Ground-Based Remote Sensing of Cloud Properties ...
343
13.6 Signal Statistics
13.6.1 I and Q Voltage statistics
Now that we have the I and Q voltages, and hence the received power p.nt , in hand, we can
now obtain estimates of some of their statistical properties. Recall that both the I and Q
voltages contain sums of sines and cosines with arguments that contain the term
(13.56)
We cannot emphasize the next two points strongly enough. The first point is that the remainder
of 2kcrm %271" is a random variable with a uniform distribution over the interval from 0 to 271".
That is, since the exact locations of the scatterers are random across the resolution volume,
there is no fixed relation between 2kcr from one particle to the next and the phases 2kcrm %271"
end up uniformly distributed over 0 to 271". Therefore, the cosines and sines in the sums for I and
Q are random variables distributed between -1 and 1 with a probability density function that
is symmetric about O. The second point is that a variable that is the sum of random variables of
equal magnitudes tends to be Gaussian-distributed when a large number of terms are involved.
Since the I and Q voltage amplitudes Vm are due to a well-behaved particle size distribution
where no one magnitude is significantly larger than all of the other magnitudes, a series of
measurements of either I and Q, if long enough, produces values that are Gaussian-distributed
(Figure 13.13).
300
Q) 250
u
s:::
Q)
).j
).j
200
::I
U
U
0
....
0
150
>.
u
s::: 100
Q)
6- Q)
).j
50
~
0
- 10
-5
. '
. ,
o
Volts
5
10
Figure 13.13: Frequency of occurrence of I (solid line) and Q (dashed line) for the 4096
samples of the forward simulation.
The I and Q voltages have a number of other interesting properties. At a single, isolated
instant in time one cannot predict the exact value of I because one cannot know ahead of
time the exact location of each of the particles in the resolution volume. This is not to say
that one cannot predict the value of I an instant after a measurement of I; if two samples of
343
13.6 Signal Statistics
13.6.1 I and Q Voltage statistics
Now that we have the I and Q voltages, and hence the received power p.nt , in hand, we can
now obtain estimates of some of their statistical properties. Recall that both the I and Q
voltages contain sums of sines and cosines with arguments that contain the term
(13.56)
We cannot emphasize the next two points strongly enough. The first point is that the remainder
of 2kcrm %271" is a random variable with a uniform distribution over the interval from 0 to 271".
That is, since the exact locations of the scatterers are random across the resolution volume,
there is no fixed relation between 2kcr from one particle to the next and the phases 2kcrm %271"
end up uniformly distributed over 0 to 271". Therefore, the cosines and sines in the sums for I and
Q are random variables distributed between -1 and 1 with a probability density function that
is symmetric about O. The second point is that a variable that is the sum of random variables of
equal magnitudes tends to be Gaussian-distributed when a large number of terms are involved.
Since the I and Q voltage amplitudes Vm are due to a well-behaved particle size distribution
where no one magnitude is significantly larger than all of the other magnitudes, a series of
measurements of either I and Q, if long enough, produces values that are Gaussian-distributed
(Figure 13.13).
300
Q) 250
u
s:::
Q)
).j
).j
200
::I
U
U
0
....
0
150
>.
u
s::: 100
Q)
6- Q)
).j
50
~
0
- 10
-5
. '
. ,
o
Volts
5
10
Figure 13.13: Frequency of occurrence of I (solid line) and Q (dashed line) for the 4096
samples of the forward simulation.
The I and Q voltages have a number of other interesting properties. At a single, isolated
instant in time one cannot predict the exact value of I because one cannot know ahead of
time the exact location of each of the particles in the resolution volume. This is not to say
that one cannot predict the value of I an instant after a measurement of I; if two samples of