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

The filter is causal and recursive, so the outputs are computed sequentially starting from \(n=0\), taking all values before the start of the record (\(x_{-1}\) and \(y_{-1}\)) to be zero (the filter starts at rest).

Step 1 (n = 0):

\[y_0 = x_0 - 2x_{-1} + 3y_{-1} = 1.2 - 2(0) + 3(0) = 1.2\]

Step 2 (n = 1):

\[y_1 = x_1 - 2x_0 + 3y_0 = -0.8 - 2(1.2) + 3(1.2) = -0.8 - 2.4 + 3.6 = 0.4\]

Step 3 (n = 2):

\[y_2 = x_2 - 2x_1 + 3y_1 = 2.3 - 2(-0.8) + 3(0.4) = 2.3 + 1.6 + 1.2 = 5.1\]

\(\boxed{y_2 = 5.1}\)