Linear Systems

Most DSP techniques are based on a divide-and-conquer strategy called superposition. The
signal being processed is broken into simple components, each component is processed
individually, and the results reunited. This approach has the tremendous power of breaking a
single complicated problem into many easy ones. Superposition can only be used with linear
systems, a term meaning that certain mathematical rules apply. Fortunately, most of the
applications encountered in science and engineering fall into this category. This chapter presents
the foundation of DSP: what it means for a system to be linear, various ways for breaking signals
into simpler components, and how superposition provides a variety of signal processing
techniques.


Signals and Systems
A signal is a description of how one parameter varies with another parameter.
For instance, voltage changing over time in an electronic circuit, or brightness
varying with distance in an image. A system is any process that produces an
output signal in response to an input signal. This is illustrated by the block
diagram in Fig. 5-1. Continuous systems input and output continuous signals,
such as in analog electronics. Discrete systems input and output discrete
signals, such as computer programs that manipulate the values stored in arrays.
Several rules are used for naming signals. These aren't always followed in
DSP, but they are very common and you should memorize them. The
mathematics is difficult enough without a clear notation. First, continuous
signals use parentheses, such as: x(t) and y(t) , while discrete signals use
brackets, as in: x[n] and y[n] . Second, signals use lower case letters. Upper
case letters are reserved for the frequency domain, discussed in later chapters.
Third, the name given to a signal is usually descriptive of the parameters it
represents. For example, a voltage depending on time might be called: v(t) , or
a stock market price measured each day could be: p[d] .
Signals and systems are frequently discussed without knowing the exact
parameters being represented. This is the same as using x and y in algebra,
without assigning a physical meaning to the variables. This brings in a fourth
rule for naming signals. If a more descriptive name is not available, the input
signal to a discrete system is usually called: x[n] , and the output signal: y[n] .
For continuous systems, the signals: x(t) and y(t) are used.
There are many reasons for wanting to understand a system. For example, you
may want to design a system to remove noise in an electrocardiogram, sharpen
an out-of-focus image, or remove echoes in an audio recording. In other cases,
the system might have a distortion or interfering effect that you need to
characterize or measure. For instance, when you speak into a telephone, you
expect the other person to hear something that resembles your voice.
Unfortunately, the input signal to a transmission line is seldom identical to the
output signal. If you understand how the transmission line (the system) is
changing the signal, maybe you can compensate for its effect. In still other
cases, the system may represent some physical process that you want to study
or analyze. Radar and sonar are good examples of this. These methods
operate by comparing the transmitted and reflected signals to find the
characteristics of a remote object. In terms of system theory, the problem is to
find the system that changes the transmitted signal into the received signal.
At first glance, it may seem an overwhelming task to understand all of the
possible systems in the world. Fortunately, most useful systems fall into a
category called linear systems. This fact is extremely important. Without the
linear system concept, we would be forced to examine the individual characteristics of many unrelated systems. With this approach, we can focus
on the traits of the linear system category as a whole. Our first task is to
identify what properties make a system linear, and how they fit into the
everyday notion of electronics, software, and other signal processing systems.

Requirements for Linearity
A system is called linear if it has two mathematical properties: homogeneity
(hÇma-gen-~-ity) and additivity. If you can show that a system has both
properties, then you have proven that the system is linear. Likewise, if you can
show that a system doesn't have one or both properties, you have proven that
it isn't linear. A third property, shift invariance, is not a strict requirement
for linearity, but it is a mandatory property for most DSP techniques. When
you see the term linear system used in DSP, you should assume it includes shift
invariance unless you have reason to believe otherwise. These three properties
form the mathematics of how linear system theory is defined and used. Later
in this chapter we will look at more intuitive ways of understanding linearity.
For now, let's go through these formal mathematical properties.
As illustrated in Fig. 5-2, homogeneity means that a change in the input signal's
amplitude results in a corresponding change in the output signal's amplitude.
In mathematical terms, if an input signal of x[n] results in an output signal of
y[n] , an input of k x[n] results in an output of k y[n], for any input signal and
constant, k.

