a question a bout Matlab
1
High Angular Resolution Techniques (Ch 12)
Phased Arrays Doppler Beam Sharpening Synthetic Aperture
PHASED ARRAYS
G. Brooker, Introduction to Sensors for Ranging and Imaging: SciTech, 2008. http://www.fas.org/programs/ssp/man/us wpns/air/special/e3.html
2
Resolution of a Single Aperture For imaging systems (not null steering trackers), the angular
resolution is limited by the beam divergence.
Beam divergence 3dB (beamwidth) is a function of the wavelength and the aperture size
where 3dB – 3dB Beamwidth k – Constant (70 for and 1.22 for rad) - Wavelength (m) d – Aperture diameter (m)
The for a weighted aperture, the cross range resolution, xr, is the product of the beamwidth (rad) and range (m)
d
k dB
3
d
R Rxr dB
22.1 3
Requirement for Phased Arrays The wavelength is fixed by atmospheric window,
propagation effects or physical size constraints. It is difficult to make the antenna diameter, d, arbitrarily
large to obtain the required angular resolution because of manufacturing limitations
Use two or more transceivers in an array to synthesise an effective linear aperture equal to the array baseline.
Uses Phased array radars Phased array sonar (linear or 2D) underwater or in the air Long baseline radio telescopes
http://www.a-research.com.au/sodar.html
3
Transmitter Beam Synthesis
Individual elements radiate precisely in phase to produce wave crests that move forward in phase
Interfere constructively to produce a strong narrow beam directed straight ahead
If a linear phase shift is applied across the array, then the beam is reconstructed at an angle
Receiver Beam Forming The power received by each element is the sum of the received power
scattered by target P from all the transmit elements
The elements outputs are summed via lines of equal length to give Ea
ddd
Incoming Wave
123N N-1
E1E2EN
Ea
N
k ka tE
1
)sin(
4
Phase Shift Between Elements For an incoming signal at an angle to the array The phase shift between adjacent elements is (rad) where d (m) is
the element spacing
rad 2
Sin d
Moving Phase Reference to Geometric Centre of Array
The equation that describes the output voltage from the array for phase shifts starting with element 1 at B is
It is usual to measure the phase shift relative to the geometric centre of the array, not element 1.
This displaces the phase front represented by line AB, down to CD
N
k a ktE
1
1sin
5
Revised Equation Counting from Centre
The equation that describes the output voltage for phase shifts displaced to the geometric centre is
N
k
N
k a
N kt
N ktE
1
1
2
1 sin
2
1 1sin
Two Point Array
Ea = sin(t+/2) + sin(t-/2) Ea = sin(t).2cos(/2)
sinA+sinB=2sin[(A+B)/2]cos[(A-B)/2]
sin2A = 2sinAcosA
)2/sin(
)2/2sin( )sin(
tEa
The field intensity pattern, |Ea()| for the antenna is the magnitude of the amplitude factor
6
Four Point Array
sin 3 / 2 sin / 2 sin / 2 sin 3 / 2aE t t t t (1) (2) (3) (4)
See your book for the trig identities used to manipulate this equation
sin 4 / 2 sin
sin / 2 aE t
The General Case
N Point Array
1
sin / 21 sin sin
2 sin / 2
N
a k
NN E t k t
These equations give identical results if you plug in the numbers, but I have never managed prove the relationship rigorously
7
The Field Intensity Pattern Substituting for =2dsin/
The field intensity pattern Ea() is equal to the magnitude of the Amplitude factor
Sin
d Sin
Sin dN
Sin tSinEa )..(
Sin
d Sin
Sin dN
Sin Ea )(
Amplitude factor
The Field Intensity Pattern Continued
Nulls where the numerator is zero = 0, +/-, +/-2 etc
The denominator is zero at = 0, +/-, +/-2 etc
Applying L’Hopitals rule where Ea = 0/0 we find maxima each with value N where sin = +/-nλ/d
Sin
d Sin
Sin dN
Sin Ea )(
Sin dN
Sin d
8
The Field Intensity Pattern cont… • The maximum where sin = 0 is
called the Main lobe, all the other lobes are called Grating lobes.
• If d/ = 0.5, the grating lobe does not appear for n = +/-1 in real space because sin>1 which is not possible.
• If d/ = 1 the grating lobes appear at +/-90, however as most real radiating elements do not radiate much at = 90, the grating lobes are suppressed.
