# The scaling factor

### The Scaling factor

Scaling factor is linked with the constellation mapping operation for normalizing the average energy to one.

Consider a typical 16-QAM modulation scheme where are used.

From Figure 1 we observed that the numbers of used constellation points in all the four quadrants are same. Therefore the average energy computed over one quadrant is similar to energy over all quadrants. The mean power can be calculated over constellation points. The same alphabet set (j) is used to define real and imaginary axis. Hence energy for real and imaginary components are the same. The elements of each alphabet is used times by real and imaginary part respectively in each quadrant.

The average energy of 16-QAM can be compute as:

The constellation points of 16-QAM are normalized with the factor to ensure that average energy over all symbols is one.

### 64-QAM modulation scheme:

From Figure 2 we can see that each quadrant has 16 constellation points where the energy of real and imaginary components is the same. The elements of each alphabet is used times by real and imaginary part respectively.

Therefore the constellation points of 64-QAM are normalized with the factor to ensure that the average energy over all symbols is one.

### In 16-QAM modulation, the symbols are used.

The average energy of 16-QAM is. It is shown in Figure 3 below.

### Noise Model

Assuming that the additive follows the Gaussian probability distribution function, With mean and variance

The first step that we have to do to compute the probability error is by considering the symbol inside. Using as an example from Figure 3, the conditional probability distribution function of given was transmitted is:

From Figure 3, the symbol is decoded correctly only if falls in the area in the black hashed region. For example:

Using the symbol error probability of 4-PAM as the reference equation, The probability of being decoded incorrectly is:

The second step we will consider as an example for the symbol in the corner. The conditional probability function of given was transmitted is:

From Figure 3, the symbol is decoded correctly if falls in the area in the red hashed region for example:

Again using symbol error probability of 4-QAM equation as reference:

The probability of being decoded incorrectly is:

The third step is we consider the symbol which is not in the corner or not in the inside. We will take as an example:

Again as we see in Figure 3, the symbol is decoded correctly only if falls under the blue hashed region for example;

Using the above two cases as our reference, The probability of being decoded incorrectly is:

The final step will conclude the total symbol error probability by assuming that all symbols are equally likely. That means 4 in the middle, 4 in the corner and the rest is 8.

The total probability of symbol error is:

### 64-QAM

SER for 64-QAM is calculated using the M-QAM methods.

In 64-QAM modulation, the symbols are used.

The average energy of 64-QAM is. It is shown in the Figure 4 below:

Constellation points in the inside which is magenta diamonds. The number of constellation points in the inside is:

Constellation points neither at the corner nor at the centre which is the blue stars. The number of constellation point can be calculated as:

### Noise Model

Let the received symbol be:, where is the energy, is the normalizing factoris the transmit symbol and is the noise.

Assume that the additive noise follows the Gaussian probability distribution function With mean and variance

The first step that we have to do to compute the probability error is by considering the symbol inside which is labelled as the magenta-diamond. The conditional probability distribution function of given that the transmitted symbol is:

From Figure 4, we can see the symbol in the inside is decoded correctly only if real part of lies between 0 to 2 and the imaginary part of lies in between 0 to 2. The correct probability of demodulation is

The probability of the real component from 0 to 2 can be achieved by integrating the probability distribution function of two parts.

The first probability is that the real component lies from 2 to +8 and the second probability is the real component lies from -8 to 0. It is given that the total probability is always 1. The probability where the real component lies within 0 to 2 can be found by subtracting the sum of the first and second probabilities above from the first probability.

### The complementary error function is:

The probability of the symbol decoded being error is:

The second step we will consider symbol in the corner which is labelled as red square in Figure 4. The conditional probability distribution function of given that the transmitted symbol is:

From Figure 4 above, it is observed that the symbol in the inside is decoded correctly only if real part of lies from 6 to 8 and the imaginary part of lies from 6 to 8

The probability of the real component that lies from 6 to 8 can be found by integrating the probability distribution function of the received symbol:

Similarly, The probability that the decoded symbol is correct, given is transmitted is:

The probability of the symbol decoded being in error is:

The third step is to consider the symbol neither at the corner nor inside which is labelled as the blue star in Figure 4. The symbol in the inside is decoded correctly only if real part of lies from 6 to 8 and the imaginary part of lies from 0 to 2.

The probability of the real component that lies from 6 to 8 can be found by integrating the probability distribution function of the received symbol:

The probability of the imaginary component falling within 0 to 2 is computed by integrating the probability distribution function of two parts. First is the probability that the imaginary component lies from 2 to +8 and the second probability is where the imaginary component lies from -8 to 0. The sum of the first and second probability is subtracted from 1 to find the probability of the imaginary component that lies within 0 to 2.

The probability that the symbol is decoded correctly is:

The probability of the symbol decoded being in error is:

The final step will conclude the total symbol error probability. The average error is compute to find the joint symbol error rate.

### Connection between Bit Error and Symbol Error probability

From Figure 5 above, adjacent constellation symbols differ by only one bit with Gray coded bit mapping. Only 1 out of bits will be in error if the noise causes the constellation to cross the decision threshold.

The relation between bit error and symbol error is:

Approximate symbol is used above since the noise may cause the constellation to fall near a diagonally located constellation point when very low value of is reached. This event is negligible if the value of is reasonably high.

The BER of 16-QAM is achieved by combining:

where

And the 16-QAM SER equation

The BER for Gray coded 16-QAM in Additive White Gaussian Noise (AWGN) is:

64-QAM

The BER of 64-QAM is estimated using the SER divided by the number of bits per symbol

where

And the 64-QAM SER equation

We get the BER approximation for 64-QAM is:

### Bandwidth Efficiency

16-QAM has the largest distance between points, but it requires very linear amplification. M-ary schemes or 64-QAM are more bandwidth efficient but more susceptible to noise. This is shown in Figure 6 below:

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Figure 6: Impact of modulation on bandwidth efficiency

### Power Efficiency

64-QAM may have higher bandwidth efficiency, 16-QAM on the other hand have higher power efficiency. This is shown in the Figure 7 where the graph Bandwidth Efficiency vs. Power Efficiency of various modulation techniques is shown.

### System Complexity

System complexity refers to the total of circuits involved and the technical difficulty of the system. Cost manufacturing is the major part played in system complexity, which plays a main part in choosing a modulation technique.

Normally the demodulator is more complex than the modulator. Going deeper, coherent demodulator is much more complex than non-coherent demodulator since carrier recovery is required. All of these are the roots for complexity comparison.

As we compared before between 16 and 64-QAM, high level M-ary schemes which means the 64-QAM have higher bandwidth efficiency. That is traded with the scheme is more susceptible to noise which will require linear amplification thus increasing the complexity of the system.

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