# Total bandwidth in a network calculation

I have data on the number of devices in a wired network and the data rates on links among these devices. I also know the positions of these devices from which I can compute the distances between the devices.

The idea is that if I were replace the wired network with wireless infrastructure, how much bandwidth am I looking for. Would let's say 200MHz be sufficient for such a case.

Assuming that the BER is very good, how should I go about in calculating the total bandwidth in such a network.

• Can you please clarify "total bandwidth in such a network"? Are you interested in the average rate between any two nodes of the network or throughput? – A_A Aug 31 '16 at 13:46
• @A_A, edited my question. – smyslov Aug 31 '16 at 14:05

There are many variables involved, but you may proceed as follows. Assume the bit rate in a link is $R_b$. First of all, you choose a constellation; let's say you go with 16-QAM, which transmits four bits per symbol. Now you have a symbol rate $$R_1=R_b/4.$$ Next, you choose an encoder that gives you the BER you need. This encoder will have a certain code rate $R_c$. After encoding, you now have a symbol rate $$R_2=\frac{R_1}{R_c}.$$ Next, you need to take into account the overhead: you need framing and training data. Let's say the overhead is $x\%$. Then your final symbol rate is $$R_3=(1+x)R_2.$$ Now you can calculate the bandwidth you need: $$B=\gamma R_3,$$ where $\gamma$ is an "excess bandwidth" over the theoretical $B=R_3$. The excess bandwidth is typically between 1 and 2.

In summary: the symbol rate is given by the data rate, the constellation, the encoder, and the overhead. The bandwidth is determined by the symbol rate and the excess bandwidht factor.

Note that each of these variables affects the others. For example, if you choose QPSK instead of 16-QAM, you'll have better BER, but $R_1$ will be larger. However, now you need an encoder with smaller rate, so $R_2$ will be smaller. And so on. So, you need to find a combination of variables that you can implement within your constraints and that gives you the BER and data rate you need.

• Assuming that I would want to choose Zigbee which guarantees 250kbps and 2MHz bandwidth and I need to calculate the spectrum necessary for the scenario mentioned in my question, should I still be calculating the way you mentioned? – smyslov Sep 1 '16 at 7:23
• You asked about the bandwidth needed to transmit a given data rate. If you use ZigBee, the bandwidth is fixed and the data rate (250kbps) has already been calculated for you (the ZigBee designers followed a procedure similar to what I described). You can of course still use your own protocols, error correction and framing on top of ZigBee, which would reduce the effective data rate below the nominal 250kbps. – MBaz Sep 1 '16 at 12:46

I am still a bit unclear but I am happy to adjust the response if more data becomes available.

Assuming that what you have now resembles a weight matrix $W$, where the weights are the transmission rates of each link, then, the total bandwidth is simply the sum of all entries of $W$.

This doesn't say much about the network.

If, instead, you were to calculate a suitable average of $W$, then that would tell you the average bandwidth between any two nodes. A better approximation to that would be an all-to-all "shortest" path calculation which would take $W$ into account.

EDIT: It has subsequently come to my attention through this question that the phrase "...the weights are the transmission rates of each link, then, the total bandwidth is simply the sum of all entries of $W$." contains an error when it practically equates bandwidth with transmission rate. This is not correct even for something as simple as BPSK. The "differences" are due to the Baud Rate and encoding schemes employed.