ECEN5633 Radar TheoryECEN5633 Radar TheoryLecture #17 10 March 2015Lecture #17 10 March 2015Dr. George ScheetsDr. George Scheetswww.okstate.edu/elec-eng/scheets/ecen5633www.okstate.edu/elec-eng/scheets/ecen5633

ECEN5633 Radar TheoryECEN5633 Radar TheoryLecture #17 10 March 2015Lecture #17 10 March 2015Dr. George ScheetsDr. George Scheetswww.okstate.edu/elec-eng/scheets/ecen5633www.okstate.edu/elec-eng/scheets/ecen5633 Read 12.2Read 12.2 Problems 11.5, 8, & 12.5Problems 11.5, 8, & 12.5 Corrected quizzes due 1 week after returnCorrected quizzes due 1 week after return

Live: 12 MarchLive: 12 March Exam #2, 31 March 2014 (Exam #2, 31 March 2014 (<< 4 April DL) 4 April DL)

ECEN5633 Radar TheoryECEN5633 Radar TheoryLecture #18 12 March 2015Lecture #18 12 March 2015Dr. George ScheetsDr. George Scheetswww.okstate.edu/elec-eng/scheets/ecen5633www.okstate.edu/elec-eng/scheets/ecen5633

ECEN5633 Radar TheoryECEN5633 Radar TheoryLecture #18 12 March 2015Lecture #18 12 March 2015Dr. George ScheetsDr. George Scheetswww.okstate.edu/elec-eng/scheets/ecen5633www.okstate.edu/elec-eng/scheets/ecen5633

Read 13.1 & 2Read 13.1 & 2 Problems 12.7, 8, & Web 3Problems 12.7, 8, & Web 3 Corrected quizzes due 1 week after returnCorrected quizzes due 1 week after return

Live: 12 MarchLive: 12 March Exam #2, 31 March 2014 (Exam #2, 31 March 2014 (<< 4 April DL) 4 April DL)

Coherent Detection (PLL), Single Pulse, Fixed Pr

Coherent Detection (PLL), Single Pulse, Fixed Pr

Noise PDFGaussianMean = 0

Variance = kTºsysWn

Echo PDFGaussian

Mean = Pr0.5

Variance = kTºsysWn

r (volts)Matched Filter

Outputat Optimum Time

γ

Coherent Detection (PLL)M Pulse Integration

Fixed Pr

Coherent Detection (PLL)M Pulse Integration

Fixed Pr

Noise PDFGaussianMean = 0

Variance = MkTºsysW n

Signal PDFGaussian

Mean = MPr0.5

Variance = MkTºsysW n

r (volts)Matched Filter

Outputat Optimum Timeγ

Coherent Detection, Single PulseRCS Exponential PDF

Coherent Detection, Single PulseRCS Exponential PDF

Noise PDFGaussianMean = 0

Variance = kTºsysW n

Echo PDFGaussian☺Rayleigh

Mean = Pr0.5

Variance = Var(sig) + Var(noise)= 0.2734Pr

+ kTºsysW n

r (volts)Matched Filter

Outputat Optimum Time

γ

Coherent DetectionM Pulse Integration

RCS Exponential PDF

Coherent DetectionM Pulse Integration

RCS Exponential PDF

Noise PDFGaussianMean = 0

Variance = MkTºsysW n

Signal PDFGaussian

Mean = MPr0.5

Variance = MkTºsysW n

+ MPr0.2734

r (volts)Matched Filter

Outputat Optimum Timeγ

Variance of MFD voltage (Rayleigh) PDF

Integral Result Integral Result

Stephen O. RiceStephen O. Rice

Born 1907Born 1907 Died 1986Died 1986 Bell Labs 1930 – 1972Bell Labs 1930 – 1972 IEEE FellowIEEE Fellow Paper "Mathematical Analysis of Paper "Mathematical Analysis of

Random Noise" discusses Rice PDFRandom Noise" discusses Rice PDF

Source: http://www.ieeeghn.org/wiki/index.php/Stephen_Rice

Friedrich BesselFriedrich Bessel Born 1784Born 1784 Died 1846Died 1846 German MathematicianGerman Mathematician In 1820's, while studyingIn 1820's, while studying

"many body" gravitational"many body" gravitational systems, generalized systems, generalized solutions forsolutions for

Rice PDFRice PDF

x

Starts to look somewhat Gaussian when v/σ2 > 2

Coherent DetectionCoherent Detection Previous Equations are IdealPrevious Equations are Ideal

