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Topics in Communication System Design: Carrier Triple Beats
Analysis of carrier triple beats that create thirdorder intermodulation interference when two or more carriers are present in one channel
Technical Feature
Topics in Communication System Design: Carrier Triple Beats
Howard Hausman
Miteq Inc.
Hauppauge, NY
Intermodulation interference is typically calculated based on two signals present in the RF or IF bandwidth. When the bandwidth is less than an octave, evenorder intermodulation products, such as second and fourthorder intermodulation interference, are out of band and can be filtered out. Oddorder intermodulation, such as third and fifthorder products, are inband interference, with sidebands as close to the carrier as the spacing of the desired carriers. In general, these signals cannot be filtered and must be dealt with by linearizing the system such that the interference is suppressed below the desired dynamic range. Thirdorder intermodulation, which is created by the mixing of the fundamental of one signal and the second harmonic of the other signal, is usually higher in amplitude than the fifthorder intermodulation product and therefore of primary concern to the system designer.
When more than two carriers are present in a channel, thirdorder intermodulation interference can be created by the multiplication of three fundamental carriers; this is called carrier triple beats (CTB). These spurious signals (CTB) are inband at a level 6 dB higher than the thirdorder intermodulation products created from two signals because there is no second harmonic involved in the production of the interference signal. The level of CTB interference is further enhanced by the fact that multiple CTB signals can occur in the same frequency band. The number of interference signals that can be superimposed on any particular channel is related to the number of desired carriers present. Statistically, more CTB interference occurs in the center of the band.
Determining Intermodulation Interference Levels
CTB interference is a spurious signal created from the interaction of three or more signals summed together in a nonlinear device. The level of the interference signal is related to the levels of the input signals and the nonlinearity of the device. In designing a system, the required operating signal levels and the respective acceptable spurious levels determine the acceptable nonlinearity of the equipment. Given the system's nonlinear characteristics, the acceptable spurious level and the output level of the carriers, the maximum number of carriers can be determined.
To determine the maximum operating signal, the nonlinear characteristics of the components are defined and the resultant spurious responses are evaluated. The same analysis is performed in reverse to define the acceptable nonlinearity given an operating signal range. A generalized nonlinear system can be represented by a Taylor series expansion of the nonlinear transfer characteristic.
S_{o} = a_{0} + a_{1} S_{i} + a_{2} S_{i} ^{2} + a_{3} S_{i} ^{3} + a_{4} S_{i} ^{4} + … (1)
where
S_{o} = output signal
S_{i} = input signal
a_{n} = coefficients of the device
(n = 0, 1, 2, 3, 4, …)
For a linear system
a_{n} = 0 for n > 1
If the device is AC coupled
a_{0} = 0
If S_{i} consists of three signals of equal amplitude, then
S_{i} = E1cos(w_{1} t) + E1cos(w_{2} t) + E1cos(w_{3} t)
where
E1 = peak amplitude
w1, w2 and w3 = respective radian frequencies
When S_{i} is applied to a nonlinear system, intermodulation products are created in all the higher order terms, proportional to the coefficients of the respective term.
Secondorder Intermodulation of CTB
The secondorder intermodulation product of three inband signals is usually almost an octave away from the desired carriers. In a narrow band system, these signals can easily be filtered and are therefore not considered in the spurious analysis. Appendix A is a detailed derivation of the resultant spectrum of the secondorder intermodulation from three inband signals.
The secondorder intermodulation term is expressed as
Secondorder Intermodulation
=2nd order=
a_{2} S_{i} ^{2} = a_{2} [E1cos(w_{1} t) + E1cos(w_{2} t) + E1cos(w_{3} t)]^{2}
Expanding the terms
2nd order =
a_{2} {[E1cos(w_{1} t)]^{2} + [E1cos(w_{2} t)]^{2}
+ [E1cos(w_{3} t)]^{2} + 2[E1cos(w_{1} t)
· E1cos(w_{2} t)] + 2[E1cos(w_{1} t)
· E1cos(w_{3} t)] + 2[E1cos(w_{3} t)
· E1cos(w_{2} t)]} (2)
All of the terms are the sum or difference frequency of two carriers closely spaced (narrow band). A trigonometric expansion would put all of the interference signals almost an octave away. For systems with a bandwidth less than an octave, these products can be filtered and therefore their effects on the system performance are negated. For the purpose of this analysis the secondorder effects of three carriers beating with each other will be considered negligible.
