Topics in communication system design carrier triple beats

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Home » Topics in Communication System Design: Carrier Triple Beats
Industry News Semiconductors / Integrated Circuits
Topics in Communication System Design: Carrier Triple Beats
Analysis of carrier triple beats that create third-order
intermodulation interference when two or more carriers are present
in one channel
January 1, 2002

Howard Hausman
No Comments
Technical Feature
Topics in Communication System Design: Carrier Triple Beats
Related Resources
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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, even-order intermodulation products, such as second- and fourth-
order intermodulation interference, are out of band and can be filtered out. Odd-order intermodulation, such as
third- and fifth-order products, are in-band 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. Third-order 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 fifth-order intermodulation product and therefore of primary concern to the system
designer.
When more than two carriers are present in a channel, third-order intermodulation interference can be created
by the multiplication of three fundamental carriers; this is called carrier triple beats (CTB). These spurious
signals (CTB) are in-band at a level 6 dB higher than the third-order 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.
So = a0 + a1 Si + a2 Si
2 + a3 Si
3 + a4 Si
4 + … (1)
where
So = output signal
Si = input signal
an = coefficients of the device
(n = 0, 1, 2, 3, 4, …)

For a linear system
an = 0 for n > 1
If the device is AC coupled
a0 = 0
If Si consists of three signals of equal amplitude, then
Si = E1cos(w1 t) + E1cos(w2 t) + E1cos(w3 t)
where
E1 = peak amplitude
w1, w2 and w3 = respective radian frequencies
When Si is applied to a nonlinear system, intermodulation products are created in all the higher order terms,
proportional to the coefficients of the respective term.
Second-order Intermodulation of CTB
The second-order intermodulation product of three in-band 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 second-order
intermodulation from three in-band signals.
The second-order intermodulation term is expressed as
Second-order Intermodulation
=2nd order=
a2 Si
2 = a2 [E1cos(w1 t) + E1cos(w2 t) + E1cos(w3 t)]2
Expanding the terms
2nd order =
a2 {[E1cos(w1 t)]2 + [E1cos(w2 t)]2
+ [E1cos(w3 t)]2 + 2[E1cos(w1 t)
· E1cos(w2 t)] + 2[E1cos(w1 t)
· E1cos(w3 t)] + 2[E1cos(w3 t)
· E1cos(w2 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 second-order 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 a3
coefficient of the Taylor series expansion. A trigonometric expansion of this term confirms that the interference
signals are in-band and therefore cannot be filtered out.
An Analysis of Third-order Intermodulation of CTB Signals
If Si consists of three signals of equal amplitude:

Si = E1cos(w1 t) + E1cos(w2 t) + E1cos(w3 t)
Third-order CTB =3rd order=
a3 Si
3 = a3 [E1cos(w1 t) + E1cos(w2 t) + E1cos(w3 t)]3 (3)
Expanding the terms
3rd order = a3 [E1cos(w1 t) + E1cos(w2 t) + E1cos(w3 t)]
· {[E1cos(w1 t)]2 + [E1cos(w2 t)]2 + [E1cos(w3 t)]2
+ 2[E1cos(w1 t) E1cos(w2 t)]+ 2[E1cos(w1 t) E1cos(w3 t)]
+ 2[E1cos(w3 t) E1cos(w2 t)]} (4)
Multiplying out and combining terms (see Appendix B for the detailed derivation)
3rd order = a3 E13 {[cos(w1 t)]3 + [cos(w2 t)]3 + [cos(w3 t)]3
+ 3/2[2cos(w1 t) + 2cos(w2 t) + 2cos(w3 t)
+ 1/2cos((2w1 -w2 )t) + 1/2cos((2w1 +w2 )t)
+ 1/2cos((2w1 -w3 )t) + 1/2cos((2w1 +w3 )t)
+ 1/2cos((2w2 -w1 )t) + 1/2cos((2w2 +w1 )t)
+ 1/2cos((2w3 -w1 )t) + 1/2cos((2w3 +w1 )t)
+ 1/2cos((2w2 -w3 )t) + 1/2cos((2w2 +w3 )t)
+ 1/2cos((2w3 -w2 )t) + 1/2cos((2w3 +w2 )t)
+ (6/4)cos((w1 -w2 +w3 )t) + (6/4)cos((w1 -w2 -w3 )t)
+ (6/4)cos((w1 +w2 +w3 )t)
+ (6/4)cos((w1 +w2 -w3 )t)} (5)
Assuming that all carrier frequencies (w1 , w2 , w3 …) are located in a narrow band (much less than an octave)
and considering only the in-band terms, other than the fundamental
3rd order (in-band) =
a3 E13 · {1/2cos((2w1 -w2 )t)
+ 1/2cos((2w1 -w3 )t)
+ 1/2cos((2w2 -w1 )t)
+ 1/2cos((2w3 -w1 )t)
+ 1/2cos((2w2 -w3 )t)
+ 1/2cos((2w3 -w2 )t)
+ (6/4)cos((w1 -w2 +w3 )t)
+ (6/4)cos((w1 -w2 -w3 )t)
+ (6/4)cos((w1 +w2 -w3 )t)} (6)
The resultant frequencies are 2 Sig 3rd order (2w2 -w1 ) and CTB. The relative amplitudes are (1/2)a3 E13 and
(6/4)a3 E13 , respectively.
The CTB signals are three times higher than the third-order intermodulation products when more than two
carriers are present.
Two Signal, Third-order Intermodulation Distortion
The resultant frequencies are (see Appendix B for the derivation) 2w1 -w2 and 2w2 -w1 . The relative
amplitudes are (3/4)a3 E13 for both.

