Harmonic Analysis of Nonlinear Devices for Virtual Bass System

Audio Engineering Society
Convention Paper

Presented at the 124th Convention
2008 May 17–20
Amsterdam, The Netherlands
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Journal of the Audio Engineering Society.
Harmonic and Intermodulation Analysis
of Nonlinear Devices Used in Virtual
Bass Systems
NAY OO, AES Student Member and WOON-SENG GAN, AES Member
DSP Lab, School of EEE, Nanyang Technological University, Singapore
{e070001,ewsgan}@ntu.edu.sg




ABSTRACT
Nonlinear devices (NLD) are used in the virtual bass system. NLD generates harmonics which in turn
create the pitch perception and are used in the audio bass enhancement systems using psychoacoustics. This
paper presents the mathematical derivations and analyses of five different NLD devices, together with
intermodulation analysis of harmonics generated by these NLDs. The five NLDs are half-wave rectifier,
full-wave rectifier, square wave, polynomial function and exponential function. The derivation of harmonic
analysis equations are based on Fourier Theorems, Chebyshev Polynomials, and Taylor Series expansions.
Besides the harmonics, intermodulation components are also resulted from NLDs. Both mathematical
analysis and simulation results are presented for the intermodulation effects of harmonics generated by
NLDs.

reproduction bandwidths of these devices are very
limited. Normally, bass frequencies are below 250 Hz,
1. INTRODUCTION
and the small loudspeakers’ or low-quality headphones’
cut-off frequencies are higher than 250 Hz. The audio
Small loudspeakers, embedded in the consumer portable
frequencies below the cut-off frequencies cannot be
electronic devices, cannot reproduce rich bass reproduced or attenuated severely.
frequencies, because of psychical size limitation.
Similarly, low-quality headphones also have very poor
bass frequency response. Therefore, the audio

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Nay Oo and Woon-Seng Gan
Harmonics and IM distortion analysis of NLDs

To extend the low-frequency audio bandwidth, without
using dynamical systems which have memory or static
pushing the physical limit of audio reproduction memoryless nonlinearities [22]. The five NLDs
devices, we can make use of a psychoacoustic presented in this paper are static memoryless nonlinear
phenomenon, called the “Missing Fundamental” [10-
systems and can be constructed as a parallel connection
12]. The “Missing Fundamental” phenomenon states of different orders of weighted nonlinearities, as shown
that human can perceive the virtual pitch at the in Figure 1.
fundamental frequency when the harmonics are present,
even if the fundamental frequency itself is removed or
Sy
Linear
stem
not present. This phenomenon can be used to create the
virtual bass system or low frequency psychoacoustic
h1
audio bandwidth extension systems [1-9].
1
(x)
This topic has been well-researched by various
researchers, both in academic and industry. A variety of
h2
bass enhancement systems using psychoacoustics have
2
(x)
been implemented, and reported in the literatures [1-9].
Basically, there are two approaches to implement the
x
h
y
1
psychoacoustic bandwidth extension system or virtual
3
(x)
bass system, such as time-domain approach and
frequency-domain approach [21]. In time-domain
...
approach, NLD is used as a central processing block to
hn
generate the harmonics based on incoming audio signal.
By generating harmonics, the virtual bass system can
n
(x)
create virtual pitch at the bass frequencies, which our
human ear can perceive, even though those frequencies
Stati
c Non - linear Sy

stems
may not be psychically present [1-9].

