FUNDAMENTALS OF PASSIVE AND ACTIVE SONAR                                         Instructor:                              ...
www.ATIcourses.comBoost Your Skills                                             349 Berkshire Drive                       ...
DECISION: H1/H0 ?ACTIVE SONAR DETECTION MODEL                                  POST-DETECTION                             ...
DEFINITIONSAcoustic pressure, p: The difference between the total pressure, ptotal, and                      the hydrostat...
ACOUSTIC INTENSITY LEVELS EXPRESSED IN DECIBELS                                                       (prms)2             ...
ACOUSTIC INTENSITY REFERENCE VALUES: AIR AND WATER                                                           (prms)2  Ave...
DECIBELS MEASURED IN AIR AND WATER                                                     X watts/meter2 100 dB water re 10-6...
REPLICA CORRELATION (1 of 6)o     A cross-correlation sequence can be calculated for any pair of causal      sequences x[k...
REPLICA CORRELATION (2 of 6)                                                                                     40       ...
REPLICA CORRELATION (3 of 6)                                                                                              ...
REPLICA CORRELATION (4 of 6)                                                                    40   x[k],                ...
REPLICA CORRELATION (5 of 6)              SIGNAL                             k = +∞      INPUT      +              NOISE  ...
REPLICA CORRELATION (6 of 6)              SIGNAL                             k = +∞      INPUT      +              NOISE  ...
REPLICA CORRELATION FOR A CONTINUOUS CW WAVEFORM (1 of 4)                                         TRANSMISSION            ...
REPLICA CORRELATION FOR A CONTINUOUS CW WAVEFORM (2 of 4)                                                       ECHO      ...
REPLICA CORRELATION FOR A CONTINUOUS CW WAVEFORM (3 of 4)                               INCREASING                        ...
REPLICA CORRELATION FOR A CONTINUOUS CW WAVEFORM (4 of 4)                           INCREASING                          RE...
REPLICA CORRELATOR OUTPUT FOR A CW              WAVEFORM MASKED BY NOISE (1 of 3)                     15                  ...
REPLICA CORRELATOR OUTPUT FOR A CW                                          WAVEFORM MASKED BY NOISE (2 of 3)             ...
REPLICA CORRELATOR OUTPUT FOR A CW                                          WAVEFORM MASKED BY NOISE (3 of 3)             ...
RANDOM WALK SAMPLE AND A GENERAL RESULT (1 of 2)                                              INDEPENDENT NORMALLY DISTRIB...
NOISE SAMPLE MULTIPLIED BY A REPLICA                     INDEPENDENT NORMALLY DISTRIBUTED NOISE, σ = 1           3VALUES  ...
RANDOM WALK SAMPLE AND A GENERAL RESULT (2 of 2)                       POINT-BY-POINT PRODUCTS OF REPLICA AND NOISE       ...
PEAK REPLICA CORRELATOR OUTPUT WHEN ECHO MATCHES REPLICA                          ECHOES OF INCREASING DURATION           ...
SUMMARYIf the number of ‘matched’ sampled data points in a receivedsignal and replica is N, and if the noise masking the s...
HYPOTHETICAL TRANSMISSION, RECEIVED DATA, AND            INSTANTANEOUS POWER OUTPUT OF A REPLICA CORRELATORARBITRARY      ...
THRESHOLD SETTING DETERMINES (WITHIN LIMITS) THE     RESULTING PROBABILITIES OF DETECTION AND FALSE ALARM                 ...
PROBABILITIES OF FOUR POSSIBLE OUTCOMES                                                           SEPARATE AND DISTINCT   ...
DECISION: H1/H0 ?    ACTIVE SONAR DETECTION MODEL(DISCUSSED SO FAR, ALONG WITH SONAR EQUATION)                            ...
DECISION: H1/H0 ?ACTIVE SONAR DETECTION MODEL         (NEXT STEPS)                                  POST-DETECTION        ...
REPLICA CORRELATION FOR A CONTINUOUS CW WAVEFORM                                                               Replica mat...
EFFECT SHIFTING A CW WAVEFORM’S FREQUENCY BY φ    USING THE ‘NARROWBAND’ APPROXIMATION                                    ...
TRANSMITTED AND RECEIVED CW FREQUENCIES                                    AFTER APPLYING A NARROWBAND DOPPLER SHIFTInstan...
TRANSMITTED AND RECEIVED CW FREQUENCIES                                    FOR AN ECHO PRODUCED BY A CLOSING TARGET,      ...
