Introduction to Radar Basics - MATLAB Simulations for Radar Systems Design - page 8

leading
edge

trailing
edge

incident pulse

v

L τ =

at time t = t 0

s = ∆ t

s = ∆ t

d ∆ t =

at time t = t 0 + ∆ t L ' = c τ'

reflected pulse

leading
edge

trailing
edge

Figure 1.6. Illustrating the impact of target velocity on a single pulse.

c τ' = c ∆ tv ∆ t –

(1.11)

Dividing Eq. (1.11) by Eq. (1.10) yields,

c τ'
c τ

c ∆ tv ∆ t –

c ∆ t ∆ t

------ = ----------------------- (1.12)

+

which after canceling the terms and from the left and right side of Eq.
(1.12) respectively, one establishes the relationship between the incident and
reflected pulses widths as

c ∆ t

–
cv

cv

----------- τ

τ′ = (1.13)

+

In practice, the factor is often referred to as the time dilation ( cv – ) ⁄ ( cv + )

factor. Notice that if , then . In a similar fashion, one can compute v = 0 τ′ = τ

for an opening target. In this case, τ′

vc +

cv

----------- τ

τ′ = (1.14)

–

To derive an expression for Doppler frequency, consider the illustration
shown in Fig. 1.7 . It takes the leading edge of pulse 2 seconds to travel a
distance to strike the target. Over the same time interval, the leading
edge of pulse 1 travels the same distance

∆ t

( cf r ⁄ ) d –

. More precisely, c ∆ t

d ∆ t = (1.15)

c
f r

--- – d = c ∆ t (1.16)

solving for yields ∆ t