Figure 1: A block diagram showing a polynomial NLD
can be constructed using different orders of
Since NLD is used in the central processing block of the
nonlinearities, connected in parallel.
time-domain approach virtual bass system, five types of

NLDs have been studied in this paper. Harmonic The usefulness of nonlinear system for the virtual bass
analysis using single tone as an input and system is that it can generate new frequencies, which is
intermodulation distortion analysis using logarithmic not possible for linear system. The nonlinear system
multitones as inputs are also investigated. Virtual bass
generates new frequencies as harmonics, that consist of
system research is an interdisciplinary research, desired components to enhance the bass in virtual bass
containing psychoacoustics, non-linear system theory system, and intermodulation components that are
and signal processing. In this paper, we attempted to
undesired components and cause unpleasant distortion.
relate these three fields to understand the nature of To understand the nature of harmonic generation and
NLDs for virtual bass system. Therefore, in Section 2,
intermodulation contamination, we can use two tests.
we present the harmonic and intermodulation distortion
The first test is the single tone test and the second test is
analysis of static memoryless nonlinearities up to sixth
the multitones test. Subsequent sections present the
order polynomial. These polynomials can in turn be
single tone harmonic analysis, multitones harmonic and
viewed as a parallel connection of static memoryless
intermodulation component analysis, and four
nonlinearities as shown in Figure 1 [22-23]. Section 3
measurement metrics to perform objective comparisons
presents the detailed analysis of five types of NLDs,
among the NLDs. All the results presented in this paper
including harmonic analysis and intermodulation are obtained using MATLAB.
distortion analysis. Section 4 discusses the NLD
simulation results. Section 5 concludes this paper.
2.1. Single Tone Harmonic Analysis
2.
STATIC MEMORYLESS NONLINEARITIES
A single sine tone with an adjustable amplitude is fed
into the NLD (or nonlinearity) under investigation, as
NLD is a nonlinear device which can be implemented
shown in Figure 2. The output harmonics amplitudes are
either digital or analog means. It can be constructed
measured using Discrete Fourier Transform (DFT) or
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Harmonics and IM distortion analysis of NLDs

respective derived mathematical formulas, such as Fourier coefficients can then be computed from
Schaefer-Suen equations.
polynomial series coefficients. Base on this idea,
Generated Ha

rmonics
Schaefer derived a generalized equation which produces
A
Fourier series (2) coefficients{c , c , c , c ,...} from the
ST
1
c
0
1
2
3
2
c
polynomial series coefficients{h , h , h , h ,...} . This
3
c
0
1
2
3
c4
c
equation is powerful in the sense that it can calculate the
L n
n
x
magnitudes of harmonics produced by any order of
frequency
frequency
static nonlinearities individually, as well as any order of
polynomial based NLD, excited by a single tone. The
a stat

ic n
onlinear sy
stem
Schaefer’s equation for a single tone harmonic analysis

(
nonline it
ar y)
is shown as follows:
Figure 2: A diagram showing the single tone input

∞
harmonic analysis of a static nonlinear system.
1
hk 2 j ⎛ k + 2 j ⎞
c =
, (6)
k
∑ +
k 1
2 j
⎜
⎜

−
⎟
⎟
2
j=0 2
⎝
j
⎠
As described in previous section, any static memoryless

nonlinear system can be approximated by polynomial
where binomial coefficients are
n
series. Referring to Figure 1, nonlinearity x is a sub

block of polynomial NLD. The polynomial based NLD
⎛k + 2 j ⎞
(k + 2 j)!
=
single tone analysis is based on the works by R. A.
⎜
⎜
⎟
⎟
. (7)
j
⎝
⎠ (k + j (
)! j )
!
Schaefer [14] and C. Y. Suen [24]. Schaefer derived the
mathematical relationship between polynomial power
series (1) and Fourier series (2) using a relationship
However, Schaefer’s original equation has one
from Chebyshev polynomial of the first kind [15].
limitation that it fixes the input tone to unity. To

overcome this limitation, C. Y. Suen [24] derived a
generalized harmonic analysis equation with varying
y = f (x) = h + h x +
2
h x +
3
h x +
(1)
0
1
2
3
K
single tone amplitude, A as a parameter using a different
approach. Suen’s equation took into consideration of
y = 1 c + c cosθ + c cos θ
2 + c cos θ
3 +
, (2)
0
1
2
3
K
2
the adjustable input amplitude tone. By comparing the

two equations and rearranging the terms, we arrive at
where y and x denotes the output and input of the NLD,
the Schaefer-Suen equation for the generalized
respectively. A single tone cosine wave is fed into the
harmonic equation as
NLD for the harmonic analysis. This can be expressed

