Fundamentals of Passive and Active Sonar Technical Training Short Course Sampler

FUNDAMENTALS OF PASSIVE AND ACTIVE SONAR

Instructor:

Duncan Sheldon, Ph.D.

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DECISION: H1/H0 ?
ACTIVE SONAR DETECTION MODEL
POST-DETECTION
PROCESSOR

TRANSMISSION
DETECTOR

PRE-DETECTION
FILTER

DELAY AND
ATTENUATION AMBIENT NOISE RECEPTION

TIME-VARYING
MULTIPATH

TARGET
TIME-VARYING DELAY AND
MULTIPATH ATTENUATION
REVERBERATION 2

DEFINITIONS

Acoustic pressure, p: The difference between the total pressure, ptotal, and
the hydrostatic (or undisturbed) pressure po.
ptotal = ptotal(x,y,z;t) Pascals
po = po(x,y,z) or
Newtons/(meter)2
p = p (x,y,z;t) = ptotal - po

  
Acoustic intensity, I : A vector whose component I •n in the direction of any

unit vector n is the rate at which energy is being

transported in the direction n across a small plane

element perpendicular to n , per unit area of that plane
element.
  
I ≡ pu ; where u is the fluid velocity at (x,y,z;t).
 
I = I (x,y,z;t) Watts/(Meter)2
3

ACOUSTIC INTENSITY LEVELS EXPRESSED IN DECIBELS
(prms)2
For a plane wave, Iref = ρ c
ref
PRESSURE
1 Pascal = 1 Newton/(1 Meter)2
prms = Root -mean -square pressure

ENERGY ρ = Density
1 Joule = 1 Newton x 1 Meter c = Sonic velocity
POWER prms, ref water = 10- 6 Pascals
1 Watt = 1 Joule/(1Second)
ρ c water = 1.5 x 10 + 6 kgm/(meter 2second)
INTENSITY (or ENERGY FLUX)
Watts/(Meter)2 prms, ref air = 20 x 10- 6 Pascals
Joules/Second /(Meter)2
ρ c air = 430 kgm/(meter 2second)
Intensity level, I, usually refers to a normalized intensity magnitude,

( | I |average /Iref), expressed in decibels. For example, I = 60 dB means

60 dB = 10 log10 (| I | average/ Iref)

6 = log10 (| I | average/ Iref)

I = 10 +6 x Iref [ Watts/(meter2) ] 4

ACOUSTIC INTENSITY REFERENCE VALUES: AIR AND WATER
 (prms)2
Average acoustic intensity magnitude for a plane wave, | I | =
ρc .
prms = Root -mean -square of acoustic pressure
ρ = Density
c = Sonic velocity

For water: Iref water ≡ | I |ref water

ρ c = 1.5 x 10 + 6 kgm/(meter 2 second)

Iref water ≡ [10 − 6 Pascals] 2 [ρ c]
/ = 0.67 x 10 −18 Watts / (meter)2
water

For air: Iref air ≡ | I |ref air

ρ c = 430 kgm/(meter 2 second)

Iref air ≡ [ 20 x 10 − 6 Pascals] 2 [ρ c]
/ ≈ 10 −12 Watts / (meter)2
air
5

DECIBELS MEASURED IN AIR AND WATER

X watts/meter2
100 dB water re 10-6 Pa = 10log10 ( I ref water watts/meter2
)
X watts/meter2
= 10log10 ( )
6.76 x 10-19 watts/meter2
Y watts/meter2
100 dB air re 20 x 10-6 Pa = 10log10 ( )
I ref air watts/meter2

Y watts/meter2
= 10log10 ( )
10-12 watts/meter2

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

For air: Y watts/meter2 = 10-2 watts/meter2
10-2 watts/meter2
10log10 ( ) = 61.7 dB difference in intensity level.
6.76 x 10-9 watts/meter 2

6

REPLICA CORRELATION (1 of 6)
o 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:
k = +∞
xcorr[x,y;n] = ∑x[k] y[k -n] ≡ g[n]
k = −∞
where g[n] is used to denote a cross-correlation sequence whose
constituent sequences are understood but not explicitly stated.

k = +∞
INPUT x[k] X ∑x[k] y[k -n] g[n] OUTPUT
k = −∞

REPLICA y[k-n] REPLICA CORRELATOR

o 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:
 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. 7

40
x[k], 30 k(2) NON-ZERO
INPUT 20 VALUES OF x[k]
(SIGNAL) 10
k(2) = 4
k(1), FIXED BULK DELAY
k
0
k(1) k(1) + k(2)
y[k], k(2) NON-ZERO
REPLICA VALUES OF y[k]
4
3
2
1 k(2) = 4
k
0 1 2 3

y[k-n],
SHIFTED n INCREASES Each n > 0 produces a
4
REPLICA 2
3 different (shifted)
1
y[k-n] sequence.
k
0 n n+ k(2)
y[k-k(1)],
SHIFTED
REPLICA 4
3
ALIGNED 2
n = k(1) 1
WITH INPUT
k
0 k(1) k(1) + k(2) 8