CTB Intermodulation Interference Signals
CTB intermodulation products are spurious signals due to the cube of the input signal multiplied by the a_{3} coefficient of the Taylor series expansion. A trigonometric expansion of this term confirms that the interference signals are inband and therefore cannot be filtered out.
An Analysis of Thirdorder Intermodulation of CTB Signals
If S_{i} consists of three signals of equal amplitude:
S_{i} = E1cos(w_{1} t) + E1cos(w_{2} t) + E1cos(w_{3} t)
Thirdorder CTB =3rd order=
a_{3} S_{i} ^{3} = a_{3} [E1cos(w_{1} t) + E1cos(w_{2} t) + E1cos(w_{3} t)]^{3} (3)
Expanding the terms
3rd order = a_{3} [E1cos(w_{1} t) + E1cos(w_{2} t) + E1cos(w_{3} t)]
· {[E1cos(w_{1} t)]^{2} + [E1cos(w_{2} t)]^{2} + [E1cos(w_{3} t)]^{2}
+ 2[E1cos(w_{1} t) E1cos(w_{2} t)]+ 2[E1cos(w_{1} t) E1cos(w_{3} t)]
+ 2[E1cos(w_{3} t) E1cos(w_{2} t)]} (4)
Multiplying out and combining terms (see Appendix B for the detailed derivation)
3rd order = a_{3} E1^{3} {[cos(w_{1} t)]^{3} + [cos(w_{2} t)]^{3} + [cos(w_{3} t)]^{3}
+ 3/2[2cos(w_{1} t) + 2cos(w_{2} t) + 2cos(w_{3} t)
+ 1/2cos((2w_{1} w_{2} )t) + 1/2cos((2w_{1} +w_{2} )t)
+ 1/2cos((2w_{1} w_{3} )t) + 1/2cos((2w_{1} +w_{3} )t)
+ 1/2cos((2w_{2} w_{1} )t) + 1/2cos((2w_{2} +w_{1} )t)
+ 1/2cos((2w_{3} w_{1} )t) + 1/2cos((2w_{3} +w_{1} )t)
+ 1/2cos((2w_{2} w_{3} )t) + 1/2cos((2w_{2} +w_{3} )t)
+ 1/2cos((2w_{3} w_{2} )t) + 1/2cos((2w_{3} +w_{2} )t)
+ (6/4)cos((w_{1} w_{2} +w_{3} )t) + (6/4)cos((w_{1} w_{2} w_{3} )t)
+ (6/4)cos((w_{1} +w_{2} +w_{3} )t)
+ (6/4)cos((w_{1} +w_{2} w_{3} )t)} (5)
Assuming that all carrier frequencies (w_{1} , w_{2} , w_{3} …) are located in a narrow band (much less than an octave) and considering only the inband terms, other than the fundamental
3rd order (inband) =
a_{3} E1^{3} · {1/2cos((2w_{1} w_{2} )t)
+ 1/2cos((2w_{1} w_{3} )t)
+ 1/2cos((2w_{2} w_{1} )t)
+ 1/2cos((2w_{3} w_{1} )t)
+ 1/2cos((2w_{2} w_{3} )t)
+ 1/2cos((2w_{3} w_{2} )t)
+ (6/4)cos((w_{1} w_{2} +w_{3} )t)
+ (6/4)cos((w_{1} w_{2} w_{3} )t)
+ (6/4)cos((w_{1} +w_{2} w_{3} )t)} (6)
The resultant frequencies are 2 Sig 3rd order (2w_{2} w_{1} ) and CTB. The relative amplitudes are (1/2)a_{3} E1^{3} and (6/4)a_{3} E1^{3} , respectively.
The CTB signals are three times higher than the thirdorder intermodulation products when more than two carriers are present.