∞
mathematically as follows.
1
h
k
j
k +
2
2 j
k +2 j ⎛
+
⎞
c =
∑ A
×
. (8)
k 1
2 j
⎜
⎜

k
−
⎟
⎟
2
j=0
2
⎝
j
⎠
x = cosθ , θ = 2 ft
π , (3)


With (8), we can adjust the amplitude of the input
where f is the frequency in Hz, and t is the time in tone, A and calculate the generated harmonic amplitudes
seconds. If cosθ is fed into the NLD in (1), output directly from polynomial coefficients, h ’s. The infinite
k
becomes
series in (8) is convergent, and for the nth order

nonlinearity, we only need to calculate up
y = f (x)
to k = n or ck= . This equation is powerful in the sense
n
= f (cosθ )
that without going through DFT, we can calculate the
generated harmonics amplitudes using simple algebraic
= h + h cosθ +
2
h cos θ +
3
h cos θ +
(4)
0
1
2
3
K
formulas. Using (8), we can also count the total number

of harmonics generated by a particular nonlinearity and
Equation (2) and (4) can be related by Chebyshev list the results in Table 1 as follows.
polynomials of the first kind:



T (cosθ ) = cos θ
k . (5)
k


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Harmonics and IM distortion analysis of NLDs

Table 1. Relationship between harmonic numbers, total
Amplitude
Amplitude
number of harmonics and order of nonlinearity.
n
DC and Harmonics Numbers
Total number of
Harmonic (H

)
T
T
harmonics, H
1
2
NLD
L
2
DC, 2
1
3 1,3
2
f f
1
2
f − f
2 f
2
1
1
4
DC, 2, 4
2
Intermodu io
t n components (I
M)

5 1,3,5
3
Figure 3: A diagram showing the two input tones
6
DC, 2, 4, 6
3
intermodulation analysis.
7 1,3,5,7
4
8
DC, 2, 4, 6, 8
4
In virtual bass system, the input signal to the NLD are
9 1,3,5,7,9
5
low-pass filtered [1-6]. The cut-off frequency of the
10
DC, 2, 4, 6, 8, 10
5
low-pass filter is near the loudspeaker cut-off frequency

or resonance frequency. Therefore, the input signal
From Table (1), the following observations are made.
spectrum to the NLD is assumed to be less than 200 Hz

(audio bass frequencies). From 20 Hz to 200 Hz, we
• The number of harmonics produced by the calculate five logarithmic tones as follows [22]. We use
nonlinearity increases when the order of logarithmic-multitones stimulus as input because
nonlinearity increases.
previous research findings by others showed that only
• The odd order nonlinearity can produce only odd logarithmically equal space multitones can generate
harmonics, and the even order nonlinearity can more intermodulation distortion components of the
produce only even order harmonics.
device under test [22]. The calculation of the
• The odd order nonlinearity always reproduces the frequencies of five logarithmic multitones is listed in
fundamental which is the first harmonic number.
Table 2.
• The even order nonlinearity always produces DC
component.
Table 2. Five logarithmic multitones calculations from
• The maximum harmonic number is always equal to
20 Hz to 200 Hz.
the order of nonlinearity.
• For the total number of harmonics generated, H can No. Calculation Rounded
be formulated as follows:

