# SINR averaging in communication system

Under the simulation of a communication system suffering from noise (AWGN) and co-channel interference, what is the appropriate way of calculating SINR? Could one calculate it at a packet level as:

$$SINR_{packet} = \frac{\sum{\left \| x \right \|^{2}}}{\sum \left \| n \right \|^{2}+\sum \left \| z \right \|^{2}}$$

where $x$ is the total useful waveform for that packet, $n$ is the corresponding noise waveform and $z$ is the waveform of the interferer, or is it necessary to calculate it on a per bit basis as

$$SINR_{bit} = \frac{\sum{\left \| x_{b} \right \|^{2}}}{\sum \left \| n_{b} \right \|^{2}+\sum \left \| z_{b} \right \|^{2}}$$

where $x_{b}$ is the useful waveform for that particular bit, $n_{b}$ is the corresponding noise waveform for that bit and $z_{b}$ is the waveform of the interferer for that bit and then take the average across all bits of the same packet like so:

$$SINR_{packet} = \frac{1}{k}\sum_{bit=1}^{bit=k}{SINR_{bit} = \frac{1}{k}\sum_{bit=1}^{bit=k}{\frac{\sum{\left \| x_{bit} \right \|^{2}}}{\sum \left \| n_{bit} \right \|^{2}+\sum \left \| z_{bit} \right \|^{2}}}}$$

Which one is supposed to converge more quickly and give more realistic results? Which one would be better related to bit error rate (BER) and which to packet error rate (PER)? Finally could Central Limit Theorem be of any use proving that the first and last expression converge to the same number for large number of bits in a packet?

I believe real systems calculate SNR as in the first formula, except of course for the $z$ term.

Signal-to-interference-plus-noise ratio (SINR) is by definition

• $SINR = \frac{P}{I+N}$,

where $P$ - is useful signal power, $I$ - interference signal power and $N$ - noise power in the band of interest. Look

http://en.wikipedia.org/wiki/Signal-to-interference-plus-noise_ratio


for example.

So the first formula you've presented seems to be rigth one. Time span of calculation $SINR$ depends on your needs and interference nature. The common way is using duration of $x$. In general you have to distinguish both $x$ and $i$ at first, because $i$ is non-stationary random process so you can't think it as receiver noise. So in most common case in real life you can estimate $SINR$ only if you are able to combat such an interference in you system.

If inteference appears as short pulses in time you can do measurement with relatively short sliding window to estimate $SINR$ in dynamics. If signal power is quite stationary in short averaging you can use previously estimated $P$ in formula above to find out $SINR$ even if demodulator fails to defeat it.

You can combine measurement of $SINR$ at the input of demodulator, $MSE$ at its output and $BER$ at the decoder's output to make some sophisticated modem quality control.

• I know the definition of SINR. My question in short is should I average it per packet as a whole (eq. 1 in my post) or per bit in the packet (eq. 3 in my post)? As I said it's a simulation and therefore I know the exact waveforms for x, n and z. – user113478 Jul 5 '14 at 7:20
• You probably should use the first formula. The second formula seems to give the same result but it's odd. Find signal power $P$ and interference power $I$ at first. But if interference doesn't exist at all duration of the packet use its real duration instead to find its power out. – Serj Jul 5 '14 at 8:27