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
non-linear 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 high-power
multichannel cable system spectrum.
The number of distorting intermodulation products created by these inline-amplifying 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 third-order non-linearity in a wideband multichannel system.
This article describes a general algorithm to calculate the frequency locations of third-order
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 non-linear
third-order systems and a discussion of where the most important intermodulation distortion
products lay. A practical example of a four-tone test is then performed on a standard IRC
cable TV system with third-order non-linearity.
on-linearity of amplifiers in broadband applications greatly contributes to system performance
degradation because of interfering distortion signals. Broadband systems containing many equal-power
channels produce intermodulation distortion signals when amplified by line amplifiers and
LNAs.
Narrowband systems readily use simple two-tone tests to analyze third-order 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 multi-tone tests to analyze third-order
intermodulation distortion.
Third-Order Broadband System Distortion Analysis
Broadband systems are comprised of many equally spaced channels.
A broadband system with equally spaced multi-channels is described by equation 1,
fi = f1 + (i -1) · fch 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 non-linear third-order distorting
amplifier produces four types of tones: the primary; the third harmonic; the third-order intermodulation
products; and frequency sum products.
The terminology, "third-order 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, third-order 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 third-order
intermodulation distortion products produced by the amplifier is given by equation 2.
N3rd = 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.
Nsum = 2N-1 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.
NTOT = 2 · N + 2 · N ·(N-1) + 2N-1 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 third-order 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 third-order non-linearity 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 N-channel system are defined using equation 1 as
fa0 = f1 + (a0 - 1) · fch
fan = f1 + (an -1) · fch
Eq.5
where an is a positive non-zero 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 non-linear 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 third-order intermodulation products (see equation 9),
{2 · fan ± fam} |
N |
for m ≠ n |
m=0 |
Eq.9
The definition of the third-order 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, · · ·, 2N-1 - 1, 2N-1 - 2, 2N-1
- 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 N-l 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 third-order non-linearity is subjected to a four-tone
test using equations 5 through 17. The four equally spaced tones are defined using equation
5 as
fa0
= f1 + (a0 - 1) · fch
Eq. 18
fa1
= f1 + (a1 - 1) · fch
Eq.19
fa2
= f1 + (a2 - 1) · fch
Eq. 20
and
fa3
= f1 + (a3 - 1) · fch
Eq.21
A total of 40 tones are generated by the non-linear system: four primary tones; four third-harmonic
tones; 24 third-order intermodulation products (from equation 2); and eight frequency sum
products (from equation 3). The primary tones, third-harmonic and third-order intermodulation
tones using equations 7, 8 and 9 are given by
{fa0, fa1, fa2, fa3}
Eq.22
{3 · fa0, 3 · fa1, 3 · fa2, 3 · fa3}
Eq.23
and
{2 · fa0 ± fa1, 2 · fa0 ± fa2, 2 · fa0 ± fa3, 2 · fa1
± fa0, 2 · fa1
± fa2, 2 · fa1
± fa3, 2 · fa3
± fa0, 2 · fa3
± fa1, 2 · fa3
± fa2, }
Eq.24
The sum product frequency locations for the four-tone test are found using equations 10
through 17. The first and last frequency sum product using equations 10 and 11 are given by
4 · f1 + (a0 + a1 + a2 + a3)
· fch - 4 · fch
Eq.25
and
2 · f1 + (a3 + a2 + a1 - a0)
· fch - 2 · fch
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
qxy = |
0 0 0 |
1 0 0 |
1 0 0 |
2 1 0 |
2 1 0 |
3 1 0 |
Eq.27
and
Bxy = |
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
Cx =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 · f1 + (a0 - a1 + a2 +
a3) · fch - 2 · fch 2 · f1 + (a0
+ a1 - a2 + a3) · fch - 2 · fch
(a0 - a1 - a2 + a3) · fch
2 · f1 + (a0 + a1 + a2 - a3) · fch
- 2 · fch (a0 - a1 + a2 - a3)
· fch (a0 + a1 - a2 - a3)
· fch}
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 third-order non-linearity. 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 non-constant 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 second-order 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 third-order products that appear as second-order 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
third-order non-linearity. 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 Radio-Frequency 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 MTN-109, Matrix Test Inc., 2/1998 Chris Arnott, RF
Micro Devices 7628 Thorndike Rd. Greensboro, N.C. 27409
(336)-931-7375 carnott@rfmd.com
NOTE: Chris is no longer with RFMD
1 |
2 |
3 |
4 |
121.25 |
6 |
521 |
242.5 |
254.5 |
266.5 |
278.5 |
2 |
3 |
4 |
5 |
|
|
545 |
254.5 |
266.5 |
278.5 |
290.5 |
3 |
4 |
5 |
6 |
|
|
569 |
266.5 |
278.5 |
290.5 |
302.5 |
4 |
5 |
6 |
7 |
|
|
593 |
278.5 |
290.5 |
302.5 |
314.5 |
5 |
6 |
7 |
8 |
|
|
617 |
290.5 |
302.5 |
314.5 |
326.5 |
6 |
7 |
8 |
9 |
|
|
641 |
302.5 |
314.5 |
326.5 |
338.5 |
7 |
8 |
9 |
10 |
|
|
665 |
314.5 |
326.5 |
338.5 |
350.5 |
8 |
9 |
10 |
11 |
|
|
689 |
326.5 |
338.5 |
350.5 |
362.5 |
9 |
10 |
11 |
12 |
|
|
713 |
338.5 |
350.5 |
362.5 |
374.5 |
10 |
11 |
12 |
13 |
|
|
737 |
350.5 |
362.5 |
374.5 |
386.5 |
|