Two Signal, Thirdorder Intermodulation Distortion
The resultant frequencies are (see Appendix B for the derivation) 2w_{1} w_{2} and 2w_{2} w_{1} . The relative amplitudes are (3/4)a_{3} E1^{3} for both.
CTB Levels Compared to Thirdorder Intermodulation
A tabulation of the thirdorder intermodulation is listed in Table 1 .
Table 1  
 Two Signals  CTB 
Relative  (3/4)a_{3} E1^{3}  (6/4)a_{3} E1^{3} 
The CTB intermodulation is 6 dB higher than the twotone, thirdorder intermodulation products. The level of a single carrier triple beat product can therefore be determined by using the familiar equations for twotone, thirdorder intermodulation product interference.

Fig. 1 Thirdorder intermodulation levels for a typical 10 dB gain amplifier. 
The twotone, thirdorder intermodulation product interference is determined by noting the relative single carrier power with respect to the thirdorder intercept point (usually 10 db above the 1 dB compression point), as shown in Figure 1 .
IP_{3} = 2(I_{3rd} A)
CTB = IP_{3} +6 dB
where
IP_{3} = relative thirdorder intermodulation level (dBc)
CTB = relative carrier triple beat intermodulation level (dBc)
A = signal 1 amplitude
= signal 2 amplitude (dBm)
I_{3rd} = thirdorder intercept point (dBm)
dBc = relative level of the intermodulation with respect to the single carrier amplitude (A)

Fig. 2 Twotone, thirdorder intermodulation levels. 
An example for two output signals of equal amplitude A, as shown in Figure 2 , is
A = +3 dBm
I_{3rd} = +20 dBm
IP_{3} = 34 dBc
(absolute level = 31 dBm)
Three carriers of equal amplitude A will have a CTB interference of CTB = IP_{3}  6 dB = 34 dBc + 6 dB = 28 dBc (25 dBm), as shown in Figure 3 . The three interference signals each down 28 dBc are products of the three carriers at frequencies w_{1} , w_{2} and w_{3} . The lowest interference signal is at frequency w_{1} +w_{2} w_{3} . In the center the interference signal is at frequency w_{3} +w_{1} w_{2} and the highest frequency is at w_{3} +w_{2} w_{1} .
Calculating CTB Intermodulation Levels for N Equal Amplitude Signals

Fig. 3 Carrier triple beat intermodulation diagram. 
Unlike thirdorder intermodulation interference, CTB signals can overlap each other and add noncoherently. This considerably increases the overall spurious in any given channel. The total spurious interference is related to the number of carriers and the position of the carrier, that is, carriers at the ends of the bandwidth have less interference products than channels in the center of the band. The number of interference carriers (beats) in any channel is given by
where
beats = number of interference carriers in the measured channel
N = number channels
M = number of the measured channel, 1 ≤ M ≤ N
The maximum number of interference carriers occurs in the center of the band (M » N/2). For N >> 1, the maximum number of beats (beat_{max} ) is given by
The worstcase level of CTB can be arrived at by calculating the level of each CTB (which is the thirdorder intermodulation level + 6 dB), adding noncoherently the number of beat signals that will fall into the respective band (the worstcase being in the center of the band). The CTB interference can therefore be determined using
CTB = 2(I_{3rd} carrier) + 6 + 10log(beat_{max} ) (8)
where
I_{3rd} = thirdorder intercept point (dBm)
carrier = single carrier output signal level (dBm)
beat_{max} = number of interference products in any signal channel
In terms of the total number of carriers N
For example,
I_{3rd} = +15
carrier = 20 dBm
N = 24
CTB = 45.5 dBc
Calculating the Number of Carriers in a Given Channel
Inversely, the total number of carriers that can be multiplexed into a single channel, knowing the required CTB interference level, can be calculated assuming that all of the interference signals are noncoherent and the bandwidth is wide enough for all of the carriers to exist with acceptable adjacent channel interference.
