Intermodulation noise related to THD in dynamic nonlinear wide-band amplifiers

Abstract

In this work it is shown that the power of the intermodulation
noise of a wide-band amplifier with dynamic nonlinearities can be
estimated by the total harmonic distortion (THD) with a sinusoid input
signal of appropriate amplitude and frequency. The THD is, as opposed to
the intermodulation noise, easy to measure and use as a design parameter

Full text

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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: ANALOG AND DIGITAL SIGNAL PROCESSING, VOL. 45, NO. 7, JULY 1998 873
Fig. 7.
2 ?? ? ?????.
BER comparison of the DLMS and LMS algorithms over channel
performance was similar to the DLMS algorithm. In these cases the
reduced step size and more severe channel distortion resulted in very
little performance variation between the three algorithms.
In Fig. 7, the performance of the LMS and DLMS algorithms
are compared in terms of the bit-error rate (BER) over channel
2 for a (6,6) DFE (500 training symbols were allocated). Perfect
decision feedback and imperfect decision feedback were compared.
The performance loss in this case is only marginal, since the delay
in adaptation is not very large. This is expected since, for both
algorithms, convergence was reliably achieved within the 500-symbol
training period.
IV. CONCLUSION
In this paper pipelined transversal filter-based DFE’s employing the
DLMS training algorithm have been described. An order-recursive
DFE structure was developed which allows a DFE of arbitrary
length to be constructed by cascading a series of identical processing
modules. Alternative filtering structures were chosen for the FFF’s
and FBF’s in order to minimize the global communication. The per-
formance of the new DFE’s were compared using simulated channels
to introduce ISI and were found to be only marginally inferior to
those for the conventional DFE. However, the pipelined DFE’s more
than double the throughput rate of conventional structures and are
very suitable for VLSI implementation. A pipelined version of the
DNLMS algorithm was also proposed for a DFE, which removes the
dependency of the convergence speed on the input signal power.
ACKNOWLEDGMENT
The authors would like to thank the other members at the Centre
for Communications, University of Bristol, and the reviewers of this
paper for their helpful suggestions and for pointing out some of the
references.
REFERENCES
[1] A. Nix, M. Li, J. Marvill, T. Wilkinson, I. Johnson, and S. Barton,
“Modulation and equalization considerations for high performance radio
LAN’s (HIPERLAN),” in Proc. PIMRC, vol. 3, The Hague, The
Netherlands, Sept. 1994, pp. 964–968.
[2] S. P. Smith and H. C. Torng, “A fast inner product processor based
on equal alignments,” J. Parallel Distrib. Comput., vol. 2, no. 4, pp.
376–390, 1985.
[3] G. Long, F. Ling, and J. G. Proakis, “The LMS algorithm with delayed
coefficient adaptation,” IEEE Trans. Acoust., Speech, Signal Processing,
vol. 37, pp. 1397–1405, Sept. 1989.
[4] M. D. Meyer and D. P. Agrawal, “A high sampling rate delayed LMS
filter architecture,” IEEE Trans. Signal Processing, vol. 40, pp. 727–729,
Nov. 1993.
[5] J. G. Proakis, Digital Communications, 2nd ed.
1989.
[6] N. J. Bershad, “Analysis of the normalized LMS algorithm with Gauss-
ian inputs,” IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-
34, pp. 793–806, Aug. 1986.
[7] S. Haykin, Adaptive Filter Theory, 2nd ed.
Prentice-Hall, 1991.
[8] R. Perry, D. Bull, and A. Nix, “An adaptive DFE for high data rate
applications,” in IEEE Proc. Vehicular Technology Conf. (VTC 1996),
vol. 2, Atlanta, GA, Apr. 1996, pp. 686–690.
[9] J. Thomas, “Pipelined systolic architectures for DLMS adaptive filter-
ing,” J. VLSI Signal Processing, vol. 12, pp. 223–246, June 1996.
[10] H. Herzberg, R. Haimi-Cohen, and Y. Be’ery, “A systolic array realiza-
tion of an LMS adaptive filter and the effects of delayed adaptation,”
IEEE Trans. Signal Processing, vol. 40, pp. 2977–2803, Nov. 1992.
