EEL 3135: Signals and Systems
Dr. Fred J. Taylor, Professor
Lesson Title: Impulse Invariant FIR
Lesson Number: 12 (Section 56 to 58)
Background:
In Chapter 5, elementary FIR filters are discussed. The Chapter ends with a discussion of some
of the basic properties of FIRs and demonstrations.
Recall that an FIR implements the
convolution sum given by:
[
]
[
]
∑
=

=
M
k
k
k
n
x
h
n
y
0
1.
which defines the process by which input samples x[k] and converted into output samples y[k].
The convolution process describe in Equation 1 has a short hand notation and it is:
y=h x
`
2
Example:
Convolve the N = 4 point time series
x={x[0], x[1], x[2], x[3]} = [1, 1, 1, 1] with the L = 4 tap FIR
having an impulse response h={h[0], h[1], h[2], h[3]} = [ 1, 1, 1, 1].
The convolution sum, y=h x,
is of length (N+L1) = 7 and is given by:
y[0]=h[0]x[0] = 1
y[1]=h[0]x[1] + h[1]x[0] = 2
y[2]=h[0]x[2] + h[1]x[1] + h[2]x[0] = 3
y[3]=h[0]x[3] + h[1]x[2] + h[2]x[1] + h[2]x[0] = 4
h[0]
h[1]
h[2]
h[3]
x[3]
x[2]
x[1]
x[0]
h[0]
h[1]
h[2]
h[3]
x[3]
x[2]
x[1]
x[0]
h[0]
h[1]
h[2]
h[3]
x[3]
x[2]
x[1]
x[0]
h[0]
h[1]
h[2]
h[3]
x[3]
x[2]
x[1]
x[0]
1
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Dr. Fred J. Taylor, Professor
y[4]=h[0]0 + h[1]x[3] + h[2]x[2] + h[2]x[1] = 3
y[5]=h[0]0 + h[1]0 + h[2]x[3] + h[2]x[2] = 2
y[6]=h[0]0 + h[1]0 + h[2]0 + h[2]x[3] = 1
The outcome is graphically interpreted in Figure 1.
Figure 1:
Convolution of two pulses.
 end of example
The example problem motivates an observation that the convolution sum values can be defined
in terms of an inner product (dot product), y=h
T
x
k
=
∑
h
i
x
k
i
, where h is a vector of impulse
response coefficients and
x
k
is a reversed in time listing of sample values, namely:
(
29
1
[
],.
..,
1
[
],
[



=
L
k
x
k
x
k
x
x
K
k
3.
Time Invariance
An important class of filters is called
timeinvariant
FIR filters.
A timeinvariant has a predictable
response when signal delays are introduced. Assume a signal s(t) is presented to system S and
shown in Figure 2.
If a timeinvariant system is presented with an input x[k], producing an
output y[k], then when presented with an input x[kk
0
], the output is y[kk
0
],
The study illustrated
in Figure 3 can be used as a test for timeinvariance. Timeinvariant systems have constant
coefficient and behave in the manner shown in Figure 2 and 3.
If the systems coefficients are
time varying, then the behavior shown in Figure 2 cannot be guaranteed in general.
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 Spring '08
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 Digital Signal Processing, LTI system theory, Finite impulse response, Dr. Fred J. Taylor, Dr. Fred J.

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