A General Algorithm to Calculate Third Order Intermodulation Product Locations for any Number of Tones  by Chris Arnott A web exclusive from CED Magazine Note: This paper used to be available on the CED (Communications Engineering & Design) website, but has been removed. So, I scanned the copy provided to me by the author, Chris Arnott, when we worked together at RFMD. I will remove the article at the request of CED.
Chris Arnott, RF Micro Devices Cable operators offering digital communication services on their systems provide customers with Internet access, digital video and business network solutions to add flexibility and profitability to their systems. A major system consideration for successful implementation of a modem digital cable system is system linearity. Inadequate system linearity distorts the channel information and can lead to low system operability or reliability. Amplifying components placed within the system for signal amplification or frequency conversation contribute to system distortion. All amplifiers and frequency conversion components exhibit nonlinear amplification and produce distortion, causing intermodulation products. This distortion corrupts the channels and can lead to high bit error rates. The problem is more severe in these wideband cable systems because each amplifying component input sees the entire highpower multichannel cable system spectrum. The number of distorting intermodulation products created by these inlineamplifying components is very large. Worse, many of these intermodulation products fall within the same channels and the distortion power accumulates with the number of products. This accumulated distorting power is the main reason why poor system linearity can cause low system reliability. Therefore, care must be taken when selecting amplification components for wideband cable systems in order to ensure adequate system linearity. This paper shows a general algorithm to calculate the number of distortion products created by a thirdorder nonlinearity in a wideband multichannel system. This article describes a general algorithm to calculate the frequency locations of thirdorder intermodulation distortion products produced by a broadband amplifier for any number of test tones. It also defines a broadband system with equally spaced channels. The analysis includes a calculation of total primary and intermodulation product signals produced by nonlinear thirdorder systems and a discussion of where the most important intermodulation distortion products lay. A practical example of a fourtone test is then performed on a standard IRC cable TV system with thirdorder nonlinearity. onlinearity of amplifiers in broadband applications greatly contributes to system performance degradation because of interfering distortion signals. Broadband systems containing many equalpower channels produce intermodulation distortion signals when amplified by line amplifiers and LNAs. Narrowband systems readily use simple twotone tests to analyze thirdorder intermodulation distortion [1]. Narrowband intermodulation analysis is simplified by ignoring distortion products not located within close proximity of the desired channel. These simplifications cannot be readily utilized in broadband system intermodulation distortion analysis many intermodulation products lie within the system bandwidth. Intermodulation distortion analysis is more difficult in broadband systems because all intermodulation products can interfere with many channels simultaneously Therefore, broadband systems require multitone tests to analyze thirdorder intermodulation distortion. ThirdOrder Broadband System Distortion Analysis Broadband systems are comprised of many equally spaced channels. A broadband system with equally spaced multichannels is described by equation 1, f_{i} = f_{1} + (i 1) · f_{ch} for i = 1,2,3,···, N + 1 Eq. 1 where i represents the channel number, f1 represents the beginning channel frequency, represents the number of channels, and fch represents the channel spacing. A nonlinear thirdorder distorting amplifier produces four types of tones: the primary; the third harmonic; the thirdorder intermodulation products; and frequency sum products. The terminology, "thirdorder intermodulation product" is traditionally used to define the important intermodulation products in narrowband analysis. This terminology, though technically incorrect, is used to define the same products in broadband intermodulation analysis and will be clarified later. The third harmonic, thirdorder products, and frequency sum products are interference signals that degrade desired channel