Spurious signals created by the presence of multiple carriers in a given bandwidth are a limiting factor in determining a signal channel's dynamic range. These spurious outputs due to system nonlinearities are the result of mixing of fundamental and harmonics of each of the signals (i.e., intermodulation interference). The easiest form of intermodulation interference to measure is two-tone, third-order intermodulation. This only requires the presence of two carriers at equal power levels closely spaced in frequency. The results of this measurement is used to determine the third-order intermodulation intercept point, a theoretical level used to calculate third-order intermodulation levels at any total power level significantly lower than the intercept point. When more than two signals are present in a single communications channel, the dominant interference is due to carrier-triple-beat (CTB) interference, which is the mixing of the fundamental of three carriers producing an interference signal in the same frequency band as the desired carriers. What follows is a review of the types of interference generated by multiple carriers and how to estimate their levels.

Third-Order Intercept Point

The third-order intercept point (Fig. 1) can be used to calculate carrier triple-beat interference for N carriers. In the limit as the number N gets large, multiple carriers exhibit a noise-like characteristic and can be described as the noise spectral density in a given resolution bandwidth. This suggests a practical alternative to measuring channel interference when a large number of carriers are active. Instead of loading the channel with signal sources, the communication channel is loaded with Gaussian noise covering the entire effective bandwidth. The interference is measured by placing a stopband filter in a similar resolution bandwidth and noting the resulting noise level in the stopband with respect to the noise in the passband. The factor is called the noise power ratio (NPR), a measure of communication-channel performance developed many years ago by the telephone companies and being resurrected today by satellite-communications systems to test the viability of multiple-carrier transmissions through a common satellite transponder.

If the total output power is kept constant, the relationship between the two-tone third-order intermodulation interference and the CTB interference for N signals is predictable. Since the results of NPR tests are similar to that of a system with a large number of carriers, these results can be extended to predict the NPR performance of a given system.

The following analysis assumes that all interference is caused by third-order intermodulation, which is usually dominant with higher-order intermodulation having secondary and tertiary effects. Therefore, the results presented are in general a good first-order approximation and should only be used as a convenient tool rather than a exact analytical result.

For system bandwidths less than an octave, even-order intermodulation products are out of band. Odd-order intermodulation products fall in-band, with sidebands close to the carrier. The level of interference is related to the system nonlinearity which can be defined by the theoretical intercept points for each higher-order nonlinearity (e.g., third-order intermodulation is defined by a third-order intercept point, fifth-order nonlinearity is defined by a fifth-order intercept point, etc.).

For simple (nonlinearized) systems, the third-order nonlinearity is usually the most prominent. In-band third-order distortion is the mixing of a fundamental of one signal and the second harmonic of another signal. The presence of more than two carriers in a nonlinear channel creates a spurious response, consisting of the mixing of three fundamental carriers, i.e., CTB. These spurious CTB signals fall in-band at a level 6 dB higher than two-tone, intermodulation products because there are no second harmonics involved in the production of the interference signal. The level of CTB interference is further enhanced by multiple CTB signals occurring in the same frequency band. Figure 2 shows three fundamental signals and the resultant carrier triple beats. The interference signals each down −30 dBc, are products of the three carriers at frequencies, W1, W2, and W3.

Third-order intermodulation interference is caused when two signals are present in the same non-linear communications channel. Measurement of this interference at a carrier signal level below the system compression levels gives the design engineer the ability to calculate the third-order intercept point. This, in turn, allows a prediction of intermodulation interference at any level below the system compression levels. This information can also be used to calculate the worst-case CTB performance for any number (N) of signals in the channel.

The relationship between third-order intercept point, carrier level, and third-order intermodulation interference (Fig. 3) is as follows:

dBc3rd = −2(I_{3rd} − Carrier)

where:

dBc3rd = the third-order intermodulation level with respect to a single carrier (dBc),

Carrier = the single carrier output level (dBm), and

I_{3rd} = the third-order intercept point (dBm).