DOPPLER PARAMETER S AND THE                WIDEBAND DOPPLER TRANSFORMATIONo   s ≡ (1 − ν / c) / (1 + ν / c) = 1 − (2ν / c ...
EFFECT SHIFTING A CW WAVEFORM’S FREQUENCY BY φ USING THE WIDEBAND DOPPLER TRANSFORMATION                                C ...
REPLICA CORRELATOR OUTPUT FOR A                     FREQUENCY-MODULATED WAVEFORM (1 of 3)           EchoPRESSURE          ...
REPLICA CORRELATOR OUTPUT FOR A                     FREQUENCY-MODULATED WAVEFORM (2 of 3)           EchoPRESSURE          ...
REPLICA CORRELATOR OUTPUT FOR A                         FREQUENCY-MODULATED WAVEFORM (3 of 3)                             ...
THE LINEAR FREQUENCY MODULATED (LFM)              WAVEFORM OF UNIT AMPLITUDE                              Instantaneous fr...
TIME-FREQUENCY DIAGRAM REFERRED TO REPLICA WAVEFORM (1 of 2)      Instantaneous        frequency                          ...
TIME-FREQUENCY DIAGRAM REFERRED TO REPLICA WAVEFORM (2 of 2) Instantaneous                                      Frequency ...
TIME DELAY DIAGRAMS                                                                Echo data moves with        Instantaneo...
TIME DELAY DIAGRAMS                                                                Echo data moves with        Instantaneo...
TIME DELAY DIAGRAMS                                                                Echo data moves with        Instantaneo...
TIME DELAY DIAGRAMS                                                                Echo data moves with        Instantaneo...
TIME DELAY DIAGRAMS                                                                        Echo data moves with        Ins...
TIME DELAY DIAGRAMS                                                                Echo data moves with        Instantaneo...
TIME DELAY DIAGRAMS                                                                Echo data moves with        Instantaneo...
TIME DELAY DIAGRAMS                                                                Echo data moves with        Instantaneo...
USING THE TIME-FREQUENCY DIAGRAM TO ESTIMATE THE MAXIMUM           OUTPUT OF A REPLICA CORRELATOR WHEN THE ECHO           ...
TIME-FREQUENCY DIAGRAM REFERRED TO REPLICA WAVEFORM Instantaneous                                      Frequency shift is ...
USING THE TIME-FREQUENCY DIAGRAM TO ESTIMATE THE         MAXIMUM OUTPUT OF A REPLICA CORRELATOR Instantaneous   frequency(...
HYPERBOLIC FREQUENCY MODULATED (HFM) WAVEFORMS,  THEIR TIME-PERIOD AND TIME-FREQUENCY DIAGRAMS                            ...
USING THE TIME-FREQUENCY DIAGRAM TO ESTIMATE THE         MAXIMUM OUTPUT OF A REPLICA CORRELATOR Instantaneous             ...
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 ...
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 ...
ANALYTIC SIGNAL WITHOUT APPROXIMATION (1 of 2)                   |U(f)|                                                Q(f...
ANALYTIC SIGNAL WITHOUT APPROXIMATION (2 of 2)              ο ψ (t)                is the analytic signal of real waveform...
APPROXIMATING THE ANALYTICAL SIGNAL WITH A        NARROWBAND COMPLEX WAVEFORM (1 of 6)                |U (f)|             ...
ANALYTIC SIGNAL AND COMPLEX ENVELOPE OF REAL WAVEFORM u(t,fo)                                                   ψ(t,fo) IS...
FACTORS OF THE COMPLEX NARROWBAND WAVEFORM                        ψ (t)                                                   ...
COMPLEX PLANE FOR THE NARROWBAND ψ (t ) WAVEFORM                                  ˆ                             Im        ...
DEFINITION OF INSTANTANEOUS FREQUENCY                 j[ Φ(t) ]ψ (t) = A(t) e ˆ                           is a particularl...
AMPLITUDE, FREQUENCY, AND PHASE MODULATIONο In the expression ψ (t )                     ˆ       = A(t ) e       [ ∫      ...
IMPORTANCE OF THE COMPLEX ENVELOPEο In the expressions            ψ(t) = [a(t) + j b(t)] exp(+2π j fo t) , and            ...
OBTAINING THE ANALYTIC SIGNAL AND REAL WAVEFORM               o General form of the analytic signal is ψ (t ) = A(t )     ...
OBTAINING THE ANALYTIC SIGNAL AND REAL WAVEFORM               o General form of the analytic signal is ψ (t ) = A(t )     ...
