# How Non-linearities Create Harmonics and Intermodulation Products

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)= A0 + A1.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 = A0 + A1.x1 + A2.x2 + A3.x3 +... + Ai.xi +... An.xn                          (Eq. 8)

With x(t) = K ej(ωt+φ)                                                                                                                                 (Eq. 9)

Where:

y =   A0 +

A1. K ejt+φ) +                      (contains ω)

A2. (K ejt+φ))² +                   (contains 2ω)

A3. (K ejt+φ))3 +                   (contains 3ω)

Ai. (K ej(ωt+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 = xa + xb                                                                                             (Eq. 10)

Where xa has frequency ωa and xb has frequency ωb. Here x is also called a two-tone signal.

By injecting this x in the polynomial:

y = A0 + A1.x1 + A2.x2 + A3.x3 +... + Ai.xi +... An.xn                               (Eq. 11)

We obtain:

1. Order-0 Product or Constant Part: A

2. First-Order Products or Linear Parts: A1.x

A1 * x = A1.(xa + xb)                                                                                     (Eq. 12)

This contains the two original frequencies, ωa, and ωb, as expected

3. Second-Order Products or Quadratic Parts: A2.x2

A2.x² = A2.(xa + xb)² = A2.(xa² + xb² + 2. xa xb)                                         (Eq. 13)

The term xa² contains frequency 2.ωa, and the term xb² contains frequency 2.ωb; they are the second-order harmonics.

The term xa*xb (product of 2 sinewaves) contains 2 new frequencies ωa + ωb and Іωa - ωbІ.

The 2 “parasitic” frequencies ωa + ωb and ωb – ωa can easily be filtered out.

4. Third-Order Products: A3.x3

A serious problem appears here:

By developing (xa + xb)3, you find xa3, xb3, 3xa² xb, and 3xa xb². With Xa and Xb containing sinewaves at different frequencies, their multiplications will generate other frequencies such as 3wa, 3wb , 2wa + wb , 2wa - wb, wa+2wb, and 2wa-wb. The mixtures between the original frequencies, 2ωa + ωb, 2ωa - ωb, and 2ωb – ωa, are also called third-order intermodulation products (IM3).

While the terms 3wa, 3wb, 2wa + wb , and wa+2wb are easy to eliminate, this is no longer true with the terms 2ωa - ωb and 2ωb – ωa ! These have the annoying particularity of occurring in the same frequency range as ωa and ωb. 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 2ωa - ωb and 2ωb – ωa 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: A4.x4 and Nth-Order Products: An.xn

A similar pattern also applies to frequencies 4wa, 4wb , 2wa + 2wb , 2wa - 2wb, wa + 3wb , 3wa + wb, 3wb - wa, and 3wa - wb. 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.