Bit error rate curves, theoretical and simulation The BER is the average number of errors that would occur in a sequence of 
$n$ bits.  When $n = 1$, we can 
think of the BER as the probability that any given bit will be received in error. Basically, it lies between 0 and 1 -- 


when BER = 0, we can say no error in the received bits. 
If at least one error is seen, we can say with 100 
percent certainty that the BER is definitely not zero.
as long as at least one bit was successfully received, we can state that the BER is most definitely less than 1. 


When we measure the performance of a digital communication system with and without equalization, with that of the theoretical BER for a particular kind of modulation, we prefer the simulation to be exact match with the theoretical. 
My Questions are :

(1) Is theoretical BER the upper or lower performance bound?

(2) Is it desired that the BER tends to zero? Is it reasonable to say that  since this may not happen in real communication, so there is always a quest to come up with an equalizer designed with an objective that the BER tends to zero?

(3) What if the simulation curve with and without equalizer falls below the theoretical? What can be said about the equalization method playing a role and if it is desired that the simulation curve - with and without equalizer is below the theoretical.

(4) What can be said about performance of the system with and without equalizer if the simulated BER is above the theoretical?

(5) Does an equalizer essentially help to improve the BER? By improving -how much improvement is desired? Tradeoff? Do we wish the simulated curve to be an exact match to the theoretical and then say that the equalizer is functioning properly in removing intersymbol interference and noise?

 

What is the interpretation for the bit error performance?
Note: usually, the "theoretical BER" is called $P(e)$, the probability of bit error (sometimes with other symbols, like $P_b$). A rate implies experiment; the probability implies theory.

1) $P(e)$ is a lower bound for the BER: since practice is messier than theory, the BER will be higher than $P(e)$ (assuming the theory is correct; that is, that it models reality properly).

2) I'm not sure I understand the question. We always want $P(e)$ to be as low as possible, but it is ultimately bounded by the channel capacity.

3) If by "below the theoretical" you mean "better than $P(e)$", I'd say something is wrong with your simulation or your assumptions.

4 and 5) You use an equalizer when you have a wideband channel with intersymbol interference. Such a system will always have worse BER than a narrowband channel. The equalizer tries to eliminate the ISI; if the equalizer is perfect, the BER of the wideband system will be the same as that of a narrowband channel, assuming equal $E_b/N_0$. In practice, the equalizer will not be perfect. In this case, theory can help predict the equalizer's performance, and how much BER improvement to expect.