Require instantaneous phase lock to echoRequire instantaneous phase lock to echo Won't happen in realityWon't happen in reality

Will effectively lose part of echo pulse…Will effectively lose part of echo pulse…• … … Till PLL or Phase-Frequency detector locksTill PLL or Phase-Frequency detector locks

Lock can be obtained on Doppler Shifted echoesLock can be obtained on Doppler Shifted echoes Could use bank of PLL's, free running at different freqsCould use bank of PLL's, free running at different freqs

Coherent Detection not used a lotCoherent Detection not used a lot But equations give feel as to processBut equations give feel as to process

Have somewhat easily digestible derivations Have somewhat easily digestible derivations

Non Coherent Radar DetectionNon Coherent Radar Detection Fixed PFixed Prr & Random Noise & Random Noise

Single Range Bin Single Range Bin Noise has Rayleigh DistributionNoise has Rayleigh Distribution

Mean = 1.253 Mean = 1.253 σσnn

Variance = 0.4292Variance = 0.4292 σ σnn22

σσnn22 = kTº = kTºsyssysWWnn (if calculations off front end) (if calculations off front end)

Signal + Noise has Ricean DistributionSignal + Noise has Ricean Distribution≈ Gaussian if ≈ Gaussian if αα//σσnn

22 = P = Prr0.50.5//σσnn

22 > 5 > 5 Mean = PMean = Prr

0.50.5

Variance = kTºVariance = kTºsyssysWWnn

Noncoherent (Quadrature) Detection, Single Pulse, Fixed Pr

Noncoherent (Quadrature) Detection, Single Pulse, Fixed Pr

Noise PDFRayleigh

Mean = 1.253(kTºsysWn)0.5

Variance = 0.4292kTºsysWn

Echo PDF≈ GaussianMean = Pr

0.5

Variance = kTºsysWn

r (volts)Matched Filter

Outputat Optimum Time

γ

Ex) P(Hit | Coherent) = 0.3253 & P(Hit | Noncoherent) = 0.1692

Noncoherent Detection, M Pulse Integration(Envelope Detection, fixed Pr)

Noncoherent Detection, M Pulse Integration(Envelope Detection, fixed Pr)

Sample envelope M times, sum resultsSample envelope M times, sum resultsMake decision based on sumMake decision based on sumNoise and Signal PDF's approximately GaussianNoise and Signal PDF's approximately Gaussian

P(Hit) = P(Hit) = Q[0.6551QQ[0.6551Q-1-1[P(FA)] + 1.253M[P(FA)] + 1.253M0.50.5 – (M*SNR) – (M*SNR)0.50.5]]

Noncoherent (Quadrature) DetectionM Pulse Integration

Fixed Pr

Noncoherent (Quadrature) DetectionM Pulse Integration

Fixed Pr

Noise PDF≈ Gaussian

Mean = M1.253(kTºsysWn)0.5

Variance = M0.4292kTºsysWn

Signal PDFGaussian

Mean = MPr0.5

Variance = MkTºsysWn

r (volts)Matched Filter

Outputat Optimum Timeγ

Ex) P(Hit | Coherent) = Q(-8.848) & P(Hit | Noncoherent) = Q(-6.523)

CommentComment Noncoherent Integration GainNoncoherent Integration Gain

Sometimes stated as MSometimes stated as M0.5 0.5

P(Hit) ≈ Q[ QP(Hit) ≈ Q[ Q-1-1[P(FA)] – (M[P(FA)] – (M0.50.5*SNR)*SNR)0.50.5 ] ] "Noncoherent Integration Gain, and it's Approximation" "Noncoherent Integration Gain, and it's Approximation"

Mark Richards, GaTech, May 2013Mark Richards, GaTech, May 2013 Has an example where gain is MHas an example where gain is M0.83330.8333

P(Hit) ≈ Q[ QP(Hit) ≈ Q[ Q-1-1[P(FA)] – (M[P(FA)] – (M0.8330.833*SNR)*SNR)0.50.5 ] ] EX) P(Hit) ≈ Q[4.753 – 10EX) P(Hit) ≈ Q[4.753 – 100.8330.833*18.5) *18.5) 0.50.5

= Q[4.753 – 11.22] = Q[-6.469]= Q[4.753 – 11.22] = Q[-6.469] Safer to say gain is MSafer to say gain is Maa; 0.5 < a < 1.0; 0.5 < a < 1.0