A simple resistor provides a good example of both homogenous and nonhomogeneous
systems. If the input to the system is the voltage across the
resistor, v(t) , and the output from the system is the current through the resistor,
i(t) , the system is homogeneous. Ohm's law guarantees this; if the voltage is
increased or decreased, there will be a corresponding increase or decrease in
the current. Now, consider another system where the input signal is the voltage
across the resistor, v(t) , but the output signal is the power being dissipated in
the resistor, p(t) . Since power is proportional to the square of the voltage, if
the input signal is increased by a factor of two, the output signal is increase by
a factor of four. This system is not homogeneous and therefore cannot be
linear.
The property of additivity is illustrated in Fig. 5-3. Consider a system where
an input of x produces an output of . Further suppose that a different 1[n] y1[n]
input, x , produces another output, . The system is said to be additive, 2[n] y2[n]
if an input of x results in an output of , for all possible 1[n] % x2[n] y1[n] % y2[n]
input signals. In words, signals added at the input produce signals that are
added at the output.

The important point is that added signals pass through the system without
interacting. As an example, think about a telephone conversation with your
Aunt Edna and Uncle Bernie. Aunt Edna begins a rather lengthy story about
how well her radishes are doing this year. In the background, Uncle Bernie is
yelling at the dog for having an accident in his favorite chair. The two voice
signals are added and electronically transmitted through the telephone network.
Since this system is additive, the sound you hear is the sum of the two voices
as they would sound if transmitted individually. You hear Edna and Bernie,
not the creature, Ednabernie.
A good example of a nonadditive circuit is the mixer stage in a radio
transmitter. Two signals are present: an audio signal that contains the voice
or music, and a carrier wave that can propagate through space when applied
to an antenna. The two signals are added and applied to a nonlinearity,
such as a pn junction diode. This results in the signals merging to form a
third signal, a modulated radio wave capable of carrying the information
over great distances.
As shown in Fig. 5-4, shift invariance means that a shift in the input signal will
result in nothing more than an identical shift in the output signal. In more
formal terms, if an input signal of x[n] results in an output of y [n], an input
signal of x[n% s] results in an output of y[n% s] , for any input signal and any
constant, s. Pay particular notice to how the mathematics of this shift is
written, it will be used in upcoming chapters. By adding a constant, s, to the
independent variable, n, the waveform can be advanced or retarded in the
horizontal direction. For example, when s’ 2 , the signal is shifted left by two
samples; when s’ &2 , the signal is shifted right by two samples.

Shift invariance is important because it means the characteristics of the
system do not change with time (or whatever the independent variable
happens to be). If a blip in the input causes a blop in the output, you can
be assured that another blip will cause an identical blop. Most of the
systems you encounter will be shift invariant. This is fortunate, because it
is difficult to deal with systems that change their characteristics while in
operation. For example, imagine that you have designed a digital filter to
compensate for the degrading effects of a telephone transmission line. Your
filter makes the voices sound more natural and easier to understand. Much
to your surprise, along comes winter and you find the characteristics of the
telephone line have changed with temperature. Your compensation filter is
now mismatched and doesn't work especially well. This situation may
require a more sophisticated algorithm that can adapt to changing
conditions.
Why do homogeneity and additivity play a critical role in linearity, while shift
invariance is something on the side? This is because linearity is a very broad
concept, encompassing much more than just signals and systems. For example,
consider a farmer selling oranges for $2 per crate and apples for $5 per crate.
If the farmer sells only oranges, he will receive $20 for 10 crates, and $40 for
20 crates, making the exchange homogenous. If he sells 20 crates of oranges
and 10 crates of apples, the farmer will receive: 20×$2 % 10×$5 ’ $90 . This
is the same amount as if the two had been sold individually, making the
transaction additive. Being both homogenous and additive, this sale of goods
is a linear process. However, since there are no signals involved, this is not
a system, and shift invariance has no meaning. Shift invariance can be thought
of as an additional aspect of linearity needed when signals and systems are
involved.
Static Linearity and Sinusoidal Fidelity
Homogeneity, additivity, and shift invariance are important because they
provide the mathematical basis for defining linear systems. Unfortunately,
these properties alone don't provide most scientists and engineers with an
intuitive feeling of what linear systems are about. The properties of static
linearity and sinusoidal fidelity are often of help here. These are not
especially important from a mathematical standpoint, but relate to how humans
think about and understand linear systems. You should pay special attention
to this section.
Static linearity defines how a linear system reacts when the signals aren't
changing, i.e., when they are DC or static. The static response of a linear
system is very simple: the output is the input multiplied by a constant. That
is, a graph of the possible input values plotted against the corresponding
output values is a straight line that passes through the origin. This is shown
in Fig. 5-5 for two common linear systems: Ohm's law for resistors, and
Hooke's law for springs. For comparison, Fig. 5-6 shows the static
relationship for two nonlinear systems: a pn junction diode, and the
magnetic properties of iron.