reception. The amplifier passes all N primary tones and generates a total of N third harmonic distortion tones. The number of thirdorder intermodulation distortion products produced by the amplifier is given by equation 2. N_{3rd} = 2 · N · (N 1) for N ≥ 2 Eq. 2 The number of frequency sum distortion products generated by the amplifier is given by equation 3. N_{sum} = 2^{N1} for N ≥ 2 Eq.3 Combining the number of primary and third harmonic tones and equations 2 and 3, the total number of tones at the amplifier's output is given by equation 4. N_{TOT} = 2 · N + 2 · N ·(N1) + 2^{N1} for N ≥ 2 Eq.4 The total number of output tones increases dramatically as the number of channels increases, which indicates the importance of good system linearity because each distortion tone can potentially distort a channel. The frequency sum products are the largest number and strongest interfering distortion products and contribute more to desired signal degradation [23]. The power frequency sum distortion products are 6 dB higher than the thirdorder products [3]. The higher power and greater number is the reason the frequency sum products are considered the most important. The frequency location of intermodulation distortion interferers in a broadband system with thirdorder nonlinearity is investigated by applying N arbitrary channel frequency tones. Each tone is assumed to be of equal amplitude and have zero degree correlated phase. The N tones in an arbitrarily spaced Nchannel system are defined using equation 1 as f_{a0} = f_{1} + (a_{0}  1) · f_{ch} f_{an} = f_{1} + (a_{n} 1) · f_{ch } Eq.5 where an is a positive nonzero successively increasing sequence of integers given by Eq.6 Using a composite input signal formed by a summation of cosine functions with frequencies a broadband system intermodulation analysis can be performed. The nonlinear amplifier is simulated as a third degree monomial with a coefficient of one or gain of one. Cubing the composite function and using trigonometric identities the frequency locations of the output tones are found as four distinct sequences as given by, for the primary tones (see equation 7)
for the third harmonic tones (see equation 8), Eq.8 and thirdorder intermodulation products (see equation 9), {2 · f_{a0} ± f_{a}_{n}}  ^{N}  ^{m=0} 
{2 · f_{an} ± f_{a}_{m}}  ^{N}  for m ≠ n  ^{m=0} 
Eq.9 The definition of the thirdorder intermodulation given in equation 9 is the same as for tones used to describe intermodulation distortion in narrowband analysis. Three sequences describe the frequency sum products. The first and last sum products are unique where the first product location is given by equation 10,
Eq.10 and the last term is located at (see equation 11),
Eq. 11 The sequence for the frequency locations of the remaining sum products is given by (see equation 12),
Eq.12 where x = 0, 1, 2, · · ·, 2^{N1}  1, 2^{N1}  2, 2^{N1}  3, Eq. 13 and y = 0, 1, 2, · · ·, N  4, N  3, N  2 Eq.14 Exponent bxy is an element in Bxy that describes the base 2 binary digits of x + 1 for each x state with N 1 significant bits and Cx is the sum of the binary digits represented by row elements bxy. The sequence described in (12) locates the sum product frequency locations by negating coefficients a1  an for all possible x states. Matrix Bxy ~d vector Cx are found with numerical base 2 conversion techniques. First a calculation of an x by y matrix containing the quotients of state x divided by 2 in column 0 is given by equation 15.
Eq.15 The x by y matrix Bxy contains elements with the remainders of state x+ 1 divided by 2 in column 0 is given by equation 16, Eq.16 where the row elements represent base 2 binary digits of x+ 1 to Nl significant bits. The vector Cx is the summation of the row elements of Bxy as given by equation 17 Eq.17 As an example, a system exhibiting thirdorder nonlinearity is subjected to a fourtone test using equations 5 through 17. The four equally spaced tones are defined using equation 5 as f_{a0} = f_{1} + (a_{0}  1) · f_{ch} Eq. 18 f_{a1} = f_{1} + (a_{1}  1) · f_{ch} Eq.19 f_{a2} = f_{1} + (a_{2}  1) · f_{ch} Eq. 20 and f_{a3} = f_{1} + (a_{3}  1) · f_{ch} Eq.21 A total of 40 tones are generated by the nonlinear system: four primary tones; four thirdharmonic tones; 24 thirdorder intermodulation products (from equation 2); and eight frequency sum products (from equation 3). The primary tones, thirdharmonic and thirdorder intermodulation tones using equations 7, 8 and 9 are given by {f_{a0}, f_{a1}, f_{a2}, f_{a3}} Eq.22 {3 · f_{a0}, 3 · f_{a1}, 3 · f_{a2}, 3 · f_{a3}} Eq.23 and {2 · f_{a0} ± f_{a1}, 2 · f_{a0} ± f_{a2}, 2 · f_{a0} ± f_{a3}, 