Fig. 1 Third-order
intermodulation levels for a
typical 10 dB gain amplifier.
Fig. 2 Two-tone, third-order
intermodulation levels.
CTB Levels Compared to Third-order Intermodulation
A tabulation of the third-order intermodulation is listed in Table 1 .
Table 1
Third-order Intermodulation
Distortion
Two Signals CTB
Relative
amplitude
(3/4)a3 E13 (6/4)a3 E13
The CTB intermodulation is 6 dB higher than the two-tone, third-order intermodulation products. The level of a
single carrier triple beat product can therefore be determined by using the familiar equations for two-tone, third-
order intermodulation product interference.
The two-tone, third-order intermodulation product interference is determined by
noting the relative single carrier power with respect to the third-order intercept
point (usually 10 db above the 1 dB compression point), as shown in Figure 1 .
IP3 = -2(I3rd -A)
CTB = IP3 +6 dB
where
IP3 = relative third-order intermodulation level (dBc)
CTB = relative carrier triple beat intermodulation level (dBc)
A = signal 1 amplitude
= signal 2 amplitude (dBm)
I3rd = third-order intercept point (dBm)
dBc = relative level of the intermodulation with respect to the single carrier amplitude (A)
An example for two output signals of equal amplitude A, as shown in Figure 2 , is
A = +3 dBm
I3rd = +20 dBm
IP3 = -34 dBc
(absolute level = -31 dBm)
Three carriers of equal amplitude A will have a CTB interference of CTB = IP3 - 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 w1 , w2 and w3 . The lowest interference signal is at frequency w1 +w2
-w3 . In the center the interference signal is at frequency w3 +w1 -w2 and the
highest frequency is at w3 +w2 -w1 .
Calculating CTB Intermodulation Levels for N Equal Amplitude Signals
Unlike third-order 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

Fig. 3 Carrier triple beat
intermodulation diagram.
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 (beatmax ) is given by
The worst-case level of CTB can be arrived at by calculating the level of each CTB (which is the third-order
intermodulation level + 6 dB), adding noncoherently the number of beat signals that will fall into the respective
band (the worst-case being in the center of the band). The CTB interference can therefore be determined using
CTB = 2(I3rd -carrier) + 6 + 10log(beatmax ) (8)
where
carrier = single carrier output signal level (dBm)
beatmax = number of interference products in any signal channel
In terms of the total number of carriers N
For example,
I3rd = +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)
carrier = single carrier output signal level (dBm)
N = number of modulated carriers
As an example,
CTB = ≤ -57.6 dBc
I3rd = +20 dBm
carrier = -15 dBm
The relative third-order intermodulation level is
IP3 (dBc) = -2 (I3rd -A)
where
A = signal 1 amplitude = Signal 2 amplitude (dBm)
For N output signals
A = +3 dBm
I3rd = +20 dBm
IP3 = -34 dBc (absolute level = -31 dBm)
First check to see that the two-tone, third-order intermodulation interference is below the required specification
where A = +3 dBm
IP3 = -2(I3rd - 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 worst-case 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 third-order intercept point. To the left is
the acceptable interference level.
Table 2
Number of Carriers in a Single Channel
Level Below
Third-order
Intercept
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 worst-case 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
in-band carriers, the interference level is more than 6 dB above that calculated for two-signal, third-order
intermodulation.
It must be noted that this is a worst-case 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 Multi-signal 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 second-Order Intermodulation of Three In-Band Signals
This analysis is presented to show that the second-order effects are out of band (assuming a narrow band
system) even when there are more than two carriers present.
Second-order intermodulation of three in-band 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 second-order intermodulation on three in-
band signals.
Second-order intermodulation of CTB
Second-order Intermodulation = 2nd order = a2 Si
Expanding the terms:
2nd order = a2 {[E1cos(w1 t)]2 + [E1cos(w2 t)]2 + [E1cos(w3 t)]2 + 2[E1cos(w1 t) · E1cos(w2 t)] + 2[E1cos(w1
t) · E1cos(w3 t)] + 2[E1cos(w3 t) · E1cos(w2 t)]}
If all of the carriers are in a narrow frequency band the second-order 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 second-order effects of three carriers beating with each other
will be considered negligible.
Appendix B
Derivation of the Carrier Triple Beat (CTB) Equations and the Third-Order 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 two-tone intermodulation terms (below the CTBs) and fundamental