• For a non scanning array, the best element spacing is d =
• For a scanned array the best spacing is d < /2
Grating Lobes for N=10 and d/ = 1
9
No Grating Lobes for N=20 and d/ = 0.5
Radiation Pattern: Linear Array The radiation pattern is defined as the normalised square of the
amplitude factor
For Nd = L (the length of the array) and for sin = (small angle)
Sin
d SinN
Sin dN
Sin
N
E G
a
a 22
2
2
2
)(
2
2
)(
Sin
L
Sin L
Sin Ga
10
Linear Array cont…
For d = /2, the half power beamwidth 3dB (rad) is as follows
If N is sufficiently large, the antenna will be equivalent to a uniformly illuminated aperture, and the first sidelobe will be 13.2dB down.
For directive elements, the antenna pattern is the product of the element factor Ge() and the array factor Ga().
G() = Ge().Ga()
rad 73.1
3 N
dB
The beamwidth is smaller than the normal 1.22/d which gives 2.44/N. This is because the sidelobes are so high in this case. See eqn 5.14
Arrays of Directive Elements
Directive Element Pattern
Isotropic Array Pattern
Directive Element Array pattern
11
2D Rectangular Array
Radiation Pattern: 2D Rectangular Array
The radiation pattern may be approximated as the product of the patterns of the two planes that contain the principle axes of the antenna
G(,) = G()G()
Sin
d SinSin
d Sin
Sin dM
SinSin dN
Sin
G 22
22
),(
12
4x4 Rectangular Array of Isotropic Elements
Antenna Gain For large arrays, the non-scanned antenna gain can be
approximated by the gain of a uniformly illuminated aperture
For a scanned array, the gain is reduced by the scan angle o because the projected aperture is reduced in size.
2
4
A
Go
2
cos4 )(
oo A
G
13
Beam Steering
•If the same phase is applied to all the elements of the array, then the main beam will be broadside to the array and = 0 •The direction of the main beam will be o if the relative phase difference between elements is
o
d
sin 2
A nt
e n n a
A nt
e n n a
(a)
(b)
Generators Phase Shifters
Radiating Elements
Steered Radiation Pattern
The radiation pattern is then
Grating lobes will occur at
For a scan over +/-90, the element spacing should be d=/2 For a practical array that can scan over +/-60, the spacing d>0.54
)(
)( )(
22
2
2
2
o
o a
a
SinSin d
SinN
SinSin dN
Sin
N
E G
nSinSin d
og )(
14
Corrections to Improve Range Resolution
1
2
3
4
5
6
Phase Shift
1
2
3
4
5
6
Time Delay
1
2
Phase Shift
Time Delay
1
2
1
2
1
2
5ns
10ns
Sum Signal
5ns
Sum Signal
Sum Signal
6.5ns
5ns
(a) (b) (c)
Phase shift only Delay adjust only Delay adjust And phase shift
Half Power Beamwidth The half power beamwidth of a scanned array can be approximated
by the following formula (not valid near end-fire)
Taper is generally used to reduce the sidelobe levels. Using cosine on a pedestal An = a0 + 2a1Cos(2n/N) where 0 < 2a1 < a0
rad 886.0
3 o
dB NdCos
rad 2
636.01 886.0
2
0
1 3
a
a
NdCos o dB
15
Active and Passive Arrays With the advent of low-cost MMICs, it is now practical to
manufacture individual transceiver modules to build active arrays Some of the best in the world are produced by CEA in Canberra,
Australia
Scalable in size and power to meet a broad range of applications, suitable from 'Corvettes to Cruisers'; Full 3D multifunction capabilities; Advanced classification capabilities; Optimised for littoral and open ocean; Evolves to meet changing requirements; Very high reliability, no in-mission maintenance.
