Transcript ppt - Oklahoma State University

ECEN5633 Radar Theory
Lecture #8
5 February 2015
Dr. George Scheets
www.okstate.edu/elec-eng/scheets/ecen5633
 Read 8.1, 8.2, 8.4
Skim 8.3
 Problems 2.19, 26, & 30
 Corrected Quizzes due 1 week after return
 100 point Test #1
 17
February (Live)
 No later than 24 February (DL)
Time Averaged Autocorrelation
of
Infinite Length Baseband Pulse Stream
RXX(τ)
Pave
…
-T
-Tp 0
…
Tp
τ (sec)
T
Can be thought of as a triangle
convolved with an
infinite length stream of delta functions.
F.T. of Triangle
Peak value is
area under the
triangle
= Pavg*Tp
Nulls are at
integer
multiples of
1/pulse width
= 1/Tp Hz
F.T. of Infinite Length Time
Domain Stream of Delta Functions

Is an infinite length stream of delta
functions in the frequency domain.
 δ(t)
in time domain → δ(f)/T in freq domain
 T seconds spacing in time domain →
1/T Hz spacing in frequency domain
Convolution in Time Domain

A triangle convolved with infinite length
train of delta functions =
RXX(τ)
Pave
…
-T

-Tp 0
…
Tp
τ (sec)
T
Convolution in Time Domain =
Multiplication in Frequency Domain
 Power
Spectrum = (sinc2)(infinite train of δ)
Convolution in Time Domain
Is multiplication in Frequency Domain
 Power spectrum of baseband pulse
stream…

Delta
Functions
are
= 1/T Hz
apart
Peak value is
= PavgTp/T
… …
f(Hz)
Radar Cross Section

Complicated Function of target…
 Size
 Material
 Shape
& Orientation
reflectors have large σ
 Antennas can have large σ
 Corner
 Frequency
RCS of WWII A-26 Invader

source: Wikipedia
Small angle changes can
cause big change to σ
due to…
 Reflections
Scatter Directions
Phase Cancellations
 Absorption
 Thru Transmission
σ is a Random Variable
If target or radar is moving
 Target has many scatterers?
None dominate?

σ
is Exponentially Distributed
 Pr
is Exponentially Distributed
 Receiver
echo voltage is Rayleigh Distributed
Mapping of 1 RV to Another
PDF fX(x) & mapping y = g(x) known
Need fY(y)
 Can find fY(y) via fX(x)/|g'(x)|
 Then substitute x = g -1(y)
 Note bounds of y (may differ from x)
 We will focus on 1 to 1 mappings

 Specific
x map to a single value of y
John William Strutt
3rd Baron Rayleigh
English Physicist
 Born 1842
 Died 1919
 Won Noble Prize in 1904

 Discovery
of Argon
 Researched EM waves

Rayleigh PDF's named
after him
source: Wikipedia
Snell's Law

Should have been named after Ibn Saul
940 – 1000
 Persian Mathematician & Optics Engineer
 Showed up in his 984 paper
"On Burning Mirrors & Lenses"
 Circa

Named after Willebrord Snellius
 Born
1580, Died 1626
 Dutch Astronomer & Mathematician
 Derived equivalent version in 1621
Atmosphere
Slows down EM waves
 Bends EM waves
 4/3 Earth Model

 Radar
Horizon function of 4/3 Earth Radius
Edwin Armstrong
Born 1890
 Died 1954
 Army Officer & Professor
at Columbia University
 Credited with inventing

 Superheterodyne
Receivers (1918)
 FM Radios (Patented in 1933)

Winner of 1st IEEE Medal of Honor
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
Phase Locked Loop
cosωct
(from antenna)
X
sin(ωvcot)
Active
Low Pass
Filter
Voltage
Controlled
Oscillator
LPF with
negative gain.
-sin((ωvco -ωc)t)
VCO frequency and phase locked.
ωvco-ωc = 0 & θ = 0
Input to VCO ≈ 0 volts.
Phase Locked Loop
cosωct
(from antenna)
X
Active
Low Pass
Filter
Voltage
Controlled
sin((ωvcot )+θ) Oscillator
-sinθ
VCO on frequency & positive θ?
VCO phase is slightly ahead & needs to slow down.
Negative voltage momentarily applied.
Phase Locked Loop
cosωct
(from antenna)
X
sinωvcot
Active
Low Pass
Filter
Voltage
Controlled
Oscillator
LPF with
negative gain.
-sin(ωvco -ωc)t
VCO off frequency?
Oscillating input voltage moves VCO frequency up & down.
If close enough to input, system will lock.
VCO Input Voltage