oFFWT
tFFWT
jTtt
HFM
2/2/ TtT +≤≤−
76
( ) ][][ 222
11
21 ln2exp21rect)( |
+−
−=π
θπψ oFFWT
tFWT
jTtt
HFM
+1
+T/2-T/2
][rectTt
+t-t+1
][ 21rect −
Tt
+t-t
T0
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
11
21 ln2exprect)( |
+−+
=π
θπψ oFFWT
tFFWT
jTtt
HFM
2/2/ TtT +≤≤−
and the above is not a simple time shift of
77REVERBERATION
TARGET
TRANSMISSION
DELAY ANDATTENUATION
TIME-VARYINGMULTIPATH
DELAY ANDATTENUATION
AMBIENT NOISE
ACTIVE SONAR DETECTION MODEL
TIME-VARYING MULTIPATH
RECEPTION
PRE-DETECTIONFILTER
REPLICA CORROR DFT
POST-DETECTIONPROCESSOR
DECISION: H1/H0 ?
(NEXT STEPS)
78
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.
∆T
T
fff
f
f
1
f23456
TIME
FRE
QU
EN
CY
W
∆T
T
fff
f
f
1
f23456
TIME
FRE
QU
EN
CY
WWW
∆T
T
fff
f
f
1
f23456
TIME
FRE
QU
EN
CY
WWWB∆
T
fff
f
f
1
f23456
TIME
FRE
QU
EN
CY
WWWW
∆T
T
fff
f
f
1
f23456
TIME
FRE
QU
EN
CY
WWW
∆T
T
fff
f
f
1
f23456
TIME
FRE
QU
EN
CY
WWWW
∆T
T
fff
f
f
1
f23456
TIME
FRE
QU
EN
CY
WWWWB∆B∆
T
fff
f
f
1
f23456
TIME
FRE
QU
EN
CY
WWWW
79
The power spectral density of any pulse is ‘spread’ about it's carrier frequency:
∆F is a measure of this spread.
∆T
∆ B ∆ F
f6f5f4f3f2f1
FREQ
UEN
CY
W
TIMET
SPECTRUM OF PULSED WAVEFORMS
80
RAN
GE
TIME
Impulsivesource R
ANG
E
TIME
Rectangular∆T pulse
R1
∆T ∆T
RAN
GE
TIME
OpeningDoppler
∆T (∆Ts)
R1(t)
RAN
GE
TIME
ClosingDoppler
∆T (∆Ts)
R1(t)
s < 1s > 1
s = 1
TRANSMITTED AND RECEIVED DURATION TIMES
81
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
Si..
..
.
82
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
o
o
SCHEMATIC DIAGRAM FOR A SEMI-COHERENT PROCESSOR MATCHED TO STRETCH FACTOR Si
Echo
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
83
1 2 3 4 5 6 7 8 9 10
NUMBER OF PULSES, N
PRO
CES
SIN
G G
AIN
1 2 3 4 5 6 7 8 9 10
NUMBER OF PULSES, N
PRO
CES
SIN
G G
AIN
NUMBER OF PULSES, N
PRO
CES
SIN
G G
AIN
EXPECTED PROCESSING GAIN FOR FULLY COHERENT PROCESSING OVER N PULSES
Lower frequency
Higher frequency
84REVERBERATION
TRANSMISSION
DELAY ANDATTENUATION
TIME-VARYINGMULTIPATH
DELAY ANDATTENUATION
TIME-VARYING MULTIPATH
POST-DETECTIONPROCESSOR
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
85
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
o
For a linear, time invariant filter:
o
o
THE ‘MATCHED FILTER’
~ ∑∑∑+∞=
−∞=
+∞=
−∞=
+∞=
−∞==−=−=
k
k
k
k
k
k
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,
∑+∞=
−∞=−
k
kk]h[n[k]f
∑+∞=
−∞=−
k
kk]h[k][nf
=n]g[~
86
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.
o
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.
o
87
Instantaneous frequency
Time scale referred to replica waveform
Echo data moves withrespect to replica data
established in correlator
φ
Instantaneous frequency
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.
Time scale referred to replica waveform
τd 0 τd > τo
Replica data
=
t~+
t~+
0t =~
0t =~
t~−
t~−
Overlap region producing maximum correlator power output for frequency shift φ
τd = τd^
88
φ
Instantaneous frequency
Time scale referred to replica waveformt~+0t =~t~−
φ = 0
Instantaneous frequency
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
Overlap region producing maximum correlator power output for frequency shift φ
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.