n
1
20 Hz
20 Hz
H =

2
( n is even), (9)
2
20 ×
−
log 1 1
( 4) = 35.565 Hz
36 Hz
+1
= n
H
( n is odd). (10)
2
3
.
35 565 ×
−
log 1 1
(
)
4 =
.
63 246 Hz
63 Hz
2.2. Multitones Harmonic Analysis
4
63 246
.
×
−
log 1 1
(
)
4 = 112 468
.
Hz
112 Hz
In a virtual bass system, the generated harmonics are
desired frequency components which create virtual bass
5
1
×
−
=
perception in the human auditory system, whereas
112.468 log
1
( 4)
200 Hz
200 Hz
generated intermodulation components are heard as
audio distortion. The intermodulation components are
To simulate the intermodulation effect, the
formed by addition or subtraction of two or more input
“interharmonic analysis formula” for nonlinear system
frequency components, as shown in Figure 3.
identification was modified to suit the two tones
harmonic analysis [20]. X ( ω
j ) denotes two input tones,
and Y ( jω) denotes the nonlinearity output components
of DC, fundamental frequencies, harmonics of even and
odd order and intermodulation components. They can be
expressed as
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Nay Oo and Woon-Seng Gan
Harmonics and IM distortion analysis of NLDs

10
algorithm is used. A MATLAB command, sortrows, is
X ( jω
φ
) = ∑
j (k )
(
A k )e
δ (ω − ω (k)) (11)
0
used in our simulation. The third stage, Harmonic and
k =1
Intermodulation Components Separation decouples
harmonics and intermodulation frequencies and put into
with
the separate vectors. By this stage, the harmonic
A 5
( + l) =
(
A l) ,
components and intermodulation distortion components
ω
can be found. The final stage, Frequency Overlapping
5
( + l) = ω
− (l) , (12)
0
0
overlaps the redundant frequencies. At the end of the
φ 5
( + l) = φ
− (l) ,
process, we can obtain two vectors with harmonics and
intermodulation components separately.
l =
,
1
{
,
3
,
2
}
5
,
4
,
Table 3. Frequency combinations and order of
where A , ω and φ are amplitude, frequency and phase.
0
nonlinearity for the first stage of Figure 4.
The output of the second order nonlinearity can be
expressed as
Nonlinearity
Total Number of Frequency
Order, n
Combinations, C
10 10
φ
φ
Y ( jω) = ∑ ∑
j( (n)+ (m))
[ (
A n) (
A m)]e
× (13)
2
100
n=1m=1
3 1000
δ (ω − [ω (n) + ω (m)]).
4 10000
0
0
5 100000
The output of the third order nonlinearity can also be
6 1000000
expressed as

10 10 10
Simulation F
ramework
j(φ (n) φ
+ (m) φ(k))
Y ( jω
+
) = ∑ ∑ ∑[ (
A n) (
A m) (
A k)]e
(14)
n 1
= m 1
= k 1
=
Two
Frequenc s
ie
δ (ω − [ω (n) + ω (m) + ω (k)]) .
0
0
0
Input
comb
n
inatio
Extending the expansion equation, the frequency
Tones
combination equation for an nth order nonlinearity can
be constructed with n nested summations. A simulation
was performed up to 6th order nonlinearity. The
Sorting
simulation framework is shown in Figure 4 which can
be used for simulating any nonlinearity block in Figure
1. The results of individual nonlinear system simulation
Harmonics
can be combined together to construct the NLDs
IM

and

simulation. As for this later case, individual
Separation
nonlinearities are simulated first and the resultant
harmonics are combined to get the overall system
response. For the first stage, Frequency Combination,
Frequenci s
e
Ou
ma
tput
gnitues
the total number of frequency contributions, C [20] can
Overlappi g
n
(H + IM) co
mponents
be formulated as
in vectors

n
C = (2F ) , (15)
Figure 4: Simulation framework for multitones
where F is the number of multitones inputs and n is the
harmonic and intermodulation components analysis
order of nonlinearity. In this paper, five logarithmic
tones (F = )
5 , as shown in Table 2, are used. The The simulation framework in Figure 4 is for a single
frequency combinations for the first stage can be nonlinearity of order, n . As for the NLD case, where
calculated, as described in Table 3. The second stage,
there are multiple order of different nonlinearities up to
Sorting, sorts the generated frequencies in ascending
n = 6 , weighted and connected in parallel, as shown in
order. At this stage, frequencies are not overlapping yet.
Figure 1. Therefore, we can link up the simulation
This stage can take up a long time if not an efficient
framework for individual nonlinearity, add the vectors
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Harmonics and IM distortion analysis of NLDs