CEAFAR Active phased Array
16
Matching and Mutual Coupling The impedance of the array elements varies with the scan angle Spurious lobes may appear due to the miss-match This is a difficult problem to solve analytically, and is often
determined experimentally by exciting a single element and terminating all of the surrounding ones
Coupling is proportional to 1/d for d=/2, so the pattern and impedance are drastically altered by surrounding elements. Generally the surrounding 5x5 or even 9x9 elements must be considered.
http://www.zonamilitar.com.ar/foros/threads/rinc%C 3%B3n-de-aviones-especializados.23364/page-7
http://www.fas.org/programs/ssp/man/uswpns/air/special/e3.html
17
The Saab Erieye system uses an active phased array radar mounted in a two sided array geometry contained in a large beam shaped structure carried above the fuselage. The limitation of the two sided array is that it can only cover two 120 degree sectors abeam of the aircraft, leaving 60 degree blind sectors over the nose and tail of the aircraft
Erieye video
http://www.youtube.com/watch?v=BOaxi2G0aNI
Thinned Arrays
Reducing the number of elements leaves the main lobe unaltered but degrades the sidelobe level
Thinning to 10% reduces the main lobe level down to 10%, but leaves the sidelobe level at 90%
If the removed elements are replaced by matched dummy elements, then the pattern remains unchanged, only the gain is decreased
M. Skolnik, Radar Handbook: McGraw Hill, 1970.
18
Advantages of Phased Arrays
Inertialess rapid beam steering
Multiple, independent beams
Potential for large peak and average powers
Control of radiation pattern
Graceful degradation
Convenient aperture shape
Electronic beam stabilisation
Courtesy Eli Brookner
MMIC Based Phased Arrays
Courtesy Eli Brookner
19
Courtesy Eli Brookner
Pave Paws AN/FPS-115 Early Warning Radar
Frequency 433MHz Tx Power 284-440W 1792 radiating elements Each array scans +/-60 30m diameter 6000km range for 10m2
target
Note the 1980s graphical interface
E. Brookner. (1985) Phased-Array Radars. Scientific American. 76-84.
20
Pave Paws Radar Coverage
http://www.globalsecurity.org/space/systems/pavepaws.htm
Sea Based X-Band Radar
Tracking and discrimination for ground-based midcourse defence (Starwars mk II)
Mass 50,000 tons, tall as a 24 story building Mechanically scanned X-Band phased array, 65%
populated 45,000 Tx/Rx elements
http://www.mda.mil/news/11news0001.html
21
Acoustic Phased Array: Paul Thompson
16 radiating elements
16 receiver elements
3D imaging from a single “ping”
Target Acoustic Image
P. Thompson, "Design and Construction of an Ultrasound Imaging System Using Phased Arrays," BE Honours Thesis, ACFR/AMME, University of Sydney, Sydney, 2003.
Sidescan Sonar & Towfish
Sidescan Array
Operational Principle
G. Brooker, Introduction to Sensors for Ranging and Imaging: SciTech, 2008.
22
Operational Principle A side-scan sonar antenna a short (50) linear transducer array
made of a piezo-electric material that is towed behind a ship
The transducer is excited by a short ( 3s) high voltage sinusoidal stimulus at a frequency close to the resonant frequency of the array which the array converts to vibrations and radiates into the water.
The operational frequency is generally between 50kHz and 500kHz with some short-range units operating up to 1MHz.
The same array is used to receive any echoes. These are then amplified and recorded to form an image. In modern systems, the signal is digitised in the tow-fish and transmitted to the surface for processing and display.
The operational range for low frequency units (f 100kHz) is about 500m, this decreases to 50m at a frequency of 1MHz due to the increasing attenuation of water with frequency.
Sound Attenuation in Water
Attenuation increases by a factor of 10 from 100kHz to 1MHz
23
Beam Pattern
Because of its shape, the array produces a fan beam pattern narrow azimuth beamwidth
(typ. 0.75 to 1.5) determined by the length of the array
wide elevation beamwidth (typ. 35 to 65) determined by the vertical aperture of each element.
Arrays are placed on either side of the tow-fish and angled slightly downward to produce the patterns shown
Sidescan Image: The Port Hunter
Bang Pulse
Water Krill & Fish etc
Ocean Floor
http://uboat.net/allies/mer chants/1925.html
Sidescan Playback
http://www.youtube.com /watch?v=AJoqh2SdQZ k
24
Effect of Shadowing
Because of the shallow grazing angles, shadows can add significantly to the information available from a sidescan image
J. Fish and A. Carr. (2001, March). Acoustics and Sonar (AUSS Ltd). Available: http://www.marine-group.com/acoustic.htm
Signal Processing
The standard matched-filtering principles developed for radar are applied to sonar systems to ensure that the maximum SNR is achieved.
Most side-scan systems are real beam in that their cross range resolution is a function of range xr = R.az
Digital techniques can be applied to correct for phase front curvature. This is known as focussing, and it can be used to achieve a fairly constant linear beamwidth with range (this is similar to SAR processing).
