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Q RF Propagation RADAR HORIZON RF Propagation TARGET VISIBILTY Detection & Estimation Probability CRAMER RAO LOWER BOUND Detection & Estimation Probability MAX LIKELIHOOD ESTIMATION Detection & Estimation Probability BINOMIAL Antennas ANTENNA BEAMWIDTH Antennas ANTENNA DIRECTIVITY Antennas ANTENNA GAIN Fourier Relationships CONTINUOUS-TIME FOURIER TRANSFORMATION Fourier Relationships FILTERING Radar Processing RADAR CROSS SECTION Fourier Relationships MODULATION PROPERTY Detection & Estimation Probability RICIAN Detection & Estimation Probability ERROR FUNCTIONS Detection & Estimation Probability NORMAL Detection & Estimation Probability RAYLEIGH RF Propagation WAVELENGTH RF Propagation DOPPLER SHIFT P r =P t G t G r 4πR λ 2 Pr: Received Power Pt: Transmit Power Gt: Transmit Gain Gr: Receive Gain R: Range c: Speed f: Frequency H: Horizon Re: Earth Radius ~ 6,371 km H: Horizon Re: Earth Radius ~ 6,371 km x: Observations p: Probability distribution function (or joint) θ: Distribution parameters can be vectors p: Success probability of each trial k: Number of successes n: Number of trials λ: Wavelength d: Antenna Diameter θ 1d : Half-power beamwidth in one principal plane (degrees) θ 2d : Half-power beamwidth in the other principal plane (degrees) A e : Effective Aperture Area λ: Wavelength μ: Mean σ: Standard Difference A: Distance between the reference point and the center of the bivariate distribution I 0 : Bessel Function of the first kind with order zero μ: Mean σ: Standard Difference A: Distance between the reference point and the center of the bivariate distribution RF Propagation FRIIS TRANSMISSION EQUATION D h = 2HR e λ = f c f d = –2v r / λ Target Height 2Re (Target Range - 2HRe) 2 = CRB = ∂θ ∂ ln p(x, θ) [ ] E ∂θ ∂ ln p(x, θ) [ ] ( } T { ) -1 f(k; n, p)= Pr(X =k) = ( ) p k (1−p) n−k k n p(r)= { r σ 2 e r 2 2σ 2 − 0 (r < 0) (0≤r≤∞) 1.2 1 0.8 0.6 0.2 0.4 0 0 2 4 6 8 10 σ = 0.5 σ =3 σ =1 σ = 4 σ = 2 p(r)= { − I 0 ( ) for (r < 0) for (A ≥ 0, r ≥ 0) σ 2 r e 0 2σ 2 (r 2 + A 2 ) σ 2 Ar 0.6 0.5 0.4 0.3 0.1 0.2 0.0 0 2 4 6 8 v = 0.0 v = 2.0 v = 0.5 v = 4.0 v = 1.0 σ = 1.00 μ: Mean σ: Standard Difference A: Distance between the reference point and the center of the bivariate distribution 1 σ 2π p(x) = (μ z =0; σ x =1.0) e − (x−μ) 2σ 2 Standard Normal Curve f(z) 0.4 0.1 0.2 0.3 -3 -2 -1 0 3 2 1 68.27% 95.45% 99.73% 3-σ 2-σ 1-σ z 1 2π z 2 2 e[ - ] Joint Density Function Π L(θ; x 1 , ..., x n )= f (x 1 , x 2 , ..., x n | θ)= f (x i | θ) n i = 1 Likelihood Σ ln L (θ; x 1 , ..., x n )= ln f (x i | θ) n i = 1 Log-Likelihood x i : Observations n: Number of Samples f: Is one, or joint, probability distribution(s) θ: Distribution parameters can be vectors μ: Mean σ: Standard Difference A: Distance between the reference point and the center of the bivariate distribution erfc(z)=1−erf(z)= 2 π ∫ ∞ e -t 2 d t erfc(x) 2 1.5 0.5 -2 -4 -2 -4 4 2 1 z erf(z)= 2 π ∫ z 0 e -t 2 d t ±1-σ: P (-1 ≤ z ≤ 1) = 0.6827 ±2-σ: P (-2 ≤ z ≤ 2) = 0.9545 ±3-σ: P (-3 ≤ z ≤ 3) = 0.9973 θ BW 3dB ∼ 0.886 b Nd cos θ 0 λ Phased Array, Radians θ BW null ∼ 1.22 d λ d λ θ BW 3dB ∼ 0.88 Parabolic, Radians D ≈ 4π ≈ θ 1d θ 2d 180 π ( ) 2 40000 θ 1d θ 2d G ant = λ 2 4πA e s(τ) = e j2π(f c τ+ bτ 2 ) , - ≤ τ ≤ 2 τ p 2 τ p 2 1 B p = b τ p s(): Transmitted Signal Waveform f c : Center Frequency τ: Range Time (fast time) τ p : Pulse Length b: Chirp Rate B p : Pulse Bandwidth γ: Range Frequency * “u” stands for unabsorbed or under K; “a” stands for absorption region or above K Electronic Warfare NOISE JAMMING Sidelobe J self : Self Protect Jammer Power J/S: Jam to Signal Ratio at Radar Receiver S: Radar Received Signal Power P t jam : Jammer Transmit Power G t jam : Jammer Transmit Gain R jr : Range between Jammer and Radar R: Range between Radar Target and Radar λ: Jammer Transmit Wavelength G r radar : Radar Receiver Gain L r radar : Radar Receiver Losses P t radar : Radar Transmit Power G t radar : Radar Transmitter Gain σ: Radar Target Radar Cross Section BW Radar : Radar Transmit Bandwidth BW Jam : Jammer Transmit Bandwidth J: Jammer Power Rmax jammed : Jammed Radar Range (Burn through Range) R max : Max Radar Range J/N: Jammer to Noise Ratio N: Total Noise k: Boltzmann’s constant T s : Receiver Temperature B N : Receiver Noise Bandwidth SNR: Radar Signal to Noise Ratio N f : Receiver Noise Figure (>1) J S = EIRP jam EIRP radar ( ) 4πR 2 σ ( ) ( ) J S = EIRP jam EIRP radar 4πR 2 σ ( ) BW radar ( ) BW jam P t jam G t jam ( ) 2 λ 4πR jr J self = L r radar } EIRP jam If BW jam ≥ BW radar 10 1 -150 -140 -130 -120 -110 -100 -90 -80 -70 -60 Reduction in Radar Detection Range due to JNR Normalized Maximum Radar Range Range (km) 10 2 10 3 J S Burn- through range for SNR = 13 dB J/N ~ ( ) 4 R max jammed R max Assume: J >> N BW Jam = BW Radar R max jammed 4 = P t G' t G' r λ 2 (4π) 3 (kT s B N N f +J) * SNR * L r* L t Mainlobe Reduction in Normalized R max 1 0.8 0.6 0.4 0.2 Rmax Rmax Jammed Main Beam ↓ Reduction in Radar Detection Range due to JNR Normalized Maximum Radar Range Jammer to Noise Ratio (dB) 0 5 10 15 20 25 30 35 40 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x(t) ↔ X(ω) 2π 1 x(t) = ∫ X(ω)e jωt dω +∞ -∞ Synthesis X(ω) = ∫ x(t)e -jωt dt +∞ -∞ Analysis h(t) * x(t) ↔ H(ω) X(ω) X(ω) x(t) H(ω) h(t) H(ω) X(t) h(t) * x(t) 1 δ(t) H(ω) h(t) H(ω) h(t) e jω ο t H(ω) e jω ο t H(ω ο ) H(ω): Frequency Response : Convolution operation Convolution Property Ideal Lowpass Filter Differentiator Fourier Relationships PARSEVAL’S RELATION 2π 1 ∫ |x(t)| 2 dt = ∫ |X(ω)| 2 dω +∞ -∞ +∞ -∞ ∫ T o |x(t)| 2 dt = ∑ |a k | 2 +∞ k=-∞ ~ T o 1 H(ω) -ω c ω c ω y(t) = =>H(ω) = jω dt dx(t) |H(ω)| ω x(t-t o ) ↔ e -jωt o X(ω) Time Shifting Differentiation ↔ jω X(ω) dt dx(t) jω 1 ∫ t x(τ)dτ ↔ X(ω) + πX(0) δ(ω) -∞ Integration Linearity ax 1 (t)+bx 2 (t) ↔ aX 1 (ω)+bX 2 (ω) Modulation Duality Property 2π 1 s(t) p(t) ↔ [S(ω)P(ω)] Convolution h(t) * x(t) ↔ H(ω)X(ω) x(t) t 1/a 1 1 e - a a 1/a √2 1/a |X(ω)| ω ↓ < X(ω) π/2 π/4 - π/4 - π/2 ω - a X(ω) - w w ω 1 X(ω) t π T1 sin ωT1 ω 2 2T1 x(t) - T1 T1 t 1 x(t) t π w w π sin wt 2πt 2 S∝σ, range Radar Cross Section (RCS, σ) Scattering σ = = lim 4πr 2 Incident Power Density / 4π Reflected Power to Receiver / Solid Angle |E i | 2 |E s | 2 ( ) P t P r or S σ Radar Processing TYPICAL VALUES OF RCS Radar Processing RADAR AMBIGUITY FUNCTION Radar Processing NOISE POWER Radar Processing SPEED OF LIGHT Radar Processing MAX UNAMBIGUOUS RANGE Radar Processing SIGNAL TO NOISE RATIO S(t): Complex Baseband Pulse τ: Time Delay f: Doppler Shift x(τ, t) =∫ ∞ s(t)s * (t-τ)e i2πft dt −∞ = kT s B N N f Noise Power in Receiver Speed of Light (approx) 3x10^8 300 1.62x10^5 1x10^9 1x10^3 Units m/sec m/usec NM/sec Ft/sec Ft/usec R max = c 2PRF PRF High Medium Low PRF 100 kHz 25 kHz 10 kHz Unambiguous Range 1.5 km 6 km 15 km Range Ambiguous Ambiguous Unambiguous Doppler Unambiguous Ambiguous Ambiguous c: Speed of Light PRF: Pulse Repetition Frequency Pr: Received Power Pt: Transmit Power Gt: Transmit Gain Gr: Receive Gain R: Range No: Noise Power L: Losses P R SNR= = P t G t G r σλ 2 G p L (4π) 3 R 4 k B T s B n N f N o ELECTRONIC WARFARE QUICK REFERENCE GUIDE Military Standard Bands U.S. Industry Standard Bands (IEEE Radar Designation) HF VH L F UHF S W * a Millimeter * u International Standard Bands G H G H C M A G H G H G H G H F F F F B B C C C C J 250 Frequency (MHz) Frequency (GHz) 20 30 100 200 300 500 400 300 200 100 80 60 40 30 20 15 10 8 6 5 4 3 2 1.5 12 18 27 110 7 (HF) 8 (VHF) 9 (UHF) 10 (SHF) 12 11(EHF) Band Designation HF VHF UHF L S C X K u K K a V W Frequency Range 3–30 MHz 30–300 MHz 300–1,000 MHz 1–2 GHz 2–4 GHz 4–8 GHz 8–12 GHz 12–18 GHz 18–27 GHz 27–40 GHz 40–75 GHz 75–110 GHz F F F F F F F F RF Propagation Detection & Estimation Probability Antennas f z (z)= -∞<x<∞ RADIO Wavelength (Meters) Frequency (Hz) 10 3 10 4 10 8 10 12 10 15 MICROWAVE 10 -2 INFRARED 10 -5 VISIBLE 10 -6 ULTRAVIOLET 10 -8 X-RAY 10 -10 GAMMA RAY 10 -12 THE ELECTROMAGNETIC SPECTRUM 10 16 10 18 10 20 = ln L n 1 =(θ|x) = ln f (x i | θ) Σ n i = 1 n 1 Average Log-Likelihood ˆ ˆ P t radar G t radar G r λ 2 S = (4π) 3 R 4 } EIRP radar σ G r radar Radar Processing LINEAR FM WAVEFORM σ f (x 1 , x 2 , ..., x n | θ) = f (x 1 | θ) x f (x 2 | θ) x ... x f (x n | θ) .0001 -40 Insects Birds Human Small Car Fighter Aircraft Bomber: Transport Aircraft Ships -30 -20 -10 0 10 20 30 40 .001 .01 0.1 1.0 10 100 1000 10000 dBsm m 2 Electronic Warfare Fourier Relationships Radar Processing X-band 300 m/s 0.03 m 20 kHz S-band 300 m/s 0.1 m 6 kHz Wavelength 3.00 m 0.10m 0.05m 0.03m f 100 MHz 3 GHz 6GHz 10GHz Band VHF S C X Velocity Wavelength Doppler Shift kT s : = -174 dBm K: Boltzmann’s constant = 1.38*10 -23 J/K B n : Noise Bandwidth T s : System Noise Temperature T s usually set to T 0 = 290K N f : Noise figure of receiver r ∞ K: Boltzmann’s constant = 1.38*10 -23 J/K B n : Noise Bandwidth T s : System Noise Temperature T s usually set to T 0 = 290K N f : Noise figure of receiver σ
