First order speckle statistics

Magnitude (using Trahey's notation)

Given a monochromatic carrier signal described with the phasor form $e^{j2\pi f_{c}t}$, where $f_{c}$ is the center frequency, we introduce a formulation of the RF speckle pattern in the analytic form:

$$P(x,y,z;t)=A(x,y,z)e^{j2 \pi f_{c}t}$$ (5.1)

where $A(x,y,z)$ is the complex phasor amplitude. $A(x,y,z)$ can be decomposed into magnitude and phase:

$$A(x,y,z)=\vert A(x,y,z)\vert e^{j\theta(x,y,z)}$$ (5.2)

The intensity of the phasor is given by:

$$I(x,y,z)=\vert A(x,y,z)\vert^2$$ (5.3)

Each scatterer contributes to the complex phasor amplitude:

$$A(x,y,z)=\sum_{k=1}^{N}\frac{1}{\sqrt{N}}a_{k}(x,y,z) = \frac{1}{\sqrt{N}}\sum_{k=1}^{N}\vert a_{k}\vert e^{j\theta_{k}}$$ (5.4)

Assumptions:

• The amplitude $a_{k}/\sqrt{N}$ and phase $\theta_{k}$ of the $k^{th}$ scatterer are statistically independent of each other and of those of all other scatterers.
• The phases of the scatterers are uniformly distributed over $[-\pi,\pi]$.

Now we can calculate an assortment of expected values:

$$\begin{split}A^r =\text{Real}(A) &= \frac{1}{\sqrt{N}}\sum_{k... ...t\big{\rangle}^2}{2}\ \langle A^r A^i \rangle &= 0 \end{split}$$ (5.5)

Through the application of the Central Limit Theorem, $A(x,y,z)$ has a complex Gaussian PDF, with joint real and imaginary parts:

$$\begin{split}p_{r,i}(A^rA^i)&=\frac{1}{2\pi\sigma^2}\exp\bigg... ...ac{1}{N} \sum_{k=1}^{N}\frac{\vert a_{k}\vert^2}{2} \end{split}$$ (5.6)

For large $N$ the Rayleigh PDF of the phasor magnitude is found to be:

$$p(V)= \begin{cases}\frac{V}{\sigma^{2}N}\exp\Big{(}-\frac{V^2}{2\sigma^{2}N}\Big{)}& V \geq 0,\ 0 &\text{otherwise.} \end{cases}$$ (5.7)

The first order statistics for magnitude are:

$$% latex2html id marker 4901 \begin{split}\mu_{V} &= \langle ... ...\sigma_{V} &= 1.91\text{, equivalent to 5.6 dB SNR} \end{split}$$ (5.8)

Intensity (using Goodman's notation)

$\vert A(x,y,z)\vert^2$ has the PDF:

$$p(I)=p(V^2)= \begin{cases}\frac{1}{2 \sigma^{2}}\exp\Big{(}-\frac{I}{2\sigma^{2}}\Big{)}& I \geq 0,\ 0 &\text{otherwise.} \end{cases}$$ (5.9)

and statistics: $\sigma_{I} = \mu_{I} = 2 \sigma^{2}$

A review of random variables

Given two random variables $x$ and $y$:

$$\begin{split}\text{The mean of } x &= \mu_{x} = \langle x \ra... ... x y \rangle &= \langle x \rangle \langle y \rangle \end{split}$$ (5.10)