of harmonics and IM components and go through again
2.3.3. Harmonic to Multitones Ratio ( Δ )
H
Sorting and Frequency Overlapping stage for overall
NLD. By doing this, we can get the harmonics and
Harmonic to Multitones Ratio ( Δ ) measures the ratio
intermodulation components of polynomial
H
approximated NLDs which are described in Section 4.
between the power of generated harmonics and the
summation of the power of input multitones.
Δ
2.3. Harmonic Richness and Intermodulation
The
formula for multitones analysis is given as
H
Distortion Measurement Metrics
∑L (H 2
)
k =1
k
2.3.1. Total Harmonic Richness ( THR )
Δ =
, (18)
H
∑N T 2
( )
i=1
i
In this paper, we define a term known as the total
harmonic richness ( THR ) which is the ratio between the
where ∑N
2
( ) is the summation of the power of input
i=1
i
T
powers of NLD generated harmonics (from 1st to 6th
order) to the power of the fundamental input tone. We
multitones and N is the number of multitones.
include only the first six harmonics due to the findings
from the region of dominance in psychoacoustic pitch
2.3.4. IM Distortion to Multitones Ratio ( Δ
)
IM
perception researches [8-10, 17-19]. The THR formula
for single tone analysis is given as
Intermodulation Distortion to Multitones Ratio( Δ
)
IM
measures the ratio between the power of generated IM
2
2
2
(H ) + (H ) +
+ (H )
1
2
6
components and the summation of the power of input
THR =
L
, (16)
2
(ST )
multitones. The Δ
formula for multitones analysis is
IM
given as
where the numerator is the summation of generated
harmonics power from first to sixth harmonics, and the
M
2
∑ (IM )
k 1
=
k
denominator is the power of the single tone which is the
Δ
=
, (19)
IM
N
2
∑ (T )
fundamental frequency.
i 1
=
i
2.3.2. Harmonic to IM Distortion Ratio ( HIDR )
where HIDR , Δ and Δ
are objective measurement
H
IM
metrics for multitones input harmonic and
Harmonic to Intermodulation Distortion Ratio ( HIDR )
intermodulation distortion analysis. THR metric is used
measures the ratio between the summation of the power
for objective measurement index of the harmonic
of generated harmonics and the summation of the power
richness of a particular NLD under investigation. In this
of Intermodulation components. The HIDR formula for
paper, these four metrics are used to compare the static
multitones analysis is
memoryless NLDs of the virtual bass system.
∑L (H 2
)
k =
3. NLD HARMONIC ANALYSIS
HIDR =
1
k
,
∑M (IM 2
)
k =1
k
In this section, harmonic and intermodulation distortion
(17)
analysis of five types of static nonlinear memoryless
NLDs are presented. For each NLD, we presented the
where ∑L (H 2
) is the summation of the power of L
k =1
k
original system transfer functions, followed by
polynomial approximated transfer functions. These
harmonic components, and ∑M IM 2 is the summation
k =1
k
transfer functions can be used to implement the NLD
of the power of M intermodulation distortion both in analog or digital systems.
components, generated by the NLD or nonlinearity
under investigation.
To approximate the original system transfer function,
we use MATLAB commands, such as polyfit and
polyval. In this paper, half-wave rectifier, full-wave
rectifier and limiter are approximated using these
commands, and the resultant plots are presented. For the
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Nay Oo and Woon-Seng Gan
Harmonics and IM distortion analysis of NLDs