The total number of carriers N is given by
where
CTB = maximum acceptable CTB interference level (dBc)
I_{3rd} = thirdorder intercept point (dBm)
carrier = single carrier output signal level (dBm)
N = number of modulated carriers
As an example,
CTB = ≤ 57.6 dBc
I_{3rd} = +20 dBm
carrier = 15 dBm
The relative thirdorder intermodulation level is
IP_{3} (dBc) = 2 (I_{3rd} A)
where
A = signal 1 amplitude = Signal 2 amplitude (dBm)
For N output signals
A = +3 dBm
I_{3rd} = +20 dBm
IP_{3} = 34 dBc (absolute level = 31 dBm)
First check to see that the twotone, thirdorder intermodulation interference is below the required specification where A = +3 dBm
IP_{3} = 2(I_{3rd}  carrier) = 2(20  (15)) = 70 dBc
This obviously meets the desired criteria.
The total number of carriers considering CTB interference is
N = 5.909 = 5 carriers
It should be noted that this analysis is valid for CW carriers, which should be considered a worstcase signal. Modulated carriers exhibit spectrum spreading, which in effect will lower the intermodulation interference.
Table 2 is convenient for estimating the number of carriers that a given bandwidth could sustain (neglecting the bandwidth of the carrier, interchannel interference and available system bandwidth). Across the top is the level of each carrier (assuming all the carriers are the same level) below the thirdorder intercept point. To the left is the acceptable interference level.
Table 2  
Level Below  20 dB  25 dB  30 dB  35 dB  40 dB 
CTB (dBc)  Number of Carriers  
20  14  44  141  448  1420 
25  7  25  79  252  797 
30  4  14  44  141  448 
35  2  7  25  79  252 
40  1  4  14  44  141 
45  0  2  7  25  79 
50  0  1  4  14  44 
55  0  0  2  7  25 
60  0  0  1  4  14 
65  0  0  0  2  7 
70  0  0  0  1  4 
75  0  0  0  0  2 
80  0  0  0  0  1 
Conclusion
Determining the capacity of a channel is more involved than allocating enough bandwidth. CTB interference is an important factor to consider when a channel is loaded with many carriers. It has been shown that the interference level increases as the square of the increase in the number of carriers. The worstcase interference is in the center of the band where there are more combinations of frequencies in a given channel. At the ends of the band the interference levels go down, but unless the power level of the carriers in the center are lower than the carrier powers at the band edges, it is prudent to assume the worst interference for the system design.
Although the problem is critical as the number of carriers increase, it should be noted that even with only three inband carriers, the interference level is more than 6 dB above that calculated for twosignal, thirdorder intermodulation.
It must be noted that this is a worstcase analysis. Most modulated carriers exhibit a band spreading that lowers the average spectral density, which will somewhat lower the respective interference.
References
1. S. Winder, "Single Tone Intermodulation Testing," RF Design , December 1993.
2. M. Leffel, "Intermodulation Distortion in a Multisignal Environment," RF Design , June 1995.
3. R. Hawkins, "Combining Gain, Noise Figure and Intercept Points for Cascaded Circuit Elements," RF Design , March 1990.
4. J. Waltrich, "Compute CTB in Hybrid Fiber/Coax Systems," Microwaves & RF , December 1998.
5. D. Henkes and S. Kwok, "Intermodulation: Concepts and Calculations," Applied Microwaves & Wireless , July/August 1997.
Howard Hausman received his BSEE and MSEE degrees from Polytechnic University and is currently chief technology engineer at Miteq Inc. During his career he has designed microwave systems and components for satellite communications, radar and reconnaissance, which includes receivers, transmitters and synthesizers. Mr. Hausman is also an adjunct professor at Polytechnic University, where he teaches graduate courses in electrical engineering, and has been an adjunct professor at Hofstra University. He has presented lectures and authored papers relating to microwave and communication systems.
Appendix A
Derivation of secondOrder Intermodulation of Three InBand Signals
This analysis is presented to show that the secondorder effects are out of band (assuming a narrow band system) even when there are more than two carriers present.
Secondorder intermodulation of three inband signals is usually almost an octave away from the desired carriers. In a narrow band system, these signals can be easily filtered and therefore not considered in the spurious analysis. This is a derivation of the resultant spectrum of secondorder intermodulation on three inband signals.