Echo data moves withrespect to replica data
established in correlator
PARTIAL AND COMPLETE OVERLAP
τd = τd^
^
89
Instantaneous frequency
Time scale referred to replica waveform
Echo data moves withrespect to replica
established in correlator
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
90
0
0.2
0.4
0.6
0.8
1
2|),(| φτχ
NARROWBAND AMBIGUITY FUNCTION FOR A TYPICAL LFM WAVEFORM
91
NARROWBAND AMBIGUITY FUNCTION FOR AN LFM WAVEFORM
T=1 sec W=10 Hz Volume=0.99dB
dB
f T
92
RAN
GE
TIME
Impulsivesource R
ANG
E
TIME
Rectangular∆T pulse
R1
∆T ∆T
RAN
GE
TIME
OpeningDoppler
∆T (∆Ts)
R1(t)
RAN
GE
TIME
ClosingDoppler
∆T (∆Ts)
R1(t)
s < 1s > 1
s = 1
TRANSMITTED AND RECEIVED DURATION TIMES
93
-60 -40 -20 0 20 40 60
0.9
1.0
1.1
1.2
0
0.2
0.4
0.6
0.8
1
Stretch Parameter, S
TIME DELAY IN SAMPLING INTERVALS
WIDEBAND AMBIGUITY FUNCTION FOR A TYPICAL HFM WAVEFORM
AM
PLI
TUD
E
94
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 IN SECONDS WITH RESPECT TO WAVEFORM CENTERS
CLO
SIN
G K
NO
TSC
LOS
ING
KN
OTS
95
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!
φ
τ
T
LFM ambiguity function
W1
T1
W
Time delay resolution is W -1
Frequency resolution is T -1
φ
τT1
T
CW ambiguity function
Time delay resolution is T
Frequency resolution is T -1
96
φ
τ
φ
τ
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
τ
For το and φο
97
RAN
GE
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.
98
FREQUENCY
B
∆fDOPPLER
PO
WE
R S
PE
CTR
UM
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.( )
MEDIUM FREQUENCY DISPERSION OF A LONG CW WAVEFORM (2 of 3)
99
TIME
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
MEDIUM FREQUENCY DISPERSION OF A LONG CW WAVEFORM (3 of 3)
100
CONVOLUTION OF CW TRANSMISSION AND
FREQUENCY SPREAD
CONVOLUTION OF CW TRANSMISSION AND
FREQUENCY SPREAD
B
T-1
f = fcarrier
∆f = 0
f = fcarrier
+f
+∆f
+f
-∆f
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
T-1
f = fcarrier
f = fcarrier
∆f = 0
+f
+∆f
+f
-∆f
CONVOLUTION OF CW TRANSMISSION AND
FREQUENCY SPREAD
SINC FUNCTION OF CW TRANSMISSION
OF DURATION T.
MEDIUM’S FREQUENCY SPREAD WITH MEAN B
101
RA
NG
E
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.
0
Bulk time delay(or just ‘time delay’)
MEDIUM TIME DISPERSION OF TRANSMITTED GAUSSIAN PULSE
L
102
0
TIME
f2
f1
to
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
1.0
∆T
Proportional to L-1
CW echo f2 experiences a local maximum but echo f1 does not.
TIME-DELAY SPREAD AFFECTS SIGNAL FADING
103
BL
EFFECT OF TIME-DELAY & FREQUENCY SPREAD ON AN LFM WAVEFORM, TO = 1 Sec, W = 400 Hz
L = 5 ms, B = 2 Hz
dB
dB
- 0.01 - 0.005 0 0.005 0.01
Time Delay, τ Seconds
Time Delay, τ Seconds
Fre
quency S
hift, H
ert
zFre
quency S
hift, H
ert
z
Ambiguity Function, B = 0 and L = 0
104
0 2 4 6 8 10
0
12.5
25
0
- 5
-10
dB
B, Frequency Spread, Hz
L, T
ime-
dela
y Sp
read
, m
illis
econ
ds
PEAK RESPONSE LOSS FOR LFM WAVEFORM SUBJECTED TO REPLICA CORRELATION
To = 1 Second, W = 400 Hz
dB
dB
105
Recall the results for the ratio of signal output power to interference output power for a replica correlator:
=SN
OUTPUT
SoT2(1 cycle) oΝ
=SR
OUTPUT
SW2(1 cycle) oR
INPUT
INPUT
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:
W
W
