Modeling Nonlinear Amplifiers

Modeling Non­Linear Amplifiers and the Mystery of
the Complex Fourier Series
By Robert Hembree

Introduction
Statement of Purpose and Reason behind that
purpose and the Goal.
What is a Non­Linear Amplifier
What does it do
How does it work physically
Mathematical Model
Types of Functions
Linear Gain
In real life there is no clipping!!!
Example sounds

Introduction Continued
Mathematical Analysis
Complex exponential and its Fourier Transform
General Taylor Series applied to Complex
Exponential
Find Taylor Series of my model Functions
What does that do to the signal.
Plug in complex representation of Cosine
Complex representation of Fourier Series!
Future work
References

Purpose
The purpose is to model a Non­Linear Amplifier,
more specifically it is to model the response
characteristics of a tube guitar amplifier.
Recreate their warm sound.
Create a cheap, effective, and eventually
marketable computer based model of this
phenomenon.

Reason
Most solutions to this that other people have
created are expensive and of low quality

Reason
Most of the Cheap solutions use a simple
Bicubic filter.
This produces a harsh sound
Many solutions will just set a hard clipping to
create the distortion

What Is a Non­Linear Amplifier

What it is
Gain is how much a signal is amplified.
In a Non­Linear Amplifier the gain is variable
with respect to the input volume.
Instead of the gain increasing in a linear fashion
it increases in a nonlinear fashion

How Does it physically work.
http://en.wikipedia.org/wiki/Image:Clipping_compared_to_limiting.svg

Mathematical Model

Mathematical Models
The functions need to be:
Odd
Bounded on top and bottom by horizontal
asymptotes
And for sound purposes it helps for them to be
analytic (Bicubic lacks this feature)
So I use:
f x ,c=2/arctanc x
f x ,c=tanhc x

Tanh
Vout
Vin

Arctan
Vout
Vin

Things to note
As you approach zero the slope is linear. This is
known as the linear gain
As your voltage input gets higher the output
asymptotically approaches 1 or ­1
We get no clipping. We never actually go
outside of the physical parameters
This is realistic.
We can approximate clipping by setting the gain
to a very high value

Linear Gain
This is the amount of amplification that you get
at input voltages close to zero.
This can be determined by taking the derivative
at zero
For the arctan model we get:
And for the tanh we get that gain is:
linear gain = c
linear gain= 2c
This is a measure of how we are amplifying the
signal. This value is like the percentage of
signal boost.

The Model

How well does it work

More

More

Some of My results
This is the Clean Signal that I started out with.

Arctan

Arctan filter Applied

Tanh applied

Mathematical Analysis

The Complex Exponential and its
Fourier Transform
The Complex Exponential has the form:
g x=e 2 i f 0
x
When we take the Fourier Transform we get:
F f =∫ e 2 i f 0 x e −2 f x dx= f − f 0

Taylor Series and Its Effect on the
Complex Exponential
So what Happens when we square the
Complex Exponential:
e 2 i f 0 x 2 =e 2 i 2f 0 x
Meaning that the Fourier Transform becomes:
F f =∫ e 2 i 2f 0 x e −2 f x dx= f −2f 0

Graphical Example

The Taylor Series
In general The Taylor series is:
∞
g x=∑
n=0
g n 0 x n
n !
So that the effect is:
F f =∫ ge 2 i f x 0
e −2i f x dx=∫∑
∞
n=0
g n 0e 2i f 0 x n
n !
e −2i f x dx=∑
∞
n=0
g n 0
n !
f −nf 0

Taylor Expansion of my models

Tanh

Atan

A Final Curiosity
Combine Cos with the Taylor series:
∞
gcos2 f 0
x=∑
n=0
g n 0cos n 2 f 0
x
n!
∞
=∑
n=0
gn 0 e2 i f x 0
e −2 i f x 0 n
2 n
n !
Essentially what we get is the Fourier Series on
expansion.