LDI DIGISONDE 4D - Figure 1-14: Resolution of Overlapping Complementary Coded Pulses

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DIGISONDE-4D
SYSTEM MANUAL
VERSION 1.2.11
SECTION 1 - GENERAL SYSTEM DESCRIPTION 1-23
Figure 1-14: Resolution of Overlapping Complementary Coded Pulses
1:43. The phase of the received signal is detected by quadrature sampling; but, how is the complex quantity,
α
i
, or a
i
·exp[
i
], related to the RF phase (
i
) of each individual multipath component? It can be shown that this
phase represents the phase of the original RF signal components exactly. As shown in Equations 1-11 and
1-12, the down-converting (frequency translation) of r(t) by an oscillator, exp[j2πf
0
t] results in:
     
ii
P
i
iii
P
i
i
jtpatfjjtfjtpatr 
exp)(2exp2exp)()(
0
00
0
1
−=−−=

==
1-11
or
)()(
0
1 i
P
i
i
tptr

−=

=
,
where
 
iii
ja

exp=
is a complex amplitude
1-12
This signal maintains the parameter 
i
which is the original phase of each RF multipath component. Note that
the oscillator is defined as having zero phase (exp[j2f
0
t]).
1:44. Due to many possible mechanisms the pulse compression process will have imperfections, which may
cause energy reflected from any given height to leak or spill into other heights to some degree. This leakage is
the result of channel induced Doppler, mathematical imperfection of the phase code (except in the Complemen-
tary Codes which are mathematically perfect) and/or imperfection in the phase and amplitude response of the
transmitter or receiver. Several codes were simulated and analyzed for leakage from one height to another and
for tolerance to signal distortion caused by band-limiting filters. All of the pulse compression algorithms used
are cross-correlations of the received signal with a replica of the unit amplitude code known to have been sent.
Therefore, since Equation 1-10 represents a “cross-correlation” (the unit amplitude function p(t) is cross-
correlated with the complex amplitude weighted version) of p(k) with itself, it is the leakage properties of the
autocorrelation functions which are of interest.
1:45. The autocorrelation function of the Complementary phase code is shown in Figure 1-15.
VIS1-8
Odd #'d Pulses
Even #'d Pulses
—
–1 1 –4 2 –3 –3
—
Code 1
Matched Filter
Code 2
Matched Filter
— — — — — —
—
–1 1 –1 –1
—
— — — — — —
—
1 1 1 –1
—
—
1
—
2 4 –4
12 –2
—
3
—
+
— — — —
8
—
24
— — — —
=
—
1 1 4 2 3 –3
—
—
–1
—
–2 4 4
12 2
—
–3
—