Question

A causal recursive filter is given by \(y_n = x_n - 2x_{n-1} + 3y_{n-1}\). If the input values, \(x_0\), \(x_1\) and \(x_2\) are 1.2, -0.8 and 2.3, respectively, the output, \(y_2\), of the filter is __________ (rounded off to one decimal place).

Hint:

Evolve the recursion one sample at a time from rest (x(-1) = y(-1) = 0); you need y0 and y1 before you can get y2.

Answer 1

Correct Answer: 5.1

Set up a small running table and evolve the difference equation \(y_n = x_n - 2x_{n-1} + 3y_{n-1}\) one step at a time, treating the record as starting at rest (\(x_{-1}=y_{-1}=0\)):

n	x_n	x_(n-1)	y_(n-1)	y_n = x_n - 2x_(n-1) + 3y_(n-1)
0	1.2	0	0	1.2
1	-0.8	1.2	1.2	-0.8 - 2.4 + 3.6 = 0.4
2	2.3	-0.8	0.4	2.3 + 1.6 + 1.2 = 5.1

Each row only needs the previous input and the previous output, which is the defining feature of a first-order recursive (IIR) filter - the feedback term \(3y_{n-1}\) gives the filter memory, so its output at any sample depends on the entire past history, not just the current input.

Reading off the last row, \(y_2 = 5.1\).

\(\boxed{y_2 = 5.1}\)