# How to desgin an RF system? non-linearity

In the first part of how to design an RF system we talked about the RF system parameters that one needs to know. There mainly we discussed about gain(power gain), matched gain parameters and stability. There are other important parameter and terms in RF design- non-linearity parameters, noise figure, phase noise and dynamic range. Here some aspect of non-linearity are explained.

### Non-Linearity

A RF system block or component such as amplifier, mixer etc can be classified as linear or non-linear component. Amplifier are non-linear component because the the output is not linearity scaled with input. Generally if x(t) is the input signal and y(t) is the output signal from the component then for a memoryless and linear system they are related by the following equation.

If the component is memoryless but non-linear then the input and output relationship is,

where a0, a1, a2 are coefficients which depends on time.

If we have input signal which is sinusoid x(t) = A cos(wt) then the above equation is expanded as follows and rewritten as follows-

From the above equation we can characterize a non-linear RF component via different non-linearity phenomenons- Gain Compression, Desensitization, Cross Modulation and Intermodulation.

The gain g of a non-linear and symmetric system is due to primarily the third term and is given by,

In the above gain expression, a1 can be called linear gain term and a3 the non-linear term

1 dB  compression point

The 1dB compression point is defined as the input signal level at which the gain g above is reduced by 1dB compared to the linear gain term a1.
From the above expression of gain g we can define the 1 dB compression point mathematically as,

Zero Gain

A non-linear RF component may be run by bias supply voltage. Thus the output signal from such non-linear RF component is the combined effect of non-linearity and bias condition. If the input signal is very large compared to its bias operating point, the gain can become zero, that is there is no gain. Hence zero gain may be lead because of either the linearity coefficient a3 < 0 or bias condition.

Blocking

Blocking phenomenon occurs if there is additional signal to the desired signal at the input and because the additional signal is stronger it gets modulated instead of the desired signal. Suppose A1cos(w1t) is the desired signal and A2cos(w2t) is another signal(the interfering signal) that is passed along with the desired signal into the RF component. The input signal to the RF component is the combined signal as follows.

Now we can calculate the gain of the non-linear component substituting this signal into the following,

And we end up with,

If the magnitude of the interfering signal A2 is sufficiently large then the gain becomes zero. This is called blocking.

In similar term, since it appears that the interfering signal is being modulated by the desired signal when passing through the component this phenomenon is also called cross modulation.

Intermodulation

A signal is usually made up of not just single frequency but a range of frequencies. Such signal if passed through a non-linear circuit component, each frequency gets modulated such that the output signal contents harmonics. Not only that the harmonics gets mixed up and produces signal with different signal of different frequencies altogether. This process of mixing is called intermodulation.

Let say the input signal is,

Then after passing through a non-linear component the output signal can be grouped into the followings,

Thus the effect of intermodulation leads to generation of so called second order intermodulation distortion and third order intermodulation distortions. This is really due to imperfect circuit design and due to the non-linear frequency response of the component. The second order distortion is usually seen and is important where the high frequency signal is downconverted to baseband such as in case of homodyne receivers. The signal component that leads to third order distortion is near the desired component and therefore they are difficult to filter out. These second order distortion component and third order distortion components leads to graphical point on the power diagram referred as to second order intercept point and third order intercept point.