[11] H. Samueli, B. Daneshrad, B. C. Wang, and H. T. Nicholas, “A 64-
tap CMOS echo canceller/decision feedback equalizer for 2B1Q HDSL
transceivers,” IEEE J. Select Areas Commun., vol. 9, pp. 839–847, Aug.
1991.
[12] K. K. Parhi, C-Y. Wang, and A. P. Brown, “Synthesis of control circuits
in folded pipelined DSP architectures,” IEEE J. Solid-State Circuits, vol.
27, pp. 29–43, 1992.
[13] N. R. Shanbhag and K. K. Parhi, Pipelined Adaptive Digital Filters.
Norwell, MA: Kluwer, 1994.
[14] A. Gatherer and T. H.-Y. Meng, “A robust adaptive parallel DFE using
extended LMS,” IEEE Trans. Signal Processing, vol. 41, pp. 1000–1005,
Feb. 1993.
[15] A. P. Clark and S. F. Hau, “Adaptive adjustment of receiver for distorted
digital signals,” Proc. Inst. Elect. Eng., vol. 131, no. 5, pp. 526–536,
Aug. 1984.
New York: Macmillan,
Englewood Cliffs, NJ:
Intermodulation Noise Related to THD in
Dynamic Nonlinear Wide-Band Amplifiers
Henrik Sj¨ oland and Sven Mattisson
Abstract—In this brief it is shown that the power of the intermodulation
noise of a wide-band amplifier with dynamic nonlinearities can be
estimated by the total harmonic distortion (THD) with a sinusoid input
signal of appropriate amplitude and frequency. The THD is, as opposed
to the intermodulation noise, easy to measure and use as a design
parameter. This brief is an extension of our paper [1], which treats static
nonlinearities.
Index Terms—Distortion, intermodulation, wide-band amplifiers.
I. INTRODUCTION
In [1] it was shown that the intermodulation noise power due to
a static nonlinearity can be estimated by a total harmonic distortion
Manuscript received September 5, 1996; revised July 7, 1997. This paper
was recommended by Associate Editor V. Porra.
The authors are with the Department of Applied Electronics, Lund Univer-
sity, S-22100 Lund, Sweden (e-mail: hsd@tde.lth.se).
Publisher Item Identifier S 1057-7130(98)05062-9.
1057–7130/98$10.00  1998 IEEE
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874IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: ANALOG AND DIGITAL SIGNAL PROCESSING, VOL. 45, NO. 7, JULY 1998
(THD) measurement if the input amplitude is selected appropriately.
If the variance of the wide-band signal is ?, the input amplitude ?
is to be selected as
? ? ?????
(1)
Equation (1) was derived assuming a Gaussian input signal and sim-
ilar distortion contribution from even- and odd-order nonlinearities.
If odd-order nonlinearities dominate, the amplitude is to be selected
higher, and if even-orders dominate, it has to be lower. The variance
? is chosen to keep the clipping distortion power below the required
level [1].
In the derivation of (1) a novel approach was used. Using the
probability densities of a sinusoid and a Gaussian, the error (dis-
tortion) was calculated assuming that the amplifier characteristic
could be represented by a low-order polynomial. Clipping distortion
was treated separately. By comparing the distortion in the sinusoid
case with the Gaussian, (1) could be derived. This approach is now
augmented to allow dynamic nonlinearities. The result is that we can
also determine an appropriate frequency for the THD measurement.
The error of the amplifier output depends on the signal and its time
derivatives and their history. This is too complex to use directly. To
be able to handle each derivative by itself and ignore their history,
we average the error contribution from each time derivative.
Let ? be the input; the average output error is then given by
? ? ? ????? ? ?????? ? ??????? ? ???????? ? ???
where ????? models the static nonlinearity. Note that all ??functions
give mean errors. As in [1], we assume these mean-error functions
to be soft before clipping.
Amplifiers can be designed to be very linear at a certain frequency,
but highly nonlinear at other frequencies. Such amplifiers can not be
modeled well by (2) because they rely on cancellation of derivative
terms, thus making it inappropriate to handle each derivative by itself.
The required balance conditions are typically narrow-band and are
thus not used in wide-band amplifiers.
