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

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and the solution is:

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

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

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

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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.

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