Derivations Flashcards
Superheterodyne Reciever
multiplication of two signals
carrier signal sc(t) and local oscillator slo(t)
sc(t) * slo(t) = sin(2πfct) * sin(2πflot)
sin(x) = (e^ix - e^-ix) / 2i
answer
sc(t) * slo(t) = 1/2 cos(2π(fc-flo)t) - 1/2cos(2π(fc+flo)t)
Radar Equation - Simple
Isotropic antenna and no attenuation loss
Power density = Pt/4πR^2
Including gain
Power density = PtGt/4πR^2
Encapsulate reflectivity by assuming an isotropic source
Pr = PtGtσ/4πR^2
Returns the observed echo power flux at the recieving antenna
Pr = PtGtσ/(4πR^2)^2
G = 4πAe/λ^2
Pr = PtGtσAe/(4πR^2)^2
Rearrange for R
Rmax occurs when Pr = Smin
Monostatic Radar when Gt = Gr = G
Radar Equation - Including System Noise
Reciever Noise
N = k To βn
βn = ∫ |H(f)|^2 df / |H(f0)|^2
Noise figure
Fn = Nout/kToβnGa
or Fn = Sin/Nin / Sout/Nout
Rearrange for Sin and subsitute Nin = k To βn
Smin occurs when we have (Sout/Nout)min
Subsitute this into our Radar equation
Radar Equation - Pulse Integration
Integration efficiency factor
Ei(n) = (S/N)1 / n(S/N)n
integration improvement factor
Ii(n) = nEi(n)
putting into Radar equation
Surveillance Radar Equation
Draw diagram
L1 x L2 = Ae
Tscan = Ti . φ/θ
rearrange for θ
n = fp Ti
θ = λ^2/Ae
substitute for θ and rearrange for Ae (1)
The average power of a pulse train is
Pav = Pt τ fp = Pt fp/β (2)
recall simple Radar equation in the form of Ae^2 substitute (1) and (2)
cancel and rearrange as required
Doppler frequency shift
phase length of the two-way path
φ = 2π/λ . 2R
φ = 4π/λ . R
moving target -> rate of change is
dφ/dt = ω = 4π/λ . dR/dt
dR/dt = vr
fd = ω/2π
=> fd = 4π/2πλ . vr
fd = 2vr/λ
Matched Filter
Noisy input to the IF is the signal to be recovered plus additive Gaussian noise
r(t) = s(t) + n(t)
where n(t) has spectral height N0/2
Output of the filter is given by the convolution integral
y(t) = (t ∫ 0) r(τ) h(t-τ) dτ
signal part
ys(t) = (t ∫ 0) s(τ) h(t-τ) dτ
noise part
yn(t) = (t ∫ 0) n(τ) h(t-τ) dτ
SNR = ys^2(t)/E[yn^2(t)]
denominator
E[yn^2(t)] = E{[(t ∫ 0) n(u) h(t-u) du][(t ∫ 0) n(v) h(t-v) dv]}
= N0/2 (t ∫ 0) h^2(t-u) du
numerator
Cauchy-Schwartz inequality
<S,Q>^2 <= |S|^2 |Q|^2
=> [(t ∫ 0) s(u) h(t-u) du]^2 <= (t ∫ 0) s^2(u) du (t ∫ 0) q^2(u) du
equality is reached when q(u) = cs(u)
=> [c (t ∫ 0) s(u) du]^2
putting it back together we get the SNR maximised when h(t-u) = cs(u)
SNR = 2/N0 (T ∫ 0) s^2(u) du
SNR = 2εs/N0
Matched Filter Frequency Response
Take the Fourier Transform of the Impulse response function
H(f) = c (T ∫ 0) s(T-u) exp(-i2πfu) du
r = T - u
u = T - r
du = -dr
u = 0 -> r = T and u = T -> r = 0.
= cexp(-i2πfT) (T ∫ 0) s(r)exp(i2πfr) dr
= cexp(-i2πfT) [ S(f)]*
=> |H(f)| = c|S(f)|
Radiometry
L = d^2φ/dAdΩ
φ = LAsΩd
Where
Ωd = Adcosθd/r^2
It is θd or θs depending on what is given
If the source and detector are parallel θd = θs =θ
θs = tan ^(-1)(r/h)
use trig to find r based on the variables given
For off axis detection include another cos term
For a point source φ = I ΩL
ΩL = AL/r^2
E = dφ/dA
Lambertian Source
φh = ( ∫ h) LAscosθsdΩd
using spherical coordinates
φh = LAs2pi 1/2
M = dφ/dA
M = Lπ
Fly-past dynamics
yac = x0 yac(dot) = 0
xac = x0/tan(A)
xac(dot) = -V = d/dt(xac)
use the chain rule
xac(dot) = -V = -x0/sin^2(A) dA/dt
rearrange for dA/dt
for second derivative use the chain rule again and subsitute for dA/dt
use double angle trig sin2x = 2sinxcosx
Delay line canceller
V1 = ksin(2πfdt - Φ0)
V1 = ksin(2πfdt - Φ0)
V2 = ksin[2πfd(t-Tp) -Φ0]
V1-V2
sinA-sinB = 2sin[(A-B)/2]cos[(A+B)/2]
determine A-B/2 and A+B/2
Substitute back into V1-V2
Only consider the sign term and divide by k
Fourier transform tutorial 1
Doppler shift for a pulse Doppler radar for a single delay line canceller.
r12 = r1 - r2
||r12|| = sqrt{(x1-x2)^2 + (y1-y2)^2}
Take time derivative
d||r12||/dt = [(x1-x2)(x1(dot)-x2(dot)) + (y1-y2)(y1(dot)-y2(dot)]/||r12||
x1(dot) = v1 sin φ1
y1(dot) = v1 cos φ1
x2(dot) = v2 sin φ2
y2(dot) = v2 cos φ2
fd = -||r12||/lambda