THREE COMMON WAVEFORMS     The following slides give complex representations,     i.e., ψ (t ) for three common waveforms:...
FT          sin ( π f T )COMPLEX NARROWBAND                       u (t) = rect (t /T )        ←  →                      ...
THE LINEAR FREQUENCY MODULATED (LFM)                     WAVEFORM OF UNIT AMPLITUDE                                       ...
Fundamentals of Passive and Active Sonar Technical Training Short Course Sampler
Fundamentals of Passive and Active Sonar Technical Training Short Course Sampler
Fundamentals of Passive and Active Sonar Technical Training Short Course Sampler
Fundamentals of Passive and Active Sonar Technical Training Short Course Sampler
Fundamentals of Passive and Active Sonar Technical Training Short Course Sampler
Fundamentals of Passive and Active Sonar Technical Training Short Course Sampler
Fundamentals of Passive and Active Sonar Technical Training Short Course Sampler
Fundamentals of Passive and Active Sonar Technical Training Short Course Sampler
Fundamentals of Passive and Active Sonar Technical Training Short Course Sampler
Fundamentals of Passive and Active Sonar Technical Training Short Course Sampler
Fundamentals of Passive and Active Sonar Technical Training Short Course Sampler
Fundamentals of Passive and Active Sonar Technical Training Short Course Sampler
Fundamentals of Passive and Active Sonar Technical Training Short Course Sampler
Fundamentals of Passive and Active Sonar Technical Training Short Course Sampler
Fundamentals of Passive and Active Sonar Technical Training Short Course Sampler
Fundamentals of Passive and Active Sonar Technical Training Short Course Sampler
Fundamentals of Passive and Active Sonar Technical Training Short Course Sampler
Fundamentals of Passive and Active Sonar Technical Training Short Course Sampler
Fundamentals of Passive and Active Sonar Technical Training Short Course Sampler
Fundamentals of Passive and Active Sonar Technical Training Short Course Sampler
Fundamentals of Passive and Active Sonar Technical Training Short Course Sampler
Fundamentals of Passive and Active Sonar Technical Training Short Course Sampler
Fundamentals of Passive and Active Sonar Technical Training Short Course Sampler
Fundamentals of Passive and Active Sonar Technical Training Short Course Sampler
Fundamentals of Passive and Active Sonar Technical Training Short Course Sampler
Fundamentals of Passive and Active Sonar Technical Training Short Course Sampler
Fundamentals of Passive and Active Sonar Technical Training Short Course Sampler
Fundamentals of Passive and Active Sonar Technical Training Short Course Sampler
Fundamentals of Passive and Active Sonar Technical Training Short Course Sampler
Fundamentals of Passive and Active Sonar Technical Training Short Course Sampler
Fundamentals of Passive and Active Sonar Technical Training Short Course Sampler
Fundamentals of Passive and Active Sonar Technical Training Short Course Sampler
Fundamentals of Passive and Active Sonar Technical Training Short Course Sampler
Fundamentals of Passive and Active Sonar Technical Training Short Course Sampler
Fundamentals of Passive and Active Sonar Technical Training Short Course Sampler
Fundamentals of Passive and Active Sonar Technical Training Short Course Sampler
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Fundamentals of Passive and Active Sonar Technical Training Short Course Sampler

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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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Fundamentals of Passive and Active Sonar Technical Training Short Course Sampler

  1. 1. FUNDAMENTALS OF PASSIVE AND ACTIVE SONAR Instructor: Duncan Sheldon, Ph.D.ATI Course Schedule: http://www.ATIcourses.com/schedule.htmATIs Passive & Active Sonar: http://www.aticourses.com/Fundamentals_of_Passive_and_Active_Sonar.htm
  2. 2. www.ATIcourses.comBoost Your Skills 349 Berkshire Drive Riva, Maryland 21140with On-Site Courses Telephone 1-888-501-2100 / (410) 965-8805Tailored to Your Needs Fax (410) 956-5785 Email: ATI@ATIcourses.comThe Applied Technology Institute specializes in training programs for technical professionals. Our courses keep youcurrent in the state-of-the-art technology that is essential to keep your company on the cutting edge in today’s highlycompetitive marketplace. Since 1984, ATI has earned the trust of training departments nationwide, and has presentedon-site training at the major Navy, Air Force and NASA centers, and for a large number of contractors. Our trainingincreases effectiveness and productivity. Learn from the proven best.For a Free On-Site Quote Visit Us At: http://www.ATIcourses.com/free_onsite_quote.aspFor Our Current Public Course Schedule Go To: http://www.ATIcourses.com/schedule.htm
