A Fourier wave is expressed in the sensor coordinate system with spatial frequencies of $\omega x$ in the scan direction and $\omega y$ in the normal direction, and its amplitude is denoted as $A(\omega x,\omega y)$. The radiance at a point $x$, $y$ is $A(\omega x,\omega y)ei(\omega xx+\omega yy)$. The response of the sensor is written as $A(\omega x,\omega y)Ts(\omega x,\omega y)$, where $Ts(\omega x,\omega y)$ is the transfer function (TF) of the sensor. The system TF is the product of the TFs of the individual blocks of Fig. 4. The blur error of this wave is $A(\omega x,\omega y)[1\u2212Ts(\omega x,\omega y)]$. For a two-dimensional field, the variance of blur error is the expected contributions of all frequencies within the domain, which is sampled. Display Formula
$\sigma b2=\u222b\u221212Y12Y\u222b\u221212X12Xd\omega xd\omega yS(\omega x,\omega y)|1\u2212Ts(\omega x,\omega y)|2,$(2)
and the spectrum Display Formula$S(\omega x,\omega y)=E{A(\omega x,\omega y)A*(\omega x,\omega y)}$(3)
is the expected value of the squared amplitude of all waves. The rectangular region in the Fourier domain over which this integration is performed is denoted $R$ and the remainder of the plane is denoted as $R\u2032$. The variance of aliasing error is the power of the measurement beyond $R$. Display Formula$\sigma a2=\u222cR\u2032d\omega xd\omega yS(\omega x,\omega y)|Ts(\omega x,\omega y)|2.$(4)