Re: [Discuss-gnuradio] benchmark_tx and rx : too many FALSE packet

[Marcus Müller]

Hi SangHyuk

On 16.03.2016 02:21, SangHyuk Kim wrote:
> Dear Nathan,
>
> I know QAM is the fastest modulation because it can carry more bits
> per symbol. This is a reason why I use QAM modulation.
>
> As bits per symbol level is higher, bit error rates will be increased.
> I expected it returns feasible error rate. However, error rates were
> too high. So I wondered "is this normal case or what is it I'm missing".
Well, how do you figure it is "too high"?

I mean, the math is all there: A packet of length N symbols for which we
assume that the error probability is identical and independent fails
when at least one of these flips (assuming CRC is collision free, which
we can easily do).
The packet error probability is then the complementary to the packet
success probility, which is the product of the individual symbol error
probabilities. These being IID (as assumed above), this packet success
probability is instantly clear:
$P_\text{packet good} = (P_\text{symbol good})^N$.

Now, let's assume we're foing 16-QAM. One can, based on a noise model
(which typically has Gaussian noise amplitude) derive the error
probabilities for all 16 QAM points (using the Q function), average
that, and come to a formula that gives one symbol error probabilty
$P_\text{symbol wrong} = 1-P_\text{symbol good}$. Not going to do that
here. Every digital communication textbook does that. Usually a shortcut
is taken and only the probabilities of switching to the nearest
neighboring constellation points are considered. That approximation
holds well for QAM.

The approximate solution given in such calculation is typically (you
should really do some literature research on your own)

$P_\text{symbol wrong} \approx 3\, Q\left(\frac{2}{\sqrt
5}\sqrt{\frac{E_b}{N_0}}\right)$

with $Q(x)$ being the probality that a normal Gaussian random variable
takes a value larger than $x$,
$\frac{E_b}{N_0}$ being the bit energy to noise power density ratio;
that should be defined as
$\frac{E_b}{N_0} = \text{SNR} \cdot
\frac{B_\text{signal}}{r_\text{bits}}$, i.e. the SNR on the channel
times the spectrum-per-bit-rate.

You do the math; is the packet error rate really so much higher than
what you'd expect under these formulas? If it really is by more than a
factor of 2, this is worth looking into any further. If it is not: well,
you won't be much better with an SNR like that.

Just a quick example: If your channel's SNR and your modulation / pulse
shaping choices lead to a symbol error probability of $10^{-4}$ (which
is pretty good), and your packets are 1500B long, and you're using
16QAM, i.e. 4b/Symbol, then your packet has $N=3000$ symbols:

$P_\text{packet good} = (P_\text{symbol good})^N =
(1-10^{-4})^3000\approx 74\%$


Best regards,
Marcus