Radar P(Hit), Fixed PrRadar P(Hit), Fixed Pr

Single Pulse, CoherentSingle Pulse, Coherent P(Hit) = Q[ Q P(Hit) = Q[ Q-1-1[P(FA)] – SNR[P(FA)] – SNR0.50.5]] Equation 12.19 in textEquation 12.19 in text

M Pulse Integration, CoherentM Pulse Integration, Coherent P(Hit) = Q[ Q P(Hit) = Q[ Q-1-1[P(FA)] – (M*SNR)[P(FA)] – (M*SNR)0.50.5 ] ] See equation 13.3 in textSee equation 13.3 in text

Radar P(Hit), Exponential PrRadar P(Hit), Exponential Pr

Single Pulse, CoherentSingle Pulse, CoherentNoise is GaussianNoise is GaussianSignal (echo) Voltage is RayleighSignal (echo) Voltage is Rayleigh Evaluate 2nd Order PDF f(n,s) or f(n)☺f(s)Evaluate 2nd Order PDF f(n,s) or f(n)☺f(s)

M Pulse Integration, CoherentM Pulse Integration, CoherentP(Hit) ≈ Q{[QP(Hit) ≈ Q{[Q-1-1[P(FA)][P(FA)]σσnn

– (M*P– (M*Psignal_1signal_1))0.50.5 ]/ ]/σσsumsum}} where where σσsum sum = (= (σσ22

nn + + σσ22ss))0.50.5

σσ22nn = noise power = noise power

σσ22s s = variance of noise free signal (echo) voltage= variance of noise free signal (echo) voltage

= 0.2734*M*P = 0.2734*M*Psignal_1signal_1

Radar P(Hit), Fixed PrRadar P(Hit), Fixed Pr Single Pulse, NoncoherentSingle Pulse, Noncoherent

Noise is Rayleigh DistributedNoise is Rayleigh DistributedSignal is Ricean Distributed → GaussianSignal is Ricean Distributed → GaussianP(Hit) ≈ Q[P(Hit) ≈ Q[γγ//σσnn – SNR – SNR0.50.5]]

where where γγ = {ln[1/P(FA)]2 = {ln[1/P(FA)]2σσnn22}}0.50.5

Equation 12.49 in TextEquation 12.49 in Text M Pulse Integration, NoncoherentM Pulse Integration, Noncoherent

P(Hit) ≈ Q[ QP(Hit) ≈ Q[ Q-1-1[P(FA)] – (M[P(FA)] – (MaaSNR)SNR)0.50.5 ] ]

P(Hit) ≈ Q[ 0.655QP(Hit) ≈ Q[ 0.655Q-1-1[P(FA)] +1.253M[P(FA)] +1.253M0.50.5 - (M*SNR)- (M*SNR)0.50.5 ] ]

Noncoherent DetectionFluctuating Pr

Noncoherent DetectionFluctuating Pr

Will not be derived in classWill not be derived in class Text has calculations for several casesText has calculations for several cases Below is PDF of SignalBelow is PDF of Signal

Sum of S.I. Gaussian noise & Rayleigh echoSum of S.I. Gaussian noise & Rayleigh echo

Need PDF of I sumNeed PDF of I sum22 added to another SI Q sum added to another SI Q sum22, then take , then take square root.square root.

Peter SwerlingPeter Swerling

Born 1929Born 1929 Died 2000Died 2000 PhD in Math at UCLAPhD in Math at UCLA Worked at RANDWorked at RAND

Entrepreneur (founded 2 consulting companies)Entrepreneur (founded 2 consulting companies) Developed & analyzed Swerling Target Models Developed & analyzed Swerling Target Models

in 1950's while at RANDin 1950's while at RAND

SwerlingModel

Performance

M = 10Noncoherent Integration

P(FA) = 10-9

SwerlingModel

Performance

M = 10Noncoherent Integration

P(FA) = 10-9

Source: Merrill Skolnik's Introduction to Radar Systems, 3rd Edition

Receiver Phase Locked LoopReceiver Phase Locked Loop

XActive

Low PassFilter

VoltageControlledOscillator

cosωct(from antenna)

sin((ωvcot +θ) -sin((ωvco -ωc)t+θ)

VCO set to free run at ≈ ωc

VCO output frequency = ωc + K * input voltage

LPF withnegative gain.

2 sinα cosβ = sin(α-β) + sin(α+β)

PPI with clutterPPI with clutter

Source: www.radartutorial.eu