Linear Filters, Part 4

In this part, we are going to see how to predict which frequencies are passed though a linear filter unchanged. If our linear filter has the equation

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

then we are looking to solve the equation

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

which is the same as the homogeneous equation

y_k+n + a₁ y_k+n-1 + .... + a_n-1 y_k+1 + (a_n - 1) y_k = 0.

Thus, we can use the methods of the previous part to also solve our current problem.

Find as large a linearly independent set of signals as you can that are passed through unchanged by the linear filter

y_k+2 + y_k = z_k.

Discuss the significance of what you have found.
Find as large a linearly independent set of signals as you can that are passed through unchanged by the linear filter

y_k+4 - y_k+2 + 2 y_k = z_k
from Part 2. Compare your answers to the observations you made there.