Equations

Question 1

Range

Answer 1

Range = c Δt/2

Question 2

Super heterodyne receiver

Answer 2

sc(t) * slo(t) = sin(2πfct)*sin(2πflot)

Question 3

Intermediate frequency

Answer 3

fIF = fc - flo

Question 4

Antenna effective aperture

Answer 4

Ae = pa A

pa = efficiency term

Question 5

Receiver gain

Answer 5

Gr = 4πAe/λ^2

Question 6

Simple radar equation

Answer 6

Smin = Pr

Monastic radar Gt = Gr = G

Question 7

Thermal or Johnson noise

Answer 7

N = kTo βn

Question 8

Noise bandwidth

Answer 8

βn = (-∞ ∫ ∞) |H(f)|^2 df / |H(f0)^2

Question 9

Noise figure

Answer 9

Fn = Nout/kToβnGa

Question 10

Noise figure in terms of signal to noise ratio

Answer 10

Fn = (Sin/Nin) / (Sout/Nout)

Question 11

Number of pulses

Answer 11

n = θbfp/θ(dot)s = θbfp/6ωm

Question 12

Pre detection integration

Answer 12

(S/N)n = (S/N)1/n

where n is the number of pulses

Question 13

Integration efficiency factor

Answer 13

Ei(n) = (S/N)1/n(S/N)n

Question 14

Integration improvement factor

Answer 14

Ii(n) = nEi(n)

Question 15

Average power of a pulse train waveform

Answer 15

Pav = Pt τ/Tp = Pt τ fp

Question 16

Duty cycle

Answer 16

duty = τ/Tp = τfp = Pav/Pt

Question 17

Maximum unanambigous range

Answer 17

Run = cTp/2 = c/2fp

where fp is the PRF

Question 18

Resolving ambiguities

Answer 18

n = ΔRapp/ΔRun

Question 19

Multiple PRFs

Answer 19

Run,mprf = Run,1 (Run,2/(Run,2 - Run,1))(Run,3 / (Run,3 - Run,2))

Question 20

Range resolution

Answer 20

resolution = cτ/2

Question 21

Main lobe solid angle

Answer 21

θ = λ^2/Ae

Question 22

Number of pulses for integration

Answer 22

n = fp Ti

Question 23

Phase length

Answer 23

φ = 2π/λ x 2R

Question 24

A moving target, the rate of change of phase

Answer 24

dφ/dt = ω = 4π/λ . dR/dt = 4π/λ . vr

Question 25

Pulse compression ratio

Answer 25

τ/τ(comp) = ΔF/Δf

Question 26

Time bandwidth product

Answer 26

τΔF

Question 27

Solid angle of a small flat area tilted to the LoS

Answer 27

Ω = A /r^2cosθ

Question 28

Radiant flux

Answer 28

Φ = dQ/dt

Question 29

Radiant exitance

Answer 29

M = dΦ/dA

Question 30

Irradiance

Answer 30

E = dΦ/dA

Question 31

Radiant intensity

Answer 31

I = dΦ/dΩ

Question 32

Radiance

Answer 32

L = d^2Φ/d(Acosθ)dΩ

Question 33

Lambertian source

Answer 33

M = Lπ

Question 34

Off axis detector causes the detector flux to be

Answer 34

φd = cos^3 θs

Question 35

Flux for parallel surfaces

Answer 35

φd = cos^4 θ

Question 36

Conservation of power in terms of transmission, absorption, reflection and emission

Answer 36

Φ0 = Φa + Φr + Φt normalising 1 = α + p + τ

Question 37

Emissivity

Answer 37

ε(λ) = M(λ)/M^BB(λ)

Question 38

Directional spectral emissivity

Answer 38

ε(λ, θ, φ) = L(λ, θ, φ)/L^BB(λ)

Question 39

Thin lens

Answer 39

1/f = 1/p + 1/q

Question 40

Optical magnification

Answer 40

M = hi/ho = -q/p

Question 41

Instantaneous field of view

Answer 41

IFoV = |tan^-1(hi/q)|

Question 42

Plancks equation

Answer 42

Eg = hc/λ

Question 43

Spectral responsivity

Answer 43

R(λ) = Vdet/φdet

Question 44

Noise equivalent bandwidth

Answer 44

Δf = (inf ∫ 0) | R(f)/R(f=0)|^2 df

Question 45

Noise equivalent power

Answer 45

NEP = φdet/SNR

Question 46

Specific detectivity

Answer 46

D* = √(Ad)√(Δf)/NEP

Question 47

Range performance

Answer 47

R = (sf . Tsize)/Ncyc

Question 48

Benedict-bordner equation

Answer 48

β = α^2/(2-α)

Question 49

Discrete white noise acceleration errors

Answer 49

β = 2(2-α) - 4sqrt(1-α)

Question 50

Fly past dynamics Azimuth spin rate

Answer 50

dA/dt = Vsin^2A/x0

Question 51

Fly-past dynamics Angular acceleration

Answer 51

d^2A/dt^2 = V^2/x0^2 sin2Asin^2A

Question 52

Error transmittance or sensitivity

Answer 52

T_E(s) = 1 - Tcl(s)

Question 53

Final value theorem

Answer 53

lim (t -> inf) f(t) = lim (s-> 0) sF(s)

Question 54

Time to scan a field of regard

Answer 54

Tscan = Ti φ/θ where θ is the main lobe solid angle and φ is the total solid angle

Question 55

Post detection

Answer 55

Include sqrt(n) in radar equation

Question 56

Physical extent

Answer 56

Physical extent = cτ

Question 57

Frequency response of a single delay line canceller

Answer 57

H(f) = 2sin(πfdTp)

Question 58

Square error of the measurement

Answer 58

ε = 1/N ( N Σ n = 1) (xn - yn)^2

Question 59

Square error of the filter output

Answer 59

ε = 1/N ( N Σ n = 1) (xn - xn(hat))^2

Question 60

Attenuation

Answer 60

Attenuation = 1- εf/εm

Question 61

Kalman Gain

Answer 61

Kk = P’k H^T(HP’k H^T+R)^-1

Question 62

Update estimation

Answer 62

x(hat)k = x’(hat) k + Kk(zk-Hx’( hat)k)

Question 63

Update covariance

Answer 63

Pk = (I-KkH)P’k

Question 64

Project into k+1

Answer 64

x(hat){k+1} = Ax(hat)k P{k+1} = APkA^T+Q

Question 65

Number of bins

Answer 65

n = carrier frequency/range resolution

Question 66

x^(-)1 =

Answer 66

y1

Question 67

x(hat)1

Answer 67

= 0

Question 68

Blind speed

Answer 68

fd = n/Tp = nfp where n = 0,1,2,3 ….

Question 69

Radial velocity (blind speed)

Answer 69

vn = n λfp/2 where n = 0,1,2,3….