ECEN5633 Radar TheoryECEN5633 Radar TheoryLecture #8 5 February 2015Lecture #8 5 February 2015Dr. George ScheetsDr. George Scheetswww.okstate.edu/elec-eng/scheets/ecen5633www.okstate.edu/elec-eng/scheets/ecen5633

ECEN5633 Radar TheoryECEN5633 Radar TheoryLecture #8 5 February 2015Lecture #8 5 February 2015Dr. George ScheetsDr. George Scheetswww.okstate.edu/elec-eng/scheets/ecen5633www.okstate.edu/elec-eng/scheets/ecen5633

Read 8.1, 8.2, 8.4Read 8.1, 8.2, 8.4Skim 8.3Skim 8.3

Problems 2.19, 26, & 30Problems 2.19, 26, & 30 Corrected Quizzes due 1 week after returnCorrected Quizzes due 1 week after return 100 point Test #1100 point Test #1

17 February (Live)17 February (Live) No later than 24 February (DL)No later than 24 February (DL)

Time Averaged Autocorrelation of

Infinite Length Baseband Pulse Stream

Time Averaged Autocorrelation of

Infinite Length Baseband Pulse Stream

τ (sec)

RXX(τ)

-Tp 0 Tp

Pave

T-T

……

Can be thought of as a triangle convolved with an

infinite length stream of delta functions.

F.T. of TriangleF.T. of TrianglePeak 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 FunctionsF.T. of Infinite Length Time Domain Stream of Delta Functions

Is an infinite length stream of delta Is an infinite length stream of delta functions in the frequency domain.functions in the frequency domain. δ(t) in time domain → δ(t) in time domain → δδ(f)/T in freq domain(f)/T in freq domain T seconds spacing in time domain →T seconds spacing in time domain →

1/T Hz spacing in frequency 1/T Hz spacing in frequency domaindomain

Convolution in Time DomainConvolution in Time Domain A triangle convolved with infinite length train of delta functions =A triangle convolved with infinite length train of delta functions =

Convolution in Time Domain = Multiplication in Frequency DomainConvolution in Time Domain = Multiplication in Frequency Domain Power Spectrum = (sincPower Spectrum = (sinc22)(infinite train of )(infinite train of δδ))

τ (sec)

RXX(τ)

-Tp 0 Tp

Pave

T-T

……

Convolution in Time DomainConvolution in Time Domain Is multiplication in Frequency DomainIs multiplication in Frequency Domain Power spectrum of baseband pulse Power spectrum of baseband pulse

stream…stream…

Peak value is= PavgTp/T

Delta Functions

are= 1/T Hz

apart

f(Hz)… …

Radar Cross SectionRadar Cross Section

Complicated Function of target…Complicated Function of target… SizeSize MaterialMaterial Shape & OrientationShape & Orientation

Corner reflectors have large Corner reflectors have large σσ Antennas can have large Antennas can have large σσ

FrequencyFrequency

RCS of WWII A-26 InvaderRCS of WWII A-26 Invader

source: Wikipedia

Small angle changes can Small angle changes can cause big change to cause big change to σσ due to…due to… ReflectionsReflections

Scatter DirectionsScatter DirectionsPhase CancellationsPhase Cancellations

AbsorptionAbsorption Thru TransmissionThru Transmission

σ is a Random Variableσ is a Random Variable

If target or radar is movingIf target or radar is moving Target has many scatterers?Target has many scatterers?

None dominate?None dominate? σ is Exponentially Distributedσ is Exponentially Distributed

Pr is Exponentially DistributedPr is Exponentially Distributed

Receiver echo voltage is Rayleigh DistributedReceiver echo voltage is Rayleigh Distributed

Mapping of 1 RV to AnotherMapping of 1 RV to Another

PDF fPDF fXX(x) & mapping y = g(x) known(x) & mapping y = g(x) known

Need fNeed fYY(y) (y)

Can find fCan find fYY(y) via f(y) via fXX(x)/|g'(x)|(x)/|g'(x)|

Then substitute x = g Then substitute x = g -1-1(y)(y) Note bounds of y (may differ from x)Note bounds of y (may differ from x) We will focus on 1 to 1 mappingsWe will focus on 1 to 1 mappings

Specific x map to a single value of ySpecific x map to a single value of y

John William Strutt3rd Baron RayleighJohn William Strutt3rd Baron Rayleigh

English PhysicistEnglish Physicist Born 1842Born 1842 Died 1919Died 1919 Won Noble Prize in 1904Won Noble Prize in 1904

Discovery of ArgonDiscovery of Argon Researched EM wavesResearched EM waves

Rayleigh PDF's namedRayleigh PDF's namedafter himafter him

source: Wikipedia

Snell's LawSnell's Law Should have been named after Ibn SaulShould have been named after Ibn Saul

Circa 940 – 1000Circa 940 – 1000 Persian Mathematician & Optics EngineerPersian Mathematician & Optics Engineer Showed up in his 984 paperShowed up in his 984 paper

"On Burning Mirrors & Lenses" "On Burning Mirrors & Lenses"

Named after Willebrord SnelliusNamed after Willebrord Snellius Born 1580, Died 1626Born 1580, Died 1626 Dutch Astronomer & MathematicianDutch Astronomer & Mathematician Derived equivalent version in 1621Derived equivalent version in 1621

Atmosphere Atmosphere

Slows down EM wavesSlows down EM waves Bends EM wavesBends EM waves 4/3 Earth Model4/3 Earth Model

Radar Horizon function of 4/3 Earth RadiusRadar Horizon function of 4/3 Earth Radius

Edwin ArmstrongEdwin Armstrong

Born 1890Born 1890 Died 1954Died 1954 Army Officer & Professor Army Officer & Professor

at Columbia Universityat Columbia University Credited with inventingCredited with inventing

Superheterodyne Receivers (1918)Superheterodyne Receivers (1918) FM Radios (Patented in 1933)FM Radios (Patented in 1933)

Winner of 1Winner of 1stst IEEE Medal of Honor IEEE Medal of Honor

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)

Phase Locked LoopPhase Locked Loop

XActive

Low PassFilter

VoltageControlledOscillator

cosωct(from antenna)

sin(ωvcot) -sin((ωvco -ωc)t)

VCO frequency and phase locked.ωvco-ωc = 0 & θ = 0

Input to VCO ≈ 0 volts.

LPF withnegative gain.

Phase Locked LoopPhase Locked Loop

XActive

Low PassFilter

VoltageControlledOscillator

cosωct(from antenna)

-sinθ

VCO on frequency & positive θ?VCO phase is slightly ahead & needs to slow down.

Negative voltage momentarily applied.

sin((ωvcot )+θ)

Phase Locked LoopPhase Locked Loop

XActive

Low PassFilter

VoltageControlledOscillator

cosωct(from antenna)

sinωvcot -sin(ωvco -ωc)t

LPF withnegative gain.

VCO off frequency?Oscillating input voltage moves VCO frequency up & down.

If close enough to input, system will lock.

VCO Input Voltage VCO Input Voltage