exponential function NLD, we can use Taylor’s series
magnitude of the DC and harmonic
approximate to obtain the polynomial form which is
components,{c ,c ,
,c }
0
1 L
can be formulated as follows:
6
presented in the later section.
2
4
6
The main idea is to approximate the static memoryless
A
3A
5A
c = 2 × [h + (
)h + (
)h + (
)h ] , (22)
0
0
2
4
6
NLDs using polynomials, reuse the polynomial
2
8
16
harmonic and intermodulation analysis equations, and to
study the effect of nonlinearity. We limit the maximum
3
5
3A
5A
=
+
+
order to sixth order nonlinearity because of two reasons.
c
Ah
(
)h
(
)h , (23)
1
1
3
5
4
8
The first reason is that according to psychoacoustic
pitch perception researches findings [10][17-19], human
1
are more sensitive up to sixth harmonics to create
c =
[ 2
4
A h + A h ] , (24)
2
2
4
virtual bass effect. This is called dominance region in
2
pitch perception researches. From the previous findings
in Table 1, to generate up to sixth harmonics, we need
1
3
5 5
A
c =
[ A h +
h ] , (25)
up to sixth order nonlinearity. The seventh and more
3
3
5
4
4
orders will generate the higher order harmonics which
are not needed to create bass perception. The second
6
reason is that the higher the order is, the lesser the
1
4
3A
c =
[ A h +
h ] , (26)
4
4
6
contribution to harmonic generation and more
8
4
intermodulation components are generated.
1
c =
[ 5
A h ] , (27)
To obtain the amplitudes of DC and harmonics
5
5
16
components from a NLD, we can make use of DFT or
Schaefer-Suen equation (8) or directly applying the
3
6
Fourier series to the single tone signal.
c =
[ A h ] . (28)
6
6
16
While using DFT, we have to select the frequency 3.1. Half-wave
Rectifier
NLD
resolution, f
Δ which must be sharp enough to
distinguish between two adjacent frequencies The half-wave rectifier NLD system has a transfer
components. Another important factor to take into function that can be expressed as
account is the maximum frequency, f
it can capture
max
by the DFT. These two relations can be described as
1
y =
(x + x ) . (29)
follows:
2
f
Equation (29) can be approximated using polynomials
f
s
Δ =
,
(20)
N
as follows (up to 6 order)
f
6
4
2
y = .
0 6535x − .
1 3296x + .
1 1390x (30)
s
f
=
, (21)
max
2
+ .
0 5x + .
0 0419 .
where f denotes the sampling frequency and N denotes
s
the DFT points.
Since we are approximating all the NLDs, described in
this paper as polynomial series, we can also make use of
Schaefer-Suen equation (8) from Section 2.1. However,
we limit the highest order up to six for the previously
mentioned reasons. Therefore, the generated harmonics
components can be computed easily using a calculator
or MATLAB program. Therefore, using (8), the
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Nay Oo and Woon-Seng Gan
Harmonics and IM distortion analysis of NLDs


Figure 5: Half-wave Rectifier NLD original system
transfer function and polynomial approximated transfer
function.
Figure 5 shows the half-wave rectifier original and

approximated system transfer functions. By using up to
sixth order, the polynomials of (30) can approximate
Figure 7: Half-wave Rectifier NLD single tone
quite well to the original system function of (29). Figure
magnitude responses of original transfer function and
6 shows the NLD output in time domain using the
approximated transfer function.
original and approximated function. Figure 7 shows the
frequency domain responses.
The upper plot of Figure 7 shows the harmonics,
produced by the half-wave rectifier original system
transfer function equation (29) and the lower plot shows
the harmonics produced by the polynomial
approximated equation (30). Half-wave rectifier
produces DC, fundamental and even harmonics. The
reason it can reproduce fundamental frequency can be
linked to the polynomial approximated equation (30). In
(30), due to the linear term, x , the fundamental
frequency can be reproduced. Since we approximate up
to the sixth order, the maximum harmonic number is
six.
3.2. Full-wave
Rectifier
NLD
The full-wave rectifier NLD system transfer function

can be expressed as
Figure 6: Half-wave Rectifier NLD single tone input
y = x . (31)
responses for original transfer function and
approximated transfer function.
Equation (31) can be approximated using polynomials

as
y = .
1 3070 6
x − 2 6593
.
4
x + .
2 2781 2
x + .
0 0838 . (32)
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Nay Oo and Woon-Seng Gan
Harmonics and IM distortion analysis of NLDs