40
(SIGNAL) 10
k(2) = 4
k(1) FIXED
k
k(1) k(1) + k(2)
y[k-n],
SHIFTED
n INCREASES 4
y[k-n] when n=k(1)
REPLICA 3 4
2 2 3
1 1

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

g[n],
k(1) 300
OUTPUT OF
REPLICA 200 200
CORRELATOR n = k(1)-k(2) 110 110 n = k(1)+k(2)
40
40
n
n= k(1)

The cross-correlation g[n] of x[k[ and y[k] is the output of the replica correlator, and
k ≥ k(1)+ k(2)
g[n] = ∑
k=
x[k]y[k − n] where n is the shift of y[k] with respect to x[k].
9
0


40
(SIGNAL) 10
k(2) = 4
k(1) FIXED
k
k(1) k(1) + k(2)

Instead of writing
k ≥ k(1)+ k(2)
g[n] = ∑
k=
x[k]y[k − n]
0
we can write
k = +∞
g[n] = ∑∞ y[k −n]
k=−
x[k]

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

10

SIGNAL k = +∞
INPUT +
NOISE
x[k] X ∑x[k] y[k -n] g[n] OUTPUT
k = −∞


SIGNAL+NOISE
REPLICA
2 k = +∞ 2
| g[n]| = | ∑∞ y[k −n] |
k=−
x[k]

o 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.

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

INSTANTANEOUS POWER
OUTPUT OF REPLICA THRESHOLD
CORRELATOR,
SIGNAL + NOISE
2
|g[n]|

n
n = k(1)
BULK DELAY, k(1) 11

SIGNAL k = +∞
INPUT +
NOISE
x[k] X ∑x[k] y[k -n] g[n] OUTPUT
k = −∞


SIGNAL+NOISE
REPLICA
2 k = +∞ 2
| g[n]| = | ∑∞ y[k −n] |
k=−
x[k]

INSTANTANEOUS POWER THRESHOLD
OUTPUT OF REPLICA
CORRELATOR,
SIGNAL + NOISE
2
|g[n]|
n
n = k(1)
BULK DELAY, k(1)

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.

12

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

TRANSMISSION
s(t) t
BULK DELAY

t =τ BD ECHO
s ( t − τ BD ) t
AMBIENT NOISE

n (t ) t
RECEIVED DATA

s ( t − τBD ) + n ( t ) t
REPLICAS OF THE

t = τ RD
TRANSMISSION FOR
s(t-τRD)
(1)
(1) DIFFERENT REPLICA DELAYS

REPLICA
t
t = τ RD
DELAYS
s(t-τRD)
(2)
(2)
t
t=0 13


ECHO
t =τ BD RECEIVED DATA

s ( t − τBD ) + n ( t )
t
REPLICAS OF THE
t = τ RD
(1) TRANSMISSION FOR
DIFFERENT REPLICA DELAYS
s(t-τRD)
(1)
t
t = τ RD
REPLICA (2)
DELAYS
s(t-τRD)
(2) t
t=0
Shifted replicas, each with the same shape but different delays,
τRD, i=1,2, …, n,
(i)
are under consideration,

Only one record of the received data is under consideration.

14

INCREASING
REPLICA DELAY REPLICA OF THE
TRANSMISSION FOR
τ RD
Replica
SOME REPLICA DELAY

Time, t
τ BD
Signal

τd τd = τBD - τRD
Output of the replica correlator is the product of the
echo and the replica for time delays τd.
Note triangular
shape of upper
Peak value is peaks.
at τ d= 0

−τd Time delay, τd

Replica correlator output
and its envelope
τd =0 for a CW waveform 15


INCREASING
REPLICA DELAY REPLICA OF THE
TRANSMISSION FOR
τ
Replica
SOME REPLICA DELAY RD

Time, t
Received
data

τd

Time delay, τd
−τd

τd =0 for a CW waveform
plus ambient noise. 16

REPLICA CORRELATOR OUTPUT FOR A CW
WAVEFORM MASKED BY NOISE (1 of 3)
15
10
5 Independent normally
0 distributed noise only.
-5
INSTANTANEOUS
AMPLITUDES

-10
-15 0 200 400 600 800 1000 1200
SAMPLED DATA POINTS
Sine wave 240 samples
15
long with zero-padding on
10 each side.
5

0
Normally distributed
-5 noise plus sine wave.
-10

-15
Replica of sine wave
0 200 400 600 800 1000 1200
SAMPLED DATA POINTS (offset by -5).
150

Sine wave
CORRELATOR

100
σNOISE = 4.0 amplitude = 1.0
REPLICA

OUTPUT

50

0
Sine wave power
-50
= 0.516 = 132
-100 Noise power
-150

10 log( 132) = -15 dB
-600 -400 -200 0 200 400 600
17
TIME DELAY τd IN DATA SAMPLES