Beam scanning techniques using phase shifters in the arrays can be used to spotlight particular areas
Simultaneous multi-frequency operation eg. 100kHz and 600kHz is possible for high resolution short range operation and lower resolution long range operation.
25
Pseudo 3D Images
If views are made from more than one perspective, they can be combined into a pseudo 3D image as shown here for the Fritzen
S. Hultqvist. (2007). Swedish East Coast Wrecks. Available: http://www.abc.se/~m10354/uwa/wreck-se.htm
3D Sonar Imaging
Sophisticated 2D sonar arrays such as the one developed by Thomson Marconi Sonar (TMS) in Sydney can produce short range 3D images with voxel resolutions down to 1x1x1mm
A group of 3 uniformly spaced transmitters illuminates the target with high frequency (>1MHz) sound pulses.
A sparse phased array made up of 84 tiles each made up of a random pattern of 32 hydrophone receivers receives the echo.
26
Sonar Images
Synthetic Aperture Radar
27
Definition
Synthetic Aperture Radar (SAR) and Doppler beam- sharpening (DBS) are techniques that use the forward motion of an aircraft carrying a radar to improve the cross-range resolution
Both these techniques can be used for sonar applications as well
Space Based SAR
Doppler Beam Sharpening
Doppler beam-sharpening uses the decreasing radial velocity (hence Doppler shift) across the beam footprint to synthesise improved cross range resolution
For an aircraft flying at 250m/s, the isovel (and isodop) lines at 1.25m/s spacing are shown
h
slant range
ground
elevation beamwidth
beam footprint
PLAN VIEW
SIDE VIEW
azimuth beamwidth
range gates
lin es
o f c
on st
an t
D op
pl er
s hi
ft (is
od op
s)
real aperture resolution
Doppler beam-sharpened resolution
Decrease in Velocity (m/s)
Azimuth Angle (deg)
1.25 5.73
2.5 8.11
3.75 9.93
5 11.48
6.25 12.84
7.5 14.07
28
Doppler Beam Sharpening
Limitations include a trade off between the “sharpening” and the observation time
At 10GHz, 1.25m/s isodop lines are 83Hz apart requiring an observation time of 12ms to resolve them, 3m travel time at 250m/s
At 94GHz, the lines are 800Hz apart requiring an observation time of 1.2ms (0.3m)
The beam is scanned physically to the one side of the direction of travel
A reflectivity image is built up using the higher cross-range resolution
Real Image Sharpened Image
Developing a Synthetic Aperture
Azimuth
Range
Doppler
Pulse Width
29
Generation of the Synthetic Aperture
The term “synthetic aperture” refers to the distance that the sensor travels during the time that the reflectivity data are collected from a single point
Energy from each point is made to arrive in phase at the output of the processor for all of the samples to realise the narrow beamwidth.
Good range resolution is obtained using one of the pulse compression techniques discussed in the previous lecture
Azimuth
R a n ge
Point target signature from a moving target, before and after pulse compression
Sampling the Aperture
30
Synthesizing a Beam The process to determine the radiation pattern is similar to that used
for the fixed array The primary difference is that the signal received by each element is
due only to the received power scattered by target P from one transmitter element
This results in a slightly different radiation pattern for SAR. The beamwidth is narrower, but the sidelobes are higher than that for the equivalent phased array. For Le the synthetic array length
The half power beamwidth can be found by solving for GSAR()=0.5 and solving graphically (or using Newton)
2
2
)(
Sin L
Sin L
Sin
G e
e
SAR
39.1 2
886.0
Sin Le
Unfocussed SAR Aircraft motion that deviates from a
straight line and “range walk” is compensated for.
One limiting condition for the largest aperture Lmax is the point where the round trip phase error reaches /4 as determined in the diagram
for
)2/( 28 max
Sin
L
R
L Sin
2 )2/( max
R
L
48
2 max
2/max RL
Azimuth
R a n g e
Point target signature corrected for range walk
31
Cross Range Resolution: Unfocussed SAR A second limiting condition is that the beamwidth is sufficiently wide
to illuminate the target at point P.