Figure 8: Full-wave Rectifier NLD original system
transfer function and polynomial approximated transfer
Figure 10: Full-wave Rectifier NLD single tone
function.
magnitude responses of original transfer function and
approximated transfer function.
Figure 8 shows the full-wave rectifier original system
transfer function and polynomial approximated system
3.3. Square Wave Function NLD (Limiter)
transfer function. Figure 9 shows the single tone input
response of full-wave rectifier NLD. From Figure 10,
The hard limiter or square wave function NLD transfer
the original full-wave rectifier produces infinite series
function can be mathematically described as
of harmonics, whereas the sixth order polynomial
approximated function can produce up to sixth order
⎧+ ,
1
x ≥ 1
harmonics, which are enough to create the virtual bass.
⎪
y = ⎨ ,
0
x = 0 . (33)
Since the polynomial equation (32) has only even order
⎪− ,1
x ≤ 1
nonlinearities, the generated harmonics are all even
⎩
order harmonics, including DC component.
The transfer function can be approximated as
y = 4
x
4421
.
5 − .
7
x
2621 3 + 3 9244
.
x . (34)

Figure 9: Full-wave Rectifier NLD single tone input
responses for original transfer function and

approximated transfer function.
Figure 11: Square-wave function NLD original system
transfer function and polynomial approximated transfer
function.
AES 124th Convention, Amsterdam, The Netherlands, 2008 May 17–20
Page 9 of 18

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Nay Oo and Woon-Seng Gan
Harmonics and IM distortion analysis of NLDs

3.4. Exponential
Function
NLD
The exponential function can be expanded to a
polynomial function by using Taylor’s series. The
expansion can be mathematically described as
∞
k
y =
x
f (x) = x
e = ∑ ( ) . (35)
k =0
k!
Schaefer has previously derived the generalized single
tone harmonic analysis equation for this exponential
function based on his works on harmonic analysis for
the bipolar junction transistor, and published his results

in [16]. We modified his equation for the analysis of
exponential function NLD used in the virtual bass
Figure 12: Square-wave function NLD single tone input
system. The Schaefer’s equation for the exponential
responses for original transfer function and
function harmonic analysis can be expressed as
approximated transfer function.
k ∞
⎛ A ⎞
2
( A 2) j
Figure 11 shows the original and approximated transfer
c =
. (36)
k
⎜ ⎟ ∑
⎝ 2 ⎠ j j (
! k
j)!
functions of limiter. In MATLAB simulation, we can
=0
+
use sign function for the original transfer function
simulation. The polynomial approximated transfer Since
x
x ln b
b = e
and using (35), the exponential
function up to 6th order is obtained as (34). The function with different base function can be converted
produced harmonics are all odd order harmonics, to the polynomial form as
including fundamental component as shown in Figure
13.
∞
k
y = x
b = ∑ (xlnb) . (37)
k =0
k!
To derive the harmonic analysis equation for (37), we
modify the Schaefer’s exponential function harmonic
analysis equation (36) as
k ∞
⎛ Alnb ⎞
2 j
c =
A
b
. (38)
k
⎜
⎟ ∑ ( ln
)
2
⎝ 2 ⎠ j=0 j (
! k + j)!
Since we are approximating up to the sixth order, (38)
can be expanded as
2
3
4
(ln b)
2
(ln b)
3
(ln b)
4

y = 1 + (ln b)x +
x +
x +
x
2
6
24
Figure 13: Square-wave function NLD single tone
magnitude responses of original transfer function and
5
6
(ln b)
(ln b)

5
6
+
x +
x . (39)
approximated transfer function.
120
720
Therefore, we have two ways to perform harmonic
analysis for exponential function NLD of base .
b The
first way is directly using (38) in infinite series form,
AES 124th Convention, Amsterdam, The Netherlands, 2008 May 17–20
Page 10 of 18

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