15
10
Independent
5
normally
0
distributed
-5 noise only.
INSTANTANEOUS

-10
AMPLITUDES

-15
0 500 1000 1500 2000 2500 Sine wave 480
SAMPLED DATA POINTS samples long with
15 zero-padding on
10 each side.
5
Normally
0 distributed noise
-5 plus sine wave.
-10
Replica of sine
-15
0 500 1000 1500 2000 2500 wave (offset -5).
SAMPLED DATA POINTS
REPLICA CORRELATOR

300
200 As on previous
slide,
OUTPUT

100
0 Sine wave power
-100 Noise power
-200
=> -15 dB
-300
0
-1000 -500 0 500 1000 18

15
10
Independent
5
normally
0
distributed
-5 noise only.
INSTANTANEOUS

-10
AMPLITUDES

-15
0 500 1000 1500 2000 2500 Sine wave 480
SAMPLED DATA POINTS samples long with
15 zero-padding on
10 each side.
5
Normally
0 distributed noise
-5 plus sine wave.
-10 Note
Replica of sine
-15
0 500 1000 triangular 2000
1500 2500 wave (offset -5).
SAMPLED DATA POINTS shape
REPLICA CORRELATOR

300
200 As on previous
slide,
OUTPUT

100
0 Sine wave power
-100 Noise power
-200
=> -15 dB
-300
0
-1000 -500 0 500 1000 19

RANDOM WALK SAMPLE AND A GENERAL RESULT (1 of 2)
INDEPENDENT NORMALLY DISTRIBUTED NOISE, σ = 1
3
NOISE VALUES
2
1
0
-1
-2
-3
-4
0 100 200 300 400 500 600 700 800 900 1000
SAMPLED DATA POINTS
N INCREASING N
STEPS REMOVED FROM

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

20
10
0
-10
-20
-30
-40
-50
0 100 200 300 400 500 600 700 800 900 1000
NUMBER OF STEPS, N

Root-mean-square departure from starting position is N . 20

NOISE SAMPLE MULTIPLIED BY A REPLICA

INDEPENDENT NORMALLY DISTRIBUTED NOISE, σ = 1
3
VALUES 2
1
NOISE

0
-1
-2
-3
-4 0 100 200 300 400 500 600 700 800 1000
900
SAMPLED DATA POINTS
REPLICA
1
REPLICA
VALUES

0

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

POINT-BY-POINT PRODUCTS OF REPLICA AND NOISE
2
PRODUCT
VALUES

1
0
-1
-2
0 100 200 300 400 500 600 700 800 900 1000
SAMPLED DATA POINTS
21

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

POINT-BY-POINT PRODUCTS OF REPLICA AND NOISE
(FROM PREVIOUS SLIDE)
2
PRODUCT
VALUES

1
0
-1
-2
0 100 200 300 400 500 600 700 800 900 1000
SAMPLED DATA POINTS
N INCREASING N
SUM OF ABOVE PRODUCT VALUES OUT TO N POINTS
SUM OF PRODUCT

10
5
0
VALUES

-5
-10
-15
-20
-25
-30
-35
0 100 200 300 400 500 600 700 800 900 1000
N, NUMBER OF PRODUCTS SUMMED

Root-mean-square departure from starting position (zero) is proportional to N .
22

PEAK REPLICA CORRELATOR OUTPUT WHEN ECHO MATCHES REPLICA
ECHOES OF INCREASING DURATION
1

0
N
-1
0 100 200 300 400 500 600 700 800 900 1000
SAMPLED DATA POINTS

REPLICAS OF INCREASING DURATION
1

0
N
CORRELATOR OUTPUT

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

SAMPLED DATA POINTS
500
400
300
200
100
0
0 100 200 300 400 500 600 700 800 900 1000
LENGTH OF REPLICA CORRELATION IN SAMPLED DATA POINTS, N

Peak replica correlator output with matched inputs is (nearly) proportional to N,
not N . 23

SUMMARY

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:

1) The expectation of the root-mean-square noise
1
output of the replica correlator increases as N , 2

2) The peak output of the replica correlator due to
the signal increases (nearly) linearly with N.

3) The ratio of the peak signal output to the root-mean-square
N
noise output is expected to increase as 1 ,
N 2

4) The peak signal-to-noise instantaneous power output of
  2

 N 

=
the replica correlator is expected to increase as 
 1

 N.



N2 


24

HYPOTHETICAL TRANSMISSION, RECEIVED DATA, AND
INSTANTANEOUS POWER OUTPUT OF A REPLICA CORRELATOR
ARBITRARY N SAMPLES
UNITS REPLICA OF TRANSMISSION
FOR TIME SHIFT τd
TIME

RECEIVED DATA,
SIGNAL MATCHING REPLICA PLUS NOISE
ARBITRARY
UNITS

TIME

τd
OUTPUT OF REPLICA OUTPUT AT τd = 0
ARBITRARY

CORRELATOR DUE TO SIGNAL ~ N
UNITS

TIME DELAY
- τd +τd
OUTPUT DUE TO ROOT-MEAN SQUARE VALUE
NOISE ALONE OF NOISE ALONE ~ N1/2

OUTPUT AT τ=0 INSTANTANEOUS POWER OUTPUT OF
ARBITRARY

DUE TO SIGNAL ~ N2 REPLICA CORRELATOR AND ITS PEAK
UNITS

POWER ENVELOPE
TIME DE;AY
- τd +τd
τd = 0 MEAN VALUE OF NOISE
25
POWER ALONE ~ N
(Too weak to be seen on this scale.)