  3. 3. DECISION: H1/H0 ?ACTIVE SONAR DETECTION MODEL POST-DETECTION PROCESSORTRANSMISSION DETECTOR PRE-DETECTION FILTER DELAY ANDATTENUATION AMBIENT NOISE RECEPTION TIME-VARYING MULTIPATH TARGETTIME-VARYING DELAY AND MULTIPATH ATTENUATION REVERBERATION 2
  4. 4. DEFINITIONSAcoustic 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
  5. 5. ACOUSTIC INTENSITY LEVELS EXPRESSED IN DECIBELS (prms)2 For a plane wave, Iref = ρ c ref PRESSURE1 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
  6. 6. 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
  7. 7. 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/meter2100 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
  8. 8. 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 CORRELATORo 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
  9. 9. REPLICA CORRELATION (2 of 6) 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) 1WITH INPUT k 0 k(1) k(1) + k(2) 8
  10. 10. REPLICA CORRELATION (3 of 6) 40 x[k], 30 k(2) NON-ZERO INPUT 20 VALUES OF x[k] (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 200CORRELATOR 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
  11. 11. REPLICA CORRELATION (4 of 6) 40 x[k], 30 k(2) NON-ZERO INPUT 20 VALUES OF x[k](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
  12. 12. REPLICA CORRELATION (5 of 6) SIGNAL k = +∞ INPUT + NOISE x[k] X ∑x[k] y[k -n] g[n] OUTPUT k = −∞ REPLICA y[k-n] REPLICA CORRELATOR 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
  13. 13. REPLICA CORRELATION (6 of 6) SIGNAL k = +∞ INPUT + NOISE x[k] X ∑x[k] y[k -n] g[n] OUTPUT k = −∞ REPLICA y[k-n] REPLICA CORRELATOR 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
  14. 14. 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 DATAs ( 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
  15. 15. REPLICA CORRELATION FOR A CONTINUOUS CW WAVEFORM (2 of 4) ECHO t =τ BD RECEIVED DATAs ( 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
  16. 16. REPLICA CORRELATION FOR A CONTINUOUS CW WAVEFORM (3 of 4) 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
  17. 17. REPLICA CORRELATION FOR A CONTINUOUS CW WAVEFORM (4 of 4) INCREASING REPLICA DELAY REPLICA OF THE TRANSMISSION FOR τ Replica SOME REPLICA DELAY RD Time, t Received data τd Time delay, τd−τd Replica correlator output τd =0 for a CW waveform plus ambient noise. 16
  18. 18. 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 waveCORRELATOR 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
  19. 19. REPLICA CORRELATOR OUTPUT FOR A CW WAVEFORM MASKED BY NOISE (2 of 3) 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 POINTSREPLICA 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 TIME DELAY τd IN DATA SAMPLES
  20. 20. REPLICA CORRELATOR OUTPUT FOR A CW WAVEFORM MASKED BY NOISE (3 of 3) 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 shapeREPLICA 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 TIME DELAY τd IN DATA SAMPLES
  21. 21. 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 NSTEPS 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
  22. 22. NOISE SAMPLE MULTIPLIED BY A REPLICA INDEPENDENT NORMALLY DISTRIBUTED NOISE, σ = 1 3VALUES 2 1 NOISE 0 -1 -2 -3 -4 0 100 200 300 400 500 600 700 800 1000 900 SAMPLED DATA POINTS REPLICA 1REPLICAVALUES 0 -1 0 100 200 300 400 500 600 700 800 900 1000 SAMPLED DATA POINTS POINT-BY-POINT PRODUCTS OF REPLICA AND NOISE 2PRODUCT VALUES 1 0 -1 -2 0 100 200 300 400 500 600 700 800 900 1000 SAMPLED DATA POINTS 21
  23. 23. 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 POINTSSUM 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
  24. 24. 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 NCORRELATOR 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
  25. 25. SUMMARYIf the number of ‘matched’ sampled data points in a receivedsignal and replica is N, and if the noise masking the signal isuncorrelated from sample-to-sample, then:1) The expectation of the root-mean-square noise 1 output of the replica correlator increases as N , 22) 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 24) The peak signal-to-noise instantaneous power output of   2   N   = the replica correlator is expected to increase as   1   N.    N2    24
  26. 26. HYPOTHETICAL TRANSMISSION, RECEIVED DATA, AND INSTANTANEOUS POWER OUTPUT OF A REPLICA CORRELATORARBITRARY N SAMPLES UNITS REPLICA OF TRANSMISSION FOR TIME SHIFT τd TIME RECEIVED DATA, SIGNAL MATCHING REPLICA PLUS NOISEARBITRARY UNITS TIME τd OUTPUT OF REPLICA OUTPUT AT τd = 0ARBITRARY 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 OFARBITRARY 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.)