The total output power, P_{tot} (in dBm), assuming both carriers have equal levels, can be written as:

P_{tot} = Carrier + 10log(2) = Carrier + 3 dB

Conversely, the individual carrier power is:

Carrier = P_{tot} − 3 dB

Solving for the third-order intermodulation interference (dBc3rd) in terms of total output power yields:

dBc3rd = −2(I_{3rd} − P_{tot} + 3 dB)

= −2(I_{3rd} − P_{tot}) − 6 dB

The third-order intercept point in terms of total power and third-order intermodulation is:

I_{3rd} = P_{tot} − 3 dB − (dBc3rd/2)

For convenience, Table 1 offers interference level (dBc) versus total power below the third-order Intercept point for two tones.

The carrier-triple-beat (CTB) interference level for three carriers (CTB_{3}) is similar to two-tone intermodulation except that a factor of 6 dB is added to account for the fact that no second-harmonic content is needed to create the in-band interference:

CTB_{3} = −2(I_{3rd} − Carrier ) + 6 dB

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Unlike two-tone intermodulation, CTB signals can overlap (Fig. 2), adding noncoherently. When more than three carriers are present in a single channel, the total interference is related to the number of carriers and the spectral position of the carrier. Carriers in the center of the band exhibit the worst-case interference levels, i.e., the highest number of beat frequencies in a given bandwidth (channel). The number of interference carriers (Beats) in any channel for equally spaced carriers is as follows:

Beats = N^{2}/4 + /2

where:

N = the number of carriers, and

M = the carrier position in the channel, 1 ≤ M ≤ N (typically M = 1 is the lowest frequency carrier).

The maximum interference occurs in the center of the band (M ~ N/2) where there is the maximum number of beat signals. For N >> 1 the beats (Beat_{max}) in the center of the band is:

Beat_{max} = 3N^{2}/8

Beat_{max} = the number of noncoherent interference carriers in the same channel.

The total CTB level is determined by calculating the level of each CTB and adding non-coherently the number of beat signals that will fall into the respective band:

CTB = −2(I_{3rd} − Carrier ) + 6 dB + 10log(Beats)

The 6 dB is added because the intermodulation is due to the mixing of three fundamental carriers (there are no second harmonics present in the mixing process), and Beats is given as:

Beats = N^{2}/4 + /2

and the total CTB interference is given as:

CTB = −2(I_{3rd} − Carrier) + 6 + 10log{(N^{2}/4) + /2}

The worst-case interference occurs in the center of the band (M = N/2). The total power, assuming all of the carriers have equal power, can be written as:

P_{tot} = Carrier + 10log(N)

where:

P_{tot} = the total output power for N carriers are of equal amplitude (dBm).

Solving for the individual carrier power yields:

Carrier = P_{tot} − 10log(N)

Substituting total output power for single carrier power and solving for the total CTB gives the following results:

CTB = −2*3rd − {Ptot −10log(N)}> + 6 +10log{N ^{2}/4 + /2} and *

CTB = −2*3rd − Ptot> −20log(N) + 6 +20log{N ^{2}/4 + /2}.*

Worst-case CTB interference (M ~ N/2) is plotted in Table 2 and shown in Fig. 4 as a function of total power back-off from the third-order intercept point for various numbers of carriers (N). It is obvious that the CTB interference increases 2 dB for every 1-dB increase in power. For N >> 1 the maximum interference in the center of the band with all carriers of equal amplitude can be solved:

CTB = −2(I_{3rd} − P_{tot}) −20log(N) + 6 + 10log(3N^{2}/8)

Factoring out the N^{2} term,

CTB = −2(I_{3rd} − P_{tot}) − 20log(N) + 6 + 10log(3/8) + 10log(N^{2})

CTB = −2(I_{3rd} − P_{tot}) − 20log(N) + 6 + 10log(3/8) + 20log(N)

