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Subsections
- Cross-correlation function of voltage outputs
and
from antenna pair
:
![$\displaystyle R_{ij}({\tau}) = \langle v_{\rm i}(t) v_{\rm j}(t+\tau) \rangle = \lim_{T\rightarrow\infty}{\frac{1}{T}\int_0^T{v_{\rm i}(t) v_{\rm j}(t+\tau)dt}}$](img796.png) |
(6.18) |
- Covariance of two jointly random variables:
![$\displaystyle \mu = \langle xy \rangle - \langle x \rangle \langle y \rangle$](img797.png) |
(6.19) |
For signals of zero mean, and again identifying
and
,
![$\displaystyle \mu = R_{ij}({\tau})$](img800.png) |
(6.20) |
- Cross-correlation coefficient of two jointly random variables
of variance
and
:
![$\displaystyle \rho = \frac{\mu}{\sigma_{\rm x}^2 \sigma_{\rm y}^2}$](img804.png) |
(6.21) |
For jointly normal random variables of zero mean and of variance
,
and with
and
, the cross-correlation
function
and the cross-correlation coefficient are related by
![$\displaystyle R_{ij}({\tau}) = \rho\sigma^2$](img807.png) |
(6.22) |
- Bivariate Gaussian Probability Distribution:
Assume two Gaussian random variables
and
, both of
zero mean, and variance
. The probability
that the value of
is between
and
,
and that simultaneously the value of
between
and
,
is given by the jointly gaussian probability distribution
![$\displaystyle p(x_0,y_0) = \frac{1}{2\pi\sigma^2\sqrt{1-\rho^2}} \exp{\left(\frac{-(x_0^2+y_0^2-2\rho x_0 y_0)}{2\sigma^2(1-\rho^2)}\right)}$](img814.png) |
(6.23) |
In our case, the variable
is assigned to the output voltage of antenna
at time
, and
the output voltage of antenna
at time
.
The following determination of the clipping correction is due to
[Hagen et al. 1973]:
Given two jointly normal random variables
and
with covariance
,
and given some arbitrary function
, Price's theorem states that
![$\displaystyle \frac{\partial^m\langle g(x,y) \rangle}{\partial \mu^m} = \langle...
...y}^\infty{ \frac{\partial^{2m} g(x,y)}{\partial x^m \partial y^m} p(x,y) dx}dy}$](img816.png) |
(6.24) |
For random signals of zero mean, the covariance
is identical with the
cross-correlation function
defined in Eq.6.1. As
shown by Eq.6.1, we need to accumulate products of the voltage
outputs of two antennas
, but using the quantized signals rather than
the continuous ones. Thus, with
the identification
and
, and using
and
for the quantized signals, we can apply Price's theorem to the
4-level cross-correlation function
such that
![$\displaystyle \frac{dR_4}{d\rho} = \sigma^2 \frac{dR_4}{dR} = \sigma^2\frac{d \...
...l\tilde{x}}{\partial x}\cdot \frac{\partial\tilde{y}}{\partial y} p(x,y) dx}dy}$](img823.png) |
(6.25) |
(
denotes the continuous cross correlation function,
for the sake of simplicity, antenna indices are omitted). The partial
derivatives in the integrand are easily found by using the transfer function
shown in Fig.6.5:
![$\displaystyle \tilde{x} = \Theta(x)+(n-1)\left[\Theta(x-v_0)-\Theta(-x-v_0) \right] \Theta(-x)$](img825.png) |
(6.26) |
where
for
, and 0 else. Thus,
![$\displaystyle \frac{\partial \tilde{x}}{\partial x} = 2\delta(x)+(n-1)\left[ \delta(x-v_0)+\delta(x+v_0)\right]$](img828.png) |
(6.27) |
Re-writing Price's theorem, we find
Inserting the jointly normal distribution
, and evaluating the
integral yields
or, alternatively, the integral form given in Eq.6.11.
Next: 7. LO System and
Up: 6. Cross Correlators
Previous: 6.4 The correlator on
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Anne Dutrey