87
CHAPTER
5
Linear Systems
Most DSP techniques are based on a divideandconquer 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. 51.
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:
and
, while
discrete
signals use
x
(
t
)
y
(
t
)
brackets, as in:
and
.
Second, signals use lower case letters.
Upper
x
[
n
]
y
[
n
]
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:
, or
v
(
t
)
a stock market
price
measured each
day
could be:
.
p
[
d
]
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View Full DocumentThe Scientist and Engineer's Guide to Digital Signal Processing
88
Continuous
System
Discrete
System
x(t)
y(t)
x[n]
y[n]
FIGURE 51
Terminology for signals and systems.
A system is any process that generates an output signal in
response to an input signal. Continuous signals are usually represented with parentheses, while
discrete signals use brackets.
All signals use lower case letters, reserving the upper case for the
frequency domain (presented in later chapters).
Unless there is a better name available, the input
signal is called:
x
(
t
) or
x
[
n
], while the output is called:
y
(
t
) or
y
[
n
].
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:
, and the output signal:
.
x
[
n
]
y
[
n
]
For continuous systems, the signals:
and
are used.
x
(
t
)
y
(
t
)
There are many reasons for wanting to understand a
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 Summer '09
 Digital Signal Processing, Linear Systems, Nonlinear system, Sine wave

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