2 · f_{a1} ± f_{a0}, 2 · f_{a1} ± f_{a2}, 2 · f_{a1} ± f_{a3}, 2 · f_{a3} ± f_{a0}, 2 · f_{a3} ± f_{a1}, 2 · f_{a3} ± f_{a2}, } Eq.24 The sum product frequency locations for the fourtone test are found using equations 10 through 17. The first and last frequency sum product using equations 10 and 11 are given by 4 · f_{1} + (a_{0} + a_{1} + a_{2} + a_{3}) · f_{ch}  4 · f_{ch} Eq.25 and 2 · f_{1} + (a_{3} + a_{2} + a_{1}  a_{0}) · f_{ch}  2 · f_{ch} Eq.26 The first step in finding the remaining sum product frequency locations is calculating matrices qxy and Bxy with x and y having six states and three significant bits. Using equation 15, qxy and Bxy are given by q_{xy} =  0 0 0  1 0 0  1 0 0  2 1 0  2 1 0  3 1 0 
Eq.27 and B_{xy} =  1 0 0  0 1 0  1 1 0  0 0 1  1 0 1  0 1 1 
Eq.28 Using equation 17, the vector Cx is given by C_{x} =1 1 2 1 2 2 Eq.29 Knowing bxy and Cx the remaining sum product frequency location sequence using equation 12 is given by {2 · f_{1} + (a_{0}  a_{1} + a_{2} + a_{3}) · f_{ch}  2 · f_{ch} 2 · f_{1} + (a_{0} + a_{1}  a_{2} + a_{3}) · f_{ch}  2 · f_{ch} (a_{0}  a_{1}  a_{2} + a_{3}) · f_{ch} 2 · f_{1} + (a_{0} + a_{1} + a_{2}  a_{3}) · f_{ch}  2 · f_{ch} (a_{0}  a_{1} + a_{2}  a_{3}) · f_{ch} (a_{0} + a_{1}  a_{2}  a_{3}) · f_{ch}} Eq.30 An investigation of the frequency sum distortion products in a real broadband system is investigated by applying four arbitrary, successively increasing frequency tones to a system with thirdorder nonlinearity. The tones are assumed to have equal amplitude and a correlated phase of zero degrees. Phase correlation between the test tones causes correlation between the intermodulation products. This correlation between the intermodulation products causes products falling on the same channel to add as voltages and is considered worst case. Each intermodulation product falls within close proximity to a channel in systems with no correlation between tones and appears like a noise signal. This noise like signal is the sum root mean square power of each distortion product and causes less distortion compared to a correlated system. The beginning channel frequency fl is chosen as 121.25 MHz, standard IRC cable TV video carrier frequency with 6 MHz carrier spacing. The fourth tone is 18 MHz greater than fl. The bandwidth of the system for investigation of the distortion products is 624 MHz. Tones a0  a3 are chosen as four successively increasing frequencies for cases a0 equal 1 through 10. A result of (30) for an even number of tones is constant distortion product frequencies for products with an equal number of positive and negative an coefficients. These constant product frequencies can be ignored because they do not lie above the beginning channel. The frequency locations of the nonconstant frequency sum distortion products are shown in table 1. Each distortion sum product in table 1 lies within a channel in a typical cable TV system, and will interfere with desired channels. Interestingly, the frequency sum products in the last four columns fall in at the same frequency as secondorder intermodulation (composite second order) products at +1.25 MHz offset form the carrier. This indicates a system with four much higher power carriers relative to the remaining carriers can produce thirdorder products that appear as secondorder products. Conclusion Results show that broadband intermodulation distortion analysis cannot utilize assumptions used in narrowband system analysis. The ignored frequency sum products in narrowband analysis can distort channels in broadband systems. Results of equation 3 show the number of interfering sum products greatly increase as the number of channels increase in a broadband system with thirdorder nonlinearity. The algorithm correctly calculates the distortion product frequency locations for the general case of any number of test tones. Acknowledgements The author thanks Dr. Bruce Schmukler, Greg Schramm, and Jennifer Ameling of RF Micro Devices for many useful comments and discussions related to this article. References [1] T. H. Lee, The Design of CMOS RadioFrequency Integrated Circuits. Cambridge, U.K.: Cambridge Univ. Press 1998. [2] Some Notes on Composite Second and Third Order Intermodulation Distortions, Matrix Technical Notes MT 108, Matrix Test Incorporated, 12/1998 [3] The Relationship of Intercept Points and Composite Distortions, Matrix Technical Notes MTN109, Matrix Test Inc., 2/1998 Chris Arnott, RF Micro Devices 7628 Thorndike Rd. Greensboro, N.C. 27409 (336)9317375 carnott@rfmd.com NOTE: Chris is no longer with RFMD 
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