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In this part, we are going to see how to predict which frequencies are filtered out by a linear filter. We need to recall some facts from the Difference Equations module.
As in Part 1, let S be the set of all doubly-infinite sequences. Then the equation
describes a linear transformation T: S -> S with T({yk}) = {zk} . The null space (or kernel) N of T is the set of solutions of the homogeneous equation
N is an n-dimensional vector space, and an exponential sequence, yk = rk , is in N if and only if
This polynomial equation is called the auxiliary equation or characteristic equation. Its solutions r1, r2, ..., are called characteristic roots of the difference equation. If the auxiliary equation has n distinct roots, then these exponential sequences constitute a basis for the null space N of T. Once again, we will consider only situations in which the characteristic roots are distinct. However, we now wish to consider cases where the characteristic roots are complex.
Construct the
auxiliary equation and find the characteristic roots. You should
get two imaginary roots r1 and
r2.
However, these are
complex solutions, and for the purpose of signal processing we need
to know about the real solutions.
A famous law
of complex numbers known as de
Moivre's Rule
then tells us that
then the
sequences of real numbers {Re yk} and {Im
yk} are also solutions to the same linear
difference equation. (Remember that the coefficients
ak are all real.)
Use your
computer algebra system to verify that these are in fact
solutions.
You now know how to find
the real sequences that are solutions to the homogeneous
equation
If we consider once again the linear filter
then we can regard the solutions of the homogeneous equation above as the input signals that are taken to zero by the filter; that is, those that are filtered out.
from Part 2. Compare your answers to the observations you made there.
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