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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