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

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

ARBITRATY UNITS
POWER OUTPUT,

TOO HIGH A THRESHOLD LEADS
TO MISSED DETECTIONS TOO LOW A THRESHOLD
LEADS TO EXCESSIVE
FALSE ALARMS

0
-100
-1000 -500 0 500 1000

-200 SAMPLED DATA POINTS FORWARD AND BACKWARD IN TIME
FROM EXACT OVERLAP OF CW WAVEFORM AND ITS REPLICA

26

PROBABILITIES OF FOUR POSSIBLE OUTCOMES
SEPARATE AND DISTINCT
ENSEMBLES
pd is the probability a threshold
crossing will occur when a target
TRUTH
is actually present. COLUMNS
pfa is the probability a threshold
crossing will occur when a target
TARGET IS TARGET IS
is not present. PRESENT ABSENT

THRESHOLD IS CORRECT FALSE
CROSSED,
DECIDE TARGET DETECTION ALARM
INSTANTANEOUS
PRESENT pd pfa
POWER OUTPUT OF
REPLICA THRESHOLD NOT NO
CORRELATOR
MISSED
CROSSED, ACTION
DECIDE TARGET
DETECTION
ABSENT 1 - pd 1 - pfa

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

27

DECISION: H1/H0 ?
(DISCUSSED SO FAR, ALONG WITH SONAR EQUATION)
POST-DETECTION
PROCESSOR

REPLICA
TRANSMISSION FOR CW
CORRELATOR,
TRANSMISSIONS DFT DETECTOR

PRE-DETECTION
FILTER
(Beamformer)

DELAY AND RECEPTION
ATTENUATION AMBIENT NOISE
(Receive array)

TIME-VARYING
MULTIPATH

TARGET
REVERBERATION 28

DECISION: H1/H0 ?
(NEXT STEPS)
POST-DETECTION
PROCESSOR

TRANSMISSION FOR FREQUENY REPLICA
MODULATED CORRELATOR,
TRANSMISSIONS DFT DETECTOR

PRE-DETECTION
FILTER
(Beamformer)

DELAY AND RECEPTION
ATTENUATION AMBIENT NOISE
(Receive array)

TIME-VARYING
MULTIPATH

TARGET
REVERBERATION 29

REPLICA CORRELATION FOR A CONTINUOUS CW WAVEFORM
Replica matches echo

Replica
PRESSURE except for time delay.

Time, t
Echo

+τ d
and its envelope
for a CW waveform.
- τd +τd

τd = 0

τd is the time delay of the echo with respect
to the replica established in the correlator. 30

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

Frequency = fo + φ
Echo
PRESSURE

Time, t
Replica

Frequency = fo

Envelope and its envelope
narrows
for a CW waveform
- τd +τd
τd = 0

φ is the frequency difference (shift) of the echo with respect to
the replica established in the correlator. The greater | φ |, the
narrower the envelope of the correlator’s output
31

TRANSMITTED AND RECEIVED CW FREQUENCIES
AFTER APPLYING A NARROWBAND DOPPLER SHIFT
Instantaneous frequency

g6
Received CW frequencies
f6 g1 , g 2 , g 3 , … , g 6
g5
are each the result
f5 of the same narrowband
g4
Doppler shift with respect
f4 to equally spaced transmitted
g3
CW frequencies
f
g2 3 f1 , f 2 , f 3 , … , f 6

g1 f 2

f1 T TRANSMIT
Time
T RECEIVE
32

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

g6

g5 Received CW frequencies
f6 g1 , g 2 , g 3 , … , g 6
are each the result
g 4 f5 • of a different narrowband
Doppler shift with respect
f4 to equally spaced transmitted
g3
CW frequencies
f3 f1 , f 2 , f 3 , … , f6
g2
f2
g1
f1 T TRANSMIT
Time
T RECEIVE
33

DOPPLER PARAMETER S AND THE
WIDEBAND DOPPLER TRANSFORMATION

o s ≡ (1 − ν / c) / (1 + ν / c) = 1 − (2ν / c ) if ν / c << 1

where
ν = Constant range-rate of a reflector, positive closing, and
c = Sonic velocity in the medium.

