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DIGISONDE-4D
SYSTEM MANUAL
VERSION 1.2.11
SECTION 1 - GENERAL SYSTEM DESCRIPTION 1-21
where the carrier phase of each of the multipath components is now represented by a complex amplitude
i
which carries along the RF phase term, originally defined by
i
in Equation 1-7, for each multipath. Since the
pulse compression is a linear process and contributes no phase shift, the real and imaginary (i.e., in-phase and
quadrature) components of this signal can be pulse compressed independently by cross-correlating them with
the known spreading code p(t). The complex components can be processed separately because the pulse com-
pression (Equation 1–10) is linear and the code function, p(n), is all real. Therefore the phase of the cross-
correlation function will be the same as the phase of r
1
(t).
1:39. The classical derivation of matched filter theory [e.g., Thomas, 1964] creates a matched filter by first
reversing the time axis of the function p(t) to create a matched filter impulse response h(t) = p(–t). Implement-
ing the pulse compression as a linear system block (i.e., a “black box” with impulse response h(t)) will again
reverse the time axis of the impulse response function by convolving h(t) with the input signal. If neither rever-
sal is performed (they effectively cancel each other) the process may be considered to be a cross-correlation of
the received signal, r(t) with the known code function, p(t). Either way, the received signal, r
2
(n) after matched
filter processing becomes:
r
2
(n) = r
1
(n) * h(n) = r
1
(n) * p(–n)
or by substituting Equation 1-8 and writing out the discrete convolution, we obtain the cross-correlation ap-
proach,
( ) ( ) ( )
= ==
−=−−=
M
k
P
i
iii
P
i
i
nMnkpkpnr
1 11
2
)(
where n is the time domain index (as in the sample number, n, which occurs at time t = nT where T is the sam-
pling interval), P is the number of multipaths, k is the auxiliary index used to perform the convolution, and M is
the number of phase code chips. The last expression in Equation 1-10, the (n), is only true if the autocorrela-
tion function of the selected code, p(t), is an ideal unit impulse or “thumbtack” function (i.e., it has a value of M
at correlation lag zero, while it has a value of zero for all other correlation lags). So, if the selected code has
this property, then the function r
2
(n), in Equation 1-9 is the impulse response of the propagation path, which
has a value α
i,
(the complex amplitude of multipath signal i) at each time n =
i
(the propagation delay attributa-
ble to multipath i).
1:40. Figure 1-13 illustrates the unique implementation of Equation 1-10 employed for compression of
Complementary Sequence waveforms. A 4-bit code is used in this figure for ease of illustration but arbitrarily
long sequences can be synthesized (the DPS4D uses a Complementary Code 16-chips long). It is necessary to
transmit two encoded pulses sequentially, since the Complementary Codes exist in pairs, and only the pairs to-
gether have the desired autocorrelation properties. Equation 1-8 (the received signal without its sinusoidal
carrier) is represented by the input signal shown in the upper left of Figure 1-13. The time delay shifts (in-
dexed by n in Equation 1-10 are illustrated by shifting the input signal by one sample period at a time into the
matched filter. The convolution shifts (indexed by k in Equation 1-10 sequence through a multiply-and-
accumulate operation with the four 1 tap coefficients. The accumulated value becomes the output function
r
2
(n) for the current value of n. The two resulting expressions for Equation 1-10 (an r
2
(n) expression for each
of the two Complementary Codes) are shown on the right with the amplitude M=4 clearly expressed. The non-
ideal approximation of a delta function, (n-τ
i
), is apparent from the spurious a and –a amplitudes. However,
by summing the two r
2
(n) expressions resulting from the two Complementary Codes, the spurious terms are
cancelled, leaving a perfect delta function of amplitude 2M.
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