-In this sequence, we will have a look at the bit error rate for the different modulations previously studied. In the previous episode, we saw that for a given modulation the bit error rate increased when the signal-to-noise ratio was decreased. We will assume that when the transmission and reception filters are correctly chosen, the bit error rate only depends on the signal-to-noise ratio for a given modulation. We will use the value of the signal-to-noise ratio in decibels, that is to say 10log10(SNR). To calculate the power of noise, we will consider the Rb bandwidth. We have seen in the previous episode that in this case the SNR is equal to Eb/N0, Eb being the energy per received bit. Let us have a look at the impact of the number of modulation symbols on the bit error rate. Here we have two modulations. A QPSK modulation and a 8-PSK modulation that have the same signal-to-noise ratio. We place here the decision zones for the QPSK modulation. We assume that the transmitted symbol is at the center of the blue cloud. We do not see any erroneous symbols. For the 8-PSK modulation, we have these decision zones. In fact the plane is divided into eight equal sectors. We note that in the case of the 8-PSK modulation, we have blue samples outside the decision zone. So we will have erroneous symbols. The decision zones are narrower for this 8-PSK modulation. So the bit error rate will be higher for a given signal-to-noise ratio. Let us compare the bit error rate for three modulations, the QPSK, the 8-PSK and the 16-PSK. The value of Eb/N0 in decibels is drawn in abscissa, the value of the bit error rate is drawn in ordinate. The bit error rate specified in communication systems is very low. It is below 0.01. On a linear scale, the three curves cannot be distinguished. So we will use a logarithmic scale to compare these different modulations for low BER values. Here is the BER curve with a logarithmic scale. These three curves are now separated for low values of the bit error rate. First we compare these three modulations by setting the value of the signal-to-noise ratio. Here for example Eb/N0 equals 8 dB. We determine the obtained bit error rate. Here is the bit error rate for the QPSK, for the 8-PSK, and for the 16-PSK. What do we notice? When we set the signal-to-noise ratio, the bit error rate increases with the value of M. But it is not the SNR value that is specified in communication systems. It is the target bit error rate value. For example, here we choose a 10 to the power of -4 target bit error rate. So we search the required Eb/N0 for each modulation with this target bit error rate. Here we have the Eb/N0 value for the QPSK, the 8-PSK, and the 16-PSK. What do we notice? When the bit error rate is set, the required signal-to-noise ratio increases when the value of M is increased. The values for the different modulations are shown on the slide. So we compared three phase modulations with a different number of states. We will now compare the two types of modulations previously seen, that is to say phase and quadrature amplitude modulations. To compare them, we will take an example, the 16-PSK and the 16-QAM. On the right-hand curve, we set the bit error rate to 10 to the power of -4. The required Eb/N0 for the 16-PSK is much larger than the required Eb/N0 for the 16-QAM. We can conclude that the 16-QAM modulation is better than the 16-PSK modulation in terms of required signal-to-noise ratio and thus in terms of received power. As a conclusion, what can we say? The bit error rate decreases when the signal-to-noise ratio increases. If we set the bit error rate for a given modulation, a phase modulation for example, the required signal-to-noise ratio increases when we increase the value of M. QAM-type modulations require a lower signal-to-noise ratio than PSK-type modulations for a given number of symbols.