The 20logN terms cancel, leaving a simplified equation for CTB (in dBc):

CTB = −2(I_{3rd} − P_{tot}) + 6 +10log(3/8)

Evaluating 10log(3/8) results in the following equation, relating CTB interference, third-order intercept point, and total output power for interference in the center of the band and the number of carriers much greater than one (N >> 1):

CTB = −2(I_{3rd} − P_{tot}) + 1.74 dB

The equations for two-tone intermodulation interference and CTB interference are related and a minor amount of manipulation results in a direct relationship between the expected respective interference for equal total output powers. This is significant because a relatively simple measurement (two-tone, third-order intermodulation) can be used to predict performance for a more complex multiple carrier system. These results although valid are only approximations of the actual performance. In deriving this relationship, the channel passband is assumed ideal and higher-order intermodulation effects must be ignored.

The equation for two-tone (N = 2) third-order intermodulation interference was given previously as:

dBc3rd = −2( I_{3rd} − P_{tot}) − 6 dB

Rearranging the equation results in:

dBc3rd + 6 dB = −2(I_{3rd} − P_{tot})

Using the CTB equation (for multiple tones, with N >> 1)

CTB = (−2) (I_{3rd} − P_{tot}) + 1.74 dB

and substituting (dBc_{3rd} + 6 dB) from the two-carrier equation for −2(I_{3rd} − P_{tot}) in the CTB equation results in:

CTB = dBc3rd + 6 dB + 1.74 dB

Combining terms yields:

CTB = dBc3rd + 7.7 dB

which directly relates the results of the two-tone intermodulation test results with the expected results for N carriers and is valid under the following conditions:

- The total output power is the same in both measurements.
- The number of carriers is N >> 1.
- Higher-order effects are not significant.
- The channel characteristics are the same for all frequencies of interest.
- The level of each carrier is equal.

As N gets large and approaches infinity, the individual carriers behave as a noise spectral density function and the CTB interference exhibits the first approximation of the results obtained from a NPR test. Extrapolating from this, a first-order approximation of NPR can be obtained from two-tone, third-order intermodulation data, whereas the NPR is approximately 7.7 dB above the result obtained from a two-tone, third-order intermodulation test. A word of caution must be inserted to prevent this equation from becoming any more than a rough approximation. The two-tone, third-order intermodulation measurement is made in a relatively small frequency band while the NPR can apply over a considerably larger bandwidth with variable band characteristics making this approximation even less accurate than expected.

When the total output power is equal, the interference level with respect to the carrier for multiple signals N, where N >> 1, is 7.7 dB greater than the interference level found for two-tone, third-order intermodulation (Fig. 5). Although this is a convenient closed form, in actual practice this is only a first-order approximation of the expected results. Not considered, but usually not necessary for a first-order approximation, are the effects of higher-order intermodulation, nonideal signal-channel characteristics, and the effects of signal modulation (the current analysis was based on CW signals).

REFERENCES

- Michael Leffel, "Intermodulation Distortion in a Mult-Signal Environment,"
*RF Design*, June 1995. - Joseph B. Waltrich, "Compute CTB in Hybrid Fiber/Coax Systems,"
*Microwaves & RF*, December 1998. - Oleksandr Gorbachov, "Determine IMD Products For Digital Communication Standards,"
*Microwaves & RF*, August 2000. - Steve Winder, "Single Tone Intermodulation Testing,"
*RF Design*, December 1993. - Philip M. Lally, "Determining Power Requirements for Multi-Signal Amps,"
*Microwaves & RF*, September 1994. - John H. Jacobi, "IMD: Still Unclear After 20 Years,"
*Microwaves & RF*, November 1986. - Nubar Ayrandjian, "Simple Computation of Spurious-Free Dynamic Range,"
*RF Design*, January 1987. - Manfred Bartz, "Designing Effective Two-Tone Intermodulation Distortion Test System,"
*RF Design*, November 1987.