Theory of GNSS Reflectometry - GNSS Remote Sensing: Theory, Methods and Applications - page 202

8.7 Noise and Coherence Issues

The thermal noise of the correlation measurement depends on the temperature of the
system, T , the integration bandwidth B i , as well as the coherent integration time i
(e.g. Cardellach 2002 ):

2 > D i kTB i

< j Y n j (8.54)

where k is the Boltzmann's constant. The power signal-to-thermal noise ratio (SNR)
is thus the result of dividing Eq. 8.49 by the power of the noise above. The resulting
expression cancels out the explicit dependency on the coherent integration time
i , but it keeps implicit dependency through the factor 1=B i D i , as well as the

j S 2 factor j . Therefore, the signal-to-noise level seems to increase linearly when

the integration is coherently performed during a longer period of time i ,asfar
as the signal itself is coherent during this integration interval. However, the Doppler
stripe which defines the delay-Doppler cell narrows with longer i (effect implicit
in j S 2

j ). As a result, increasing the integration time has the combined effect of 1=B i
(increasing SNR) and the reduction of the area from where the scattered signal is
contributing to the power (decreasing the SNR).
Besides the thermal noise, the reflected signal also might suffer fading or speckle
under diffuse scattering conditions. The received reflected signal at every instant of
time comes from a certain surface area which consists of many scattering points,
each generating an elementary contribution to the total received field, and all of
them summed up vectorially (amplitude and phase, Eq. 8.2 ). As seen in Eq. 8.2 and
Sect. 8.6 , the phase of each individual contribution has a significant geometric term,
related to the distance between the scatter and the receiver. As a consequence, if
the receiver moves a little amount, all these individual phases will change, and the
total vectorial sum of the fields might lead to different values of the total amplitude.
The sum might become constructive as well as destructive, leading to the effect
called fading: as the receiver moves over a rough surface, the time series of the
measured amplitudes result in a series of different values, even if the statistical
conditions of the surface have not changed. When imaging sensors are used, this
same effect happens within each resolution element (pixel), resulting in different
total amplitudes in different pixels, even if their statistical characteristics are the
same. It is then called speckle, and it corresponds to the same phenomena as fading,
but on the space domain. Elachi ( 1987 ) shows that the total amplitude have an
oscillation spectrum with frequencies from zero to an upper bound frequency f M .
This higher frequency is given by the size of the illuminated area from where the
contributions are being vectorially summed, D, the electromagnetic wavelength ,
the altitude and speed of the receiver h and v respectively:

D v
h

f M D (8.55)