I am what I am: December 2010

At times, finding the correct SNR (Signal to Noise Ratio) for the desired BER (Bit Error Rate) in a communication channel becomes a pain. Its easy to tell that the BER goes on improving as we increase the SNR level of the input signal before it makes out on the channel.
The Bit Error Probability is given as

$P_{b|h}=\frac{1}{2}erfc\left({\sqrt{\frac{|h|^2E_b}{N_0}}}\right)=\frac{1}{2}erfc\left(\sqrt{\gamma}\right)$
where $\gamma = \frac{|h|^2E_b}{N_0}$

The BER now is the function of the Probability Density Function and can be summed as the integral of the product of Bit Error Probability and probability density function.

If anyone does remember the Gaussian Distribution, the probability density function extends from the mid-half to either sides towards infinity. Applying similar theory we get,
$P_{b}=\int_0^\infty\frac{1}{2}erfc\left(\sqrt{\gamma}\right)p\left(\gamma\right)d\gamma$
where,
$p\left(\gamma\right) = \frac{1}{\bar{\gamma}}e^{\frac{-\gamma}{\bar{\gamma}}},\ \gamma \ge 0$ is the Probability density Function of $\gamma$ and
$\bar{\gamma} = \frac{E_b}{N_0}$ is our Energy of the Bits to Noise Ratio or the plain old SNR would do. Now, I am not going much deeper into the integral calculations to find out what's the result going to be because I am weak at maths. But the result from the books would show the BER as

$\begin{eqnarray}P_{b} & = & \frac{1}{2\bar{\gamma}}\int_0^{\infty} erfc (\sqrt{\gamma}) e^{-\frac{\gamma}{\bar{\gamma}}}d\gamma \\ & = & \frac{1}{2\bar{\gamma}}\left[\bar{\gamma}erfc(\sqrt{\gamma})e^{-\frac{\gamma}{\bar{\gamma}}} + \bar{\gamma}\sqrt{\frac{\bar{\gamma}}{\bar{\gamma}+1}}erf\left(\sqrt{\frac{\bar{\gamma}}{\bar{\gamma}+1}}\sqrt{{\gamma}}\right) \right]_\infty^0\\ & = & \frac{1}{2} \lef(1-\sqrt{\frac{\bar{\gamma}}{\bar{\gamma}+1}}\right)\\ & = & \frac{1}{2} \lef(1-\sqrt{\frac{(E_b/N_0)}{(E_b/N_0)+1}}\right)\end{eqnarray}$

Anyways here is the catch. If you want to compare how these signals behave as a function of SNR in Rayleigh Channel or in plain old AWGN (Additive White Gaussian Noise) channel, here is the matlab code for it.
http://dl.dropbox.com/u/15048456/Final Code/task3.m
http://dl.dropbox.com/u/15048456/Final Code/rayleigh_modulator.m

Just put those files together in the same active or working directory while compiling and executing them. Watch and learn

I am what I am

Thursday, December 30, 2010

BER as a function of SNR in Rayleigh Channel

Wednesday, December 29, 2010

Three Lions hunt down Kangaroos