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ECE108HW%2310sol_06

ECE108HW%2310sol_06 - ECE 108 Hw#10 Solution 10.7 Find the...

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ECE 108 Hw#10 — Solution 10.7 Find the poles and zeros of the transfer function. In general to determine the possible Regions of Convergence and signals it is not necessary to consider the zeros. However, there is always the possibility that one of the numerator factors (zero) will cancel one of the denominator factors (poles). Thus it is necessary to factor both the numerator and denominator. ( ) 2 2 1 2 1 1 4 1 5 3 1 1 4 4 8 z X z z z z =  + + +   To find the roots we must write this in positive powers of z. This can be done by multiplying the numerator and denominator by , i.e., 4 z ( ) 2 2 2 2 1 4 1 5 3 4 4 8 z z X z z z z =  + + +   The roots of the numerator are . These are the locations of the zeros. The roots of 1 0, 2 ± the first term in the denominator are , and the roots of the second term in 2 1 4 z + 1 2 j ± the denominator, are . Thus the apparent poles are at 2 5 1 4 8 z z + + 1 3 and 2 4 1 1 1 3 , , , and 2 2 2 4 j j However, the pole at -½ is canceled by the zero at -½. This leaves only three poles located in the z-plane as shown below

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2 Now we see there are three different signals that could have the same transform 1 2 3 1 2 4 3 4 Signal #1. Its ROC is . This is a left-sided signal Signal #2. Its ROC is . This is a two-sided signal Signal #3.
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ECE108HW%2310sol_06 - ECE 108 Hw#10 Solution 10.7 Find the...

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