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Next: Pulse Pattern Optimization Up: ACF Determination Previous: ACF Determination

Double-Pulse

 

The autocovariance function is the basis for the determination of the Doppler velocity power spectrum of the scattering region or equivalently, assuming that only a single velocity component is present, the determination of the mean Doppler velocity of the scatterers.

To determine the complex ACF values at a specific lag-time, two pulses that are separated by a time interval tex2html_wrap_inline4464 are considered gif. Since the analysis that follows deals with proportional values only, constants have been dropped. As shown in Section 3.1, voltages at the receiver depend on the Fourier component of the density fluctuation with wave vector tex2html_wrap_inline4812 (or tex2html_wrap_inline4714 in short). Therefore the voltages of the two measurements are (see Figure 3.4):

equation1101

equation1108

where tex2html_wrap_inline4942 is a variable of integration and tex2html_wrap_inline4586 is the frequency of the radar wave.

   figure1115
Figure 3.4: Range-time diagram for two pulses seperated by a time tex2html_wrap_inline4464

The backscattered signal from two pulses can contain the three components tex2html_wrap_inline4948 , tex2html_wrap_inline4950 and tex2html_wrap_inline4952 as indicated in Figure 3.4.

tex2html_wrap_inline4954
is the electron density spectrum illuminated by the first pulse and measured at time t or illuminated by the second pulse and measured at time tex2html_wrap_inline4958
tex2html_wrap_inline4960
is the electron density spectrum illuminated by the second pulse and measured at time t
tex2html_wrap_inline4964
is the electron density spectrum illuminated by the first pulse and measured at time tex2html_wrap_inline4958
We attempt to gain information about the spectrum tex2html_wrap_inline4948 .

The received voltages can be mixed with two phase-coherent transmitted frequency components that differ by tex2html_wrap_inline4632 in their phase. As an example the voltage measurements at time t are described in Equation 3.36 and 3.37. The voltages at time tex2html_wrap_inline4958 are derived the same way. These voltages are taken to be different and correspond to the real and imaginary component of the signal.

  equation1134

  equation1145

Low-pass filtering can then be used to remove the HF part corresponding to a frequency of tex2html_wrap_inline4976 . We are left with the two voltages tex2html_wrap_inline4978 and tex2html_wrap_inline4980 , one corresponding to the real and one to the imaginary component of the complex voltage measurement tex2html_wrap_inline4982 :

equation1159

The voltages are averaged over several measurements. We denote the averaging process by tex2html_wrap_inline4984 . Furthermore a ratio between two voltages differing by the lag time tex2html_wrap_inline4464 , that is tex2html_wrap_inline4982 and tex2html_wrap_inline4990 , is formed. Using the following short-hand notation for the electron density spectra

equation1166

equation1172

equation1178

the ratio of the two time separated and averaged voltage measurements can be expanded and written as follows:

eqnarray1184

where * denotes a complex conjugate value. The information about the Doppler shift is contained in the term tex2html_wrap_inline4992 . It is not possible to determine this term by a single measurement since all terms in the above ratio are of the same order. But since we are dealing with averaged values uncorrelated values will go to zero in the mean. Thus, we assume that the mean values of the received voltages are zero, and the densities tex2html_wrap_inline4948 , tex2html_wrap_inline4950 and tex2html_wrap_inline4952 are independent. The term defining the ACF of the power spectrum belonging to tex2html_wrap_inline4948 at lag tex2html_wrap_inline4464 , that is tex2html_wrap_inline5004 , is then separable easily from the ratio [And80, p. 11-15,]. To determine a complete ACF as a function of tex2html_wrap_inline4464 , this procedure has to be repeated for different tex2html_wrap_inline4464 .


next up previous
Next: Pulse Pattern Optimization Up: ACF Determination Previous: ACF Determination

Andreas Schiffler
Wed Oct 9 10:05:17 CST 1996