What are the frequencies of the second-order and third-order distortion tones given two frequency peaks?

In general, the third order distortion tones are understood to exist as in-band distortion at frequencies 2ω21 and 2ω12 in a two tone intermodulation test. Third order distortion also exists at frequencies ω1 and ω2. Second order distortion tones are found outside of a narrowband system at 2ω2, 2ω1, and ω12.  

Consider the two-tone input of a non-linear system with frequencies ω1 and ω2:

Vin = A[cos(ω1t)+cos(ω2t)]

The second order and third order distortion tones are calculated on the following page. In summary, the tones are shown in the table below. This shows that third order distortion tones are found not only in the positions mentioned above, but also contribute to the fundamental tone frequencies.  In a spurious-free system, all third order tones will be below the noise floor. This is verified in MATLAB with ω1, ω2 at 500kHz, 501kHz.

FrequencyComponents
DCa0+A2a2
ω1Aa1+2A3a3
ω2Aa1+2A3a3
2 ω1A2a2/2
2 ω2A2a2/2
ω1+ ω2A2a2
ω1 – ω2A2a2/2
ω2– ω1A2a2/2
3 ω1A3a3/4
3 ω2A3a3/4
2 ω1+ ω23A3a3/4
2 ω1– ω2A3a3/2
2 ω2+ ω13A3a3/4
2 ω2– ω1A3a3/2
– ω2A3a3/4
– ω1A3a3/4
ω1-2 ω2A3a3/4
ω2-2 ω1A3a3/4

What does the term “Spurious-free” mean in Spurious-free Dynamic Range (SFDR)?

In the term spurious-free dynamic range (SFDR), spurious-free means that non-linear distortion is below the noise floor for given input levels. The system is spurious when non-linear distortion is present above the noise floor. The system is spurious-free when non-linear distortion is below the noise floor. SFDR therefore is the range of output levels whereby the system is undisturbed by non-linear distortion or spurs.

 SFDR contrasts with compression dynamic range (or linear dynamic range (LDR)) which is the range of output levels whereby the fundamental tone is proportional to the input, irrespective of distortion tone levels. The fundamental tone is no longer considered to be linear beyond the 1dB compression point, after which the output fundamental tones do not increase at the same rate as the input fundamental tones.

Image credits (modified): Pozar, Microwave Engineering, 2nd Edition

Spurs are non-linear distortion tones generated by non-linearities of a system. The output of a non-linear system can be modeled as a Fourier series.

The first term a0 is a DC component generated by the non-linear system. The second term a1Vin is the fundamental tone with some level of gain a1. The third term a2Vin2 is a second order non-linear distortion tone. The fourth term a3Vin3 is the third-order non-linear distortion tone. Further expansion of the Fourier series generates more harmonic and distortion tones. Even order harmonic distortion tones are usually outside of the band of interest, unless the system is very wideband. Odd order distortion tones however are found much closer to the fundamental tone in the frequency domain. SFDR is usually taken with respect to the third order intermodulation distortion, however it may also occasionally be taken for the fifth order (or seventh).

IMD3: Third Order Intermodulation Distortion

We’ll begin a discussion on the topic of analog system quality. How do we measure how well an analog system works? One over-simplistic answer is to say that power gain determines how well a system operates. This is not sufficient. Instead, we must analyze the system to determine how well it works as intended, which may include the gain of the fundamental signal. Whether it is an audio amplifier, acoustic transducers, a wireless communication system or optical link, the desired signal (either transmitted or received) needs to be distinguishable from the system noise. Noise, although situationally problematic can usually be averaged out. The presence of other signals are not however. This begs the question, which other signals could we be speaking of, if there is supposed to be only one signal? The answer is that the fundamental signal also comes with second order, third order, fourth order and higher order distortion harmonic and intermodulation signals, which may not be averaged from noise. Consider the following plot:

We usually talk about Third Order Intermodulation Distortion or IMD3 in such systems primarily. Unlike the second and fourth order, the Third Order Intermodulation products are found in the same spectral region as the first order fundamental signals. Second and fourth order distortion can be filtered out using a bandpass filter for the in-band region. Note that the fifth order intermodulation distortion and seventh order intermodulation distortion can also cause an issue in-band, although these signals are usually much weaker.

Consider the use of a radar system. If a return signal is expected in a certain band, we need to be able to distinguish between the actual return and differentiate this from IMD3, else we may not be able to trust our result. We will discuss next how IMD3 is avoided.

IP3 Distortion and Linearity

RF/Photonics Lab
November 2019
Michael Benker

IP3 Distortion & Linearity

Linearity is the measure of a system’s performance as an output signal being proportional to the input signal level. Not every system can be expected to perform ideally and thus linearly. Devices such as diodes and transistors are examples of non-linear systems.

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The intercept point of the third order, IP3 is a measure of the linearity of a system. IP3 is the third order of a Taylor series expansion of the input signal’s presence in the frequency domain. Being third order, this term in a Taylor series expansion is understood as distortion since it is different from the sought output signal. In contrast to the second order harmonics, which fall outside of the frequency band of the first order signal, the third order is found in the same frequency band as the original or first order signal. Similarly, consecutive even orders (4, 6, 8, etc) are found outside of the frequency band of the first order signal. Consecutive odd orders beyond the third order such as IP5 and IP7 also cause distortion but are not of primary focus since the amplitude of these order signals are weaker after consequent exponentiation.

The meaning of an intercept point of an nth order (IPn) on a dBm-dBm axis is the point at which the first-order and nth-order powers would be equal for a given input power. In the case of IP3, this indicates the power level needed for a third-order power to potentially drown out the first-order signal with distortion. The 1 dB compression point defines the range of linear operation for a system.

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