ECEN5633 Radar Theory
Lecture #17
10 March 2015
Dr. George Scheets
www.okstate.edu/elec-eng/scheets/ecen5633
 Read 12.2
 Problems 11.5, 8, & 12.5
 Corrected quizzes due 1 week after return
 Live:

12 March
Exam #2, 31 March 2014 (< 4 April DL)
ECEN5633 Radar Theory
Lecture #18
12 March 2015
Dr. George Scheets
www.okstate.edu/elec-eng/scheets/ecen5633




Read 13.1 & 2
Problems 12.7, 8, & Web 3
Corrected quizzes due 1 week after return
 Live: 12 March
Exam #2, 31 March 2014 (< 4 April DL)
Coherent Detection (PLL),
Single Pulse, Fixed Pr
γ
Noise PDF
Gaussian
Mean = 0
Variance = kTºsysWn
Echo PDF
Gaussian
Mean = Pr0.5
Variance = kTºsysWn
r (volts)
Matched Filter
Output
at Optimum Time
Coherent Detection (PLL)
M Pulse Integration
Fixed Pr
γ
Noise PDF
Gaussian
Mean = 0
Variance = MkTºsysW n
r (volts)
Matched Filter
Output
at Optimum Time
Signal PDF
Gaussian
Mean = MPr0.5
Variance = MkTºsysW n
Coherent Detection, Single Pulse
RCS Exponential PDF
γ
Noise PDF
Gaussian
Mean = 0
Variance = kTºsysW n
Echo PDF
Gaussian☺Rayleigh
Mean = Pr0.5
Variance = Var(sig)
+ Var(noise)
= 0.2734Pr
+ kTºsysW n
r (volts)
Matched Filter
Output
at Optimum Time
Coherent Detection
M Pulse Integration
RCS Exponential PDF
γ
Noise PDF
Gaussian
Mean = 0
Variance = MkTºsysW n
r (volts)
Matched Filter
Output
at Optimum Time
Signal PDF
Gaussian
Mean = MPr0.5
Variance = MkTºsysW n
+ MPr0.2734
Variance of MFD voltage (Rayleigh) PDF
Integral Result
Stephen O. Rice
Born 1907
 Died 1986
 Bell Labs 1930 – 1972
 IEEE Fellow
 Paper "Mathematical Analysis of
Random Noise" discusses Rice PDF

Source: http://www.ieeeghn.org/wiki/index.php/Stephen_Rice
Friedrich Bessel
Born 1784
 Died 1846
 German Mathematician
 In 1820's, while studying
"many body" gravitational
systems, generalized
solutions for

Rice PDF
Starts to look
somewhat
Gaussian when
v/σ2 > 2
x
Coherent Detection

Previous Equations are Ideal
 Require
instantaneous phase lock to echo
 Won't happen in reality
 Will
effectively lose part of echo pulse…
• … Till PLL or Phase-Frequency detector locks
 Lock
can be obtained on Doppler Shifted echoes
 Could

use bank of PLL's, free running at different freqs
Coherent Detection not used a lot
 But
equations give feel as to process
Have somewhat easily digestible derivations
Non Coherent Radar Detection

Fixed Pr & Random Noise
Single Range Bin
 Noise
has Rayleigh Distribution
= 1.253 σn
 Variance = 0.4292 σn2
 σn2 = kTºsysWn (if calculations off front end)
 Mean
 Signal
+ Noise has Ricean Distribution
≈ Gaussian if α/σn2 = Pr0.5/σn2 > 5
= Pr0.5
 Variance = kTºsysWn
 Mean
Noncoherent (Quadrature) Detection,
Single Pulse, Fixed Pr
γ
Noise PDF
Rayleigh
Mean = 1.253(kTºsysWn)0.5
Variance = 0.4292kTºsysWn
r (volts)
Matched Filter
Output
at Optimum Time
Echo PDF
≈ Gaussian
Mean = Pr0.5
Variance = kTºsysWn
Ex) P(Hit | Coherent) = 0.3253 & P(Hit | Noncoherent) = 0.1692
Noncoherent Detection, M Pulse Integration
(Envelope Detection, fixed Pr)