Here is the important part: the output signal obtained by this method is
identical to the one produced by directly passing the input signal through the
system. This is a very powerful idea. Instead of trying to understanding how
complicated signals are changed by a system, all we need to know is how
simple signals are modified. In the jargon of signal processing, the input and
output signals are viewed as a superposition (sum) of simpler waveforms. This
is the basis of nearly all signal processing techniques.
As a simple example of how superposition is used, multiply the number 2041
by the number 4, in your head. How did you do it? You might have imagined
2041 match sticks floating in your mind, quadrupled the mental image, and
started counting. Much more likely, you used superposition to simplify the
problem. The number 2041 can be decomposed into: 2000 % 40 % 1 . Each of
these components can be multiplied by 4 and then synthesized to find the final
answer, i.e., 8000 % 160 % 4 ’ 8164 .
Common Decompositions
Keep in mind that the goal of this method is to replace a complicated problem
with several easy ones. If the decomposition doesn't simplify the situation in
some way, then nothing has been gained. There are two main ways to
decompose signals in signal processing: impulse decomposition and Fourier
decomposition. They are described in detail in the next several chapters. In
addition, several minor decompositions are occasionally used. Here are brief
descriptions of the two major decompositions, along with three of the minor
ones.
Impulse Decomposition
As shown in Fig. 5-12, impulse decomposition breaks an N samples signal into
N component signals, each containing N samples. Each of the component
signals contains one point from the original signal, with the remainder of the
values being zero. A single nonzero point in a string of zeros is called an
impulse. Impulse decomposition is important because it allows signals to be
examined one sample at a time. Similarly, systems are characterized by how
they respond to impulses. By knowing how a system responds to an impulse,
the system's output can be calculated for any given input. This approach is
called convolution, and is the topic of the next two chapters.
Step Decomposition
Step decomposition, shown in Fig. 5-13, also breaks an N sample signal into
N component signals, each composed of N samples. Each component signal
is a step, that is, the first samples have a value of zero, while the last samples
are some constant value. Consider the decomposition of an N point signal,
x[n] , into the components: x . The component 0[n], x1[n], x2[n], þxN&1[n] k th
signal, x , is composed of zeros for points 0 through , while the k[n] k&1
remaining points have a value of: x[k] & x[k&1]. For example, the 5 th
component signal, x , is composed of zeros for points 0 through 4, while 5[n]
the remaining samples have a value of: x[5] & x[4] (the difference between
ATRIAL FIBRILLATION

The atria are the chambers that connect to the ventricles of the
heart. In the figure below they are highlighted in blue and are
labeled. The two large chambers that they are connected to are called
the ventricles.  The heart has an internal electrical system that
controls the rhythm and beating of the heart. During each beat a
signal is sent from the top to the bottom of the heart. This causes
the heart to contract and pump blood. In a healthy adult at rest the
signal to begin a new heartbeat occurs 60 to 100 times per minute
During atrial fibrillation there are disorganized electrical signals
to the atria of the heart. The heart normally has these signals to
cause a rhythmic heartbeat. But with atrial fibrillation the heart
beats become obscured. It may beat up to 100-180 times per minute.
The electrical signals do not travel the normal pathways, but may
spread throughout the atria in a rapid and random fashion. This can
cause the atria to beat rapidly and irregular. This type of beating of
the atria is called fibrillation.  The signals can also cause the
beating of the ventricles to be off as well. During this heart problem
the atria and ventricles no longer beat in a coordinated way.

