Project Cyclops, A Design... - Department of Earth and Planetary ...

Define Nowlets(t) be a sinusoidal signal such that
Qn(x) = s_x +(2n-1)b/2] (b)V rect
S(_)= _(co- _o).
and
Then
Pn(V) =
{y - _h - (2_ - l)c.] }, rect (-_)
G(u,v) = Vab sine (_) sine b(v _-_)
X
Then
exp
(-]nhv._
N
n=l
[ booo ncv._
expj 2.-1) -- V+
G(u,v) = _ Qn(x)exp Xf ux dx
n=l
The displayed intensity pattern is
x f_f_ en0')exp -121r V vy ay
l(u'v)=lG(u'v)lZ=(Vab)2 sine2 (_'_) sinc2_-_ -b_°)27rV
N
N
The first integral yields
X _.. exp_(2n-1)_]
n=l
n=l
exp [-j(2n - 1)_ ]
Vb sine(_), { expln(2n_f l,bu] Skxf(27rVu)l/I
where
and the second
gives
asine exp _ [h-(2n-l)c]v
}
where sinc(_) = sin n/j/rr_ and S(w) is the spectrum of
S(t).
Thus
G(u,v) =
N
Vab sinc (_f) _
n=l
o
2V
The product of the two series is equal to sin2NO/sin20
which has narrow (for large n) peaks at 0 = _r,
= 0, -+1,... The u coordinate of the bright spots is
u = (?vc/2rrV)6o0 and the width of the spot between first
zeros is 2_f[b. The u coordinate of these spots gives
coarse frequency information. In the v direction a comb
function with a sinc 2 envelope is obtained. The distance
between zeros of the sine 2 function is 2hf/a; the comb
peaks occur at
c
2zrV/
exp _ [h-(2n- l)clv
and have a spacing of _f/c. Since a < c, the envelope
width is greater than the comb spacing. Thus, within
Ivl < hf/2c there is one comb spike at a time. The location
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