o Each closing range-rate νi maps into a ‘stretch’ parameter s : νi si
i
1 to 1
o Each transmitted CW pulse of frequency fk becomes a
received CW pulse of frequency gki that depends upon si:

gki = f 1 + 2ν i / c = f k /si k = 1,2, …, N
k
o The difference between the received and transmitted
CW frequencies depends on both si and fk:

s
gki _ fk = fk 1 - i
si 34

EFFECT SHIFTING A CW WAVEFORM’S FREQUENCY BY φ
USING THE WIDEBAND DOPPLER TRANSFORMATION
C
L
Frequency = fo + φ

Echo
PRESSURE

Time, t
- τd
Replica

Frequency = fo

C
L
Envelope of replica
correlator output
- τd +τd

τd = 0
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. 35

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

Echo
PRESSURE
No
frequency
Time, t
Replica
shift

- τd
and its envelope for a
frequency-modulated waveform

- τd +τd

τd = 0


Echo
PRESSURE
No
frequency
Time, t
Replica
shift

- τd
and its envelope for a
frequency-modulated waveform

- τd +τd
Time resolution of the
envelope is ~W-1 where
τd = 0 W is the bandwidth of
the waveform.


C
L

Echo
PRESSURE
With echo
+τˆd frequency
Replica

shift

τˆd is the time delay when C
L
the replica correlator’s output +τˆd and its envelope for a
envelope is a maximum. frequency-modulated echo
experiencing a frequency shift.
- τd +τd

τd = 0
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. 38

THE LINEAR FREQUENCY MODULATED (LFM)
WAVEFORM OF UNIT AMPLITUDE

Instantaneous frequency, Hz

Bandwidth, W fc , Carrier
frequency >> W

-t +t
-T/2 +T/2
Time, Seconds

39

TIME-FREQUENCY DIAGRAM REFERRED TO REPLICA WAVEFORM (1 of 2)
Instantaneous
frequency
Echo φ , frequency shift
Replica

~ ~
-t ~
+t
t=0 Time

T
Echo

~
PRESSURE

+t
Replica

~
+t

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.
40

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

Instantaneous Frequency shift is not uniform
frequency (closing target produces
Echo
(no narrowband increased echo frequencies)
assumption) Replica

~ ~
-t +t
~
t=0 Time

T Contraction if
frequency increases
Echo

~
PRESSURE

+t
Replica

~
+t

In the wideband case, the echo’s frequency shift is not uniform
over its duration, and any frequency shift brings about a dilation
or contraction in the duration of the waveform. 41

TIME DELAY DIAGRAMS

Echo data moves with
Instantaneous respect to replica
frequency Replica established in correlator
(narrowband
assumption)
Increasing clock time
τd
~ ~
-t +t
~
t=0 Time scale referred to replica waveform

BULK DELAY
t =τ BD ECHO
s ( t − τ BD ) t
τd > 0 Clock time

s(t-τRD )
(1) t = τ RD
(1) REPLICA

REPLICA
t
DELAY t=0
42

TIME DELAY DIAGRAMS

(narrowband
assumption)

~ ~
-t +t
~

BULK DELAY
t =τ BD ECHO
s ( t − τ BD ) t
τd > 0 Clock time

s(t-τRD )
(1) t = τ RD
(1) REPLICA

REPLICA
t
DELAY t=0
43

TIME DELAY DIAGRAMS

(narrowband
assumption)

~ ~
-t +t
~

BULK DELAY
t =τ BD ECHO
s ( t − τ BD ) t
τd > 0 Clock time

s(t-τRD )
(2) t = τ RD
(2) REPLICA
t
REPLICA
DELAY t=0
44

TIME DELAY DIAGRAMS

(narrowband
assumption)

~ ~
-t +t
~

BULK DELAY
t =τ BD ECHO
s ( t − τ BD ) t
τd > 0 Clock time

s(t-τRD )
(3) t = τ RD
(3) REPLICA
t
REPLICA
DELAY t=0
45

TIME DELAY DIAGRAMS

Instantaneous τ d = τˆd when overlap is greatest respect to replica
frequency established in correlator
(narrowband
assumption) Replica

~ ~
-t +t
~

BULK DELAY
t =τ BD ECHO
s ( t − τ BD ) t
τ d = τˆd > 0 Clock time

s(t-τRD )
(4) t = τ RD
(4) REPLICA
t
REPLICA
DELAY t=0
46

TIME DELAY DIAGRAMS

(narrowband
assumption) Replica

~ ~
-t +t
~

BULK DELAY
t =τ BD ECHO
s ( t − τ BD ) t
τd > 0 Clock time

s(t-τRD )
(5) t = τ RD
(5) REPLICA
t
REPLICA
DELAY t=0
47

TIME DELAY DIAGRAMS

(narrowband
assumption) Replica

~ ~
-t +t
~

BULK DELAY
t =τ BD ECHO
s ( t − τ BD ) t
τd = 0 Clock time

s(t-τRD )
(6) t = τ RD
(6) REPLICA
t
REPLICA
DELAY t=0
48

TIME DELAY DIAGRAMS

(narrowband
assumption) Replica

~ ~
-t +t
~

BULK DELAY
t =τ BD ECHO
s ( t − τ BD ) t
τd< 0 Clock time

s(t-τRD )
(7) t = τ RD
(6) REPLICA
t
REPLICA
DELAY t=0
49

USING THE TIME-FREQUENCY DIAGRAM TO ESTIMATE THE MAXIMUM
OUTPUT OF A REPLICA CORRELATOR WHEN THE ECHO
HAS A FREQUENCY SHIFT φ
Instantaneous
frequency τd = 0 τd > τ^d
Replica Echo data moves with
data respect to replica data
established in correlator
−~
t ~ =0 +~
t
t Time scale referred to replica waveform
Instantaneous
frequency τd = τd
^

