5.4 Resolution Capability of Radar Location and Radio Navigation Systems
115
Introduce new relative shift variables of reference signal and received one,
reflected from a target:
τ = t s − t d ,
(5.79)
ν = F d − f s .
(5.80)
We introduce a variable of integration as well:
t
= t − t d .
(5.81)
Due to evident transformations, we obtain (omitting a stroke in notation t’):
ψ s = a
⎧
⎨
⎩
∞
−∞
˙
S(t) ˙
S
∗
(t − τ )e
j2πντ dt
⎫
⎬
⎭
exp j 2π [( f 0 + f s )τ ]dt.
(5.82)
Function in curved brackets in (5.82) formula, we designate as:
s (τ, ν) =
∞
−∞z
˙
S(t) ˙
S
∗
(t − τ )e
j2πντ dt.
(5.83)
Function s was named as uncertainty function of a signal. Parameters τ and ν
are connected with range differences and radial velocities by relations:
τ =
2(D − D s )
c
; ν =
2(V − V s )
λ
,
(5.84)
where D and V —distance to a target and relative radial velocity correspondingly.
Values D s i V s —distance and velocity, on which signals processing system is
adjusted (matched filter or correlation device). Consequently, a function ψ s can be
written as follows:
ψ s (τ, ν) = a
.
s
(τ, ν)e
jθ(τ,ν)
,
(5.85)
where θ (τ, ν)—phase of complex-valued function ψ s , equals to:
θ (τ, ν) = 2π [( f 0 + f c )τ ] = 2π [ f 0 + F d − ν]τ.
(5.86)
In Fig. 5.25, the coordinate systems of variables are depicted t s i F d , and τ and
ν for a certain point of target position (TP 1 ) t s = t d1 i f s = F d , where τ = 0, ν =
0. At evaluation of resolution capability, a position of second-order chain relatively
to the first one corresponds to τ and ν shifts.