# Drawbacks of Non-linear System: Gain Compression

Gain compression is the most important problem that is faced in a non-linear system. Read about the effects of Non-Linearity in RF systems. Harmonic distortion is one of the drawbacks of having a non-linear system that is discussed in the previous section, __harmonic distortion__.

If we have a linear system with input x(t)=Acos *w*t, then we will have y=a_{1}.x, where ‘a’ is the gain, so the output for this linear system would be a_{1}Acos *w*t. In the non-linear system, the output would be y(t)=a_{1}.x +a_{2}.x^{2}+a_{3}.x^{3}, and we can find the gain for this system by defining the gain. We have different kinds of amplitudes in a non-linear system as it produces harmonics. So gain can be defined here as the output amplitude A_{0} of the fundamental frequency by the input amplitude ‘ A.’ Using the equations for input and output, we could find out the gain in the system. The output equation can be further written as:

In a non-linear system, we have a_{1}.x, a_{2}.x^{2} & a_{3}.x^{3}, where most of the systems, the a_{1} and a_{3,} have opposite signs. If a_{1} is positive, then a_{3} will be negative. This is true for most differential systems. In the section effects of Non-Linearity in RF systems, we showed that in the MOS differential pair, the equation for V_{out} with a_{1} and a_{3} have opposite signs, and there is no a_{2}.

When the system is linear, the gain is constant, but in a non-linear system, gain depends on input amplitude. So, assuming a_{1} is positive and a_{3} is negative, we can write the gain as:

As we increase, the amplitude gain will decrease as a_{3} is negative. So when the system is linear, the gain is a_{1}, which is constant; in a non-linear system, our gain will change and depend on amplitude ‘A.’

This example graph shows a theoretical comparison between a linear and non-linear amplifier to explain gain compression. The graph shows a region called a linear region. The input amplitude in this region is small; therefore, the part (3a_{3}A^{2}/4) in the equation can be neglected. In the linear region, the gain for both linear and non-linear amplifiers remains the same. In a linear amplifier, the gain is constant, and when the input amplitude increases, the output increases at the same rate. However, for a non-linear amplifier with negative a_{3}, the gain decreases as we increase the input, which is called compression. The gain decreases due to the negative value, and increasing the amplitude higher will lead to 0 gain. Therefore, having a highly non-linear amplifier with a very high amplitude will result in no gain.

Also, notice that as we increase ‘A,’ the gain starts decreasing, and after a certain point, the output becomes constant for some time before decreasing, falling to 0.

There is a point shown in the graph called A_{-1dB}. It is called the input compression point. It is a point in the graph where the difference in linear and non-linear amplifier’s output amplitude in fundamental frequency is 1dB. It can also be defined as the point where the non-linear amplifier’s output is 1dB less than the linear amplifier’s output. This compression point is used to measure non-linearity. A_{-1dB} estimates the non-linearity of our system, if it is high, then it is linear, and if it is low, then it is non-linear. It can be calculated using logarithmic equations as shown below:

In this equation, a linear system a_{3} will be 0; therefore, A-1dB will be infinity, which means the system is completely linear. If the system is non-linear, then a_{3} increases and A_{-1dB} decreases, reaching a point after which the gain starts falling.

In summary, if we have a system with very low input amplitude, then there will be no problem; however, when the input increases, it reaches a point called A_{-1dB, }after which increasing the input will result in decreased gain, and this is called gain compression.