φ Overlap region producing
maximum correlator power
output for frequency shift φ
−~
t ~ =0 +~
t
t Time scale referred to replica waveform

The greater the overlap region, the larger the maximum replica correlator output.
In this example an LFM waveform has experienced a narrowband Doppler shift.
50

TIME-FREQUENCY DIAGRAM REFERRED TO REPLICA WAVEFORM

Instantaneous Frequency shift is not uniform
frequency (closing target produces
Echo
(no narrowband increased echo frequencies
assumption) Replica at higher frequencies)

~ ~
-t +t
~
t=0 Time

T Contraction if
frequency increases
Echo

~
PRESSURE

+t
Replica

~
+t

In the wideband case, the echo’s frequency shift is not uniform
over its duration, and any frequency shift brings about a dilation
or contraction in the duration of the waveform. 51

USING THE TIME-FREQUENCY DIAGRAM TO ESTIMATE THE
MAXIMUM OUTPUT OF A REPLICA CORRELATOR
Instantaneous
frequency
(no narrowband
assumption)
Replica Echo data moves with
~ respect to replica ~
-t +t
~ Time scale referred to replica waveform
t=0
Instantaneous
frequency
τd
Overlap region producing
maximum correlator output
~ ~
-t +t
t=0
Here an LFM echo has experienced a wideband Dopper transformation
rather than a narrowband Doppler shift. The overlap with the replica is
reduced, and the corresponding replica correlator output is reduced. 52

HYPERBOLIC FREQUENCY MODULATED (HFM) WAVEFORMS,
THEIR TIME-PERIOD AND TIME-FREQUENCY DIAGRAMS
Instantaneous period τ (t), linear
τ1
∆τ το
seconds
per cycle
τ2
-t +t
-T/2 0 +T/2
Instantaneous frequency (hyperbolic)

F2 = (τ2 )−1
W
F1 = (τ1 )−1
-t +t
-T/2 0 +T/2
HFM waveforms and linear period modulated
53
(LPM) waveforms are the same.

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

Instantaneous Higher frequency echo
frequency data moves with
(no narrowband τd respect to replica
assumption)
Overlap region producing
maximum correlator output
Replica
~ ~
-t +t
t=0

Here an HFM echo has experienced a wideband Dopper transformation
and the HFM’s time-frequency distribution adjusts itself to remain ‘more’
overlapped than in a similar case for an LFM waveform.

54

ANALYTIC SIGNAL OF A REAL WAVEFORM (1 of 2)
ο 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).
ο 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.
|U(f)|

ENVELOPE
Q(f)
-f +f
+1
- fo2 + fo2 -f +f

|Ψ(f)|
FT
ENVELOPE u(t) ← → U ( f )


Ψ(f) = 2 Q(f) U(f)
FT
ψ (t ) ← → Ψ( f )
-f +f
55
+ fo2

ANALYTIC SIGNAL OF A REAL WAVEFORM (2 of 2)

ο 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).
ο 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 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.
|U (f)|
Q(f)

+1
-f +f -f +f

- fo + fo
FT
ENVELOPES |Ψ (f)| u(t) ← → U ( f )


Ψ(f) = 2 Q(f) U(f)
-f FT
+f ψ (t ) ← → Ψ( f )
56
+ fo

ANALYTIC SIGNAL WITHOUT APPROXIMATION (1 of 2)

|U(f)| Q(f)
ENVELOPE
+1
-f +f
-f +f

- fo2 + fo2 FT
u(t) ← → U ( f )


|Ψ(f)| Ψ(f) = 2 Q(f) U(f)
ENVELOPE
FT
ψ (t ) ← → Ψ( f )
+∞
u(τ ) dτ
-f +f
ψ (t) = u(t) + j
∫
− ∞ π (t −τ )

+ fo2 u(t) = Re (ψ (t))

+∞ is the Hilbert transform of u(t)
u(τ ) dτ
∫
− ∞ π (t −τ ) ˆ
and is often denoted by u(t).

57

ANALYTIC SIGNAL WITHOUT APPROXIMATION (2 of 2)

ο ψ (t) is the analytic signal of real waveform u(t), and is always given by
+∞
u(τ ) dτ
ψ (t) = u(t) + j u(t) = u(t) + j
ˆ
∫
− ∞ π (t −τ )
even if u(t) is not narrowband! However, it may be difficult to find ˆ
u(t).