(2)
II. DERIVATIVES WITH GAUSSIAN AND SINUSOIDAL INPUT
Assume the input signal to be Gaussian with constant spectral
density ????? from zero up to ????. Above ????, we assume the
spectral density to be zero. If we take an ?th-order derivative of
the Gaussian input signal, it is also Gaussian. The spectral density
is given by
??
? ?????????????
(3)
The variances of the derivatives can now be calculated by inte-
grating the spectral densities
??
?
?
?
?
??
????????????????
??????????????
?
?
???
?? ? ?? ??
????????????
?? ? ?
?
(4)
If we take a derivative of a sinusoid, the result will also be a sinusoid
but with a different phase and amplitude. The phase does not affect
the distribution, so we just have to consider the amplitude, where the
amplitude of the ?th derivative is
?? ? ? ? ???????
(5)
We want to find the frequency ??, where the distortion due to the
?th derivative is similar for the sinusoid and the wide-band signal.
As the derivatives of a sinusoid are sinusoids and the derivatives
of a Gaussian signal are Gaussian, we can use (1) that relates the
distortion with a sinusoid input signal to that with a Gaussian one
?? ? ?????
? ?(4, 5)? ? ??????????
????????????
?? ? ?
? ??????
? ? ?????
? ???
????
??? ? ??
(6)
Equation (6) is the main result in this brief, and it gives the frequency
to use in the THD test. Depending on the order of the dominating
derivative, the frequency is to be selected differently. The amplitude
is determined by the static nonlinearity.
III. SLEW-RATE CLIPPING
Equation (1) and, hence, also (6), are only valid if the nonlinearity
??is soft. As in [1], we therefore handle clipping separately. Clipping
can occur in any derivative, but as slew-rate (SR) clipping dominates
in most cases, we concentrate on that. The static clipping has already
been examined in [1].
The maximum value of the time-derivative of the output is called
the SR. If the demanded derivative exceeds SR, there will be large
distortion called SR clipping [2].
Let ? be the derivative of the input signal normalized with SR
? ?
??
???
(7)
To estimate the SR clipping distortion power we use the integral
??? ???????
?
??
??
???????????????? ?
?
?
????????????????
? ?Gaussian, Symmetry?
?
? ?
?
?
??????
?
? ??? ??????????
(8)
If we compare (8) with [1, eq. (9)], which describes static clipping,
we see a similarity. The exponential function determines the order
of magnitude in both equations. The result is that if the function
??? ??????? is not much different from the polynomial of [1, eq.
(9)], ?? is to be selected approximately equal to ?? for the same
amount of SR clipping as static clipping distortion
??? ?????? ?? ? ??? ?? ?????
??
? ?? ? ?? ?????
??
?
(9)
Equation (9) gives approximately the required SR capability of a
wide-band amplifier when the input signal is Gaussian and has a
constant spectral density. The amplifier has to be capable of producing
a sinusoid with the maximum amplitude required at ???? divided by
the square root of three, without SR clipping. This is independent of
the required dynamic requirements of the amplifier, as the power of
the signal will be selected small enough to keep the static clipping
below the required level.
It remains to show that ??? ??????? behaves as stated. Fig. 1
illustrates a typical SR clipping scenario.
The error-voltage-time product due to the demanded SR ?? ? ???
for the time interval ?? centered around ? is
?????? ?? ? ?? ? ??? ? ?????
(10)
The average error voltage due to the demanded SR ? then becomes
??? ??????? ??????
??
? ?? ? ??? ? ???? ? ??? ???????
? ??? ????????? ?? ? ???? ???? ??????
(11)
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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: ANALOG AND DIGITAL SIGNAL PROCESSING, VOL. 45, NO. 7, JULY 1998 875
Fig. 1.An SR clipping scenario.
Fig. 2.Amplifier model for the numerical experiment.
??? ??????? behaves almost as the polynomial in [1, eq. (9)]. The
only difference is the ????????term that has a value of about one
and always between zero and four.
IV. RESULTS
The amplitude of the THD test is determined by the static nonlin-
earity [1], and the frequency is determined by (6). When the first-order
derivative (SR) dominates the dynamic distortion, the frequency is to
be selected as ???????. At this frequency, the amplifier must also be
able to reproduce a sinusoid of full amplitude without SR clipping (9).
