1164 Chapter 31
The average sums values together starting M 1
samples forward from the current point and moving M 2
samples back from the current point and divides by the
number of points that were summed together to form an
average that smooths out the signal. The moving
average is a digital filter that removes high frequency
information through averaging.
31.3.3 System Properties
System properties are a convenient way to describe
broad classes of systems. Important system properties
include linearity, shift invariance, causality, and stabil-
ity. These properties are important because they lead to
a representation of systems that can be readily analyzed.
31.3.3.1 Linearity
A linear system is one where the output of a sum of lin-
early scaled input signals is equal to the sum of the lin-
early scaled output signals. Mathematically, a system,
T{•}, is linear when
and
Then
(31-7)This means that when the input to a linear system is
a sum of signals, the output is the sum of the signals
transformed individually.
As an example, consider a system that performs a
scalar multiply y[n]=Dx>n@(when D!, y[n] is a louder
version of x[n], and when D , y[n] is a quieter version
of x[n]). This system is linear because
An example of a nonlinear system would be a
compressor/limiter because the output of a
compressor/limiter to a sum of signals is generally not
equal to the sum of the compressor/limiters applied to
the signals individually.
31.3.3.2 Time InvarianceA time-invariant system is one where a delay in the
input signal causes the output to be delayed by the same
amount. Mathematically, a system, T{•}, is time invari-
ant if when then(31-8)When the input, x[n], to a linear system is delayed,
the output, y[n], is delayed correspondingly. There is no
absolute time reference associated with the system. The
combination of time invariance and linearity makes the
design and analysis of a large class of DSP theory and
applications much simpler due to the convolution opera-
tion and Fourier analysis tools.^131.3.3.3 CausalityA causal system is one where the output of the system at
a given time only depends on the present and past val-
ues of the input signal. No future data can be required to
produce an output signal at the present time in a causal
system. In the moving average system of Eq. 31-6, the
system is causal only if M 1 =0.31.3.3.4 StabilityA system is bounded input/bounded output stable if and
only if every bounded input sequence produces a
bounded output sequence. A sequence is bounded if
each value in the sequence is less than infinity. For real
applications, stability is critically important because a
system would stop operating properly should it ever
become unstable.31.3.4 Linear Time-Invariant SystemsWhen the linearity property is combined with the
time-invariance property to form a linear time-invariant
(LTI) system, then the analysis of systems is very
straightforward. Because a sequence can be represented
as a sum of weighted delayed impulses as shown in Eq.
31-2, and an LTI system response is the sum of the com-
ponent responses of the sequence components as shown
in Eq. 31-7, the response of an LTI system is completely
determined from its response to an impulse. Since an
input signal can be represented as a collection of
delayed and scaled impulses, the response to the full
sequence is known. The response of a system to an
impulse is commonly referred to as the impulse
response of the system. Mathematically,y 1 >@n =Tx^` 1 >@ny 2 >@n =Tx^` 2 >@nTax^` 1 >@n +bx 2 >@n =Tax^` 1 >@n +Tbx^` 2 >@n
aT x^` 1 >@n +bT x^` 2 >@n ay 1 >@n += by 2 >@nyn>@ D= ax 1 >@n +bx 2 >@ n= Dax 1 >@ Dn + bx 2 >@nyn>@=Txn^`>@Txn N^`>@– =yn N>@–