terms that cause the familiar signal compression as the signal level increases.
Third-Order intermodulation of CTB
Third-order CTB = 3rd order = a3 Si
3rd order = a3 [E1cos(w1 t) + E1cos(w2 t) + E1cos(w3 t)]
· {[E1cos(w1 t)]2 + [E1cos(w2 t)]2 + [E1cos(w3 t)]2 + 2[E1cos(w1 t)
· E1cos(w2 t)] + 2[E1cos(w1 t) · E1cos(w3 t)] + 2[E1cos(w3 t) E1cos(w2 t)]}
Multiplying out the cube term
3rd order CTB = a3 {[E1cos(w1 t)]3 + [E1cos(w2 t)]3
+ [E1cos(w3 t)]3 + 3[E1cos(w1 t)]2 · [E1cos(w2 t)] + 3[E1cos(w1 t)]2
· [E1cos(w3 t)] + 3[E1cos(w1 t)] · [E1cos(w2 t)]2 + 3[E1cos(w1 t)]
· [E1cos(w3 t)]2 + 3[E1cos(w2 t)]2 · [E1cos(w3 t)] + 3[E1cos(w2 t)]
· [E1cos(w3 t)]2 + 6[E1cos(w1 t)] · [E1cos(w2 t)] · [E1cos(w3 t)]}
Using the respective trigonometric identities for the resultant square term:
+ {3[1/2 + (1/2)cos(2w1 t)] cos(w2 t) + 3[1/2 + (1/2)cos(2w1 t)] cos(w3 t) +
3[1/2 + (1/2)cos(2w2 t )] cos(w1 t) + 3[1/2 + (1/2)cos(2w3 t)] cos(w1 t) +
3[1/2 + (1/2)cos(2w2 t)] cos(w3 t)+ 3[1/2 + (1/2)cos(2w3 t)] cos(w2 t)
+ 6[cos(w1 t) cos(w2 t) cos(w3 t)]}
Multiplying out the cube term:
+ [3/2 cos(w2 t) + (3/2)cos(2w1 t) cos(w2 t) + (3/2)cos(w3 t)
+ (3/2)cos(2w1 t ) cos(w3 t) + 3/2 cos(w1 t) + (3/2)cos (2w2 t ) cos(w1 t)
+ (3/2)cos(w1 t) + (3/2)cos (2w3 t) · cos(w1 t) + (3/2)cos(w3 t)
+ (3/2)cos(2w2 t )cos(w3 t) + (3/2)cos(w2 t) + (3/2)cos(2w3 t) cos(w2 t)
+ 6[cos(w1 t) cos(w2 t)cos(w3 t)]}
Factor out 3/2
+ {3/2[cos(w2 t) + cos(2w1 t) cos(w2 t) + cos(w3 t) + cos(2w1 t ) cos(w3 t)
+ cos(w1 t) + cos(2w2 t) cos(w1 t) + cos(w1 t) + cos(2w3 t ) · cos(w1 t)
+ cos(w3 t) + cos(2w2 t) cos(w3 t) + cos(w2 t) + cos(2w3 t) cos(w2 t)]
+ 6[cos(w1 t) cos(w2 t) cos(w3 t)]}
Combine 1st order terms
+ 3/2[2 cos(w1 t) + 2cos(w2 t) + 2cos(w3 t) + cos(2w1 t) cos(w2 t)

