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),
and thirdorder intermodulation products (see equation 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, and the last term is located at (see equation 11), The sequence for the frequency locations of the remaining sum products is given by (see equation 12), where x = 0, 1, 2, · · ·, 2^{N1}  1, 2^{N1}  2, 2^{N1}  3, and y = 0, 1, 2, · · ·, N  4, N  3, N  2 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. 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,
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
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} f_{a1} = f_{1} + (a_{1}  1) · f_{ch} f_{a2} = f_{1} + (a_{2}  1) · f_{ch} and f_{a3} = f_{1} + (a_{3}  1) · f_{ch} 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}} {3 · f_{a0}, 3 · f_{a1}, 3 · f_{a2}, 3 · f_{a3}} and {2 · f_{a0} ± f_{a1}, 2 · f_{a0} ± f_{a2}, 2 · f_{a0} ± f_{a3}, 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} and 2 · f_{1} + (a_{3} + a_{2} + a_{1}  a_{0}) · f_{ch}  2 · f_{ch} 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
and
Using equation 17, the vector Cx is given by C_{x} =1 1 2 1 2 2 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} 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. NOTE: Chris is no longer with RFMD


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