  27. 27. 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 NOISEREPLICA 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
  28. 28. 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 pfaPOWER 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
  29. 29. DECISION: H1/H0 ? ACTIVE SONAR DETECTION MODEL(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 TIME-VARYING DELAY AND MULTIPATH ATTENUATION REVERBERATION 28
  30. 30. DECISION: H1/H0 ?ACTIVE SONAR DETECTION MODEL (NEXT STEPS) POST-DETECTION PROCESSORTRANSMISSION FOR FREQUENY REPLICA MODULATED CORRELATOR, TRANSMISSIONS DFT DETECTOR PRE-DETECTION FILTER (Beamformer) DELAY AND RECEPTIONATTENUATION AMBIENT NOISE (Receive array) TIME-VARYING MULTIPATH TARGETTIME-VARYING DELAY AND MULTIPATH ATTENUATION REVERBERATION 29
  31. 31. REPLICA CORRELATION FOR A CONTINUOUS CW WAVEFORM Replica matches echo Replica PRESSURE except for time delay. Time, t Echo +τ d Replica correlator output 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
  32. 32. EFFECT SHIFTING A CW WAVEFORM’S FREQUENCY BY φ USING THE ‘NARROWBAND’ APPROXIMATION Frequency = fo + φ Echo PRESSURE Time, t Replica Frequency = fo Replica correlator outputEnvelope and its envelopenarrows 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
  33. 33. TRANSMITTED AND RECEIVED CW FREQUENCIES AFTER APPLYING A NARROWBAND DOPPLER SHIFTInstantaneous 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
  34. 34. TRANSMITTED AND RECEIVED CW FREQUENCIES FOR AN ECHO PRODUCED BY A CLOSING TARGET, A WIDEBAND DOPPLER TRANSFORMATION g6Instantaneous frequency 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
  35. 35. DOPPLER PARAMETER S AND THE WIDEBAND DOPPLER TRANSFORMATIONo 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 1o 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 ko The difference between the received and transmitted CW frequencies depends on both si and fk: s gki _ fk = fk 1 - i si 34
  36. 36. 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 = 0φ is the frequency shift of the echo with respect to the replicaestablished in the correlator. In the ‘wideband’ case the echoundergoes a Doppler transformation and not simply a Doppler shift. 35
  37. 37. REPLICA CORRELATOR OUTPUT FOR A FREQUENCY-MODULATED WAVEFORM (1 of 3) EchoPRESSURE No frequency Time, t Replica shift - τd Replica correlator output and its envelope for a frequency-modulated waveform- τd +τd τd = 0 τd is the time delay of the echo with respect to the replica established in the correlator. 36
  38. 38. REPLICA CORRELATOR OUTPUT FOR A FREQUENCY-MODULATED WAVEFORM (2 of 3) EchoPRESSURE No frequency Time, t Replica shift - τd Replica correlator output 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. τd is the time delay of the echo with respect to the replica established in the correlator. 37
  39. 39. REPLICA CORRELATOR OUTPUT FOR A FREQUENCY-MODULATED WAVEFORM (3 of 3) C L Echo PRESSURE With echo +τˆd frequency Replica shiftτˆd is the time delay when C L Replica correlator outputthe replica correlator’s output +τˆd and its envelope for aenvelope 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
  40. 40. THE LINEAR FREQUENCY MODULATED (LFM) WAVEFORM OF UNIT AMPLITUDE Instantaneous frequency, HzBandwidth, W fc , Carrier frequency >> W -t +t -T/2 +T/2 Time, Seconds 39
  41. 41. 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
  42. 42. 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
  43. 43. 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 waveformBULK 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
  44. 44. TIME DELAY DIAGRAMS Echo data moves with Instantaneous respect to replica frequency Replica established in correlator (narrowband assumption) Increasing clock time ~ ~ -t +t ~ t=0 Time scale referred to replica waveformBULK 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
  45. 45. TIME DELAY DIAGRAMS Echo data moves with Instantaneous respect to replica frequency Replica established in correlator (narrowband assumption) Increasing clock time ~ ~ -t +t ~ t=0 Time scale referred to replica waveformBULK 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
  46. 46. TIME DELAY DIAGRAMS Echo data moves with Instantaneous respect to replica frequency Replica established in correlator (narrowband assumption) Increasing clock time ~ ~ -t +t ~ t=0 Time scale referred to replica waveformBULK 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
  47. 47. TIME DELAY DIAGRAMS Echo data moves with Instantaneous τ d = τˆd when overlap is greatest respect to replica frequency established in correlator (narrowband assumption) Replica Increasing clock time ~ ~ -t +t ~ t=0 Time scale referred to replica waveformBULK 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