Secondorder intermodulation of CTB
Secondorder Intermodulation = 2nd order = a_{2} S_{i} ^{2} = a_{2} [E1cos(w_{1} t) + E1cos(w_{2} t) + E1cos(w_{3} t)]^{2}
Expanding the terms:
2nd order = a_{2} {[E1cos(w_{1} t)]^{2} + [E1cos(w_{2} t)]^{2} + [E1cos(w_{3} t)]^{2} + 2[E1cos(w_{1} t) · E1cos(w_{2} t)] + 2[E1cos(w_{1} t) · E1cos(w_{3} t)] + 2[E1cos(w_{3} t) · E1cos(w_{2} t)]}
If all of the carriers are in a narrow frequency band the secondorder products are approximately an octave away. Systems less than an octave wide can filter these products and therefore negate their effects on the system performance. For the purpose of this analysis the secondorder effects of three carriers beating with each other will be considered negligible.
Appendix B
Derivation of the Carrier Triple Beat (CTB) Equations and the ThirdOrder Intermodulation Equations
The following trigonometric expansion shows how CTB interference is formed. It should be noted that in addition to carrier triple beat terms there are twotone intermodulation terms (below the CTBs) and fundamental terms that cause the familiar signal compression as the signal level increases.
ThirdOrder intermodulation of CTB
Thirdorder CTB = 3rd order = a_{3} S_{i} ^{3} = a_{3} [E1cos(w_{1} t) + E1cos(w_{2} t) + E1cos(w_{3} t)]^{3}
Expanding the terms:
3rd order = a_{3} [E1cos(w_{1} t) + E1cos(w_{2} t) + E1cos(w_{3} t)]
· {[E1cos(w_{1} t)]^{2} + [E1cos(w_{2} t)]^{2} + [E1cos(w_{3} t)]^{2} + 2[E1cos(w_{1} t)
· E1cos(w_{2} t)] + 2[E1cos(w_{1} t) · E1cos(w_{3} t)] + 2[E1cos(w_{3} t) E1cos(w_{2} t)]}
Multiplying out the cube term
3rd order CTB = a_{3} {[E1cos(w_{1} t)]^{3} + [E1cos(w_{2} t)]^{3}
+ [E1cos(w_{3} t)]^{3} + 3[E1cos(w_{1} t)]^{2} · [E1cos(w_{2} t)] + 3[E1cos(w_{1} t)]^{2}
· [E1cos(w_{3} t)] + 3[E1cos(w_{1} t)] · [E1cos(w_{2} t)]^{2} + 3[E1cos(w_{1} t)]
· [E1cos(w_{3} t)]^{2} + 3[E1cos(w_{2} t)]^{2} · [E1cos(w_{3} t)] + 3[E1cos(w_{2} t)]
· [E1cos(w_{3} t)]^{2} + 6[E1cos(w_{1} t)] · [E1cos(w_{2} t)] · [E1cos(w_{3} t)]}
Using the respective trigonometric identities for the resultant square term:
3rd order = a_{3} E1^{3} {[cos(w_{1} t)]^{3} + [cos(w_{2} t)]^{3} + [cos(w_{3} t)]^{3}
+ {3[1/2 + (1/2)cos(2w_{1} t)] cos(w_{2} t) + 3[1/2 + (1/2)cos(2w_{1} t)] cos(w_{3} t) +
3[1/2 + (1/2)cos(2w_{2} t )] cos(w_{1} t) + 3[1/2 + (1/2)cos(2w_{3} t)] cos(w_{1} t) +
3[1/2 + (1/2)cos(2w_{2} t)] cos(w_{3} t)+ 3[1/2 + (1/2)cos(2w_{3} t)] cos(w_{2} t)
+ 6[cos(w_{1} t) cos(w_{2} t) cos(w_{3} t)]}
Multiplying out the cube term:
3rd order = a_{3} E1^{3} {[cos(w_{1} t)]^{3} + [cos(w_{2} t)]^{3} + [cos(w_{3} t)]^{3}
+ [3/2 cos(w_{2} t) + (3/2)cos(2w_{1} t) cos(w_{2} t) + (3/2)cos(w_{3} t)
