Digital Signal Processing Reference
In-Depth Information
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Chapter 5
Discrete-Time Filters
The previous Chapters gave us a thorough overview on both the nature of
discrete-time signals and on the tools used in analyzing their properties. In
the next few Chapters, we will study the fundamental building block of any
digital signal processing system, that is, the linear filter. In the discrete-time
world, filters are nothing but procedures which store andmanipulate math-
ematically the numerical samples appearing at their input and their out-
put; in other words, any discrete-time filter can be described procedurally
in the form of an algorithm. In the special case of linear and time-invariant
filters, such an algorithm can be concisely described mathematically by a
constant-coefficient difference equation.
5.1
Linear Time-Invariant Systems
In its most general form, a
discrete-time system
can be described as a black
box accepting a number of discrete-time sequences as inputs and produc-
ing another number of discrete-time sequences at its output.
In this topic we are interested in studying the class of
linear
time-invariant
(LTI) discrete-time systems with a single input and a single
output; a system of this type is referred to as a
filter
. A linear time-invariant
system
canthusbeviewedasanoperatorwhichtransformsaninput
sequence into an output sequence:
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