1.5 u(t) = - sin(2πfot)
u (t ) -0.5 < t < +0.5, Seconds
Instantaneous Amplitude

1
ˆ
u (t ) u(t) = 0; t > |0.5| Seconds
0.5 Envelope
fo = 3 Hertz; T = 1 Second
0
The envelope of u(t)
-0.5 is given by:

u 2 (t) + u 2 (t)
ˆ
-1

-1.5
-1.5 -1 -0.5 0 0.5 1 1.5 2
Time, seconds 58

APPROXIMATING THE ANALYTICAL SIGNAL WITH A
NARROWBAND COMPLEX WAVEFORM (1 of 6)
|U (f)|
Q(f)
ENVELOPE
+1
-f +f -f +f

- fo + fo
FT
|Ψ (f)| u(t) ← → U ( f )

ENVELOPE
Ψ(f) = 2 Q(f) U(f)
-f FT
+f ψ (t ) ← → Ψ( f )

+ fo
+∞
u(τ ) dτ
The analytic signal is always given by ψ (t) = u(t) + j
∫
− ∞ π (t −τ )

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

ψ (t) ≅ ψ (t) = [a(t) + j b(t)] exp[ 2π j fo t ]
ˆ
where a(t) and b(t) are real, +fo is a central frequency of U(f),
and a(t) and b(t) vary slowly compared to 2π fot.
59

ANALYTIC SIGNAL AND COMPLEX ENVELOPE OF REAL WAVEFORM u(t,fo)
ψ(t,fo) IS THE INVERSE FOURIER
U(f,fo) IS FOURIER TRANSFORM OF u(t.fo)
TRANSFORM OF Ψ (f,fo)
|U ( f , fo ) | | Ψ( f , fo ) | FT
ψ (t, fo ) ←→ Ψ( f , fo )
ENVELOPE
ENVELOPE
OF |U( f , fo )|
OF |Ψ ( f , fo )|
-f +f

-fo 0 fo -f 0 fo +f
DOWNSHIFT BY fo

ψ(t,fo) is the analytic signal υ(t) IS THE INVERSE FOURIER
~
of real waveform u(t.fo). TRANSFORM OF Ψ(f)
~
| Ψ( f ) | FT ~
υ (t ) ←→ Ψ( f )
υ(t) is the complex envelope
ENVELOPE
of real waveform u(t,fo). ~
OF |Ψ ( f )|

60
-f 0 +f

FACTORS OF THE COMPLEX NARROWBAND WAVEFORM ψ (t)
ˆ

Im Im
exp(+2π j fo t) |ψ (t)| =
ˆ
1/2
[a(t)2 + b(t)2 ]
b(t)
Re Re
a(t)

unit exp(-2π j fo t)
circle absent
a(t) and b(t) are real and
ψ (t) = [a(t) + j b(t)] exp(+2π j fo t)
ˆ vary slowly with respect
u(t) = Re (ψ (t) )
ˆ to 2π fot.

[a(t) + j b(t)]is the
complex envelope of ψ (t ).
ˆ 61

COMPLEX PLANE FOR THE NARROWBAND ψ (t ) WAVEFORM
ˆ
Im
ψ(t), rotates

‹
Im[ψ(t)]

‹
counter clockwise
Φ(t)
Re
Re[ψ(t)] = u(t)

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

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)
‹

|ψ(t)| = a(t)2 + b(t)2 = A(t )
‹

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

‹
‹

j[ Φ(t) ]
ψ (t) = A(t) e
ˆ , Φ(t ) = 2π fo t + θ (t )

62

DEFINITION OF INSTANTANEOUS FREQUENCY

j[ Φ(t) ]
ψ (t) = A(t) e
ˆ is a particularly convenient form of the

analytic signal because Φ(t) is the argument of ψ (t ) and the
ˆ
instantaneous phase of u(t) in radians.

The instantaneous frequency of u(t) in Hertz is

finstantaneous(t ) ≡ fi (t ) ≡ 1 d [Φ(t)]
2 π dt
Given fi(t), the argument of ψ (t) and instantaneous phase of
ˆ
u(t) is given by
Φ(t) = 2π
∫ fi (t) dt + θ o

And we can write ψ (t ) as
ˆ

ψ (t) = A(t) e
ˆ [ ∫ ]
j 2π fi (t) dt + θ o
63

AMPLITUDE, FREQUENCY, AND PHASE MODULATION

ο In the expression ψ (t )
ˆ = A(t ) e [ ∫
j 2π fi (t) dt + θ o]
 Variations in A(t) are amplitude modulation or AM
( A(t) = a(t )2 + b(t )2 ) ,
 Variations in fi(t) are frequency modulation or FM.

ο Most active sonar waveforms are either frequency or amplitude
modulated; frequency modulation is more common.