+ (3/2)cos(2w_{1} t ) cos(w_{3} t) + 3/2 cos(w_{1} t) + (3/2)cos (2w_{2} t ) cos(w_{1} t)
+ (3/2)cos(w_{1} t) + (3/2)cos (2w_{3} t) · cos(w_{1} t) + (3/2)cos(w_{3} t)
+ (3/2)cos(2w_{2} t )cos(w_{3} t) + (3/2)cos(w_{2} t) + (3/2)cos(2w_{3} t) cos(w_{2} t)
+ 6[cos(w_{1} t) cos(w_{2} t)cos(w_{3} t)]}
Factor out 3/2
3rd order = a_{3} E1^{3} {[cos(w_{1} t)]^{3} + [cos(w_{2} t)]^{3} + [cos(w_{3} t)]^{3}
+ {3/2[cos(w_{2} t) + cos(2w_{1} t) cos(w_{2} t) + cos(w_{3} t) + cos(2w_{1} t ) cos(w_{3} t)
+ cos(w_{1} t) + cos(2w_{2} t) cos(w_{1} t) + cos(w_{1} t) + cos(2w_{3} t ) · cos(w_{1} t)
+ cos(w_{3} t) + cos(2w_{2} t) cos(w_{3} t) + cos(w_{2} t) + cos(2w_{3} t) cos(w_{2} t)]
+ 6[cos(w_{1} t) cos(w_{2} t) cos(w_{3} t)]}
Combine 1st order terms
3rd order = a_{3} E1^{3} {[cos(w_{1} t)]^{3} + [cos(w_{2} t)]^{3} + [cos(w_{3} t)]^{3}
+ 3/2[2 cos(w_{1} t) + 2cos(w_{2} t) + 2cos(w_{3} t) + cos(2w_{1} t) cos(w_{2} t)
+ cos(2w_{1} t) cos(w_{3} t) + cos(2w_{2} t) cos(w_{1} t) + cos(2w_{3} t) · cos(w_{1} t)
+ cos(2w_{2} t )cos(w_{3} t) + cos(2w_{3} t) cos(w_{2} t)] + 6[cos(w_{1} t) cos(w_{2} t)cos(w_{3} t)]}
Expand the thirdorder terms
3rd order = a_{3} E1^{3} {[cos(w_{1} t)]^{3} + [cos(w_{2} t)]^{3} + [cos(w_{3} t)]^{3}
+ 3/2[2 cos(w_{1} t) + 2cos(w_{2} t) + 2cos(w_{3} t) + (1/2)cos((2w_{1} w_{2} )t)
+ (1/2)cos((2w_{1} +w_{2} )t) + (1/2)cos((2w_{1} w_{3} )t) + (1/2)cos((2w_{1} +w_{3} )t)
+ (1/2)cos((2w_{2} w_{1} )t) + (1/2)cos((2w_{2} +w_{1} )t) + (1/2)cos((2w_{3} w_{1} )t)
+ (1/2)cos((2w_{3} +w_{1} )t) + (1/2)cos((2w_{2} w_{3} )t) + (1/2)cos((2w_{2} +w_{3} )t)
+ (1/2)cos((2w_{3} w_{2} )t) + (1/2)cos((2w_{3} +w_{2} )t) + 6[(1/2)cos((w_{1} w_{2} )t)
+ (1/2)cos((w_{1} +w_{2} )t)]cos(w_{3} t)}
Further expand the thirdorder terms
3rd order = a_{3} E1^{3} {[cos(w_{1} t)]^{3} + [cos(w_{2} t)]^{3} + [cos(w_{3} t)]^{3}
+ 3/2[2 cos(w_{1} t) + 2cos(w_{2} t) + 2cos(w_{3} t) + (1/2)cos((2w_{1} w_{2} )t)
+ (1/2)cos((2w_{1} +w_{2} )t) + (1/2)cos((2w_{1} w_{3} )t) + (1/2)cos((2w_{1} +w_{3} )t)
+ (1/2)cos((2w_{2} w_{1} )t) + (1/2)cos((2w_{2} +w_{1} )t) + (1/2)cos((2w_{3} w_{1} )t)
+ (1/2)cos((2w_{3} +w_{1} )t) + (1/2)cos((2w_{2} w_{3} )t) + (1/2)cos((2w_{2} +w_{3} )t)
+ (1/2)cos((2w_{3} w_{2} )t) + (1/2)cos((2w_{3} +w_{2} )t) + 6[(1/4)cos((w_{1} w_{2} +w_{3} )t)
+ (1/4)cos((w_{1} w_{2} w_{3} )t) + (1/4)cos((w_{1} +w_{2} +w_{3} )t) + (1/4)cos((w_{1} +w_{2} w_{3} )t)]}
Thirdorder Intermodulation Interference Signals
Thirdorder intermodulation products are spurious signals due to the cube of the input signal multiplied by the a_{3} coefficient of the Taylor series expansion.
