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DIGISONDE-4D
SYSTEM MANUAL
VERSION 1.2.11
1-26 SECTION 1 - GENERAL SYSTEM DESCRIPTION
where S
i
is the input signal amplitude, Q
i
the input noise amplitude, S
p
the processed signal amplitude, and Q
p
the processed noise amplitude. Q is chosen for the random variable to represent the noise amplitude, since N
would be confusing in this discussion. This coherent summation is similar to the pulse compression processing
described in the preceding section, where N, the number of pulses integrated is replaced by M, the number of
code chips integrated.
1:52. Another perspective on this process is achieved if the signal is normalized during integration, as is of-
ten done in an FFT algorithm to avoid numeric overflow. In this case S
p
is nearly equal to S
i
, but the noise am-
plitude has been averaged. Thus by invoking the central limit theorem [Freund, 1967 or any basic text on prob-
ability], we would expect that as long as the input noise is a zero mean (i.e., no DC offset) Gaussian process,
the averaged RMS noise amplitude,
np
(p for processed) will approach zero as the integration progresses, such
that after N repetitions:
np
2
=
ni
2
/
N (the variance represents power)
1:53. Since the SNR can be improved by a variable factor of N, one would think, we could use arbitrarily
weak transmitters for almost any remote sensing task and just continue integrating until the desired signal to
noise ratio (SNR) is achieved. In practical applications the integration time limit occurs when the signal under-
goes (or may undergo, in a statistical sense) a phase change of 90
. However, if the signal is changing phase
linearly with time (i.e., has a frequency shift, ), the integration time may be extended by Doppler integration
(also known as, spectral integration, Fourier integration, or frequency domain integration). Since the Fourier
transform applies the whole range of possible phase shifts needed to keep the phase of a frequency shifted sig-
nal constant, a coherent summation of successive samples is achieved even though the phase of the signal is
changing. The unity amplitude phase shift factor, e
–jt
, in the Fourier Integral (shown as Equation 1–18) varies
the phase of the signal r(t) as a function of time during integration. At the frequency () which stabilizes the
phase of the component of r(t) with frequency over the interval of integration (i.e., makes r(t) e
–jt
coherent)
the value of the integral increases with time rather than averaging to zero, thus creating an amplitude peak in
the Doppler spectrum at the Doppler line which corresponds to :
dtetrRtr
tj
−
==
)()()(F
1:54. Does this imply that an arbitrarily small transmitter can be used for any remote sensing application,
since we can just integrate long enough to clearly see the echo signal? To some extent this is true. There is no
violation of conservation of energy in this concept since the measurement simply takes longer at a lower power;
however, in most real world applications, the medium or environment will change or the reflecting surface will
move such that a discontinuous phase change will occur. Therefore a system must be able to detect the re-
ceived signal before a significant movement (e.g., a quarter to a half of a wavelength) has taken place. This
limits the practical length of integration that will be effective.
1:55. The discrete time (sampled data) processing looks very similar (as shown in Equation 1-19). For a
signal with a constant frequency offset (i.e., phase is changing linearly with time) the integration time can be
extended very significantly, by applying unity amplitude complex coefficients before the coherent summation is
performed. This stabilizes the phase of a signal which would otherwise drift constantly in phase in one direc-
tion or the other (a positive or negative frequency shift), by adding or subtracting increasingly larger phase an-
gles from the signal as time progresses. Then when the phase shifted complex signal vectors are added, they
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