ECEN5633 Radar Theory Lecture #8 5 February 2015 Dr. George Scheets n Read 8.1, 8.2, 8.4 Skim 8.3 n Problems 2.19, 26, & 30 n Corrected Quizzes due 1 week after return n 100 point Test #1 u 17 February (Live) u No later than 24 February (DL)

Time Averaged Autocorrelation of Infinite Length Baseband Pulse Stream τ (sec) R XX (τ) -T p 0 T p P ave T -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 = P avg *T p Nulls are at integer multiples of 1/pulse width = 1/T p Hz

F.T. of Infinite Length Time Domain Stream of Delta Functions n Is an infinite length stream of delta functions in the frequency domain. u δ(t) in time domain → δ(f)/T in freq domain u T seconds spacing in time domain → 1/T Hz spacing in frequency domain

Convolution in Time Domain n A triangle convolved with infinite length train of delta functions = n Convolution in Time Domain = Multiplication in Frequency Domain u Power Spectrum = (sinc 2 )(infinite train of δ) τ (sec) R XX (τ) -T p 0 T p P ave T -T … …

Convolution in Time Domain n Is multiplication in Frequency Domain n Power spectrum of baseband pulse stream… Peak value is = P avg T p /T Delta Functions are = 1/T Hz apart f(Hz) … …

Radar Cross Section n Complicated Function of target… u Size u Material u Shape & Orientation F Corner reflectors have large σ F Antennas can have large σ u Frequency

RCS of WWII A-26 Invader source: Wikipedia n Small angle changes can cause big change to σ due to… u Reflections Scatter Directions Phase Cancellations u Absorption u Thru Transmission

σ is a Random Variable n If target or radar is moving n Target has many scatterers? None dominate? u σ is Exponentially Distributed F Pr is Exponentially Distributed u Receiver echo voltage is Rayleigh Distributed

Mapping of 1 RV to Another n PDF f X (x) & mapping y = g(x) known Need f Y (y) n Can find f Y (y) via f X (x)/|g'(x)| n Then substitute x = g -1 (y) n Note bounds of y (may differ from x) n We will focus on 1 to 1 mappings u Specific x map to a single value of y

John William Strutt 3 rd Baron Rayleigh n English Physicist n Born 1842 n Died 1919 n Won Noble Prize in 1904 u Discovery of Argon u Researched EM waves n Rayleigh PDF's named after him source: Wikipedia

Snell's Law n Should have been named after Ibn Saul u Circa 940 – 1000 u Persian Mathematician & Optics Engineer u Showed up in his 984 paper "On Burning Mirrors & Lenses" n Named after Willebrord Snellius u Born 1580, Died 1626 u Dutch Astronomer & Mathematician u Derived equivalent version in 1621

Atmosphere n Slows down EM waves n Bends EM waves n 4/3 Earth Model u Radar Horizon function of 4/3 Earth Radius

Edwin Armstrong n Born 1890 n Died 1954 n Army Officer & Professor at Columbia University n Credited with inventing u Superheterodyne Receivers (1918) u FM Radios (Patented in 1933) n Winner of 1 st IEEE Medal of Honor

Receiver Phase Locked Loop X Active Low Pass Filter Voltage Controlled Oscillator cosω c t (from antenna) sin((ω vco t +θ) -sin((ω vco -ω c )t+θ) VCO set to free run at ≈ ω c VCO output frequency = ω c + K * input voltage LPF with negative gain. 2 sinα cosβ = sin(α-β) + sin(α+β)

Phase Locked Loop X Active Low Pass Filter Voltage Controlled Oscillator cosω c t (from antenna) sin(ω vco t) -sin((ω vco -ω c )t) VCO frequency and phase locked. ω vco -ω c = 0 & θ = 0 Input to VCO ≈ 0 volts. LPF with negative gain.

Phase Locked Loop X Active Low Pass Filter Voltage Controlled Oscillator cosω c t (from antenna) -sinθ VCO on frequency & positive θ? VCO phase is slightly ahead & needs to slow down. Negative voltage momentarily applied. sin((ω vco t )+θ)

Phase Locked Loop X Active Low Pass Filter Voltage Controlled Oscillator cosω c t (from antenna) sinω vco t -sin(ω vco -ω c )t LPF with negative gain. VCO off frequency? Oscillating input voltage moves VCO frequency up & down. If close enough to input, system will lock.

VCO Input Voltage