Thirdorder Intermodulation = 3rd order =
a_{3} S_{i} ^{3} = a_{3} [E1cos(w_{1} t) + E1cos(w_{2} t)]3
Expanding the terms:
3rd order = a_{3} [E1cos(w_{1} t) + E1cos(w_{2} t)] · {[E1cos(w_{1} t)]^{2}
+ [E1cos(w_{2} t)]^{2} + 2[E1cos(w_{1} t) · E1cos(w_{2} t)]}
Factoring out the cube term and using the respective trigonometric identities for the resultant square term:
3rd order = a_{3} E1^{3} {cos(w_{1} t) + cos(w_{2} t)} {[1/2 + (1/2)cos(2w_{1} t)]
+ [1/2 + (1/2)cos(2w_{2} t)] + [cos(w_{1} tw_{2} t) + cos(w_{1} t+w_{2} t)]}
Multiplying out the cube term:
3rd order = a_{3} E1^{3} {cos(w_{1} t) + cos(w_{2} t) + [[(1/2)cos(2w_{1} t)]
· [cos(w_{1} t)] + cos(w_{2} t)] + [[(1/2)cos(2w_{2} t)] · [cos(w_{1} t) + cos(w_{2} t)]]
+ [[cos(w_{1} tw_{2} t)] · [cos(w_{1} t) + cos(w_{2} t)]]
+ [[cos(w_{1} t+w_{2} t)] · [cos(w_{1} t) + cos(w_{2} t)]]}
If the system is less than an octave in bandwidth, all of the frequency terms that are added (w_{1} + w_{2} ) are filtered out and can be discarded in this analysis:
3rd order = a_{3} E1^{3} {cos(w_{1} t) + cos(w_{2} t) + (1/4)cos(w_{1} t )
+ (1/4) [cos(2w_{1} tw_{2} t)] + (1/4)cos(2w_{2} tw_{1} t) + (1/4)cos(w_{2} t)
+ (1/2)[cos(2w_{1} tw_{2} t) + cos(w_{2} t) + cos(w_{1} t2w_{2} t) + cos(w_{1} t)]
+ 1/2[cos(w_{2} t) + cos(w_{1} t)]}
Note: cos(w_{2} t) = cos(w_{2} t) and cos(w_{1} t2w_{2} t) = cos(2w_{2} tw_{1} t)
Combining terms:
3rd order = a_{3} E1^{3} {(9/4)cos(w_{1} t) + (9/4)cos(w_{2} t)
+ (3/4)cos(2w_{1} tw_{2} t) + (3/4)cos(2w_{2} tw_{1} t)}
The resultant spectral frequencies are at:
Frequency w_{1} , w_{2} , 2w_{1} w_{2} , 2w_{2} w_{1}
Relative Amplitude (9/4) a_{3} E1^{3} (9/4) a_{3} E1^{3} (3/4) a_{3} E1^{3} (3/4) a_{3} E1^{3}
The thirdorder intermodulation produces signals at the input frequencies and two side bands on either side of the input frequencies.
The coefficient a_{3} is typically 180° out of phase with the coefficient a1. In cases where the input is a single signal, this out of phase relationship effectively reduces the output level when the input signal becomes large. The effective reduction is related to the a_{3} coefficient. This suggests a relationship between thirdorder intermodulation levels and single signal compression points (the 1 dB output gain compression level is typically 10 dB below the thirdorder output intercept point).