 
	Since the beating of these chambers are off, the amount of
blood being pumped through each chamber is irregular as well. The
ventricles will receive random amounts of blood depending how much the
atria has pumped to them. It is possible that blood can pool in the
atria.

For my animation I plan to have the ventricles beat faster, as well as
having the atria quivering and beating rapidly. I also plan to animate
the blood circulating in an irregular pattern.

ATRIAL FLUTTER

Similarly to atrial fibrillation, atrial flutter is a
tachyarrhythmia. Meaning that the heart beats faster than
normal. Unlike atrial fibrillation, during atrial flutter the heart
beats rhythmically, where in atrial fibrillation the beating is more
randomly executed.

The fast beating atrial muscles causes the beating of the ventricles
to be out of sync with the atrial beats.

Since the atrial muscles are not beating correctly blood can pool up
in the atrium and stay longer than usual. The atrial muscle can not
pump blood forward to the ventricle.

For this animation I plan to incorporate much of the same idea as the
atrial fibrillation animation, but make rapid beating become regular
and more rhythmic.

Cardiomyopathy Animation
By: Mohamad Haidar

Cardiomyopathy is a heart muscle disease. The heart enlarges and does
not effectively pump blood. This disease usually progresses to the
point where patients develop life-threatening heart failure. There are
two main types of cardiomyopathy and I will attempt to animated both
of them. (Dilated and Hypertrophic)

The first type is Dilated cardiomyopathy: the heart muscle is damaged
and becomes weak which leads the heart chambers to enlarge. The heart
stretches as it tries to compensate for it's weakened pumping ability.

I will enlarge the Left ventricle in our heart model and create a
thinner outside ventricle wall. The left ventricle will expand more
than the right ventricle and will not contract fully.

Dilated Cardiomyopathy: 

Blood Flow: If the heart muscle is weak, it is unable to pump out the
same portion of blood that it could at normal strength. Rather than
simply "accepting" the limitations of decreased pumping ability, the
heart undergoes changes to compensate. Blood flows more slowly through
an enlarged heart, so blood clots may form.  A clot can break free,
circulates in the bloodstream and block a small blood vessel.  I will
slow down the blood flow and not fully empty out the enlarged
ventricle after the contraction. This is what nurse Judy advised me to
do. This will leave some blood to show that an enlarged ventricle is
too weak to keep the blood flowing normally. I will also create a
blood clot that will flow in the bloodstream.

The second type of cardiomyopathy is Hypertrophic cardiomyopathy:
Heart muscle fibers enlarge abnormally. The heart wall thickens,
leaving less space for blood in the chambers.

 The disease is characterized by a disorderly growth of heart muscle
fibers, causing the heart chambers to become thick-walled and
bulky. All the chambers are affected, but the thickening is generally
most striking in the walls of the left ventricle. Most commonly, one
of the walls, the septum, which separates the right and left
ventricles, is asymmetrically enlarged.

In one form of the disease, the wall (septum) between the two
ventricles becomes enlarged and obstructs the blood flow from the left
ventricle. The syndrome is known as hypertrophic obstructive
cardiomyopathy.

I will have to create a thicker heart wall while making the ventricles
smaller. The wall will be so thick it will not leave a lot of space
for the ventricles which will also have to be modified to look small
and abnormal. The septum wall will be asymmetrically enlarged.

Hypertrophic Cardiomyopathy: Blood flow: Hypertrophic cardiomyopathy
can impair blood flow both into and out of the heart. Since the heart
does not relax correctly between beats, less blood fills the chamber
and is pumped from the heart. Besides obstructing blood flow, the
thickened wall sometimes distorts one leaflet of the mitral valve,
causing it to leak.

The distorted left ventricle contracts, but the supply of blood to the
brain and other vital organs may be inadequate because blood is
trapped within the heart during contractions.

This overgrowth creates a bulge that protrudes into the ventricular
chamber and impedes the flow of blood from the heart to the aorta and
the rest of the body.  The problem is not that the heart muscle is
weak but that the overgrown heart muscle impedes the flow of blood
through and out of the heart.

I will make less blood fill the chamber and less blood will be pumped
out. Some blood will stay in the heart after in contracts to show that
it is being obstructed by the thickened wall. I might also create a
leak in the mitral valve but I have to discuss this with nurse Judy.