# Is there a relationship between Average energy of transmission and power saving

As mentioned in many books, the average energy of transmitted symbol is given by:

$$E_s = (2/3)(M - 1)$$

where $$M$$ indicated the modulation order. Suppose that we are using in the first scenario $$SISO$$ system, and we need to transmit 8bit/channel use, so the needed modulation is $$256QAM$$ which is equivalent to $$E_s = 170$$. now suppose that we are using $$MIMO$$ system with two transmitters antennas and two receiver antennas, so to transmit the data rate of 8bit/channel use, we need to transmit 4bit/channel use in each antenna, we will need to use $$16QAM$$, so the needed $$E_s = 10$$ in each antennas, which is in total 20.

can we say, the power saving for transmitting 8bit/s/Hz in case of using MIMO system is 170/20 ?

Is that right? thank you

## 1 Answer

First of all, 8 b/s/Hz is not a bit rate, but rather a spectral efficiency.

Second, you don't transmit $$\log_2(M)$$ bits per second, but rather $$\log_2(M)$$ bits per channel use, which is roughly $$T_s$$ seconds, where $$T_s$$ is the symbol time. So, the bit rate is $$\frac{\log_2(M)}{T_s}$$.

If you adjust your wording to the above by saying 8 bits/channel use, then, yes, you are right, you save energy by using MIMO, but the saving is $$170/20$$ not the reciprocal.

• Thank you .. Yes I totally agree with you. However I supposed that $T_s$ equals one second for simplification. But that means we have power saving almost 90% !! is that right? – Gze Jan 13 '19 at 6:19
• On average, yes, I think that's true. However, in the MIMO case, there will be interference between the two streams of data, which will affect the performance negatively. It's interesting to compare the two systems in terms of bit error rate and complexity (because you will need an equalizer at the receiver to separate the streams) as well. – BlackMath Jan 13 '19 at 12:08