# Impulse response of a system in z domain

The question 3-23 in the "Discrete-Time Signal Processing - Second Edition" is:

and the solution is:

I cannot understand the solution. In the second row of the answer when I multiply (-4) with this denominator:

and then add the result to: it gives a result different from the numerator of the first row. This multiplication must be equal to

Which is not. Is this solution true? If yes, can anyone help me to understand if?

## 1 Answer

You are right.

In the solution, the second line which performs the long division

$$H(z) = -4 + \frac{ 5 + \frac{7}{2} z^{-1} }{1 - \frac{3}{4}z^{-1} + \frac{1}{8} z^{-1} }$$

is wrong and should be corrected as:

$$H(z) = -4 + \frac{ 5 - 3 z^{-1} }{1 - \frac{3}{4}z^{-1} + \frac{1}{8} z^{-1} } .$$

However, the partial fraction expansion at the following third line is correct; indicating there was a typo in the second line. Finally, the impulse response (which is based on line three) is also correct.