Equations Flashcards
Range
Range = c Δt/2
Super heterodyne receiver
sc(t) * slo(t) = sin(2πfct)*sin(2πflot)
Intermediate frequency
fIF = fc - flo
Antenna effective aperture
Ae = pa A
pa = efficiency term
Receiver gain
Gr = 4πAe/λ^2
Simple radar equation
Smin = Pr
Monastic radar Gt = Gr = G
Thermal or Johnson noise
N = kTo βn
Noise bandwidth
βn = (-∞ ∫ ∞) |H(f)|^2 df / |H(f0)^2
Noise figure
Fn = Nout/kToβnGa
Noise figure in terms of signal to noise ratio
Fn = (Sin/Nin) / (Sout/Nout)
Number of pulses
n = θbfp/θ(dot)s = θbfp/6ωm
Pre detection integration
(S/N)n = (S/N)1/n
where n is the number of pulses
Integration efficiency factor
Ei(n) = (S/N)1/n(S/N)n
Integration improvement factor
Ii(n) = nEi(n)
Average power of a pulse train waveform
Pav = Pt τ/Tp = Pt τ fp
Duty cycle
duty = τ/Tp = τfp = Pav/Pt
Maximum unanambigous range
Run = cTp/2 = c/2fp
where fp is the PRF
Resolving ambiguities
n = ΔRapp/ΔRun
Multiple PRFs
Run,mprf = Run,1 (Run,2/(Run,2 - Run,1))(Run,3 / (Run,3 - Run,2))
Range resolution
resolution = cτ/2
Main lobe solid angle
θ = λ^2/Ae
Number of pulses for integration
n = fp Ti
Phase length
φ = 2π/λ x 2R
A moving target, the rate of change of phase
dφ/dt = ω = 4π/λ . dR/dt = 4π/λ . vr
Pulse compression ratio
τ/τ(comp) = ΔF/Δf
Time bandwidth product
τΔF
Solid angle of a small flat area tilted to the LoS
Ω = A /r^2cosθ
Radiant flux
Φ = dQ/dt
Radiant exitance
M = dΦ/dA
Irradiance
E = dΦ/dA
Radiant intensity
I = dΦ/dΩ
Radiance
L = d^2Φ/d(Acosθ)dΩ
Lambertian source
M = Lπ
Off axis detector causes the detector flux to be
φd = cos^3 θs
Flux for parallel surfaces
φd = cos^4 θ
Conservation of power in terms of transmission, absorption, reflection and emission
Φ0 = Φa + Φr + Φt
normalising
1 = α + p + τ
Emissivity
ε(λ) = M(λ)/M^BB(λ)
Directional spectral emissivity
ε(λ, θ, φ) = L(λ, θ, φ)/L^BB(λ)
Thin lens
1/f = 1/p + 1/q
Optical magnification
M = hi/ho = -q/p
Instantaneous field of view
IFoV = |tan^-1(hi/q)|
Plancks equation
Eg = hc/λ
Spectral responsivity
R(λ) = Vdet/φdet
Noise equivalent bandwidth
Δf = (inf ∫ 0) | R(f)/R(f=0)|^2 df
Noise equivalent power
NEP = φdet/SNR
Specific detectivity
D* = √(Ad)√(Δf)/NEP
Range performance
R = (sf . Tsize)/Ncyc
Benedict-bordner equation
β = α^2/(2-α)
Discrete white noise acceleration errors
β = 2(2-α) - 4sqrt(1-α)
Fly past dynamics
Azimuth spin rate
dA/dt = Vsin^2A/x0
Fly-past dynamics
Angular acceleration
d^2A/dt^2 = V^2/x0^2 sin2Asin^2A
Error transmittance or sensitivity
T_E(s) = 1 - Tcl(s)
Final value theorem
lim (t -> inf) f(t) = lim (s-> 0) sF(s)
Time to scan a field of regard
Tscan = Ti φ/θ
where θ is the main lobe solid angle
and φ is the total solid angle
Post detection
Include sqrt(n) in radar equation
Physical extent
Physical extent = cτ
Frequency response of a single delay line canceller
H(f) = 2sin(πfdTp)
Square error of the measurement
ε = 1/N ( N Σ n = 1) (xn - yn)^2
Square error of the filter output
ε = 1/N ( N Σ n = 1) (xn - xn(hat))^2
Attenuation
Attenuation = 1- εf/εm
Kalman Gain
Kk = P’k H^T(HP’k H^T+R)^-1
Update estimation
x(hat)k = x’(hat) k + Kk(zk-Hx’( hat)k)
Update covariance
Pk = (I-KkH)P’k
Project into k+1
x(hat){k+1} = Ax(hat)k
P{k+1} = APkA^T+Q
Number of bins
n = carrier frequency/range resolution
x^(-)1 =
y1
x(hat)1
= 0
Blind speed
fd = n/Tp = nfp
where n = 0,1,2,3 ….
Radial velocity (blind speed)
vn = n λfp/2
where n = 0,1,2,3….
