# Energy per symbol calculation

I am having trouble calculating my Energy per symbol and Es/No. Using these parameters and equation

$$N_s =$$Number Symbols

$$R_s =$$Symbol Rate

$$T_s =$$Symbol Time

$$sps =$$samples per symbol

x = signal = QAM-16 with no noise added

Then using these, two equations:

$$E_s = E_{total} / N_s = sum(abs(x)^2) / N_s$$

$$E_s=P_{avg} / (sps/R_s) = P_{avg} sps T_s$$

This second equation comes from Energy = Power x Time. So Energy per symbol = Power x Time for Symbol, but my symbol spans more samples because its upsampled, one symbol spans sps samples. So Energy per symbol = Power x time for symbol x sps.

Where $$P_{avg} = sum(abs(x)^2) / N$$

N = Number Samples

I have tried both equations after the RRC transmit and after the RRC receive filters, both times these equations give two different answers.

First equation gives a very large number and the second is incredibly small number in comparison. Ive tried removing sps from the second equation but does not get anywhere close to first equation.

Ultimately I need to work out Es/No and I cant get Es right. Any thoughts please? :/

• I've tried to help you before, with little success; I will just provide a hint. If the RRC filter response $p(t)$ is normalized so that its energy is 1, then the energy of $ap(t)$ is $|a|^2$. This is true for any linear modulation.
– MBaz
Mar 13 '21 at 20:16
• I don't think so with this, you provided answers to setting up RRC which was great as I have that working. But the energy per symbol. I can't normalise the RRC filter, because I am trying to calculate energy per symbol of any signal that has gone through the RRC and then has channel effects like impairments. So I need to derive energy per symbol Mar 14 '21 at 16:21
• You are trying to calculate $\frac{E_s}{N_0}$ from the time domain signal after the RRC is applied? Mar 15 '21 at 12:45
• Just Es at the moment, I've tried the two equations after transmit RRC is applied and also after the recieve RRC is applied. I just find it peculiar that the two equations I listed above seem fundamental the way to do it but both give different answers whenever in the chain I use them. Mar 15 '21 at 16:54

The first equation, $$\frac{1}{N_s}\sum |x[n]|^2$$, is correct. Applying the RRC filter kind of smears the energy of the symbol over several samples so that when you add them all back up again you get $$E_s$$, and of course divide by $$N_s$$ for multiple symbols. The RRC filter should be normalized as someone in the comments noted.

The second equation I can't say much as it is not clear how you compute $$P_{avg}$$. Secondly, dimensional analysis is a fancy way of saying writing out the units and seeing what cancels out. For example, in the second equation:

$$\text{sps }T_s\rightarrow \frac{\text{samples}}{\text{symbol}}\frac{\text{seconds}}{\text{symbol}}=\frac{\text{(samples)(seconds)}}{(\text{symbol})^2}$$

I've found this method useful as a "sanity check" for myself many times. The above units should suggest to you that something is amiss (also why do you need two equations for the same quantity?).

Edit The question was updated to include definition of $$P_{avg}$$. The OP is computing average power per sample and trying to convert it to average power per symbol. Again, write out the units it becomes clear:

$$\frac{\text{energy}}{\text{sample}}\frac{\text{samples}}{\text{symbol}}=\frac{\text{energy}}{\text{symbol}}$$

sps = 4;
Ns = 100;
Es = 10;
rrc = rcosdesign(0.25, 6, sps, 'sqrt');
s = sqrt(Es)*qammod(randi([0, 15], Ns, 1), 16, 'UnitAveragePower', true);
x = upfirdn(s, rrc, sps);

equation1 = sum(abs(x).^2)/Ns;
equation2 = sum(abs(x).^2)/length(x)*sps;

• I added to post where I get average power. Also, Average power x Time = Energy . Average power x Symbol Time = Energy per symbol. However my symbol time is upsampled by sps, so the symbol spans sps*Ts. Symbol Energy = Power x Symbol Time x sps Mar 18 '21 at 11:38
• @NatalieJohnson How are you upsampling a continuous variable (time)? The symbol lasts for the symbol duration $T_s$, that is it. See my edit about your equation 2. Mar 18 '21 at 12:29