Copyright by Michael G. Ellis, Ph.D.

**The word “Predistortion”
originated from filter theory**.

Lowpass filter with perfect inductors

Response of our perfect filter

But the inductors are not infinite Q's, so that act
like they have resistors in series with them. This can cause the response to be
unacceptable.

A really unacceptable frequency response due to using
real inductors with finite Q's.

Predistortion involves modifying the values of the
inductors and capacitors so that the resulting frequency response is as close
as possible to the desired response using real inductors with finite Q's.

The results
are never quite as good as using ideal inductors, but usually the results are
acceptable.

Within the past 2 decades, a group of
optics engineers have temporarily "borrowed" the use of the word

"PREDISTORTION"

to mean something else.

Definition:

A system is linear if the output voltage is
proportional to the input voltage.

Figure: A linear system

If a system is non-linear, it can produce both even
and odd order distortion.

The sine wave below has NO even order distortion
because the top and the bottom half of the sine wave is distorted in the same
manner.

Sine wave with only odd
order distortion

The sine wave below is distorted differently on the
top and bottom half cycles, so it contains even order distortion.

Sine wave with even order distortion

A typical CATV laser transmitter contains a laser,
Mach-Zehnder modulator, and 50-870 MHz RF input. The Mach-Zehnder is not
linear.

_{}

The Mach-Zehnder produces both even and odd order
distortion.

How do we apply the input signal so that the
Mach-Zehnder has no even order distortion?

_{}

or

_{}

or

_{}

This is called the Composite Second Order (CSO)
Control Loop

_{}

Now the even order distortion problem is solved.

The Entire Block Diagram

_{}

The 3rd order predistortion circuit must have the
inverse function of the Mach-Zehnder to cancel it out.

_{}

Notice that the 3rd-order predistortion network
cascaded with the Mach-Zehnder will produce a linear transfer function.

We need a circuit that provides more gain at the
positive and negative peaks of the RF input signal than at the middle of the RF
input signal.

How do we do this?

With diodes, or course. A diode has less resistance as
the current through it is increased.

A working predistortion network.

The top 2 diodes have less and less resistance as the
input signal goes more and more positive, and the bottom two diodes have less and
less resistance as the input signal goes more and more negative.

The Diode Equation

_{}

With a source resistance of R_{s }and a load
resistance of R_{L}, and M=2 for the number of diodes, the transfer function becomes

_{}

where

_{}

Deriving this may take an hour, a day, or longer,
depending on your training in high school algebra

We must

(1) solve the previous
equation iteratively,

(2) cascade the output
voltage, VL, with

_{}, and

(3) take the Fourier
transform of the Light Out to look at any remaining 3rd, 5th, 7th, and 9th
order distortion.

This tells us if we have
cancelled the 3rd order distortion (linearized the Mach-Zenhder transfer function)
without increasing the 5th, 7th, and 9th order products too much.

The code for the Fourier
transform

For harmon = 1 to 20

twopi = 2*3.141592654

sine = 0: cosine=0

towpipt = twopi/pt

for i = 1 to pt

if i = 1 then

oldcos = 1

oldsin = 0

end if

x = i*twopipt

newcos = cos(x)

newsin=sin(x)

sine=-(newcos-oldcos)*b(i)+sine

cosine=(newsin-oldsin)*b(i)+cosine

oldcos=newcos

oldsin=newsin

next i

engy(harmon)=(sine^2+cosine^2)/(2*twopi)

next harmon

Test Your Knowledge

Linearizer

Mach-Zenhder

Predistorting

3rd order

Composite Second Order

Staff of 20 people

2 years development time

Intermission.

How long was the hundred years war?

116 years.

What was King George IV's first name?

Albert

Where are Panama hats made?

In Equador

Where do Chinese gooseberries come from?

New Zealand

What is a camel's hair brush made from?

Squirrel's hair

Intermission over.

**A subtractor**

An example of expansive predistortion

(This is what we want).

An example of compresive predistortion

(This is what we don't want).

Expansive or Compressive?

____________

Expansive or Compressive?

__________

This is a hyperbolic arctangent circuit