The beamwidth is obtained by equating GSAR()=0.5 as before
The cross-range resolution cr = R = R.sin for small angles
Substituting for Le and simplifying
dBRL 3max
39.1 2
886.0
Sin Le
eL Sin
2
886.0
e cr
L
R
4
886.0
2/max RLLe Rcr 3.0
Focussed SAR
Azimuth
R a ng
e
Removal of the range curvature from the returns from a point target into a single range gate to allow correlation in azimuth that results in the improved cross- range resolution
32
A Doppler Perspective A point scatterer enters the forward
edge of the beam. It will have Doppler frequency:
For small beamwidths, the Doppler frequency decreases linearly to 0 and then increases again.
The angle to the target as a function of time is
The Doppler frequency as a function of time will then be
Target just enters beam
Target just leaves beam
Target
Synthetic Aperture
)2/cos( 22
3dB r
d
vv f
R
vt
R
vtvv tf rd cos
22 )(
Doppler Perspective cont… Taking the derivative to obtain the rate of change of Doppler
frequency, or the Doppler slope
At t=0
The total Doppler shift over time Td = time within the beam for = -3dB/2 to +3dB/2, assuming a linear change in frequency
By analogy to the linear FM range resolution, the signal can be passed through a matched filter to give a spectral resolution f = 1/Td
R
vt
R
vv
dt
df d sin. 2
R v
dt
df d 22
dd T R
v f .
2 2
33
Doppler Perspective cont… The cross range resolution is then the optimised cross range resolution
of the real beam b = Le scaled by the ratio of the spectral resolution to the whole Doppler shift
Substituting
But Le = v.Td = Rb = R/D where D = antenna aperture
The cross range resolution for focussed SAR is independent of the range R
d e
d bcr
f
f L
f
f
.
222 2
1
2 d
e dd
ecr Tv
R L
TTv
R L
2 .
22
D
R
DR
L
R
e cr
Resolution Comparison
Frequency f = 94GHz Aperture D = 120mm
34
Layover Weak Return
Shadow Shadow Shadow
ForeshorteningLayover
Terrain with slopes steeper than these tangentials will be imaged with layover
Depression angle
(a) (b) (c) (d)
Ground range projection
Distortion in SAR Images
layover, when the range to the top of an object is less than the distance to its base
foreshortening, when the near side of elevated objects appears steeper than it actually is
shadowing, when a tall opaque object blocks the signal path behind it, and no returns are received
35
Distortion in SAR Images: Stretching
Depression angle
Ground range image plane
Slant range image plane
Shadowing in SAR Imagee
Sandia. (2004). X-band Synthetic Aperture Imagery, Sandia Labs, . Available: http://www.sandia.gov/RADAR/imageryx.html
36
Measurement Coherence:
Speckle
Sandia Lab’s Miniature SAR Specifications Total mass 10kg
Range 15km
Resolution 10cm
37
China Lake Airfield 3m
Sandia. (2004). X-band Synthetic Aperture Imagery, Sandia Labs, . Available: http://www.sandia.gov/RADAR/imageryx.html
Another Airport that isn’t Hong Kong
Sandia. (2004). X-band Synthetic Aperture Imagery, Sandia Labs, . Available: http://www.sandia.gov/RADAR/imageryx.html
38
Piers and a River 1mSandia. (2004). X-band Synthetic Aperture Imagery, Sandia Labs, . Available: http://www.sandia.gov/RADAR/imageryx.html
Pipeline Crossing a River 1mSandia. (2004). X-band Synthetic Aperture Imagery, Sandia Labs, . Available: http://www.sandia.gov/RADAR/imageryx.html
39
T-72 Tanks in Formation 10cm
Space Based SAR To achieve good angular resolutions
from real aperture space-borne radars is impossible at lower frequencies because the size of the antenna becomes prohibitively large.
With a SAR, the large synthetic aperture results in a cross range resolution independent of range cr = D/2 where D is the antenna aperture
The good range resolution r = c/2f is achieved by transmitting a wide bandwidth chirp.
Because the trajectory of the satellite or shuttle is precisely known and stable, motion compensation is not required and exceptionally high quality images can be produced.
40
Interferometry
Because SAR is concerned with the phase relationships between scatterers on the ground Two similar images are produced using offset
antennas, or on subsequent passes over the same area,
Interference patterns can be used to determine the true height of the objects on the ground.