ο 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

ψ (t) = A(t) [ exp ( j (2π fo t + θ (t) ) )]
ˆ
i.e., θ(t) ‘switches’ or ‘flips’ the phase of the carrier frequency.
64

IMPORTANCE OF THE COMPLEX ENVELOPE

ο In the expressions

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

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

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

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

 Amplitude modulation is a(t)2 + b(t)2 = A(t ) , and

 Frequency modulation is 1 d [θ (t ) ] where θ (t) = Arg ( a(t) + j b(t))
2π dt
ο All the properties of the waveform (favorable and unfavorable!)
independent of fo are embedded in the complex envelope.

ο 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.
65

OBTAINING THE ANALYTIC SIGNAL AND REAL WAVEFORM

o General form of the analytic signal is ψ (t ) = A(t ) e j[Φ(t)]
d
Start with fi(t) for the
 finstantaneous(t ) ≡ fi (t ) ≡ 1 [Φ(t)]
waveform you want. 2 π dt

∫
 Φ(t) ≡ 2π fi (t) dt
Next integrate to Need not be
find the phase. narrowband.
Finally, insert Φ(t) in the Φ(t) = 2π fo t + θ (t) + θo
general form for ψ(t). If narrowband.
where d[θ(t)]/dt << 2π fo


( A(t) = a(t )2 + b(t )2 ) ,

o A(t) is often explicitly stated, i.e., rect(t/T).
o Re[ψ(t)] gives the real waveform, u(t). 66

OBTAINING THE ANALYTIC SIGNAL AND REAL WAVEFORM

o General form of the analytic signal is ψ (t ) = A(t ) e j[Φ(t)]
d
Start with fi(t) for the
 finstantaneous(t ) ≡ fi (t ) ≡ 1 [Φ(t)]
waveform you want. 2 π dt

∫
 Φ(t) ≡ 2π fi (t) dt
Then integrate to An indefinite integral with
find the phase. a constant of integration.
Finally, insert Φ(t) in the Φ(t) = 2π fo t + θ (t) + θo
general form for ψ(t). If narrowband.
where d[θ(t)]/dt << 2π fo


( A(t) = a(t )2 + b(t )2 ) ,

o A(t) is often explicitly stated, i.e., rect(t/T).
o Re[ψ(t)] gives the real waveform, u(t). 67

THREE COMMON WAVEFORMS

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

o CW (continuous wave)

o LFM (linear frequency modulation)

o HFM (hyperbolic frequency modulation)

68

FT sin ( π f T )
COMPLEX NARROWBAND u (t) = rect (t /T ) ←  →
 T = U( f )
CW WAVEFORMS OF πfT
FT
UNIT AMPLITUDE u (t) exp ( 2π j fc t) ←  →
 U ( f − fc )

ψ (t) = 1T

rect (t/T ) exp [ j 2 π ( fc t + θo )] 0.8T Shift to
2π frequency fc
0.6T
where T fc >> 1 cycle
0.4T to get U(f - fc )

U(f)
1 if |t/T| < 1/2
rect (t/T ) = 0.2T
0 if |t/T| > 1/2
0T
θο radians is a constant
that determines the phase -0.2T
of the CW waveform.
-0.4T-10 -8
∫
-6 -4 -2 0 2 4 6 8 10
(Φ(t) ≡ 2π fc dt = 2π fc t + θo ) T T T T T T T T T T T
FREQUENCY
Cycles /Second 69


Bandwidth, W Carrier
fc frequency >> W

-t +t
-T/2 +T/2
Instantaneous phase (radians) = Φ(t) = 2 π fc t + θ (t) , t < |T/2| and TW >> 1

and instantaneous frequency (Hertz) = f (t ) ≡
1 d[ Φ(t) ]
i 2π dt
= 1 d [ 2 π fc t + θ (t) ] = fc + 1 d [θ (t) ]
2π dt 2π dt

Linear frequency modulation means
1 d [θ (t) ] = k t so θ (t) = 2 π k t 2 + θ
o
2π dt 2
i.e., the instantaneous frequency is a linear function of time, and k = W/T from the figure.

Therefore the LFM’s instantaneous phase = 2 π ( fc t + k t 2 + θo ) (radians)
2 2π 70


W fc >> W

-t +t

-T/2 +T/2
j( fc t + k t 2 + θo )
t
ψ (t) = rect [] [T
exp 2 π
2 2π ]
j[ Φ(t) ] t
Given: ψ (t) = A(t) e and A(t) = rect [] T
Step 1 fi (t) = fc + k t

Step 2
Φ(t) = 2π
∫ fi (t) dt

Φ(t) = 2 π ( fc t + k t 2 + θo )
2 2π 71

HYPERBOLIC FREQUECY MODULATED (HFM) WAVEFORMS,
THEIR TIME-PERIOD AND TIME-FREQUENCY DIAGRAMS
Instantaneous period (linear) = τ(t)
τ1
∆τ το
τ2
-t +t
-T/2 0 +T/2
Instantaneous frequency (hyperbolic) = Finst(t) = 1/τ(t)

F2 = 1/τ2
W
F1 = 1/τ1
-t +t
-T/2 0 +T/2
HFM waveforms and linear period modulated
72
(LPM) waveforms are the same.

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