**Introduction **

In my **previous blog**, we have seen how the polynomial expression giving output y(t) versus input x(t) preserves the frequency integrity: y(t) contains the same frequencies entered at x(t) as wanted for linear devices!

y(t)= A_{0 }+ A_{1}.x(t)

**Now, What happens When the Device Is Not Linear?**

Let’s use the polynomial expression established earlier. We start with x containing only one frequency, ω:

**y = A _{0} + A_{1}.x^{1} + A_{2}.x^{2} + A_{3}.x^{3} +... + A_{i}.x^{i} +... A_{n}.x**

^{n }

^{ }(Eq. 8)

With x(t) = K e^{j}^{(}^{ω}^{t+}^{φ}^{) }(Eq. 9)

Where:

y = A_{0} +** **

A_{1}. K e^{j}^{(ω}^{t+}^{φ)} + (contains ω)

A_{2}. (K e^{j}^{(ω}^{t+}^{φ)})² + (contains 2ω)

A_{3}. (K e^{j}^{(ω}^{t+}^{φ)})^{3} + (contains 3ω)

A_{i}. (K e^{j}^{(}^{ω}^{t+}^{φ}^{)})^{i} + (contains i.ω)

The device generates multiple frequencies that were not present originally. The fundamental is the term with ω; all the multiple terms of ω, are called harmonics. These create distortion. The higher the Ai, the stronger the harmonics and so the distortion. However, the situation is not that dramatic yet; those harmonics are far from the signal bands (often) and thus easy to filter out (Figure 3).

The problem comes when you have an input signal containing several frequencies; which is so with a data band. Let’s consider two frequencies:

x = x_{a} + x_{b} (Eq. 10)

Where x_{a} has frequency ω_{a} and x_{b} has frequency ω_{b. }Here x is also called a two-tone signal.

By injecting this x in the polynomial:

** ** y = A_{0} + A_{1}.x^{1} + A_{2}.x^{2} + A_{3}.x^{3} +... + A_{i}.x^{i} +... A_{n}.x^{n} (Eq. 11)

We obtain:

**1. Order-0 Product or Constant Part: A**

**2. First-Order Products or Linear Parts: A _{1}.x**

A_{1} * x = A_{1}.(x_{a} + x_{b}) (Eq. 12)

This contains the two original frequencies, ω_{a, }and_{ }ω_{b}**, **as expected

**3. Second-Order Products or Quadratic Parts: A _{2}.x^{2}**

A_{2}.x² = A_{2}.(x_{a} + x_{b})² = A_{2}.(x_{a}² + x_{b}² + 2. x_{a} x_{b}) (Eq. 13)

The term x_{a}² contains frequency 2.ω_{a}, and the term x_{b}² contains frequency 2.ω_{b}; they are the second-order harmonics.

The term x_{a}*x_{b} (product of 2 sinewaves) contains 2 new frequencies **ω**_{a}** + ω**_{b}_{ }and **І****ω**_{a}** - ω**_{b}**І. **

The 2 “parasitic” frequencies ω_{a} + ω_{b} and ω_{b} – ω_{a }can easily be filtered out.

**4. Third-Order Products: A _{3}.x^{3}**

A serious problem appears here:

By developing (x_{a} + x_{b})^{3}, you find x_{a}^{3}, x_{b}^{3}, 3x_{a}² x_{b,} and 3x_{a} x_{b}². With X_{a} and X_{b} containing sinewaves at different frequencies, their multiplications will generate other frequencies such as 3w_{a}, 3w_{b }, 2w_{a }+ w_{b },_{ }2w_{a }- w_{b, }w_{a}+2w_{b}, and 2w_{a}-w_{b. }The mixtures between the original frequencies, 2ω_{a} + ω_{b}, 2ω_{a} - ω_{b}, and 2ω_{b} – ω_{a, }are also called third-order intermodulation products (IM3).

While the terms 3w_{a}, 3w_{b}, 2w_{a }+ w_{b }, and w_{a}+2w_{b} are easy to eliminate, this is no longer true with the terms 2ω_{a} - ω_{b }and 2ω_{b} – ω_{a} ! These have the annoying particularity of occurring in the same frequency range as ω_{a }and_{ }ω_{b}. If one of these latter terms carries information (modulated), then you must be sure that the other terms will not interfere with the intermodulation terms. As we said earlier, they fall in the same bands as the useful signal bands and thus cause unrecoverable jamming and interferences, and your information can be just lost or corrupted.

**Figure 4** shows that even with strong expensive filters, it will not be easy (even impossible) to remove the IM3 terms 2ω_{a} - ω_{b }and 2ω_{b} – ω_{a} because they are embedded in the useful band! This is precisely why in RF the third-order terms are so critical and must be known, measured, and minimized everywhere in the signal chain.

*Figure 4.** View of the different frequencies generated by a two-tone input signal applied to a nonlinear device.*

**5. Fourth-Order Products: A _{4}.x^{4} and N^{th}-Order Products: A_{n}.x^{n}**

A similar pattern also applies to frequencies 4w_{a}, 4w_{b }, 2w_{a }+ 2w_{b },_{ }2w_{a }- 2w_{b}, w_{a }+ 3w_{b} , 3w_{a }+ w_{b}, 3w_{b }- w_{a}, and 3w_{a }- w_{b}. As with the second-order terms, all the frequencies here are far from the fundamentals. The problematic IM products are in an odd order of n (i.e., IP3, IP5, IP7, etc.).

Hopefully, for practical devices, the higher-order terms vanish rapidly and can be neglected.

We could continue the discussion by considering x with more than two frequencies. They will simply give us more IMs.

**Summary**

In this blog, we have seen how non-linearities create harmonics and intermodulation products: their level and their severity can be predicted. Some IM3 products are particularly concerning because they occur in the useful signal band, causing data loss or corruption.

In the **next blog**, we will learn how IM3 can be characterized and evaluated; introducing the spec IP3.