Sample envelope M times, sum results
Make decision based on sum
Noise and Signal PDF's approximately Gaussian
P(Hit) =
Q[0.6551Q-1[P(FA)] + 1.253M0.5 – (M*SNR)0.5]
Noncoherent (Quadrature) Detection
M Pulse Integration
Fixed Pr
γ
Noise PDF
≈ Gaussian
Mean = M1.253(kTºsysWn)0.5
Variance = M0.4292kTºsysWn
r (volts)
Matched Filter
Output
at Optimum Time
Signal PDF
Gaussian
Mean = MPr0.5
Variance = MkTºsysWn
Ex) P(Hit | Coherent) = Q(-8.848) & P(Hit | Noncoherent) = Q(-6.523)
Comment



Noncoherent Integration Gain
 Sometimes stated as M0.5
 P(Hit) ≈ Q[ Q-1[P(FA)] – (M0.5*SNR)0.5 ]
"Noncoherent Integration Gain, and it's
Approximation"
 Mark Richards, GaTech, May 2013
 Has an example where gain is M0.8333
 P(Hit) ≈ Q[ Q-1[P(FA)] – (M0.833*SNR)0.5 ]
 EX) P(Hit) ≈ Q[4.753 – 100.833*18.5) 0.5
= Q[4.753 – 11.22] = Q[-6.469]
Safer to say gain is Ma; 0.5 < a < 1.0
Radar P(Hit), Fixed Pr

Single Pulse, Coherent
P(Hit) = Q[ Q-1[P(FA)] – SNR0.5]
 Equation

12.19 in text
M Pulse Integration, Coherent
P(Hit) = Q[ Q-1[P(FA)] – (M*SNR)0.5 ]
 See
equation 13.3 in text
Radar P(Hit), Exponential Pr

Single Pulse, Coherent
Noise is Gaussian
Signal (echo) Voltage is Rayleigh
 Evaluate

2nd Order PDF f(n,s) or f(n)☺f(s)
M Pulse Integration, Coherent
P(Hit) ≈ Q{[Q-1[P(FA)]σn – (M*Psignal_1)0.5 ]/σsum}
σsum = (σ2n + σ2s)0.5
 σ2n = noise power
 σ2s = variance of noise free signal (echo) voltage
= 0.2734*M*Psignal_1
 where
Radar P(Hit), Fixed Pr

Single Pulse, Noncoherent
Noise is Rayleigh Distributed
Signal is Ricean Distributed → Gaussian
P(Hit) ≈ Q[γ/σn – SNR0.5]
where γ = {ln[1/P(FA)]2σn2}0.5
 Equation

12.49 in Text
M Pulse Integration, Noncoherent
P(Hit) ≈ Q[ Q-1[P(FA)] – (MaSNR)0.5 ]
P(Hit) ≈ Q[ 0.655Q-1[P(FA)] +1.253M0.5
- (M*SNR)0.5 ]
Noncoherent Detection
Fluctuating Pr
Will not be derived in class
 Text has calculations for several cases
 Below is PDF of Signal


Sum of S.I. Gaussian noise & Rayleigh echo
PDF of I sum2 added to another SI Q sum2,
then take square root.
 Need
Peter Swerling





Born 1929
Died 2000
PhD in Math at UCLA
Worked at RAND
Entrepreneur (founded 2 consulting companies)
Developed & analyzed Swerling Target Models
in 1950's while at RAND
Swerling
Model
Performance
M = 10
Noncoherent
Integration
P(FA) = 10-9
Source: Merrill Skolnik's Introduction to Radar Systems, 3rd Edition
Receiver Phase Locked Loop
cosωct
(from antenna)
X
Active
Low Pass
Filter
LPF with
negative gain.
2 sinα cosβ = sin(α-β) + sin(α+β)
Voltage
Controlled
sin((ωvcot +θ) Oscillator
-sin((ωvco -ωc)t+θ)
VCO set to free run at ≈ ωc
VCO output frequency = ωc + K * input voltage
PPI with clutter
Source: www.radartutorial.eu