The THD test gives an estimate of the intermodulation distortion
with a wide-band Gaussian input signal when both dynamic and static
nonlinearities can be significant.
A drawback with the method presented is that the amplifier
must have a small signal bandwidth much larger than the operating
bandwidth in order to get an accurate THD measurement. If this is
not the case, a two-tone test would be preferable. The two tones are
to be selected at frequencies close to the THD test frequency. To
avoid clipping we suggest the amplitudes to be ???. Trigonometric
equations [3] can be used to relate the intermodulation components of
the two-tone test to the harmonics of a THD test without bandwidth
reductions.
V. NUMERICAL EXPERIMENT
To validate the method we performed a numerical experiment in
MATLAB [4]. We used the amplifier model of Fig. 2, which models
the behavior of a typical dominant pole amplifier with feedback.
The parameters were selected so that the model should behave as
an audio power amplifier. The dominant pole was located at 1 kHz.
The direct current (dc) gain of the dominant-pole stage was 200 k?,
the transconductance of the input stage was 20 mS, the voltage gain
of the output stage was one, and ? was 1/20, resulting in a dc loop
gain of 200 and a 200-kHz bandwidth. The maximum current of the
input stage was ?2 mA, resulting in an SR of 2.5 V/?s referred to
the output. Before clipping, the nonlinearity of the input stage was
third-order compressive with a third-order intercept point of 288 mV
referred to ????. The maximum output voltage was set to ?20 V by
output stage clipping. Before clipping, the nonlinearity of the output
stage was also third-order compressive, but with an intercept point
of 56 V referred to ??.
The amplifier can just handle an output signal, at 20 kHz, of 20
V before SR clipping occurs, resulting in the SR clipping distortion
being smaller than the output stage clipping distortion for a wide-band
Gaussian signal with ???? up to ?? kHz ? ???? ? ???? kHz.
A Gaussian signal with constant spectral density between dc and
???? ? ?? kHz and ? ? ??? was generated and sent through the
amplifier model with the nonlinearities present and an identical model
without the nonlinearities. The power of the difference between the
outputs, which is equal to the power of the intermodulation noise
from the nonlinear amplifier, was then calculated.
A sinusoid with the amplitude ???? ? ???? V and the frequency
?? kHz????? ? ????? kHz was then generated and fed to the
amplifier model. The amplitude was selected higher than (1) because
the third-order nonlinearity dominates [1]. We got the THD figure by
using a fast Fourier transform (FFT) on the output signal.
To make the MATLAB program simple, we used Forward Euler
as integration method and generated an input signal with sufficiently
small time steps to make the integration numerically stable.
The distortion related to maximum amplitude was 0.041% for the
Gaussian signal and 0.074% for the sinusoid. To demonstrate the
importance of correct frequency, we also tested a sinusoid with the
same amplitude but different frequencies. The THD was 0.099% at
20 kHz and 0.014% at 2 kHz. This indicates that the test frequency
is approximately correct. The estimation of the intermodulation was,
in this case, pessimistic, but less than a factor of two too large.
VI. CONCLUSION
In this brief the statistical approach for estimating intermodulation
noise in static nonlinearities of [1] has been augmented to include
dynamic nonlinearities. The method results in a simple relation
between THD and intermodulation distortion, which was validated
by a numerical experiment. The static nonlinearity determines the
amplitude of the signal in the THD test, and the dynamic nonlinearity
determines the frequency.
REFERENCES
[1] H. Sj¨ oland and S. Mattisson, “Intermodulation noise related to THD
in wide-band amplifiers,” IEEE Trans. Circuits Syst. I, vol. 44, pp.
180–183, Feb. 1997.
[2] E. M. Cherry, “Transient intermodulation distortion—Part I: Hard non-
linearity,” IEEE Trans. Acoust, Speech, Signal Processing, vol. ASSP-
29, pp. 137–146, Apr. 1981.
[3] B. Westergren and L. R˚ ade, BETA Mathematics Handbook.
Sweden: Studentlitteratur, 1993.
[4] MATLAB Reference Guide, The MathWorks, Inc., Natick, MA, 1994.
Lund,
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