**Introduction**

In the **previous blog** post, we learned how the various intermodulation products are generated by non-linearities with multi-tone signals. In this blog post, we will show the method to evaluate the IPn; especially the one of the 3d order: IP3.

**Short Refresh About IPn Meanings**

Figure 10. IPn as crossing points between nth-order and first-order intermodulation product curves.

In Figure 10 we see, for example, that IP2 is the point where first-order and second-order lines cross. IP3 is the point where first-order and third-order lines cross and so on. Values are read on the x or y axis. There are thus two actual values measuring the IP point: input and output intercept point. They are noted:

**IIPn**for nth-order input intercept point, measured on the input power axis (x)**OIPn**for nth-order output Intercept point, measured on the output power axis (y)

**Relationship Between IIP and OIP**

LOG Y = Log (A1 X) = Log A1 + Log X (Eq. 15)

At the intercept point:

LOG Y = OIP and LOG X = IIP (Eq.16)

Therefore:

OIP = Log A1 + IIP (Eq. 17)

Log A1 is usually the gain specified for the device. Therefore, before it gets saturated, we have simply:

OIPdBm = GdBm + IIPdBm (Eq. 18)

**Intercept Point Evaluation and Real Situation**

In practice, IPn points are virtual because they occur at high Pout; so high the device saturates well before the signals reach the crossing points. All these straight lines are, in fact, asymptotes extrapolated from smaller values of x and y. This implies that we will need an indirect method to measure IP points.

**Intercept Points – Graphic Interpretation and Construction**

Since we cannot apply and, therefore, measure signals that approach an IP point (because the device would be saturated well before), we need to apply a signal with smaller amplitudes. We can take the x-y figure with the axis in dB (or dBm) (Figure 11) and consider the first-order and the nth-order straight lines:

We apply an input signal, PIN; it must be small enough to not saturate the device. It will give the corresponding output, POUT. These points appear in the X and Y axis, respectively (**Figure 11**):

Figure 11. Power levels with straight lines for first-order and nth-order and their intercept points.

PIN is the applied input signal (from the generator), POUT is the output signal at the first-order (measured), and POUT_n is the output at the nth-order (measured). We can call ΔP = POUT - POUT_n, which is the difference between measured powers at the first-order and nth- order frequencies.

Using a spectrum analyzer, one can discriminate among the various powers appearing at various frequencies. We can now determine the relationship between the applied and measured signals versus intercept points (IPs). **Figure 12** shows that one can see two triangles inside the rectangle of Figure 7.

Figure 12. IPn computation via a graphical method.

Their vertical sides must be in the same ratio as their hypothenuse slopes. Where:

n/1 = (OIPn - POUT + ΔP)/(OIPn – POUT) (Eq. 19)

n = 1 + ΔP/(OIPn - POUT) (Eq. 20)

(OIPn – POUT) . (n-1) = ΔP (Eq. 21)

OIPn – Pout = ΔP/(n-1) (Eq. 22)

Therefore, in conclusion:

**OIPn = P****OUT**** + ΔP/(n-1)** (Eq. 23)

In particular, for IP3, we have:

**OIP3 = P****OUT**** + ****ΔP/2** (Eq. 24)

Since POUT = PIN + G, with all terms in dBm, and since OIPn = IIPn + G, we have:

OIPn = PIN + G + ΔP/(n-1) (Eq. 25)

IIPn + G = PIN + G + ΔP/(n-1) (Eq. 26)

Therefore:

**IIPn = P****IN**** + ****Δ****P/(n-1)** (Eq. 27)

**Particular Case: IP3**

The PIN gives a linear output power part, POUT1. This is observed on the Spread Analyzer (SA) at the fundamental frequencies. Also, POUT3 will be the level reached by the intermodulation products of the third order. They are located at the sum and difference of the 2 original frequencies. You see two triangles formed from OIP3, IP3, POUT1 and POUT3.

Slope 3 progresses three times more than Slope 1. Using the similar triangles law:

IIP3 = PIN – ΔP/2; where ΔP is POUT1 - POUT3

Figure 13. Graphic determination of IP3

**Summary**

A graphic method has been built to effectively measure the intercept points IPn in general and the IP3 in particular. One needs simply a dual tones generator, a mixer, and a spectrum analyzer to quasi-immediately evaluate the intercept points at any order IPn, and in particular the third order IP3. In the **next blog post**, we will detail more about the lab setup for the IPn measurement. Link with other related parameters such as the 1dB compression point, the ACPR… will be mentioned.

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