Derivations Flashcards
(14 cards)
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
y1 = x0 y1(dot) = 0
x1 = x0/tan(A)
x1(dot) = -V = d/dt(xac)
use the chain rule
x1(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
g(t) = Arect(t/T)cos(2πfct)
rect(t) = { 1 , -1/2 < t < 1/2
{ 0 , t > 1/2
Fourier transform
H(f) = (inf int -inf) f(t) exp(-2πft) dt
Substitute rect into g(t) and change the limits this is f(t)
Convert cos to exponential
Take the integral using the limits
Convert to sin upon completion of the integral
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