In addition to being useful for mapping ground features, this technology has a number of other uses: Local deformation of the earth’s crust as an early
warning of earthquakes or volcanoes. Ground subsidence due to mining activities or
excessive use of ground-water
Interferometric SAR Image of the San Francisco Area
41
Oil rigs
Mississippi Delta
Ship
42
Magellan SAR map of Venus
(26/11/04). Magellan Mission to Venus. Available: http://www2.jpl.nasa.gov/magellan/
43
Sif Mons 2km High and 300km in diameter 3D Image produced by combining SAR and altimeter data
(26/11/04). Magellan Mission to Venus. Available: http://www2.jpl.nasa.gov/magellan/
Phased Array Application
Performance of Sidescan Sonar System
44
Sidescan System Evaluation ITC 5202 Transducer
Size of Unit 68.5x3.8cm
Array Dimensions 1.27x53cm shaded active array
Resonance Frequency 117kHz
Useable frequency range 111-126kHz
Beam pattern: 53cm line
1.27cm line
1.5 at 117kHz 60 at 117kHz
Efficiency >40%
Input power <5% duty cycle 1500W
Operating Depth Unlimited
Weight 4.3kg
Housing aluminium
Transducer Array
Receive -180dB rel 1V/Pa
Transmit 170dB rel 1Pa/V at 1m
45
Worked Example
What is the smallest target that can be detected by the ITC-5202 transducer at a range of 500m?
Operational frequency f = 117kHz
Velocity of sound (assumed constant) c = 1522m/s
Wavelength = c/f = 13mm
Pulse Width and Range Resolution The quality factor is determined from the operational band
The rise time of any pulse generated by the transducer is related to the resonant frequency and the quality factor
The minimum pulse-width must be at least twice the rise time if the pulse it to reach its peak value. For a “rectangular” pulse, the total pulse-width should be 5rise = 333s. This longer pulse gives improved long range performance at the expense of the resolution
8.7 111126
117
lu
r
ff
f Q
s f
Q
r rise 7.66
10117
8.7 3
m c
R 25.0 2
103331522
2
6
46
Cross Range Resolution
The cross range resolution at 500m (for the given azimuth beamwidth of 1.5) is the product of the beamwidth in radians and the range (no focussing)
mRxr 1.13 3.57
5.1 500
Pulse Repetition Frequency (PRF) and Duty Cycle (DC)
To operate out to a maximum unambiguous range of 500m, the maximum pulse repetition frequency PRF is
The transmitter power is limited to a maximum of 1500W for a Duty Cycle of less than 5%. The duty cycle in this case is
which is much smaller than the limit, so the maximum power can be applied to the transmitter.
Hz R
c PRF 52.1
5002
1522
2 max max
%05.01033352.1100100DutyCycle 6 PRF
47
Pulse Compression
If pulse compression is applied to obtain the best possible resolution for the sonar but using the full duty cycle =5%, then
And the duration of the pulse increases to
0.05m 10152
1522
2 3
f
c Rpc
s109.32 52.1100
5
100 3
pr pc
f
Transmitter Power Density If the transducer was omnidirectional, then the power density at a
range of 1m would be the product of the electrical power Pelec , and the conversion efficiency divided by the surface area of a sphere with a radius of 1m
The antenna gain, known as the Directivity Index (DI), which is defined in terms of the power with respect to an isotropic radiator can be calculated from the elevation and azimuth beam widths (in radians)
The actual power density in the direction of the peak gain is the product of the gain and the isotropic value
2/7.47 4
4.01500
4 mW
P I eleciso
4.458 605.1
3.5744 2
G
2/218644.4587.47 mWGII iso
48
Sound Pressure Level The sound pressure level (SPL) or S is generally given in dB relative
to 1Pa at a range of 1m. This can be calculated from the power density and the acoustic impedance of the water.
The acoustic impedance, Z of water is the product of the density and the velocity
The relationship between the acoustic pressure, P in Pascals, the power density, I in W/m2 , and the impedance, Z, is
This can be re written for the acoustic pressure in Pa for the sound pressure level, S, as follows
sec/1056.115224.1026 26 mkgcZ o
IZP 2
IZPS 1226 10)10(
Sound Pressure Level in dB
This is generally written in dB form
dBS
IZPS
dB 3.2253.105120)1056.121864(log10120
)(log1010log10)10(log20log10 6
10
10 12
10 6
1010
49
Sound Pressure Level from Graphs It can be seen that at 117kHz, the
transmitter voltage response is 170dB rel to 1Pa/Volt at 1meter.
