Linear Filters, Part 2

Linear Filters

Part 2: Difference Equations as Linear Filters

Recall from the Difference Equations module that given numbers a₁, a₂, ... , a_n, with a_n different from 0, and a sequence {z_k}, the equation

y_k+n + a₁ y_k+n-1 + .... + a_n-1 y_k+1 + a_n y_k = z_k

is called a linear difference equation of order n. These behave in basically the same way for doubly-infinite sequences as they do for singly-infinite sequences. As before, if {z_k} is the zero sequence { ..., 0, 0, 0, ... }, then the equation is homogeneous. Otherwise, it is nonhomogeneous.

If our sequences represent sampled signals, as in Part 1, then we can think of a linear difference equation as a linear filter. The sequence {y_k} is the input to the filter, and the sequence {z_k} is the output of the filter.

Consider the linear filter

y_k+4 - y_k+2 + 2 y_k = z_k.
Let

$\displaystyle y_{k}= \sin(k\pi/2)$
be the input sequence. Graph both the input and the output sequences.
Now consider the same filter but let

$\displaystyle y_{k}= \sin(k\pi/6)$
be the input sequence. Graph both the input and the output sequences. Then let

$\displaystyle y_{k}= 2^{k/4} \sin \left(\frac{k \arctan \sqrt{7}}{2}\right)$
be the input sequence. Graph both the input and the output sequences. What do you observe in these two cases? Do the output sequences look like the input sequences? Do they look like some other common sequences? (You may need to look at your axes closely and take round-off error into account.)
Use your computer algebra system to verify symbolically your observations in the previous step.

A difference equation can be regarded as a filter because, as we saw, waves with some frequencies are filtered out while others are allowed to pass unchanged. Electronic circuits that act this way are very important in many types of signal processing. For example, an equalizer on a stereo processes musical signals so that high-frequency noise is filtered out but lower frequency signals are allowed to pass.

In the next part we will show how to predict how a linear filter will behave.