+ cos(2w1 t) cos(w3 t) + cos(2w2 t) cos(w1 t) + cos(2w3 t) · cos(w1 t)
+ cos(2w2 t )cos(w3 t) + cos(2w3 t) cos(w2 t)] + 6[cos(w1 t) cos(w2 t)cos(w3 t)]}
Expand the third-order terms
+ 3/2[2 cos(w1 t) + 2cos(w2 t) + 2cos(w3 t) + (1/2)cos((2w1 -w2 )t)
+ (1/2)cos((2w1 +w2 )t) + (1/2)cos((2w1 -w3 )t) + (1/2)cos((2w1 +w3 )t)
+ (1/2)cos((2w2 -w1 )t) + (1/2)cos((2w2 +w1 )t) + (1/2)cos((2w3 -w1 )t)
+ (1/2)cos((2w3 -w2 )t) + (1/2)cos((2w3 +w2 )t) + 6[(1/2)cos((w1 -w2 )t)
+ (1/2)cos((w1 +w2 )t)]cos(w3 t)}
Further expand the third-order terms
+ 3/2[2 cos(w1 t) + 2cos(w2 t) + 2cos(w3 t) + (1/2)cos((2w1 -w2 )t)
+ (1/2)cos((2w2 -w1 )t) + (1/2)cos((2w2 +w1 )t) + (1/2)cos((2w3 -w1 )t)
+ (1/2)cos((2w3 -w2 )t) + (1/2)cos((2w3 +w2 )t) + 6[(1/4)cos((w1 -w2 +w3 )t)
+ (1/4)cos((w1 -w2 -w3 )t) + (1/4)cos((w1 +w2 +w3 )t) + (1/4)cos((w1 +w2 -w3 )t)]}
Third-order Intermodulation Interference Signals
Third-order intermodulation products are spurious signals due to the cube of the input signal multiplied by the
a3 coefficient of the Taylor series expansion.
Third-order Intermodulation = 3rd order =
a3 Si
3 = a3 [E1cos(w1 t) + E1cos(w2 t)]3
3rd order = a3 [E1cos(w1 t) + E1cos(w2 t)] · {[E1cos(w1 t)]2
+ [E1cos(w2 t)]2 + 2[E1cos(w1 t) · E1cos(w2 t)]}
Factoring out the cube term and using the respective trigonometric identities for the resultant square term:
3rd order = a3 E13 {cos(w1 t) + cos(w2 t)} {[1/2 + (1/2)cos(2w1 t)]
+ [1/2 + (1/2)cos(2w2 t)] + [cos(w1 t-w2 t) + cos(w1 t+w2 t)]}
Multiplying out the cube term:
3rd order = a3 E13 {cos(w1 t) + cos(w2 t) + [[(1/2)cos(2w1 t)]
· [cos(w1 t)] + cos(w2 t)] + [[(1/2)cos(2w2 t)] · [cos(w1 t) + cos(w2 t)]]
+ [[cos(w1 t-w2 t)] · [cos(w1 t) + cos(w2 t)]]
+ [[cos(w1 t+w2 t)] · [cos(w1 t) + cos(w2 t)]]}
If the system is less than an octave in bandwidth, all of the frequency terms that are added (w1 + w2 ) are
filtered out and can be discarded in this analysis:

3rd order = a3 E13 {cos(w1 t) + cos(w2 t) + (1/4)cos(w1 t )
+ (1/4) [cos(2w1 t-w2 t)] + (1/4)cos(2w2 t-w1 t) + (1/4)cos(w2 t)
+ (1/2)[cos(2w1 t-w2 t) + cos(-w2 t) + cos(w1 t-2w2 t) + cos(w1 t)]
+ 1/2[cos(w2 t) + cos(w1 t)]}
Note: cos(-w2 t) = cos(w2 t) and cos(w1 t-2w2 t) = cos(2w2 t-w1 t)
Combining terms:
3rd order = a3 E13 {(9/4)cos(w1 t) + (9/4)cos(w2 t)
+ (3/4)cos(2w1 t-w2 t) + (3/4)cos(2w2 t-w1 t)}
The resultant spectral frequencies are at:
Frequency w1 , w2 , 2w1 -w2 , 2w2 -w1
Relative Amplitude (9/4) a3 E13 (9/4) a3 E13 (3/4) a3 E13 (3/4) a3 E13
The third-order intermodulation produces signals at the input frequencies and two side bands on either side of
the input frequencies.
The coefficient a3 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 a3 coefficient. This suggests a relationship between third-order
intermodulation levels and single signal compression points (the 1 dB output gain compression level is typically
10 dB below the third-order output intercept point).
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