The electrical power input Pelec is related to the RMS voltage V and the transducer conductance G
For a conductance G = 5.5k mho from the transducer specification table and a power Pelec = 1500W
The sound pressure level SdB for 522V applied to the transducer is
GVPelec 2
Vrms G
P V elec 522
105.5
1500 3
dBVSdB 3.224)522(log20170)(log20170 1010
Transmission Loss As the signal propagates through the water, the sound pressure
level reduces because the wave is expanding on a spherical wave- front and due to attenuation. The transmission loss in dB is H and is determined as follows:
Because the sound pressure is determined relative to the level existing at one meter from the effective centre of the sound source, the equation can be rewritten for this reference distance as follows
The attenuation in dB/m is given by the following formula where f is the frequency of the sound in kHz.
)(log20 12 1
2 10 rr
r
r H dB
rrH dB 10log20
mdBf f
f dB /0329.0117102.3
3600117
117036.0 102.3
3600
036.0 27 2
2 27
2
2
50
Target Strength T T in dB is defined by ratio of the reflected sound pressure scattered
by the target at a distance of one meter from the effective centre of the scattered sound to the incident sound pressure on the target
This target strength is determined by its size, shape and the fraction of sound that is re-radiated.
If the scattering cross section is square meters, then T in dB is given by the following formula
As with the radar case, a sphere with a radius, a, much larger than the wavelength will have a cross section equal to the projected area
i
r
P
P T 10log20
4
log10 10T
2 log20
2 log10
4 log10 10
2
10
2
10
aaa T
Applying the Sonar Range Equation for a Spherical Target
For a spherical target, the echo sound pressure level E relative to 1Pa at a range of 1m from the receiver is easily calculated as follows
THSE dBdB 2
2 log202log40 1010
a rrSE dBdBdB
120656.0log403.225 10 rrEdB
51
Range Eqn Applied to the Sea Floor Target strength will be the product of the range resolution and the
cross range resolution modified by a scaling factor to take into account the reflectivity o of the surface.
and the echo SPL 1m from the receiver will be
SdB – 225.3dB dB – 0.0328 dB/m o – 0.1 m2/m2 az – 1.5 (0.0262 rad) R – 0.25m
4
.. log10
4
.. log10
4 log10 101010
az ooo rRXRRA
T
r R
rrSE az o
dBdBdB
4
. log102log40 1010
rrrEdB 1010 log108.420656.0log403.225
Noise Floor The noise level at sea is mostly generated by wind and wave action
on the surface. It is proportional to sea-state and inversely proportional to frequency.
From the table reproduced in the notes, we will assume sea state 3 generated by a wind speed of 15 knots
Isotropic Noise pressure N1 (dB relative to 1Pa) into a 1Hz bandwidth at a frequency of 1kHz is 65dB
The frequency relationship to map the noise pressure level at 1kHz to the transducer frequency is (see Fig 9.4 in your book)
For a sea state 3 and the transducer frequency of 117kHz
kHzf fNN 101 log17
dBN f 30117log1765 10
52
Noise Floor
53
Noise Pressure Level: Pulsed System
Because the noise floor is defined for an isotropic receiver into a 1Hz bandwidth
The total noise pressure level in dB relative to 1Pa must take into account the bandwidth of the receiver (in Hz) and its directivity or gain G.
The minimum detectable signal level will be 13dB higher than this
Smin = 38+13 = 50dB
dBLN 38)4.458(log10)103(log1030 10 3
10
GNL fN 1010 log10log10
Noise Pressure Level: Pulse Compression
The receiver bandwidth for a pulse compression system can be approximated by the reciprocal of the uncompressed pulse width (signal observation time
pc = 1/pc = 30Hz
The minimum detectable signal level will be 13dB higher than this
Smin = 38+13 - 20 = 30dB
54
Signal and Noise Levels
Targets Detectable at 500m
The sea floor with a SNR = 25dB
A sphere with a diameter of 1m with a SNR = 30dB
A sphere with a diameter of 0.1m might be detectable but with an SNR = 10dB it cannot produce a Pd = 0.9 and a Pfa = 10
-6
55
Signal Level out of Transducer
The actual voltage output by the transducer is determined from the transducer specifications. The open circuit receiving response at 117kHz is –180dB rel 1V/Pa. For a signal pressure of 50dB (the sea-floor return at 500m), the output is
This is very small, and so receiver noise would be a consideration when the actual detection characteristics of the system were being considered.
dBV 13018050)(log20 10
nVV 32010 20/130