  48. 48. TIME DELAY DIAGRAMS Echo data moves with Instantaneous respect to replica frequency established in correlator (narrowband assumption) Replica Increasing clock time ~ ~ -t +t ~ t=0 Time scale referred to replica waveformBULK 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
  49. 49. TIME DELAY DIAGRAMS Echo data moves with Instantaneous respect to replica frequency established in correlator (narrowband assumption) Replica Increasing clock time ~ ~ -t +t ~ t=0 Time scale referred to replica waveformBULK 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
  50. 50. TIME DELAY DIAGRAMS Echo data moves with Instantaneous respect to replica frequency established in correlator (narrowband assumption) Replica Increasing clock time ~ ~ -t +t ~ t=0 Time scale referred to replica waveformBULK 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
  51. 51. 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
  52. 52. 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
  53. 53. 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 ~ Time scale referred to replica waveform 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
  54. 54. 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.
  55. 55. 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 ~ Time scale referred to replica waveform t=0Here 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
  56. 56. 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
  57. 57. 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
  58. 58. 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
  59. 59. 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, SecondsInstantaneous 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
  60. 60. 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 ) dτThe 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
  61. 61. ANALYTIC SIGNAL AND COMPLEX ENVELOPE OF REAL WAVEFORM u(t,fo) ψ(t,fo) IS THE INVERSE FOURIERU(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
  62. 62. 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 andψ (t) = [a(t) + j b(t)] exp(+2π j fo t) ˆ vary slowly with respectu(t) = Re (ψ (t) ) ˆ to 2π fot. [a(t) + j b(t)]is the complex envelope of ψ (t ). ˆ 61
  63. 63. 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
  64. 64. DEFINITION OF INSTANTANEOUS FREQUENCY j[ Φ(t) ]ψ (t) = A(t) e ˆ is a particularly convenient form of theanalytic signal because Φ(t) is the argument of ψ (t ) and the ˆinstantaneous 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 dtGiven fi(t), the argument of ψ (t) and instantaneous phase of ˆu(t) is given by Φ(t) = 2π ∫ fi (t) dt + θ oAnd we can write ψ (t ) as ˆ ψ (t) = A(t) e ˆ [ ∫ ] j 2π fi (t) dt + θ o 63
  65. 65. AMPLITUDE, FREQUENCY, AND PHASE MODULATIONο 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
  66. 66. IMPORTANCE OF THE COMPLEX ENVELOPEο 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 dtο 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
  67. 67. OBTAINING THE ANALYTIC SIGNAL AND REAL WAVEFORM o General form of the analytic signal is ψ (t ) = A(t ) e j[Φ(t)] dStart 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  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. o A(t) is often explicitly stated, i.e., rect(t/T). o Re[ψ(t)] gives the real waveform, u(t). 66
  68. 68. OBTAINING THE ANALYTIC SIGNAL AND REAL WAVEFORM o General form of the analytic signal is ψ (t ) = A(t ) e j[Φ(t)] dStart 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  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. o A(t) is often explicitly stated, i.e., rect(t/T). o Re[ψ(t)] gives the real waveform, u(t). 67
  69. 69. 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
  70. 70. 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) = 1Trect (t/T ) exp [ j 2 π ( fc t + θo )] 0.8T Shift to 2π frequency fc 0.6Twhere T fc >> 1 cycle 0.4T to get U(f - fc ) U(f) 1 if |t/T| < 1/2rect (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
  71. 71. THE LINEAR FREQUENCY MODULATED (LFM) WAVEFORM OF UNIT AMPLITUDE Instantaneous frequency Bandwidth, W Carrier fc frequency >> W -t +t -T/2 +T/2Instantaneous phase (radians) = Φ(t) = 2 π fc t + θ (t) , t < |T/2| and TW >> 1and 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 dtLinear frequency modulation means 1 d [θ (t) ] = k t so θ (t) = 2 π k t 2 + θ o 2π dt 2i.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

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